Available online at www.sciencedirect.com Fuzzy Sets and Systems 146 (2004) 253–269 www.elsevier.com/locate/fss Duality in linear programming with fuzzy parameters and matrix games with fuzzy pay-os C.R. Bector a , S. Chandra b; ∗ , Vidyottama Vijay b a Department of Business Administration, University of Manitoba, Winnipeg, Man., Canada R3T 5V4 b Department of Mathematics, Indian Institute of Technology, Hauz Khas, New Delhi 110016 India Received 7 August 2002; received in revised form 23 May 2003; accepted 11 June 2003 Abstract A dual for linear programming problems with fuzzy parameters is introduced and it is shown that a two person zero sum matrix game with fuzzy pay-os is equivalent to a primal–dual pair of such fuzzy linear programming problems. Further certain diculties with similar studies reported in the literature are discussed. c  2003 Elsevier B.V. All rights reserved. Keywords: Fuzzy numbers; Fuzzy matrix game; Fuzzy duality 1. Introduction One of the most celebrated and useful result in the matrix game theory asserts that every two person zero sum matrix game is equivalent to two linear programming problems which are dual to each other. Thus, solving such a game amounts to solving any one of these two mutually dual linear programming problems and obtaining the solution of the other by using linear programming duality theory. The earliest study of two person zero sum matrix game with fuzzy pay-os is due to Campos [2] which still remains the most basic reference on this topic. Later Nishizaki and Sakawa [9] extended these ideas of Campos [2] to multiobjective matrix games as well. Though these studies have been motivated by the classical (crisp) two person zero sum matrix game theory but unlike their crisp counter parts, they do not take into consideration the fuzzy linear programming duality aspects and, therefore, do not seem to fully conceptualize the fuzzy matrix game model. In this context it may be noted that although certain fuzzy linear programming duality results are available * Corresponding author. Tel.: +91-11-6591479; fax: +91-11-6862037. E-mail address: chandras@maths.iitd.ernet.in (S. Chandra). 0165-0114/$ - see front matter c  2003 Elsevier B.V. All rights reserved. doi:10.1016/S0165-0114(03)00260-4