A note on modeling multiple choice requirements for simple mixed integer programming solvers

-This note introduces the Special Ordered Inequalities (SOl) as a new technique for integer programming modeling of multiple choice requirements and piecewise linear functions. The standard modeling technique based on the use of Special Ordered Set (SOS) structure requires a branch and bound solver to be armed with the special tools for SOS processing. Otherwise, the standard branching rule applied on individual SOS variables may lead to enormously long computation process thus making impossible to solve quite small problems in a reasonable time. The proposed SO1 modeling technique allows us to eliminate the necessity of the special branching rule for handling SOS structure. The standard branching on individual variables of SO1 is equivalent to the special SOS branching. Thus the proposed modeling technique seems to be very attractive for users of simple mixed integer programming solvers as well as for those who want to build solver independent mathematical programming models. 1. I N T R O D U C T I O N In the great majority of real-life mixed integer programming models most of integer variables represent some multiple choice requirements [1]. There is a variety of applications leading to models with this structure (cf. Martin and Sweeney [2]; Hummeltenberg [3] and references therein). In particular, one may consider multi-item production scheduling [4,5,6], sales resource allocation [7], menu planning [8], and catalog space planning [9]. A multiple choice requirement is usually modeled with a generalized upper bound on a set of zero-one variables [10], thus creating the so-called Special Ordered Set (SOS). For instance, the multiple choice requirement Z E { a l , a 2 , . . . , a r } (1) tThis work was partially supported by MDA Project of The International Institute for Applied Systems Analysis (IIASA), Laxenburg, Austria. :~All correspondence should be addressed to: Wtodzimierz Ogryczak, Faculty of Mathematics & Informatics, Warsaw University, Banacha 2, 02-097 Warsaw, Poland. §Wlodzimierz Ogryczak is a Visiting Professor at The Laboratory of Operations Research of U.L.B., Belgium; on leave from Faculty of Mathematics and Informatics, Warsaw University, Poland. He received both his M.S. and Ph.D. in Mathematics and Computer Science from Warsaw University. His interests focus on the computer solution to operations research and management problems, especially in the area of interactive decision support systems for multi-criteria optimization problems and linear programming solvers. He has published two books (in Polish) on computer solution to linear and integer programming problems. His papers have appeared in Computers and Operations Research, European Journal of Operational Research, Journal of Information & Optimization Sciences, Journal of Multi-Criteria Decision Analysis, Linear Algebra and Its Applications, Mathematical Programming Study, Optimization, and others.