On duality in linear programming under fuzzy environment C.R. Bector^*, S. Chandra b aDepartment of Business Administration, University of Manitoba, Winnipeg, Man., Canada R3T 5V4 h Department of Mathematics, Indian Institute of Technology, Hauz Khas, New Delhi 110016, India Received 21 August 1998; received in revised form 1 February 2000; accepted 13 April 2000 Abstract A pair of linear primal-dual problems is introduced under fuzzy environment and appropriate results are proved to establish the duality relationship between them. Possible extensions are also suggested. Keywords: Linear primal-dual problems; Fuzzy environment 1. Introduction A number of researchers have exhibited their interest in the topic of fuzzy linear programming after Zadeh [5] developed the concept of fuzzy set theory. However, in contrast with the vast literature on modeling and solution procedures for a linear program in a fuzzy environment (see, for example, [3,7]), the studies in duality are rather scarce. The most basic results on duality in fuzzy linear programming are due to Rodder and Zimmermann [4] and Hamacher et al. [2]. In [4] a generalization of maxmin and minmax problems in a fuzzy environment is presented and thereby a pair of fuzzy dual linear programming problems is constructed. An economic interpretation of this duality in terms of market and industry is also included in [4]. The paper by Hamacher et al. [2] is mostly devoted to the study of sensitivity analysis in fuzzy linear programming. The purpose of the present paper is twofold. Firstly, to observe that there are certain inherent di:culties with the fuzzy dual formulations of [4] because when they are specialized to the crisp situation they do not lead to a standard primal-dual pair for linear programming, and secondly, to construct a modi;ed pair of fuzzy primal-dual linear programming problems. To achieve this, we divide the paper into three sections. In Section 1, we introduce the fuzzy dual formulation of Rodder and Zimmermann [4] and make certain observations on their formulation. In Section 2 a modi;ed formulation is presented, whereas its comparison and possible extensions are included in Section 3.