EUROPEAN JOURNAL OF OPERATIONAL RESEARCH ELSEVIER European Journal of Operational Research 85 (1995) 625-635 I Theory and Methodology Average shadow price in a Mixed Integer Linear programming problem Alejandro Crema Escuela de Computacidn, Facultad de Ciencias, Universidad Central de Venezuela, Apartado 47002, Caracas 1041-A, Venezuela Received April 1993; revised July 1993 Abstract Recently a new concept of shadow price, the average shadow price, based on the average and not on the marginal contribution of a resource, has been developed for pure Integer Linear Programming problems. In this paper we prove that average shadow prices can be used in a Mixed Integer Linear Programming problems and that some of its properties are analogous with the properties of shadow prices in a Linear Programming problem. A natural link with the right-hand-side parametric problem leads us to a proof of completeness for parametric analysis. Keywords: Shadow prices; Langrangean relaxation; Integer Programming 1. Introduction The concept of shadow price has been studied by many researches in Mathematical Programming and Operations Research because of its economical interpretation and its relation with duality theory. In Linear and Convex Programming the concept has been well studied (Akgull [1], Aucamp andi Steinberg [3], Gal [4], Shapiro [10, pp. 36-38] and many others). All these studies are based on the Concept of marginal contribution of a resource (or a 'package of resources') to the optimal objective value. The marginal analysis may not be useful in (pure or mixed) Integer Linear Programming because the objective function is neither concave nor convex when the availability of one or more resources changes. In 1988, Kim and Cho [9] analyzed a new concept, which they called average shadow price (a.s.p), with rich economical interpretation in pure Integer Linear Programming problems. The analysis of Kim and Cho is based on the average and not on the marginal contribution of a resource. Previous work about shadow prices in Integer Linear Programming have been published by Gomory and Baumol [6] and Alcaly and Klevorick [2]. In this paper we prove that a.s.p can be used in Mixed Integer Linear Programming problems and that some of its properties are analogous with the properties of the shadow prices in a Linear PrOgramming problem. In Section 2 we review the concept and present a procedure that finds the exact value of the a.s.p, by solving a finite sequence of Mixed Integer Linear Programming problems. Section 3 presents some relations between a.s.p and optimal lagrangean multipliers (Geoffrion [5]). Section 4 presents a procedure to obtain the net profit function. In Section 5 we establish a natural link with the right-hand-side (r.h.s) parametric problem (see the paper of Jenkins [7] for a survey of parametric problems in Integer Linear Programming). Concluding comments are given in Section 6. 0377-2217/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSD1 0377-2217(94)00003-U