Transportation Networks: Recent Methodological Advances

TRANSPORTATION NETWORKS: RECENT METHODOLOGICAL ADVANCES SELECTED PROCEEDINGS OF THE 4TH EURO TRANSPORTATION MEETING

Related Pergamon books

DAGANZO Fundamentals of Transportation and Traffic Operations ETTEMA & TIMMERMANS Activity Based Approaches to Travel Analysis GARLING, LAITILA & WESTIN Theoretical Foundations of Travel Choice Modeling STOPHER & LEE-GOSSELIN Understanding Travel Behaviour in an Era of Change Related Pergamon journals—sample copies available on request

TRANSPORTATION RESEARCH PART A: POLICY AND PRACTICE Editor. Frank A. Haight TRANSPORTATION RESEARCH PART B: METHODOLOGICAL Editor. Frank A. Haight

TRANSPORTATION NETWORKS: RECENT METHODOLOGICAL ADVANCES SELECTED PROCEEDINGS OF THE 4TH EURO TRANSPORTATION MEETING

edited by: MICHAEL G.H. BELL

1998 Pergamon An imprint of Elsevier Science Amsterdam - Lausanne - New York - Oxford - Shannon - Singapore - Tokyo

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Contents 1. Introduction MICHAEL G. H. BELL AND RICHARD E. ALLSOP 2. Descent Methods of Calculating Locally Optimal Signal Controls and Prices in Multi-Modal and Dynamic Transportation Networks MIKE SMITH, YANLING XIANG AND ROBERT YARROW 3. Analysis of Traffic Models for Dynamic Equilibrium Traffic Assignment B. G. HEYDECKER AND J. D. ADDISON

1

9 35

4. An Efficient Algorithm for the Continuous Network Loading Problem: A DYNALOAD Implementation Y. W. Xu, J. H. WUANDM. FLORIAN

51

5. Considering Travelers' Risk-Taking Behavior in Dynamic Traffic Assignment DAVID E. BOYCE, BIN RAN AND IRENE Y. Li

67

6. Price-Directive Traffic Management: Applications of Side Constrained Traffic Equilibrium Models TORBJORN LARSSON AND MICHAEL PATRIKSSON

83

7. Improved Algorithms for Calibrating Gravity Models NANNE J. VAN ZIJPP AND BENJAMIN G. HEYDECKER 8. The Stability of Stochastic User Equilibrium with a Given Set of Route Information

99

115

TOSHIHIKO MlYAGI

9. Traffic Equilibrium in a Dynamic Gravity Model and a Dynamic Trip Assignment Model XIAOYAN ZHANG AND DAVID JARRETT 10. Foundations of a Theory of Disequilibrium Network Design TERRY L. FRIESZ AND SAMIR SHAH

129 143

11. The Continuous Equilibrium Optimal Network Design Problem: A Genetic Approach N. D. CREE, M. J. MAKER AND B. PAECHTER

163

12. Continuous Equilibrium Network Design Problem with Elastic Demand: Derivative-Free Solution Methods HAI-JUN HUANG AND MICHAEL G. H. BELL

175

vi

Contents

13. A Column Generation Approach to Bus Driver Scheduling SARAH FORES, LES PROLL AND ANTHONY WREN 14. Stochastic Network Models and Solution Methods for Dynamic Fleet Management Problems RAYMOND K. CHEUNG

195

209

15. Periodic Shipping Strategies for the Minimization of the Logistic Costs LUCA BERTAZZI AND MARIA GRAZIA SPERANZA

223

16. An Algorithm for the Combined Distribution and Assignment Model

239

JAN T. LUNDGREN AND MlCHAEL PATRIKSSON

17. Assessing the Performance of Artificial Neural Network Incident Detection Models HUSSEIN DIA AND GEOFF ROSE

255

18. Reliability Measures of an Origin and Destination Pair in a Deteriorated Road Network with Variable Flows YASUO ASAKURA

273

19. Variational Inequality Model of Ideal Dynamic User-Optimal Route Choice DAVID E. BOYCE, DER-HORNG LEE AND BRUCE N. JANSON

289

20. Travel Times Computation for Dynamic Assignment Modelling C. BUISSON, J. P. LEBACQUE AND J. B. LESORT

303

Author index

319

INTRODUCTION Michael G H Bell Transport Operations Research Group University of Newcastle and

Richard E A llsop University of London Centre for Transport Studies University College London

THE EVENT From the 9th to the 11th September 1996 around 70 participants, largely academics but also including consultants and civil servants, gathered at the University of Newcastle for the 4th Meeting of the EURO (Association of European Operational Research Societies) Working Group on Transportation. They were brought together by a shared interest in the application of operational research techniques for solving transport problems. The large number of papers presented (58 appeared in the programme) and the lively debate confirmed the strength of interest in the field. The major themes of the conference were: 1. 2. 3. 4. 5. 6. 7. 8.

Traffic monitoring, management and control (13 papers) Static traffic assignment (10 papers) OD estimation and reliability (8 papers) Routing and scheduling (8 papers) Dynamic traffic assignment (8 papers) Traffic modeling and simulation (5 papers) Network design (3 papers) Planning and other (3 papers)

Traffic assignment continues to be a popular topic, reflecting its importance in practice. Somewhat surprising perhaps is the enduring interest in static assignment, despite is practical limitations.

THEORY AND PRACTICE Operational Research (OR), despite its partly military origins, has the capacity to help society to achieve tremendous improvements in human welfare. For example, the Soviet mathematician and economist Kantorovich formulated and solved a linear programming problem in 1939 and the Simplex Method of linear programming, usually attributed to Danzig in 1949 who was then working as a Mathematical Advisor to the United States Air Force, ranks among the most practically significant mathematical advances of the mid-twentieth century (see Hillier and Lieberman, 1990). 1

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M. G. H. Bell and R. E. Allsop

Linear programming has played a prominent role in transport, indeed one of its basic forms is known in the literature as the Transportation Problem. Any OR method is only as good as the practical benefits it yields, which in the case of the Simplex Method has been enormous. A good OR method, therefore, finds a good solution to an important problem, a theme that will be returned to repeatedly in this chapter. Much of operational research is concerned with finding the best solution to some problem. However, there are many problems where this is not practical on grounds of computational complexity, for example the well-known Traveling Salesman Problem. In such cases, efficient procedures for finding a good solution (referred to as heuristics), are more useful. Clarke and Wright's simple but elegant Savings Method, which finds a good solution to the Traveling Salesman Problem, has proved to be of great practical use to freight industry and has doubtless resulted in huge savings. Much of the success of the Savings Method can be attributed to the ease with which additional constraints can be embedded in it. This points to the fact that real problems rarely correspond exactly to Standard Problems, like the Traveling Salesman Problem, the Chinese Postman Problem or the classical Vehicle Routing Problem. Hence the greater the ability of an OR method to accommodate the diversity of the real world the greater its usefulness in practice. The corollary to the point that good OR methods solve real problems is that good OR is problem driven. This point is sometimes obscured in the academic arena, where survival may depend on publishing hi international refereed journals. Peer review is not end user review. The problem is at its worst where academics, driven by a fascination with some OR technique combined with a need to publish, build models, sometimes described as "extensions" or "enhancements" of earlier models, which are of little or no practical use. These models are then "tested" on small examples bearing only a superficial resemblance to a real problem, where the issue here is not the smallness of the example but rather its artificiality. In summary, theory and practice should go hand in hand. There is a Darwinian process whereby the fittest OR methods will survive in practice. It is the task of conferences, such as the 4th EURO Working Group on Transportation, to present new OR methods along with evidence about the fitness of competing OR methods.

SOUND MATHEMATICS Underlying operational research hi transport (or any other field for that matter) is a mathematical model, consisting of a system of equations quantifying the relationships between the key variables. The systems approach represented by operational research contrasts to more qualitative approaches adopted by other disciplines which also have important contributions to make in the transport field, whereby the analysis is based on logic involving qualities. Qualitative conclusions, like "more compact cities are more environmentally sustainable", have, where they are valid, useful policy implications. Indeed policy makers require a qualitative comprehension of the field. They need to understand, to pursue the same example further, what the causes of greater environmental sustainability are. However, hi seeking to justify qualitative conclusions it is often helpful if someone can quantify what is meant by concepts such as "compactness" and "greater environmentally sustainability", collect

Introduction some data, and demonstrate an association. Of course, a statistical association by itself is no proof of causation, irrespective of the amount of data collected, so a mathematical model based on a verifiable mechanism is required after all to show, for example, how "compactness" might lead to "greater environmental sustainability". Without mathematical models to quantify and lend credence to qualitative conclusions, there is a danger that policy is overtaken by ill-informed dogma, like '^privatisation will save money". Constructing a good mathematical model is as much an art as a science. A good mathematical model is a thing of beauty, being both simple and elegant. It not only provides useful results, it also furthers understanding. By facilitating a "what if' analysis of the problem, the relationships between the key variables can be explored. It is also inspirational. Consider, for example, the impact that Entropy Maximisation has had on transportation planning theory. The relevance of statistical mechanics to the modelling of trip distribution was first demonstrated by Cohen (1961), but it was the contribution of Wilson (1967) that stimulated the prodigious volume of published papers that continues, somewhat abated, to this day. However, there is a danger that the model takes over and that research becomes technique rather than problem driven. While it is right that all the possible avenues of application should be explored, there is a tendency (commented on earlier) for academics, driven by a need to publish, to explore avenues that should be recognised sooner to be blind. An interesting interface between the quantitative and qualitative approaches is represented by Fuzzy Logic. Quantities are related to qualities by fuzzy sets and membership functions, logical relations are defined between these qualities, and then a decision that is of interest is quantified. As an example, the flow of vehicles approaching a traffic signal is measured by a vehicle detector. The measurements is then mapped to levels of belief in fuzzy sets like "high flow" and 'low flow" by membership functions. The logical relationship is then "//flow is high then extend green else end green". The ultimate decision to end or extend the green signal will then depend on the levels of belief regarding the magnitude of flow. Fuzzy logic has been successfully applied in many control applications, and its potential for traffic signal control and traffic simulation is gradually being realised. A paper by Pursula and Niittymaki evaluating the use of fiizzy logic for signal control (not in these selected proceedings) was presented at the Meeting.

OPTIMISATION Optimisation can be described as making something as good as possible. Every word in this description is significant. In transport applications, the something is a representation of some part of the transport system and the social, economic and physical world served by the system. To make the representation mathematically tractable, we have to simplify and formalise reality in a way that reflects adequately those essential characteristics that must be reflected if the mathematical results we obtain are to address the real problem effectively. To make this possible, we must set boundaries to our representation that match up with areas of operational control or planning responsibility or design remit that make sense to those who are handling the real problems. Making implies influence, and our application of optimisation must recognise correctly which aspects of the real problem are open to change and which should realistically be regarded as fixed. As good implies one or more criteria of goodness, represented by an objective function - or two or more criterion functions with explicit tradeoffs between them - and these again should match the objectives and criteria that are perceived hi practice by those who are handling the real problems. As possible recognises that there are in real life many limitations to what can be done to address a typical problem - some of them physical or engineering-based, others attitudinal, perceptual, behavioural or

M. G. H. Bell and R. E. Allsop political, and all needing to be expressed appropriately as mathematical constraints. In all these respects, the key to effective application of optimisation is good mathematical judgement combined with good two-way communication with those responsible for the real-life problem being addressed. Much transport related OR work concerns optimisation. This may be system optimisation, whereby best system design variables are sought, or user optimisation, whereby the results of rational user behavior are sought. The determination of best traffic signal timings for given traffic flows is an example of system optimisation, while the assignment of traffic flows to networks so that only minimum cost paths are used is an example of user optimisation. A bilevel problem arises where a best system design is sought subject to users reacting rationally. A well-known example of a bilevel programming problem, which goes back to Allsop (1974) and Gartner (1976), is the optimisation of traffic signal tunings for a network where drivers choose minimum cost routes. The literature on the solution of convex optimisation problems is voluminous and in parts intensely mathematical. The difficulty is that many real optimisation problems are not convex, particularly where they have a bilevel structure. This has the consequence that there may be many local solutions, so a convex programming method will at best lead to a local optimum. There has been an unfortunate tendency for academics schooled in convex optimisation to cling to the methods they know best, rather than to look at methods suited to non-convex problems. Fortunately, new probabilistic search techniques, like Genetic Algorithms, are to hand. While they have quite a different character to convex optimisation procedures, and are perhaps better described as heuristics (since there is a residual chance that the global optimum is missed), they do appear to be adaptable to the diversity of real problems. While many of the Standard Problems of OR have one objective function, many real problems may have multiple objectives. An example of this is traffic signal control, where attention has recently turned to having different objectives for different parts of the network (for example, vehicle capacity for major roads and environmental emissions for minor roads) or for different road user groups (for example, delay for public transport and safety for pedestrians and cyclists). In this context, Fuzzy Logic is proving to be a very useful OR method for control problems. Although it does not offer optimisation as such, it does permit optimal control rules to be implemented on the basis of incomplete and imperfect data.

SIMULATION The exponential growth in computing power has stimulated the use of simulation as a method of analysis. Nowhere is this more evident than in the research surrounding the development of Intelligent Transport Systems (ITS). While simulation will always play an important part in transport research, there are numerous pitfalls: 1. Complex models may obscure their assumptions. There is a risk that a simulation aimed at demonstrating some effect may have that effect embedded in the assumptions, but that because of the complexity of the model this is not obvious. Moreover, the relationship between the key variables may be obscured by model complexity. 2. Complex models compound error. As the number of relationships included hi a simulation increases, so does the scope for error. The use of microscopic traffic simulation to study ITS is an example of a field where many relationships are frequently combined (for example, departure time choice, destination choice, mode choice, route choice, all affected in various ways by transport

Introduction telematics). The behavioral assumptions are therefore numerous leading to the compounding of errors. 3. Complex models have an increased chance of chaotic (and therefore unexplainable) behavior. Even a few rather simple behavioral relationships can produce results that are not repeatable and therefore appear to be chaotic. While chaotic behavior may be a feature of some processes, a belief that the world is governed by chaos would be a counsel of despair. Prediction would be impossible and rational behavior would be pointless. 4. Complex models may yield results that are difficult to interpret, and therefore are difficult to translate into policy. When many factors are at work, causation is difficult to establish, and without causation policy is difficult to formulate. This comment also applies to examples chosen for demonstration or analysis. While examples should always resemble real problems, they should be small enough to establish causation. For the above reasons, simulation is no substitute for sound mathematics. While simulation can demonstrate that certain models are associated with certain effects, mathematical analysis may be able to prove that a certain model implies certain effects, and moreover show why.

EQUILIBRIUM OR EQUILIBRATION? One basic assumption of many mathematical models used in transport related OR work, and indeed underlying many of the papers included in this volume, is one of equilibrium. In the case of a deterministic equilibrium, this supposes that a system finds itself in an essentially tuneless state where system users are either unable or unwilling to change their behavior. This has the great attraction that it is not necessary to consider how or indeed when decisions are taken. "While a transport system may never actually be in a state of equilibrium, it is assumed that it is at least near equilibrium, tending towards equilibrium, and only prevented from attaining equilibrium by changes hi external factors ... At equilibrium, the transport system reduces to a fixed point (the equilibrium costs and flows), and powerful analytical techniques ... exist for finding the fixed point. Proponents of equilibrium theory take it as a matter of faith that, given the existence of an equilibrium, there are behavioral mechanisms that push the transport system to this fixed point." (Bell and lida, 1997) The concept of equilibrium generalises to stochastic processes, where the fixed point now refers mean costs and flows. System users are still unable or unwilling to change their behavior, but thenknowledge of system costs are imperfect. Recently, equilibrium theory has come under criticism on the grounds that: "... we are not hi such a state [of equilibrium] now, there is no guarantee that the system is moving towards it, we will ever arrive there, and even if we did we would not stay there for long. For all its importance hi the history of economic thought, the concept of equilibrium now acts as a barrier to understanding how things are changing, and how they might change." (Goodwin, 1998) The point that equilibrium models, whether of the deterministic or stochastic variety, may form a barrier to the analysis of the consequences of change is a powerful one. In general, real problems are about making changes to systems to improve them hi some way. However, adjustments to change

6

M. G. H. Bell and R. E. Allsop

generally do not occur instantaneously, and require a consideration of reaction times, inertia and habit. Long-term elasticities generally differ from short-term elasticities. What this means is that the concept of equilibrium and techniques for finding equilibria need to be supplemented by understanding, modeling and computation of the more complex process of equilibration, bearing in mind that whilst in some situations, perhaps mainly those of more operational interest, the process may be well-advanced, in others, probably including those of greatest concern to policy-makers, there is time only for the early stages of equilibration to take place before the ground rules change to shift the eventual but actually unattainable equilibrium state itself, and thus redirect the process of equilibration.

STATIC PROBLEMS OR DYNAMIC PROBLEMS? The time dimension can of course be captured explicitly in the mathematical model without relaxing the equilibrium assumption. However, in the context of dynamic models, the concept of equilibrium becomes ambiguous. In fact, many definitions are possible, all retaining the essential feature that adaptation to changes is instantaneous and therefore all system users are always either unwilling or unable to change their behavior. In the field of dynamic traffic assignment, much work has gone into modeling rational route choice behavior where choices are based on, for example, (a) perfect foresight regarding network costs, (b) perfect knowledge of current network costs, (c) perfect knowledge of past network costs, or (c) imperfect knowledge of current network costs. In reality, system users may be willing and able to change their behavior but have just not got round to doing so. The point made forcefully by Goodwin (1998) is that an understanding of system dynamics is not possible without an understanding of the leads and lags in adaptation processes. Thus while equilibrium theory can find the states to which a system would tend under certain hypothetical conditions and which can give useful pointers to real-world behavior, for a better explanation of the workings of the real world there is a need also to understand adaptation processes.

FUTURE PERPSECTTVES In the light of the preceding comments, future work will probably need to move away from the equilibrium theory which has underlain much transport-related OR work to date, as evidenced by the papers in this volume. Greater attention needs to be given to adaptation processes, and the effect that these have on system dynamics. An interesting example of the importance of adaptation processes is provided by the traffic signal optimisation problem referred to earlier, where route choice responds to signal timings If the traffic signals are optimised with respect to traffic flows, and then users respond to the signals giving new traffic flows, both ad infinitum, the system may never converge and if it does it may converge to a stationary point that is not even a local optimum to the problem of optimising traffic signal timings subject to rational choice of routes by users, as is shown in Fig. 1. An alternative form of adaptation would be the Stackelberg game, where the signals are optimised taking the route choice reactions into account, which may well lead to a better set of signal tunings (this is certainly the case in Fig. 1). In reality, there are infinitely many games that can be played, and the most appropriate depends on the objective of the analysis. Is it to explain system dynamics as observed, or is it to design optimal

Introduction

signal settings, and if the latter, under what assumptions about user behavior and how this can be influenced? This is but one example of the range, depth and fascination of the scope for further OR in the transport field.

Figure 1: The path to mutual consistency

REFERENCES ALLSOP R E (1974) Some possibilities for using traffic control to influence trip distribution and route choice. Proceedings of the 6th International Symposium on Transportation and Traffic Theory (Ed. D. J Buckley), Elsevier, New York. BELL M G H and IIDA Y (1997) Transportation Network Analysis , Chichester: Wiley. COHEN M H (1961) The relative distribution of households and places of work. In Theory of Traffic Flow - Proceedings of the Symposium held at the General Motors Research Laboratories, Warren, Michigan, December 1959, ed. R Herman, 79-84. Amsterdam: Elsevier. GARTNER N H (1976) Area traffic control and network equilibrium. In: Lecture Notes in Economics and Mathematical Systems, Vol. 118 (Ed. M.A. Florian), Berlin: Springer-Verlag,274297. GOODWIN P B (1998) The end of equilibrium, in Garling T (Ed.) The Theoretical Foundations of Travel Choice Modeling, Elsevier, Oxford (In Press). fflLLffiR F S and LEEBERMAN G J (1990) Introduction to Operations Research (5th Edition) Singapore: McGraw-Hill International. WILSON A G (1967) Entropy maximising models hi the theory of trip distribution, mode split and route split. Journal of Transportation Economics and Policy, Vol. 3, 108-126.

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DESCENT METHODS OF CALCULATING LOCALLY OPTIMAL SIGNAL CONTROLS AND PRICES IN MULTI-MODAL AND DYNAMIC TRANSPORTATION NETWORKS Mike Smith, YanEng Xiang, Robert Yarrow Yori Network Control Group Department of Mathematics University of York York Y01 5DD United Kingdom

Abstract A bilevel descent method of optimising signals and prices for a multi-modal network while taking account of travellers' choices (equilibrium) is specified within a framework which may, when developed, be efficient for large networks. Similar trilevel methods for the corresponding dynamic problem are also presented. Fairly complete proofs of convergence of the method to a local optimum are given for the steady state case but there remains a gap when we seek to prove convergence in a dynamic context; however if the method converges (in a dynamic context) to the set of equilibria then (under natural conditions) it must also converge to the set of local optima.

INTRODUCTION This paper proposes bilevel and trilevel descent methods of calculating signal timings and prices which seek to take correct account of drivers' route-choices (and travellers' modechoices), in the sense that travellers are supposed to choose routes and modes which cause the overall traffic distribution to be in equilibrium as specified essentially by Wardrop (1952).

10

M. Smith et al.

The methods are designed to produce locally optimal signal timings and road prices in steady state and dynamic models. One of the methods described here relies in a fundamental manner on Lyapunov techniques similar to those demonstrated in Smith (1984a, b, c)-

As is usual we are content to find flow, green-time, price triples which satisfy a KarushKuhn-Tucker type of necessary condition. It is important to note that the procedures outlined here may only be expected to reach a local optimum since the optimal control problems are non-convex. Allsop (1974), Maher and Akcelik (1974) and Allsop and Charlesworth (1977) were the first to consider using traffic signal timings to influence the distribution of travel. More recent work includes papers by Tobin and Friesz (1988), Friesz et al. (1990), Heydecker and Khoo (1990), Yang and Yagar (1995), Yang (1997), Smith et al. (1997) and most recently Chiou (1997). More references are contained in Smith et al. (1996). It is interesting to compare the approaches adopted by Chiou and Smith et al. Chiou begins with a nearly standard traffic control model, based on TRANSYT, and then allows variations in routeing; while Smith et al. begin with a nearly standard traffic assignment model and then allow variations in control parameters. Chiou adds assignment to a control model; Smith et al. add control to an assignment model.

A NATURAL DIRECTION FOR BILEVEL PROBLEMS Here we suppose that a transport problem has flows, costs and delays all given by a vector z and signal green times and road prices all given by a vector A. We suppose given a function £"(2, A), which is a measure of disequilibrium, and a function Z(z, A) which gives the value of an objective or criterion function. Here we give a direction for [z, A] which if followed, and all goes well, reduces E to 0 (or to a small value), so that equilibrium is approached, while taking care that Z is either reduced or caused to increase at a minimum rate. This direction seems natural and is at the heart of the bilevel method described here. However this direction suggests more complicated multilevel directions and these may well be more efficient. The vector x = [z, A] comprises both z and A ([2, A] denotes a column vector). For clarity, the natural feasibility constraints are introduced explicitly later in terms of z, A; and then negative gradients are to be projected onto or toward the feasible region. Here we suppose that there are no active constraints, that constraints are embodied in E(z, A) = 0, or that negative gradients are all appropriately projected onto the set of those x satisfying the

Descent methods of calculating optimal signal controls and prices

11

constraints. The method motivating our natural direction is, in outline, as follows. Given two smooth (Ci) objective functions E and Z follow a direction which reduces E and also does the best possible in reducing Z, or diminishing the increase in Z, subject to the over-riding necessity to reduce E. In this way we may expect to minimise Z subject to the over-riding requirement of minimising E. In our transport context, first a smooth objective function E is defined which specifies the extent to which all the variables (including the controls) depart from equilibrium. E > 0 and E = 0 if and only if we are at an equilibrium. This may be the same objective as that in Smith (1984). Assume now that, away from equilibrium, the gradient grad E of this function does not vanish and so defines at each non-equilibrium point x the half-space {x + $; 8- (grad E) < 0} of locally non-increasing E. This is a basic assumption throughout this section. The method is, essentially to follow, at each point, the direction

where Projlgrad E(—grad Z) is —(the gradient of Z) projected onto the half-space {8; (grad E)S < 0} to which grad E is normal. Following this direction reduces E and also seeks to reduce Z only by moving sideways or forwards relative to - grad E. The direction takes account of the bilevel nature of the problem since reducing E always gets priority. Steepest descent methods are not usually efficient, at least close to a solution. So it is envisaged that direction (1) will be generalised in further work to

where SE and 8z are more efficient descent directions for E and Z. Corresponding changes will be made in the directions derived below from (1). To introduce continuity, let e be positive and revise the direction (1) to

It proves beneficial to define desc E to be the direction of steepest descent of E. ~"r i- , * * ' Hgrad £||' to define desc Z similarly and to change direction (2) to (putting x in explicitly)

12

M. Smith et al.

where desc(Z, E)(x) is the steepest descent direction desc Z(x) projected onto the halfspace denned by desc E(x). This vector (3) is independent of the units used to measure E and Z provided E and e are in the same units. The point x will be called on e-Karush-Kuhn- Tucker point or a weakly locally e-optimal point if Se(x) is the zero vector. Such points have E < e and have no Z-descent directions within {x;E(x) < e}.

Evolution Equation Consider the differential equation

for all t > 0, where a:(0) = x0 and Sf(x) is given by (3). The u-limit set of a solution trajectory of an evolution equation such as (4) is the set of those x which the trajectory approaches arbitrarily closely for arbitrarily large values of time. (We suppose that this trajectory is defined for all t > 0.) Now let E and Z have continuous gradients, let grad E = 0 imply E = 0, let x follow (4) for non-negative t with x(0) = x0 for some fixed a;0 and let the solution trajectory { x ( t ) ; 0 < t < 00} be bounded. The u;-limit set of this trajectory,

is, under natural conditions, a non-empty subset of e-Karush-Kuhn-Tucker points. This observation suggests the following algorithm.

An Algorithm for the Bilevel Problem of Minimising Z Subject to E Being a Minimum of Zero This algorithm is rigorous in the sense that local optimality is certainly achieved if certain natural conditions hold. See the Appendix presented at the end of the paper for an outline justification of the algorithm - this justification comprises the main elements of a convergence proof. Indeed the algorithm is designed so as to permit a fairly ready proof of convergence. The proposed algorithm depends on the two functions

Descent methods of calculating optimal signal controls and prices

13

where E and Z are as above and (for all x)

The direction 5£(x) in (4) is related to these two functions (under natural conditions) as follows. If E(x) > e then E declines toward e in direction 8f(x] while if E(x) < e then Z declines in direction 8£(x) toward a minimum of Z in {x; E(x) < e} and so D(x) tends to zero. Let x0 be any given starting point, let E0 — E(x0) and D0 = D(xQ). From x0 follow (4) with e = Eo/16 until E(x(t)) < E0/8. Then follow (4) with e = £0/4 until D ( x ( t ) ) < D0/4:. Suppose that xi is a point reached by this two-part trajectory. Then E(x) < -E0/4 throughout the second part of this trajectory and so

Repeat the above two-part procedure with E0 and DQ replaced by E0/4: and D0/4 (and so on) to obtain a sequence of points XQ, Xi, x 2 , . . . , satisfying

. Thus E(xn) —> 0 and D(xn) —>• 0 as n —> oo and so (under natural conditions) xn converges to the set {x; E(x) — 0} of equilibria and D(xn) -> 0 so the gradient of Z is increasingly opposite to the gradient of E. It seems natural to measure the lack of weak local-optimality of x by the sum E(x) + D(x). Then the lack of weak local-optimality E(xn) + D(xn) —> 0 as n —> oo for our sequence above. It further seems natural to agree that x is asymptotically weakly locally optimal iff x is the limit of a sequence of points whose lack of weak local optimality tends to zero. With these agreements any limit point of our sequence is asymptotically weakly locally optimal. The fractions (| and |) have been selected so as to leave room for discretizing the smooth trajectory described above, and an appropriate discretization seems likely to generate a sequence {xn} of points such that (for all n)

Thus any limit point of this more practical sequence will be, according to our agreements, asymptotically weakly locally optimal. Bilevel, Trilevel and Multilevel Optimisation. The idea here may be extended to include more constraints than the single E = 0 and more levels than two by employing

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projections onto cones rather than just half-spaces.

BILEVEL OPTIMISATION IN A LINK FLOW FRAMEWORK WITH CAPACITY CONSTRAINTS The previous sections are here applied to the optimisation of signal timings and road prices taking account of travellers' choices of route and mode within a steady state model. There are to be two networks - the first is the base network model of the actual road network representing road links and junctions and the second is a multi-copy version of the base network, with one network-copy for each commodity, and also one signal-copy which represents the signal stages at each single junction. Within each copy the commodity flowing on that copy must have a single specific destination node. If there are K commodities the multicopy network has K copies of the basic network. The simplest commodity comprises all travellers with the same destination. Then if there are K destinations there are K commodities. We may also consider different types of vehicle as different commodities. Here, for example, we may consider all heavy goods vehicles with a specific destination. In this case the copy of the basic network may have no links which correspond to link i in the basic network - the heavy vehicles may be prohibited from certain road links. Thus the copies may in fact essentially be copies of subsets of the base network. In what follows all the copies are considered together as a single multi-copy network. (At a first reading think of an and ar as 1.) Multicopy networks were introduced by Charnes and Cooper (1961).

Notation Variables to be found and controlled. Xr = flow along link r in the multicopy network; Cn = least cost of reaching the destination from node n in the multi-copy network and Cn is to be zero if n is that destination; and 6, =

bottleneck delay at the exit of link i.

Control variables. Vfc = proportion of time stage k is green (a stage is a collection of links with the same tail node and shown green at the same time); and

Descent methods of calculating optimal signal controls and prices

PT = price to be paid to traverse link r in the multi-copy network. Fixed given data. 5r =

positive free-flow cost of traversing link r in the multicopy network;

S,- =

saturation flow at exit of link i in the base network;

or =

passenger car unit equivalent of a unit of flow in that commodity whose network contains link r; and

an =

passenger car unit equivalent of a unit of flow in that commodity whose network contains node n.

Multicopy network structure data. Anr = Qr = an if node n is the tail node of link r and n is not the destination (n after r) and 0 otherwise; Bnr = or = an if node n is the head node of link r (n before r) and 0 otherwise; Mir = ar if link i corresponds to or "belongs to" link r in the multi-copy network and 0 otherwise (link r may be a route and so may "contain" several links z); Nik — Sf if link i is in stage k and 0 otherwise; and Jmk =

1 if links in stage k have tail node m in the base network and 0 otherwise.

Immediately derived variables. Xi =

passenger car unit, or normalised, flow along link i in the base network so that Xi = 53r MirXr (it is this normalised flow which causes congestion);

Hi =

proportion of time link i is green, so that the nominal capacity of link z, y,-Sj, =

Performance and demand functions. Wn(C) = flow generated at node n if cost to destination vector is C (with inelastic demand Wn(C) = constant,,). Wn(C) = 0 for all C if n is a destination;

15

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fi(xi,Siyi) = congestion cost of traversing link i when the normalised link flow is x, and the green time is y,-; and 0. The point x* is weakly locally e-optimal if E(x*) < e and there is no descent direction d for Z such that the straight line segment

Remarks. Under natural conditions x* is weakly locally ^-optimal if and only if Sc(x*) = 0.

Descent methods of calculating optimal signal controls and prices

31

Also, under natural conditions, x* is weakly locally e-optimal if and only if there is a sequence {xn} converging to x* and with E(xn) > 0 for all n such that max(J5(x n ) — e,0) + D(xn) —>• 0 as n —>• oo. (Such points must satisfy E(x*} < e.) Definition. The point x* is asymptotically weakly locally optimal if there is a sequence {xn} of points with xn —>• x* and E(xn] > 0 such that E(xn) + D(xn) —> 0 as n -)• oo. Remark. If desc E(x) converges as x —>• x* (with E(x) > 0) to a single direction d*, and x* is asymptotically weakly locally optimal then there is no descent direction d for Z at x* with [x*, x* + d] C EQ.

Part 2. Showing that, Under Natural Conditions, the Two-Part Trajectory Divides E and D by a Factor of 4 Here we show that the two part trajectory may in fact be arranged to divide E(x) and D(x) by 4 as required. This two-part trajectory begins at XQ, and we shall put EQ = E(XQ) and DO = D(XQ). The end point of the first part of the trajectory and the initial point of the second part will be called XQ. The assumptions needed are reasonable and are: 1. For each e > 0 there is 7y(e) > 0 such that

This is a slight strengthening of the basic requirement that grad E(x) is non-zero if E(x) is non-zero; it follows immediately if {x; e < E(x) < E0} is compact. 2. There is (9 > 0 such that ||grad Z(x)\\ > 9 > 0 if E(x) < EQ. Dividing E by 4. The first part of the trajectory begins at XQ and follows (4) for t > 0 where e = E0/W. To show that this part of the trajectory divides EQ by 8 we suppose that E(x(t)) > Eo/8 for all t. Then:

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Hence, for all i,

This is not possible since E ( x ) > 0 for all x. Hence it cannot happen that E(x(t}} > E0/8 for all t and there are values of t such that E ( x ( t ) ) < E0/8. Let T be the least value of t for which E ( x ( t ) ) < E0/8, and let x0 = X(T). Dividing D by 4. The second part of the trajectory begins at XQ and follows (4) for t > 0 with e = Eo/4. To show that this part of the trajectory divides D0 by 4 we suppose that D(x(t)) > Do/4 for all t. Then:

Hence, for all t,

This is not possible since we may assume that Z ( x ) > 0 for all x such that -E(z) < E0. Hence it cannot happen that

and there must be a time t for which D(x(t)) < D0/4. This appendix shows that, under certain assumptions, the algorithm does generate a sequence of points which converges to the set of points x which are asymptotically weakly locally optimal; and also in outline justifies this as a reasonable objective.

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33

REFERENCES Addison J D and Heydecker B G (1996), An exact expression of dynamic equilibrium. Proceedings of the Thirteenth International Symposium on Transportation and Traffic Theory, Pergamon (Editor J B Lesort), Elsevier, 359-384. Allsop R E (1974), Some possibilities for using traffic control to influence trip distribution and route choice. Proceedings of the Sixth International Symposium on Transportation and Traffic Theory, Sydney, Australia (Editor D J Buckley), Elsevier, New York and Amsterdam, 345-374. Allsop R E and Charlesworth J A (1977), Traffic in a signal-controlled road network: an example of different signal timings inducing different routeings. Traffic Engineering and Control, Vol. 18, 262-4. Charnes A and Cooper W W (1961), Multi-copy traffic network models. Proceedings of the Symposium on the Theory of Traffic Flow, held at General Motors Research Laboratories, 1958 (Editor: R Herman), Elsevier, Amsterdam. Chiou S (1997), Optimisation of area traffic control subject to user equilibrium traffic assignment, Proceedings of the 25th European Transport Forum annual meeting, Sept 1997, 53-62. Friesz T L, Tobin R L, Cho H-J and Mehta N J (1990), Sensitivity analysis based heuristic algorithms for mathematical programs with variational inequality constraints, Mathematical Programming, 48, 265-284. Heydecker B G and Khoo T K (1990) The equilibrium network design problem. Proceedings of the Conference on Models and Methods for Decision Support, Sorrento, 587-602. Luo Z-Q, Pang J-S and Ralph D. (1996), Mathematical programs with equilibrium constraints, Cambridge University Press. Maher M J and Akcelik R (1975), The redistributional effects of an area traffic control policy, Traffic Engineering and Control, 16, 383-385. Smith M J (1984a), A descent algorithm for solving a variety of monotone equilibrium problems, Proceedings of the Ninth International Symposium on Transportation and Traffic Theory, The Netherlands, VNU Science Press, Utrecht, 273-297. Smith M J (1984b), The stability of a dynamic model of traffic assignment - an application of a method of Lyapunov. Transportation Science, 18, 245 - 252.

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Smith M J (1984c), A Descent Method for Solving Monotone Variational Inequalities and Monotone Complementarity Problems. Journal of Optimization Theory and Applications, 44, 485 - 496. Smith M J, Xiang Y, Yarrow R and Ghali M (1996), Bilevel and other modelling methods for urban traffic management and control, Presented at a Symposium at CRT, University of Montreal (forthcoming). Smith M J, Xiang Y, Yarrow R and Ghali M 0 (1997), Bilevel optimisation of signal timings and road prices on urban road networks, Preprints of the IFAC/IFIP/IFORS Symposium, Greece, 16-18 June, 628-633. Tobin R L and Friesz T L (1988), Sensitivity analysis for equilibrium network flow, Transportation Science, 22(4), 242-250. Wardrop J G (1952), Some Theoretical Aspects of Road Traffic Research, Proceedings of the Institution of Civil Engineers II, 235-278. Yang H and Yagar S (1995), Traffic assignment and signal control in saturated road networks, Transportation Research, 29B(4), 231-242. Yang H (1997), Sensitivity analysis for the elastic-demand network equilibrium problem with applications, Transportation Research, 31B(1), 55-70.

ANALYSIS OF TRAFFIC MODELS FOR DYNAMIC EQUILIBRIUM TRAFFIC ASSIGNMENT

B G Heydecker and J D Addison Centre for Transport Studies University College London LONDON WC1E 6BT United Kingdom

ABSTRACT Dynamic traffic assignment provides a valuable means to investigate congested road networks and hence to develop traffic management measures for them. These methods have two distinct components: a traffic model, which represents the propagation of vehicles through the network, and a route choice model, which represents the drivers' response to the conditions that they encounter. We investigate the properties and suitability of various traffic models for use in dynamic assignment using an analysis based upon a dynamic extension of Wardrop's equilibrium condition for route choice. We consider various requirements of plausible traffic behaviour, notably conservation of traffic "and dependence only on traffic downstream, and establish the crucial importance of the latter in the present context. General analytical results are complemented by calculations for simple example networks: this shows that the deterministic queueing and the kinematic wave models of traffic are suitable for this use, but that several other traffic models that are used widely give rise to dynamic assignments that have unacceptable characteristics.

INTRODUCTION Dynamic equilibrium assignment provides a method to model the formation and spread of congestion in road networks where demand varies, possibly exceeding capacity for some part of a study period. These methods are therefore important in the development and evaluation of proposed measures to manage and mitigate congestion. The calculation of dynamic traffic equilibria is of interest for these practical reasons, but relatively little is known about their dependence on the component models of traffic and travel behaviour that are needed to calculate them. In this paper, we apply a mathematical analysis to dynamic traffic equilibrium assignment in order to achieve a fuller understanding of its nature and properties. 35

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Modelling the dynamics of flows and travel times during busy periods has two distinct components. The first is to estimate how a specified temporal inflow profile to the routes through a network will propagate through it. Quantities that are of interest in this analysis include the destination arrival rates or route outflows, and the travel times. This is known as the dynamic network loading problem and has been studied in its own right (see, for example, Astarita, 1996a, 1996b; Xu, Wu, Florian, Zhu and Marcotte, 1997). In the present paper, we analyse various traffic models that have been used for this. The second component represents the way in which travellers respond to the traffic conditions that they encounter: here we investigate a dynamic version of Wardrop's (1952) equilibrium principle of route choice. A matter of principal interest in the present paper is the consequence of adopting each of the various traffic models for network loading in conjunction with this for the estimation of dynamic traffic assignments. By a combination of general analysis of route flows and explicit calculations for simple example networks, we show that the choice between the various possibilities can have a profound effect on the nature of the resulting assignments. We use these observations to identify some fundamental shortcomings in the context of dynamic traffic assignment of some kinds of models that are used widely for this. On the basis of this, we add to the growing body of literature that recommends against continuing the use of whole link models. We draw a contrast between these models, and deterministic queueing and wave-based models which represent traffic behaviour more realistically and which result in more plausible equilibrium assignments.

We base our present analysis on a generalisation of Wardrop's (1952) equilibrium principle of route choice to dynamic user equilibrium (DUE). Thus we suppose that the travel costs incurred by traffic on all routes between each origin-destination pair that are entered by traffic at each instant are equal and less than those that would be on any unused route at that instant. Throughout our analysis, we suppose that the relevant travel costs are those that would be experienced whilst travelling along a route so that they are determined by the conditions encountered rather than any calculated on the basis that the state of the network at the instant of entry will persist throughout the journey. The DUE principle can be stated mathematically after Beckmann (1956) as

where e _(s)is the instantaneous rate at which traffic enters route p at time s , P^ is the set of all routes from o to d, Cp(s) is the cost incurred on route p by a vehicle entering it at time s , and C^(s) is the minimum cost of travel from o to d for journeys that start at time s . Following Heydecker and Addison (1996), we differentiate with respect to time the first of the two cases in (1). Thus for an assignment to remain in equilibrium, we require that the flows satisfy

where /^(s) is the common rate of change of costs on minimum cost routes.

Traffic models for dynamic equilibrium traffic assignment

37

Under first-in first-out (FIFO) traffic discipline, the time taken to traverse a route through the network is determined by the accumulated entry and exit flows, £ (s) and G (t) respectively: the exit time tp(s) for traffic entering route p at time 5 satisfies Ep(s) = G PC (s)l. Differentiating this with respect to entry time s gives ep(s) = 5p["c(s)]T^(s). The instantaneous inflow ep(s) is determined by the assignment proportion \ip(s) for the corresponding route and the total inflow T^s) according to e (s) = n (s) T^s). Together, these lead to

We now suppose that the cost criterion for route choice is a weighted linear combination of travel time with other factors, such as distance, that are constant for each route, so that Cp(s) = a\ip(s)-s\ + P^,. Using this together with (2) shows that the left-hand side of (3) is independent of route p, so that the value of the right-hand side is too. Finally we use the definition of \ip(s) as a proportion to give (4)

This expression (4) shows that in order to maintain a dynamic traffic equilibrium, the proportion of traffic that is assigned to each route at each instant is equal to the proportionate outflow from that route at the corresponding time of arrival at the destination. This result expresses formally the fact that at equilibrium, traffic must arrive at the destination on the various routes in use in the same proportion as it departs from the origin. This condition applies equally to partial routes and can therefore be applied to determine splitting proportions at nodes throughout a network where overlapping routes diverge: these quantities are used in the analyses of Papageorgiou (1990), and Wisten and Smith (1996). Condition (4) is a necessary condition for dynamic traffic equilibrium. Provided that the quantities on the right-hand side are all determined at an earlier time than is the assignment proportion on the left, assignments at each instant will depend on earlier ones: we refer to this desirable property as causal determinism. However, if this does not obtain, then assignments will depend on those in the past, which is clearly unsatisfactory. Whether or not causal determinism obtains depends on the traffic model that is used for dynamic network loading: it will do so whenever traffic behaviour at each point depends on conditions downstream but not upstream of it. This analysis shows that downstream dependence of a traffic model is a crucial requirement for its satisfactory use in dynamic assignment modelling. When a well-formulated traffic model is used, causal determinacy will indeed obtain. In these circumstances, condition (4) has been shown (Heydecker and Addison, 1996) to be a valuable adjoint to (1) for the calculation of dynamic equilibrium traffic assignments. However, we note that satisfaction of this condition is not sufficient to ensure that an assignment is a dynamic equilibrium; rather it is necessary to ensure that the difference between travel times on the route in use remains constant over time. Because this condition requires that outflow be proportional to inflow, it will be satisfied by any all-or-nothing assignment of all demand to a single route. In the present paper, we continue to establish the value of condition (4) for the mathematical analysis of dynamic traffic equilibrium assignments.

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ANALYSIS OF TRAFFIC MODELS The exact result (4) will now be used to analyse the effect on dynamic traffic equilibrium assignments of adopting each of a range of models that could be used to describe the propagation of traffic along the links of a network. In this analysis, we consider various characteristics of the assignments that result in each case and also consider the requirement that assignments should respect causal determinism. We consider in turn three distinct kinds of traffic model: queueing models, wave models, and whole link models.

Queueing Models In queueing models, traffic is supposed to flow freely along links and then possibly to incur delay in a queue at the downstream end. Conservation of traffic in the queue requires that the outflow be the difference between the lagged inflow and the rate at which the amount of traffic in the queue L(t) changes, so that (5)

where <J> is the free-flow travel time. This equation provides the mechanism to incorporate any queueing model in the analysis provided that the dynamics of queue length variation can be specified. Consider first the case of deterministic queueing, which has been used in the present context by Newell (1987), Arnott, de Palma and Lindsey (1990), and Ghali and Smith (1993) amongst others. This model can be used conveniently in overload conditions to represent congestion, but queue lengths and delays are zero unless the queue has been overloaded in the recent past. With this model, the outflow g(t) is given as

where Q is the capacity, or service rate, of the queue. The time of completion of service can be calculated directly as i(s) = s + fy + L(s + fy)jQ so that the delay d(s) incurred due to congestion on a journey that starts at time s is (?) According to this, the outflow at the time of egress either is equal to the inflow at the time of entry or is equal to the capacity due to the presence of earlier demands that have not been served by that time. In either case, the outflow at the time of egress is determined at or before the time of entry to the link. Using the outflows (6) in the necessary condition (4) leads to a closed-form rule for dynamic equilibrium assignments which is that traffic is assigned to congested routes according to their capacity. This rule can be applied directly to solve for dynamic equilibrium assignments in

Traffic models for dynamic equilibrium traffic assignment

39

networks that have a distinct bottleneck for each route; with some ingenuity, it can also be applied in more general networks (Gardner, 1996). We distinguish between two cases according to the state of the routes in use. In the first case, there is a queue on each route in use at the time of egress (or one is forming), so that the outflow from each route will be equal to its capacity. The assignment that will maintain equilibrium is then given according to (4) by

where P^s) is the set of routes in use at time s . In the second case, at least one route has no queue and the sum of the capacities of all the routes in use exceeds the demand. The outflow from those routes on which there is no queue at the time of egress will be equal to the inflow at the time of entry, whilst that on routes with queues will be equal to their capacity. The assignment that will maintain equilibrium is given in this case by

Because the absence or presence of queueing vehicles depends on earlier flows and hence earlier assignments, the assignment at time s is also determined by earlier events. Accordingly, use of queueing models of this kind in dynamic equilibrium assignment models will result in causally determinate assignments. These assignments have a step discontinuity whenever the set of routes in use changes, as occurs when a new route comes into use. In the case that the addition of a new route still leaves inadequate capacity for the demand, the assignment to the new route immediately it comes into use is in accordance with (8) and hence is greater than the capacity of that route. In the case that the capacity of the newly augmented set of routes exceeds the demand, the assignment to the new route p + immediately it comes into use is in accordance with (9) and hence is equal to the difference between the demand and the capacity of the routes P^s ') that were already in use, so that

For continuous demands, this is strictly positive because the demand exceeds the combined capacity of the routes already in use. In either case, the new route that comes into use has a step increase in assignment whilst the others that were already in use have corresponding proportionate step decreases in assignment. Payne and Thompson (1975) developed a steady-state analysis that identifies the condition (9) directly for cases where the demand is stable over a sufficiently long period, but this approach excludes reference to transients and the resulting dynamic behaviour of the queues and flows.

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Consider now the analysis developed by Whiting of transient oversaturated queues with stochastic behaviour (see, for example, Kimber and Hollis, 1979). This approach has the attraction that realism might be improved by the non-zero estimates of queue lengths and delays for undersaturated queues. According to this approach, the queue length during an interval with constant arrival rate and capacity varies over time according to the initial queue length and these two rates. Here, we consider the instantaneous application of this approach which gives the total derivative of queue length with respect to time as a function of the values of arrival rate, capacity and queue length at that time. Thus the dependence of queue length change can be expressed as

However, using this in the general equation (5) for conservation of traffic leads to the expression for outflow rate

which does not satisfy the downstream dependence requirement for causal determinacy: according to this, the outflow rate g ( t ) at time t depends on the arrival rate at the back of the queue at that time. We therefore conclude that this queueing analysis is unsuitable for use in dynamic traffic assignment.

Wave Models Highly detailed models of traffic dynamics have the advantage that they offer plausible descriptions of flow, including the propagation of any congestion through the network. However, they have the disadvantage of being both analytically and computationally demanding. An example of models of this kind is provided by Lighthill's and Whitham's (1955) wave model which can be developed using techniques of partial differential equations to describe the assignments spatial dynamics of traffic on each link in a way that respects causal determinism. Models of this kind have been developed for dynamic traffic assignment problems by Newell (1988), Addison and Heydecker (1995), and by Heydecker and Addison (1996). Other models that we do not consider here include the hydrodynamic ones introduced by Payne (1971) and developed since then by Kerner and Konhauser (1993). According to the wave model, the speed v(x,t) and flow q(x,t) at position x and time t are determined entirely by the corresponding density k(x,t). The key feature of this model to which we appeal in the present analysis stems from the propagation of regions of constant density k along waves with a speed w(k) that is determined by the density alone; because of this, the trajectories of the waves have constant speed. Ughthill and Whitham show that w(k) = dq/dk which, together with q = kv leads to w(k) = v(k) + kdvjdk 0 so that dxjds > 0 and hance gx(s+) < T(s^ : using this in (17) gives \i^ < 1 . However, \i{(s) = 1 (s<st\ so the assignment proportions, and hence the assigned flows, are discontinuous at time s =s# when the new route comes into use. Few authors have published extensive detail of dynamic assignments calculated using whole link models. Amongst the results that have been published, we note that those of Papageorgiou (1990), Jayakrishnan et al (1995), and Lam and Huang (1995) show intervals throughout which the assignments are discontinuous; this suggests that the discontinuity at the instant of initial use of a new route is not an isolated phenomenon but rather is typical of dynamic equilibrium assignments calculated using whole link models. Earlier observations concerning the slow dynamic behaviour of whole link models (Addison and Heydecker, 1995) provide some insight into this behaviour. In order for the assignment to remain in equilibrium, the cost of travel on the second route to be used should start immediately to increase at the same rate as that on the first route. However, the slow dynamic behaviour of whole link models makes this impossible with any continuous assignment. By contrast, the assignments reported by Wie et al (1995) are continuous and smooth in time but inspection of the corresponding costs of travel on the routes in use reveals that they are not close to an equilibrium, especially during the period when costs are increasing.

DISCUSSION AND CONCLUSIONS On the basis of the analysis presented here, we find that the character of the network loadings differ substantially according to the traffic model that is used for network loading. This leads to estimates of costs of travel and outflows that also differ substantially. The novel condition (4) that is necessary in order to maintain equilibrium shows that the calculated outflows are of crucial importance in an equilibrium assignment, so that the different character of the network loadings will lead to substantially different assignments.

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The issue of causality is of fundamental importance in dynamic models: in any plausible model, we require current behaviour to be determined by that in the past rather than by future events. Both the deterministic queueing and the wave models satisfy this requirement, but the transient stochastic queueing and whole link models do not. This lack of appropriate causal behaviour invalidates use of these models in dynamic traffic assignment. The character of dynamic equilibrium assignments differs substantially according to the traffic model that is used in their calculation. We have shown that those calculated using the wave model are continuous when a new route comes into use, though they can exhibit discontinuities at other times because of the formation of shock waves. Dynamic equilibrium assignments calculated using deterministic queueing models are relatively simple in character, with assignment proportions mostly determined by capacities, leading to jump discontinuities when new routes come into use. We have shown that dynamic equilibrium assignments calculated using whole link models are also discontinuous at the instant when a new route comes into use; evidence from the literature shows that in order to approach equilibrium, they can be discontinuous throughout intervals of the study period during which several routes are in use. Comparison of the various traffic models considered here shows that dynamic equilibrium assignments calculated using either deterministic queueing or wave models have plausible interpretations whilst those calculated using more sophisticated transient queueing and whole link models do not. Deterministic queueing models provide a convenient means to estimate travel times but provide plausible approximations only when flows exceed capacity, whilst the wave model is most useful when link flows are each within the corresponding capacities.

ACKNOWLEDGEMENTS The authors are grateful to Richard Allsop for his encouragement and several stimulating discussions on the topic of this paper and to an anonymous referee for his helpful suggestions. The work reported here was funded by the UK Engineering and Physical Science Research Council.

REFERENCES ADDISON, J.D. and HEYDECKER, E.G. (1995) Traffic models for dynamic assignment. In: Urban Traffic Networks: dynamic flow modelling and control (eds N.H. Gartner and G. Improta). London: Springer, 213-231. ARNOTT, R., de PALM A, A. and LINDSEY, R. (1990) Departure time and route choice for the morning commute. Transportation Research, 24B(3), 209-28. ASTARITA, V. (1996a) A continuous time link model for dynamic network loading based on travel time functions. In: Transportation and Traffic Theory (ed J-B Lesort). Oxford: Pergamon, 79-102. ASTARITA, V. (1996b) The continuous time link based dynamic network loading problem: some theoretical considerations. Proceedings of the 24th PTRC European Transport Forum, Seminar D. London: PTRC, P404(l). BECKMANN, M., McGUIRE, C.B. and WINSTEN, C.B. (1956) Studies in the economics of transportation. New Haven: Yale University Press.

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BERNSTEIN, D., FRIESZ, T.L., TOBIN, R.L. AND WIE, B.-W. (1993) A variational control formulation of the simultaneous route and departure-time choice equilibrium problem. In: Transportation and Traffic Theory (ed. C.F. Daganzo). London: Elsevier, 107-26. BOYCE, D.E., RAN, B. and LEBLANC, L.J. (1995) Solving an instantaneous dynamic useroptimal route choice model. Transportation Science, 29(2), 128-42. DAGANZO, C.F. (1995) Properties of link travel time functions under dynamic loads. Transportation Research, 29B(2), 95-98. GARDNER, D.C. (1996) Dynamic traffic assignment. Thesis for MSc, University of London. GHALI, M.O. and SMITH, M.J. (1993) Traffic assignment, traffic control and road pricing. In: Transportation and Traffic Theory (ed. C.F. Daganzo). London: Elsevier, 147-69. HEYDECKER, E.G. and ADDISON, J.D. (1996) An exact expression of dynamic traffic equilibrium. In: Transportation and Traffic Theory (ed J-B Lesort). Oxford: Pergamon, 359-83. HURDLE, V.F. (1986) Technical note on a paper by Andre de Palma, Moshe Ben-Akiva, Claude Lefevre, and Nicolaos Litinas entitled "stochastic equilibrium model of peak period traffic congestion." Transportation Science, 20(4), 287-9. JAYAKRISHNAN, R., TSAI, W.K. and CHEN, A. (1995) A dynamic traffic assignment model with traffic-flow relationships. Transportation Research, 3C(1), 51-72. KERNER, B.S. and KONHAUSER, P. (1993) Cluster effect in initially homogeneous traffic flow. Physics Review, 48E, 2335-38. KIMBER, R.M. and HOLLIS, E.M. (1979) Traffic queues and delays at road junctions. TRRL Laboratory Report, LR 909. Crowthorne: TRL. LAM, W.H.K. and HUANG, H.-J. (1995) Dynamic user optimal traffic assignment model for many to one travel demand. Transportation Research, 29B(4), 243-259. LIGHTHILL, M.J. and WHITHAM, G.B. (1955) On kinematic waves: II. A theory of traffic flow on long crowded roads. Proceedings of the Royal Society, 229A, 317-45. MERCHANT, O.K. and NEMHAUSER, G.L. (1978) A model and an algorithm for the dynamic traffic assignment problem. Transportation Science, 12(3), 183-99. NEWELL, G.F. (1987) The morning commute for nonidentical travellers. Transportation Science, 21(1), 74-88. NEWELL, G.F. (1988) Traffic Flow for the morning commute. Transportation Science, 22(1), 47-58. NEWELL, G.F. (1993) A simplified theory of kinematic waves in highway traffic. Part I: General Theory. Transportation Research, 27B(4), 281-7. PAPAGEORGIOU, M. (1990) Dynamic modelling, assignment, and route guidance in traffic networks. Transportation Research, 24B(6), 471-95. PAYNE, H.J. (1971) Models of freeway traffic and control. In: Proceedings of Mathematical Models of Public Systems, 1, 51-61. La Jolla, CA: Simulation Councils. PAYNE, H.J. and THOMPSON, W.A. (1975) Traffic assignment on transportation networks with capacity constraints and queueing. Paper presented to the 47th National ORSA/TIMS North American Meeting. RAN, B. and BOYCE, D.E. (1996) A link-based variational inequality formulation of ideal dynamic user-optimal route choice problem. Transportation Research, 4C(1), 1-12. VYTHOULKAS, P. (1990) A dynamic stochastic assignment model for the analysis of general networks. Transportation Research, 24B(6), 453-69. WARDROP, J.G. (1952) Some theoretical aspects of road traffic research. Proceedings of the Institution of Civil Engineers, 1(2), 325-62.

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WIE, B.-W., TOBIN, R.L. and FRIESZ, T.L. (1994) The augmented Lagrangian method for solving dynamic network traffic assignment models in discrete time. Transportation Science, 28(3), 204-220. WIE, B.-W., TOBIN, R.L., FRIESZ, T.L. and BERNSTEIN, D. (1995) A discrete time, nested cost operator approach to the dynamic network user equilibrium problem. Transportation Science, 29(1), 79-92. WISTEN, M. and SMITH, MJ. (1996) A distributed algorithm for the dynamic traffic equilibrium assignment problem. In: Transportation and Traffic Theory (ed J-B Lesort). Oxford: Pergamon, 385-408. XU, Y.W., WU, J.H., FLORIAN, M., ZHU, D.L. and MARCOTTE, P. (1997) New advances in the continuous dynamic network loading problem. In: Transportation Networks: Recent Methodological Advances (ed MGH Bell). Oxford: Pergamon.

APPENDIX: CALCULATION OF VEHICLE TRAJECTORIES We consider the trajectory of a vehicle that enters a route at time s . We use analysis of accumulated flows to establish closed-form expressions for the time t(k) and position.x(fc) at which the vehicle passes a wave of density k that entered at time Sk>S. This analysis can be used to calculate the trajectory along a link of a vehicle parametrically in the density of entering traffic. The flow entering the route during the interval (sk,S\ is E(s) - E(s^. According to the FIFO principle, when this amount of flow has passed the wave that enters at time Sk, then the vehicle that entered at time s will pass it. Thus we have to solve for t(k) the equation

(AD where q+(k) is the rate at which traffic passes a wave of density k . Now

so that

This wave travels at speed w(k) so its position on the route at that time is

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AN EFFICIENT ALGORITHM FOR THE CONTINUOUS NETWORK LOADING PROBLEM : A DYNALOAD IMPLEMENTATION Y. W. Xu, J.H. Wu andM. Florian Center for Research on Transportation Universite de Montreal Montreal CP 6128 Sue. Centre-ville Quebec CANADA H3C 3J7 Abstract The continuous dynamic network loading problem(CDNLP) consists in determining, on a congested network, time-dependent arc volumes, together with arc and path travel times, given the time varying path (departure) flow rates over a finite time horizon. This problem constitutes an intrinsic part of the dynamic traffic assignment problem. In this paper, we present a modified DYNALOAD algorithm for the continuous dynamic network loading problem. An efficient implementation is developed. Numerical examples are provided. Keywords: Continuous Dynamic Network Loading. Running head: Continuous Dynamic Network Loading.

1

Introduction

The continuous dynamic network loading problem (CDNLP in short) aims at finding, on a congested network and over a fixed time period, time-dependent arc volumes, arc travel times and path travel times corresponding to given path (departure) flow rate functions. This problem has recently received considerable attention, due to its importance in dynamic transportation models that assume either full information at departure nodes and the equilibrium principle ([3], [8], [17]) or take explicitly into account en-route real time information ([1], [5], [2], [4], [6] ). Indeed, the CDNLP is a "within-day" model, in the sense of Cascetta and Cantarella [2]. Whenever disequilibrium situations are encountered ([4], [7]), one is confronted with both "day-to-day" and "within-day" effects. There are two main solution procedures to this problem. The methods of the first type are mainly based on simulation (DYNASMART [6], CONTRAM [15]). One advantage of the simulation approach is its detailed description of traffic phenomena. However, it is computationally intensive, especially if it is used within an interactive environment. Moreover, simulation provides no clue about the nature of the model under study. In particular, it is difficult to establish some analytical properties of the relationship between "This research was supported in part by NSERC individual operating grants OGP0157735, OGP0007406 and the NSERC strategic grant "Parallel software for IVHS". 51

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the path (departure) flow rates and the path travel times. The methods of the second type are based on the analytical analysis of the problem. Recently, Wu et al (1995) propose a functional equation method and solve it by using an optimization method and Xu et al (1996) propose a theoretical algorithm DYNALOAD with a implementation. One important feature of the algorithm DYNALOAD is that the solution method solves the problem in finite steps, if the problem has a solution, that is, the arc exit time functions are invertible. The solution method is based on the observation that the functional forms of the arc entry flow rate, arc exit flow rate and arc exit time functions are only computed at a finite set of instants and that, given the arc inverse functions, all these functions are computable. It is important to obtain a good computation of the arc inverse functions during the discretization process. In this paper, we introduce a particular implementation of DYNALOAD in order to reduce the discretization errors by eliminating some of intermediate variables and computing the arc volumes using path flow rate functions directly. This can be a great benefit for a large network where many discretizations for the computation will be carried. It is interesting to mention that the computational results from our implementation here in C are almost same as that from the implementation [18] in GAMS. This paper is organized as follows. In Section 2, we present the model of the dynamic network loading problem and provide a proposition which is used in our implementation. Section 3 is devoted to the implementation of DYNALOAD. In Section 4, we report the computational results for three examples. And we conclude the paper in Section 5.

2

Model

Let G — (N, A) be a transportation network, composed of a set of nodes N and a set of arcs A. Let / C N x N denote the set of all O-D pairs, Ki the set of paths connecting the O-D pair i e /, and K = Uje/^Q. GJ and rfj is the origin node and the destination node of the O-D pair i respectively. The allowable departure time period of the users is a finite interval [0, T], although the time period under study (system period) will be a longer time interval. The following notations are used in this paper. We let The basic indices: / be the set of all O-D pairs; A; be a path index; Ki be the set of paths for O-D pair i; K be the set of all paths, that is, K = U^Ki] Td — [0, T] be the departure time period, that is, the fixed time period under the consideration for departures only; Ka be set of all paths containing arc a; ka— be arc preceding arc a on path k; The given data: hk(t] be the path flow rate for path k departing at instant t € [0, T]; The variabels: Tka be the latest path-fc flow arrival instant at the tail of arc a; Ta be the latest flow arrival instant at the tail of arc a: Ta = maxkeKa{Tka}; bka(t) be arc entry flow rate of path k into arc a at instant t ; 6ka(t} be arc exit flow rate of path k out of arc a at instant t ; ba(t) be arc entry flow rate into arc a at instant t : ba(t) = J2keKa ^fca(*)j ea(t) be arc exit flow rate from arc a at instant t : ea(t) — J2k£K« eka(t)', va(t) be arc volume (vehicles) on arc a at instant t; sa(v) be arc travel delay function of arc a; ra(t) be arc exit time of flow entering arc a at instant t ; r~l(t) be arc inverse function of Ta(t}; Sk(t) be path travel time for users departing at instant t £ [0,T]). For clearity, we call the relationship between the arc travel time and the arc volume the arc travel delay function and that between the arc travel time and the system time the

An efficient algorithm for the continuous network loading problem arc travel time function (or simply the arc travel time). The mathematical formulation of the CDNLP is proposed in [18] and is given as follows. (CDNLP)

Fanally, for a path k — ai — a2 — ... — a nfc , the path travel time of path k departing at instant t, Sk(t), can be calculated by the following recursive equations

for all time t e [0, T], and for all path k G K where tj is the instant that flows arrive at the tail of arc a., along path k, Hk is the number of arcs on the path. And in an alternative expression, we may write

We indicate that for evaluating ra at instant t, the algorithm first determines bka(t] in an appropriate interval for all path k e Ka, then computes, one by one, ba(t), va(t), ra(t), T~l(t) and ejfc a (t) in appropriate intervals. As indicated in [18], whenever obtaining efc a (t), we can compute bka(t) for the next time interval. In this paper, we consider the case where this CDNLP is solvable with the unique solution. In [18], we identify a boundedness condition as a sufficient condition for the existence of the unique solution. Now we prove one useful proposition which is used in our implementation. Let A; be an arbitrary path in K^i e I. Suppose node j is on path k. Denote by Tikj(t) (path-node exit time) the flows exit time instant at node j, for flows departing GJ at instant t along path k.

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Y. W. Xu et

It is clear that, by the definition, for arc a = (p, q) and any t e [0, T], it holds (9)

The following proposition indicates that if the CDNLP is solvable, that is, the arc FIFOs are satisfied, then (i) all path FIFOs and the FIFOs for all the corresponding subpaths within the paths are satisfied; (ii) the path-node exit time functions are invertible; (Hi) basic relationships between the arc entry flow rates and the path flow rates are established. Proposition 1 Suppose that the CDNLP is solvable. We have (i) For any path k in K and any node j on k, the path-node exit time function H^ is absolutely continuous over its domain (thus differentiate almost everywhere) with

(ii) For any arc a = (p, q) on path k, the path-node exit time function Tl^g is invertible with

(Hi) For any arc a = (p, q) on path k, it holds

Proof, (i) This is a direct result of (3) and (9). (ii) This assertion follows from (i) and (9). (iii) Suppose k = ai — 0,1 — • • • — ai with aj = (PJ-I,PJ). Then Tlkp0(t) — t,Vt and bkai(t) = hk(t) by definition. (10) is correct for ai. For j = 2 we have

Set £ = Tail(i) — n fepi (i), i.e., t = r ai (f) = Ukpl(£) for £ > 0 and rename £ as t, we obtain

(10) is correct for j = 2. Suppose now for n = j > 2 it holds

(11) We consider n — j + 1. It follows that

An efficient algorithm for the continuous network loading problem

55

Set t = HkPj(£) = Taj(^kPj-.i(^)} for £ > 0, and rename £ as t, we obtain

where the second equality follows from (11) and the third from (9). Therefore (10) is also correct for n = j + 1. This completes the proof. HI Denote by Vka(t) the arc partial volume of arc a at instant t contributed by path k. By this definition, we have «a(y) - £ w te (y).

(12)

keK"

By using (10), we obtain for any y >

vka(y) = f" M£R = /n7(y) 6*«(n*P(0)dn*P(*) = F"^ hk(t)dt

(13)

where x = 11^(0) if the flows of path k have not yet exited from arc a at instant y, and a; = r~l(y) otherwise. In the first case, we have U^(x) = 0. And in the second case we get from Proposition l(ii) U^(x) = U^(r-l(y)} = U^(y). We notice that this expression for Vka is identical to the one used in Friesz et al (1993) and Wu et al (1995). We next address a modified DYNALOAD algorithm for solving the CDNLP. In Xu et al (1996), the CDNLP is solved, by using DYNALOAD, by it- ations in which variables are calculated in the following order: e^a, bka, ba, va, ra and T~' By obtaining r~l at the present iteration, we may compute eka and other variables for the next iteration. In the modified algorithm stated below, instead of using bka, we may directly use hf. to calculate Vka using (13). The order for computing variables in each iteration are: Vka, va, Ukq and

3

Implementation

When implementing DYNALOAD, we do not construct the desired functions in continuous forms fully (over their domains) but, instead, in discrete forms. Evaluation of the functions at other points are done by interplation. As described in the previous section, for each arc a = (p, g), the corresponding functions vfca and IIfcg are obtained from one time interval to the other time interval. In this implementation, we take four points in each time interval as observation points. At the four points, equations (2), (9), (12) and (13) are exactly calculated. While at rest points, values of the functions are approximated by polynomial functions with degree three by using the cubic spline technique (see Appendix A). Before presenting the implementation of the algorithm, we need some notations used in the algorithm. The computation of the algorithm is an iterative process. At each iteration i, there is a particular system time Ti and arc current time interval [l~, /+] for each arc a where l~ , 1+ are the arc lower and upper bound of the interval.

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We notice that in the course of computing dynamic functions, it is not necessary to compute the inverse function ILj^1 over [T a (/~),r a (/+)] instantly. Instead, the inverse function will be computed at certain particular points when calculating vka. Because IL-9 is cubic, the value of H^1 at any point could be evaluated by using an algebra formula. This simplification makes the modified algorithm more efficient than the original DYNALOAD in both computing time and memory space. For executing DYNALOAD, one needs to compute and save ra and eka- While for executing the modified algorithm, one only needs to compute and save nfcg. If the departure time period is pretty long, then the number of iterations for each arc dynamics could be pretty large. Thus savings in computing time and memory space for the modified algorithm are significant. As stated before, Tka is the latest path-/; flow arrival instant at the tail of arc a. We also define teka as the earliest path-A; flow arrival instant at the tail of arc a. We should have teka = Ilfca(O) and Tka = Hfc a (T) at the end of the computation. Given a time interval [a;, y], we say that path k £ Ka is active on arc a during [x, y] if some flows of path k will pass arc a during [x,y\. It is clear that path k £ Ka is active on arc a during [x, y] if and only if teka < y and Tka > x. For a given time interval [x,y], arc a is said to be active during [x,y] if at least one path k £ Ka is active during [x,y]. We further define the active degree of arc a in [x,y], denoted by pa, to be the number of active paths on arc a during [x, y]. Let A be the active arc set where all the arcs are of positive active degree, Ai is a subset of A where the upper bounds of the arc current time intervals, that is, /+, are equal to the system time at iteration i, that is, Tj. The proposed implementation includes two separate parts: an initialization and a main loop. In INITIALIZATION, we set for each arc a a starting arc current time interval [0, sa(0)]. We also determine an initial active arc set. In MAIN LOOP, we first determine an arc (or arcs) in the active arc set with the minimum of upper bounds of the arc current time intervals. Then for each active k £ Ka, we first need to determine a working interval, say, [x,y], for path k. The working interval for path k may be smaller than the arc current time interval. This will happen when flows of path k arrive at an intermediate arc for the first time, or last time. We then determine the departure time interval, say, [£0,^3], corresponding to the working interval and equally divide the departure time interval with four points (resulting in three subintervals). Corresponding vka,va,Ta and Ukq value will be calculated at these points. The active degree of each arc, pa, Va £ A is updated. These computations are performed in the subroutine ARC DYNAMIC (a, k). The arc current time interval [/~, /+] of a £ A is updated based on the new system time 7$. The active arc set will be updated as well according to the updated active degrees. The system time is increased to the minimum of upper bounds of arc current time intervals for all the arcs in A, that is, Tj = min{/+ : a £ A}. The following is the detail for the implementation.

AN IMPLEMENTATION FOR THE MODIFIED DYNALOAD ALGORITHM 1. INITIALIZATION i := 0, To := 0 for a in A do I- := 0

It •= «.(0) pa := the number of paths k £ Ka which take a as the departure arc

An efficient algorithm for the continuous network loading problem

57

endfor A := {a e A : pa > 0} 2. MAIN LOOP while Ti < T or A ^ 0 do i:=i + l Ti := min{/+ : a e A} Ai := argmin{/+ : a € A} for a in Ai do for fc in Ka do if A; is active on a over [l~, 1+] then ARC DYNAMICS (a,k) over [l~,l+] endif endfor update /- = Ti and 1+ = Ta(Ti) endfor if pa — 0 then A:= A — {a} endif endwhile

3. ARC DYNAMICS (a,fc) over [£,/+] determine working interval [x, y] determine corresponding path-A; departure time interval [to, is]: to = Ilj^a;), t^ = Yl^ determine t\ and ti to make (^0,^1,^2,^3) equally spaced for n = 0,1,2,3 do compute £n = n fcp (t n ) using (9) for fc' e £Ta do if fc' is active on a at £„ then compute ^fc'a(^n) using (13) endif endfor compute va(£n) using (12) compute r a (£ n ) using (2) compute II fcg (t n ) using (9) enffor if a is not the last arc of path k then A := A U {the next arc of a on &} endif ify = nfcp(T)thenpa:=pa-l endif

4

Numerical examples

We provide three numerical examples in this section: example 1 is a small network with three arcs and three paths; the second has 12 arcs and 14 paths cited from Xu et al (1996); and the third example is the city of Sioux Falls, which consists of 24 nodes and 76 arcs. It is the same network used by Suwansirikul et al (1987) and Wie et al (1995).

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The path flow rate functions for all three examples are of the form hk(t) = 9k(T-t)t where T is the departure time horizon, and 9k is a parameter for path k. Arc travel delay functions are all quadratic functions. For the first two examples, the arc travel delay functions are of the form SaK) =ttaO+ Oia\Va + aa2V^.

And the functional form for arc travel delay functions for the third example is V -r-f

\'2

250aao/

where aa0, aai and aa2 are constant. In the above equation, 250aa0 partly represents the "capacity" of arc a. In dynamic traffic problems, the capacity of an arc should be interpreted as the maximum number of occupants of the arc. Suppose that the free-flow speed is 60 kilometers an hour (1,000 meters a minite), and a vehicle occupies at least, say, four meters in the most congested case. Then the maximum number of occupants of the arc is 250aao. The network for example 1 is shown in Figure 1. The network data is given in Table 3. Table 4 shows parameters for the path flow rate functions and arc travel delay functions. The computational results are drawn in Figure 2 where v.a, s.a, taoM are the arc volume va, arc travel time sa and arc exit time ra respectively for a — 1, 2,3 and S-k is the path travel time of path k for k = 1,2,3. The CPU time of solving the problem is summarized in Table 1. We notice that the FIFO conditions are respected. The network for example 2 is given in Figure 2. Table 5 gives out parameters for network paths and arcs respectively. The computational results are shown in Figure 4 where v_a, s_a, tao-a are the arc volume va, arc travel time sa and arc exit time ra respectively for a = 1,2,..., 12 and SJc is the path travel time of path k for A; = 1,2,..., 14. The CPU time of solving the problem is summarized in Table 1. We notice that the FIFO conditions are respected as well. We have compared the results with that in [18] and concluded that the computed results from both implementations are the same. In example 3, we use the network of Sioux Falls (Suwansirikul et al, 1987). Undirected links of the network should be interpreted as two directed arcs with opposite directions. Parameters aao and ao2 for arc travel delay functions are shown in Table 6, which are cited from Suwansirikul et al (1987). aai were not provided there. We set in this example aai = 0.01 for all a e A. There are 24 origin and destination nodes on the network. The number of total O-D pairs is 552. We conducted several computations using different departure time horizons and different path flow rates. We use two sets of paths. For the first set, there are total 552 paths, one path for each O-D pair, with 9k being 0.0005. These paths are obtained by applying a shortest path algorithm to the network of the city of Sioux Fall with aa0 being the arc length of arc a £ A. There are total 1104 paths in the second set. We select for each O-D pair two paths in this computation. Two paths for each O-D pair are obtained in the following way. The first path is the shortest path between the two nodes with the arc length for arc a being a:ao, while the second path is the shortest path with the arc length for arc a being -~. The parameter 9k is set to be either 0.0002 or 0.00005 depending on the first path or the second path. The computational results of the third example are shown in Figure 5. CPU times (in second) of solving this problem

An efficient algorithm for the continuous network loading problem

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Table 3: Network data for the first example O-D PAIR

PATH (1,2) (2,3) (3,1)

1: 2: 3:

(M) (c,6) (a,c)

Table 4: Function parameters for the first example

a + {£;p + r Aep

(13) Definitional constraints:

IC(o=«a(o, sc(o=va(o, S*:;,(o=

wpm

r.spm

r.spm

(14) Nonnegativity conditions: >0, Vm,a,p,r,s (15)

(16) Boundary conditions:

£;,,(0) = o, yp,w,r,s (17)

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D. E. Boyce et al.

(18) For each class of travelers, the constraints expressed in (9) - (18), including flow propagation and conservation constraints, are applicable. These constraints are used to generate path and link flows when route departure flows are determined. The path departure flow /"(O is determined by the stochastic loading function, explained in Section 2.5. The link flow propagation constraints (13) are implemented for each link a, each route p, each O-D pair rs, and each time t, regardless of traveler classes. Therefore, the FIFO requirement can be ensured. 2.4

Link Travel Time Functions

For the travelers in a stochastic network, the perceived travel time is used. To simplify the computation, it is assumed that the actual travel time depends on the number of vehicles and the inflow rate. Equation (19) shows the link travel time function chosen for the numerical results of this study:

(19) where a and p are coefficients, Ta f is the free flow travel time on link a, Ca is its capacity, and xa>max ls its maximum holding capacity. There are various link travel time functions for different link types, such as freeway and arterial. The computational implementation with these functions will be reported in subsequent papers. 2.5

Route Choice Conditions and The VI Formulation

Route Choice Conditions: As shown in Sheffi (1985), a satisfaction function can be defined as follows:

(20) This satisfaction function has the following property:

(21) The SDUO route choice conditions are then defined as follows: (22) Note that the mean actual route travel time 77™ (0 is increasing with path departure flow

n >0

(23) For each path p and each O-D pair rs, define an auxiliary cost term as follows:

Travelers' risk-taking behavior in dynamic traffic assignment

73

tfp (0 It is obvious that the above equality states the SDUO route choice conditions, since dr] " (/) / cf™ (t) > 0 . As shown in Nagurney (1993), the above system of equations is equivalent to the following variational inequality for each time instant / e [0,+oo) :

(25) where superscript * denotes that path departure flow/has an optimal value. Since F"(t) = 0, the above inequality is also equivalent to the integral form:

(26) 3. THE SOLUTION ALGORITHM 3.1

The Discrete VI

To convert our continuous time VI problem into a discrete time VI problem, the time period [0, T] is subdivided into K small time intervals. Each time interval is regarded to be one unit of time. Then, ua(k) represents the inflow into link a during interval k, va(k) represents the exit flow from link a during interval k, xa(k) represents the number of vehicles at the beginning of interval k, andfp(k) represents the departure flow from path;? during interval k. To simplify the formulation, we round the estimated actual travel time on each link in the following way so that each estimated travel time is equal to a multiple of the time interval: ?a(k) = i if / -0.5