Fuzzy Measurement of Sustainability

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FUZZY MEASUREMENT OF SUSTAINABILITY No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form or by any means. The publisher has taken reasonable care in the preparation of this digital document, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained herein. This digital document is sold with the clear understanding that the publisher is not engaged in rendering legal, medical or any other professional services.

FUZZY MEASUREMENT OF SUSTAINABILITY

YANNIS A. PHILLIS AND

VASSILIS S. KOUIKOGLOU

Nova Science Publishers, Inc. New York

Copyright © 2009 by Nova Science Publishers, Inc.

All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Library of Congress Cataloging-in-Publication Data Phillis, Yannis A., 1950Fuzzy measurement of sustainability / Yannis A. Phillis and Vassilis S. Kouikoglou. p. cm. ISBN 978-1-60741-900-6 (E-Book) 1. Sustainable development. 2. Fuzzy systems. I. Kouikoglou, Vassilis S., 1961- II. Title. HC79.E5P5134 2009 338.9'2701511313--dc22 2008042820

Published by Nova Science Publishers, Inc.   New York

CONTENTS

Preface

vii

Chapter 1

Introduction

1

Chapter 2

Introduction to Fuzzy Logic

11

Chapter 3

Sustainability Indicators

65

Chapter 4

Fuzzy Assessment

73

Chapter 5

Sustainability of Organizations

113

Appendix

Programming Hints and Tips

143

References

165

Index

169

PREFACE People from many disciplines talk about sustainable development (SD) without having a concrete definition or idea what it is. Politicians and decision-makers, biologists and environmentalists, engineers and scientists, philosophers and sociologists often mean different things when they mention SD. To take an extreme example, a neoclassical economist might define SD as continual economic growth without regard to the environmental situation, whereas at the other end of the spectrum, a deep ecologist might define it as the preservation and restoration of the ecosystem, ignoring the effect on the economy completely. It has become clear throughout the last few decades leading up to the time this book was written that our society is facing environmental problems of global magnitude such as species extinction, population explosion, global warming, water shortages, exhaustion of fisheries, and deforestation, among others, which have economic and social repercussions. Many people doubt that the world we shall bequeath to our descendants will be all healthy, supportive, and beautiful as the one we inherited from our parents. The fundamental questions then of the SD discussion are: Can we continue our present course without destroying part or most of our environment and society? And if we have to change course (which is currently the consensus opinion), how do we do it? This book addresses one of the most fundamental questions of SD: How do we define and measure in the mathematical sense sustainability? In effect it summarizes our research in this field for the last 10 years. The mathematical model which we use to evaluate sustainability serves both as a definition and as a measuring scheme. This is done with the help of fuzzy logic, which is well suited to perform reasoning in fields where concrete mathematical models do not exist and concepts such as sustainability are multifaceted and often subjective. Fuzzy reasoning, in some respects, emulates human thinking and relies on expert opinion and knowledge as well as subjective evaluations. This multistage model is developed step-by-step, in the end providing sustainability assessments of various aspects of a society on a scale from zero to one, such as the environmental or societal situation of a country. All nations of the world for which reliable data exist are then ranked according to their sustainability as measured by the model. A sensitivity analysis reveals the most important components of sustainability for each country. The book is intended for undergraduate students, graduate students, and practitioners working in the field of sustainability. It is based on a course Yannis Phillis has taught at the University of California at Los Angeles (UCLA) and the Technical University of Crete (TUC) at the senior/graduate level. The mathematical knowledge needed to follow the book is

viii

Yannis A. Phillis and Vassilis S. Kouikoglou

minimal and is all contained in the text. Computational guidance is provided for those who would like to write their own model. The book is organized as follows: Chapter 1 discusses sustainability, its problems and meanings. Chapter 2 provides an introduction to fuzzy logic which is brief but sufficient for the needs of the book. Chapter 3 introduces certain indicators of sustainability. Chapter 4 is the main part of the book, which describes the fuzzy model in detail and provides results. Chapter 5 extends the concepts of previous chapters to the measurement of the sustainability of organizations such as corporations. This last subject of corporate sustainability is very much in fashion nowadays: many major companies as part of their social responsibility publish annual reports that show their contribution to the sustainability of the region where they operate. We are indebted to several people who have patiently contributed their knowledge and work in this field. We thank our former PhD students Lucas Andrianos and Victor Kouloumpis who worked very hard for several years with us to make the model functional. Professor Nikos Tsourveloudis assisted us with Chapter 2. Nili Phillis edited the English text. Ben Davis of UCLA edited the text and contributed data for the corporate sustainability model. Chania, June 2008 Yannis A. Phillis Vassilis S. Kouikoglou

Chapter 1

INTRODUCTION 1.1. WHO IS INTERESTED IN SUSTAINABILITY? Something is wrong with the way we humans treat our earth. An environmental crisis is unfolding before our very eyes in a period of just a few decades. Such time spans are equal nearly to zero when compared to the 4.5 billion years planet earth has been in existence. Lately it is common for environmental problems to figure prominently in the media. We have become accustomed to daily stories about unusual weather phenomena attributed to global warming, toxic spills, depletion of fisheries, deforestation, species extinction, desertification, air, soil, and water pollution and so on. Most importantly, these problems have achieved their prominence due to an ever increasing volume of scientific research, which, in most cases, has performed a commendable job in exposing the facets of the environmental predicament. The present human population of 6.7 billion is expected to reach almost 8 billion by 2025 (PRB 2008). The environmental impact of an additional 1.3 billion humans is not known precisely, but it is certain that it will worsen the pressure on the ecosystems and their dwindling resources such as water, fisheries, species, etc. The strain of overpopulation is not only environmental. Many societal aspects such as health, economy, education and the political system itself will be tried. We live in a fast changing world with ever increasing uncertainty about what is in store in the future. Dramatic changes occur in only a few years or just decades and so the time the society is given to adjust to these changes is very short. In the past, climate change progressed mostly over thousands of years, allowing humans and other species enough time to evolve or move to more suitable habitats. In the present global warming era, species have only 50 to 100 years to find new hospitable places. But some plant and animal species move slowly or don’t have where to go and these are doomed. A question thus arises as to the consequences of our actions to us and the environment: How sustainable is our present course? Such a question begs a definition and a reliable scheme for assessing sustainability. This is the central subject of this book. Sustainability is a concept discussed nowadays by politicians, environmentalists, biologists, sociologists, economists, engineers, philosophers, and ordinary citizens. This happens because sustainability concerns all humans and can be used by many disciplines, scientific or other.

2


Politicians love catch phrases such as sustainable development. There is even talk in some countries of changing the ministries of economy into ministries of sustainable development (at least the latter sounds more attractive to the voters). Politics play a central role in the discussion of sustainable development. After all, it is economic activity coupled with political decisions that affect the environmental or the social system. The European Union (EU), for example, has decided that its member countries should cut down their fossil fuel use in an effort to reduce greenhouse gases emissions. To achieve this end, a concerted effort is made to install large numbers of wind turbines in all EU member countries: this process involves primarily political decisions. It is quite natural for environmentalists to be involved in the sustainability debate since the environment already is under enormous strain. The understanding is now settling in that without a robust environment we cannot have a healthy economy over a long period. The economy relies on such things as clean air, water, soil, minerals, and other resources, which are products of the physical and biological environment. Biologists measure populations of animals and plants, find correlations, statistics and relationships between and within species, and provide scientific explanations of such phenomena as exploding or shrinking populations. They develop indicators of sustainable fish or plant harvesting, indicators of biodiversity, etc. They also explain the inner workings of species and ecosystems which then lead to explanations of human environmental impact, albeit often a-posteriori. No society exists without an economy nowadays, excluding perhaps tiny groups of hunter-gatherers in the Amazon or Africa. An economy ordinarily requires labor, capital, matter, and energy to function. We know from the laws of physics that matter and energy can neither be created nor destroyed but they may be transformed into other forms of matter and energy or into each other, although the transformation of matter into energy is not of interest here. An economy then takes this useful matter and energy and, with the aid of technology, transforms it into finished products. These products, after their useful life, become garbage or are partially recycled at the expense of energy: in either case, the products generate pollution. Take the example of a car, which is made of iron, copper, nickel, lead, plastics and other materials. Mining of these materials results in pollution. The mined matter is then transferred to a factory where a purified form of the metals is extracted using energy. Next, the auto maker combines all the materials to produce the car. Each and every stage so far has polluted the environment. Finally the user of the car, after some time, takes the car to a dump where it is partially recycled and partially left to rot. Once again the end product is pollution taken up by the environment. If this pollution is qualitatively and quantitatively within the absorbing capacity of the environment, the problem ends there. If it’s not, however, we end up with contamination in the form of unwanted substances in the air, water, and soil. In any case, as shown in Figure 1.1, the economic activity, with the aid of technology, inevitably generates pollution. Of course, this is not done just for the fun of it: the economic process is a sine qua non condition for the existence of currently 6.7 billion people. Otherwise we would go back to a few hundreds of thousands of vulnerable and primitive huntergatherers. The tool of this economic process is technology, the child of science, and its end product is utility and pollution. Technology changes all the time. It provides tools for the detection and measurement of pollutants as well as systems that clean up the environment when possible. The aim of new

Introduction

3

technologies is to produce goods with minimal adverse environmental impact. It therefore becomes clear that both economists and engineers have a stake in the sustainability debate. Based on ideology, economists generate ideas about growth and development which influence both society and the environment. Sustainable development is an idea that means different things to capitalists, ecological economists or Marxists. Sociologists, on the other hand, ask questions about the importance of sustainability to society and the mechanism of raising awareness among people. Equally important is the question of finding motivations to mobilize people in the direction of improving sustainability. Sustainability looks to the future and in some way guarantees an acceptable social and ecological system for future generations. Questions about the rights of as yet unborn people belong to the realm of philosophy as do questions about the rights of other species. Finally, we the common citizens are the ones who enjoy highly polluting sports utility vehicles, electronic gadgets and toys, and air conditioned houses. We consume, often irrationally, we pollute and thus, we breathe dirty air, drink water laced with toxic substances, and eat unhealthy food. We also vote, enjoy life and, when bad things happen, we suffer. Without doubt we have the greatest stakes in the sustainability case. We all could develop sustainably provided: 1) We develop a definition or description of sustainable development 2) We possess mechanisms and we have the political will to do so. This book deals with the first condition and provides some mathematical tools to answer it. The second question, although of the utmost importance and complexity, is outside the scope of this book. The main reason is that question 2 is not scientific but has to do with values, ethics, and politics.

Material recycling

Product remanufacturing

Raw materials

Production

Pollution

Environment

Figure 1.1. The economic process.

Product recycling

Consumption

4


1.2. DIFFICULTIES OF ACHIEVING SUSTAINABILITY People are speaking about sustainability more now because the environmental crisis has become obvious to most of them. A society is an extremely complex system and the possibilities of action within this system many. When a problem arises, its solutions depend on the knowledge, values, and the willingness of the decision makers and the citizens to undertake a certain course of action. Thus complex societal problems such as sustainability are not only technical but primarily cultural and political (Prugh et al. 2000). Roughly speaking, a society is sustainable if it provides basic necessities and happiness to the present generation and generations to come over a very long time. We shall come back to this. In the past, sustainability was not a major issue if at all, because the size of human population and the level of consumption, on average, were such that the terrestrial or aquatic ecosystems were not threatened by human actions on a global scale. The scales of economies were small and there was always a frontier where humans could expand. Today, in contrast, we have reached or exceeded the capacity of the earth to supply resources and absorb pollution. Anything we do has environmental consequences which, when multiplied by all people and then added up, explain for example why global warming is occurring. In the past, human society relied on the consumption of solar energy to satisfy its needs for food and cover. Plants and animals are the products of this solar energy through photosynthesis and they provide food, building materials, clothing, medicines, a stable climate, a stable atmosphere, esthetic pleasure and so on. Fossil fuels are also products of this solar energy: plants and microorganisms trapped in the strata of the earth provide coal, oil, and natural gas, which in essence represent the nuclear energy of the sun which these species received via photosynthesis thousands of years ago. Today we consume the ecosystem itself to support our growth. We remove forests and fish beyond replenishment, pump water beyond recharging, drive species to extinction, poison the air, water, and soil, sometimes irreversibly. To see our impact on the environment numerically, let us present two examples: First, suppose you live in Greece around the time this book was written. • You have a GNI PPP (gross national income in purchasing power parity, which is gross national income converted to international dollars—international dollars correspond to goods and services one could buy in the U.S. with a certain amount of money) of U.S. $32,520 (PRB 2008). • You are expected to live on average 77 years if you are male or 81 years if you are female (nature also loves women). • You wake up in the morning in a room with central heating in winter or air conditioning in summer, you turn on a few lights, make your coffee, toast your bread and then you drive or ride a bus or train to work. All in all by the end of the day your consumption has led directly or indirectly to the emission of 26.4 kilograms (kg) of carbon dioxide (CO2) or equivalently 7.2 kg of carbon (C). In a year you have burdened the environment with over 2.6 tons of C which partly contributes to global warming (International Atomic Energy Agency 2003).

Introduction

5

• If you wear a gold wedding ring you have contributed 3 tons of discharge at a mine in South Africa or the U.S. (47% of gold is recycled, thus polluting less). Make this 10 to 20 tons of discharge if you wear a gold watch (Gardner and Sampat 1998). • For lunch you eat a 250 gram (g) beef steak. If this meat was produced in Brazil, it destroyed 5 m2 of tropical forest, making a small contribution to global warming. If the animal was one year old, it had produced 27 kg of methane (CH4), a very potent greenhouse gas. To obtain a further picture add 750 liters (lt) of water and 1.7 kg of grain to make the 250 g steak (Durning and Brough 1991). If you top the steak with 100 g of rice, then your lunch dish required about 90 g of nitrogen fertilizer, of which 80 g escaped directly into the environment contributing to global warming, eutrophication, water and air pollution. • Your car requires 160 tons of water for its steel parts and tires. Its battery contains about 8 kg of lead which has generated about 310 kg of pollution at a mine in Australia or the U.S. (73% is recycled). The car has about 10 kg of copper which has generated 990 kg of discharge somewhere in Chile or the U.S. (60% is recycled) (Gardner and Sampat 1998). • For dinner you decide to eat a 300 g farmed trout and a lettuce salad. You then contribute to the release of 400 g of particles, mostly organic, in a river in Epirus (Northern Greece), 19 g of ammonia, 4 g of nitrites and nitrates, 5 g of phosphorus, and small quantities of antibiotics, formaldehyde and other disinfectants. This fish consumed 700 g of grain whereas your lettuce was grown with nitrogen and pesticide inputs (Papoutsoglou 1996). • By the end of the day you have generated 0.8 kg of garbage (50% organic, 20% paper) which usually ends up in some dump, often illegal (Eurostat 2005). Now suppose you live in the U.S. • You have a GNI PPP of U.S. $45,850. • You are expected to live on average 75 years if you are male, 2 years less than a Greek (perhaps the latter enjoys life more), or 81 years if you are female. • You have a 55% chance to be overweight or obese. • You consume 200 g or 53 teaspoons of sweeteners per day, mostly through processed foods. • You generate 5.5 tons of C per year or 15 kg per day, twice that of a Greek. • Your car costs more environmentally because you use it more: In the U.S., 3 kg of C releases per day are due to transportation. • Every day you use materials such as newspapers, paper and plastic bags, cars, appliances, gravel, sand, nitrogen, copper, zinc, etc. All in all, excluding fuel and food, you have consumed 101 kg of materials or 36,850 kg per year, or 2,838 tons per lifetime. Every day you generate 2 kg of garbage, 1/3 of which is packaging (Gardner and Sampat 1998). It has been estimated (Power 1996) that 90% of the economic activity in rich countries satisfies or aims to satisfy wants, which later become needs. Such is the story of cable TV, cellular phones, fax machines, SUV’s, electronic gadgets, large houses, and air travel. In

6


some sense we are all trapped in an endless cycle of consumption. Still, consumption patterns differ, despite a tendency to homogenization aided by the enormous advances in communication, transportation, and international trade. People exhibit different consumption behavior in the U.S., Greece, Nepal, Chad, or Peru; this is due to their different economic power and in a great measure to their personal and social values. Thus sustainability, while dealing with the economic activity, is also about values. The environmental behavior of a nation is also affected by values among other things. The environmental ethic in Northern Europe for example, is quite advanced, although one may argue that the Northern Europeans enriched themselves by destroying enough of their environment so they can afford now to protect whatever remains of it. Others argue that poor countries supply rich countries with cheap labor, resources, and dumps for their toxic or radioactive garbage, becoming heavily polluted in the process. However, the point that sustainability is mainly a matter of values remains valid. There are many nongovernmental organizations, scientists, or citizens who raise awareness about unfair treatment of poor countries and press for legislation and effective measures against this type of exploitation. In all western type democracies, a small group of politicians, advisers, and lobbyists influence or directly make all the decisions of the state, whereas the rest of us are just “consumers.” The great ancient Athenian leader Pericles in his famous speech “Epitaph,” honoring the dead Athenians of the Peloponnesian War, called anyone who does not participate in politics “useless.” Of course personal participation in public affairs is possible when population is small, just a few hundred of thousands. Even in Ancient Athens, however, women were excluded from the public gatherings where debates took place and decisions were made. In the end a few thousand people were deciding for about 300,000 but this is a far cry from the present “democratic” practice where often a handful decide for hundreds of millions. Sustainability is about values, opinions, arguments, consensus or in one word politics which, of course, needs to be based on good science to be correct and beneficial. There are many examples in history, and even today, of more or less directly democratic systems besides Athens. Such systems might play a very important role in the process of transforming us from consumers into informed citizens who strive for sustainability. The reader may find interesting ideas about the subject of politics and sustainability in Prugh et al. (2000).

1.3. WHAT IS SUSTAINABILITY? The biological and physical environment, in two words the global ecosystem, provides the economy with: a) Resources such as wood, metals, minerals, fuels, food, drugs, water, air, fiber and so on. b) Services as for example the cycles of H2O, C, CO2, N, O2; photosynthesis; soil formation. c) Mechanisms to absorb waste.

Introduction

7

According to an attempt to monetize all three global services (Costanza et al. 1997), their total value in 1997 was $33 trillion/year. This number could be disputed on various grounds, but its enormity gives us an idea of the importance of the environment. Economic growth is based on these three services and since the global ecosystem does not grow, economic growth cannot continue indefinitely. If this is so, and since the human mind cannot stop inventing new things and improving existing systems and processes, what should be done to preserve a basic level of satisfaction of necessities that would provide happiness to our society for a long time? In other words, how can we achieve sustainability? It is claimed by several economists that substituting one form of capital for another solves the problem of finiteness of the ecosystem and the concomitant existence of limits. A worker can be replaced by a robot, wood for the shipping industry by iron, or metals for aircraft by synthetic materials. But substitution has its limits too, which result from ecological limits; in the long run, total output will go down. Replacing harpoon whaling by modern ships depletes the population of whales and substituting traditional nets with driftnets depletes fisheries. More importantly, substitution often is impossible owing to the scale of the global ecosystem or the laws of nature. There is no substitute for a stable climate which we are changing rapidly with dire consequences as well as there are no substitutes for extinct species and their services to nature. One of the most basic economic indicators of the state of an economy is the gross national product or GNP, which is the total value of all final goods and services produced in a country in one year, plus income earned by its citizens abroad, minus income earned by foreigners in the country. The total value of final goods and services produced within a country is the gross domestic product (GDP). GNP is a useful indicator, but also one-sided and deceptive; if someone cuts down all forests or depletes fisheries, the GNP goes up even though economic catastrophy is near. Such growth of the GNP is unsustainable. In fact, GNP growth and consumerism go hand in hand. There have been attempts to correct the single-sidedness of GNP or GDP. One such attempt (Venetoulis and Cobb 2004) defines the Genuine Progress Indicator (GPI). GPI includes the value of household work, parenting, volunteer work, services of customer durables, highways and streets, which GDP ignores. It also subtracts defensive expenses such as auto accidents, social costs such as cost of crime, and depreciation of environmental assets. As an example, the GDP and GPI per capita for the U.S, in the year 2000 were $35,000 and $11,554 respectively. Sustainable development does not necessarily mean growth, but improvement of the various societal sectors, as for example health, education, or the state of the environment. So what is sustainable development (SD)? To some people it is a contradiction in terms. There are economists who believe that SD means: • Sustaining economic growth indefinitely while others see it as: • The ability to maintain desired social values, institutions, cultures or other social characteristics or as: • Development (improvement) that can be continued for a long time.

8


In 1980, the International Union for the Conservation of Nature and Natural Resources (IUCN) wrote in its World Conservation Strategy (WCS) (Lélé 1991) that it espouses “the overall aim of achieving sustainable development through the conservation of living resources.” Since then the concept of SD gained momentum as the environmental degradation intensified and became obvious to many. Today most proponents of SD take it to mean: • The existence of environmental and economic conditions needed to sustain human wellbeing at a given level over a very long time. In more general terms, SD according to the Brundtland Report (WCED 1987) is: • Development that meets the needs of the present without compromising the ability of future generations to meet their own needs. In a slightly different way, IUCN defined SD as: • Development that improves the quality of human life within the carrying capacity of supporting ecosystems. SD was the central subject of the Earth Summit held in Rio de Janeiro in 1992. Agenda 21, which resulted from the meeting, gives a comprehensive list of actions needed to achieve SD. Leaders from over 150 states committed themselves to undertaking those actions that will render development sustainable. It is common to classify SD as strong and weak: • Strong sustainability is primarily focused on the environment and ignores economics such as the cost of achieving sustainability. It considers pollution, emissions, biodiversity, soil erosion etc. • Weak sustainability is basically economic sustainability. It ignores environmental inputs to the economy and considers consumption, economic growth, financial value etc. In this book sustainability integrates both aspects, environmental and economic. Two questions arise in the context of SD: 1. What is the space over which sustainability is considered? Answer. Ecosystem, a region, a country, and since pollution often has no borders, the globe. As the scale becomes smaller, the boundaries of the system which we examine become more uncertain. The sustainability of California, for example, depends on regional attributes such as local pollution, water resources, coastal areas, agriculture, education, health etc. but also external factors, e.g., climate, ozone depletion, water resources of the Colorado river, foreign imports and so on. In this book the most common scale of SD is that of a country. 2. What is the time horizon of sustainability? Answer. It depends on the specific attribute. For example, the climate exhibits several periodicities due to a change of the earth’s orbit from almost circular to elliptic with a periodicity of 100,000 years. Also the axis of motion of the earth has two periodicities of

Introduction

9

40,000 and 20,000 years respectively. Thus, the climate naturally oscillates between cold and warm with a variation in average global temperature of 5 °C or more. Incidentally, such climatic changes occur over periods of thousands of years. Today’s global warming occurs within only a few decades. The fluidity of time scales in the context of SD is shown in two more examples: pest problems have time scales of the order of 20 years. The life expectancy of species on the other hand is between 1 and 5 million years on average, although from the fossil record we know of species that survived only a few thousand years before becoming extinct. Perhaps the most important aspect of time is that of uncertainty. We simply cannot plan our society for thousands of years into the future. Often even a few decades are beyond our predictive and planning capabilities. We have only long-term goals and intentions and all the time we make adaptive decisions which bring us as close to our goals as possible. In the sequel a mathematical methodology will be developed whereby sustainability will be assessed via fuzzy logic. In effect the model that assesses sustainability serves also as a definition of the concept.

1.4. PROBLEMS-QUESTIONS 1.1. Give an example where substitution is impossible. Give one more where substitution leads to output reduction. 1.2. Outline two changes of national accounting that would make GDP more representative of the sustainability of a country. 1.3. List five values that improve sustainability. Provide brief explanations. 1.4. Energy consumption is the main contributor to CO2 emissions. Energy is used in the following areas: • transportation • households • industries – agriculture • food production • recreation • tourism Outline 10 actions that would make these sectors more sustainable. 1.5. Atmospheric concentrations of CO2 in ppm (parts per million) in volume followed an almost linear trajectory between 1985 and 2000. The corresponding concentrations were

10


345.9 ppm and 369.5 ppm. If this trend continues what will the corresponding concentrations be in 2050 and 2100?

Chapter 2

INTRODUCTION TO FUZZY LOGIC Fuzzy logic was introduced by Zadeh (1965) as an extension of the classical two-valued logic, in which a proposition is either true or false and an object either belongs or does not belong to a set. Zadeh studied the concept of vagueness by assuming that propositions and set memberships are true with degrees ranging from 0 (100% false) to 1 (100% true). This method can handle incomplete knowledge and inexact or vague data in a systematic way. In this book we develop fuzzy models to assess the sustainability of nations, geographical regions, and organizations.

2.1. WHY ASSESS SUSTAINABILITY VIA FUZZY LOGIC? Sustainability is a multifaceted concept for which there is no widely accepted definition or measurement method. To reveal its dimensions and acquire a better understanding of the concept, we analyze it by following a top-down approach. As discussed in the previous chapter, sustainability has two broad dimensions, ecological and human. In the next chapters we shall explain how these two components break down into a number of secondary components and variables. In defining the sustainability of nations, for example, water availability and water quality are considered to be basic components of the ecological sustainability. Water sustainability is in turn assessed from data about annual availability of renewable water resources, water withdrawals, releases of water pollutants, availability of wastewater treatment plants, and so on. Such variables are used as basic inputs in sustainability calculations. Therefore, mathematically, sustainability is a composition of functions of several variables which, in turn, are also composite functions of more primitive variables. There are two reasons why it is not possible to determine these functions explicitly: 1) Sustainability is an inherently vague and complex concept and cannot be described, let alone measured, by traditional mathematics. Policy makers and scientists often prefer natural language expressions rather than equations or numerical values in assessing sustainability. For example, degrees of law enforcement, the state of civil liberties, and the state of human rights in a country, which are important components of human sustainability, are often obtained by subjective assessments.

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Yannis A. Phillis and Vassilis S. Kouikoglou 2) Statistics and system identification are used to build models for systems whose structure is not known. These methods require a number of input-output measurements, a collection of candidate models, and a criterion to select the best model based on these measurements. The main problem with assessing sustainability using these methods is the lack of output data. Although many of the inputs are measurable, it is impossible to estimate the output, e.g., the sustainability of a country in a given year.

Fuzzy logic, on the other hand, is suitable for assessing sustainability because it can model complex systems about which we have only partial knowledge as to their dynamics, the parameters or inputs that affect them, and the values of these inputs. Fuzzy logic can handle knowledge and data represented in various ways such as mathematical models, linguistic rules, numerical values, or linguistic expressions. In this chapter we briefly review fuzzy logic and its application to the construction of knowledge-based systems.

2.2. FUZZY SETS AND LINGUISTIC VARIABLES 2.2.1. Definitions Definition 2.1. A set is a collection of elements. The empty set ∅ has no elements. We write x ∈ X when we want to say that an element x belongs to the set X. Classical set theory deals with crisp sets, where every element is either member or nonmember of a set. Analogously, every proposition (or statement) in classical two-valued logic is either true or false. This principle, known as the law of the excluded middle, was introduced by Aristotle who developed the earliest formal system of logic. The law of the excluded middle has been strongly criticized by several philosophers and mathematicians as too narrow for describing real-life situations. Heraclitus suggested that propositions could be simultaneously true and not true (for example, light is both wave and particle). In fuzzy logic an element belongs to a fuzzy set with some degree of membership in the interval [0, 1]. This degree represents a subjective assessment of belonging and is called the membership grade of the element. Definition 2.2. Let X be a set and let x be an element of X. A fuzzy set A in X is a collection of ordered pairs [x, μΑ(x)] for x ∈ X. X is called the universe of discourse of A and μΑ(x) is a number in [0, 1] which represents the membership grade of x to A (degree to which x belongs to A). We write A = {[x, μΑ(x)], x ∈ X }. Example 2.1. Consider human height and the fuzzy set A = “tall height.” Let X = {1.60, 1.65, 1.70, 1.75, 1.80, 1.85, 1.90} be a set of heights in meters. One possible definition of A could be


13

A = 0/1.60 + 0/1.65 + 0/1.70 + 0.2/1.75 + 0.8/1.80 + 1/1.85 + 1/1.90 which reads as follows: 1.60 is tall with 0 degree of membership, 1.65 is tall with 0 degree of membership and so on up to 1.90 is tall with degree of membership 1. The notation above of fractions of membership grades and particular values connected by summation signs is quite standard and does not imply division or summation. Now if X is the same but the context is different, A could be represented differently. For example, if A is redefined as “tall height for boys,” a possible representation could be A = 0.2/1.60 + 0.5/1.65 + 1/1.70 + 1/1.75 + 1/1.80 + 1/1.85 + 1/1.90. But if A = “tall height for basketball players,” then A = 0/1.60 + 0/1.65 + 0/1.70 + 0/1.75 + 0/1.80 + 0/1.85 + 0.2/1.90. This example serves as an illustration of the subjectivity of certain situations and how this can be expressed in the context of fuzzy logic. When X is a countable set, e.g., X = {x1, …, xn}, it is customary to denote the fuzzy set A as A = μA(x1)/x1 + μA(x2)/x2 + … + μA(xn)/xn =

n

∑ μ A ( xi )/xi ; i =1

when X is continuous, A is denoted as A = ∫ μ A ( x)/x . X

The sum and integral signs, ∑ and ∫, in the above expressions denote collections of objects “μA(x)/x,” in which the slash acts as a comma. Example 2.2. (a) The fuzzy set A = “integers close to 2” can be defined by assigning a membership grade of 1.0 to the integer value 2 and smaller membership grades to the other integers depending on their distance from 2. For example A = … 0.0/−1 + 0.0/0 + 0.5/1 + 1.0/2 + 0.5/3 + 0.0/4 + …, or, by omitting all elements with zero membership grades, A = 0.5/1 + 1.0/2 + 0.5/3.

14


(b) Different universes of discourse or notions of distance give rise to different definitions of A. For example, the fuzzy set of real numbers close to 2 can be defined by ∞

B = ∫ e −| x − 2| /x . −∞

= (−∞, ∞), and its membership function is

Its universe of discourse is the real line, plotted in Figure 2.1.

μ B (x)

1 0,8 0,6 0,4 0,2

0 -5 -4 -3 -2 -1 0

1

2 x

3

4

5

6

7

8

9

Figure 2.1. Real numbers close to 2.

(c) Crisp sets are special cases of fuzzy sets. Take the English alphabet Y = {a, …, z} and the fuzzy set C = “English alphabet” in Y. Then C = 1.0/a + … + 1.0/z, that is, a crisp set is a fuzzy set with membership grades 1 everywhere. Definition 2.3. The function μA: X → [0, 1] is the membership function of the fuzzy set A in X. Triangular, trapezoidal, Gaussian, and sigmoid functions are the most commonly used membership functions of fuzzy sets. The choice depends on the particular application and is made subjectively or by trial and error. A few more definitions follow. Definition 2.4. The support of A is the set of all elements of X with nonzero membership in A, or symbolically support(A) = {x ∈ X | μΑ(x) > 0}. For the fuzzy set A of Example 2.2a we have support(A) = {1, 2, 3}.


15

Definition 2.5. A fuzzy singleton, denoted μ(x)/x, is a fuzzy set whose support contains only one element x. Fuzzy singletons are the simplest fuzzy sets. They are also at the boundary between fuzzy logic and two-valued logic. For example, every crisp singleton, that is, a set with a sole element, is also a fuzzy singleton (with membership grade 1). For this reason, some methods, to be described in subsequent sections, use fuzzy singletons as “interfaces” to transform fuzzy variables into crisp ones and vice versa. Definition 2.6. The height of a fuzzy set A in X is the maximum value of its membership function. We write height(A) = max μA(x). x∈X

Definition 2.7. A peak value of A is a point x ∈ X at which μA(x) is maximum, i.e.,

μA(x) = height(A). Definition 2.8. A fuzzy set A is normal if height(A) = 1, otherwise it is subnormal. The set of all peak values of a normal fuzzy set A is called core or nucleus of A. Thus, core(A) = {x ∈ X | μA(x) = 1} The fuzzy set B = “real number close to 2” shown in Figure 2.1 is normal and has only one peak value, 2. Such sets are called fuzzy numbers. Definition 2.9. Given two fuzzy sets A and B in the same universe of discourse X, we say that A is a fuzzy subset of B and write A ⊆ B if μΑ(x) ≤ μΒ(x) for every x ∈ X. The fuzzy sets A and B are equal if μA(x) = μΒ(x) for every x ∈ X; we write A = B. Definition 2.10. Fuzzy sets in X are sometimes referred to as fuzzy subsets of X. Since any crisp set X is equivalent to the fuzzy set {[x, μ(x) = 1], x ∈ X }, a fuzzy set A in X satisfies μA(x) ≤ 1 for all x. Hence we may write A ⊆ X. Fuzzy sets are commonly used to express the way humans extract qualitative information from numerical, categorical or linguistic data, and the way they rate, summarize, and process this information to make decisions and assessments. To this end, a fundamental concept of fuzzy logic is the notion of linguistic variable introduced by Zadeh (1973, 1975).

16


Loosely speaking, a linguistic variable is a variable “whose values are words or sentences in a natural or artificial language,” as Zadeh has put it. More precisely, a linguistic variable is a fuzzy partition of some physical domain X into possibly overlapping regions. Each region is represented with a fuzzy set in X called linguistic value. Take for example the variable “water sustainability” which can be assessed numerically on a scale from 0 to 1. We represent it with the linguistic values “bad,” “average,” and “good.” These values are fuzzy sets and are called the primary terms as opposed to modifier terms such as “very,” “fairly,” “more or less,” “not,” “and,” “or,” etc. Modifier terms are used to generate additional linguistic values, such as “very bad,” “very very bad,” “fairly good,” “not very good,” “bad or average,” and so on. Proceeding in this manner we may end up with a vast number of fuzzy sets. An alternative is to introduce syntactic rules of some natural language to generate different linguistic values. For example, consider the rule “bad → very bad” whereby the first term “bad” is replaced by the term “very bad.” Starting from “bad” and applying this rule successively produces an infinite sequence of linguistic values: “very bad,” “very very bad,” “very very very bad,” and so on. Systems which can generate words and phrases according to a given grammar are studied in the context of formal languages and theoretical machines. The theory of such systems is beyond the scope of this book. We also need a rule which associates with each linguistic value its membership function in [0, 1]. As previously, when the number of linguistic values is very large, we can define appropriate operators for the modifiers “very,” “not,” “or,” etc., to change the membership functions of the primary terms. For example, given the membership function of S = “small number” we can define

μvery S(x) = [μS(x)]2 μmore or less S(x) = [μS(x)]1/2 for every x ∈ [0, 1]. Thus, if the number 3.14 is considered to be a small number with degree of certainty 0.5, it is very small with membership grade 0.25 and more or less small with membership grade about 0.7. Definition 2.11. A linguistic variable N is a quadruple [T(N ), X, G, μ ] in which T(N ) is the set of names of linguistic values of N, X is a universe of discourse, G is a context-free grammar used to generate elements of T(N ), and μ is a mapping which assigns a membership function to each linguistic value of T(N ). T(N ) is referred to as the term set of N. For simplicity, it is customary to use the same symbol to denote both the linguistic variable (the quadruple above) and its name. The terms “fuzzy set,” “fuzzy subset,” and “linguistic value” bear the same meaning when they refer to linguistic variables. Example 2.3. Excessive use of water reduces water sustainability. Water use intensity is calculated by the ratio of the annual amount of water withdrawn in a region to the amount of renewable water resources in that region. According to the Organization of Economic


17

Cooperation and Development (OECD 2004), water use intensity is classified as “low,” “moderate,” “medium-high,” or “high” according to the following scale: L = “low intensity of water use” = {x | x < 0.1} M = “moderate intensity of water use” = {x | x ∈ (0.1, 0.2]} MH = “medium-high intensity of water use” = {x | x ∈ (0.2, 0.4]} H = “high intensity of water use” = {x | x > 0.4}. A particular fuzzy representation of these sets (with no additional modifier terms) is shown in Figure 2.2. intensity of water use L

M

MH

Linguistic variable (N ) Linguistic values (T )

H

Membership functions (μ ) 1

μ(x)

0 0

0.4

1 x

Figure 2.2. The linguistic variable “intensity of water use.”

The membership functions of the fuzzy sets L, M, MH, and H are given by

1, x ≤ 0.05 ⎧ ⎪ 0.15 − x ⎪ μL(x) = ⎨ , 0.05 ≤ x ≤ 0.15 ⎪ 0.1 0, otherwise ⎪⎩

0, ⎧ ⎪ x − 0.05 ⎪ , ⎪ 0.1 μM(x) = ⎨ ⎪ 0.30 − x , ⎪ 0.15 ⎪ 0, ⎩

x ≤ 0.05 0.05 ≤ x ≤ 0.15 0.15 ≤ x ≤ 0.30 otherwise

18


0, ⎧ ⎪ x − 0.15 ⎪ , ⎪ 0.15 μMH(x) = ⎨ ⎪ 0.45 − x , ⎪ 0.15 ⎪ 0, ⎩

x ≤ 0.15 0.15 ≤ x ≤ 0.30 0.30 ≤ x ≤ 0.45 otherwise

0, x ≤ 0.30 ⎧ ⎪ x − 0.30 ⎪ , 0.30 ≤ x ≤ 0.45 μH(x) = ⎨ ⎪ 0.15 1, otherwise. ⎩⎪ In set theory, two approximately equal objects or elements may belong to different sets. In the above example, an annual water use of 0.4 per amount of renewable resources belongs to the crisp set MH. The value 0.400001, which is slightly higher than 0.4, belongs to the set H. In fuzzy sets, however, the transition from membership to non-membership is gradual: both 0.4 and 0.400001 belong to the fuzzy set MH with membership grade about 1/3 and to the fuzzy set H with membership grade about 2/3 (see Figure 2.2).

2.2.2. Convex Fuzzy Sets Convexity in the usual sense refers to the shape of geometric bodies, such as line segments, planar regions, solids, or vector spaces of higher dimensions. As illustrated in Figure 2.3a, convexity means that for every pair of points x and y in X, all points of the straight line between x and y are also elements of X or, equivalently,

λx + (1 − λ)y ∈ X for all λ ∈ (0, 1). In fuzzy set theory, convexity is taken to mean that the membership function has no “dips.” Thus the fuzzy sets A and C of large and medium-sized numbers shown in Figure 2.4 are convex while the sets B and D are not. A condition for convexity is the following. Definition 2.12. A fuzzy set A in X is convex if, for every pair of points x, y ∈ X,

μA[λx + (1 − λ)y] ≥ min[μA(x), μA(y)] for all λ ∈ (0, 1).

Introduction to Fuzzy Logic convex sets x

z

x

z

x

z

19 nonconvex sets

z = λx + (1 − λ)y

y

x

y

z

x

z

x

z

y

(a)

y

y

y

(b)

Figure 2.3. Crisp sets (a) convex; (b) nonconvex.

Where do we need convex fuzzy sets? In most practical applications, the fuzzy sets used are linguistic values of some measurable characteristic as, for example, “light weight” or the sets A = “large numbers” and C = “medium-sized numbers” of Figure 2.4a. The set B of Figure 2.4b is not suitable for describing large numbers, because the number 5 belongs to B with a smaller membership grade than the number 4. Do the fuzzy sets C and D qualify as fuzzy sets of medium-sized numbers? The answer requires more delicate reasoning: If we believe that the numbers 2 and 4 are medium-sized with some degrees of certainty, then any intermediate number should be considered to be medium-sized with some intermediate degree of certainty. This is exactly the condition of Definition 2.12. We see in Figure 2.4 that μC(3) > min[μC(2), μC(4)] whereas μD(3) < min[μD(2), μD(4)]. Therefore, the fuzzy set C agrees with our conception of medium-sized numbers but the set D does not agree. A = “large numbers” 1

B 1

0

0 4

0

5

6

4

0

C = “medium-sized numbers” 1

5

6

D 1

0

0 0

2

3

4

(a)

0

2

3

4

(b)

Figure 2.4. Fuzzy sets: (a) convex; (b) nonconvex.

Despite their apparent differences, the notions of convexity of crisp and fuzzy sets are related mathematically by means of the α-level sets which are defined as follows.

20


Definition 2.13. The crisp set of all elements of X with grade of membership to A at least α is called the α-level set; symbolically Aα = {x ∈ X | μΑ(x) ≥ α}. For example, the α-level set of the fuzzy set A shown in Figure 2.5a is the interval Aα = [x1, x2] marked with a heavy line. We observe that A is a convex fuzzy set in the sense of Definition 2.12 and its level sets Aα are also convex, in the usual sense, for every α. The fuzzy set B in Figure 2.5b is not convex; its α-level set Bα is also not convex, because it contains two separate segments, [y1, y2] and [y3, y4], with λy2 + (1 − λ)y3 ∉ Bα. A

B

α

α

x2

x1

y 1 y2 y 3

Aα

y4

Bα

(a)

(b)

Figure 2.5. Fuzzy sets: (a) convex; (b) nonconvex.

We see from the above examples that the convexity of fuzzy sets is equivalent to the convexity of their level sets. Hence, an alternative definition of convexity for fuzzy sets is Definition 2.14. A fuzzy set A is convex if all its α-level sets are convex, i.e., if x and y are any points in Aα, then

λx + (1 − λ)y ∈ Aα for all λ ∈ (0, 1) and all α ∈ [0, 1].

2.2.3. Algebra of Fuzzy Sets The basic operations of crisp sets are complementation, intersection, and union. In classical logic, complementation corresponds to the logical negation “not,” intersection corresponds to conjunction (connective “and”) and union corresponds to disjunction (connective “or”). These operations can be extended to fuzzy sets as follows (Zadeh 1975): Definitions 2.15. Let A be a fuzzy set in X. – • The complement A; of A is a fuzzy set in X with membership function

μA; –(x) = 1 − μA(x), x ∈ X.


21

• The intersection A ∩ B is a fuzzy set in X with membership function

μA∩B(x) = min[μA(x), μB(x)], x ∈ X. • The union A ∪ B is a fuzzy set in X with membership function

μA∪B(x) = max[μA(x), μB(x)], x ∈ X. Example 2.4. Consider the crisp sets X = “intensity of water use” = {x | x > 0} and H = “high intensity of water use” = {x | x > 0.4}, as defined in Example 2.3. Crisp sets satisfy the following principles – law of contradiction: H ∩ H; = ∅, – law of the excluded middle: H ∪ H; = X. The law of contradiction ensures that there are no elements of X which are members and non-members of H and the law of the excluded middle says that each element of X is either member of H or non-member of H. Let us examine these principles in the context of fuzzy sets. Take the membership functions 0, x ≤ 0.30 ⎧ ⎪ x − 0.30 ⎪ μH;–(x) = 1 − μH(x) and μH(x) = ⎨ , 0.30 ≤ x ≤ 0.45 ⎪ 0.15 1, otherwise ⎪⎩ – – as shown in Figure 2.6a. The membership functions of H ∩ H; and H ∪ H; are shown in Figs 2.6b, 2.6c. 1

– H

H

μ(x)

– H∪H – H∩H

0

(a)

(b)

(c)

– Figure 2.6. Fuzzy sets: (a) H and H; ; (b) their intersection; (c) their union.

It is worth noting that, contrary to what holds in set theory, when H is a fuzzy set in – – X, then H ∩ H; ≠ ∅ and H ∪ H; ≠ X.

22


As in crisp sets, the operations of union and intersection can be extended to several fuzzy sets. For example, the intersection of three fuzzy sets is defined by μA∩B∩C(x) = min[μA(x), μB(x), μC(x)]. The binary operations min and max are special cases of triangular norms and triangular conorms. Fuzzy intersections and unions may be defined generally via triangular norms and triangular conorms as follows: Definition 2.16. A triangular norm or T-norm for short is a bivariate function T: [0, 1]×[0, 1]→[0, 1] satisfying T(0, 0) = 0

(boundary condition)

T(1, x) = x

(1-idempotent law)

T(x, y) = T(y, x)

(symmetry)

T(x, z) ≤ T(y, z) whenever x ≤ y

(monotonicity)

T[T(x, y), z] = T[x, T(y, z)]

(associativity).

The following are examples of fuzzy intersections defined by T-norms: minimum:

μA∩B(x) = min[μA(x), μB(x)]

algebraic product: μA∩B(x) = μA(x)μB(x) bounded difference: μA∩B(x) = max[0, μA(x) + μB(x) − 1] Hamacher:

μA∩B(x) =

μ A ( x)μB ( x) μ A ( x) + μB ( x) − μ A ( x)μB ( x)

Einstein:

μA∩B(x) =

μ A ( x)μB ( x) 1 + [1 − μ A ( x )][1 − μ B ( x )]

Dubois-Prade:

μA∩B(x) =

μ A ( x)μB ( x) , p < 1. max[ μ A ( x ), μ B ( x ), p ]

Definition 2.17. A triangular conorm or S-norm for short is a bivariate function S: [0, 1]×[0, 1]→[0, 1] satisfying S(1, 1) = 1

(boundary condition)

Introduction to Fuzzy Logic S(0, x) = x

(0-idempotent law)

S(x, y) = S(y, x)

(symmetry)

S(x, z) ≤ S(y, z) whenever x ≤ y

(monotonicity)

S[S(x, y), z] = S[x, S(y, z)]

(associativity).

23

S-norms have the same properties as T-norms, except that their neutral element is 0 instead of 1. The reader can easily verify that if T(x, y) is any T-norm, then the function S(x, y) = 1 − T(1 − x, 1 − y) is an S-norm. If we define: i) the complement of a fuzzy set by μA; –(x) = 1 − μA(x) ii) the intersection of two fuzzy sets by μA∩B(x) = T[μA(x), μB(x)] and iii) the union of two fuzzy sets by μA∪B(x) = S[μA(x), μB(x)], then Eq. (2.1) is consistent with De Morgan’s law A∪B = A ∩ B . The following S-norms are derived from the T-norms given above using Eq. (2.1): maximum:

μA∪B(x) = max[μA(x), μB(x)]

algebraic sum: μA∪B(x) = μA(x) + μB(x) − μA(x)μB(x) bounded sum: μA∪B(x) = min[1, μA(x) + μB(x)] Hamacher:

μA∪B(x) =

μ A ( x) + μB ( x) − 2μ A ( x) μB ( x) 1 − μ A ( x)μB ( x)

Einstein:

μA∪B(x) =

μ A ( x) + μB ( x) 1 + μ A ( x)μB ( x)

Dubois-Prade: μA∪B(x) =

[1 − μ A ( x )][1 − μ B ( x )] , p < 1. max{[1 − μ A ( x )], [1 − μ B ( x )], p}

(2.1)

24


2.2.4. The Extension Principle So far we have considered operations of fuzzy sets in the same universe of discourse. As discussed in Section 2.1, sustainability is a function of heterogeneous variables. For example, water sustainability, which is a component of the overall sustainability, depends on “water use intensity,” “release of water pollutants,” etc., which have different physical domains. In this section we define fuzzy functions and Cartesian products of fuzzy sets having different universes of discourse. This is done with the use of the so-called extension principle. A function f: X→Y is a relation which associates each element x of X with a single element y = f (x) of Y. X is called the domain of f and Y is its range. Some elements of Y may not be associated with elements of X. These elements are of no interest in studying f. The collection of all interesting elements y such that y = f (x) for at least one x in X is denoted f (X ). Thus, the function f: X→Y maps X into the set Y and onto the set f (X ). Each element y of f (X ) corresponds to one or more elements of X. For example, y = | x | corresponds to both x and −x. We write y = f (x), x ∈ X, y = f (x' ), x' ∈ X, y = f (x" ), x" ∈ X, …. A proposition which summarizes the above conditions is “y belongs to f (X ) whenever x belongs to X or x' belongs to X or x" belongs to X …” where “or” denotes the logical union. Now let A be a fuzzy set in X with membership function μA(x). By taking “or” to mean the usual fuzzy union in the conditions given above, we can define the fuzzy set f (A ) in f (X ) as follows:

μf (A )(y) = degree to which y belongs to f (A ) = max[μA(x), μA(x' ), μA(x" ), …] = max μ A ( x) . x∈ X f ( x )= y

(2.2)


25

Example 2.5. (a) Let f (x) = x2 and A = 0.65/−1 + 1.0/0 + 0.5/1. Then X = {−1, 0, 1}, f (X ) = {0, 1}, and from Eq. (2.2)

μf (A )(0) = 1 and μf (A )(1) = max[μA(−1), μA(1)] = 0.65. (b) Next consider the same function f (x) = x2 but a different fuzzy set A, as shown in Figure 2.7a. A

f(A)

1

1

μA(x)

μf(A)(y)

0 0

0.3

0.6

x

0 0

0.09

(a)

0.36

y

(b)

Figure 2.7. Fuzzy sets: (a) A; (b) f (A).

The membership function of A is

1, 0 ≤ x ≤ 0.3 ⎧ ⎪ 0.6 − x ⎪ , 0.3 ≤ x ≤ 0.6 μA(x) = ⎨ ⎪ 0.3 0, otherwise. ⎪⎩ Applying Eq. (2.2) we obtain

μf (A )(y) = max μ A ( x) x∈ X x2 = y

⎧⎪max μ A ( x), y = x 2 , x ∈ [0, 0.6) =⎨ x ⎪⎩ 0, otherwise because μA(x) = 0 for all x outside [0, 0.6). Also μA(x) = 1 for x ∈ [0, 0.3] and one-to-one in (0.3, 0.6); thus

⎧ 1, y = x 2 ∈ [0, 0.09] ⎪ μf (A )(y) = ⎨ μ A ( x ), y = x 2 ∈ (0.09, 0.36) ⎪ 0, otherwise; ⎩

26


1, y ∈ [0, 0.09] ⎧ ⎪ = ⎨ μ A ( y ), y ∈ (0.09, 0.36) ⎪ 0, otherwise ⎩

1, y ∈ [0, 0.09] ⎧ ⎪ ⎪ 0 .6 − y =⎨ , y ∈ (0.09, 0.36) ⎪ 0.3 0, otherwise ⎪⎩ as shown in Figure 2.7b. Functions of several variables can be viewed as functions of one composite variable or vector. In set theory, several sets can be combined using the Cartesian product. The Cartesian product X1 × … × Xn of n crisp sets X1, …, Xn is a set whose elements are the ordered pairs (x1, …, xn) for all x1 ∈ X1, …, xn ∈ Xn. Symbolically, X1 × … × Xn = {(x1, …, xn) | x1 ∈ X1, …, xn ∈ Xn }. When all sets are the same, i.e., X1 = … = Xn = X, the Cartesian product is denoted Xn. Consider now the fuzzy sets A1 and A2 in X1 and X2 respectively. To define the Cartesian product of A1 and A2 we must determine the degree to which the pair (x1, x2) belongs to A1 × A2. In ordinary sets, (x1, x2) ∈ X1 × X2 is interpreted as “x1 belongs to X1 and x2 belongs to X2.” The above expression can be made fuzzy if we take “and” to mean the usual fuzzy intersection. Therefore, the Cartesian product A1 × A2 of fuzzy sets A1 and A2 is a fuzzy set with membership function

μA1×A2(x1, x2) = min[μA1(x1), μA2(x2)], x1 ∈ X1, x2 ∈ X2.

(2.3)

Finally, suppose the function f: X→Y maps elements x = (x1, …, xn) of a vector space X = X1 × … × Xn to elements y of f (X ). Take the Cartesian product A = A1 × … × An of the fuzzy sets Ai in Xi and construct the fuzzy set f (A ) as follows: First, use Eq. (2.3) to compute the membership function of the fuzzy set A, i.e.,

μA(x1, x2, …, xn) = min[μA1(x1), …, μAn(xn)]; then apply Eq. (2.2) to get


μf (A )(y) =

=

max

( x1 , x2 , …, xn ) ∈ X f ( x1 , x2 , …, xn ) = y

max

( x1 , …, x n ) ∈ X f ( x1 , …, xn ) = y

27

μ A ( x1 , … , x n )

min[μA1(x1), …, μAn(xn)].

(2.4)

This is the extension principle of fuzzy functions. The function μf (A )(y) is determined by solving a nonlinear programming problem. Example 2.6. Consider the function y = f (x1, x2) = x1 + x2 in i = 1, 2, with membership functions

2

and the fuzzy sets Ai in ,

−ix

⎧e μAi(x) = ⎨ x ≥ 0; 0 ⎩ otherwise. Then, for A = A1 × A2,

μf (A )(y) = max min[μA1(x1), μA2(x2)]. ( x1 , x2 ) x1 + x2 = y

It follows from Eq. (2.4) and the definitions of μAi(x), i = 1, 2, that μf (A )(y) is zero for negative values of y, x1, and x2. It then remains to consider cases in which xi ≥ 0 and y ≥ xi. The membership function of f (A ) is written

μf (A )(y) = max min[e − x1 , e −2 ( y − x1 ) ] . x1 y ≥ x1 ≥ 0

Now fix y and consider the exponential terms in the right side of the above equation. Both are continuous functions of x1. As x1 increases from 0 to y the first term, e−x1, is continuous and decreasing in x1, taking values in the interval [1, e−y]. The second term, e−2(y − x1), is increasing in x1 and equals e−2y for x1 = 0 and 1 for x1 = y. Both terms equal e−2y/3 at the point x1 = 2y/3. It then follows by the monotonicity properties that e−2y/3 is greater than the second term for x1 < 2y/3 and also greater than the first term for x1 < 2y/3. Hence, for each x1 ∈ [0, y] we have e−2y/3 ≥ min[e−x1, e−2(y − x1)] and, therefore, the membership function is given by

μf (A )(y) = max min[e − x1 , e −2( y − x1 ) ] = e−2y/3. x1 y ≥ x1 ≥ 0

28


For functions defined in n with n > 2, a general method to solve nonlinear programming problems arising in the application of the extension principle is presented in Baas and Kwakernaak (1977, Theorems 1–3). T-norms and S-norms other than min and max can be used in the extension principle to simplify the problem of computing μf (A )(y). A common choice is

μf (A )(y) =

n

∏ μ A ( xi ) , ( x , …, x ) ∈ X max

f

1

n ( x1 , …, xn )

i

= y i =1

(2.5)

where Π denotes the algebraic product.

2.2.5. Fuzzy Relations and Compositions of Fuzzy Relations A relation R(X1, …, Xn) among n sets X1, …, Xn is a subset of their Cartesian product X = X1 × … × Xn. We write R(X ) ⊆ X. Put otherwise, a relation R on X is a rule for determining whether an n-vector (x1, …, xn) ∈ X satisfies a certain condition. A relation between two sets is called binary. Relations are more general than functions, as shown in Figure 2.8. Relation (x, y): • each x m ay correspond to several y’s • each y m ay correspond to several x’s

Function: • each x corresponds to a single y • each y m ay correspond to several x’s

y

y x

x

Figure 2.8. Illustration of the difference between a relation and a function.

A relation R(X ) among n sets, can be represented by an n-dimensional array μR(x) where x = (x1, …, xn) ∈ X and

⎧1

μR(x) = ⎨ x ∈ R(X ); 0

⎩ otherwise.

For instance, the binary relation R(x1, x2) = “x1 is less than x2” with x1 ∈ X1 = {0, 5} and x2 ∈ X2 = {0, 5, 10, 15, 20} may be expressed by the 2 × 5 matrix

Introduction to Fuzzy Logic 0

5

10

15

29

20

⎡0 1 1 1 1 ⎤ ⎢0 0 1 1 1 ⎥ ⎣ ⎦

0 5

If we allow set membership to take values in [0, 1], then R(X ) becomes a fuzzy relation in the Cartesian product X with membership function μR(x). We write R(X ) = {[x, μR(x)] | x ∈ X }. The following example presents an array representation of a fuzzy binary relation. Example 2.7. Take the variables x1 = “annual withdrawal of water resources per capita” and x2 = “annual renewable water resources per capita” (in m3) corresponding to a given country. Assume for simplicity that both variables have a common finite universe of discourse X = {0, 5, 10, 15, 20, 25}. The fuzzy relation R = “annual withdrawal is much less than renewable resources” defined on X 2 can be represented by the matrix

x1 \ x 2 0 5 10 15 20 25

0

⎡0 ⎢0 ⎢ ⎢0 ⎢ ⎢0 ⎢0 ⎢ ⎣0

5

10

15

20

25

1 1

1 0

1 0.1

1 0.2

1 0.5

0 0 0 0

0 0 0 0

0.05 0 0 0

0.1 0.03 0 0

⎤ ⎥ ⎥ 0.15 ⎥ ⎥ 0.04 ⎥ 0.02 ⎥ ⎥ 0 ⎦

where the choice of μR(x1, x2) depends on the meaning of “much less” according to R. Two or more fuzzy relations in the same Cartesian product can be combined using the operations of union, intersection, and complementation. For example, if R1 = “x much greater than y” and R2 = “x much smaller than y” are fuzzy relations, then the fuzzy relation R = R1 ∪ R2 = “x much different from y” can be represented by the max operation:

μR(x, y) = max[μR1(x, y), μR2(x, y)]. Now consider the fuzzy relations R1(X, Y ) and R2(Y, Z ) in different Cartesian products X × Y and Y × Z that have a common set Y. The composition of R1 and R2 is a fuzzy relation R(X, Z ) = R1 ° R2 on the product space X × Z such that the proposition “(x, z) belongs to R” has the same degree of validity as the proposition

30


“(x, y) belongs to R1 and (y, z) belongs to R2 for all elements y of Y.” The following diagrams are schematic representations of the composition of R1 and R2. X o

R1

R2

Y y1

o

o

o o z

x o y2

μR1(x, y1)

Y y1

R2

o

μR2(y1, z)

o

μR2(y2, z)

o

o

o

Z o o z

x o

o o

R1

X

Z

μR1(x, y2)

y2

o

o

(a)

(b)

Figure 2.9. Composition of relations R1 and R2: (a) crisp; (b) fuzzy.

It follows by inspection of Figure 2.9a that the previous proposition can be written as follows: “[(x, y1) belongs to R1 and (y1, z) belongs to R2] or [(x, y2) belongs to R1 and (y2, z) belongs to R2] or … , for all elements y1, y2, … of Y.” When maximum is used for “or” and minimum is used for “and” the above decodes to the max-min composition:

μR(x, z) = max min[μR1(x, y), μR2(y, w)]. y ∈Y

(2.6)

Max-min composition for relations is the analog to the extension principle for functions. It is a special case of max-star composition, which is defined by

μR(x, z) = max T[μR1(x, y), μR2(y, w)] y ∈Y

(2.7)

where T( · , · ) is any T-norm. A fuzzy set and a fuzzy relation can also be combined into a composite fuzzy set. Suppose we would like to derive a relation by combining a fuzzy set A in X and a fuzzy relation R(X, Y ). The composition of A and R is the fuzzy set B = A ° R in Y such that the proposition “y belongs to B” has the same degree of validity as the proposition


31

“x belongs to A and (x, y) belongs to R for all elements x of X.” Reasoning as before, we deduce that the membership function of B is given by

μB(y) = max T[μA (x), μR (x, y)], x ∈X

(2.8)

which is the max-star composition of a fuzzy set and a fuzzy relation. Example 2.8. A positive forest change improves land sustainability y, which is measured on a scale between zero and one. Suppose that only the values y ≥ 0.7 are considered to be high values for land sustainability. Therefore the fuzzy set “high land sustainability” has a support Y = [0.7, 1]. Working with the support instead of the universe of discourse reduces the number of calculations, because the support sets do not contain elements with zero membership grades. For a given country, say, ABC consider the fuzzy set A = “annual forest change in country ABC” = 0.2/(3%) + 1/(0%) + 0.1/(−6%) with support X = {3%, 0%, −6%} and the fuzzy relation in X × Y R = “when forest change is positive, land sustainability is usually high” = 0.4/(3%, 1) + 0.5/(3%, 0.9) + 0.6/(3%, 0.8) +0.5/(3%, 0.7) + 0/(0%, 1) + 0/(0%, 0.9) + 0.1/(0%, 0.8) +0.5(0%, 0.7) + 0/(−6%, 1) + 0/(−6%, 0.9) + 0/(−6%, 0.8) +0.05/(−6%, 0.7) The composition B = A ° R is the fuzzy set “land sustainability in ABC is usually high.” Use of the algebraic product for T( · , · ) in Eq. (2.8) yields

μB(y) = max [μA(x)μR(x, y)]. x ∈X

For y = 0.7, the above gives

μB(0.7) = max(0.2 × 0.5, 1 × 0.5, 0.1 × 0.05) = max(0.1, 0.5, 0.005) = 0.5. Working similarly, we obtain

μB(0.8) = 0.12, μB(0.9) = 0.1, and μB(1) = 0.08.

32


2.3. FUZZY LOGIC 2.3.1. Overview of Propositional Logic Classical logic is the formal study of reasoning and proof that deals with propositions which are either true or false. Propositional logic is the subject of classical logic that deals with the construction of propositions and the study of their truth. Simple propositions such as L = “water use intensity is low” G = “water sustainability is good” are called atomic propositions. By combining atomic propositions together with connectives “not,” “or,” “if … then,” etc., we obtain composite propositions, such as “if L then G” = “if water use intensity is low, then water sustainability is good.” Propositional logic uses the five connectives shown below:

Table 2.1. The five propositional connectives Symbola –

Name Negation Conjunction Disjunction Implicationb Equivalence

Examples of use – L; = “not L” L ∩ G = “L and G” L ∪ G = “L or G” L → G = “if L then G” = “L implies G” L ↔ G = “L is equivalent to G” = “G if and only if L”

∩ ∪ → ↔

Other symbols are often used in the literature: ¬, ~ (negation); ∧, & (conjunction); ∨ (disjunction); ⊃, ⇒ (implication); ≡, ≈, ~, ⇔ (equivalence). b Implication is equivalent to set inclusion, i.e., L ⊆ G. a

Let μL = 0, 1 be the truth values of proposition L (1 for true, 0 for false). The truth values of composite propositions can be derived from the truth values of the atomic propositions using the following truth table: Table 2.2. Definitions of propositional connectives by means of the truth table

μL

μG

μG;–

μL∩G

μL∪G

μL→G

μL↔G

1 1 0 0

1 0 1 0

0 1 0 1

1 0 0 0

1 1 1 0

1 0 1 1

1 0 0 1


33

Truth tables are used to prove the validity of tautologies. A tautology is a composite proposition which is always true. The most common tautologies of propositional logic are: Law of the excluded middle: L ∪ L; De Morgan’s laws:

–

( L ∪ G ) ↔ (L ∩ G ) ( L ∩ G ) ↔ (L ∪ G )

Transitivity of implication (syllogism): [(L → G) ∩ (G → H)] → (L → H) Tautologies of implication: (L → G) ↔ ( L ∩ G ) (2.9) –

(L → G) ↔ ( L; ∪ G) Rules of inference: [L ∩ (L → G)] → G (modus ponens) – – [(L → G) ∩ G; ] → L; (modus tollens). Example 2.9. Let us show the tautologies of implication (2.9) using truth tables L 1 1 0 0

L 1 1 0 0

G 1 0 1 0

G 1 0 1 0

L→G 1 0 1 1

L→G 1 0 1 1

– G; 0 1 0 1

– L; 0 0 1 1

– L ∩ G; 0 1 0 0

– L; ∪ G 1 0 1 1

L∩G 1 0 1 1

(L → G) ↔ ( L ∩ G ) 1 1 1 1

– (L → G) ↔ ( L; ∪ G) 1 1 1 1

Implications and rules of inference play an important role in model building and knowledge representation. An implication or “if-then” rule has the form if (proposition 1 is true/false) and/or … and/or (proposition n is true/false), then (proposition n + 1 is true/false) or simply

34


if premise (antecedent part), then conclusion (consequent part). The most common rule of inference is modus ponens. It is written as [L ∩ (L → G)] → G or Premise 1: L is true Premise 2: if L then G is also true Conclusion: G is true. Note that Premise 2, being a valid implication, has its own premise (“if L”) and conclusion (“then G”).

2.3.2. Approximate Reasoning Approximate reasoning is the process of combining propositions and implications with certain degrees of truth to derive conclusions. For example, a rule of inference with fuzzy premises and conclusion is a form of approximate reasoning.

Fuzzy Implication Let L and G be two fuzzy sets with corresponding universes of discourse X and Y. Consider the implication L → G defined by “if x is L, then y is G,” where x ∈ X and y ∈ Y. This implication is a fuzzy relation in the product space X × Y. Various implication membership functions can be obtained from the tautologies of implication of the previous section (Eqs. (2.9)) using appropriate T-norms and S-norms:

μL→G(x, y) = 1 − μL∩G;–(x, y) = 1 − T[μL(x), 1 − μ G(y)]

(2.10)

μL→G(x, y) = μL;–∪G(x, y) = S[1 − μL(x), μ G(y)].

(2.11)

Yet another class of implications is

μL→G(x, y) = T[μL(x), μG(y)].

(2.12)

A special case of the above is the algebraic product μL(x)μG(y), which is reminiscent of the statistical correlation. Therefore, Eq. (2.12) can be interpreted as a generalized measure of dependence between two fuzzy sets. The most commonly used fuzzy implications are based on Eq. (2.12). These are Mamdani (minimum) implication: μL→G(x, y) = min[μL(x), μG(y)] and


35

Larsen (product) implication: μL→G(x, y) = μL(x)μG(y). Note that the same (T-norm) operation is used for the intersection and the Cartesian product. Although (L → G) ↔ (L ∩ G) is not a tautology in the two-valued logic (it is not consistent with the definition of Table 2.2), Eq. (2.12) with a minimum or product T-norm provides more comprehensive and intuitive inference results than Eqs. (2.10) and (2.11) in engineering applications (see, e.g., Mendel 1995). Example 2.10. Consider the fuzzy statements “Desertification of land is medium (M) with grade of membership 0.4.” “Current forest is weak (W) with grade of membership 0.7 and strong (S) with grade of membership 0.2.” Also consider the following rules which result from expert knowledge “If desertification of land is M and current forest is W, then state of land is bad (B).” “If desertification of land is M and current forest is M, then state of land is average (A).” Using the algebraic product implication we obtain “State of land is B with grade 0.4 × 0.7 = 0.28 and A with grade 0.4 × 0.2 = 0.08.”

Compositional rule of inference In fuzzy logic, modus ponens is extended to generalized modus ponens (GMP): Premise 1: x is L* Premise 2: if x is L, then y is G Conclusion: y is G*. Premise 1 is the input of Premise 2. Contrary to the ordinary modus ponens, the fuzzy set L* in the first premise of the generalized modus ponens is not necessarily the same as the fuzzy set L of the second premise; as a result, the fuzzy set G* of the conclusion is not necessarily the same as G. Example 2.11. Premise 1: Student J has a very high IQ Premise 2: if a student has a high IQ, then he is academically good Conclusion: Student J is academically very good.

36


Generalized modus ponens is similar to evaluating a function at a given point: Given a function G(x) (Premise 2) and the point x = L* (Premise 1), we can infer that y = G(L*). This is reminiscent of the extension principle of functions. Likewise, Premises 1 and 2 of GMP can be viewed as fuzzy relations, with Premise 1 being a unary relation, because the set L* is not a Cartesian product. This approach, due to Zadeh (1973, 1975c), leads to the compositional rule of inference (CRI) whereby the generalized modus ponens is represented by the composition of the fuzzy set L*, viewed as a fuzzy relation, and the fuzzy relation L→G. Symbolically, G* = L*° (L → G). The membership function of the fuzzy set G* is obtained by max-star composition (Eq. (2.6)):

μG*(y) = max T[μL*(x), μL→G(x, y)]. x ∈X

(2.13)

As previously, the minimum and the product are the most common T-norms used in Eq. (2.13). In some cases the term “compositional rule of inference” is used to mean the special case in which the minimum is used for T( · , · ) in Eq. (2.13), whereas the term “generalized modus ponens” refers to Eq. (2.13) with the function1 T(a, b) = min(1, 1 − a + b). Example 2.12. Consider the implication H → B = “if H then B” where H = “intensity of water use is high” = 0.5/0.4 + 1/0.5 +1/0.6 + 1/0.7 + 1/0.8 + 1/0.9 + 1/1 B = “water sustainability is bad” = 1/0 + 0.5/0.1 + 0.1/0.2 are fuzzy sets in X and Υ respectively. We want to compute the degree to which water sustainability is bad, after a value x = 0.4 is observed for the intensity of water use.

1

Since min(1, 1 − a + b) is not symmetric in a and b it is not a T-norm. It is derived from the second implication tautology of Eq. (2.9) – (L → G) ↔ ( L; ∪ G) by applying the algebraic operations of complementation and bounded summation for S-norms.


37

The generalized modus ponens reads as follows Premise 1: x is H* Premise 2: if x is H, then y is B Conclusion: y is B*. Let us represent Premise 1 with the fuzzy set H* = 1/0.4. H* is a fuzzy singleton that conveys the same information as the observation x = 0.4 since μH*(0.4) = 1 and μH*(x) = 0 for x ≠ 0.4. Also, let us represent Premise 2 with the Larsen (product) implication: H→B=

∑μ

H ( x , y )∈X ×Y

( x ) μ B ( y ) / ( x, y )

= 0.5 × 1/(0.4, 0) + 0.5 × 0.5/(0.4, 0.1) + 0.5 × 0.1/(0.4, 0.2) + 1 × 1/(0.5, 0) + 1 × 0.5/(0.5, 0.1) + 1 × 0.1/(0.5, 0.2) + 1 × 1/(0.6, 0) + + 1 × 0.5/(0.6, 0.1) + 1 × 0.1/(0.6, 0.2) +… = 0.5/(0.4, 0) + 0.25/(0.4, 0.1) + 0.05/(0.4, 0.2) + 1/(0.5, 0) + 0.5/(0.5, 0.1) + 0.1/(0.5, 0.2) + 1/(0.6, 0) + 0.5/(0.6, 0.1) + 0.1/(0.6, 0.2) + ….

(2.14)

Generalized modus ponens is equivalent to B* = H*° (H → B). Applying max-star composition yields

μB*(y) = max T[μH*(x), μH→B(x, y)] x ∈X

= T[1, μH→B(0.4, y)] = μH→B(0.4, y) where the last equality follows from the fact that 1 is the identity of T-norms. The membership grades of the pairs (0.4, y) are given by Eq. (2.14). Therefore, we have

38


B* = 0.5/0 + 0.25/0.1 + 0.05/0.2. The representation of numerical inputs with fuzzy singletons is known as singleton fuzzification. In the previous example, we represented the rule H → B with a fuzzy set and then we applied max-star composition to derive the conclusion B*. With singleton fuzzification, however, we need not perform all these computations since they are equivalent to the following algorithm: 1) From H = 0.5/0.4 + 1/0.5 +1/0.6 + … we can infer that x = 0.4 is H with grade 0.5. 2) We multiply by 0.5 the membership grades of B = 1/0 + 0.5/0.1 + 0.1/0.2. This yields B*. Step 1, known as fuzzification, computes the degree to which a numerical input x belongs to the linguistic value H of rule H → B. In other words, fuzzification gives the degrees of consistency of inputs with rules. The principle behind the above algorithm is that, since the observation x = 0.4 is consistent with the premise of rule H → B with degree 0.5, it must also match the conclusion B with the same degree.

2.4. FUZZY SYSTEMS A fuzzy rule-based system or simply fuzzy system is a rule-based system that maps an input vector of numerical or linguistic variables into a numerical output. The mapping is represented with “if-then” rules (implications) relating the output to the inputs of the system. In the previous section we saw how to derive a fuzzy conclusion from a single rule. In this section we shall describe how to combine several rules and conclusions into a composite conclusion and how to calculate the corresponding numerical output. Depending on the particular application, the output of a fuzzy system may be a control action, decision, diagnosis, or assessment. Fuzzy systems are used as complements of conventional mathematical models to enhance their effectiveness or as stand-alone applications when mathematical models become inefficient or do not exist. The most popular applications of fuzzy systems are in process control, fault diagnosis, pattern recognition, operations research, and decision support systems.

2.4.1. Description The main ingredient of a fuzzy system is knowledge. Knowledge is incorporated into a fuzzy system in various forms: •

A linguistic variable and a number of linguistic values are assigned to each input and output variable. The membership functions are selected by experts or determined by trial and error so as to provide a complete and concise representation of each variable.

Introduction to Fuzzy Logic • • •

•

39

Numerical inputs are converted into linguistic variables by means of the fuzzification operation. The model of the system is encoded in its rule base. The rule base contains “if-then” rules relating output to input linguistic variables. An inference engine combines fuzzy inputs and “if-then” rules using appropriate composition operations and computes a fuzzy output. The fuzzy output is represented by the degrees (membership grades) to which the output belongs to the corresponding linguistic values. Finally, defuzzification combines the membership grades of the fuzzy output into a single numerical value.

A block diagram of a fuzzy system with one inference engine and one numerical output is shown in Figure 2.10. Multistage fuzzy systems are generalizations of single stage fuzzy systems. These systems are built by interconnecting several inference engines. They are used to approximate processes involving a large number of input variables, and will be discussed later in this chapter. FUZZY SYSTEM numerical inputs: input 1 = x input 2 = x' …

Fuzzification L = linguistic values

fuzzy inputs: μL(x)

Inference Engine Λ= L° R

fuzzy output: μΛ(y)

Defuzzification

numerical output y

Λ = linguistic values

μR(x,y)

Rule Base rules R: “if input 1 is L and input 2 is L' and …, then output is Λ”

Figure 2.10. Block diagram of a single-output, single-stage fuzzy system.

The basic functions of a fuzzy system are fuzzification, inference, and defuzzification. Fuzzy systems admit various mathematical representations, which depend on the choice of Tnorms and S-norms of the inference engine and the defuzzification method.

2.4.2. Fuzzification Consider some input i of a fuzzy system and let xi denote its numerical value. Fuzzification determines the degrees μL(xi) to which the numerical value xi belongs to the fuzzy sets L of the term set T(i). In general, the universe of discourse and the term set of i are different from those of the other inputs of the system. In Example 2.2 and Figure 2.2, we saw that a water use intensity of 0.4 belongs to the linguistic values “medium-high” and “high” with membership grades μMH(0.4) = 0.33 and μH(0.4) = 0.67. Consider also the following example of fuzzification:

40


Example 2.13. LAND is a component of the ecological sustainability ECOS. PRESSURE(LAND) or PR(LAND) for short, is an aggregate measure of the changing forces human activities exert on land. To compute PR(LAND) we use several inputs but, for simplicity, here we consider only two of them: Municipal waste generation per capita and year (WASTE) and population growth rate as percentage of the current population of a country (GROWTH). The inputs are represented by three linguistic values, L (low), M (medium), and H (high). Although we use the same term set for both inputs, the linguistic variables have different universes of discourse, as shown in Figure 2.11. 1

L

M

H

1

L

M

H

0.6 0.4

0.5 0

0 300

800

1.95

0

WASTE

3

GROWTH

Figure 2.11. Membership functions of L, M, and H for WASTE and GROWTH.

For observed values xWASTE = 400 and xGROWTH = 1.95, it follows by inspection of the figure above that WASTE is low with grade 0.5 and medium with grade 0.5, and GROWTH is medium with grade 0.6 and high with grade 0.4. We write

μL(xWASTE) = μM(xWASTE) = 0.5, μH(xWASTE) = 0 μL(xGROWTH) = 0, μM(xGROWTH) = 0.6, μH(xGROWTH) = 0.4.

2.4.3. Inference The inference engine is the mechanism whereby the input rules are combined to produce a control output. There are two basic types of inference engines. • Composition inference: This engine aggregates all rules in one fuzzy relation. Then the fuzzified inputs and the aggregated fuzzy relation are combined with the aid of the composition operation to obtain the fuzzy control output. This type of engine is not so common in fuzzy systems. • Individual rule firing: Here each rule is fired individually and the corresponding output is computed. The outputs are combined into a composite fuzzy set. Individual rule firing is the most common implication in fuzzy systems. It is also the one we shall use in the sequel to compute sustainability. In the next sections we will review the two most popular mathematical representations of fuzzy systems, the Mamdani fuzzy system and the Takagi-Sugeno-Kang fuzzy system. Both


41

systems use the same fuzzification method, “if-then” rules with “and” connectives, and individual rule firing.

2.4.4. Mamdani Fuzzy Systems In a Mamdani fuzzy system T-norms and S-norms are represented by the minimum and maximum operations.

Inference Consider an inference engine that combines n inputs with values xi, i = 1, …, n, to compute a numerical value xn+1 for the output. A rule Rp of the knowledge base has the form Rp :

⎪⎧ ⎨ ⎩⎪

If [input 1 is L(1, p)] and … and [input n is L(n, p)] (Premise); then [xn+1 is L(n + 1, p)]

(Consequence)

where L(i, p) is the linguistic value of i in the premise of Rp. We use the notation μi,p(xi) for membership functions and μi,p for membership grades as a shorthand notation of the usual μL(i, p)(xi). However, when we refer to a particular fuzzy set L we shall write μL(xi) or μL as usual. The value xi belongs to L(i, p) with grade μi,p, which is obtained by fuzzification2. The intersection operation “and” that connects the propositions “input i is L(i, p),” i = 1, …, n, of Rp is represented algebraically by the minimum of the individual truth values. Alternatively, the premise of Rp can be interpreted as the Cartesian product of the fuzzy sets corresponding to the individual propositions. In either case, the firing strength of rule Rp (degree to which Rp is applicable or overall degree of consistency of inputs with Rp) is

μn+1,p = min(μ1,p, ..., μn,p).

(2.15)

We say that rule Rp is active or fires if μn+1,p > 0. Example 2.13 (continued). Let the linguistic values of PR(LAND) be the fuzzy sets VB (very bad), B (bad), A (average), G (good), and VG (very good). Their membership functions will be specified later. Consider a knowledge base that relates PR(LAND) to the inputs WASTE and GROWTH by means of the rule base of Table 2.3 (recall that PR(LAND) gets worse as WASTE or GROWTH increases). The total number of rules in a rule base equals the product of the numbers of the input linguistic values. Here we have a total of 3 × 3 = 9 rules. Equivalently, if all inputs 2

This is a standard assumption in most fuzzy systems and in this book. In some cases, however, the only information we have about some input i is that it belongs to a set L, but there is no rule Rp in the rule base whose fuzzy set L(i, p) matches L exactly. An approach to this problem (Zadeh 1988) involves the use of the compositional rule of inference to compute the degree of consistency of L with each rule Rp, i.e.,

μi,p = maxx min[μL(x), μi,p(x)]. When L is a fuzzy singleton, this approach simplifies to the inference method we shall describe next (see also the discussion following Example 2.12 in Section 2.3.2).

42

Yannis A. Phillis and Vassilis S. Kouikoglou have the same number of linguistic values m, the total number of rules is nm. Here m = 3 values, L, M, and H, and n = 2 input variables and the number of rules is 32 = 9. Table 2.3. Tabular representation of the rule base of Example 2.13

Rule Rp

if and then if and then Rule WASTE GROWTH PR(LAND) WASTE GROWTH PR(LAND) Rp is is is is is is

R1 R2 R3 R4 R5

L L L M M

L M H L M

VG G A G A

R6 R7 R8 R9

M H H H

H L M H

B A B VB

Table 2.4 provides a more concise representation of the knowledge base. Table 2.4. More compact representation of the rule base of Example 2.13

WASTE

GROWTH L M H L

VG

G

A

M H

G A

A B

B VB

We now determine the rule firing strengths. We know that WASTE is L or M with grade 0.5 and GROWTH is M with grade 0.6 or H with grade 0.4. There are 2 × 2 = 4 firing rules. Application of Eq. (2.15) yields (the membership grades are shown in parentheses): R2:

if WASTE is L(0.5) and GROWTH is M(0.6), then PR(LAND) is G(min(0.5, 0.6) = 0.5)

R3:

if WASTE is L(0.5) and GROWTH is H(0.4), then PR(LAND) is A(min(0.5, 0.4) = 0.4)

R5:

if WASTE is M(0.5) and GROWTH is M(0.6), then PR(LAND) is A(min(0.5, 0.6) = 0.5)

R6:

if WASTE is M(0.5) and GROWTH is H(0.4), then PR(LAND) is B(min(0.5, 0.4) = 0.4).

In the previous example, the rules R3 and R5, assign the same linguistic value, A, to the output. A rule base is considered to be a union of different rules. Therefore, two or more rules with the same consequent linguistic value L can be combined into a single rule with a firing


43

strength equal to the maximum of the individual rule firing strengths. The overall membership grade of the output to the fuzzy set L is given by

μL =

max

p: L ( n +1, p ) = L

μn+1,p

(2.16)

where “p: L(n + 1, p) = L” is an abbreviation for the compatibility condition “all rules Rp such that their consequences assign the linguistic value L to the output.” Example 2.13 (concluded). By applying Eq. (2.16) to rules R3 and R5, we find

μA[PR(LAND)] = max(0.4, 0.5) = 0.5. The membership grades of PR(LAND) to the other fuzzy sets are

μVB[PR(LAND)] = 0, μB[PR(LAND)] = 0.4, μA[PR(LAND)] = 0.5, μG[PR(LAND)] = 0.5, and μVG[PR(LAND)] = 0. Suppose we have computed the membership grades μL for all fuzzy sets L of the output term set T(n + 1). A numerical value for the output is obtained by defuzzification. As we shall see in the next section, some defuzzification methods require only the membership grades μL of the output to the individual linguistic values L. Other methods use an aggregate output fuzzy set, which represents the union of all L ∈ T(n + 1). The aggregate fuzzy set is obtained in two steps: 1) According to the generalized modus ponens, the fuzzy sets L specified in the rules are modified during the process of approximate reasoning because the inputs do not match exactly the rule premises. The overall conclusion regarding membership to L is represented with a new fuzzy set L* with membership function

μL*(x) = min[μL, μL(x)],

(2.17)

where μL is computed from Eq. (2.16). 2) The modified output fuzzy sets L* are combined into an aggregate fuzzy set Λ by means of an S-norm. The most common choice is maximum S-norm

44


μΛ(x) = * max μL*(x).

(2.18)

L : L∈T ( n +1)

1

L

M

H

L

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

M

VB

H

B

Rule 2: if WASTE = L(0.5)

1

VG

min ⇒ 0.5

L

M

800

0

and

GROWTH = M(0.6),

H

⇒ G(0.5)

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0 0.2 0.4 0.6 0.8

3

L

1

0

M

1

then PR(LAND) = G(0.5) VB

H

B

A

G

VG

min ⇒ 0.4

0 300

Rule 3: if WASTE = L(0.5)

1

L

M

800

0

and

GROWTH = H(0.4),

H

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.2 0.4 0.6 0.8

3

L

1

0

M

max ⇒ A(0.5)

1

then PR(LAND) = A(0.4) VB

H

B

A

G

VG

min ⇒ 0.5

0 300

800

Rule 5: if WASTE = M(0.5)

1

L

M

0

H

L

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0 0.2 0.4 0.6 0.8

3

GROWTH = M(0.6),

and

0

M

1

then PR(LAND) = A(0.5) VB

H

B

A

G

VG

min ⇒ 0.4

⇒ B(0.4)

0 300

800

Rule 6: if WASTE = M(0.5)

and

0

0 0.2 0.4 0.6 0.8

3

GROWTH = H(0.4),

1

then PR(LAND) = B(0.4)

VB 1

B

A

G

VG

1

0.8

0.8

0.6

0.6

0.2

G

0 300

0.4

A

Λ

PR(LAND) = B(0.4) or A(0.5) or G(0.5) = B* ∪ A* ∪ G* =Λ

0 0 0.2 0.4 0.6 0.8

1 P(LAND)

Figure 2.12. Mamdani inference for Example 2.13.

*

0.4 0.2

*

B

*

A

G

0 0 0.2 0.4 0.6 0.8

1 PR(LAND)


45

The membership functions μL*(x) and μΛ(x) are determined by evaluating Eqs. (2.17) and (2.18) for a large number of points x of the physical domain of the output. A graphical representation of Mamdani inference for the system of Example 2.10 is shown in Figure 2.12. The fuzzy output PR(LAND) is assessed in [0, 1], where the value 0 belongs to the fuzzy set VB (very bad) and 1 belongs to VG (very good), both with membership grades 1.

Defuzzification A crisp value xn+1 for the output is computed by defuzzification. We present the most popular defuzzification methods starting from those that consider a single output set L* in X with membership function μL*(x) given by Eq. (2.17). • Maximum defuzzification: The numerical output is a value xn+1 for which μL*(xn+1) is maximum. If μL*(x) has several maximizers, then xn+1 is the midpoint of these values. • Centroid defuzzification: The output is the center of gravity of μL*(x), which is given by

∫ xμ L ( x )dx *

xn+1 =

X

∫ μ L ( x )dx

.

*

X

• Center-of-area defuzzification: The output is the point xn+1 that separates the area under μL*(x) into two equal parts, i.e. xn +1

∞

−∞

xn +1

∫ μ L* ( x )dx =

∫ μ L ( x )dx . *

For discrete sets X, the integrals in the above formulas are replaced by sums. In particular, the center of area of the fuzzy set μ1/a1 + … + μm/am with ai ∈ X and ai < ai+1 can be calculated as follows: 1) 2) 3) 4)

Define the partial sums Mi = μ1 + … + μi, for i = 1, …, m. Calculate the half-area h = Mm/2. Determine the index i such that Mi ≤ h < Mi+1. Interpolate between ai and ai+1; thus

xn +1 − ai h − Mi = ai +1 − ai M i +1 − M i and, finally,

xn +1 = ai +

h − Mi ( ai +1 − ai ) . M i +1 − M i

46


Example 2.14. For the fuzzy set L* = 0.5/1 + 0.8/2 + 0.9/3 + 0.2/4 we have h = (0.5 + 0.8 + 0.9 + 0.4)/2 = 2.4/2 = 1.2, M1 = 0.5 < h, M2 = 0.5 + 0.8 = 1.3 > h. Hence, by interpolation,

x n +1 = 1 +

1.2 − 0.5 (2 − 1) = 1 + 0.7/0.8 = 1.875. 1.3 − 0.5

If the term set T(n + 1) of the output has several fuzzy sets, then there are two approaches to defuzzification: • The modified output fuzzy sets L* are combined into a single fuzzy set Λ using an S-norm, usually the maximum

μΛ(x) = * max μL*(x), L : L∈T ( n +1)

and then one of the previous defuzzification methods is applied. • Each fuzzy set L* is defuzzified individually to a numerical value yL, using one of the three methods described above, and then the output is computed from

xn+1 =

∑ yL μL

L∈T ( n +1)

∑ μL

(2.19)

L∈T ( n +1)

where μL is given by Eq. (2.16). When the output linguistic values L, L ∈ T(n + 1), are fuzzy singletons 1/yL, the two approaches described above give the same results. Singleton defuzzification, known as height defuzzification, is the most common defuzzification method, because it has similar properties with the other methods and minimal computational requirements (see Driankov et al. 1996, for a comparison of various defuzzification methods). Example 2.15. Consider the variable PR(LAND) of Example 2.13 with membership grades μVB = 0, μB = 0.4, μA = 0.5, μG = 0.5, and μVG = 0. Take yVB = 0, yB = 0.3, yA = 0.5, yG = 0.7, and yVG = 1. Then Eq. (2.19) gives xPR(LAND) =

0 × 0 + 0.3 × 0.4 + 0.5 × 0.5 + 0.7 × 0.5 + 1 × 0 0.72 = 0 + 0.4 + 0.5 + 0.5 + 0 1.4


47

= 0.514.

2.4.5. Takagi-Sugeno-Kang (TSK) Fuzzy Systems A Takagi-Sugeno-Kang (TSK) fuzzy system uses a) product S-norms for fuzzy intersections and Cartesian products, b) sums, which are not T-norms, for fuzzy unions, and c) height defuzzification in which the yL’s are functions of the input (x1, …, xn) rather than constants. In a TSK system, the firing strength of rule Rp is given by n

μn+1,p = ∏ μi , p ,

(2.20)

i =1

where μi,p is the membership grade of input i to the fuzzy set L(i, p). If several rules assign the same fuzzy set L to the output variable xn+1, then the overall membership grade μL of xn+1 to L is given by the sum of the individual firing strengths. Thus,

μL =

∑μ

n +1, p p: L ( n +1, p ) = L

,

(2.21)

where, as previously, “p: L(n + 1, p) = L” stands for the compatibility condition “all rules Rp such that their consequences assign the linguistic value L to the output.” A crisp value for the output is computed using Eq. (2.19):

xn+1 =

⎛

n

⎝

i =1

⎞

∑ ⎜ y L( n+1, p ) ∏ μi , p ⎟ p

⎛ n ⎞ ∑ ⎜ ∏ μi , p ⎟ p ⎝ i =1 ⎠

∑ yL μL ⎠ = L∈T ( n +1) ∑ μL

(2.22)

L∈T ( n +1)

where the weights yL are functions of the numerical inputs and T(n + 1) is the output term set. If yL is a polynomial with terms of the form x1k1x2k2… xnkn for ki = 0, 1, …, then Eqs. (2.20)−(2.22) define a Takagi-Sugeno-Kang (TSK) fuzzy system. If the weights are affine functions, that is, linear functions of the vector (x1, …, xn) increased by a constant, then we have a first-order TSK system; if they are constant, we have a zero-order TSK system. A specific form of Eqs. (2.19) and (2.22) will be used in the computations of sustainability where the weights yL will be the peak values of the corresponding fuzzy sets. For brevity, any fuzzy system that uses these equations is referred to as TSK even when the weight functions are not polynomials.

48


2.4.6. Design Criteria For Fuzzy Systems A number of issues arise in the context of building a fuzzy system. These are:

Completeness of Fuzzy Partitions Each input of a fuzzy system has a number of linguistic values which form a fuzzy partition of the physical domain. This partition should cover the entire physical domain, otherwise the fuzzy system will not respond to certain numerical inputs. A linguistic variable is a complete partition if any point of its universe of discourse belongs to at least one linguistic value with grade > 0. In many applications and in this book, the fuzzy sets are ordered sets in which a certain binary relation “” is defined. For example, the fuzzy sets labeled L = “low,” M = “medium,” and H = “high” satisfy L < M < H. It is convenient to define “A ≤ B” to mean “either A = B or A < B.” Consistency of Fuzzy Unions When sums are used to represent fuzzy unions, as in TSK inference, the membership functions must be selected so that any point of the universe of discourse belongs to the fuzzy sets with a total degree that does not exceed 1. If this condition is satisfied as equality for all x, then we have the so-called Ruspini partition (Ruspini 1969). Thus for a partition with fuzzy sets L, M, and H we must have degree to which x belongs to L ∪ M ∪ H = μL(x) + μM(x) + μH(x) = 1. If the sum exceeds 1 for some x, then the union can be represented by some S-norm, such as the maximum or the bounded sum min[1, μL(x) + μM(x) + μH(x)].

Convexity of Linguistic Values As we have discussed in Section 2.2.2, every fuzzy set that describes something measurable, such as height, intensity, etc., must be a convex fuzzy set. In general, if L is a linguistic value of an ordered fuzzy partition and x, y, and z are three points of the universe of discourse of L such that x < y < z, then the intermediate point must belong to L with a degree at least as high as the minimum of the membership grades of the extreme points or

μL(y) ≥ min[μL(x), μL(z)]. Completeness of Rule Bases A rule base is a matrix of linguistic values of the output given linguistic combinations of the inputs. There might be combinations of inputs which produce a null output. Then we say that the rule base is not complete. A rule base is complete if all combinations of linguistic inputs produce nonnull outputs.


49

Consistency of Fuzzy Rules A rule base should be designed in such a way that no contradictions ensue. A rule base is consistent if it contains no rules with the same antecedents and different consequents. Take for example the rules if WASTE is L and GROWTH is L, then PR(LAND) is VG if WASTE is L and GROWTH is L, then PR(LAND) is G. These rules are inconsistent. It should be stressed that all the rule bases in this book are formulated so that no two rules have the same antecedents and thus they are consistent.

Complexity For a system with n inputs and m fuzzy sets for each input, there are mn different combinations of linguistic inputs. If the rule antecedents are connected with “and,” which is the most common format, then the number of rules in the rule base is mn as well. This number increases geometrically with the number of inputs and polynomially with the number of linguistic values. Both parameters contribute to rule explosion but the first one is more important. For example, combining 6 inputs with 5 linguistic values requires 56 = 15,625 rules, whereas combining 5 inputs with 6 linguistic values requires 65 = 7,776 rules. A given numerical input usually belongs to two or three consecutive linguistic values. For example, a daily municipal waste of 2 kg per capita, could correspond to the proposition “WASTE is L with membership grade 0, M with grade 0.2 and B with grade 0.8” for L = low, M = medium, and B = big. In general, suppose that each numerical input belongs to k linguistic values, where k is a positive integer less than or equal to m. Then, the inference engine fires a total of kn rules by executing minimum (Mamdani) or product (TSK) operations. Each one of these operations involves n input membership grades. This requires a total of (n − 1)kn operations since, for example, one multiplication gives the product of two membership grades. The number (n − 1)kn also increases geometrically with n and polynomially with k. Usually it does not depend on m: When we place more fuzzy sets in the same universe of discourse to enhance precision, we simultaneously decrease the ranges of their supports. Fuzzy systems with a small number of fuzzy values and inputs have low storage and computational requirements but they are not as informative and accurate as their counterparts. The design of systems with many inputs calls for a trade-off between conflicting objectives. There are two approaches to get around the problem of rule explosion and long execution times: 1) Rule base compression: Take for example the rules if WASTE is L and GROWTH is L, then PR(LAND) is VG if WASTE is L and GROWTH is M, then PR(LAND) is VG. These rules have the same consequences and can be combined into a composite one

50


if WASTE is L and GROWTH is (L or M), then PR(LAND) is VG. 2) Hierarchical decomposition: Consider the function f = (x + y)2 + (x + y + z)2 + (x + y − 3z)2 to be evaluated at a given point (x, y, z). We can evaluate f1 = x + y and substitute it into the three terms in parentheses; then we square each term and add them. This defines a three-level hierarchy of calculations: At the bottom level, we have the evaluation of f1. The middle level involves the calculations f2 = f1 + z, f3 = f1 − 3z and the top level f = f12 + f22 + f32. The function f is a composite function with a hierarchical structure. We shall discuss this issue in the context of fuzzy systems in a separate section.

Monotonicity In many applications, a natural requirement is that the output of a fuzzy system (assessment, control action, decision) be monotonic with respect to its inputs. For example, a fuzzy controller for an air conditioner increases the motor speed as the room temperature deviates from a target value. Also the sustainability of a nation increases when its basic inputs, e.g., freshwater quality/availability, air quality, indices of economic performance, etc., improve. Consider a single-stage TSK fuzzy system with inputs x1, …, xn and output xn+1. Won et al. (2002) proved that xn+1 is an increasing function of the input variables under the following conditions: Condition MC1: The membership functions assigned to the inputs are piecewise differentiable, in the sense that they are continuous on the corresponding domains and differentiable at all but a countable number of points. Moreover, for any pair of linguistic values A and B belonging to the same fuzzy partition, if A < B then

1 dμ A ( x ) 1 dμ B ( x ) ≤ , μ A ( x) dx μ B ( x) dx for all x where μA(x) and μB(x) are differentiable and nonzero.

(2.23)


51

Condition MC2: The rule bases are increasing, that is, for any pair of rules Rp and Rq the premises of which “input j is L(j, p)” and “input j is L(j, q)” are identical for j ≠ i and satisfy L(i, p) < L(i, q), the corresponding fuzzy sets for the output satisfy L(n + 1, p) < L(n + 1, q). Condition MC3: The weights used in the defuzzification are piecewise differentiable and increasing, that is, the weight yL(x1, …, xn) in Eq. (2.22) assigned to the fuzzy set L of xn+1 is an increasing function. Moreover, for two distinct fuzzy sets L and Λ of xn+1, L < Λ implies that yL(x1, …, xn) ≤ yΛ(x1, …, xn) for all input vectors (x1, …, xn). It should be emphasized that the above are sufficient conditions for monotonicity, that is, there are monotonic TSK systems which do not satisfy MC1–MC3. MC1 holds for various types of membership functions such as trapezoidal and Gaussian. 1

A

μA, μB

MC1: ai ≤ bi

B

1

A

B

MC1: a ≤ b, σ > 0

μA, μB

μ A ( x) =

⎛ x −a ⎞ −⎜ ⎟ e ⎝ σ ⎠

μ B ( x) = e 0

a1 a2 a3

a4

0

b4

a

(a)

⎛ x −b ⎞ −⎜ ⎟ ⎝ σ ⎠

2

2

b

(b)

Figure 2.13. Membership functions satisfying MC1: (a) trapezoidal; (b) Gaussian.

Conditions MC2 and MC3 agree with intuition: a non-decreasing fuzzy system should assign large values, linguistic and numerical, to large inputs. Unfortunately, no monotonicity results exist for Mamdani fuzzy systems. Below, we give an example of a Mamdani fuzzy system which satisfies MC1–MC3 but its output is a decreasing function in some region of the input domain. Example 2.16. Consider a Mamdani fuzzy system with inputs x1 and x2, output x3, linguistic values L (low), M (medium), and H (high), and an increasing rule base:

x1

L M H

x2 L L L M

M L M M

H M M H

The fuzzy sets L, M, and H are defined in [0, 10]. Their membership functions are shown in Figure 2.14 and their weights are yL = 0, yM = 5, and yH = 10.

52

Yannis A. Phillis and Vassilis S. Kouikoglou 1 0.8

L

M

H

1 0.6 0.4

0.2 0

0 4

0

6 7

10

xi Figure 2.14. Membership functions of L, M, and H .

Suppose that x1 = 4 and x2 = 6. Then x1 is L with grade 0.2 and M with grade 0.8, and x2 is M with grade 0.8 and H with grade 0.2. Application of Eq. (2.15) yields (the membership grades are shown in parentheses): if x1 is L(0.2) and x2 is M(0.8), then x3 is L(min(0.2, 0.8) = 0.2) if x1 is L(0.2) and x2 is H(0.2), then x3 is M(min(0.2, 0.2) = 0.2) if x1 is M(0.8) and x2 is M(0.8), then x3 is M(min(0.8, 0.8) = 0.8) if x1 is M(0.8) and x2 is H(0.2), then x3 is M(min(0.8, 0.2) = 0.2). Thus, x3 is L with grade 0.2 and M with grade max(0.2, 0.8, 0.2) = 0.8. Using Eq. (2.19) we obtain a crisp value for x3, thus: x3 =

0 × 0.2 + 5 × 0.8 = 4. 0.2 + 0.8

Next, we increase the second input to x2 = 7. Then, x2 is M with grade 0.6 and H with grade 0.4. The same rules are activated with the following firing strengths: if x1 is L(0.2) and x2 is M(0.6), then x3 is L(min(0.2, 0.6) = 0.2) if x1 is L(0.2) and x2 is H(0.4), then x3 is M(min(0.2, 0.4) = 0.2) if x1 is M(0.8) and x2 is M(0.6), then x3 is M(min(0.8, 0.6) = 0.6) if x1 is M(0.8) and x2 is H(0.4), then x3 is M(min(0.8, 0.4) = 0.4). In this case, x3 is L with grade 0.2 and M with grade max(0.2, 0.6, 0.4) = 0.6 with a crisp value

Introduction to Fuzzy Logic x3 =

53

0 × 0.2 + 5 × 0.6 3 = = 3.75. 0.2 + 0.6 0.8

We see that although the rule base is increasing and the input x2 increases by 1, the output of the Mamdani system decreases. The reader can verify that the output of a TSK system is 0.42 for x2 = 6 and 0.44 for x2 = 7. Hence, a TSK system is monotonically increasing for this example. In the rest of this chapter we shall see that conditions MC1–MC3 ensure monotonicity of the outputs of multistage TSK fuzzy systems.

2.5. A MONOTONIC HIERARCHICAL FUZZY SYSTEM 2.5.1. Introduction Multistage inference is a process of sequential reasoning whereby the consequence of one inference stage is passed as input to other stages. Hierarchical inference is a multistage inference without feedback. A hierarchical fuzzy system with three inputs and two inference engines is depicted in Figure 2.15. The variables labeled 1, 2, and 3 are the basic inputs. The first inference engine combines inputs 1 and 2 into a composite variable, 4. The second inference engine computes the fuzzy output 5 by combining inputs 3 and 4. A numerical output is calculated by defuzzification.

Numerical input (1) FUZZY RULES

Fuzzy intermediate output/input (4)

Fuzzy output (5)

FUZZY RULES

Numerical input (2)

Defuzzification

Numerical output (5)

First inference engine Numerical input (3)

Second inference engine

Fuzzification

Figure 2.15. A hierarchical fuzzy system with two inference engines.

Hierarchical reasoning is common. Often public lectures, articles, and books begin with an informal presentation of the main points and basic concepts of a subject, and then they go through each point and concept in more detail. Also, human intelligence employs both hierarchical and approximate reasoning to decompose complex concepts into their simple parts, or large scale problems into smaller ones. Hierarchical inference has found numerous applications in control systems, approximation theory, pattern recognition, medicine, and environmental systems. Compared to single-stage inference systems, hierarchical fuzzy systems can cope better with the curse of rule explosion in problems involving many variables and changing environments (Torra 2002).

54


In the previous chapter and Section 2.1, we gave a brief account of the human and environmental dimensions of sustainability, their components (water, land, economy, society, etc.), and other more elementary variables which describe the state and dynamics of each component (availability of resources, quality, intensity of use, etc.). In the rest of this chapter, we shall describe a hierarchical network of TSK fuzzy systems which has the property of being monotonic, i.e., an improvement of an elementary variable leads to an improvement of sustainability. We begin with a few definitions and examples.

2.5.2. Hierarchical Fuzzy Systems Definitions and Examples At its first level, multistage inference uses fuzzy inputs which we call basic, in order to distinguish them from the outputs of intermediate inference stages which are used as inputs to subsequent stages; we shall refer to the latter as composite inputs or composite variables. A multistage fuzzy system is represented by a directed graph whose nodes correspond to the inference engines. Two such systems are shown in Figure 2.16. Inference engines are denoted ei; system variables (basic inputs, composite variables, and output), numerical or linguistic, are denoted xi. x1

e1

x1

x4

e2

x2

x3

x5

e1

x4

e2

x2

x5

x3

(a)

(b)

Figure 2.16. Compact representation: (a) system of Figure 2.15; (b) a system with feedback.

A directed arc from ei to ej indicates that the output of engine ei is used as input to ej (ej may have other inputs as well, basic and composite). A path from ei to ek formed by consecutive arcs (ei, ej), (ej, ej'), …, (ej", ek) indicates that the output of ek depends on the output of ei; we say that ei precedes ek. Of practical interest are multistage inference systems without feedback paths, called hierarchical systems. More formally, a hierarchical fuzzy system is a network of inference engines that has an acyclic structure, i.e., if engine ei precedes engine ek, then there is no path from ek to ei. The system shown in Figure 2.16a is hierarchical, while that in Figure 2.16b is not. By convention, if ei precedes ek, then we assign ek to a higher hierarchical level relative to ei. The levels determine the sequence in which the inference engines are activated. This sequence must ensure that an engine is activated after its preceding engines. For example, in the system of Figure 2.17 engines e1 and e2 must be activated before e3. Because e1 and e2 use basic inputs, either engine can be activated first. Engines e1 and e2


55

belong to the bottom level, engine e3 belongs to the intermediate level, and e4 belongs to the top level.

x1

e1 e3

e4

e2 x2

Figure 2.17. A system with two inputs and four engines.

There is a simple way to determine the levels of a hierarchical network (Knuth 1977, pp. 263−268): First we assign all engines not preceded by any other engine to the lowest level U1. Next, we remove these engines from the network and repeat the same process to determine the second level U2. We continue until all engines are assigned to levels. Example 2.17. In Figure 2.18 we illustrate the decomposition of the four-engine network of Figure 2.17. Engines e1 and e2 have only basic inputs, so they belong to the lowest level U1. After removing e1 and e2 we see that U2 = {e3} is the second level and U3 = {e4} is the third level.

x1

e1 e3

e4

e2 x2 U1

U2

U3

Figure 2.18. Levels of the system of Figure 2.17.

Multistage Fuzzy Inference In a hierarchical fuzzy system, each inference stage a) b) c)

fuzzifies its input variables, computes the membership grades of the output using Mamdani inference (Eqs. (2.15)–(2.18)) or TSK inference (Eqs. (2.20) and (2.21)), and computes a numerical output by defuzzification.

Steps (a) and (c) need not be included in every inference stage. For example, in the system of Figure 2.18, after fuzzifying the basic inputs x1 and x2 we can activate the engines

56


e1 and e2 and pass their fuzzy outputs directly to e3 without defuzzification and fuzzification. Similarly, we can transfer the fuzzy output of e3 directly to e4 which computes the fuzzy output of the system and its numerical value by defuzzification. Choosing between numerical and fuzzy intermediate inputs is a problem of finding a trade-off between information loss and explosion of fuzziness (Maeda 1996, Torra 2002). Using numerical inputs requires defuzzification and fuzzification which are not inverse operations. This introduces an amount of distortion in the process of transferring information from one inference stage to another. On the other hand, when reasoning is repeated in multiple stages, the use of fuzzy composite inputs (no defuzzification and new fuzzification is performed at the output of each intermediate stage) may render the final conclusion quite uncertain and uniform across several linguistic values of the output variable. The model of SD to be described in subsequent chapters does not defuzzify or fuzzify intermediate outputs. In summary, hierarchical inference is carried out as follows: 1) Decompose the hierarchical network of inference engines into disjoint levels U1, …, UV, as described in the previous section. 2) Trace the levels downstream: For j = 1, …, V, activate each engine in level Uj, thus, a) fuzzify the numerical inputs if needed b) compute the membership grades of the output, and c) defuzzify the output if needed. In the next section we consider systems with numerical or fuzzy intermediate inputs and give sufficient conditions under which their outputs are monotonic functions of the basic inputs.

Monotonicity In Section 2.4.6 we presented conditions under which the numerical output of a singlestage TSK fuzzy system is an increasing function of its inputs. We now consider hierarchical fuzzy systems with inference engines of the TSK type. For simplicity, we only consider zero-order TSK systems, in which the weight functions used in defuzzification are constant, although the results we shall derive apply to piecewise differentiable weight functions as well. We restate conditions MC1–MC3 with changes marked in italics: Condition MC1: For a system with fuzzy inputs, the membership functions assigned to every basic input are piecewise differentiable and satisfy inequalities (2.23). For a system with numerical intermediate inputs, the membership functions of every input (basic or composite) are piecewise differentiable and satisfy these inequalities. Condition MC2: The rule bases are increasing. Condition MC3: For each inference engine that performs defuzzification, the weights assigned to the output fuzzy sets are constant. For any two output fuzzy sets L and Λ with weights yL and yΛ, the relation L < Λ implies that yL ≤ yΛ.


57

The next two theorems show that these conditions ensure monotonicity in hierarchical fuzzy systems with either crisp or fuzzy intermediate inputs. Theorem 1. In a hierarchical TSK fuzzy system that uses numerical intermediate inputs, if conditions MC1–MC3 are satisfied for each inference engine, then the output of each stage is increasing with respect to each basic input. Proof. Let U1, …, UV be the levels of the fuzzy system. Consider all inference engines of U1. These are single-stage TSK systems satisfying MC1–MC3, so their numerical outputs are increasing functions of the basic inputs. Each inference engine of U2 accepts basic and composite numerical inputs. Conditions MC1–MC3 are satisfied again, so the outputs of all engines in U2 are increasing functions of basic and composite inputs. But the latter are increasing functions of the basic inputs and the composition of functions preserves the property of monotonicity. Therefore, the outputs of all engines of U2 are increasing functions of the basic inputs. The proof proceeds by induction for U3, …, UV using the same arguments. Theorem 2. In a hierarchical TSK fuzzy system that uses fuzzy intermediate inputs, if each basic input satisfies MC1, the rule bases are increasing (MC2), and the weights of the last engine satisfy MC3, then the output is nondecreasing with respect to each basic input. Proof. Here an inference engine computes the membership grades of its output variable and passes them directly to subsequent inference stages. First we prove that this system is equivalent to a single-stage system, in the sense that both systems produce the same logical and numerical outputs. Then we show that the assumptions of the theorem ensure that this equivalent system satisfies MC1–MC3 and, therefore, its output is increasing. The proof of the general case is tedious because it involves the transformation of composite functions into simple ones. To avoid superfluous notation, without loss of generality, we consider the system of Figure 2.19 with two inference engines and three basic inputs, x1, x2, and x3. This is the simplest hierarchical system with multi-input inference engines. x1

e1 x2

x4

e2

x5

x3

Figure 2.19. A two-stage fuzzy system with two inputs per stage.

Let Rq be a rule of e2 which corresponds to the proposition Rq: “if [x3 is L(3, q)] and [x4 is L(4, q)], then [x5 is L(5, q)]”

58


and Rp a rule of e1 which corresponds to the proposition Rp: “if [x1 is L(1, p)] and [x2 is L(2, p)], then [x4 is L(4, p)].” By applying Eqs. (2.20) and (2.21), we calculate the firing strength of rule Rq as follows:

μ5,q = μ3,qμ4,q = μ 3, q =

∑μ

μ 2, p

1, p p: L ( 4 , p ) = L ( 4 ,q )

∑μ

μ 2 , p μ 3, q .

1, p p: L ( 4 , p ) = L ( 4 ,q )

(2.24)

The last sum corresponds to a union of rules involving only basic inputs. These rules define an equivalent single-stage TSK system whose inputs have the same membership functions as the original two-stage system. The rule base of the equivalent system is the Cartesian product of the rule bases of e1 and e2, and its rules have the form Rp,q: “if [x1 is L(1, p)] and [x2 is L(2, p)] and [x3 is L(3, q)], then [x5 is L(5, q)] where, by Eq. (2.24), p and q satisfy the compatibility condition L(4, p) = L(4, q). The equivalent single-stage system satisfies condition MC1 since its inputs are the basic inputs of the original system. Next, we prove that it satisfies MC2 and MC3 so Theorem 1 applies. Consider the rule Rp,q and a rule Rr,s of the form Rr,s: “if [x1 is L(1, r)] and [x2 is L(2, r)] and [x3 is L(3, s)], then [x5 is L(5, s)]” where s and r are such that L(4, r) = L(4, s). Suppose that the premises of Rp,q and Rr,s are identical except for L(1, p) < L(1, r). We verify that the rules of the equivalent system are monotonic as follows: L(1, p) < L(1, r) (by assumption) L(2, p) = L(2, r) and L(3, q) = L(3, s) (by assumption) L(4, p) ≤ L(4, r) (by monotonicity of the rules Rp and Rr of inference engine e1) L(4, q) ≤ L(4, s) (by the compatibility conditions L(4, p) = L(4, q) and L(4, r) = L(4, s)) L(5, q) ≤ L(5, s) (by monotonicity of the rules of inference engine e2).


59

By similar arguments, we can show that if the premises of Rp,q and Rr,s are identical except for either L(2, p) < L(2, r) or L(3, q) < L(3, s), then L(5, q) ≤ L(5, s). Therefore, the equivalent system satisfies MC2. We now complete the construction of the equivalent single-stage system by defining its weight functions so that they satisfy MC3. Let T(5) be the term set of output linguistic values. The original two-stage system gives the numerical output by height defuzzification

∑ yL μL

x5 =

L∈T (5)

∑ μL

,

L∈T (5)

where the weights yL satisfy MC3. But these are also weights of the equivalent single-stage system since, by L(5, q) = L and Eq. (2.24),

∑

L∈T (5)

x5 =

∑

∑ y μ μ 2, p μ 3, q

L 1, p p, q : L ( 4 , p )= L ( 4 ,q ) L ( 5, q ) =L

L∈T (5)

∑ μ1, p μ 2, p μ 3,q

.

(2.25)

p, q : L ( 4 , p )= L( 4 , q ) L (5 , q )= L

Therefore, the equivalent system satisfies MC3. The above conclusion holds for any two-level hierarchical system of the type shown in Figure 2.19, since the set or algebraic products in Eqs. (2.24) and (2.25) can include any number of terms. Therefore, for a general hierarchical system with V levels U1, …, UV, all the outputs of the second lowest level U2 can be obtained by aggregating the corresponding subsystems of U1 and U2 into equivalent single-stage systems. This will give a new hierarchical system with V − 1 levels which is equivalent to the original system. Continuing in the same manner we obtain the single-stage equivalent of the original system which is monotonic because it satisfies MC1–MC3.

2.6. PROBLEMS 2.1. The triangular membership function is defined by

0, x < a ⎧ ⎪x−a ⎪ , a≤ x≤b ⎪b − a μ(x) = triangle(a, b, c) = ⎨ ⎪c − x , b ≤ x ≤ c ⎪c −b ⎪ 0, x > c. ⎩

60


The fuzzy sets A and B in (−∞, ∞) have membership functions μA(x) = triangle(0, 10, 20) and μB(x) = triangle(7, 18, 30). a) Plot μA(x) and μB(x). b) Derive the membership functions of A ∩ B using minimum, algebraic product, and bounded difference T-norms. c) Derive the membership functions of A ∪ B using maximum, algebraic sum, and bounded sum S-norms. 2.2. The trapezoidal membership function is defined by

0, x < a ⎧ ⎪x−a ⎪ , a≤ x≤b ⎪b − a ⎪ μ(x) = trapezoid(a, b, c, d) = ⎨ 1, b ≤ x ≤ c ⎪d − x ⎪ , c≤x≤d ⎪d − c ⎪⎩ 0, x > d . The fuzzy sets A and B in (−∞, ∞) have membership functions μA(x) = trapezoid (0, 5, 8, 13) and μB(x) = trapezoid(5, 10, 12, 17). a) Plot μA(x) and μB(x) . b) Derive the membership functions of A ∩ B using minimum, algebraic product, and bounded difference T-norms. c) Derive the membership functions of A ∪ B using maximum, algebraic sum, and bounded sum S-norms. 2.3. The bell membership function is defined by

μ(x) = bell(a, b, c) =

1 ⎛ x −c⎞ 1+ ⎜ ⎟ ⎝ a ⎠

2b

.

a) Plot the membership functions μA(x) = bell(4, 4, 0) and μB(x) = bell(6, 4, 10). b) Compute the membership function μC(x) = min[μA(x), μB(x)] and its complement μC;– (x). 2.4. Show that the following functions are valid T-norms:

⎧min( x, y ), if max( x, y ) = 1 0, otherwise ⎩

a) T1(x, y) = ⎨

Introduction to Fuzzy Logic b) T2(x, y) =

xy x + y − xy

c) T3(x, y) =

xy 1 + (1 − x )(1 − y )

[ (

)]

d) T4(x, y) = max 0, x r + y r − 1

1 r

61

, r ≠ 0.

2.5. Let T(x, y) be a T-norm. Prove that the function S(x, y) given by Eq. (2.1), S(x, y) = 1 − T(1 − x, 1 − y), is an S-norm. 2.6. Show that the following functions are valid S-norms:

⎧max( x, y ), if min( x, y ) = 0 a) S1(x, y) = ⎨ 1, otherwise ⎩ x + y − 2 xy b) S2(x, y) = 1 − xy c) S3(x, y) =

x+ y 1 + xy

[

]

1

d) S4(x, y) = 1 − max[0, (1 − x) r + (1 − y ) r − 1] r , r ≠ 0. Hint: Apply Eq. (2.1) to the T-norms of Problem 2.4. 2.7. Consider the function f(x) = lnx and a fuzzy set A with membership function as shown in Figure 2.7a. Compute the membership function of the fuzzy set f (A ). 2.8. Given a universe of discourse X with a crisp subset Y, the complement of Y with – respect to X is the set of elements of X not in Y, i.e., Y; = {x ∈ X | x ∉ Y}. – a) Verify the law of contradiction, Y ∩ Y; = ∅, and the law of the excluded middle, – Y ∪ Y; = X. b) Do these laws hold in fuzzy logic too? Why? (Hint: Take any fuzzy subset A of X.) 2.9. Consider the following fuzzy implications – R1: A → B = A; ∪ B – – – R2: A → B = (A ∩ B) ∪ (A; ∩ B) ∪ (A; ∩ B; )

62


– – where Α, Β represent fuzzy sets and A; , B; their complements according to Zadeh’s definition. Show that R2 ⊆ R1. (Hint: Use properties of T-norms and S-norms.) 2.10. Consider a Mamdani implication A → B = “if x is A, then y is B,” where A = 0.33/6 + 0.61/7 + 1/8 + 0.67/9 + 0.33/10 B = 0.4/4+0.67/5+1/6+0.4/7+0.6/8. If an observation for x is A* = 0.5/6 + 1/7 + 0.5/8, compute y = B* by using the max-min composition. 2.11. Consider the fuzzy relation “Bill is slightly taller than Nick” defined in {170, 172.5, 175, 177.5, 180, 182.5, 185}2 (height in centimeters), which is represented by the matrix shown in Table 2.5. Further, the observation “Bill is not very tall” may be defined as 1/170 + 0.9/172.5 + 0.9/175 + 0.7/177.5 + 0.5/180 + 0.2/182.5 + 0/185. Compute the membership function of the fuzzy set “height of Nick” by using the max-min composition.

Bill

Table 2.5. Fuzzy relation for Problem 2.11

170 172.5 175 177.5 180 182.5 185

170 1 1 1 1 1 1 1

172.5 0.8 1 1 1 1 1 1

175 0.4 0.8 1 1 1 1 1

Nick 177.5 0.2 0.4 0.8 1 1 1 1

180 0.1 0.2 0.4 0.8 1 1 1

182.5 0 0.1 0.2 0.4 0.8 1 1

185 0 0 0.1 0.2 0.4 0.8 1

2.12. A fuzzy system with inputs x1, x2, and output y, all in the same universe of discourse {0, 1, 2, 3}, has a complete rule base with rules “if x1 is A1 and x2 is A2, then y is B.” The linguistic values A1, A2, and B belong to the term set {low (L), medium (M), high (H)}, where L = 1.0/0 + 0.6/1 + 0.1/2


63

M = 0.5/0 + 1.0/1 + 0.4/2 + 0.1/3 H = 0.5/2 + 1.0/3. The system employs the following algebraic transformations: A = “A1 and A2”: μA = 0.6 max(μ1, μ2) + 0.4 min(μ1, μ2) A → B = “if A then B”: μA→B = max(1 − μA, μB). a) Suppose that x1 = 1 and x2 = 2. Compute the membership function of the output by using the max-min composition. b) Suppose now that the observation for x1 is “more or less medium,” and for x2 is “very high,” where

μvery H(x) = [μH(x)]2 μmore or less M(x) = [μM(x)]1/2. For simplicity, assume that the only rule which activates (fires) is “if x1 is H and x2 is M, then y is H.” Compute the membership function of the output. 2.13. Compute a crisp value of the fuzzy set “height of Nick” of Problem 2.11 by using the center-of-area and the centroid defuzzification formulas. Compare the results.

Chapter 3

SUSTAINABILITY INDICATORS 3.1. WHAT ARE SUSTAINABILITY INDICATORS? Sustainability indicators isolate specific attributes of sustainability quantitatively or qualitatively to provide an idea of the state of each attribute. Concentrations of pollutants, rates of harvesting biodiversity, average numbers of doctors or hospitals could serve as indicators. People love numbers when they try to measure things. As we shall see, all the indicators of the fuzzy model of the next chapter will be expressed numerically with the help of fuzzy logic, although at first glance they might appear qualitative. Sustainability indicators have been used by specialists for decades or even hundreds of years. Let us see some of those.

3.2. THE SHANNON INDEX OF BIODIVERSITY Claude E. Shannon in two famous papers (Shannon 1948) developed information theory which laid the foundations of modern communication systems. His theory found applications, besides telecommunications, in physics and biology among others. The central concept of information theory is called entropy and is related conceptually to the entropy of thermodynamics. Definition 3.1. Let a random experiment have n outcomes E1, E2, …, En with probabilities P1, P2, …, Pn respectively. The entropy of this random experiment is defined as n

H = −∑ Pi log m Pi . i =1

If the base of the logarithm m is 10, the units of H are Hartleys; if m = 2, then H is measured in bits (from binary units) and if m = e, H is measured in nats (from natural units). Conceptually, the entropy has several equivalent interpretations: it is a measure of average disorder in a random situation, or a measure of average uncertainty, or average number of choices available. Take as an example the tossing of a loaded coin and let the

66


probabilities of heads and tails be P(h) = p and P(t) = 1 − p respectively. If p = 1, the outcome is always h and there is no uncertainty or choice. Indeed, the entropy is H = 1 × log21 + 0 × log20 = 0 bits. If on the other hand p = 1/2, the uncertainty or choice reaches a maximum and

H=

1 1 log 2 2 + log 2 2 = 1 bit . 2 2

Graphically: H Hmax

1

0 0

0.5

1

p

Figure 3.1. Plot of H = −plog2p − (1 − p)log2(1 − p).

In general, H reaches its maximum when P1 = P2 = … = Pn = 1/n and

H max = n ×

1 log 2 n = log 2 n bits . n

Entropy can also be viewed as the average number of binary questions (Yes-No) one asks to find the solution to a problem. For example, suppose that at a certain point we have the choices Ci, i = 1, 2, 3, 4, with the corresponding probabilities on the arcs in Figure 3.2. There are two Yes-No questions to be asked in all; the first question is: “is it group A or B?” and the second one is: “is it the upper branch or the lower?” The entropy is

H = 4× as expected.

1 log 2 4 = 2 bits 4


67

C1 1/4 A 1/4

C2

1/4 C3 1/4 B C4

Figure 3.2. Composite experiment.

Now consider the experiment in Figure 3.3. C1

1/4

A 1/4

C2

1/2 C3

Figure 3.3. Another composite experiment.

Here we ask “Is it A or C3?” If the answer is “A,” then we ask one more question “Is it C1 or C2?” If on the other hand the answer is “C3” we stop. The probability of A being true is P(A) =

1 1 1 + = . 4 4 2

If the answer of the first question is “A,” then we ask the second question with probability 1. Hence we ask 2 questions in all with probability 1/2. If the answer to the first question is “C3,” there is no other question to be asked and this answer has probability P(C3) =

1 . 2

On average we have

2× The entropy is

1 1 + 1 × = 1.5 questions. 2 2

68


1 1 H = 2 × log 2 4 + log 2 2 4 2 = 1.5 bits. These interpretations of entropy led biologists to its use as a measure of shuffledness in populations or choice of biodiversity: the higher H is the more diverse an ecosystem is. Specifically, let an ecosystem consist of n species. Each species i has Ni individuals, i = 1, …, n. The total number of individuals of all species is n

∑ Ni = N . i =1

Let

Pi =

n Ni , H = −∑ Pi log 2 Pi . N i =1

If P1 = P2 = … = Pn = 1/n, then the biodiversity index H reaches its maximum Hmax = log2n. Example 3.1. An ecosystem has n = 10 species and a total of N = 10,000 individuals. If N1 = 10,000 and N2 = … = N10 = 0, the biodiversity index is H = 1log21 + 9log20 = 0 but if N1 = N2 = … = N10 = 1,000, then H = 10 ×

1 log210 = 3.32 bits, 10

which is the maximum possible biodiversity in the sense of the Shannon index.

3.3 TOTAL FACTOR PRODUCTIVITY (TFP) Sustainability indicators that reduce a complex attribute into a number such as the Shannon index, are quite attractive but may lead to false conclusions. This is the case with the Total Factor Productivity or TFP. It is defined as (Lynam and Herdt 1989)

TFP =

value of outputs of a farming system . value of inputs into a farming system


69

Take an area of cleared rainforest where the output in the beginning is high and the inputs low. According to the above definition this system is highly sustainable. But after a few applications of chemicals and a few productive seasons, the soil deprived of its natural nutrient cycles of the rainforest becomes barren and its output goes down dramatically. To be sure, phosphorous fertilizers together with other chemical inputs could sustain such a system much longer, but the end result is the elimination of unestimable services of the tropical forest in favor of a small number of cash crops. The TFP does not account for complex biological processes. Even conventional farms, which might become productive with massive applications of chemical fertilizers and pesticides, reach a point of diminishing returns. Finally, too many chemical inputs destroy the natural processes that agriculture needs, such as humus and the beneficial microorganisms in the soil. A high TFP does not necessarily guarantee sustainability. Other considerations should be included in a more detailed manner; complex systems are rarely amenable to simplistic reductions.

3.4. MAXIMUM SUSTAINABLE YIELD (MSY) The Maximum Sustainable Yield or MSY expresses simply the idea of continuity of flow. If a system, e.g., an aquifer has an inflow X and an outflow Y, the difference X – Y, which is the accumulation, can be drawn without disturbing the system.

X

Y

X−Y

Figure 3.4. Flow through a system.

If X − Y > 0, then we have an outflow and if X – Y < 0, we have an inflow. If now X are births and Y are deaths in a biological population such as game, fish, or plants, the difference X – Y represents the maximum number of individuals we can harvest without reducing the population; this number is called MSY. MSY in this context is rather simplistic. When harvesting a population, care should be taken not to take too many females which are needed for breeding. Also, breeding seasons should be avoided to prevent reduction in births. MSY does not account for such details. To be able to compute MSY in the case of biological populations, one has to develop models of the population dynamics. Such models exist and are given in the form of differential equations. They contain parameters, usually constant, which give the carrying capacity of the ecosystem, the rate of change of the population over time, the harvesting rate etc. If the harvesting effort, e.g., tons of a given fish species over a year, is at MSY, then, supposedly the fishery is sustainable. The story of the Peruvian anchovy fishery shows otherwise. Anchovies were fished at MSY, which was about 10 million tons per year in the

70


1960’s. Then in the early 1970’s the Peruvian anchovy fishery suddenly collapsed. The problem was that MSY did not represent reality and did not guarantee sustainability. The reason for this is that population dynamics are quite complex and often random processes, rarely if at all amenable to simple differential forms. In the anchovy story for instance, factors such as the occurrence of El Niño could not be factored in. El Niño brings warmer water into the fishery and with it, perhaps large numbers of predators which tip the balance. Interactions with other populations, global warming and the associated nutrient imbalance, or other unforeseen factors cannot be expressed algebraically. Therefore, the utility of MSY is often doubtful, despite its simplicity and the fact that it is still used by the UN as a sustainability indicator. For a detailed critique of MSY, the reader may consult Bell and Morse (1999).

3.5. SUSTAINABILITY ASSESSMENT BY FUZZY EVALUATION (SAFE) The SAFE model was developed to assess the sustainability of nations (Phillis and Andriantiatsaholiniaina 2001, Phillis et al. 2003, Andriantiatsaholiniaina et al. 2004). It uses indicators of environmental integrity, economic efficiency, and social welfare as inputs and employs hierarchical fuzzy inference to provide a sustainability measure. As shown in Figure 3.5, the overall sustainability (OSUS) in the SAFE model encompasses two broad components, called primary indicators, ecological sustainability (ECOS) and human sustainability (HUMS). The ecological dimension of sustainability comprises four secondary indicators: water quality (WATER), land integrity (LAND), air quality (AIR), and biodiversity (BIOD). The human dimension of sustainability comprises another four secondary indicators: political aspects (POLICY), economic welfare (WEALTH), health (HEALTH), and education (KNOW). Each secondary indicator is evaluated by means of three tertiary indicators: Pressure (PR), State (ST), and Response (RE) indicators. State is the present state of a component such as the size of forested land. Pressure is a force tending to change State such as the deforestation rate. Response is the reaction taken to bring Pressure to a level that will guarantee a better State as, for example, protecting a given area. The Pressure-State-Response approach was originally proposed by OECD (1991) to assess the environmental component of sustainability. A detailed review and variants of this approach are presented in Spangenberg and Bonniot (1998). Finally, each tertiary indicator is a function of a number of more specialized indicators (not shown in Figure 3.5). For example, the state of biodiversity is an aggregate measure of the forest area and the numbers of plant, fish, and mammal species per square kilometer. We call these indicators basic because they are the starting point for computing all the composite indicators described above.


71

OSUS

ECOS

HUMS

PR ST RE

AIR

POLICY

PR ST RE

PR ST RE

LAND

WEALTH

PR ST RE

PR ST RE

WATER

HEALTH

PR ST RE

PR ST RE

BIOD

KNOW

PR ST RE

Figure 3.5. Dependencies of sustainability components in the SAFE model.

SAFE provides flexibility in the choice of indicators and, although the structure shown in Figure 3.5 remains the same, the basic indicators can be changed if necessary to capture the rapid changes in the global natural and socio-economical environments. The next chapters describe two applications of SAFE to the assessment of sustainability of nations or geographic regions and the sustainability of organizations.

3.6. PROBLEMS-QUESTIONS 3.1. Given the population of a country P and the average consumption per capita C, derive a simple expression of the environmental impact I of this population if we assume that I is proportional to C. C can be energy E or matter M per capita or energy per unit GDP. 3.2. The inflow to a lake is X km3/year and the outflow to the sea via a river is Y km /year. The difference Z = X − Y is withdrawn by a nearby city to satisfy the needs of people. To maximize the amount that goes to the city Y should become zero and Z = X. Is this desirable? Why? 3

3.3. List 10 services that species provide to the environment or humans.

72


3.4. Public transportation needs oil or electricity for its operation. It thus contributes to greenhouse gas emissions. Why is it desirable over the private car? 3.5. The Gini index measures the extent to which the distribution of the income in a country deviates from a perfectly equal distribution. Zero means perfect equality and 100 perfect inequality. Give a more or less desirable value of the index according to your judgment. Explain. 3.6. List 5 reasons why a good public health system can improve sustainability. 3.7. Do the same for education. 3.8. Do the same for economy.

Chapter 4

FUZZY ASSESSMENT This chapter provides a detailed description of the sustainability assessment of countries using the SAFE model. The first two sections deal with the description of the model and its inputs. Section 4.3 presents a method to make different indicators comparable using normalization and linear interpolation between the most desirable and the least desirable values. Annual indicator data are often unavailable or imprecise. Moreover, past environmental and socioeconomic pressures often continue to be effective in the future. Section 4.4 presents an approach to partially address these issues by exponentially smoothing current normalized values with past values. Sections 4.5 and 4.6 describe the linguistic values, membership functions, and rule bases employed in the SAFE model. Apart from the assessment of sustainability, SAFE provides further insights to sustainable development by identifying the most important indicators that affect sustainability. This is achieved through sensitivity analysis, which is described in Section 4.7. The last section of this chapter gives a ranking of countries from most sustainable to less sustainable according to SAFE and presents the most important indicators for selected countries.

4.1. CONFIGURATION OF THE SAFE MODEL In Section 3.5 we saw the main dependencies of sustainability components in the SAFE model. A more detailed description of the various features of SAFE is given in Figure 4.1.

74


5

Third order inference engines

PRESSURE FUZZY RULES

STATE RESPONSE

LAND

6

PRESSURE

Second order inference engines

FUZZY RULES

STATE RESPONSE

FUZZY RULES

WATER


STATE

ECOS

RESPONSE

BIOD

7


STATE

First order inference engine

RESPONSE

FUZZY RULES

AIR


STATE RESPONSE

POLIC

OSUS: numerical output by defuzzification


STATE RESPONSE PRESSURE

FUZZY RULES

STATE

8

FUZZY RULES

WEALTH

HUMS

RESPONSE

HEALTH


STATE RESPONSE

KNOW

FUZZY INFERENCE FOR PR, ST, RE 1

0

Fuzzy basic indicators

4 FUZZY RULES FOR TERTIARY VARIABLES: Pressure (PR) State (ST) Response(RE)

3 FUZZIFICATION 1

0

1

Normalized basic indicators NORMALIZATION

0

υc

[τc, Tc]:

Uc

target values

c

xc … xc'

2 c

1

1

c'

EXPONENTIAL SMOOTHING

Normalized time series

(xc,1, xc,2, …), ..., (xc',1, xc',2, …)

c'

(zc,1, zc,2, …), ..., (zc',1, zc',2, …) Time series of basic indicators

Figure 4.1. Steps 1–8 of the hierarchical assessment of sustainability. (Kouloumpis et al. 2008, Figure 1. © 2008 by IEEE. Used with permission.)

Fuzzy Assessment

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The steps of the model are outlined below. Annual data for each basic indicator c form a time series (zc,1, zc,2, …, zc,n), where the indices 1, 2, …, n correspond to the time points at which measurements were taken, e.g., 1 could be the year 2000, 2 the year 2001, etc. Each value zc,k is normalized on [0, 1] by linear interpolation between the most desirable and the least desirable values as we shall soon see. 2. The normalized time series, (xc,1, xc,2, …, xc,n), is transformed into a single normalized value xc by exponential smoothing. 3. For each value xc we compute its membership grades to various fuzzy sets, which form a complete ordered partition of the interval [0, 1]. This is called fuzzification. The fuzzy sets “bad,” “average,” and “good” are examples of ordered sets (see also the discussion of Section 2.4.6). 4. Fuzzy inference engines of the Takagi-Sugeno-Kang (TSK) type (see Section 2.4.5) use the fuzzified inputs and “if-then” rules to compute the fuzzy tertiary variables Pressure (PR), State (ST), and Response (RE). 5–7. Subsequent TSK inference engines compute fuzzy values for the secondary indicators of sustainability (LAND, WATER, POLICY, …), the primary indicators (ECOS and HUMS) and, finally, the overall sustainability (OSUS). 8. A crisp value, i.e., a single numerical value for OSUS is computed by height defuzzification. 1.

4.2. BASIC INDICATORS We begin the assessment of sustainability with the selection of basic indicators and the compilation of numerical data for each one of them. This selection requires caution so that indicators cover every possible aspect of sustainability and, at the same time, do not overlap with each other too much. Below we present a number of basic indicators used to assess the tertiary components of sustainability, Pressure (PR), State (ST), and Response (RE). Definitions of indicators are taken from Esty et. al (2005), Food and Agricultural Organization (FAO), International Union for the Conservation of Nature (IUCN 1994), Organization of Economic Cooperation and Development (OECD 2000, 2004, and website), Ordoubadi (2005), United Nations Environmental Program (UNEP), United Nations Statistics Division (2006), United Nations Development Program (UNDP 2003), United Nations Educational, Scientific, and Cultural Organization (UNESCO), United Nations Framework Convention on Climate Change (UNFCCC), World Health Organization (WHO), World Bank (1995, 2008), and the Freedom House Annual Survey (2007). Many of these references also provide annual data about basic indicators for most countries of the world.

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4.2.1. Ecosystem LAND Indicators PR(LAND) (1) Municipal waste generation (kg per capita per year): Waste collected and treated by or for municipalities. It covers waste from households (including bulky waste), similar waste from commerce and trade, office buildings, institutions and small businesses, yard and garden waste, street sweepings, the contents of litter containers, and market cleaning waste. The definition excludes waste from municipal sewage networks and treatment, as well as municipal construction and demolition waste. Reducing waste generation improves land sustainability. (2) Nuclear waste (tons of heavy metals per capita per year): Nuclear waste is primarily due to spent fuel from nuclear power plants. It is assumed that nuclear waste influences land sustainability negatively due mainly to generation of heavy radioactive metals. (3) Hazardous waste (tons of waste per capita per year): Waste found in streams to be controlled according to the Basel Convention on the Control of Transboundary Movements of Hazardous Wastes and their Disposal. Reduction of hazardous waste improves land sustainability. (4) Population growth rate (percentage): Average annual exponential rate of population change for given periods of years. Small or zero population growth rate is perceived as influencing positively land sustainability. (5) Pesticide consumption (kg of pesticide consumption per hectare of arable land): Pesticide use intensity refers to the amount of pesticide used per hectare of arable and permanent cropland. Excessive use of pesticides in agricultural activities has negative impacts on soil, water, humans and wildlife. (6) Fertilizer consumption (kg of fertilizer per hectare of arable land): Fertilizer consumption measures the quantity of plant nutrients used per unit of arable land in the form of nitrogenous, potash, and phosphate fertilizers (including ground rock phosphate). Excessive use of fertilizers from agricultural activities has a negative impact on soil and water, altering chemistry and levels of nutrients and leading to eutrophication of water bodies. ST(LAND) (7) Desertification of land (percent of dryland area): Areas with a potential hazard of desertification. All major continents face problems of land degradation in dryland areas, commonly known as desertification. Dryland areas are ‘fragile’ in that they are extremely vulnerable to land degradation resulting from over-grazing and other forms of inappropriate land use. (8) Forest area (percent of what existed in the year 2000): Forest area is land under natural or planted stands of trees, whether productive or not. Forests maintain land sustainability.

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RE(LAND) (9) Forest change (annual rate of change): Forest area change is the net change in forests and includes expansion of forest plantations and losses and gains in the area of natural forests. A positive forest change improves land sustainability. (10) Protected area (ratio to surface area): An area of land and/or sea especially dedicated to the protection and maintenance of biological diversity, and of natural and associated cultural resources, and managed through legal or other effective means (IUCN 1994). Protected area ensures land sustainability. (11, 12) Recycling rates: glass11, paper12 (percent of apparent consumption): Recycling rates are the ratios of the quantity collected for recycling to the apparent consumption. Reducing uncontrolled waste improves land sustainability.

WATER Indicators PR(WATER) (13) Total water withdrawals (percent of total renewable resources): Total annual amount of water withdrawn per amount of renewable water resources. Excessive use of water reduces water sustainability. (5) Pesticide consumption (6) Fertilizer consumption Indicators 5 and 6 have already been described. ST(WATER) (14) Organic water pollutant emissions (BOD, biological oxygen demand in kg per capita per day): Emissions of organic water pollutants are measured by biochemical oxygen demand, which is the amount of oxygen that bacteria in water will consume to break down waste. This is a standard water treatment test for the presence of organic pollutants. (15) Phosphorous concentration (mg phosphorus per liter of water). It is a measure of eutrophication, which affects the health of aquatic resources. High levels of phosphorus increase the chances of eutrophication. (16) Metals concentration (micro-Siemens per centimeter): It is a widely used bulk measure of metals concentration and salinity. Siemens is a unit of electric conductivity. High levels of conductivity correspond to high concentrations of metals. RE(WATER) (17) Public wastewater treatment plants (percent of population connected): Connected means actually connected to a waste water treatment plant through a public sewage network. Non-public treatment plants, i.e., industrial waste water plants, or individual private treatment facilities such as septic tanks are not included. High connectivity improves water sustainability.

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BIODIVERSITY Indicators PR(BIOD) (18–23) Threatened bird18, mammal19, plant20, fish21, amphibian22, and reptile23 species (percentage): Includes all species that are critically endangered, endangered, or vulnerable, but excludes introduced species, species whose status is insufficiently known, those known to be extinct, and those for which a status has not been assessed. IUCN has established detailed quantitative definitions for the above categories. Very briefly: Critically endangered is a species that faces an extremely high risk of extinction. A species in this category has experienced or will experience a population reduction of at least 80% within the next 10 years or the next 3 generations, whichever is longer and causes of extinction may not have ceased or may not be understood or may not be reversible. Endangered species face a very high risk of extinction and the corresponding population reduction as above is at least 50%. Finally, vulnerable species face a high risk of extinction and a corresponding population reduction of at least 30%. Other criteria could also apply to these definitions. ST(BIOD) (7) Desertification of land (8) Forest area RE(BIOD) (9) Forest change (10) Protected area Indicators 7–10 have already been described.

AIR Indicators PR(AIR)y (24) Ozone depleting substances per capita (consumption in ozone depleting potential metric tons): An ozone depleting substance is any substance containing chlorine or bromine, which destroy the stratospheric ozone layer that absorbs most of the biologically damaging ultraviolet radiation. Ozone depleting potential (ODP) refers to the amount of ozone depletion caused by a substance. ODP is defined as the ratio of the impact on stratospheric ozone of a substance to the impact of the same mass of CFC-11. CFC-11 has an ODP of 1. (25) Greenhouse gas (GHG) emissions per capita (tons of CO2 equivalent). Emissions of total GHG (CO2, CH4, N2O, hydrofluorocarbons (HFC’s), perfluorocarbons (PFC’s), and SF6), excluding land-use change and forestry. To convert all emissions to CO2 equivalent, the global warming potential (GWP) is used. GWP is an index

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used to translate the level of emissions of various gases into a common measure in order to compare the relative radiative forcing of different gases without directly calculating the changes in atmospheric concentrations. GWP is the ratio of the warming caused by a substance to the warming caused by the same mass of CO2 (see also Table 5.9).

ST(AIR) (26) Mortality from respiratory diseases (number of deaths per 100,000 persons). Diseases of the respiratory system generally cause irritation and reduced lung function, especially in more susceptible members of the population such as young children, the elderly and asthmatics. (27–29) Atmospheric concentrations of NO227, SO228 and total suspended particulates29 (μg/m3 of air ): The values were originally collected at the city level. The number of cities with data provided by each country varies. Within each country the values have been normalized by city population for the year 1995, and then summed to give the total concentration for the given country. High concentrations decrease air sustainability. RE(AIR) (30) Renewable resources production (percent of total primary energy supply): The higher the proportion of renewable energy sources is, the less a country relies on environmentally damaging sources such as fossil fuel and nuclear energy.

4.2.2. Human System POLICY Indicators PR(POLICY) (31) Military spending (percent of GDP): For members of the North Atlantic Treaty Organization (NATO) it is based on the NATO definition, which covers militaryrelated expenditures of the defense ministry and other ministries. Civilian-type expenditures of defense ministries are excluded. Military assistance is included in the expenditure of the donor country. Purchases of military equipment on credit are recorded at the time the debt is incurred, not at the time of payment. Data for nonNATO countries generally cover expenditure of the ministry of defense; excluded are expenditures on public order and safety, which are classified separately. (32) Ratio of refugees from a country to total population of that country: Refugees are people who are recognized as refugees under the 1951 Convention Relating to the Status of Refugees or its 1967 Protocol, the 1969 Organization of African Unity Convention Governing the Specific Aspects of Refugee Problems in Africa, people recognized as refugees in accordance with the UNHCR (United Nations High Commissioner for Refugees) statute, people granted a refugee-like humanitarian status, and people provided with temporary protection.

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Yannis A. Phillis and Vassilis S. Kouikoglou (33) Poverty rate (percent of population): National poverty rate is the percentage of population living below the national poverty line. The latter is usually estimated by finding the total cost of all the essential resources that an average human adult consumes in one year.

ST(POLICY) (34) Political Rights: The Freedom House Annual Survey employs the Political Rights checklist to help determine the degree to which people can participate in the political process of their country. Each country is then rated on a seven-category scale, 1 representing the most free and 7 the least free. (35) Civil Liberties: The Freedom House Annual Survey employs a Civil Liberties checklist to help monitor the progress and decline of human rights worldwide. As previously, each country is rated on a seven-category scale, 1 representing the most free and 7 the least free. (36) Gini index: It measures the extent to which the distribution of income among individuals or households within an economy deviates from a perfectly equal distribution. A Gini index of zero would represent perfect equality and an index of 100 would imply perfect inequality—a single person or household accounting for all income or consumption. (37) Corruption Perceptions Index: International Transparency, the coalition against corruption, gathers data for the last two years for all countries to compute the Corruption Perceptions Index (CPI). CPI (Transparency International 2007) ranges from 1 to 10 or from the most to the least corrupt countries and it expresses a degree of misuse of power by public officials and politicians for private gain such as bribes, favoritism, embezzlement of money, etc. RE(POLICY) (38) Environmental laws and enforcement: This index ranges: from zero to one and is obtained by a subjective assessment on the basis of various world reports and experts’ knowledge. National environmental laws are included in the context of this indicator as well as international agreements such as the Convention on Biological Diversity, the Ramsar Convention on Wetlands of International Importance, the Convention on International Trade of Endangered Species (CITES), national environmental laws, etc. (39) Tax revenue (percent of GDP): Tax revenue refers to compulsory transfers (payments) to the central government for public purposes. Certain compulsory transfers such as fines, penalties, and most social security contributions are excluded. Refunds and corrections of erroneously collected tax revenue are treated as negative revenue.

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WEALTH Indicators PR(WEALTH) (40) GDP implicit deflator (average annual percent growth rates): Reflects changes in prices for all final demand categories, such as government consumption, capital formation, and international rate, as well as the main component, private final consumption. It is derived as the ratio of current to constant-price GDP. It is known as inflation indicator affecting the sustainability of a national economy. (41) Imports (percent of GDP): Shows the cost plus insurance and freight value in U.S. dollars of goods purchased from the rest of the world. (42) Unemployment: Unemployment refers to the share of the labor force that is without work but available for and seeking employment. Definitions of labor force and unemployment differ by country. (43) Unemployment equality gap between genders: This variable shows the absolute difference between unemployment rate for female and male labor force. ST(WEALTH) (44) Central government debt (percent of GDP): Debt is the entire stock of direct government fixed-term contractual obligations to others outstanding on a particular date. It includes domestic and foreign liabilities such as currency and money deposits, securities other than shares, and loans. It is the gross amount of government liabilities reduced by the amount of equity and financial derivatives held by the government. Because debt is a stock rather than a flow, it is measured as of a given date, usually the last day of the fiscal year. (45) GNI per capita PPP (based on PPP, purchasing power parity): PPP GNI is gross national income converted to international dollars using purchasing power parity rates. An international dollar has the same purchasing power over GNI as a U.S. dollar has in the United States. GNI is the total market value of all final goods and services produced within a country (also called gross domestic product or GDP), plus income received from other countries such as interest and dividends, minus similar payments made to other countries. PPP equalizes the purchasing power of different currencies for a given set of goods. Thus GNI PPP (U.S.$) is national income converted to international dollars using a conversion factor. International dollars correspond to the amount of a given basket of goods and services one could buy in the U.S. with a given sum of money. Data are in current international dollars. This indicator is commonly used to evaluate the status of wealth sustainability at the national level. (33) Poverty rate (indicator has already been described) RE(WEALTH) (46) Exports (percent of GDP): Exports of goods and services represent the value of all goods and other market services provided to the rest of the world. Exports create wealth. (47) Foreign direct investment, net inflows (percent of GDP): Foreign direct investment is net inflows of investment to acquire a lasting management interest (10 percent or

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Yannis A. Phillis and Vassilis S. Kouikoglou more of voting stock) in an enterprise operating in an economy other than that of the investor.

HEALTH Indicators PR(HEALTH) (48) Infant mortality rate: Number of infants who die before reaching one year of age, expressed per thousand live births in a given year. (49) Maternal mortality rate: Annual number of deaths from pregnancy or childbirth related causes per 100,000 live births. A maternal death is defined by WHO as the death of a woman while pregnant or within 42 days of the termination of pregnancy from any cause related to or aggravated by the pregnancy, including abortion. (50) HIV/AIDS prevalence (percent of population aged 15–49): Prevalence of HIV refers to the percentage of people ages 15–49 who are infected with HIV. (51) Tuberculosis prevalence (per 100,000 population): It refers to people with all forms of TB, including TB in people with HIV infection. (52) Standardized reported malaria cases per 1,000: Standardized cases are derived from the total reported number of cases and an appreciation of the proportion of these cases that were laboratory-confirmed. Reported cases per country for the most recent year for which WHO/RBM (World Health Organization/Roll Back Malaria) received data. The standardized case reporting rate (per 1,000 per year) is calculated by dividing the standardized cases by the national population size estimated by the United Nations Population Division for the middle of the year under consideration. (53) Solid fuel household use (percent of total energy): The use of solid fuels in households is associated with increased mortality from pneumonia and other acute lower respiratory diseases among children as well as increased mortality from chronic obstructive pulmonary disease and lung cancer (where coal is used) among adults. It is also a Millennium Development Goal indicator. National energy statistics on the proportion of population using solid fuels are based either on data from surveys or censuses, or on modeling where no survey or census data are available. ST(HEALTH) (54) Life expectancy: Number of years a newborn infant would live if patterns of mortality prevailing at the time of its birth were to stay the same throughout its life. (55, 56) Infants immunized against severe diseases: Percent of one-year-old infants immunized against measles55 and diphtheria-pertussis-tetanus (DPT)56. (57) Daily per capita calorie supply (percent of total requirements): Data taken from the Food and Agriculture Organization (FAO) food balance sheets. The calories and protein actually consumed may be lower than the figure shown, depending on how much is lost during home storage, preparation, and cooking, and how much is fed to pets and domestic animals or discarded.

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RE(HEALTH) (58) Number of doctors (per thousand people): The term doctors includes physicians that are defined as graduates of any facility or school of medicine who are working in the country in any medical field (practice, teaching, research). (59) Hospital beds (per thousand people): Hospital beds include inpatient beds available in public, private, general, and specialized hospitals and rehabilitation centers. In most cases beds for both acute and chronic care are included. (60) Public health expenditure (percent of GDP): Consists of recurrent and capital spending from government budgets, external borrowings and grants, and social health insurance funds. (61, 62) Access to improved water sources61 and to improved sanitation62 (percent of population): The percentage of population with access to the facilities that can provide them with safe water and sanitation. Access to the above is a fundamental need and a human right vital for the dignity and health of all people.

KNOWLEDGE Indicators PR(KNOW) (63–65) Ratio of students to teaching staff (primary63, secondary64, and tertiary65 education): Teaching staff includes (OECD 2000) professional personnel involved in direct student instruction: classroom teachers, special education teachers, other teachers who work with students as a whole class, and chairpersons of departments; it does not include nonprofessional personnel who support teachers. ST(KNOW) (66, 67) Expected years of schooling; male66 and female67: Average number of years of formal schooling that a child is expected to receive, including university education and years spent in repetition. It may also be interpreted as an indicator of the total educational resources, measured in school years, a child will require over the course of schooling. (68, 69) Net school enrollment ratio; primary68 and secondary69: Number of children of official school age, as defined by the education system, enrolled in primary or secondary school, expressed as percentage of the total number of children of that age. (70) Literacy rate, adult total (percent of people with ages 15 and above): Adult literacy rate is the percentage of people ages 15 and above who can, with understanding, read and write a short, simple statement on their everyday life. (71) World Bank’s Knowledge Economy Index (KEI): KEI measures the degree to which a country uses knowledge efficiently to improve its economical development. It is an aggregate index that represents the overall level of development of a country or region towards the knowledge economy. RE(KNOW) (72) Public expenditure on R&D (percent of GDP): Expenditures for research and

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(73)

(74)

(75)

(76)

development are current and capital expenditures (both public and private) on creative, systematic activities that increase the stock of knowledge. Included are fundamental and applied research and experimental development work leading to new devices, products, or processes. Public expenditure on education: Percentage of GNP accounted for by public spending on public education plus subsidies to private education at the primary, secondary, and tertiary levels. It may exclude spending by religious schools, which play a significant role in many developing countries. Data for some countries and for some years refer to spending by the ministry of education of the central government only, and thus exclude education expenditures by other central government ministries and departments, local authorities, and others. Personal computers (per thousand people): Estimated numbers of self-contained computers used by a single person. Access to personal computers promotes knowledge development and educational sustainability. Internet users (per thousand people): Number of computers directly connected to the worldwide network of interconnected computer systems, per 10,000 people. Access to the Internet facilitates knowledge acquisition. Information and communication technology expenditure (percent of GDP): Information and communications technology expenditures include computer hardware (computers, storage devices, printers, and other peripherals); computer software (operating systems, programming tools, utilities, applications, and internal software development); computer services (information technology consulting, computer and network systems integration, web hosting, data processing services, and other services); communications services (voice and data communications services); and wired and wireless communications equipment.

4.3. NORMALIZATION Basic indicators come in a variety of scales and units. Lower values mean better sustainability performance for some indicators but worse performance for others. For example, land sustainability improves when municipal waste generation decreases but weakens when forest area decreases. Also SO2 concentration is measured in parts per million by volume (ppmv), economic indicators in monetary units, and population data in absolute numbers or percentages. To make indicators comparable and to facilitate analysis, the data are normalized on a 0–1 scale by assigning the value 0 to the least desirable indicator values and the value 1 to the most desirable indicator values or targets, which are determined by experts, standards, laws, etc. For example, HIV/AIDS prevalence rate per cent of population had a maximum value of 37.3% over all countries in 2005. Given its significant potential for rapid spread, even a value of 2% for this indicator is considered to be very bad. The Joint United Nations Programme on HIV/AIDS provides an upper bound of 0.9% on the average HIV prevalence rate (UNAIDS 2007). The least desirable value U is chosen as twice the upper bound. All HIV/AIDS prevalence rates greater than or equal to 1.8% are assigned the value 0. The rate 0%, which is the target for this indicator, corresponds to 1.

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Let c be an indicator and zc its value for the country whose sustainability we want to assess. In general, the target of c can be a single value Tc or an interval on the real line of the form [τc, Tc] representing a range of equally desirable values for the indicator. Least desirable values can be sole points or sets of values below or above some critical threshold. Critical values are denoted υc and Uc, so that all values zc ≤ υc or zc ≥ Uc are assigned a normalized value 0. In practice υc is the minimum value of zc over all countries under examination and Uc its maximum. In some cases though we choose these numbers differently. For example, we have UAIDS = 1.8% whereas the maximum HIV/AIDS prevalence rate worldwide is 37.3%. Thus, if an indicator must be at most equal to Tc to be sustainable, then we have the case of Figure 4.2. Here we do not need υc and τc. An HIV/AIDS prevalence rate of 0.9% is assigned the normalized value 0.5 because it is halfway between the target 0% and the critical threshold 1.8% of least desirable values.

xc =

U c − zc U c − Tc

1 normalized value, xc 0

zc

Tc

Uc undesirable values

target set

Figure 4.2. Normalization by linear interpolation: smaller is better (SB).

Similarly, if an indicator must be at least equal to τc to be sustainable, we have the case of Figure 4.3 and we do not need Tc and Uc.

xc =

zc − υ c τ c − υc


υc undesirable values

zc

τc target set

Figure 4.3. Normalization by linear interpolation: larger is better (LB).

Finally, if an indicator must lie in [τc, Tc] to be sustainable, then we have the full diagram of Figure 4.4.

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υc

τc

undesirable values

Tc

target set

Uc

zc

undesirable values

Figure 4.4. Normalization by linear interpolation: nominal is best (NB).

A normalized value xc for zc is calculated as follows:

0, ⎧ ⎪ ⎪ zc − υc , ⎪τ c − υ c ⎪ 1, xc = ⎨ ⎪U − z c ⎪ c , ⎪U c − Tc ⎪ 0, ⎩

zc ≤ υc

υc < zc < τ c τ c ≤ z c ≤ Tc

(4.1)

Tc < z c < U c U c ≤ zc .

The target and least desirable values for each indicator are given in Table 4.1. Table 4.1. Least and most desirable values for the basic indicators Typea Values

c

Indicator

1

Municipal waste (kg per capita per year)

2

Nuclear waste (tons per capita per year)

SB

3

Hazardous waste (tons per capita per year)

SB

SB

υ = NAb τ = NA T = 300 U = 760

υ = NA τ = NA T=0 U = 0.05926 υ = NA τ = NA T=0 U = 0.73864

Comments T is the EU (European Union) target of annual waste generation. U is the maximum over all countries (MAXc).

Main sources of data

T is the minimum over all countries (MINc). U = MAX

http://geodata.grid.unep.ch/ mod_download/download_ xls.php?selectedID=1543

T = MIN, U = MAX.



Fuzzy Assessment Typea Values

c

Indicator

4

Population growth rate (percentage)

5

Pesticide consumption(k g per hectare)

6

Fertilizer consumption (kg per hectare)

SB

7

Desertification of land (percent of dryland area)

SB

8

Forest area (percent of what existed in 2000)

LB

9

Forest change (percentage)

Protected area 10 (ratio to surface area) Glass recycling (percent of 11 apparent consumption)

12 Paper recycling

SB

SB

LB

LB

LB

LB

υ = NA τ = NA T=0 U=3

υ = NA τ = NA T = 3.22 U = 20.4 υ = NA τ = NA T = 221.2 U = 337.75 υ = NA τ = NA T=0 U = 100 υ = 76.767 τ = 100 T = NA U = NA

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Comments Main sources of data Zero or negative values lead to population stabilization. Despite the World Bank (2007) CDexistence of higher ROM rates, 3% is common in many developing nations. T is the average of EU14d; U = MAX. Esty et al. (2005) T is the average of EU14; U = MAX.

T = minimum possible, Food and Agricultural U = maximum possible. Organization

υ = MIN, τ = MAX.

http://mdgs.un.org/unsd/md g/Data.aspx

T = NA U = NA

http://www.earthtrends.wri. org/searchable_db/index.ph υ = MIN (excluding p?step=countries&ccID%5 extremely small values), B%5D=0&allcountries=che ckbox&theme=9&variable_ τ = MAX. ID=299&action=select_yea rs Esty et al. (2005); Environmental Sustainability Index 2001, υ = MIN, τ = MAX. 2002. Available: www.yale.edu/esi

υ = 13 τ = 100

υ = MIN, τ = maximum

υ = −2 τ = 6.9 T = NA U = NA

υ=0 τ = 0.23

T = NA U = NA υ=2 τ = 100 T = NA U = NA

possible.

υ = MIN, τ = maximum possible.

http://geodata.grid.unep.ch/ mod_download/download_ xls.php?selectedID=1552 http://geodata.grid.unep.ch/ mod_download/download_ xls.php?selectedID=1551

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Yannis A. Phillis and Vassilis S. Kouikoglou Table 4.1. (Continued)

c

13

14

15

16

17

18

19

20

21

Typea Values υ = NA Water τ = NA SB withdrawals T = 13 (percent) U = 60 υ = NA BOD τ = NA emissions (kg SB per capita per T = 5.5 day) U = 27.85 υ = NA Phosphorous τ = NA concentration SB (mg per liter of T = 0.18 water) U = 1.63 Metals υ = NA concentration τ = NA SB (microT = 438 Siemens per U = 4,520 centimeter) Public υ=2 wastewater τ = 73 treatment LB plants (percent T = NA of population U = NA connected) υ = NA Threatened τ = NA mammals SB T=0 (percentage) U = 35.5 υ = NA Threatened τ = NA birds SB T=0 (percentage) U = 33.16 υ = NA Threatened τ = NA plants SB T=0 (percentage) U = 8.44 υ = NA Threatened τ = NA fishes SB T=0 (percentage) U = 55 Indicator

Comments


T = average of EU-14, U = average over all countries.

Earthtrends (2002)

T = average of EU-14, U = MAX.

World Bank (2007) CDROM


Esty et al. (2005)


Esty et al. (2005)

υ = MIN, τ = average of EU-14.

T = minimum possible, U = MAX.





UNEP (2005)

Fuzzy Assessment c

Indicator

Threatened 22 amphibians (percentage) Threatened 23 reptiles (percentage) Ozone depleting 24 substances (metric tons per capita) Greenhouse gas emissions 25 (tons of CO2 equivalent per capita) Mortality from respiratory diseases 26 (deaths per 100,000 population) Urban NO2 concentration 27 (μg/m3 of air)

Typea Values υ = NA τ = NA SB T=0 U = 20.72 υ = NA τ = NA SB T=0 U = 20.75

SB

SB

SB

SB

Urban SO2 28 concentration (μg/m3 of air)

SB

Urban TSP 29 concentration (μg/m3 of air)

SB

Renewable resources production 30 (percent of total primary energy supply)

LB

υ = NA τ = NA T=0 U = 1.1301

υ = NA τ = NA T = 0.0079 U = 0.0262

υ = NA τ = NA T = 42.5 U = 132.7

υ = NA τ = NA T = 13 U = 109.16 υ = NA τ = NA T = 1.33 U = 209 υ = NA τ = NA T = 4.5 U = 473

υ = 0.2 τ = 20 T = NA U = NA

Comments

89 Main sources of data




http://mdgs.un.org/unsd/md g/Data.aspx

T is set at the average of EU-14 reduced by 30% http://geodata.grid.unep.ch/ in accordance with the mod_download/download_ EU target of 2020; xls.php?selectedID=1195 U = MAX. http://geodata.grid.unep.ch/ T = minimum of EU-14, mod_download/download_ U = MAX. xls.php?selectedID=1761

T = minimum of EU-14, U = MAX. Esty et al. (2005); Environmental T = minimum of EU-14, Sustainability Index 2001, U = MAX. 2002. Available: www.yale.edu/esi T = minimum of EU-14, U = MAX.

υ = MIN, τ = EU target. IEA (2006)

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c

Indicator

Military spending 31 (percent of GDP) Refugees per 32 capita Poverty (percent of 33 population below national poverty line) Political rights 34 (values in [1, 7]) Civil liberties 35 (values in [1, 7])

36 Gini index

Corruption Perceptions 37 Index (values in [0, 10]) Environmental governance 38 (values in [0, 1]) Tax revenue 39 (percent of GDP)

Typea Values υ = NA τ = NA SB T = 2.05 U = 12.48 υ = NA τ = NA SB T=0 U = 0+

SB

SB

SB

SB

LB

LB

LB

υ = NA τ = NA T=0 U = 28.9

υ = NA τ = NA T=1 U=3 υ = NA τ = NA T=1 U=3

υ = NA τ = NA T = 27 U = 50

υ=3 τ=8 T = NA U = NA

υ = 0.1774 τ = 0.8 T = NA U = NA υ = 10 τ = 22.5 T = NA U = NA

Comments


T = average of EU-14, U = MAX. A zero value is sustainable; all values above zero are unsustainable.


T = minimum possible, U = MAX1e. Most developing countries range over [3, http://www.worldaudit.org/ polrights.htm 7]; 3 being their best value. Most developing countries range over [3, http://www.worldaudit.org/ civillibs.htm 7]; 3 being their best value. T = 27 is the maximum value of Scandinavian countries. Countries with Gini index above 50 exhibit very weakened social cohesion. Values below 3 correspond to extremely corrupt countries. Least corrupt countries range over [8, 10].


http://www.transparency.or g/policy_research/surveys_i ndices/cpi/2007

υ = MIN; countries with excellent environmental Esty et al. (2005) governance have values above 0.8.

υ = MIN1e (25% of countries lie below υ); τ = average of EU-14.


Fuzzy Assessment c

Indicator

GDP implicit 40 deflator (percent) Imports 41 (percent of GDP)

Unemployment (percent of 42 total labor force)

Typea Values υ = NA τ = NA SB T = 2.81 U = 9.34 υ = NA τ = NA SB T = 32.05 U = 52.32

NB

Unemployment 43 equality gap (percent)

SB

Central government 44 debt (percent of GDP)

SB

45

GNI per capita PPP

LB

Exports 46 (percent of GDP)

LB

Foreign direct investment 47 (percent of GDP)

LB

Infant mortality rate 48 (deaths per thousand)

SB

υ = 1.3 τ=4 T=7 U = 11.6

υ = NA τ = NA T=0 U = 3.7 υ = NA τ = NA T = 71 U = 140 υ = 17,710 τ = 26,635 T = NA U = NA υ = 6.83 τ = 43.96 T = NA U = NA υ = −0.8227 τ = 3.38 T = NA U = NA υ = NA τ = NA T = 3.42 U = 78

Comments


T = average of EU-14., U = MAX1.

υ = MIN1, τ = average of EU-14. Both extremely low and extremely high values were excluded and thresholds of least were set at υ = MIN1 and U = MAX1. Targets were set after consultation with experts. T = minimum possible, U = MAX1.

T = average of EU-14., U = MAX1.


υ = minimum of EU-14; τ = average of EU-14.

υ = MIN1, τ = average of EU-14.

υ = MIN, τ = average of EU-14. T = average of http://www.who.int/whosis/ Scandinavian countries, database/core/core_select.cf m U = MAX1

92


Typea Values υ = NA Maternal mortality rate τ = NA SB 49 (deaths per T = 74.5 100,000 births) U = 5,405 c

Indicator

HIV/AIDS 50 prevalence rate (percent)

Tuberculosis prevalence rate 51 (per 100,000 population)

Malaria cases 52 (per thousand people)

Solid fuel household use 53 (percent of total energy)

SB

SB

SB

SB

Life 54 expectancy (years)

LB

Immunization against measles 55 (percent of population)

LB

υ = NA τ = NA T=0 U = 1.8

υ = NA τ = NA T=0 U = 698

υ = NA τ = NA T=0 U = 1.2

υ = NA τ = NA T=0 U = 54.76

υ = 36.48 τ = 77 T = NA U = NA υ = 75 τ = 100 T = NA U = NA

Comments


T = average of Scandinavian countries, U = MAX1. The Joint United Nations Programme on HIV/AIDS provides an http://www.who.int/whosis/ upper bound of 0.9% on database/core/core_select.cf the average HIV m prevalence rate (UNAIDS 2007). The least desirable value U is chosen as twice the upper bound. The target T = 0 is the minimum possible. T = minimum possible, U = MAX.

http://www.who.int/whosis/ database/core/core_select.cf m

T = minimum possible. U = median of data; maximum was not used http://rbm.who.int/wmr200 5/tables/table_a22.pdf because of distortion due to extremely high values. WHO sources: “World Health Statistics 2007”; files: T = minimum possible, “WHO-2008-03-16U = average of data. 152733.csv,” “WHO-2008-03-16152929.csv”

υ = MIN, τ = average of World Bank (2007) CDScandinavian countries. ROM

υ = minimum of EU-14, τ = maximum possible.

Fuzzy Assessment c

Indicator

Immunization against DPT 56 (percent of population) Daily per 57 capita calorie supply Number of doctors (per 58 thousand people) Hospital beds 59 (per thousand people)

60

Public health expenditure

Access to improved 61 water sources (percent of population) Access to improved 62 sanitation (percent of population) Primary education ratio 63 of students to teaching staff

Typea Values υ = 84 τ = 100 LB T = NA U = NA υ = 2,272.8 τ = 3,500 LB T = NA U = NA υ = 0.0113 τ = 2.9663 LB T = NA U = NA υ = 0.1167 τ = 9.0439 LB T = NA U = NA υ = 0.0569 τ = 6.7348 LB T = NA U = NA

LB

LB

SB

Secondary education ratio 64 of students to teaching staff

SB

Tertiary education ratio 65 of students to teaching staff

SB

υ = 36 τ = 100 T = NA U = NA

υ=5 τ = 100 T = NA U = NA

υ = NA τ = NA T = 14 U = 70 υ = NA τ = NA T = 12 U = 47.46 υ = NA τ = NA T = 14.88 U = 45.50

Comments


υ = minimum of EU-14, τ = maximum possible.

υ = MIN, τ = average of http://www.fao.org/es/ess/o EU-14.

s/default.asp

υ = MIN, τ = average of Scandinavian countries. World Bank (2007) CDROM

υ = MIN, τ = average of Scandinavian countries.

υ = MIN, τ = average of Scandinavian countries.

υ = MIN, τ = maximum World Bank (2007) CDpossible.

ROM

υ = MIN, τ = maximum possible.




http://stats.uis.unesco.org/u nesco/ReportFolders/Report Folders.aspx

94


c

66

67

68

69

70

Typea Values υ = 3.21 Male expected τ = 12 LB years of T = NA schooling U = NA υ = 2.11 Female τ = 12 expected years LB T = NA of schooling U = NA Primary net υ = 34.56 school τ = 98.43 LB enrollment T = NA (percent of U = NA children) Secondary net υ = 5.06 school τ = 91.43 LB enrollment T = NA (percent of U = NA children) υ = 12.8 τ = 100 Literacy LB T = NA U = NA Indicator

Knowledge Economy 71 Index (KEI; values in [0, 10])

Public expenditure on 72 R&D (percent of GDP)

LB

LB

υ = 4.07 τ = 8.61 T = NA U = NA

υ=0 τ = 2.14 T = NA U = NA

Comments


υ = MIN, τ = typical in many developed countries.

υ = MIN, τ = typical in many developed countries.

υ = MIN, τ = average of World Bank (2007) CDEU-14.

ROM

υ = MIN, τ = average of EU-14.

υ = MIN, τ = maximum possible. The World Bank classifies countries according to GNI per capita as high, upper middle, lower middle, and low income. The least desirable value http://info.worldbank.org/et υ is chosen as ools/kam2/KAM_page5.asp the maximum KEI among countries with lower middle GNI income. The target τ = 8.61 is the average value of KEI among high income countries.

υ = MIN, T = AVEf.


Fuzzy Assessment Indicator Typea Values Public υ = 0.99 expenditure on τ = 5.52 LB 73 education T = NA (percent of U = NA GDP) υ = 0.6 Personal τ = 407 computers (per LB 74 thousand T = NA people) U = NA υ = 0.06 Internet users τ = 396 75 (per thousand LB T = NA people) U = NA Expenditure on υ = 1.22 information τ = 5.78 and LB 76 communication T = NA (percent of U = NA GDP) c

Comments

95 Main sources of data World Bank (2007) CDROM

υ = MIN, T = AVE.

υ = MIN, T = AVE.

υ = MIN, T = AVE.

υ = MIN, T = AVE.

SB = smaller is better; LB = larger is better; NB = nominal is best. “NA” indicates that the corresponding target or threshold is not applicable. c MAX (MIN) = maximum (minimum) value over all countries. d EU-14: Austria, Belgium, Denmark, Finland, France, Germany, Greece, Ireland, Italy, Netherlands, Portugal, Spain, Sweden, United Kingdom. These countries and Luxemburg are the countries in the European Union before the expansion on 1 May 2004. Luxemburg was not taken into account due to its very small population. e MAX1 (MIN1): Data of all countries are recorded from which extreme values (outliers) are excluded. The threshold of the most undesirable values is the maximum (minimum) of the remaining data. f AVE = average of EU-14, U.S.A., Canada, Australia, Japan, and Norway. a

b

Example 4.1. Table 4.2 shows the time series with annual data zc for two basic indicators for Poland: ozone depleting substances per capita and greenhouse gas emissions per capita. These indicators affect the composite indicator PR(AIR) and are numbered 24 and 25 respectively. The least desirable values for these indicators are the maxima of all countries and the target values are zero for z24 and 0.005 for z25. For each indicator value zc we compute the corresponding normalized value xc from Eq. (4.1). For example, a normalized value for z24 = 0.1296 is calculated as follows: x24 =

U c − z c 1.1301 − 0.1296 1.0005 = = 0.8853. = 1.1301 − 0 1.1301 U c − Tc

96

Yannis A. Phillis and Vassilis S. Kouikoglou Table 4.2. Two basic indicators for Poland: original and normalized time series

c = 24 c = 25 T24 = 0 T25 = 0.0079 U24 = 1.1301 U25 = 0.0262 Original value Normalized value Original value YEAR z24 x24 z25 1990 0.1296 0.8853 0.0148066 1991 0.0670 0.9407 NA 1992 0.0661 0.9415 0.0148066 1993 0.0673 0.9404 NA 1994 0.0435 0.9615 NA 1995 0.0455 0.9597 NA 1996 0.0142 0.9874 NA 1997 0.0080 0.9929 0.0108156 1998 0.0081 0.9928 NA 1999 0.0048 0.9957 NA 2000 0.0046 0.9960 0.0096 2001 0.0047 0.9959 0.0100072 2002 NA NA 0.0096840 2003 NA NA 0.0100178 2004 NA NA NA 2005 NA NA NA “NA” indicates that no data were available for the corresponding year. Indicator target value unsustainable value

Normalized value x25 0.6226 NA 0. 6226 NA NA NA NA 0.8407 NA NA 0.9071 0.8849 0.9025 0.8843 NA NA

4.4. EXPONENTIAL SMOOTHING Basic indicators describe certain aspects of the sustainability of a country in a specific year. Existing assessment methods use the most recent indicator measurements to assess sustainability. An important problem to be addressed concerns the cumulative effects of past environmental or societal pressures, which continue to be effective in the future. The availability and accuracy of data are often also problematic. The values for the basic indicators in a given year occasionally are missing or fraught with uncertainty due to measurement errors. Therefore, a systematic method is needed to improve the quality of information. To deal with such problems we use weighted sums of present and past indicator data as inputs to the model. Let xc,1, x c,2, …, x c,n be the available normalized values of indicator c in years t1, t2, …, tn, where tk < tk+1. These years need not be consecutive. Then an aggregate value xc(n) for the indicator can be computed using the weighted sum xc(n) = wnxc,n + wn−1xc,n−1 + … + w1x c,1 for some positive weights such that wn + wn−1 + ... + w1 = 1.

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97

It is reasonable to assume that, although past observations play a part in xc(n), we ought to give greater weights to those that are more recent. A simple choice then is to let the weights decrease geometrically with a power equal to the difference from the most recent year tn. This yields a single exponential smoothing model for time series (see, e.g., Hamilton 1994, pp. 440–443), in which the smoothed values are given by xc(k) =

xc ,k + xc ,k −1 β tk −tk −1 + … + xc ,1 β tk −t1 1 + β tk −tk −1 + … + β tk −t1

(4.2)

for k = 1, 2, …, n, where β is a number between zero and one. In the standard exponential smoothing model, the corresponding smoothed values are xc(k) = (1 − β )xc,k + β xc(k − 1) with xc(0) = 0. When applied recursively, this yields xc(k) = (1 − β )xc,k + β (1 − β )xc,k−1 + … + β k−1(1 − β )xc,1. This model differs from Eq. (4.2) in that (a) it assumes a complete data set, i.e., tk = k and (b) its weights do not sum up to one. However, when n→∞ the two models yield the same estimates. For each country and indictor c, we compute the smoothing parameter β which minimizes the sum of squared errors (SSE) where the error ek = xc,k − xc(k − 1) is the difference between the observation at time tk and the estimate xc(k − 1) from past data. We set xc(0) = 0 and xc(1) = xc,1, so e1 = xc,1 − 0 and e2 = xc,2 − xc,1 for all β. We also define SSE = e12 + e22 + e32 + … + en2. The optimal value of β is the one that minimizes SSE. To compute SSE efficiently we observe from (4.2) that xc(k) =

N (k ) D( k )

(4.3)

where N(k) = xc,k + N(k − 1)β tk−tk−1 and

(4.4)

98

Yannis A. Phillis and Vassilis S. Kouikoglou D(k) = 1 + D(k − 1)β tk−tk−1

(4.5)

with N(1) = xc,1 and D(1) = 1. Example 4.2. Table 4.3 shows the results of Eqs. (4.3)–(4.5) with β = 0.29, using the normalized values xk of ozone depleting substances per capita (indicator 25) for Poland. Table 4.3. Exponential smoothing of indicator 25 for Poland using β = 0.29 k

tk

1 2 3 4 5 6 7

1990 1992 1997 2000 2001 2002 2003

data x25,k 0.6226 0. 6226 0.8407 0.9071 0.8849 0.9025 0.8843

t −t tk − tk−1 0.29 k k−1 N(k)

D(k)

x25(k)

2 5 3 1 1 1

1.0000 1.0841 1.0022 1.0244 1.2971 1.3762 1.3991

0.6226 0.6226 0.8402 0.9055 0.8896 0.8990 0.8885

0.0841 0.00205 0.02439 0.29 0.29 0.29

0.6226 0.6750 0.8421 0.9276 1.1539 1.2371 1.2431

error ek = x25,k − x25(k − 1) 0.6226 0 0.2181 0.0669 −0.0206 0.0129 −0.0147

First we set x25(0) = 0, N(1) = x25,1 = 0.6226, and D(1) = 1. Using Eq. (4.3) with k = 1 we get x25(1) = N(1)/D(1) = 0.6226. The second available normalized value is x25,2 = 0.6226 (same as x25,1) and corresponds to the year t2 = 1992. We have t2 − t1 = 2, βt2−t1 = 0.292 = 0.0841. We compute N(2) from Eq. (4.4) and D(2) from Eq. (4.5) as follows N(2) = x25,2 + βt2−t1N(1) = 0. 6226 + 0.0841 × 0. 6226 = 0. 6226 D(2) = 1 + βt2−t1D(1) = 1 + 0.0841 × 1 = 1.0841. Thus, the estimate at k = 2 is x25(2) = N(2)/D(2) = 0.6226 (same as x25(1)). In the same manner we calculate the estimates of x25 in subsequent years. Finally, we set x25 equal to the most recent estimate, x25(7) = 0.8885, and use this as input to the SAFE model. The last column of Table 4.3 shows the errors in predicting xc,k using the most recent smoothed estimates xc(k − 1). The optimal value of β and the corresponding value xc of indicator c are computed as follows: 1) Initialization: Let δ be a small step size for β (for example, 0.1 or 0.01). Set β = 0. For this value, Eq. (4.2) yields xc(k) = xc,k for k = 1, …, n. Initialize the optimal

Fuzzy Assessment

99

estimates: βc = 0, SSEc = (xc,1 − 0)2 + (xc,2 − xc,1)2… + (xc,n − xc,n−1)2, and the estimate is xc = xc,n. 2) Main loop: Set β := β + δ. a) If β > 1, then stop: the optimal smoothing parameter is βc and the normalized indicator value is xc; b) otherwise, set SSE = 0; initialize the numerator and denominator of Eq. (4.3) for k = 0; thus, N = 0, D = 0, estimate = 0; and go to step 3. 3) Computation of SSE: For k = 1, …, n a) update SSE := SSE + (xc,k − estimate)2; b) update D := 1 + Dβ tk−tk−1, N := xc,k + Nβ tk−tk−1, and estimate = N/D. 4) Comparison: If SSE < SSEc, then update the optimal estimates setting xc = estimate, βc = β, and SSEc = SSE. Go to step 2. The above algorithm requires less than a second on a personal computer to compare 100 test values for β with step δ = 0.01 for a total of 128 countries and 76 basic indicators per country with at most 16 data values from 1990 to 2005.

4.5. FUZZIFICATION A fuzzy assessment of sustainability involves fuzzy inputs and fuzzy outputs. All sustainability indicators, basic and composite, are normalized. Therefore we must define appropriate fuzzy partitions in [0, 1]. There are many ways to define fuzzy partitions. One such possibility is described below. 1

W

M

S

VB

0

0.6

1

0

B

A

G

VG

EL VL

L

FL

I

FH

H VH EH

0 1

1

0

Basic indicator

Composite indicator

Overall sustainability

(a)

(b)

(c)

Figure 4.5. Membership functions used in the SAFE model.

The normalized basic indicators are fuzzified using three fuzzy sets with linguistic values “weak” (W), “medium” (M), and “strong” (S), whose membership functions are shown in Figure 4.5a. We assign the linguistic value W to low or average values of normalized indicators. Hence, the fuzzification is somewhat pessimistic. This is in agreement with widely accepted assessment practices. For example, according to OECD (2004), water stress (PRESSURE indicator) is considered to be rather high when the intensity of use of freshwater resources per capita is greater than 40% of the total renewable resources per capita. Since the interval [40%, 100%] has a length 0.6 relative to the length of [0%, 100%] and W = “weak water sustainability” in the SAFE model means “high water stress,” the fuzzy set W is defined in [0, 0.6], as shown in Figure 4.5a.

100


Example 4.3. The minimum SSE estimates of indicators 24 and 25 for Poland are x24 = 0.9959 and x25 = 0.8885. The corresponding membership grades to the fuzzy sets W, M, and S are

μW(24) = 0 μM(24) =

μS(24) =

0.9959 − 1 0.0041 = = 0.01025 0.6 − 1 0 .4

0.9959 − 0.6 0.3959 = = 0.98975 1 − 0.6 0.4

μW(25) = 0 μM(25) =

μS(25) =

0.8885 − 1 0.1115 = = 0.27875 0 .6 − 1 0.4

0.8885 − 0.6 0.2885 = = 0.72125. 1 − 0.6 0.4

As we shall see below and in the next section, when we combine two or more fuzzy inputs into a composite indicator we must use more fuzzy sets to represent the composite fuzzy variable. Thus, for composite indicators we use five fuzzy sets with linguistic values “very bad” (VB), “bad” (B), “average” (A), “good” (G), and “very good” (VG), as shown on Figure 4.5b. An even larger number of fuzzy sets must be used to represent the overall sustainability. Indeed, the two primary components ECOS and HUMS, each with five linguistic values, generate 52 = 25 different combinations. The number of linguistic values for OSUS is determined as follows: First, we assign positive weights a and b representing the relative importance respectively of ECOS and HUMS in the calculation of OSUS. We also assign the integer values 0, …, 4 to the five linguistic values, such that 0 corresponds to VB, 1 corresponds to B, and so on. To each pair of weights (ECOS, HUMS) we assign an index OSUS for the linguistic value of the overall sustainability, where OSUS = aECOS + bHUMS.

(4.6)

The minimum value of OSUS is 0a + 0b = 0 and the maximum value is 4a + 4b. We choose a = b = 1 to strike an equal balance between the environmental and the human dimensions of sustainability. Therefore OSUS is an integer between 0 and 8. Hence the overall sustainability must have at least nine fuzzy sets in order to aggregate ECOS and OSUS more precisely. These fuzzy sets are: “extremely low” (EL = 0), “very low” (VL = 1), “low” (L = 2), “fairly

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101

low” (FL = 3), “intermediate” (I = 4), “fairly high” (FH = 5), “high” (H = 6), “very high” (VH = 7), and “extremely high” (EH = 8) (see Figure 4.5c).

4.6. RULE BASES OSUS With a = b = 1, Eq. (4.6) generates the rule base shown in Table 4.4. For example, the integer values of ECOS and HUMS in rule R8 are B = 1 and A = 2. By Eq. (4.6), the integer value of OSUS is 1 + 2 = 3, which corresponds to the linguistic value FL. Table 4.4. Rules for OSUS Rule Rp

if ECOS is

and HUMS is

then OSUS is

R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13

VB VB VB VB VB B B B B B A A A

VB B A G VG VB B A G VG VB B A

EL VL L FL I VL L FL I FH L FL I

Rule Rp

if ECOS is

and HUMS is

then OSUS is

R14 R15 R16 R17 R18 R19 R20 R21 R22 R23 R24 R25

A A G G G G G VG VG VG VG VG

G VG VB B A G VG VB B A G VG

FH H FL I FH H VH I FH H VH EH

HUMS and ECOS The second order inference engines for ECOS and HUMS use a rule base with 54 = 625 rules because they have four inputs, each with five linguistic values. By analogy to Eq. (4.6), using equal weights for the four inputs of ECOS, we have ECOS = LAND + WATER + BIOD + AIR.

(4.7)

The minimum index for ECOS is 0 and the maximum is 4 × 4 = 16. Therefore, 17 fuzzy sets should be used to describe ECOS precisely. Had we used this number of fuzzy sets, the

102


number of fuzzy sets for OSUS would have been 33 (minimum value = 0, maximum value = 32). To avoid an explosion in the number of linguistic values, we use five linguistic values for all composite indicators. Each linguistic value of ECOS corresponds to a range of values, rather than a single one taken from the following sum SUM = LAND + WATER + BIOD + AIR. For each combination of inputs, we calculate SUM and determine the linguistic value of ECOS:

⎧VB, ⎪B, ⎪⎪ ECOS = ⎨A, ⎪G, ⎪ ⎪⎩VG,

0 ≤ SUM ≤ 3 4 ≤ SUM ≤ 7 8 ≤ SUM ≤ 11 12 ≤ SUM ≤ 15 SUM = 16.

The fuzzy sets of HUMS are determined similarly using the equation SUM = POLICY + WEALTH + HEALTH + KNOW.

Secondary Components The fuzzy sets for the secondary components of sustainability LAND, WATER, BIOD, AIR, POLICY, etc. are determined by summing the integer values of appropriate Pressure (PR), State (ST), and Response (RE) indicators. We have SUM = PR + ST + RE and

⎧VB, 0 ≤ SUM ≤ 1 ⎪B, 2 ≤ SUM ≤ 4 ⎪⎪ secondary component = ⎨A, 5 ≤ SUM ≤ 7 ⎪G, 8 ≤ SUM ≤ 10 ⎪ ⎪⎩VG, 11 ≤ SUM ≤ 12.

Tertiary Components The tertiary components, PR, ST, and RE, are functions of one or more basic indicators. RE(WATER) depends on a single basic indicator 17 and RE(AIR) depends solely on indicator 30, as discussed in Section 4.2.1. The membership grades of these components to the five linguistic values VB, B, …, and VG are computed by fuzzifying the corresponding basic indicator. Although, by convention, the universe of discourse [0, 1] of the basic

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103

indicators is partitioned into 3 fuzzy sets, W, M, and S, we use five fuzzy sets whenever a basic indicator serves as a tertiary component. For tertiary components with more basic inputs, say, 1, 2, …, n, the rule base is generated by assigning a linguistic value to SUM = L1 + L2 + … + Ln

(4.8)

where Li ∈ {0 = W, 1 = M, 2 = S} is the integer corresponding to the linguistic value of basic indicator i. When n = 2, we have SUM = L1 + L2 with a maximum value 2 + 2 = 4 and a minimum value zero. Hence, in this case, exactly five linguistic values are needed, as shown in Table 4.5. Table 4.5. Rules for tertiary components with n = 2

Rule Rp

if input 1 is

and input 2 is

then PR, ST, RE is

R1 R2 R3 R4 R5 R6 R7 R8 R9

W W W M M M S S S

W M S W M S W M S

VB B A B A G A G VG

Example 4.4. PR(AIR) depends on the basic indicators labeled 24 and 25. In Example 4.3 we found that, for Poland, indicator 24 belongs to the fuzzy set M with membership grade 0.01025 and to S with grade 0.98975, and indicator 25 is M with grade 0.58275 and S with grade 0.41725. There are 2 × 2 = 4 firing rules. Application of TSK inference and Eq. (2.20) yields (membership grades are shown in parentheses): R5:

if 24 is M(0.01025) and 25 is M(0.58275), then PR(AIR) is A(0.01025 × 0.58275 = 0.0059)

R6:

if 24 is M(0.01025) and 25 is S(0.41725), then PR(AIR) is G(0.01025 × 0.41725 = 0.0043)

R8:

if 24 is S(0.98975) and 25 is M(0.58275), then PR(AIR) is G(0.98975 × 0.58275 = 0.5768)

R9:

if 24 is S(0.98975) and 25 is S(0.41725),

104

Yannis A. Phillis and Vassilis S. Kouikoglou then PR(AIR) is VG(0.98975 × 0.41725 = 0.4130).

The overall membership grade of the PR(AIR) to the fuzzy set G is the sum of the firing strengths of R6 and R8 (see Eq. (4.21)):

μG[PR(AIR)] = 0.0043 + 0.5768 = 0.5811. The other membership grades are: μVB[PR(AIR)] = μB[PR(AIR)] = 0, μA[PR(AIR)] = 0.0059, and μVG[PR(AIR)] = 0.4130. For n > 2, we need more than five linguistic values to represent the tertiary variables. Alternatively, we can use only five linguistic values and specify a range of values for SUM for each value. In general, we use more pessimistic rules than those generated by Eq. (4.8) for composite indicators when these indicators are very important or whenever some basic inputs to them are missing. For example, PR(WATER) is a key indicator for sustainable development and has a distinct rule base in the SAFE model. The rule consequences of the tertiary components for various values of n > 2 are shown below. For n = 3 we have

⎧VB, 0 ≤ SUM ≤ 1 ⎪B, SUM = 2 ⎪⎪ tertiary component = ⎨A, SUM = 3 ⎪G, SUM = 4 ⎪ ⎪⎩VG, 5 ≤ SUM ≤ 6. In particular, water stress, which is described by PR(WATER), is a critical component of sustainable development with three basic inputs. For this component we use a more pessimistic rule base, which is derived from

⎧VB, 0 ≤ SUM ≤ 2 ⎪B, SUM = 3 ⎪⎪ PR(WATER) = ⎨A, SUM = 4 ⎪G, SUM = 5 ⎪ ⎪⎩VG, SUM = 6. Observe that PR(WATER) is VG only if SUM = 6, that is, all basic indicators of PR(WATER) are sustainable. The rule base for n = 4 is derived from

Fuzzy Assessment

⎧VB, ⎪ B, ⎪⎪ tertiary component = ⎨A, ⎪G, ⎪ ⎪⎩VG,

105

0 ≤ SUM ≤ 2 SUM = 3 SUM = 4

(4.9)

SUM = 5 6 ≤ SUM ≤ 8.

For ST(POLICY) we use four inputs but a more pessimistic rule base:

⎧VB, ⎪ B, ⎪⎪ ST(POLICY) = ⎨A, ⎪G, ⎪ ⎪⎩VG,

0 ≤ SUM ≤ 3 4 ≤ SUM ≤ 5 SUM = 6

(4.10)

SUM = 7 SUM = 8.

For n = 5 we have

⎧VB, ⎪ B, ⎪⎪ tertiary component = ⎨ A, ⎪G, ⎪ ⎪⎩ VG,

0 ≤ SUM ≤ 3 SUM = 4 SUM = 5 SUM = 6 7 ≤ SUM ≤ 10.

Finally, there are four tertiary components with n = 6. The corresponding rule bases are derived from the following:

⎧VB, ⎪B, ⎪⎪ ST(KNOW) = ⎨A, ⎪G, ⎪ ⎪⎩VG, ⎧VB, ⎪B, ⎪⎪ PR(HEALTH) = ⎨A, ⎪G, ⎪ ⎪⎩VG, and

0 ≤ SUM ≤ 4 SUM = 5 SUM = 6 SUM = 7 8 ≤ SUM ≤ 12, 0 ≤ SUM ≤ 5 SUM = 6 SUM = 7 8 ≤ SUM ≤ 10 11 ≤ SUM ≤ 12,

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⎧VB, 0 ≤ SUM ≤ 7 ⎪ B, SUM = 8 ⎪⎪ PR(LAND), PR(BIOD) = ⎨A, SUM = 9 ⎪G, SUM = 10 ⎪ ⎪⎩VG, 11 ≤ SUM ≤ 12. HEALTH, LAND, and BIOD are critical factors of sustainable development and so we use more pessimistic rules than KNOW.

Availability of Data For almost every country in the world there is at least one basic indicator for which no data are available. In this case, one or more tertiary indicators have fewer inputs than those specified in the corresponding rule base. Two approaches to resolve this problem are discussed below: 1) One approach is using a rule base with fewer inputs than those of the original rule base. For example, if a tertiary component has four basic inputs, of which two are not available for some country, then we can use the rule base of the previous section with n = 2 instead of n = 4. This approach seems simple but its computer implementation has a few technical difficulties. Also it requires introducing new rule bases for those tertiary components whose original rule bases are pessimistic, as well as for secondary components for which one tertiary input depends on a sole basic indicator which is missing. 2) An alternative approach is to use linguistic averages of all existing inputs in place of the missing ones. More specifically, each partial combination of linguistic values of the available inputs is completed by assigning to each missing input a linguistic value which is the “average” of linguistic values of the available inputs. Example 4.5. Consider a rule base with four basic inputs, 1, …, 4, of which only 1 and 2 are available. Let μW(i), μM(i), and μS(i) be the membership grades of i = 1, 2 to the fuzzy sets W = 0, M = 1, and S = 2. Take, for example, the combination (W, M, x, x). Replace x with the average of W and M, (0 + 1)/2 = 0.5, rounded off at 1 = M or 0 = W. For this partial combination, the unavailable inputs 3 and 4 are both set to either M or W with membership grade 1. SAFE uses integer parts so x = int(0.5) = 0 = W. Therefore, the partial combination (W, M, x, x) becomes (W, M, W, W), for which SUM = 0 + 1 + 0 + 0 = 1. According to Eq. (4.9) or (4.10) the output is VB with degree of membership

μVB = μW(1) × μM(2) × 1 × 1. In the same way, we can compute the linguistic value and the corresponding degree of membership of the output, for each combination of linguistic values of the available inputs.

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4.7. SENSITIVITY ANALYSIS Sensitivity analysis plays a fundamental role in decision making because it determines the effects of a change in a decision parameter on system performance. In this section, we attempt to provide an answer to the question of how to design policies for sustainable development. The SAFE model could aid decision makers to formulate sustainable policies by assessing sustainability for different scenarios of development. A scenario is defined by the available sustainability indicators, which largely reflect the results of policies and actions taken in a particular period. When these values are changed and the resulting changes on sustainability observed, we could identify the most important indicators contributing to sustainable development. This procedure is known as sensitivity analysis. Sensitivity analysis entails the computation of the gradients of ECOS, HUMS, and OSUS with respect to each basic indicator. A gradient gives the increase of sustainability per unit increase of some basic indicator. To perform sensitivity analysis we follow the steps: 1) Calculation of OSUS: a) For a given country, normalize and smooth all indicator data using the methods described in Sections 4.3 and 4.4. b) Fuzzify the basic inputs. c) Compute the membership grades of composite indicators to the fuzzy sets VB, B, A, G, and VG. Start from the TSK inference engines that use only basic indicators as inputs and proceed successively to the ones that use composite indicators. Finally, compute the membership grades of OSUS to the nine fuzzy sets EL, VL, …, EH and compute a crisp value for OSUS by height defuzzification. 2) Introduction of perturbation: For some basic indicator, say, c increase its normalized value xc ∈ [0, 1] by some fixed amount δ, for example, 0.1 or 10%. If the result is greater than one, then truncate it to one to avoid overshooting permissible regions of indicators. 3) Sensitivity analysis: Assess the overall sustainability using the same set of data as in step 1 except for indicator c whose value is now xc + δ. Denote the new assessment by OSUS(xc + δ ). The gradient of OSUS with respect to xc is defined by the forward difference

Δc = OSUS(xc + δ ) − OSUS.

(4.11)

Reset the basic indicator c to its original value xc. 4) Loop: Repeat steps 2 and 3 for all basic indicators. 5) Ranking: Identify the gradients with the largest values, which correspond to the basic indicators that affect OSUS the most.

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By changing several indicators simultaneously in step 3 we can compute gradients of higher orders and formulate more comprehensive environmental policies. For example, the second-order gradient of OSUS with respect to indicators c and c' is

Δc,c' = OSUS(xc + δ, xc' + δ ) − OSUS. Sensitivity analysis is biased towards indicators which belong to small groups. For example, RE(AIR) depends only on renewable resources production. Therefore, an increase in the latter directly affects the former. ST(AIR), on the other hand, depends on four basic indicators, labeled 26–29. An improvement of one of these indicators will result in a small improvement of ST(AIR). To avoid this bias, a basic indicator c is ranked according to the product Dc = (1 − xc)Δc,

(4.12)

where 1 − xc is the distance of indicator c from the sustainable value, and Δc is the gradient of OSUS with respect to xc. Thus those indicators that affect OSUS the most and are farther in the unsustainable region are pinpointed and ranked accordingly.

4.8. RESULTS 4.8.1. Sustainability of Countries The rankings of 128 countries from most sustainable to less sustainable according to SAFE are shown in Table 4.6. An inclusion of all 192 member states of the United Nations in the study was not possible due to unavailability of data. Many countries have not developed suitable data collection mechanisms. Countries with small area and population as well as countries suffering from natural disasters or war over the last decade are typical cases. Also it is impossible to obtain data for countries that became independent recently while previously they belonged to larger states. Data are available for these larger states, e.g., the Soviet Union, but not for the newly formed republics. Table 4.6. SAFE ranking: Countries 1–20, year 2005 COUNTRY 1. Sweden 2. Finland 3. Denmark 4. Norway 5. Austria 6. Switzerland 7. Germany 8. Estonia 9. Latvia 10. Lithuania

OSUS 0.8437 0.8300 0.8136 0.8096 0.7744 0.7558 0.7504 0.7427 0.7416 0.7408

ECOS 0.7500 0.7497 0.7448 0.7476 0.7699 0.7611 0.7484 0.7472 0.7500 0.7497

HUMS 0.9374 0.9102 0.8825 0.8715 0.7789 0.7505 0.7523 0.7383 0.7332 0.7319

COUNTRY 11. Czech Republic 12. Canada 13. Portugal 14. France 15. Slovakia 16. Italy 17. Netherlands 18. Belgium 19. Hungary 20. Poland

OSUS 0.7352 0.7203 0.7128 0.7124 0.7081 0.6850 0.6820 0.6746 0.6473 0.6380

ECOS 0.7215 0.6906 0.6768 0.6443 0.7442 0.6205 0.5120 0.5361 0.5448 0.7476

HUMS 0.7490 0.7500 0.7487 0.7805 0.6721 0.7496 0.8519 0.8130 0.7497 0.5284

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Table 4.6 (continued): Countries 21–108, year 2005 COUNTRY

OSUS ECOS HUMS

COUNTRY

OSUS ECOS HUMS

21. USA 22. Slovenia 23. Australia 24. New Zealand 25. UK 26. Ireland 27. Spain 28. Greece 29. Japan 30. Israel 31. Panama 32. South Korea 33. Uruguay 34. Croatia 35. Malaysia 36. Georgia 37. Chile 38. Romania 39. Thailand 40. Albania 41. Bulgaria 42. Venezuela 43. Bolivia 44. Ukraine 45. Argentina 46. Tunisia 47. Moldova 48. Brazil 49. El Salvador 50. Nicaragua 51. China 52. Russia 53. Mongolia 54. Kyrgyzstan 55. Armenia 56. Kazakhstan 57. Mexico 58. Lebanon 59. Belarus 60. Peru 61. Ecuador 62. Gabon 63. Azerbaijan 64. Paraguay

0.6352 0.6287 0.6285 0.6271 0.6262 0.6246 0.6226 0.6219 0.6205 0.6194 0.6122 0.6120 0.6004 0.5918 0.5862 0.5736 0.5719 0.5657 0.5656 0.5654 0.5572 0.5458 0.5266 0.5199 0.5198 0.5156 0.5117 0.5117 0.5110 0.5068 0.5055 0.5052 0.5021 0.5020 0.5016 0.5004 0.4990 0.4989 0.4972 0.4966 0.4958 0.4956 0.4942 0.4941

65. Jordan 66. Colombia 67. Former Yug. Rep. of Mac. 68. South Africa 69. Morocco 70. Dem. Rep. Congo 71. Uzbekistan 72. Vietnam 73. Botswana 74. Syria 75. Angola 76. Algeria 77. Zimbabwe 78. Kuwait 79. Saudi Arabia 80. Sri Lanka 81. Tanzania 82. Turkey 83. Senegal 84. Benin 85. Congo 86. Honduras 87. Tajikistan 88. Ghana 89. Côte d’Ivoire 90. Namibia 91. Malawi 92. Philippines 93. United Arab Emirates 94. Zambia 95. Papua N. Guinea 96. Indonesia 97. Oman 98. Bangladesh 99. Kenya 100. Egypt 101. Guatemala 102. Laos 103. Uganda 104. Pakistan 105. Mozambique 106. Nepal 107. Burkina Faso 108. Madagascar

0.4902 0.4899 0.4856 0.4760 0.4754 0.4717 0.4701 0.4684 0.4635 0.4604 0.4583 0.4552 0.4474 0.4287 0.4273 0.4262 0.4237 0.4216 0.4202 0.4181 0.4178 0.4124 0.4108 0.4102 0.4038 0.4025 0.4006 0.3974 0.3863 0.3778 0.3775 0.3764 0.3690 0.3643 0.3641 0.3617 0.3615 0.3498 0.3497 0.3487 0.3457 0.3393 0.3382 0.3312

0.5204 0.5074 0.5053 0.5042 0.5025 0.4989 0.5029 0.6596 0.5099 0.5016 0.7365 0.5005 0.6944 0.6489 0.5000 0.7428 0.6356 0.6363 0.6433 0.7498 0.5000 0.7355 0.7450 0.5390 0.5392 0.5125 0.5370 0.5305 0.6703 0.7500 0.5139 0.5105 0.4986 0.5237 0.5249 0.5019 0.5025 0.4997 0.5000 0.6533 0.6960 0.7416 0.5152 0.6475

0.7500 0.7500 0.7517 0.7500 0.7500 0.7502 0.7424 0.5841 0.7311 0.7372 0.4879 0.7235 0.5063 0.5347 0.6723 0.4044 0.5081 0.4950 0.4880 0.3811 0.6145 0.3561 0.3082 0.5007 0.5004 0.5187 0.4864 0.4929 0.3518 0.2636 0.4970 0.5000 0.5056 0.4802 0.4784 0.4989 0.4954 0.4982 0.4943 0.3398 0.2956 0.2495 0.4733 0.3406

0.4770 0.5839 0.5152 0.4611 0.4999 0.6937 0.4998 0.5000 0.7007 0.5000 0.6691 0.4979 0.6942 0.3866 0.4189 0.5000 0.6112 0.5093 0.6871 0.6633 0.7352 0.5603 0.5610 0.5964 0.6812 0.5537 0.6523 0.5000 0.2500 0.7367 0.5085 0.5002 0.3454 0.5008 0.6074 0.2711 0.5336 0.5005 0.4994 0.4988 0.6474 0.5499 0.6024 0.4989

0.5034 0.3959 0.4561 0.4909 0.4508 0.2497 0.4404 0.4368 0.2264 0.4208 0.2475 0.4125 0.2007 0.4707 0.4358 0.3525 0.2362 0.3339 0.1534 0.1728 0.1004 0.2645 0.2607 0.2241 0.1264 0.2512 0.1490 0.2948 0.5226 0.0190 0.2465 0.2527 0.3927 0.2278 0.1209 0.4524 0.1894 0.1991 0.2001 0.1986 0.0439 0.1288 0.0741 0.1636

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Yannis A. Phillis and Vassilis S. Kouikoglou Table 4.6 (continued): Countries 109–128, year 2005

COUNTRY

OSUS ECOS HUMS

COUNTRY

OSUS ECOS HUMS

109. India 110. Cambodia 111. Cameroon 112. Nigeria 113. Rwanda 114. Central African Rep. 115. Iran 116. Gambia 117. Togo 118. Guinea Bissau

0.3287 0.3200 0.3190 0.3109 0.3012 0.3003 0.2963 0.2958 0.2865 0.2781

119. Yemen 120. Niger 121. Sudan 122. Guinea 123. Chad 124. Mali 125. Sierra Leone 126. Ethiopia 127. Burundi 128. Mauritania

0.2781 0.2736 0.2664 0.2654 0.2611 0.2578 0.2509 0.2506 0.2495 0.2189

0.5000 0.5000 0.5389 0.5116 0.5677 0.5319 0.2791 0.5000 0.5141 0.5000

0.1574 0.1399 0.0992 0.1102 0.0347 0.0687 0.3134 0.0915 0.0588 0.0562

0.3428 0.4949 0.5000 0.4999 0.4998 0.4902 0.5000 0.5011 0.4985 0.4023

0.2134 0.0523 0.0327 0.0309 0.0223 0.0255 0.0019 0.0000 0.0005 0.0355

4.8.2. Ranking of Indicators Table 4.7 shows the two most important indicators for selected countries, obtained from Eq. (4.11) by means of a first-order sensitivity analysis of OSUS. Table 4.7. Most important indicators for selected countries, year 2005 (first-order sensitivity analysis) COUNTRY: Indicators CANADA: Forest change, Protected area. USA: Renewable energy production, Greenhouse gas emissions. IRELAND: Gini Index, Fertilizer consumption, Renewable energy production. SPAIN: Public wastewater treatment plants, Water withdrawals

COUNTRY: Indicators GREECE: Renewable energy production, Environmental laws, GNI per capita. ROMANIA: Renewable energy production, Public wastewater treatment, Protected area. BULGARIA: Imports, Unemployment, GNI per capita. VENEZUELA: Poverty rate, Unemployment, GNI per capita, GDP implicit deflator

COUNTRY: Indicators ECUADOR: Renewable energy production, Forest change. BOTSWANA: Desertification of land, Forest change, Poverty rate. CAMBODIA: Environmental laws and enforcement, Poverty rate, Refugees per capita.

The critical factors for developed countries are ecological (renewable energy production, greenhouse gas emissions, forest change, protected areas). For developing countries the most important factors are ecological as well as human (economical, political).

4.9. PROBLEMS 4.1. The value of the sustainability indicator GDP for a given country is z = 17 (thousand U.S. $). You are given the normalizing curve of figure (a) below. Normalize and then fuzzify this value using the membership functions of figure (b).

Fuzzy Assessment

1

1

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VB

B

A

G

VG

μ(x)

normalized value, x 0

0 10

20

z

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

normalized value, x

(a)

(b)

Figure 4.6. Normalizing curve and membership functions for Problem 4.1.

4.2. Carbon emissions in the U.S. are 5 tons per capita per year which is also the maximum value among many countries. If a target value for sustainable development is at most Tc = 2.5 tons per capita per year, what are the normalized and fuzzified values of zc = 2 tons/capita-year and zc = 3 tons/capita-year? Use the membership functions for normalized basic indicators of Figure 4.5a. 4.3. Consider the rule base below Table 4.8. Rule base for Problem 4.3 Desertification of Land W W W M M M S S S

Current Forest W M S W M S W M S

ST(LAND) W M M W M S M M S

for W = weak, M = medium, S = strong, and the following premises: “Desertification of Land is W(0.5)” “Current Forest is M(0.1) and S(0.15).” Derive the pertinent “if-then” rules and compute the crisp value of ST(LAND) if the fuzzy set M has a peak yM = 0.7. 4.4. Consider the following rules from a rule base that computes OSUS

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Yannis A. Phillis and Vassilis S. Kouikoglou Table 4.9. Rule base for Problem 4.4 HUMS G G

ECOS A G

OSUS A G

and the following premises: “ECOS is G(0.5) and A(0.1)” “HUMS is G(1).” Compute the firing strengths of the pertinent “if-then” rules and the final crisp value for OSUS, using TSK inference and the membership functions of Problem 4.1. 4.5. Let ECOS = 0.8 and HUMS = 0.2. Fuzzify these values using the membership functions of Figure 4.5b. Then use the rules that fire from the following rule base to compute OSUS as well as its crisp value. Table 4.10. Rule base for Problem 4.5 HUMS VB VB VB VB VB B B B B B

ECOS VB B A G VG VB B A G VG

OSUS VB VB B A G VB B B A G

Chapter 5

SUSTAINABILITY OF ORGANIZATIONS 5.1. INTRODUCTION The previous chapters of this book discussed the assessment of the sustainability of countries or geographic regions. We come now to another area where the same sustainability concepts can be applied: that of organizations. An organization is an entity or purposeful structure with boundaries that separate it from its environment. According to Webster's dictionary it is an administrative and functional structure (as a business or a political party). An organization pursues certain goals such as education (this is a university) or production of goods and services if it is a corporation. Organizations interact with their environment, physical, biological and social, they affect it and become affected by it. It is quite natural then that organizations play an important role in the sustainability of a region or country. A company might use production technologies that are environmentally friendly and continuously strive toward improving them or it might disregard such technologies to enhance short term profits. A company might resort to friendly or hostile employment practices. To be sustainable an organization should employ practices that improve the welfare of society and have a minimal environmental impact. One has then to assess societal and environmental impact of organizations quantitatively. Such assessment presupposes that organizations accept the principles of accountability and transparency. Equivalently, organizations must understand their public role and impact and voluntarily provide reliable data about their economic, social, and environmental performance. Sustainability of organizations should not be confused with self-sufficiency. Factories ordinarily need a number of external inputs to function, such as energy, matter, and labor, for example, and they transform matter into finished products while at the same time they generate pollution which is released into the environment. Sustainability of organizations is associated with their activities, emissions, impact of products, installations, policies, etc. It is desirable to improve all activities, that is, reduce emissions, improve products, build environmentally friendly installations, contribute to the economic welfare of the society, and so on. Questions about space and time over which sustainability is examined can be asked as we have already seen in previous chapters. The boundaries of an organization are defined by its physical ones but they are not necessarily limited by them. A car company for example, generates pollution locally but its supply chain and products have global environmental

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impacts and national or international economic effects. Time scales also depend on the type of organization and its products or services. Each organization has its own space and time demands when sustainability is assessed. There are several reasons why organizations assess or should assess their sustainability: 1. It is good marketing to provide society with sustainability data. People often prefer products of environmentally and socially friendly companies. 2. Sustainability is becoming the goal of local and national governments and organizations cannot ignore this trend. 3. Improving environmental performance often improves financial and economic performance of an organization by making the various stages of production more efficient. 4. A sustainability report exposes progress towards improvement and provides guidance towards new strategic decisions. 5. A sustainability report shows compliance with existing environmental and labor laws and regulations. Models that assess sustainability of organizations should be customized to the realities of each particular organization. However, general guidelines should exist to provide directions. A few corporations follow the guidelines of such organizations as the Coalition for Environmentally Responsible Economies (CERES) and the Global Reporting Initiative (GRI). In this book we follow the general fuzzy model we have already developed and adapt it to each particular case.

5.2. THE SAFE MODEL We start with the model of sustainability assessment of Chapter 4. This is the basic model of sustainability assessment of organizations. OSUS as previously is a function of HUMS and ECOS. HUMS is a function of POLIC, WEALTH, HEALTH, and KNOW, and ECOS is a function of AIR, LAND, WATER, and BIOD. The inputs to the eight secondary components are as before pressure, state, and response (PSR) but now the basic indicators are different. However, the PSR convention need not be adopted if the number of basic inputs is not too large.

Basic Indicators The choice of basic indicators depends on the type of organization under consideration. One should keep in mind when choosing basic indicators that they reflect the organization’s economic, environmental, and social performance and its contribution locally, regionally, or globally to sustainability. Norms and targets for these indicators are dictated by legal requirements and scientific or expert knowledge. Boundaries The boundaries of the organization extend from all possible inputs such as supply chains to all outputs such as products, clients, etc.


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Assessment of Indicators Values of indicators are provided by organizations or can be estimated using various techniques. For example, emissions due to the production, use, and disposal of goods can be evaluated using life cycle assessment (LCA). We shall see such techniques in more detail later in this book. An Example of Basic Indicators The following is an indicative list of possible sustainability indicators. AIR Greenhouse gas emissions (all emissions concern production and product use/disposal) NOx, SOx, CO emissions VOC (volatile organic compounds) emissions CFC (chlorofluorocarbons) emissions Renewable energy use Fossil fuel use Percentage of environmentally friendly buildings, i.e., buildings with • • • •

passive solar design economy fixtures automatic switches faucets that turn off when not used

LAND Solid and liquid waste generation (all waste concerns production and product use/disposal) Amount of solid and liquid waste treated/recycled Nuclear waste Reforested land Restored land Amount of material dumped/recycled WATER Water use for production Water treatment/recycling BOD (biological oxygen demand) COD (chemical oxygen demand) Hydrocarbon effluents Wastewater effluents BIOD Number of species/habitats affected by production and product use/disposal Protected habitats conserved or destroyed

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POLICY Compliance with international human rights and environmental agreements Compliance with national environmental laws/regulations Compliance with national labor laws/regulations Benefits to employees Percentage of employees covered by bargaining agreements Nondiscrimination policies Avoidance of child labor Respect of rights of indigenous people expressed by numbers of violations of such rights Financial and other contributions to communities WEALTH Revenue Production/operating costs Financial assistance received from government New capital investments HEALTH Injury rates Fatality rates Lost days Occupational health problems Health insurance coverage of employees KNOW R&D expenditures Percentage of employees receiving training in • • •

new technologies and methods ethics anti-corruption policies

Interaction with institutions of higher learning.

5.3. AN APPLICATION Publicly available data have been collected for a number of beverage companies operating in the same region (Phillis and Davis 2008). The indicators used in the SAFE model and their dependencies are shown in Figure 5.1. It was not possible to collect data about BIOD and KNOW.


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Basic indicators 1. Greenhouse gas emissions 2. Toxic releases 3. Solid waste generated 4. Solid waste recycled 5. Hazardous waste generated 6. Water use 7. Pension benefits paid to employees

Secondary indicators AIR

Primary indicators

LAND

ECOS

WATER

8. Employees unionized 9. Contributions to communities 10. Taxes paid to government

OSUS POLICY WEALTH

HUMS

11. Sales revenue per employee HEALTH 12. Debt ratio 13. Lost-time injury frequency 14. Total recordable injury rate 15. Lost work days 16. Health and life insurance benefits

Figure 5.1. Hierarchical assessment of corporate sustainability.

Below we give definitions of the basic indicators and their most desirable and least desirable values.

5.3.1. Basic Indicators AIR GHG: Greenhouse gas (GHG) emissions (metric tons CO2 equivalent emitted per million dollars of annual net sales) measure a company’s impact on climate change. Lower emissions of GHG imply that the company is more sustainable due to its decreased risk from new environmental regulations, carbon taxes, etc. and also due to its reduced impact on the global ecosystem. We assume that lower is better and that any value below a certain threshold is sustainable, i.e., its normalized value is one. The threshold is set at TGHG = 100 tons CO2 equivalent per million dollars annual net sales, which is the average value of beverage companies. The upper bound at which sustainability is zero is the maximum value over all years for all companies in the same region. This value is UGHG = 682. A normalized value for GHG is obtained using the formula of Figure 4.2. TX: Environmental Protection Agency (EPA) toxic releases (g per liter produced) lead to lower sustainability since more emissions to the air harm humans and the ecosystem.

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For toxic releases, similarly to GHG emissions, we assume that lower is better. The upper target value is chosen as the average over all data points and it is TTX = 6.54 g per liter of production. The maximum value is UTX = 24.85 g/lt.

LAND SW: Solid waste generation (g per liter produced) is the mass of solid waste that is dumped by the company into a landfill, rather than reused or recycled in some manner. It does not count waste that is sold to other companies or is used to make other products. A lower amount of waste dumped is better for the environment due to less pollution of the land and greater amount of land available to the ecosystem for other purposes (farming, animal habitat, etc.). Less waste is also an economic benefit, since companies that produce less waste will spend less money on raw materials, run a lower risk of environmental fines and penalties and have less land and waste removal costs. As previously, the average value TSW = 787 g/lt is considered to be the threshold for sustainability and the maximum USW = 3,013 g/lt as the smallest undesirable value. RECY: Solid waste recycled (percent of total) is a measure of how efficient the company is at limiting its ecological footprint. The more waste is reused or recycled, the lower the company’s impact on the ecosystem. A higher rate of recycling is more sustainable. A lower threshold of uRECY = 50% waste recycling is subjectively chosen as unsustainable. A higher rate of recycling increases sustainability linearly to τRECY = 99%, where it is assumed that sustainability is one. A normalized value for RECY is obtained using the formula of Figure 4.3. HW: Hazardous waste (g per liter produced) generated by the company harms the ecosystem because that waste must be treated or dumped. The less hazardous waste the company produces, the more sustainable it is. Suppose that any level of waste production below THW = 0.0147 g/lt (industry average) is sustainable with value one, with sustainability decreasing linearly to the maximum value UHW = 0.0479 g/lt.

WATER WATER: Water use (per unit of production) is a measure of the company’s impact on water resources. If less water is used to make a given amount of product, more water is available for humans and other species to use. Fresh water is an increasingly valuable and scarce resource; since production requires water as an input, a good measure of water efficiency is the ratio of water used to product generated. Lower water use is better, so we set the upper target level to the industry average TWATER = 5 liters of water per liter of product and the lower unsustainable value to the maximum over all companies, UWATER = 6. POLICY PENB: Pension benefits paid to employees are a measure of how the company’s policies are affecting the well-being of its workers after they retire. Money that the company contributes to pensions directly reduces money which must be spent by governments for social programs, contributes to the local economy, and maintains or increases the company’s


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desirability as a place of employment. All of these things contribute to the sustainability of the human system. Pension benefits to employees should not be too low or too high. We have chosen the minimum value for full sustainability as the poverty line τPENB = $10,210 for a one person household in the U.S. in 2007, as determined by the U.S. Department of Health and Human Services (available: http://aspe.hhs.gov/POVERTY/07poverty.shtml). Sustainability is one from this value up to the poverty line for a two-person household, TPENB = $13,690. Sustainability is zero at uPENB = $1,000 and is also zero at the maximum value over all data points, UPENB = $25,268. ECAU: Percentage of employees covered by bargaining agreements and unions is a measure of how well the workers of a company are organized and how active a role they are taking in maintaining their rights and compensation. More employees covered by bargaining agreements or unions means a company is more sustainable due to the lower risk of unemployment and increased quality of life for workers. It is assumed that a uECAU = 0% coverage is not sustainable and that sustainability increases linearly to one at tECAU = 58% coverage, which is the maximum value over all beverage companies. CCOM: Financial and other contributions to communities (percent of annual net sales) are another measure of the company’s benefit to society. These contributions are relatively symbolic compared to contributions via taxes, jobs, health insurance, etc., but they are a useful measure of how dedicated the company is to improving the human system within which it operates. More contributions to the surrounding communities are better since any company has an impact on its locality and the people and services in it. It is assumed that a uCCOM = 0.1% contribution, which is slightly lower than the minimum over all beverage companies, is not sustainable and that sustainability increases linearly to one at tCCOM = 0.26%, which is the maximum percentage of annual net sales contributed. TAX: Taxes paid to government are a good economic measure of a company’s benefit to society. The more they pay in taxes, within bounds, the more the companies are contributing to the stability of the country, region, or locality via infrastructure, social programs, policing and safety services, etc. However, taxes paid to the government should not be too large. We have chosen the current corporate income tax rate in the U.S. as the “ideal” value, under the assumption that it is the intention of the law that each company pay this percentage of their earnings to the government. This rate is currently 35% of income and we assume that tax rates paid below that amount are too low to support government functioning and tax rates above that amount mean that a company is not being efficient in their management of their tax obligation. A 0% or 100% tax rate is not sustainable (value zero) and sustainability increases linearly from both sides towards τTAX = TTAX = 35%.

WEALTH RPE: Sales revenue per employee (million $ per year) is a measure of a company’s economic efficiency. The more efficient a company is at producing wealth from a given amount of labor, the more sustainable it is.

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We assume that higher revenue per employee is always better and that the lower threshold value for sustainability of one is τRPE = 0.547 million U.S. dollars per employee based on the average value for this industry. Any value below the minimum over all data points, uRPE = 0.212, is not sustainable. DEBT: Debt ratio (total debt divided by total assets) per unit of income generated is an indicator of the risk involved in a company’s operations. Too much debt means the company may be overextended and risks collapse, but too little debt indicates stagnation and inefficient use of cash resources. It is assumed that the optimal debt ratio is the average over all beverage companies, τDEBT = TDEBT = 0.314 and that sustainability decreases linearly on both sides towards a debt ratio of 0 (no borrowing) and a debt ratio of 1 (too highly leveraged).

HEALTH LTIR: Lost time injury rate (number of incidences per 100 employees per year) refers to any work related injury or illness which prevents that person from doing any work day after accident. Fewer injuries where employees miss work means greater productivity of the company and therefore higher sustainability due to better performance. We assume that the only fully sustainable value is having no injuries at all, i.e., TLTIR = 0. Any value above zero is less sustainable up to the average value over all data points, which is ULTIR = 0.669 incidents per 100 employees per year. IR: Injury rate (number of incidences per 100 employees per year) measures the number of on-the-job injuries sustained by employees. It is a useful metric of how safe the company is to work, and fewer injuries mean lower insurance costs, less health care liability, and lower risk of fines or regulations due to unsafe practices. As previously, the target value for this indicator is TIR = 0 and the threshold of undesirable values is the industry average UIR = 4.24 incidents per 100 employees per year. LWD: Lost workdays (per employee per year) due to lost-time injuries are an indirect measure of the severity of nonfatal, work related injuries or illnesses. The target value for this indicator is TLWD = 0 and the threshold of undesirable values is the industry average ULWD = 0.273 lost workdays per employee per year. HLIB: Health and life insurance benefits (thousand dollars per employee per year) given to employees by a company help the human system by reducing economic risk. Higher benefits paid by the company are seen as more sustainable. We have chosen to compare health and life insurance benefits to the past history of all companies. Zero HLIB has zero sustainability, up to the average value over all data points THLIB = $1,677 per employee per year which has a sustainability of one.


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5.3.2. Normalization and Exponential Smoothing Table 5.1 shows the basic indicator time series data for a selected beverage company. Normalized values, given in parentheses, are computed by linear interpolation between most desirable (target) and least desirable indicator values, as discussed in Section 4.3. Table 5.1. Basic indicators and corresponding normalized values for a selected company Annual indicator value (normalized value) Indicator 2000 2001 2002

2003

2004

2005

GHG

NA

NA

NA

0.170 (1) 0.264 SW NA NA (1) 0.970 RECY NA NA (0.9592) 0.01923 HW NA NA (0.8635) 5.888 WATER NA NA (0.1115) 5,766 5,382 6,679 PENB (0.5180) (0.4763) (0.6173) 0.374 0.407 0.434 ECAU (0.6448) (0.7017) (0.7483)

0.157 (1) 0.266 (1) 0.970 (0.9592) 0.01310 (1) 5.699 (0.3012) 7,175 (0.6712) 0.420 (0.7241)

0.153 (1) 0.256 (1) 0.970 (0.9592) 0.01405 (1) 5.785 (0.2150) 5,440 (0.4826) 0.284 (0.4897)

0.166 (1) 0.245 (1) 0.970 (0.9592) 0.01371 (1) 5.837 (0.1632) 5,479 (0.4868) 0.580 (1)

CCOM

TX

TAX RPE DEBT LTIR IR LWD HLIB

NA NA

NA

NA

0.160 (1)

NA

NA

NA

NA

NA

0.213 (0.6086) 0.624 (1) 0.411 (0.8586) 1.030 (0) 9.910 (0) 0.316 (0) 1.349 (0.8045)

0.214 (0.6114) 0.639 (1) 0.432 (0.8285) 0.690 (0) 7.910 (0) 0.195 (0.2845) 1.541 (0.9187)

0.217 (0.6200) 0.677 (1) 0.468 (0.7760) 0.660 (0.0135) 6.660 (0) 0.398 (0) 1.795 (1)

0.216 (0.6171) 0.700 (1) 0.496 (0.7347) 0.690 (0) 5.970 (0) 0.326 (0) 2.007 (1)

0.212 (0.6057) 0.546 (0.9967) 0.512 (0.7116) 0.500 (0.2526) 5.420 (0) 0.220 (0.1934) 1.937 (1)

2006 2007 192.784 NA (0.8406) NA 0.236 (1) 0.970 (0.9592) 0.01250 (1) 5.742 (0.2581) 7,113 (0.6645) 0.530 (0.9138)

NA NA NA NA NA

6,804 (0.6309) 0.514 (0.8862) 0.001175 NA NA (0) 0.191 0.190 0.1842 (0.5457) (0.5429) (0.5257) 0.548 0.595 0.616 (1) (1) (1) 0.482 0.467 0.533 (0.7558) (0.7765) (0.6810) 0.550 0.450 NA (0.1779) (0.3274) 5.180 3.810 NA (0) (0.1004) 0.180 0.315 NA (0.3388) (0) 2.010 2.276 2.324 (1) (1) (1)

“NA” indicates that no data were available for the corresponding year. (Phillis and Davis 2008, Table 2. © 2008 by Springer. Used with permission.)

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For example, the greenhouse gas emissions were 192.784 metric tons CO2 equivalent per million dollars of annual net sales for this company in 2006. The corresponding normalized value is

682 − 192.784 489.216 = = 0.8406 682 − 100 582 as shown in Table 5.1. The normalized time series for each indicator are aggregated into a single normalized value using the method of exponential smoothing described in Section 4.4. The results are shown in Table 5.2. Table 5.2. Exponential smoothing Indicator c

Optimal smoothing parameter, β

Normalized value, xc

GHG TX SW RECY HW WATER PENB ECAU CCOM TAX RPE DEBT LTIR IR LWD HLIB

0 0 0 0 0 1 1 0.75 0 0 1 0 0.23 0 1 0

0.8406 1 1 0.9592 1 0.2098 0.5685 0.8219 0 0.5257 0.9996 0.6810 0.2939 0.1004 0.1167 1

For a smoothing parameter β = 0, the estimate xc is the most recent normalized value. For β = 1, the estimate xc is the average of the normalized time series. For other values of β, the estimate is computed from Eq. (4.3).

5.3.3. Fuzzification As in the assessment of national sustainability, the normalized basic indicators are fuzzified using three fuzzy sets, with linguistic values “weak” (W), “medium” (M), and “strong” (S), whose membership functions are shown in Figure 5.2a. For secondary and tertiary indicators we use five fuzzy sets with linguistic values “very bad” (VB), “bad” (B), “average” (A), “good” (G), and “very good” (VG), as shown on Figure 5.2b. It should be


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noted that the basic indicator “Water use per unit of production” is represented using five fuzzy sets because this is the only indicator of the secondary component WATER. For example, the crisp value xCHG = 0.8406 of Table 5.2 belongs to the fuzzy set M with grade (1 − 0.8406)/(1 − 0.6) ≈ 0.4 and to the fuzzy set S with grade (0.8406 − 0.6) /(1 − 0.6) ≈ 0.6. Also, from Figure 5.2b we see that the crisp value xWATER = 0.2098 is VB with membership grade (0.25 − 0.2098)/(0.25 − 0) ≈ 0.16 and B with grade (0.2098 − 0)/(0.25 − 0) ≈ 0.84. 1

M

W

S

VB

1

0

B

A

G

VG

EL VL

L

FL

I

FH

H VH EH

0 0

0.5 0.6

0.25

0.5

1

0

0.125

Composite indicator

Basic indicator

(a)

0.5

1


(b)

(c)

Figure 5.2. Membership functions.

Table 5.3 gives the membership grades of the basic inputs to the fuzzy sets W, M, and S. Table 5.3. Fuzzification of basic inputs Indicator c Normalized value, xc GHG TX SW RECY HW WATERa PENB ECAU CCOM TAX RPE DEBT LTIR IR LWD HLIB

0.8406 1 1 0.9592 1 0.2098 0.5685 0.8219 0 0.5257 0.9996 0.6810 0.2939 0.1004 0.1167 1

Fuzzy sets W 0.00 0.00 0.00 0.00 0.00 0.16 (VB) 0.05 0.00 1.00 0.12 0.00 0.00 0.51 0.83 0.81 0.00

M 0.40 0.00 0.00 0.10 0.00 0.84 (B) 0.95 0.45 0.00 0.88 0.00 0.80 0.49 0.17 0.19 0.00

S 0.20 1.00 1.00 0.80 1.00 0.00 (A, G, VG) 0.00 0.11 0.00 0.00 1.00 0.00 0.00 0.00 0.00 1.00

a

WATER has five fuzzy sets: VB, B, A, G, and VG.

Finally, for OSUS we use nine fuzzy sets with linguistic values “extremely low” (EL), “very low” (VL), “low” (L), “fairly low” (FL), “intermediate” (I), “fairly high” (FH), “high” (H), “very high” (VH), and “extremely high” (EH) (see Figure 5.2c).

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5.3.4. Rule Bases The rule base for OSUS is the same as the one used for the assessment of sustainability of nations (see Section 4.6). It is obtained from equation OSUS = ECOS + HUMS by assigning integer values from the set {0, 1, 2, …} to the term sets {VB, B, A, …} and {EL, VL, L, …}. For example, if ECOS = A = 2 and HUMS = VB = 0, then OSUS is 2 + 0 = 2, which corresponds to the fuzzy set L. ECOS has 3 inputs, namely, LAND, AIR, and WATER. Its fuzzy set is determined from the following equations: SUM = LAND + AIR + WATER and

⎧VB, 0 ≤ SUM ≤ 1 ⎪ B, 2 ≤ SUM ≤ 4 ⎪⎪ ECOS = ⎨A, 5 ≤ SUM ≤ 7 ⎪G, 8 ≤ SUM ≤ 10 ⎪ ⎪⎩VG, 11 ≤ SUM ≤ 12. The same rule base is used for HUMS. Table 5.4. Rules for AIR

Table 5.5. Rules for WEALTH

Rule Rp

if CHG is

and TX is

then AIR is

then and if Rule Rp RPE DEBT WEALTH is is is

R1 R2 R3 R4 R5 R6 R7 R8 R9

W W W M M M S S S

W M S W M S W M S

VB B B B A G A G VG

R1 R2 R3 R4 R5 R6 R7 R8 R9

W W W M M M S S S

W M S W M S W M S

VB B A A A G A G VG

(Phillis and Davis 2008, Tables 4 and 5. © 2008 by Springer. Used with permission.)


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The secondary indicators AIR and WEALTH have two inputs each. Their rule bases are shown in Tables 5.4 and 5.5. The rule base of AIR is more pessimistic due to the impact of air emissions on the environment. LAND has three inputs and its fuzzy rules are determined from SUM = SW + RECY + HW and

⎧VB, 0 ≤ SUM ≤ 1 ⎪ B, SUM = 2 ⎪⎪ LAND = ⎨A, SUM = 3 ⎪G, 4 ≤ SUM ≤ 5 ⎪ ⎪⎩VG, SUM = 6. As in the case of AIR, this rule base is chosen to be slightly pessimistic since the value 1 corresponds to the fuzzy set “very bad” whereas 5, which is the complement of 1 with respect to the maximum 6, is assigned to the fuzzy set “good” rather than to “very good.” Finally, POLICY and HEALTH have four inputs each. Their rule bases are generated by assigning a linguistic value to SUM = L1 + L2 + L3 + L4 where Li ∈ {0 = W, 1 = M, 2 = S} is the integer corresponding to the linguistic value of the ith input. The rules are given by

⎧VB, 0 ≤ SUM ≤ 2 ⎪ B, 3 ≤ SUM ≤ 4 ⎪⎪ POLICY, HEALTH = ⎨A, SUM = 5 ⎪G, SUM = 6 ⎪ ⎪⎩VG, 7 ≤ SUM ≤ 8. This leads to a quite pessimistic rule base since three values are assigned to VB vs. two values to VG, and two values are assigned to B vs. only one to G. To mediate this, we introduce four exceptional fuzzy rules, as we shall refer to them in the sequel: when two inputs are W = 0 and two inputs are S = 2 then, although SUM = 4, the output is A. A computer program of the SAFE model is presented in the Appendix, in which the rule bases are defined compactly using equations but the user may also introduce any number of exceptional rules.

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5.3.5. Results Table 5.6 shows the sustainability assessments for the selected company using SAFE model. Some computations are carried out below in detail (Phillis and Davis 2008). Table 5.6. Normalized values and membership grades of primary indicators and OSUS Indicator ECOS HUMS

Value 0.723 0.458

OSUS

0.591

VB0 0 0 EL0 0

B1 0 0.18 VL1 0

A2 0.11 0.8 L2 0

G3 0.89 0.02 FL3 0.02

VG4 0 0 I4 0.25

FH5 0.71

H6 0.02

VH7 0

EH8 0

Suppose we have computed the membership grades of the primary indicators to the five fuzzy sets. The membership grades of OSUS (last row of Table 5.6) are determined from the following rules A(ECOS) + B(HUMS) = 2 + 1 = 3 ⇒ OSUS is FL with grade 0.11 × 0.18 ≈ 0.02, A(ECOS) + A(HUMS) = 2 + 2 = 4 ⇒ OSUS is I with grade 0.11 × 0.8 ≈ 0.09, A(ECOS) + G(HUMS) = 2 + 3 = 5 ⇒ OSUS is FH with grade 0.11 × 0.02 ≈ 0, G(ECOS) + B(HUMS) = 3 + 1 = 4 ⇒ OSUS is I with grade 0.89 × 0.18 ≈ 0.16, G(ECOS) + A(HUMS) = 3 + 2 = 5 ⇒ OSUS is FH with grade 0.89 × 0.8 ≈ 0.71, G(ECOS) + G(HUMS) = 3 + 3 = 6 ⇒ OSUS is H with grade 0.89 × 0.02 ≈ 0.02. The final, crisp value for OSUS is computed using height defuzzification: OSUS =

0.02 × 0.375 + (0.09 + 0.16) × 0.5 + (0 + 0.71) × 0.625 + 0.02 × 0.75 0.591 = . 0.02 + (0.09 + 0.16) + (0 + 0.71) + 0.02 1

5.4. METHODS TO ESTIMATE EMISSIONS AND WASTE Data about emissions and waste generation are not always available. It is often technically very hard and economically costly to measure emissions exactly. Take for example the case of an oil refinery that has thousands of valves, flanges, various connectors, pumps, compressor seals, etc. Several methods exist whereby emissions and waste generation are estimated. Two of those are outlined below.


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5.4.1. Life Cycle Assessment (LCA) Overview This is a brief introduction to LCA. The interested reader can find a detailed exposition of this methodology in such sources as EPA (2006). LCA is a systematic way of estimating the environmental impact of industrial processes, products, or services from cradle to grave, i.e., from the stage of extraction of the raw materials associated with a product, process, or service, to the stage of product use and final disposal. In other words, LCA examines the life span of a product, process, or service (which we collectively call product) and assesses its environmental impact throughout all its stages. LCA consists of the following four phases: a. Definition of goal and scope A product is described and the context of LCA, that is, boundaries of systems to be examined and environmental effects are established. For example, LCA could be used to investigate possibilities of product improvement, to compare various products with the same use (e.g., aluminum, paper, or plastic soft drink containers), to improve planning, etc. b. Life cycle inventory All energy and matter inputs are recorded throughout the life span of a product and all emissions, effluents, and solid waste are similarly quantified and recorded. Pictorially this is seen in Figure 5.3. Energy, matter

Raw material extraction/ acquisition

Materials manufacture

Product manufacture

Product use

Product disposal Product reuse Product recycling

Air emissions, effluents, solid waste

Figure 5.3. Life cycle inventory.

c. Evaluation of environmental impact The environmental and human health impacts of the substances recorded in the life cycle inventory are assessed. This is a difficult task since there are no universally accepted methods

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to perform this assessment. According to one such method, developed by the Swedish Environmental Institute and Volvo, numerical values, called environmental indices, are assigned to various substances as shown in Table 5.7. Each index is computed subjectively, based on a number of criteria such as general impression of impact, size of affected area, regularity and intensity of problem, etc. Table 5.7. A selection of environmental indices Raw materials Cobalt Chromium Iron Manganese Molybdenum Nickel Lead Platinum Rhodium Tin Vanadium

ELU/kga 76 8.8 0.09 0.97 1,500 24.3 180 350,000 1,800,000 1,200 12

Air emissions Carbon dioxide Carbon monoxide Nitrogen oxide Nitrous oxide Sulfur oxide CFC-11 Methane

ELU/kga 0.09 0.27 0.22 7.0 0.10 300 1.0

Water emissions Nitrogen Phosphorus

ELU/kga 0.1 0.3

a

Units are environmental load units (ELU) per kg. (Allen and Rosselot 1997, Table 4-4, p. 78. © 1997 by John Wiley and Sons, Inc. Used with permission.)

d. Utilization of results to design better products for the environment and human health LCA enables specialists and decision makers to achieve the following: 1) Obtain quantitative estimates of emissions, effluents, and solid wastes related to the various stages of life cycles of a given product. 2) Assess environmental impacts of products. 3) Compare the environmental impacts and health effects of competing products or processes. 4) Assess the environmental consequences of human material consumption in a given region. 5) Focus in one or more specific environmental areas and assess material consumption impact.

Uncertainties LCA is an effective methodology to assess environmental impact of products but it is also fraught with uncertainties. First, it is often difficult and time consuming to find reliable data about emissions and waste generation. An LCA report ought to weigh the reliability and completeness of data to be of value. Second, environmental impact assessment is often subjective. There is no unique mechanism to compare the impact of, say, CO2 and NOx emissions. Polyethylene grocery


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bags are superior to paper bags when air emissions at the production stage are compared but plastics to not decompose easily in garbage dumps. Human values often play a role in assessing impact. Now let us examine all 4 stages of LCA.

A. Definition of Goal and Scope The following steps are taken to define goal and scope in LCA: 1. Define the goal This could be the assessment of the environmental impact of a product, comparison of products, improvement of product, aiding environmental decision-making, supporting certification of products, development of new products with better environmental and health performance and so on. The definition of the goal guides the analyst towards a certain level of specificity. One could for instance examine a product by a given company or a product irrespective of the company. 2. Organize and display the data Data should be organized and displayed in a way that is easily understandable and makes appropriate comparisons. If, for example, one compares plastic and paper grocery bags, the number of bags of each kind should be compared that hold the same volume of groceries. 3. Define the scope The goal of analysis dictates also the scope, i.e., whether the analysis covers all or part of the life cycle of the product.

B. Life Cycle Inventory (LCI) An LCI contains all energy and material requirements and all air/water emissions and solid wastes associated with the life cycle of a product. To compile such an inventory the following steps should be taken: 1. Draw a flow diagram For a given product draw a diagram that includes all steps of the production, use, and disposal and all inputs and outputs. 2. Collect data The flow diagram shows all pathways of a product’s life cycle. Then one identifies sources for data for these pathways such as equipment logs, government or industry reports, statistical databases, meter readings, scientific literature, etc., and records all relevant numbers.

C. Life Cycle Impact Assessment (LCIA) According to EPA (2006), LCIA has the following steps: 1. Selection and definition of impact categories Table 5.8 (EPA 2006) shows several commonly used LCIA categories.

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Table 5.8. Commonly used life cycle impact categories LCI categories

Scale

Examples of LCI Data (i.e. classification)

Global Warming Global

Global Warming Potential

Converts LCI data to carbon dioxide (CO2) equivalents Note: global warming potentials can be 50, 100, or 500 year potentials.

Stratospheric Ozone Depletion

Ozone Depleting Potential

Converts LCI data to trichlorofluoromethane (CFC11) equivalents.

Acidification Potential

Converts LCI data to hydrogen (H+) ion equivalents.

Eutrophication Potential

Converts LCI data to phosphate (PO4) equivalents.

Photochemical Oxidant Creation Potential LC50

Converts LCI data to ethane (C2H6) equivalents.

Acidification

Eutrophication

Photochemical Smog Terrestrial Toxicity Aquatic Toxicity Human Health

Resource Depletion

Land Use

Water Use

Carbon Dioxide (CO2) Nitrogen Dioxide (NO2) Methane (CH4) Chlorofluorocarbons (CFCs) Hydrochlorofluorocarbons (HCFCs) Methyl Bromide (CH3Br) Global Chlorofluorocarbons (CFCs) Hydrochlorofluorocarbons (HCFCs) Halons Methyl Bromide (CH3Br) Regional Sulfur Oxides (SOx) Local Nitrogen Oxides (NOx) Hydrochloric Acid (HCL) Hydrofluoric Acid (HF) Ammonia (NH4) Local Phosphate (PO4) Nitrogen Oxide (NO) Nitrogen Dioxide (NO2) Nitrates Ammonia (NH4) Local Non-methane hydrocarbons (NMHC)

Common Description of Possible Characterization Factor Characterization Factor

Local

Toxic chemicals with a reported lethal concentration to rodents Local Toxic chemicals with a LC50 reported lethal concentration to fish Global Total releases to air, water, and LC50 Regional soil Local Global Quantity of minerals used Resource Depletion Regional Quantity of fossil fuels used Potential Local Global Regional Local Regional Local

Quantity disposed of in a landfill or other land modifications Water used or consumed

Land Availability

Water Shortage Potential

Converts LC50 data to equivalents; uses multi-media modeling, exposure pathways. Converts LC50 data to equivalents; uses multi-media modeling, exposure pathways. Converts LC50 data to equivalents; uses multi-media modeling, exposure pathways. Converts LCI data to a ratio of quantity of resource used versus quantity of resource left in reserve. Converts mass of solid waste into volume using an estimated density. Converts LCI data to a ratio of quantity of water used versus quantity of resource left in reserve.

2. Classification If a substance contributes to only one impact category, then this impact is assessed. If, on the other hand, a substance contributes to more than one impact categories, then, according to the International Standards Organization (ISO 1998), impact is assessed according to the following:


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a) Record the portion of impact the substance has to a given category when impacts are dependent on each other from category to category. b) If impacts are independent of each other, then record the whole impact of the substance to each category. For example, certain CFC’s contribute to global warming and stratospheric ozone depletion. Then the contribution of a CFC emission is 100% to global warming and 100% to ozone depletion. 3. Characterization Characterization provides mechanisms to compare impacts directly. For example, the impact of methane emissions to global warming is different from the impact of nitrous oxide emissions. Using, however, the global warming potentials (GWP) of the Intergovernmental Panel on Climate Change (see Table 5.9) we obtain a common scale to compare impacts using the equation Impact = Emissions × GWP. Thus 10 kg of methane create an impact of: Impact = 10 × 23 = 230 whereas 10 kg of HFC-23 create an impact of: Impact = 10 × 12,000 = 120,000. In general, impact is computed from Impact = (LCA data) × (characterization factor). It should be noted that commonly accepted characterization factors exist for only a few categories of substances such as greenhouse gases or ozone depleting substances. As we have seen, impact often reflects subjective values and thus, widely accepted characterization factors are lacking. An indicative list of impact categories is shown in Table 5.9. 4. Normalization Normalization of an impact indicator is just division by a reference value to enable easy comparisons within an impact category. The following are examples of reference values. a) Total emissions in a given area b) Total per capita emissions in a given area c) Highest value. Normalization deals with one impact category only. For example ozone depletion cannot be compared with global warming.

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Yannis A. Phillis and Vassilis S. Kouikoglou Table 5.9. Global Warming Potentials (GWPs) relative to carbon dioxide

Gas

Carbon dioxide CO2 Methanea CH4 Nitrous oxide N2O Hydrofluorocarbons HFC-23 CHF3 HFC-32 CH2F2 HFC-41 CH3F HFC-125 CHF2CF3 HFC-134 CHF2CHF2 HFC-134a CH2FCF3 HFC-143 CHF2CH2F HFC-143a CF3CH3 HFC-152 CH2FCH2F HFC-152a CH3CHF2 HFC-161 CH3CH2F HFC-227ea CF3CHFCF3 HFC-236cb CH2FCF2CF3 HFC-236ea CHF2CHFCF3 HFC-236fa CF3CH2CF3 HFC-245ca CH2FCF2CHF2 HFC-245fa CHF2CH2CF3 HFC-365mfc CF3CH2CF2CH3 HFC-43-10mee CF3CHFCHFCF2CF3 Fully fluorinated species SF6 CF4 C2F6 C3F8 C4F10 c-C4F8 C5F12 C6F14 Ethers and halogenated ethers CH3OCH3 HFE-125 CF3OCHF2 HFE-134 CHF2OCHF2 HFE-143a CH3OCF3 HCFE-235da2 CF3CHClOCHF2 HFE-245fa2 CF3CH2OCHF2 HFE-254cb2 CHF2CF2OCH3 HFE-7100 C4F9OCH3 HFE-7200 C4F9OC2H5 H-Galden 1040x CHF2OCF2OC2F4OCHF2 HG-10 CHF2OCF2OCHF2 HG-01 CHF2OCF2CF2OCHF2 a

Lifetime (years)

Global Warming Potential (time horizon in years) 20 yrs 100 yrs 500 yrs

12.0b 114b

1 62 275

1 23 296

1 7 156

260 5.0 2.6 29 9.6 13.8 3.4 52 0.5 1.4 0.3 33 13.2 10 220 5.9 7.2 9.9 15

9,400 1,800 330 5,900 3,200 3,300 1,100 5,500 140 410 40 5,600 3,300 3,600 7,500 2,100 3,000 2,600 3,700

12,000 550 97 3,400 1,100 1,300 330 4,300 43 120 12 3,500 1,300 1,200 9,400 640 950 890 1,500

10,000 170 30 1,100 330 400 100 1,600 13 37 4 1,100 390 390 7,100 200 300 280 470

3,200 50,000 10,000 2,600 2,600 3,200 4,100 3,200

15,100 3,900 8,000 5,900 5,900 6,800 6,000 6,100

22,200 5,700 11,900 8,600 8,600 10,000 8,900 9,000

32,400 8,900 18,000 12,400 12,400 14,500 13,200 13,200

0.015 150 26.2 4.4 2.6 4.4 0.22 5.0 0.77 6.3 12.1 6.2

1 12,900 10,500 2,500 1,100 1,900 99 1,300 190 5,900 7,500 4,700

1 14,900 6,100 750 340 570 30 390 55 1,800 2,700 1,500

0 to each input i and compute the weighted sum SUM = w1L1 + … + wnLn.

(A. 1)

Since 0 ≤ Li ≤ m, the minimum value of SUM is 0 and the maximum is (w1 + … + wn)m. We also need k values which divide the range of SUM into k + 1 consecutive intervals. Let us denote these values T0, …, Tk−1. The output fuzzy set of each rule in the rule base can then be determined by

0 ≤ SUM ≤ T0 ⎧0, ⎪1, T0 < SUM ≤ T1 ⎪⎪ Ln+1 = ⎨ ⎪k − 1, T < SUM ≤ T k −2 k −1 ⎪ Tk −1 < SUM ≤ m( w1 + … + wn ). ⎪⎩k ,

(A. 2)

In summary, the rule base can be described by the variables n, m, k, {wi, i = 1, …, n}, and {Tj, j = 0, …, k − 1}. For five inputs with five linguistic values, this representation requires storing just 1 + 1 + 1 + 5 + 5 = 13 numbers.

A.1.3. TSK Inference Representation B is by far the most efficient encoding scheme. However, representation A can describe exceptional fuzzy rules, like the ones introduced in Section 5.3, which do not fit in representation B. Consider an inference engine with n fuzzy inputs, i = 1, …, n, and output n + 1. Each input has m + 1 linguistic values, Λ ∈ {0, …, m}, and the output has k + 1 linguistic values, L ∈ {0, …, k}. There are a total of (m + 1)n rules in the rule base, a number of which, say, N are exceptional fuzzy rules and the remaining ones admit representation B. An exceptional rule R, R = 1, …, N, is represented by the code number CR using method A of Section A.1.1. Suppose that input i belongs to the fuzzy sets Λ with membership grades μΛ(i), for Λ = 0, …, m. The following algorithm computes the membership grade of the output to each L. The algorithm traces only the fuzzy sets to which an input belongs with positive grade. TSK Inference: 1) Read n, m, k, N, {μΛ(i) for all i = 1, …, n and Λ = 0, …, m}, and {CR, for all R = 1, …, N}. 2) For each input, determine fuzzy sets with positive membership. Thus, For i = 1, …, n: Initially no fuzzy sets are assigned to i: NPOS = 0 For Λ = 0, …, m:

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Yannis A. Phillis and Vassilis S. Kouikoglou If μΛ(i) > 0, then Set NPOS := NPOS + 1 FSET(i, NPOS) = Λ Endif Next Λ NFSETS(i) = NPOS 3) 4)

5)

6) 7)

Next i Initialize the membership grades of the output: μL = 0 for all L = 0, …, k. Initialize the combination of input fuzzy sets with positive membership grades. Thus, For i = 1, …, n: First fuzzy set is CURFS(i) = 1 Integer weight of this fuzzy set is Li = FSET(i, 1) Next i TSK (product) inference: Initialize the firing strength of the rule: set μ = 1 For i = 1, …, n: Λ = Li Set μ := μ × μΛ(i) Next i Invoke the algorithm of Section A.1.1 to compute INPCODE Determine the fuzzy set of the output. Two cases to examine: A and B. Representation A: Exceptional fuzzy rules For R = 1, …, N: Determine the prefix of rule R; thus IPREFIX = int(CR/10). If INPCODE = IPREFIX, then Rule R matches with inputs; the ouput fuzzy set is the suffix of CR, thus L = CR − IPREFIX × 10 Set μL := μL + μ Go to step 8 Endif Next R Representation B: Eqs. (A.1) and (A.2) SUM = 0 For i = 1, …, n: Set SUM := SUM + wi × Li Next i For L = 0, …, k: If SUM ≤ upper threshold TL, then Ouput fuzzy set is L; thus Set μL := μL + μ Go to step 8 Endif Next i


147

8) Current combination of input fuzzy sets has been evaluated. Determine next combination without overshooting maximum values. Starting from the fuzzy set of the first input, increase it by 1, unless maximum is reached in which case increase the fuzzy set of the second input, and so on. j=1 8a) NEXTFS = CURFS(j) + 1 If NEXTFS ≤ NFSETS(j), then Current fuzzy set is CURFS(j) = NEXTFS Integer weight of this fuzzy set is Lj = FSET(j, NEXTFS) All other inputs in the new combination have the same fuzzy sets as before Go to step 5 Else all fuzzy sets of input j have been examined; reset input j and check input j + 1, thus Current fuzzy set of input j is reset to CURFS(j) = 1 Integer weight of this fuzzy set is Lj = FSET(j, 1) Examine next input; thus Set j := j + 1 If j > n, then stop since all combinations have been examined; else go to step 8a Endif The above algorithm is repeated for all inference engines of the hierarchical fuzzy system.

A.2. LINGUISTIC VALUES AND MEMBERSHIP GRADES Sections 4.3 and 4.4 give all the details needed to program the steps of normalization and exponential smoothing of the basic indicators. We therefore assume that the time series of each basic indicator has been transformed into a single value in [0, 1]. Each indicator, either basic or composite, is described by a linguistic variable. In SAFE we use three types of linguistic variables, each one with 3, 5, or 9 linguistic values. Below we describe the operations of fuzzification and defuzzification for a TSK fuzzy system.

A.2.1. Fuzzification Each linguistic value L has a triangular membership function μL(x), as shown in Figure A.1.

148 1


W

M

S

VB

0

0.6

1

0

B

A

G

VG

EL VL

L

FL

I

FH

H

VH EH

0 1

Composite indicator

Basic indicator

(a)

1

0

(b)


(c)

Figure A.1. Membership functions used in the SAFE model.

The membership functions of the leftmost linguistic values (W, VB, EL) and those of the rightmost ones (S, VG, EH) are linear segments. Each one of them can be specified by two points in the plane, (x1, μ1) and (x2 μ2), where μi = μ(xi). For example, for W we have x1 = 0, μ1 = μW(x1) = 1, and x2 = 0.6, μ2 = μW(x2) = 0. The membership grade of a crisp value x to an extreme linguistic value is computed by linear interpolation:

μ 2 − μ1 ⎧ , x ∈ [ x1 , x2 ] ⎪ μ1 + ( x − x1 ) x2 − x1 μ(x) = ⎨ ⎪0, x ∉ [ x1 , x2 ] ⎩ The remaining membership functions are triangular and can be represented by three points, (x1, 0), (x2, 1), and (x3, 0). For the fuzzy set VL shown in Figure A.1c, the three characteristic points are: (0, 0), (0.125, 1), and (0.25, 0). The membership grade of a particular value x to an intermediate fuzzy set is computed from

⎧0, ⎪ ⎪ μ1 + x − x1 , x2 − x1 ⎪ μ(x) = ⎨ ⎪ μ 2 − x − x2 , ⎪ x3 − x 2 ⎪ ⎩0,

x < x1 x ∈ [ x1 , x2 ] x ∈ [ x2 , x3 ] x > x3 .

A.2.2. Defuzzification The model provides crisp results for the overall sustainability, OSUS, and its primary components, ECOS and HUMS. Let T be the term set of one of these three fuzzy variables, that is T = {VB, …, VG} or T = {EL, …, EH}. A crisp value for the output is computed from

∑ yL μL

xn+1 = L∈T

∑ μL

L∈T

,


149

where the membership grade μL is computed using the algorithm of Section A.1.3 and the peak value, yL, is the value x for which μL(x) = 1. For the leftmost fuzzy sets W, VB, and EL of Figure A.1 we have a peak at yL = x1 while for all others the peak values are at yL = x2.

A.3. A FORTRAN 77 CODE FOR THE SAFE MODEL A.3.1. Main program The main program invokes the following four subroutines: • INPUT1 reads data about rule bases, targets and least desirable values for each basic indicator, shapes of membership functions, and dependencies among composite indicators. • INPUT2 reads data and normalizes the basic indicators for each country or organization whose sustainability is to be assessed. • FUZZIFY computes the membership grades of each fuzzy input to the corresponding fuzzy sets. • INFERENCE computes the membership grades of composite indicators to the corresponding fuzzy sets and calculates crisp values using height defuzzification. PROGRAM SAFE C SAFE: SUSTAINABILITY ASSESSMENT BY FUZZY EVALUATION IMPLICIT REAL*8(A-H,M,O-Z) COMMON/FS/XP(7,9,4),MP(7,9,4),Y(7,9),NTERM,NFS(7),NP(7,9) COMMON/BI/UMIN(100),UMAX(100),TMIN(100),TMAX(100),NT,NB COMMON/IN/X(135),M(135,9),ITERM(135),IRB(35),IIN(35,7) DIMENSION MG(9) CHARACTER*25 SYSTEM(150),SYSTOUT(150),NAME(135) WRITE(*,*)'** SUSTAINABILITY ASSESSMENT BY FUZZY EVALUATION **' CALL INPUT1(SYSTEM,SYSTOUT,NSYSTEMS,NAME,NYEARS,NSTEPS) OPEN(8,FILE='EXPSMTH.OUT') WRITE(8,*)'***************************************************' WRITE(8,*)'* EXPONENTIAL SMOOTHING OF NORMALIZED TIME SERIES *' WRITE(8,*)'***************************************************' OPEN(9,FILE='SAFEALL.OUT') WRITE(9,*)'*************************************************' WRITE(9,*)'* SUMMARY OF SUSTAINABILITY ASSESSMENTS *' WRITE(9,*)'*************************************************' WRITE(9,'(40X,3(3X,A4))')'OSUS','ECOS','HUMS' DO 5 ISYSTEM=1,NSYSTEMS C OPEN FILE 'SYSTEM(ISYSTEM)' AND READ VALUES OF BASIC INDICATORS CALL INPUT2(SYSTEM(ISYSTEM),NYEARS,NSTEPS,NAME) C OPEN OF OUTPUT FILE FOR COUNTRY OR COMPANY OPEN (7,FILE=SYSTOUT(ISYSTEM)) WRITE(7,*)'****************************************************' WRITE(7,*)'*SAFE: SUSTAINABILITY ASSESSMENT BY FUZZY EVALUATION'

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Yannis A. Phillis and Vassilis S. Kouikoglou WRITE(7,'(3H * ,16(1H-),A15,17(1H-),2H *)')SYSTEM(ISYSTEM) WRITE(7,*)'****************************************************'

C FUZZIFY BASIC INDICATORS; SET MEMB. GRADES=0 FOR UNAVAILABLE DO 15 I=1,NB IF (X(I).EQ.-123456789) THEN DO 16 J=1,NFS(ITERM(I)) 16 M(I,J)=0 ELSE CALL FUZZIFY(X(I),ITERM(I),MG) DO 17 J=1,NFS(ITERM(I)) 17 M(I,J)=MG(J) ENDIF 15 CONTINUE C ASSESS SUSTAINABILITY WRITE(*,*)'--- BASIC ASSESSMENT ---' C TRACE INFERENCE ENGINES; COMPUTE COMPOSITE INDICATORS DO 20 I=1,NT-NB CALL INFERENCE(I,NB) 20 CONTINUE WRITE(7,*)'INDICATOR VALUES AND MEM.GRADES FOR ',SYSTEM(ISYSTEM) WRITE(*,*)'INDICATOR VALUES AND MEM.GRADES FOR ',SYSTEM(ISYSTEM) DO 22 I=1,NT IF ((I.LE.NB).AND.(X(I).EQ.-123456789)) GOTO 22 WRITE(7,'(I3,1X,A25,1X,F8.5,5H MG=,9(1X,F6.4))') +I,NAME(I),X(I),(M(I,K),K=1,NFS(ITERM(I))) WRITE(*,'(I3,1X,A25,1X,F8.5,5H MG=,9(1X,F6.4))') +I,NAME(I),X(I),(M(I,K),K=1,NFS(ITERM(I))) 22 CONTINUE OSUS=X(NT) HUMS=X(NT-1) ECOS=X(NT-2) WRITE(*,120) SYSTEM(ISYSTEM),OSUS,ECOS,HUMS WRITE(9,120) SYSTEM(ISYSTEM),OSUS,ECOS,HUMS CLOSE(7) 5 CONTINUE CLOSE(8) CLOSE(9) 120 FORMAT(25H SYSTEM(country/company):,A15,3F7.4) STOP END

A.3.2. Subroutine INPUT1 INPUT1 opens file “SAFE.IN,” which contains the following blocks of data: •

Names of input and output files for each system (country or organization) to be assessed.

Programming Hints and Tips •

• •

151

Rule bases: number of fuzzy sets for the output, boundary values (thresholds) of SUM used in Eqs. (A.1) and (A.2), and exceptional rules encoded according to the method discussed in Section A.1.1. Fuzzy partitions (term sets): number of fuzzy sets, peak value and characteristic points of the corresponding membership functions, as discussed in Section A.2.1. Term set, rule base, and basic indicators used to compute each composite indicator.

Each block or sub-block of data in “SAFE.IN” is preceded by a text line with comments regarding the data that has to be typed. The program ignores this line by assigning it to a dummy character variable named DUMMY. This is done by the statement READ(4,'(A15)')DUMMY

C C C C

5

C C C C C C C C C C C

SUBROUTINE INPUT1(SYSTEM,SYSTOUT,NSYSTEMS,NAME,NYEARS,NSTEPS) IMPLICIT REAL*8(A-H,M,O-Z) COMMON/RB/NRB,NVRB(20),IWRB(20,7),ITRB(20,9),NERB(20),IERB(20,800) COMMON/FS/XP(7,9,4),MP(7,9,4),Y(7,9),NTERM,NFS(7),NP(7,9) COMMON/BI/UMIN(100),UMAX(100),TMIN(100),TMAX(100),NT,NB COMMON/IN/X(135),M(135,9),ITERM(135),IRB(35),IIN(35,7) CHARACTER*25 DUMMY,SYSTEM(150),SYSTOUT(150),NAME(135) OPEN(4,FILE='SAFE.IN') SYSTEM=INPUT FILE FOR A COUNTRY (OR ENTERPRISE) SYSTOUT=OUTPUT FILE FOR A COUNTRY (OR ENTERPRISE) NYEARS=N. OF YEARS FOR WHICH DATA ARE AVAILABLE NSTEPS=N. OF VALUES IN [0,1] OF THE EXPON. SMOOTHING PARAMETER B READ(4,'(A25)')DUMMY READ(4,*)NSYSTEMS DO 5 ISYSTEM=1,NSYSTEMS READ(4,'(A25)')SYSTEM(ISYSTEM) READ(4,'(A25)')SYSTOUT(ISYSTEM) CONTINUE READ(4,'(A15)')DUMMY READ(4,*)NYEARS,NSTEPS FOR EACH RULE BASE ENTER N.INPUTS+OUTPUT, OUT FUZZY SETS, RULE CODING NRB=N. OF DIFFERENT RULE BASES (RBs) NVRB(I)=# VARIABLES (INPUTS+OUTPUT) OF RB I IMAXFS=LABEL OF MAXIMUM FUZZY SET FOR THE OUTPUT OF RB I IWRB(I,J)=WEIGHT ASSIGNED TO THE J-TH INPUT OF RB I ITRB(I,K)=UPPER BOUND ON WEIGHTED SUMS OF INPUTS CORRESPONDING TO THE OUTPUT FUZZY SET LABELED K NOTE: TO THE FUZZY SETS LABELED 1,2,3,... WE ASSIGN INTEGER WEIGHTS 0,1,2,..., RESPECTIVELY. NERB(I)=# EXCEPTIONAL RULES EXPRESSED EXPLICITLY IN RB I IERB(I,J)=J-TH EXCEPTIONAL RULE IN RB I READ(4,'(A15)')DUMMY READ(4,*)NRB DO 10 I=1,NRB READ(4,'(A15)')DUMMY READ(4,*)NVRB(I),IMAXFS,NERB(I) READ(4,'(A15)')DUMMY READ(4,*)(IWRB(I,J),J=1,NVRB(I)-1) READ(4,'(A15)')DUMMY

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READ(4,*)(ITRB(I,K),K=1,IMAXFS) IF (NERB(I).GT.0) THEN READ(4,'(A15)')DUMMY READ(4,*)(IERB(I,J),J=1,NERB(I)) ENDIF 10 CONTINUE C FOR EACH TERM SET READ: FUZZY SETS AND SHAPE OF MEMBERSHIP FUNCTIONS C NTERM=N. OF DIFFERENT TERM SETS (FUZZY PARTITIONS) USED IN THE MODEL C NFS(I)=# FUZZY SETS OF THE TERM SET I C Y(I,J)=PEAK VALUE OF J-TH FUZZY SET OF THE TERM SET I C NP(I,J)=N. OF CHARACTERISTIC POINTS OF THE J-TH FUZZY SET OF TERM C SET I (MAX=4 POINTS, FOR TRAPEZOIDAL MEMBERSHIP FUNCTIONS) C XP(I,J,K)=X-VALUE OF K-TH CHARACTERISTIC POINT C MP(I,J,K)=MEMBERSHIP GRADE OF X-VALUE TO THE J-TH FUZZY SET OF I READ(4,'(A15)')DUMMY READ(4,*)NTERM DO 15 I=1,NTERM READ(4,'(A15)')DUMMY READ(4,*)NFS(I) DO 17 J=1,NFS(I) READ(4,'(A15)')DUMMY READ(4,*)Y(I,J),NP(I,J) READ(4,'(A15)')DUMMY READ(4,*)(XP(I,J,K),K=1,NP(I,J)) READ(4,*)(MP(I,J,K),K=1,NP(I,J)) 17 CONTINUE 15 CONTINUE C INDICATORS, UNDESIRABLE/DESIRABLE VALUES, AND CORRESPONDING TERM SETS C NB=N. OF BASIC INDICATORS C NT=TOTAL N. OF INDICATORS (BASIC + COMPOSITE) C UMIN,UMAX(I)=MAX AND MIN UNDESIRABLE VALUES FOR BASIC INDICATOR I C TMIN,TMAX(I)=MAX AND MIN TARGET VALUES FOR BASIC INDICATOR I C ITERM(I)=TERM SET (FUZZY PARTITION) USED FOR BASIC INDICATOR I READ(4,'(A15)')DUMMY READ(4,*)NB,NT READ(4,'(A15)')DUMMY C READ INDICATOR NAME; MAX-MIN VALUES,TARGETS, AND TYPE OF FUZZY SET DO 30 I=1,NB READ(4,'(A25)')NAME(I) READ(4,*)UMIN(I),UMAX(I),TMIN(I),TMAX(I),ITERM(I) 30 CONTINUE C INFERENCE ENGINES FOR COMPOSITE INDICATORS. C ENGINE LABELED K COMPUTES THE COMPOSITE INDICATOR NB+K. C INDICATOR NB+1 CORRESPONDS TO PR(LAND), NB+2 TO ST(LAND), ETC., WHILE C INDICATOR NT CORRESPONDS TO OSUS, NT-1 TO HUMS, AND NT-2 TO ECOS. C (NT-NB)=TOTAL N. OF INFERENCE ENGINES (AND COMPOSITE INDICATORS) C IRB(K)=RULE BASE USED IN ENGINE K C IIN(K,J)=J-TH IN/OUTPUT INDICATOR OF ENGINE K (MAX IIN=NVRB(IRB(K))) C ITERM(NB+K)=TERM SET USED FOR COMPOSITE INDICATOR NB+K READ(4,'(A15)')DUMMY DO 35 K=1,NT-NB READ(4,'(A15)')DUMMY READ(4,'(A15)')NAME(NB+K) READ(4,'(A15)')DUMMY


35

153

READ(4,*)IRB(K),ITERM(NB+K) READ(4,'(A15)')DUMMY READ(4,*)(IIN(K,J),J=1,NVRB(IRB(K))-1) IIN(K,NVRB(IRB(K)))=NB+K CONTINUE CLOSE(4) RETURN END

A.3.3. Subroutine INPUT2 INPUT2 opens the input file for a particular country or organization and reads the time series for each basic indicator. If an indicator value in a specific year is not available, it is entered as −123456789. As before, each block of data in the input file is preceded by a text line with comments regarding the data that has to be typed. After reading the time series for a basic indicator of a particular country or organization, INPUT2 obtains the corresponding normalized values (see function XNORM below) and invokes subroutine EXPSM which computes a single normalized indicator value using the method of exponential smoothing. SUBROUTINE INPUT2(SYSTEM,NYEARS,NSTEPS,NAME) IMPLICIT REAL*8(A-H,M,O-Z) COMMON/BI/UMIN(100),UMAX(100),TMIN(100),TMAX(100),NT,NB COMMON/IN/X(135),M(135,9),ITERM(135),IRB(35),IIN(35,7) DIMENSION XC(50) CHARACTER*25 DUMMY,SYSTEM,NAME(135) OPEN(4,FILE=SYSTEM) C FOR EACH BASIC INDICATOR K=1,2,...,NB: C READ TIME SERIES ZC C NORMALIZE ZC TO XC C COMPUTE CRISP VALUE X BY EXPONENTIAL SMOOTHING. WRITE(8,'(3H** ,A25,22H EXPONENTIAL SMOOTHING)') SYSTEM DO 10 K=1,NB READ(4,'(A15)')DUMMY WRITE(8,'(2H* ,A25)')NAME(K) DO 15 I=1,NYEARS READ(4,*) ZC C NORMALIZE INDICATOR VALUE IF (ZC.NE.-123456789) THEN XC(I)=XNORM(ZC,UMIN(K),UMAX(K),TMIN(K),TMAX(K)) WRITE(8,'(26H ANNUAL NORMALIZED VALUE=,F10.8)')XC(I) ELSE XC(I)=ZC ENDIF 15 CONTINUE CALL EXPSM(XC,NYEARS,NSTEPS,BOPT,XOPT) X(K)=XOPT 10 CONTINUE CLOSE(4) RETURN END

154


A.3.4. Function XNORM This function computes the normalized value corresponding to the exponentially smoothed estimate (ZC) of a particular indicator, for which the target (TMIN, TMAX) and least desirable (UMIN, UMAX) values are known. It uses the method described in Section 4.3. FUNCTION XNORM(ZC,UMIN,UMAX,TMIN,TMAX) C NORMALIZATION. ONE OF THE FOLLOWING CASES MUST OCCUR: C A) LOWER IS BETTER: UMIN AND TMIN ARE NOT APPLICABLE C B) HIGHER IS BETTER: UMAX AND TMAX ARE NOT APPLICABLE C C) CLOSER TO [TMIN,TMAX] IS BETTER: ALL APPLICABLE IMPLICIT REAL*8 (A-H,O-Z) C CASE A IF ((UMIN.EQ.-123456789).AND.(TMIN.EQ.-123456789)) THEN IF ((UMAX.EQ.-123456789).OR.(TMAX.EQ.-123456789)) THEN WRITE(*,*)'INPUT ERROR: Umax, Tmax SHOULD BE AVAILABLE' STOP ENDIF IF (ZC.LE.TMAX) THEN XNORM=1 ELSEIF (ZC.GE.UMAX) THEN XNORM=0 ELSE XNORM=(ZC-UMAX)/(TMAX-UMAX) ENDIF C CASE B ELSEIF ((UMAX.EQ.-123456789).AND.(TMAX.EQ.-123456789)) THEN IF ((UMIN.EQ.-123456789).OR.(TMIN.EQ.-123456789)) THEN WRITE(*,*)'INPUT ERROR: Umin, Tmin SHOULD BE AVAILABLE' STOP ENDIF IF (ZC.GE.TMIN) THEN XNORM=1 ELSEIF (ZC.LE.UMIN) THEN XNORM=0 ELSE XNORM=(ZC-UMIN)/(TMIN-UMIN) ENDIF C CASE C ELSE IF ((UMAX.EQ.-123456789).OR.(TMAX.EQ.-123456789).OR. +(UMIN.EQ.-123456789).OR.(TMIN.EQ.-123456789)) THEN WRITE(*,*)'INPUT ERROR: Umin,Umax,Tmin,Tmax SHOULD BE AVAILABLE' STOP ENDIF IF ((ZC.GE.TMIN).AND.(ZC.LE.TMAX)) THEN XNORM=1 ELSEIF ((ZC.LE.UMIN).OR.(ZC.GE.UMAX)) THEN XNORM=0


155

ELSE IF (ZC.LT.TMIN) THEN XNORM=(ZC-UMIN)/(TMIN-UMIN) ELSEIF (ZC.GT.TMAX) THEN XNORM=(ZC-UMAX)/(TMAX-UMAX) ENDIF ENDIF ENDIF RETURN END

A.3.5. Subroutine EXPSM The subroutine shown below uses the algorithm of Section 4.4 to compute an estimate and the corresponding sum of squared errors for various values of the exponential smoothing parameter β. The normalized indicator value is the estimate with the minimum sum of squared errors. SUBROUTINE EXPSM(XC,N,NSTEPS,BOPT,XOPT) C EXPONENTIAL SMOOTHING IMPLICIT REAL*8(A-H,M,O-Z) DIMENSION XC(50) C INITIALIZE CANDIDATE SMOOTHING PARAMETER B=0; C COMPUTE SUM SQ ERRORS. B=0 SSE=0 C INITIALIZE AVAILABLE DATA; INITIALIZE ESTIMATE=0 IAVAIL=0 ESTIMATE=0 DO 5 I=1,N IF(XC(I).NE.-123456789) THEN SSE=SSE+(XC(I)-ESTIMATE)**2 C WHEN B=0 THE NEW ESTIMATE EQUALS THE MOST RECENT DATA VALUE ESTIMATE=XC(I) IAVAIL=IAVAIL+1 ENDIF 5 CONTINUE C IF DATA SERIES IS NOT AVAILABLE THEN STOP IF (IAVAIL.EQ.0) THEN XOPT=-123456789 RETURN ENDIF BOPT=0 SSEOPT=SSE XOPT=ESTIMATE C INCREASE A BY STEP AND COMPUTE SUM SQ ERRORS DO 10 J=1,NSTEPS B=J*1.D0/NSTEPS C INITIALIZE SUM SQ ERRORS=0; INITIALIZE ESTIMATE=0 SSE=0 C INITIALIZE XNUMER, XDENOM IN EQUATION: ESTIMATE=XNUMER/XDENOM

156


C INITIALLY, ESTIMATE=0/0=0, BY DEFINITION. XNUMER=0 XDENOM=0 ESTIMATE=0 DO 20 I=1,N C UPDATE SUM SQ ERRORS AND SET NEW ESTIMATE=XNUMER/XDENOM IF(XC(I).NE.-123456789) THEN SSE=SSE+(XC(I)-ESTIMATE)**2 XDENOM=1+B*XDENOM XNUMER=XC(I)+B*XNUMER ESTIMATE=XNUMER/XDENOM ELSE XDENOM=B*XDENOM XNUMER=B*XNUMER ENDIF 20 CONTINUE C UPDATE BEST SMOOTHING PARAMETER AND ESTIMATE IF(SSE.LT.SSEOPT) THEN SSEOPT=SSE BOPT=B XOPT=ESTIMATE ENDIF 10 CONTINUE WRITE(8,*)' BEST BETA=',BOPT WRITE(8,*)' ESTIMATE Xc=',XOPT RETURN END

A.3.6. Subroutine INFERENCE This subroutine uses the algorithm of Section A.1.3 to compute the membership grades of a given composite indicator to the corresponding fuzzy sets and then performs defuzzification. Inputs for which no data are available are assumed to belong to the “average” fuzzy set of the available inputs. This is the approach of linguistic averages described in the last paragraph of Section 4.6. SUBROUTINE INFERENCE(I,NB) C INFERENCE ENGINE I IS USED TO ASSESS THE COMPOSITE INDICATOR NB+I IMPLICIT REAL*8(A-H,M,O-Z) COMMON/RB/NRB,NVRB(20),IWRB(20,7),ITRB(20,9),NERB(20),IERB(20,800) COMMON/FS/XP(7,9,4),MP(7,9,4),Y(7,9),NTERM,NFS(7),NP(7,9) COMMON/IN/X(135),M(135,9),ITERM(135),IRB(35),IIN(35,7) INTEGER*4 ISAVAIL(7),JAVAIL(7),IAVAIL(7),IPOSFS(7,9),ICURFS(7), +NFSIND(7),J1(7) I1=I+NB IRB1=IRB(I) NINPUTS=NVRB(IRB1)-1 C MARK INPUTS WITH MISSING OR AVAILABLE DATA: ISAVAIL(J)=0 OR 1 NAVAIL=0 DO 5 J=1,NINPUTS INDICATOR=IIN(I,J) IF (X(INDICATOR).NE.-123456789) THEN


157

NAVAIL=NAVAIL+1 ISAVAIL(J)=1 IAVAIL(NAVAIL)=INDICATOR JAVAIL(NAVAIL)=J C FUZZY SETS TO WHICH INDICATOR BELONGS WITH POSITIVE MEMBERSHIP GRADES NPOSFS=0 DO 6 IFS=1,NFS(ITERM(INDICATOR)) IF (M(INDICATOR,IFS).GT.0) THEN NPOSFS=NPOSFS+1 IPOSFS(NAVAIL,NPOSFS)=IFS ENDIF 6 CONTINUE NFSIND(NAVAIL)=NPOSFS ELSE ISAVAIL(J)=0 ENDIF 5 CONTINUE C INITIALIZE MEMBERSHIP GRADES OF OUTPUT. DO 10 L=1,NFS(ITERM(I1)) 10 M(I1,L)=0 C IF NO INPUT IS AVAILABLE, THE OUTPUT IS ALSO NA. IF (NAVAIL.EQ.0) THEN X(I1)=-123456789 RETURN ENDIF C INITIAL COMBINATION OF FUZZY SETS FOR AVAILABLE INDICATORS C CONSISTS OF FIRST (SMALLEST) FUZZY SETS; THUS ICURFS=1. DO 12 LAV=1,NAVAIL 12 ICURFS(LAV)=1 20 ISUMRULE=0 MRULE=1.D0 SUMAVAI=0.D0 C DETERMINE THE RULE THAT APPLIES: C 1) COMPUTE FIRING STRENGTH OF RULE BY COMBINING CURRENT INPUTS. C 2) DETERMINE THE FUZZY SET OF THE OUTPUT VARIABLE BY C i) COMPUTING THE WEIGHTED SUM OF FUZZY SETS OF AVAILABLE INPUTS C ii) SETTING J1(J)=LABEL OF FUZZY SET CORRESPONDING TO J-TH INPUT; C IF INPUT J IS UNAVAILABLE, THEN J1(J)=AVERAGE FUZZY SET OF C AVAILABLE INPUTS. DO 30 LAV=1,NAVAIL INDICATOR=IAVAIL(LAV) IPFS=IPOSFS(LAV,ICURFS(LAV)) MRULE=MRULE*M(INDICATOR,IPFS) JAV=JAVAIL(LAV) J1(JAV)=IPFS ISUMRULE=ISUMRULE+IWRB(IRB1,JAV)*(IPFS-1) SUMAVAI=SUMAVAI+IPFS-1 30 CONTINUE IF (NAVAIL.LT.NINPUTS) THEN C A) FOR EXCEPTIONAL RULES: EACH UNAVAILABLE INPUT IS ASSIGNED THE C 'AVERAGE' FUZZY SET (LABELED +1) OF AVAILABLE INPUTS. IAVERAGE=IDINT(SUMAVAI/NAVAIL) C B) FOR IMPLICIT RULES (WEIGHTED SUM): THE WEIGHTS OF UNAVAILABLE C INPUTS ARE MULTIPLIED WITH THE WEIGHT OF AVERAGE FUZZY SET.

158


DO 32 J=1,NINPUTS IF (ISAVAIL(J).EQ.0) THEN J1(J)=1+IAVERAGE ISUMRULE=ISUMRULE+IWRB(IRB1,J)*IAVERAGE ENDIF 32 CONTINUE ENDIF C DETERMINE THE OUTPUT FUZZY SET L CORRESPONDING TO THE CURRENT RULE; C UPDATE MEMBERSHIP GRADE M(I1,L); CHECK EXPLICIT RULES FIRST. IF (NERB(I).GT.0) THEN C ENCODE VECTOR J1 TO FORM THE INPUT PREFIX OF AN EXPLICIT RULE INPCODE=0 DO 33 J=1,NINPUTS 33 INPCODE=INPCODE+J1(J)*10**(NINPUTS+1-J) C COMPARE DERIVED PREFIX WITH PREFIX OF EACH EXPLICIT RULE; C IF THEY MATCH, THEN ASSIGN FIRING STRENGTH TO THE OUTPUT OF RULE. DO 34 IE=1,NERB(I) ICODE=IERB(I,IE) IPREFIX=ICODE/10 IF (INPCODE.EQ.IPREFIX) THEN C DETERMINE FUZZY SET OF THE OUTPUT L=ICODE-IPREFIX*10 M(I1,L)=M(I1,L)+MRULE GOTO 40 ENDIF 34 CONTINUE ENDIF C IF PREFIX DOES NOT MATCH ANY EXPLICIT RULE, THEN USE WEIGHTED SUM DO 35 L=1,NFS(ITERM(I1)) IF (ITRB(IRB1,L).GE.ISUMRULE) THEN M(I1,L)=M(I1,L)+MRULE GOTO 40 ENDIF 35 CONTINUE C DETERMINE NEXT COMBINATION OF FUZZY SETS FOR ALL AVAILABLE INPUTS; C IF ALL COMBINATIONS HAVE BEEN EVALUATED, THEN STOP. 40 LAV=1 42 NEXTFS=ICURFS(LAV)+1 IF (NEXTFS.GT.NFSIND(LAV)) THEN ICURFS(LAV)=1 LAV=LAV+1 IF (LAV.GT.NAVAIL) GOTO 50 GOTO 42 ELSE ICURFS(LAV)=NEXTFS GOTO 20 ENDIF C ALL RELEVANT RULES HAVE FIRED; DEFUZZIFY THE OUTPUT I1 50 DEFUZ=0.D0 SUMM=0.D0 DO 55 L=1,NFS(ITERM(I1)) SUMM=SUMM+M(I1,L) DEFUZ=DEFUZ+M(I1,L)*Y(ITERM(I1),L) 55 CONTINUE


159

X(I1)=DEFUZ/SUMM RETURN END

A.3.7. Subroutine FUZZIFY The steps of the subroutine below have been discussed in Section A.2.1.

20 10

SUBROUTINE FUZZIFY(XC,IT,MG) IMPLICIT REAL*8(A-H,M,O-Z) COMMON/FS/XP(7,9,4),MP(7,9,4),Y(7,9),NTERM,NFS(7),NP(7,9) DIMENSION MG(9) DO 10 I=1,NFS(IT) IF (XC.LE.XP(IT,I,1)) THEN MG(I)=MP(IT,I,1) ELSEIF (XC.GE.XP(IT,I,NP(IT,I))) THEN MG(I)=MP(IT,I,NP(IT,I)) ELSE DO 20 K=2,NP(IT,I) IF (XC.LE.XP(IT,I,K)) THEN IF ((XC.EQ.XP(IT,I,K)).OR.(XP(IT,I,K-1).EQ.XP(IT,I,K))) THEN MG(I)=MP(IT,I,K) ELSE MG(I)=MP(IT,I,K)+(XC-XP(IT,I,K))*(MP(IT,I,K-1)-MP(IT,I,K))/ +(XP(IT,I,K-1)-XP(IT,I,K)) ENDIF GOTO 10 ENDIF CONTINUE ENDIF CONTINUE RETURN END

A.4. EXAMPLES OF INPUT AND OUTPUT FILES The SAFE program reads information about rule bases, membership functions, and indicator targets and critical values from file “SAFE.IN.” In this file, the user must also specify the number of systems (countries or organizations) to be assessed and the corresponding input and output file names. We describe below sample files for the assessment of sustainability of two companies, A and B. As usual, OSUS depends on ECOS and HUMS. There are four basic indicators: SOLID WASTE and RECYCLING RATE for ECOS, and SALES REVENUE and INJURY RATE for HUMS.

160


A.4.1. File “SAFE.IN” Successive blocks and sub-blocks of data must be preceded by commenting lines. The first rows of “SAFE.IN” contain the number of systems to be assessed and the corresponding input and output files. Here we have two companies: -CORPORATE SUSTAINABILITY: # companies; input, output files 1 A.IN A.OUT B.IN B.OUT

The next two lines of the file contain the number of years for which indicator data are available and the number of values in [0, 1] to be tested for the exponential smoothing parameter β. Here we have 8 years and 100 values: -# years for time series, # points in [0,1] for smoothing beta 8 100

We use two rule bases: one for ECOS and HUMS and one for OSUS. Thus, we have -# rule bases 2

The first rule base uses two basic inputs with fuzzy sets 0(W), 1(M), and 2(S); its output has five fuzzy sets: 0(VB) if SUM = 0, 1(B) if SUM = 1, 2(A) if SUM = 2, 3(G) if SUM = 3, and 4(VG) if SUM = 4. Each input has a weight of 1 in the SUM. We now introduce two exceptions to the above rule encoding scheme. If SUM = 0 + 2 or SUM = 2 + 0 then, although SUM = 2, the output is assigned the fuzzy set 1(B). Thus two exceptional rules must be entered explicitly: 02→1, 20→1. As explained in Section A.1.1, the fuzzy set L is labeled L + 1 and, therefore, the exceptional rules are written as 132: if Input 1 is labeled 1(0 = W) and Input 2 is labeled 3(2 = S), then Output is labeled 2(1 = B)” 312: if Input 1 is labeled 3(2 = S) and Input 2 is labeled 1(0 = W), then Output is labeled 2(1 = B).” All the above are entered in “SAFE.IN” as follows: -RB1 (HUMS, ECOS): # i/o, # output fuzzy sets, # exceptional rules 3 5 2 -weights of inputs (the output depends on weighted sum of inputs) 1 1 -upper bound on sum of weighted inputs corresponding to each output FS 0 1 2 3 4 -exceptional rules expressed explicitly 132 312


161

The second rule base combines ECOS and HUMS into OSUS. All variables have five fuzzy sets. The fuzzy set of OSUS is 0 if SUM = 0 or 1, 1 if SUM = 2 or 3, 2 if SUM = 4, 3 if SUM = 5 or 6, and 4 if SUM = 7 or 8. There are no exceptional rules. The corresponding block of data is -RB2 (OSUS=ECOS+HUMS): # i/o, # output fuzzy sets, # explicit rules 3 5 0 -weights of inputs 1 1 -upper bound on sum of weighted inputs corresponding to each output FS 1 3 4 6 8

Next, the fuzzy partitions are entered. Since we have two types of fuzzy variables we set -# fuzzy variables (term sets) 2

In this application we use term sets with 3 and 5 linguistic values, whose membership functions are shown in Figure A.1a, b. The corresponding peak values and characteristic points are shown below: - TERM SET 1 (basic indicators): # fuzzy sets 3 - Fuzzy set W: peak value, # points 0 2 - x values of fuzzy set and (on next line) corresponding 0 .6 1 0 - Fuzzy set M: peak value, # points .6 3 - x values of fuzzy set and (on next line) corresponding 0 .6 1 0 1 0 - Fuzzy set S: peak value, # points 1 2 - x values of fuzzy set and (on next line) corresponding .6 1 0 1 - TERM SET 2 (composite indicators): # fuzzy sets 5 - Fuzzy set VB: peak value, # points 0 2 - x values of fuzzy set and (on next line) corresponding 0 .25 1 0 - Fuzzy set B: peak value, # points .25 3 - x values of fuzzy set and (on next line) corresponding 0 .25 .5 0 1 0

memb.grades

memb.grades

memb.grades

memb.grades

memb.grades

162


- Fuzzy set A: peak value, # points .5 3 - x values of fuzzy set and (on next line) corresponding memb.grades .25 .5 .75 0 1 0 - Fuzzy set G: peak value, # points .75 3 - x values of fuzzy set and (on next line) corresponding memb.grades .5 .75 1 0 1 0 - Fuzzy set VG: peak value, # points 1 2 - x values of fuzzy set and (on next line) corresponding memb.grades .75 1 0 1

The last block of data contains information about the indicators used in the model. It comprises three data subsets: (a) number of indicators, (b) critical and target values and term set used for each basic indicator, and (c) rule bases and inputs used by inference engines to compute the composite indicators. If a critical value υ, U, τ, or T of a basic indicator is not applicable (see Section 4.3 on normalization), then it is assigned a dummy value of −123456789. For the particular example described above we have 4 basic and 3 composite indicators. The last block of data in file “SAFE.IN” is as follows: (a)# BASIC INDICATORS, TOTAL # INDICATORS 4 7 (b)name of basic indicator; Umin,Umax,Tmin,Tmax, and term set 1SOLID WASTE GN (kg waste/ton produced) -123456789 500 -123456789 100 1 2RECYCLING RATE (%) 50 -123456789 99 -123456789 1 3SALES REVENUE (M$/yr/employee) 0.2 -123456789 0.5 -123456789 1 4INJURY RATE (injuries/100 employees/yr) -123456789 4.235 -123456789 0 1 (c) INFERENCE ENGINES: IE1: name of output 5ECOS -rule base, term set 1 2 -inputs 1 2 IE2: name of output 6HUMS -rule base, term set 1 2 -inputs 3 4 IE3: name of output 7OSUS -rule base, term set


163

2 2 -inputs 5 6

A.4.2. Data for a Particular Company The SAFE program will open files “A.IN” and “B.IN,” as specified by the user in file “SAFE.IN.” Each of these files contains time series data for the basic indicators of the corresponding company (A or B). The contents of file “A.IN” are shown below: 1 Solid waste generated (kg waste/tons produced) -123456789 150 180 -123456789 200 -123456789 160 -123456789 2 Recycling rate (%) -123456789 -123456789 85 84 88 90 97 -123456789 3 Sales revenue per employee (M$/yr/employee) 0.62446 0.639 0.67686 0.69996 0.54589 0.54799 0.59496 0.61586 4 Total injury rate (injuries per 100 employees per year) 9.91 7.91 6.66 5.97 5.42 5.18 3.81 -123456789

A.4.3. Output files After the program is executed, the following files are created:

164


• • •

“SAFEALL.OUT” contains OSUS and its primary components ECOS and HUMS for all systems. “EXPSMTH.OUT” shows the normalized time series and the optimal estimate and smoothing parameter for each basic indicator. “A.OUT” and “B.OUT”, as specified in file “SAFE.IN,” contain detailed assessments of sustainability for each system (country or company).

For the particular set of data described in the previous two sections, file “A.OUT” is as follows (some extra spaces have been removed to fit the page width of this book): **************************************************** *SAFE: SUSTAINABILITY ASSESSMENT BY FUZZY EVALUATION * ----------------A.IN ----------------- * **************************************************** INDICATOR VALUES AND MEM.GRADES FOR A.IN 1 1SOLID WASTE GN (kg waste 0.81875 MG= 0.0000 0.4531 0.5469 2 2RECYCLING RATE (%) 0.95918 MG= 0.0000 0.1020 0.8980 3 3SALES REVENUE (M$/yr/emp 1.00000 MG= 0.0000 0.0000 1.0000 4 4INJURY RATE (injuries/10 0.10035 MG= 0.8327 0.1673 0.0000 5 5ECOS 0.86121 MG= 0.0000 0.0000 0.0462 0.4627 0.4911 6 6HUMS 0.54181 MG= 0.0000 0.0000 0.8327 0.1673 0.0000 7 7OSUS 0.76091 MG= 0.0000 0.0000 0.0385 0.8794 0.0821

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INDEX A Aβ, 13, 15, 18, 20, 21, 22, 23, 24, 25, 26, 31, 42, 50, 60, 63, 103, 112, 126 abortion, 82 absorption, 136 access, 83 accidents, 7 accountability, 113 accounting, 9, 80 accuracy, 96, 133 acute, 82, 83 adjustment, 132 administrative, 113 adult, 80, 82, 83 Africa, 2, 5, 79, 109 age, 82, 83 agent, 133 aggregates, 40 agricultural, 76, 166 agriculture, 8, 9, 69 aid, 2, 40, 107, 118, 119, 143 aiding, 129 AIDS, 82, 84, 85, 92, 167 air, 1, 2, 3, 4, 5, 6, 50, 70, 79, 89, 117, 125, 129, 130, 135, 136, 137 air emissions, 125, 129, 135, 136, 137 air pollution, 5 air quality, 50, 70 air travel, 5 aircraft, 7 air-dried, 137 Albania, 109 Algeria, 109 algorithm, 38, 99, 145, 146, 147, 149, 155, 156 alternative, 16, 20, 106, 136, 137, 143 alternatives, 165 aluminum, 127

Amazon, 2 ammonia, 5 amphibians, 89 analog, 30 Angola, 109 animals, 2, 4, 82 annual rate, 77 antecedents, 49 antibiotics, 5 application, 12, 14, 28, 38, 161, 168 applied research, 84 approximate reasoning, 34 aquatic, 4, 77 Arabia, 109 Argentina, 109 Aristotle, 12 Armenia, 109 artificial, 16 asbestos, 133 assessment, 12, 38, 50, 71, 73, 74, 75, 80, 96, 99, 107, 113, 114, 115, 117, 122, 124, 128, 129, 159, 166, 167 assets, 7, 120 assumptions, 57, 133 Athens, 6, 133, 167 Atlantic, 79 atmosphere, 4 attention, 138 Australia, 5, 95, 109 Austria, 95, 108 availability, 11, 50, 54, 96 average emission factor, 138 averaging, 139 awareness, 3, 6 Azerbaijan, 109

170

Index

B bacteria, 77 balance sheet, 82 Bangladesh, 109 bargaining, 116, 119 basic indicators sustainability of nations, 75 sustainability of organizations, 115 basic input of a hierarchical fuzzy system, 53 basketball, 13 battery, 5 beef, 5 behavior, 6 Belarus, 109 Belgium, 95, 108 bell, 60 benefits, 118, 119, 120 benign, 134 beta, 160 bias, 108 biochemical, 77 biodiversity, 2, 8, 65, 68, 70, 71 biodiversity indicators, 78 Shannon index, 68 biological, 2, 6, 69, 77, 113, 115 biological processes, 69 biologically, 78 biology, 65 birds, 88 birth, 82 births, 69, 82, 92 bivariate function, 22 blocks, 150, 160 BOD emissions, 77, 88 Bolivia, 109 borrowing, 120 Botswana, 109 bounds, 119 boys, 13 Brazil, 5, 109 breeding, 69 bribes, 80 bromide, 130 bromine, 78 buildings, 76, 115 Bulgaria, 109 Burkina Faso, 109 Burundi, 110 business, 113

C California, 8 calorie, 82, 93 calorie supply, 82, 93 Cambodia, 110 Cameroon, 110 Canada, 95, 108 cancer, 82 capacity, 2, 4, 8, 69 capital, 2, 7, 81, 83, 84, 116, 165 capital expenditure, 84 carbon, 4, 111, 117, 128, 130, 132 carbon dioxide, 4, 130, 132 Cartesian product, 26 cash crops, 69 cellular phone, 5 cellular phones, 5 central government debt, 81, 91 certainty, 16, 19 certification, 129 CH4, 5, 78, 130, 132 Chad, 6, 110 changing environment, 53 chemical, 69, 115, 138, 141 chemicals, 69, 130 chemistry, 76 child labor, 116 childbirth, 82 children, 79, 82, 83, 94, 136 Chile, 5, 109 China, 109 chlorine, 78 chlorofluorocarbons, 115 chromium, 128 chronic, 82, 83 chronic obstructive pulmonary disease, 82 citizens, 1, 3, 4, 6, 7 civil liberties, 11, 80, 90 classical, 11, 12, 20, 32 classical logic, 20, 32 classification, 130 classified, 17, 79 classroom, 83 classroom teacher, 83 classroom teachers, 83 clean air, 2 cleaning, 76 clients, 114 climate change, 1, 117 clothing, 4 clustering, 167 Co, 17, 75, 166

Index CO2, 4, 6, 9, 78, 89, 117, 122, 128, 130, 132 coal, 4, 82 Coalition for Environmentally Responsible Economies, 114 coastal areas, 8 coffee, 4 cohesion, 90 Colombia, 109 Colorado, 8 commerce, 76 commercial, 137 common rule, 34 communication, 6, 65, 84, 95, 167 communication systems, 65 communities, 116, 119 compatibility, 43, 47, 58 compatibility condition, 43, 47 compensation, 119 competitor, 135 compilation, 75 complement, 20, 23, 60, 61, 125 complement of a fuzzy set, 20 complex systems, 12, 69, 168 complexity, 3 compliance, 114 components, 11, 54, 70, 73, 75, 100, 102, 103, 104, 105, 106, 114, 148, 164 composite, 11, 26, 30, 32, 33, 38, 40, 49, 50, 53, 54, 56, 57, 67, 71, 95, 99, 100, 102, 104, 107, 143, 147, 149, 151, 156, 161, 162 composition, 11, 29, 30, 31, 36, 37, 38, 39, 40, 57, 62, 63, 166 compositional rule of inference (CRI), 36 compounds, 115, 138 compression, 49 computation, 107 computer, 84, 99, 106, 125, 144 computer software, 84 computer systems, 84 computers, 84, 95 computing, 28, 71, 144 concentration, 77, 79, 84, 88, 89, 130 conception, 19 conditioning, 4 conductivity, 77 connectivity, 77 consensus, 6 conservation, 8 construction, 12, 32, 59, 76, 167 construction and demolition, 76 consulting, 84 consumerism, 7 consumers, 6

171 consumption, 4, 6, 8, 9, 71, 76, 77, 78, 80, 81, 87, 110, 128, 134, 135, 137 consumption patterns, 6 contamination, 2 continuity, 69 contributions to communities, 119 control, 38, 40, 50, 53 controlled, 76 conversion, 81 convex, 18, 19, 20, 48 cooking, 82 copper, 2, 5 corn, 136 corporations, 114 correlation, 34, 139 correlations, 2 corruption, 80, 116 Corruption Perceptions Index CPI), 80, 90 costs, 5, 7, 116, 118, 120 Côte d’Ivoire, 109 cotton, 137 coverage, 116, 119 credit, 79 Crete, 133 crime, 7 critical value, 159, 162 Croatia, 109 crops, 69 cultural, 4, 77 currency, 81 customers, 134 cybernetics, 168 cycles, 6, 69, 128 Czech Republic, 108

D data collection, 108 data communication, 84 data processing, 84 data set, 97 database, 91, 92, 166, 167, 168 death, 82 deaths, 69, 79, 82, 89, 91, 92, 136 debt, 79, 81, 91, 120 decision makers, 4, 107, 128 decision making, 107, 166 decisions, 2, 6, 9, 15, 114 decomposition, 50, 55 defense, 79 definition, 1, 3, 9, 11, 12, 20, 35, 62, 69, 76, 79, 129, 167 deflator, 81, 91, 110

172

Index

deforestation, 1, 71 defuzzification, 39 defuzzification methods center-of-area defuzzification, 45 centroid defuzzification, 45 height defuzzification, 46 maximum defuzzification, 45 degradation, 8, 76 degree, 12, 13, 16, 19, 24, 26, 29, 30, 36, 38, 41, 48, 80, 83, 106 degree of consistency, 38 demand, 77, 81, 115 Denmark, 95, 108 density, 130 Department of Health and Human Services, 119 deposits, 81 depreciation, 7 derivatives, 81 desertification, 76, 87 detection, 2 developed countries, 94, 110 developing countries, 84, 90, 110 developing nations, 87 differential equations, 69 dignity, 83 diminishing returns, 69 diphtheria, 82 direct investment, 81, 91 direct measure, 138 discourse, 12, 14, 15, 16, 24, 29, 31, 34, 39, 40, 48, 49, 61, 62, 102 diseases, 79, 82, 89 disorder, 65 distribution, 71, 80 distribution of income, 80 diversity, 77 dividends, 81 division, 13, 131 doctors, 65, 83, 93 donor, 79 download, 86, 87, 88, 89 drugs, 6

E earnings, 119 ears, 83, 92 earth, 1, 4, 8 ecological, 3, 7, 11, 40, 70, 110, 118 ecological economics, 165, 167 economic, 2, 3, 5, 7, 8, 50, 70, 84, 113, 114, 118, 119, 120 economic activity, 2, 5

economic efficiency, 70, 119 economic growth, 7, 8 economic indicator, 7, 84 economic performance, 50, 114 economic sustainability, 8 economic welfare, 70, 113 economics, 8 economies, 4 economy, 1, 2, 6, 7, 8, 54, 72, 80, 81, 82, 83, 115, 118 ecosystem, 4, 6, 7, 68, 69, 117, 118, 165 ecosystems, 1, 2, 4, 8, 136, 165 Ecuador, 109 education, 1, 7, 8, 70, 71, 83, 84, 93, 95, 113, 167 education expenditures, 84 effluents, 115, 127, 128, 133 Egypt, 109 Einstein, 22, 23 El Salvador, 109 elderly, 79 electric conductivity, 77 electricity, 71, 134, 135, 137 electronic, 3, 5, 167, 168 embezzlement, 80 emission, 4, 131, 132, 138, 139, 140 emission estimates, 137 employees, 116, 118, 119, 120, 162, 163 employees covered by bargaining agreements and unions, 119 employment, 81, 113, 119 encoding, 143, 145, 160 endangered, 78 energy, 2, 4, 71, 79, 82, 89, 92, 110, 113, 115, 127, 129, 134, 135, 136, 137 energy consumption, 134, 135 energy supply, 79, 89 engineering, 35, 166 engines, 39, 40, 53, 54, 55, 56, 57, 75, 101, 107, 147, 162 English, 14 enrollment, 83, 94 enterprise, 82 entropy, 65, 66, 67, 68 environment, 1, 2, 3, 4, 5, 6, 7, 8, 71, 113, 118, 125, 128, 133, 136, 137 environmental, 1, 2, 3, 4, 6, 7, 8, 53, 54, 70, 71, 73, 80, 90, 96, 100, 108, 113, 114, 116, 117, 118, 127, 128, 129, 134, 136, 137, 168 environmental crisis, 1, 4 environmental degradation, 8 environmental effects, 127 environmental impact, 1, 2, 3, 71, 113, 114, 127, 128, 129, 134, 137

Index environmental law, 80, 90 Environmental Protection Agency (EPA), 117, 127, 129, 134, 138, 166 environmental regulations, 117 environmentalists, 1, 2 epidemic, 167 equality, 37, 48, 71, 80, 81, 91 equipment, 79, 84, 129, 137, 138, 139, 140, 141 equity, 81 erosion, 8 estimating, 127 Estonia, 108 ethane, 130, 139, 140 ethers, 132 ethics, 3, 116 Ethiopia, 110 Europe, 6 European, 2, 6, 86, 95 European Union, 2, 86, 95 Europeans, 6 eurostat, 5, 166 eutrophication, 5, 76, 77 exceptional fuzzy rules, 125, 145 execution, 49 expenditure on health, 93 information and communication, 84, 95 public education, 84, 95 research and development, 83, 94 expenditures, 79, 84, 116 expert, 35, 114, 168 expert systems, 168 experts, 38, 80, 84, 91 exploitation, 6 exponential, 27, 75, 76, 97, 122, 147, 153, 155, 160 exponential smoothing, 96, 97, 122 exports, 81, 91 exposure, 130 extension principle, 27 extinction, 1, 4, 78 extraction, 127, 134 eyes, 1

F false, 11, 12, 32, 33, 68 farming, 118 farms, 69 fault diagnosis, 38 fax, 5 feedback, 53, 54 feedstock, 134, 137

173 females, 69 fertilization, 137 fertilizer, 5, 76 fertilizer consumption, 76, 87 fertilizers, 69, 76 fiber, 6 fines, 80, 118, 120 Finland, 95, 108 fire, 112 fires, 41, 49, 63 firing strength of a rule, 41 fish, 2, 4, 5, 69, 71, 130 fisheries, 1, 7 flexibility, 71 flow, 69, 81, 129, 134 fluorinated, 132 food, 3, 4, 5, 6, 9, 82, 136 food production, 9 foreign direct investment, 81, 91 foreigners, 7 forest area, 76, 87 forest change, 77, 87 forestry, 78 forests, 4, 7, 77 formaldehyde, 5 Fortran, 143, 149 fossil, 2, 9, 79, 130 fossil fuel, 2, 79, 130 fossil fuels, 130 France, 95, 108 freight, 81 freshwater, 50, 99, 165 fuel, 2, 5, 76, 79, 82, 92, 115, 134, 135 funds, 83 fuzzification, 38, 39, 122 SAFE model, 99 fuzzy Cartesian product, 26 composition of a set and a relation, 30 composition of relations, 29 partition, 16, 48 complete, 48 relation, 29 fuzzy logic, 9, 12, 13, 15, 35, 61, 65, 165, 166, 167, 168 fuzzy logic, 11 fuzzy number, 15 fuzzy rule-based system. See fuzzy system fuzzy set, 12 convex (first definition), 18 convex (second definition), 20 core of, 15 height of, 15

174

Index

level set of, 20 normal, 15 peak value of, 15 subset of, 15 support of, 14 fuzzy sets, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 34, 36, 39, 41, 43, 46, 47, 48, 49, 51, 56, 60, 62, 75, 99, 100, 101, 102, 103, 106, 107, 122, 123, 126, 144, 145, 146, 147, 149, 151, 156, 160, 161, 165 fuzzy singleton, 15 fuzzy system, 38 Mamdani, 41 Takagi-Sugeno-Kang (TSK), 47

G Gabon, 109 garbage, 2, 5, 6, 129 gas, 4, 5, 71, 78, 89, 95, 110, 115, 117, 122, 132, 135, 140 gases, 2, 79, 131 Gaussian, 14, 51 GDP implicit deflator, 81, 91 gene, 39 generalizations, 39 generalized modus ponens, 35 generation, 4, 40, 76, 84, 86, 115, 118, 126, 128, 135 Georgia, 109 Germany, 95, 108 Gini index, 80, 90 Global Reporting Initiative, 114 global warming, 1, 4, 5, 9, 70, 78, 130, 131, 132 global warming potential (GWP), 78, 130, 131, 132 GNI per capita PPP, 4, 81, 91 goals, 9, 113, 133 gold, 5 goods and services, 4, 7, 81, 113 governance, 90 government, 80, 81, 83, 84, 91, 114, 116, 118, 119, 129 government budget, 83 grades, 13, 14, 31, 37, 38, 39, 41, 42, 43, 45, 46, 48, 49, 52, 55, 56, 57, 75, 100, 102, 103, 104, 106, 107, 123, 126, 145, 146, 149, 156, 161, 162 grain, 5 grants, 83 graph, 54 gravity, 45 grazing, 76

Greece, 4, 5, 6, 95, 109, 133, 167 greenhouse, 2, 5, 71, 78, 89, 95, 110, 115, 117, 122, 131 greenhouse gas, 2, 5, 71, 95, 110, 122, 131 Greenhouse gas (GHG) emissions, 78, 89, 117 greenhouse gases, 2, 131 gross domestic product (GDP), 7, 9, 71, 79, 80, 81, 83, 84, 90, 91, 94, 95, 110 gross national product (GNP), 7, 84 grouping, 133 groups, 2, 108 growth, 3, 4, 7, 8, 40, 76, 81, 87 growth rate, 40, 76, 81, 87 Guatemala, 109 guidance, 114 guidelines, 114, 143 Guinea, 109, 110

H habitat, 118 halogenated, 132 happiness, 4, 7 harm, 117 harmful, 133, 137 harvest, 69 harvesting, 2, 65, 69 hazardous waste, 76, 86, 118 health, 1, 7, 8, 70, 71, 77, 83, 93, 116, 119, 120, 127, 128, 129, 133, 168 Health and Human Services, 119 health and life insurance benefits, 120 health care, 120 health effects, 128 health expenditure, 83, 93 health insurance, 83, 119 health problems, 116 heating, 4 heavy metal, 76 heavy metals, 76 height, 12, 13, 15, 46, 47, 48, 59, 62, 63, 75, 107, 126, 149 heterogeneous, 24 hierarchical fuzzy systems, 54 high risk, 78 highways, 7 HIV/AIDS, 82, 84, 85, 92, 167 Honduras, 109 horizon, 8, 132 hospital beds, 83, 93 hospitals, 65, 83 house, 75, 80, 167 household, 7, 80, 82, 92, 119

Index households, 9, 76, 80, 82 human, 1, 2, 4, 7, 8, 11, 12, 40, 53, 54, 70, 80, 83, 100, 110, 116, 119, 120, 127, 128, 133 human actions, 4 Human Development Report, 167 human dimensions, 100 human rights, 11, 80, 116 humanitarian, 79 humans, 1, 4, 15, 71, 76, 117, 118 humus, 69 Hungary, 108 hunter-gatherers, 2 hydro, 130, 136 hydrocarbons, 130, 136 hydrogen, 130

I identification, 12, 133 identity, 37 ideology, 3 if-then rule, 33 impact assessment, 128 implementation, 106 implication, 32, 33 imports, 8, 81, 91 inclusion, 32, 108 income, 4, 7, 71, 80, 81, 94, 119, 120 income tax, 119 India, 110 indicators, 2, 7, 65, 68, 70, 71, 73, 75, 84, 86, 95, 96, 99, 100, 102, 103, 104, 106, 107, 108, 110, 111, 114, 115, 116, 117, 121, 122, 125, 126, 133, 147, 149, 151, 159, 161, 162, 163, 168 indices, 50, 75, 90, 128 indigenous, 116 indirect effect, 132 indirect measure, 120 Indonesia, 109 induction, 57 industrial, 77, 127 industry, 7, 118, 120, 129 inequality, 71, 80 infant mortality, 82, 91 infants, 82 infants immunized, 82, 92 infection, 82 inference engine, 40 composition inference, 40 individual rule firing, 40 infinite, 16 inflation, 81 Information System, 165

175 information technology, 84 infrastructure, 119 injuries, 120, 162, 163, 164 injury, 120, 163 inspection, 30, 40 institutions, 7, 76, 116 instruction, 83 instruments, 140 insurance, 81, 83, 116, 119, 120 integration, 84 integrity, 70 intelligence, 53 intensity, 16, 17, 21, 24, 32, 36, 39, 48, 54, 76, 99, 128 intentions, 9 interaction, 116 Intergovernmental Panel on Climate Change, 131, 132, 165 intermediate outputs, 56 international, 4, 6, 80, 81, 114, 116, 166 International Atomic Energy Agency, 4, 166 International Bank for Reconstruction and Development, 168 International Energy Agency, 166 international trade, 6, 80 internet, 84, 95 internet users, 84, 95 interpretation, 133 intersection of fuzzy sets, 21 interval, 12, 20, 27, 75, 85, 99 intuition, 51 inventories, 136 investment, 81, 91 Iran, 110 Ireland, 95, 109 iron, 2, 7 irrigation, 137 irritation, 79 Israel, 109 Italy, 95, 108

J Japan, 95, 109 jobs, 119 Jordan, 109 judgment, 71

K Kazakhstan, 109 Kenya, 109

176

Index

knowledge acquisition, 84 knowledge economy, 83 Knowledge Economy Index (KEI), 83, 94 Korea, 109 Kuwait, 109 Kyrgyzstan, 109

L L1, 103, 125, 143, 144 L2, 103, 125, 126, 144 labeling, 144 labor, 2, 6, 81, 91, 113, 114, 116, 119 labor force, 81, 91 land, 31, 35, 40, 54, 70, 71, 76, 77, 78, 84, 87, 110, 115, 118, 130, 136 land use, 76 landfill, 118, 130, 137 landscapes, 136 land-use, 78 language, 11, 16, 143 Laos, 109 Larsen implication, 35, 37 Latvia, 108 laundering, 136, 137 law, 11, 12, 21, 22, 23, 61, 119 law enforcement, 11 law of contradiction, 21, 61 law of the excluded middle, 12, 21, 61 laws, 2, 7, 33, 61, 80, 84, 110, 114, 116 lead, 2, 5, 68, 87, 117 learning, 116 Lebanon, 109 legislation, 6 lettuce, 5 life cycle, 115, 127, 128, 129, 130, 133, 134, 136 Life Cycle Assessment (LCA), 127–37 Impact Assessment (LCIA), 129–33 Interpretation, 133 Inventory (LCI), 129 life expectancy, 9, 82, 92 life span, 127 lifetime, 5, 132 limitations, 133 linear, 9, 47, 73, 75, 85, 86, 121, 148 linear function, 47 linguistic, 12, 15, 16, 17, 19, 38, 39, 40, 41, 42, 43, 46, 47, 48, 49, 50, 51, 54, 56, 59, 62, 73, 99, 100, 101, 102, 103, 104, 106, 122, 123, 125, 143, 144, 145, 147, 148, 156, 161, 168 linguistic average, 106, 156 linguistic rule, 12

linguistic value, 16 linguistic variable, 15 complete partition, 48 literacy, 83, 94 literature, 32, 129 Lithuania, 108 loans, 81 lobbyists, 6 local authorities, 84 London, 165, 168 long period, 2 long-term, 9 losses, 77 lost time injury rate, 120 lost workdays, 120 love, 2, 65 lung, 79, 82 lung cancer, 82 lung function, 79 Luxemburg, 95

M M1, 46 machines, 5, 16 magnetic, iv maintenance, 77 malaria, 82, 92 Malaysia, 109 Mamdani fuzzy system, 41 Mamdani implication, 34 mammal, 71 mammals, 88, 136 management, 81, 119, 168 Manganese, 128 manufacturing, 137, 138 mapping, 16, 38 market, 76, 81 market value, 81 marketing, 114 Marxists, 3 maternal, 82 maternal mortality, 82, 92 mathematical, 3, 9, 12, 38, 39, 40, 167 mathematical knowledge, vii mathematicians, 12 mathematics, 11 matrix, 28, 29, 48, 62 Mauritania, 110 Maximum Sustainable Yield (MSY), 69 max-min composition, 30 max-star composition, 30, 31 in fuzzy inference, 36

Index MC1–MC3 conditions for monotonicity, 50 in hierarchical systems, 56 measles, 92 measurement, 2, 11, 96, 138 measures, 6, 71, 76, 80, 83, 120 meat, 5 media, 1, 130 median, 92 medicine, 53, 83 Mediterranean, 165 membership, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 24, 25, 26, 27, 29, 31, 34, 35, 36, 37, 38, 39, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 55, 56, 57, 58, 59, 60, 61, 62, 63, 73, 75, 99, 100, 102, 103, 104, 106, 107, 110, 111, 112, 122, 123, 126, 145, 146, 147, 148, 149, 151, 156, 159, 161, 166 membership function, 14 membership grade, 12 memory, 143 Mendel, 35, 166 metals, 2, 6, 7, 76, 77 metals concentration, 77, 88 methane, 5, 130, 131, 132, 138, 139, 140, 141 metric, 78, 89, 117, 120, 122 Mexico, 109 microorganisms, 4, 69 military, 79 military spending, 79, 90 millennium, 82 minerals, 2, 6, 130 missions, 9, 71, 110, 111, 115, 118, 122, 129, 131, 133, 140, 167 modeling, 82, 130 models, 11, 12, 38, 69, 97, 138, 141 modifier terms, 16 modus ponens rule of inference, 34 tautology, 33 modus tollens tautology, 33 Moldova, 109 momentum, 8 money, 4, 80, 81, 118 Mongolia, 109 monotonic fuzzy system, 50 morning, 4 Morocco, 109 mortality, 82, 91, 92 mortality from respiratory diseases, 79, 89 mortality rate, 82, 91, 92 motion, 8 Mozambique, 109

177 multiplication, 49 multistage fuzzy system, 39, 53, see also hierarchical fuzzy system municipal sewage, 76 municipal waste, 40, 76, 86

N Namibia, 109 nation, 6, 50 national, 4, 7, 9, 80, 81, 82, 90, 114, 116, 122 national income, 4, 81 national product, 7 natural, 2, 4, 11, 16, 50, 65, 69, 71, 76, 77, 108, 113, 165 natural capital, 165 natural disasters, 108 natural gas, 4 Nepal, 6, 109 Netherlands, 95, 108 network, 54, 55, 56, 77, 84 New York, 165, 167, 168 New Zealand, 109 newspapers, 5 Ni, 68 Nicaragua, 109 nickel, 2 Niger, 110 Nigeria, 110 nitrates, 5 nitrogen, 5, 136 nitrous oxide, 131, 132 NO2, SO2, suspended particulates, 79, 89 nodes, 54 nongovernmental, 6 nongovernmental organization, 6 nonlinear, 27, 28 normal, 15 normalization, 73, 84, 121, 147, 162 target and critical values for countries, 85 target and critical values for organizations, 117 norms, 22, 23, 28, 34, 36, 37, 39, 41, 47, 60, 61, 62 North Atlantic, 79 North Atlantic Treaty Organization, 79 Norway, 95, 108 nuclear, 4, 76, 79, 133 nuclear energy, 4, 79 nuclear power, 76 nuclear power plant, 76 nuclear waste, 76, 86 nucleus, 15

178

Index

number of doctors, 83, 93 nutrient, 69, 70 nutrient imbalance, 70 nutrients, 76

O obese, 5 obligation, 119 obligations, 81 observations, 97 oceans, 136 oil, 4, 71, 126, 133, 140 oil spill, 133 Oman, 109 operating system, 84 operations research, 38 orbit, 8 organic, 5, 77, 115, 138, 139, 140 organic compounds, 115, 138 organic water pollutants, 77 organization, 113, 114, 149, 150, 153 organizations, 6, 11, 71, 113, 114, 115, 159 outliers, 95 overpopulation, 1 overweight, 5 oxide, 128, 131, 132 oxides, 136 oxygen, 77, 115 ozone, 8, 78, 95, 98, 131 ozone, 78, 89, 130 ozone depleting substances, 78, 89

P packaging, 5, 134, 137 pain, 95 Pakistan, 109 Panama, 109 paper, 5, 87, 127, 129, 134, 135, 136, 165, 166 Paraguay, 109 parameter, 97, 99, 107, 122, 155, 160, 164 parenting, 7 Paris, 167 particles, 5 partition, 16, 48, 50, 75 passive, 115 pathways, 129, 130 pattern recognition, 38, 53 penalties, 80, 118 pension benefits, 118 pensions, 118

per capita, 7, 29, 40, 49, 71, 76, 77, 78, 81, 82, 86, 88, 89, 90, 91, 93, 94, 95, 98, 99, 110, 111, 131 performance, 50, 84, 107, 113, 114, 120, 129 periodicity, 8 personal, 6, 84, 99 personal computers, 84, 95 perturbation, 107 pertussis, 82 Peru, 6, 109 pesticide, 5, 76, 137 pesticide consumption, 76, 87 pesticides, 69, 76 petroleum, 135 pets, 82 Philippines, 109 philosophers, 1, 12 philosophy, 3 phosphate, 76, 130 phosphorous, 69 phosphorous concentration, 77, 88 phosphorus, 5, 77 photosynthesis, 4, 6 physical environment, 6 physicians, 83 physics, 2, 65 planar, 18 planning, 9, 127 plants, 2, 4, 11, 69, 76, 77, 88, 110, 137 plastic, 5, 127, 129, 134, 135, 136 plastics, 2, 129, 136 play, 2, 6, 33, 84, 97, 113, 129 pleasure, 4 pneumonia, 82 point of origin, 140 poison, 4 Poland, 95, 96, 98, 100, 103, 108 political, 1, 2, 3, 4, 70, 80, 110, 113 political aspects, 70 political rights, 80, 90 politicians, 1, 6, 80 politics, 3, 6 pollutant, 77, 136 pollutants, 2, 11, 24, 65, 77, 134, 135, 136, 137 pollution, 1, 2, 4, 5, 8, 113, 118, 133, 136, 137, 141, 167 polyethylene, 134 polynomial, 47 polynomials, 47 poor, 6 population, 1, 4, 6, 7, 40, 69, 70, 71, 76, 77, 78, 79, 80, 82, 83, 84, 87, 88, 89, 90, 92, 93, 95, 108

Index population growth, 40, 76 population growth rate, 76, 87 population size, 82 Portugal, 95, 108 potato, 136 poverty, 80, 90, 119 poverty line, 80, 90, 119 poverty rate, 80 power, 4, 6, 76, 80, 81, 97 power plant, 76 predators, 70 preference, 135, 137 pregnancy, 82 pregnant, 82 preparation, 82 pressure, 1, 114, 140 prices, 81, 136 primary terms, 16 private, 71, 77, 80, 81, 83, 84 private education, 84 probability, 67 process control, 38 production, 9, 79, 89, 108, 110, 113, 114, 115, 118, 123, 129, 135, 137, 140 productivity, 120 profits, 113 program, 125, 143, 147, 149, 151, 159, 163 programming, 27, 28, 84, 143 programming hints, 143 property, 54, 57 proposition, 11, 12, 24, 29, 30, 32, 33, 49, 57, 58 propositional logic, 32 protected area, 77, 87, 110, 165 protection, 77, 79 protein, 82 public, 6, 53, 71, 77, 79, 80, 83, 84, 113 public affairs, 6 public education, 84 public health, 71 public health expenditure, 83 pumps, 126, 139, 140 purchasing power, 4, 81 purchasing power parity, 4, 81

Q quality of life, 119

R radiation, 78 rainforest, 69

179 random, 65, 70 range, 24, 85, 90, 102, 104, 145 ranking of countries, 108–10 ranking of indicators, 110 ratio of net school enrollment, 83, 94 students to teachers, 83, 93 raw material, 118, 127, 134, 136, 137 reading, 153 real numbers, 14 reality, 70 reasoning, 19, 32, 34, 43, 53, 56, 165, 166, 168 recall, 41 recognition, 38, 53 recreation, 9 recycling, 77, 87, 115, 118, 135, 136, 137 reduced lung function, 79 reduction, 9, 69, 78 refineries, 138 refugees, 79, 90 regional, 8, 133 regulations, 114, 116, 117, 120 rehabilitation, 83 relation, 28 relationships, 2 reliability, 128 religious, 84 renewable energy, 79, 110 renewable resource, 18, 29, 77, 79, 89, 99, 108 reprocessing, 136 reptiles, 89 research, 1, 38, 83, 90, 166 research and development, 84 resources, 1, 2, 4, 6, 8, 11, 16, 18, 29, 54, 77, 79, 80, 83, 89, 99, 108, 118, 120, 165 respiratory, 79, 82, 89, 133 returns, 69 revenue, 80, 90, 119, 120, 163 rice, 5 Rio de Janeiro, 8 risk, 78, 117, 118, 119, 120 risks, 120 rodents, 130 rods, 140 Romania, 109 room temperature, 50 rule base complete, 48 consistent, 49 increasing, 51, 56 representation and compression, 143 SAFE model, 101–6 SAFE model for organizations, 124–25

180

Index

rule explosion, 49 rules of inference in propositional logic, 33 Ruspini partition, 48 Russia, 109 Rwanda, 110

S SAFE model, 70, 73 safety, 79, 119 sales, 117, 119, 122 sales revenue, 119 salinity, 77 sample, 159 sand, 5 sanitation, 83, 93 satisfaction, 7 Saudi Arabia, 109 school, 83, 84, 94 school enrollment, 83, 94 schooling, 83, 94 science, 2, 6 scientific, 1, 2, 3, 114, 129 scientists, 6, 11 seals, 126, 139, 140 securities, 81 security, 80 Senegal, 109 sensitivity, 73, 107, 110, 133, 165 sensitivity analysis, 107, 110 sentences, 16 septic tank, 77 series, 75, 95, 96, 97, 121, 122, 147, 153, 160, 163, 164 services, 4, 7, 69, 71, 81, 84, 113, 114, 119, 127, 165 set theory, 12, 18, 21, 26 severity, 120 sewage, 76, 77, 137 shape, 18 shares, 81 shipping, 7 Siemens, 77, 88 Sierra Leone, 110 sigmoid, 14 signs, 13 sine, 2 singleton fuzzification, 38 Slovakia, 108 Slovenia, 109 smoothing, 73, 75, 97, 98, 99, 122, 147, 153, 155, 160, 164 smoothing error, 97

S-norm, 22 SO2, 84, 89, 133 social, 2, 3, 6, 7, 70, 80, 83, 90, 113, 114, 118, 119 social cohesion, 90 social costs, 7 social performance, 114 social security, 80 social welfare, 70 socially, 114 society, 1, 2, 3, 4, 7, 9, 54, 113, 114, 119 socioeconomic, 73 sociologists, 1 software, 84 soil, 1, 2, 4, 6, 8, 69, 76, 130 soil erosion, 8 solar, 4, 115 solar energy, 4 solid fuel, 82, 92 solid waste, 118, 127, 128, 129, 130, 133, 136, 137 solutions, 4 sounds, 2 South Africa, 5, 109 South Korea, 109 Soviet Union, 108 Spain, 95, 109 special education, 83 specialists, 65, 128 species, 1, 2, 3, 4, 7, 9, 68, 69, 71, 78, 115, 118, 132 specificity, 129 speech, 6 speed, 50 spills, 1 sports, 3 Sri Lanka, 109 stability, 119 stabilization, 87 stages, 53, 54, 56, 57, 114, 127, 128, 129, 134, 137 standards, 84, 130, 166 statistics, 2, 82 status of refugees, 79 steel, 5 stock, 81, 82, 84 storage, 49, 82, 84 strain, 1, 2 strategic, 114 strategies, 165 streams, 76, 140 strength, 41, 43, 47, 58, 146 stress, 99, 104

Index students, 83, 93 subjective, 11, 12, 80, 128, 131, 133 subjectivity, 13, 133 subset, 15 subsidies, 84 substances, 2, 3, 78, 89, 95, 98, 127, 131 substitutes, 7 substitution, 7, 9 Sudan, 110 suffering, 108 sulfur, 136 sulfur oxides, 136 summer, 4 supply, 4, 6, 79, 82, 89, 93, 113, 114, 136 supply chain, 113, 114 surface area, 77, 87 sustainability, 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 16, 24, 31, 32, 36, 40, 47, 50, 54, 65, 69, 70, 71, 73, 74, 75, 76, 77, 79, 81, 84, 85, 96, 99, 100, 102, 107, 110, 113, 114, 115, 117, 118, 119, 120, 122, 124, 126, 143, 148, 149, 159, 164, 166, 167 sustainability assessment by fuzzy evaluation (SAFE), 70 difficulties of achieving, 4 indicators for countries, 75–84 indicators for organizations, 115–26 indicators of, 65–71 of countries, 108–10 of organizations, 114–26 strong, 8 weak, 8 what is, 6 who is interested in, 1 sustainable development, 2, 3, 7, 8, 73, 104, 106, 107, 111, 165 swallowing, 136 Sweden, 95, 108 Switzerland, 108, 166 symbolic, 119 symbols, 32 symmetry, 22, 23 syntactic, 16 synthetic, 7, 138 Syria, 109 systematic, 11, 84, 96, 127 systems, 2, 6, 7, 12, 16, 38, 39, 40, 41, 49, 50, 51, 53, 54, 56, 57, 59, 65, 69, 84, 127, 159, 160, 164, 166, 167, 168

T Tajikistan, 109

181 tanks, 77 Tanzania, 109 tar, 165 targets, 84, 114, 149, 159 tautologies, 33 implication, 33 modus ponens, 33 modus tollens, 33 tax rates, 119 tax revenue, 80, 90 taxes, 117, 119 taxes paid to government, 119 Tc, 85, 111 teachers, 83 teaching, 83, 93 technology, 2, 84 telecommunications, 65 temperature, 9, 50 temporary protection, 79 term set, 16 tetanus, 82 Thailand, 109 theoretical, 16 theory, 12, 16, 18, 21, 26, 53, 65, 167 thermodynamics, 65 threatened, 4 threatened bird, mammal, plant, fish, amphibian, and reptile species, 78, 88 threshold, 85, 95, 117, 118, 120, 146 thresholds, 91, 151 time, 1, 2, 4, 7, 8, 9, 69, 75, 79, 82, 95, 96, 97, 113, 120, 121, 122, 128, 132, 133, 147, 153, 160, 163, 164 time consuming, 128 time series, 75, 95, 96, 97, 121, 122, 147, 153, 160, 163, 164 T-norm, 22 Togo, 110 top-down, 11 total energy, 82, 92 Total Factor Productivity (TFP), 68 total injury rate, 120 total water withdrawals, 77 tourism, 9 toxic, 1, 3, 6, 117, 118 toxic releases, 117 toxic substances, 3 toys, 3 trade, 6, 49, 56, 76 trade-off, 49, 56 training, 116 trajectory, 9 transfer, 56

182

Index

transformation, 2, 57 transformations, 63 transition, 18 transparency, 90, 113, 167 transportation, 5, 6, 9, 71, 134, 135, 137 travel, 5 trees, 76 trend, 10, 114 trial, 14, 38 trial and error, 14, 38 triangular conorm. See S-norm triangular norm. See T-norm tropical forest, 5, 69 trout, 5 true/false, 33 truth table, 32 TSK fuzzy system, 47 tuberculosis, 82, 92 Tunisia, 109 Turkey, 109

U U.S. dollar, 81, 120 ubiquitous, 136 Uganda, 109 Ukraine, 109 ultraviolet, 78 uncertainty, 1, 9, 65, 66, 96, 168 unemployment, 81, 91, 119 unemployment rate, 81 uniform, 56 union of fuzzy sets, 21 unions, 22, 47, 48, 119 United Arab Emirates, 109 United Kingdom, 95 United Nations, 75, 79, 82, 84, 92, 108, 165, 167, 168 United Nations Development Program (UNDP), 75, 167 United Nations High Commissioner for Refugees (UNHCR), 79 United States, 81 universe, 12, 14, 15, 16, 24, 29, 31, 39, 48, 49, 61, 62, 102 universe of discourse, 12 university education, 83 Uruguay, 109 users, 84, 95 Uzbekistan, 109

V validity, 29, 30, 33 values, 3, 4, 6, 7, 9, 11, 12, 13, 15, 16, 19, 27, 29, 31, 32, 38, 39, 40, 41, 43, 45, 46, 47, 48, 49, 50, 51, 56, 59, 62, 73, 75, 79, 84, 85, 86, 87, 90, 91, 92, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 106, 107, 111, 112, 117, 120, 121, 122, 123, 124, 125, 126, 128, 129, 131, 132, 133, 143, 144, 145, 147, 148, 149, 151, 153, 154, 155, 159, 160, 161, 162 vapor, 140, 141 variable, 15, 16, 17, 26, 38, 46, 47, 48, 53, 54, 56, 57, 81, 87, 100, 143, 147, 151, 168 variables, 11, 15, 16, 24, 26, 29, 38, 39, 40, 42, 50, 53, 54, 55, 75, 104, 145, 147, 148, 161 variation, 9 vector, 18, 26, 28, 38, 47 vehicles, 3 Venezuela, 109 Vietnam, 109 voice, 84 volunteer work, 7 voters, 2 voting, 82

W war, 108 Washington, 165, 166, 167, 168 waste, 6, 40, 49, 76, 77, 84, 86, 115, 118, 126, 127, 128, 130, 133, 135, 136, 137, 162, 163, 164 waste water, 77, 137 wastes, 128, 129, 137 wastewater, 11, 77, 88, 110, 135 wastewater treatment, 11, 77, 88, 110, 135 water, 1, 2, 3, 4, 5, 6, 8, 11, 16, 17, 18, 21, 24, 29, 32, 36, 39, 54, 70, 76, 77, 83, 88, 93, 99, 104, 118, 129, 130, 136, 137, 140, 141, 167 water quality, 11, 70 water recycling, 137 water resources, 8, 11, 16, 29, 77, 118 wwater use, 118 water vapor, 140, 141 water withdrawals, 88 wealth, 81, 119 wear, 5 web, 84 welfare, 70, 113 well-being, 8, 118 wildlife, 76, 136

Index wind, 2 wind turbines, 2 winter, 4 wireless, 84 withdrawal, 29 women, 4, 6 wood, 6, 7 workers, 118, 119 World Bank, 75, 83, 87, 88, 90, 91, 92, 93, 94, 167, 168 World Health Organization (WHO), 75, 82, 92, 167, 168

183

Y years of schooling, 83, 94 Yemen, 110 yield, 97

Z Zimbabwe, 109 zinc, 5

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