Mathematical Programming 46 (1990) 53-60 North-Holland
53
(1, k)-CONFIGURATION FACETS FOR THE GENERALIZED ASSIGNMENT P...
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Mathematical Programming 46 (1990) 53-60 North-Holland
53
(1, k)-CONFIGURATION FACETS FOR THE GENERALIZED ASSIGNMENT PROBLEM Elsie S t e r b i n G O T T L I E B * Baruch College, The City University of New York, NY, USA M.R. R A O * * Stern School of Business, New York University, NY, USA Received 21 August 1986 Revised manuscript received April 1988 A class of facet defining inequalities for the generalized assignment problem is derived. These inequalities are based upon multiple knapsack constraints and are derived from (1, k)configuration inequalities. Key words: Generalized assignment problem, knapsack problem, special ordered sets, integer polytope, facets, (1, k)-configurations.
1. Introduction In [3] we d e r i v e various classes o f valid i n e q u a l i t i e s for the g e n e r a l i z e d a s s i g n m e n t p r o b l e m ( G A P ) . M o r e restrictive c o n d i t i o n s n e e d to be i m p o s e d in o r d e r to t r a n s f o r m these valid inequalities into facet defining inequalities. These facet defining inequalities are p a r t i c u l a r l y i m p o r t a n t since t h e y are essential for defining the c o n v e x hull o f 0-1 solutions. In this p a p e r we derive a class o f facets b a s e d u p o n the (1, k ) - c o n f i g u r a t i o n i n e q u a l i t i e s d i s c u s s e d in [3]. W e e m p l o y the s a m e n o t a t i o n as in [3] a n d refer the r e a d e r to S e c t i o n 3 a n d Section 4.3 o f that p a p e r . A d d i t i o n a l classes o f facet defining i n e q u a l i t i e s , w h i c h are s p e c i a l i z a t i o n s o f o t h e r v a l i d i n e q u a l i t i e s d e r i v e d in [3], m a y b e f o u n d in [2] a n d [4].
2. (1, k)-Configuration facets In this s e c t i o n we derive a class o f facets o f K wYH , w h e r e I wI = w a n d IHf = h, b a s e d u p o n the j o i n t valid i n e q u a l i t i e s in [3, S e c t i o n 4.3]. A s s u m e that N ' u { z } , w h e r e ]N'[ = n', is a (1, k ) - c o n f i g u r a t i o n for s o m e a r b i t r a r y k n a p s a c k p. F o r specific r, k ~< r