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Top1. Introduction
Since the inception of notion of fuzzy sets by Zadeh in 1965, the researchers have advanced it in a variety of ways and in many disciplines. Its applications can be found in many areas such as computer science, decision theory, expert systems, operations research, pattern recognition, artificial intelligence and optimization. The concept of decision making in fuzzy environment was first proposed by Bellman and Zadeh (1970). Subsequently, Tanaka et al. (1984) made use of this concept in mathematical programming. Since then the fuzzy set theory has been extensively used to capture linguistic uncertainty in optimization problems. A number of researchers have shown their interest to handle this uncertainty under the class of fuzzy mathematical programming problems (e.g., see (Hamacher et al., 1978; Zimmerman, 1978; Zimmerman, 2001; Gupta and Mehlawat, 2009; Saxena and Jain, 2014; Lai and Hwang, 1992)).
Fractional programming has attracted the attention of many researchers in the past (Schaible and Ibarki, 1983; Steuer, 1986; Stancu-Minasian, 1999). The importance of linear fractional programming comes from the fact that many real-life problems are based on optimization of ratio of physical and/or economic quantities (for example profit/cost, cost/time, cost/volume or any other quantity that measure the efficiency of a system) expressed by linear functions. Literature survey reveals wide applications of linear fractional programming in different areas ranging from engineering to economics.
A fascinating duality theory is available in the literature for linear fractional programming in crisp environment. The different authors have introduced different dual for a given linear fractional program. They developed duality theorems (i) weak duality (ii) direct duality (iii) converse duality and (iv) complementary slackness as per the dual introduced by them (Charnes and Cooper, 1962; Swarup, 1968; Kaska, 1969; Chadha, 1971; Mond and Craven, 1973; Seshan, 1980; Chadha and Chadha, 2007). For the purpose of our study, we have chosen the most appropriate dual as the dual presented by Chadha and Chadha (2007) for a given linear fractional program and corresponding duality results are established under fuzzy environment.
However, a vast literature is available on modelling and solution procedures for a linear programming problem in fuzzy environment but only a few researchers exhibit their interest in the concept of duality. The most basic results available on duality of linear programming in fuzzy environment are evaluated by Hamacher et al. (1978) and Rödder and Zimmermann (1980). Therefore, it is imperative and interesting to study duality for linear fractional programming under fuzzy environment. In recent years, several authors have worked on duality in (i) linear programming (Gupta and Mehlawat, 2009; Saxena and Jain, 2014; Bector and Chandra, 2002) (ii) quadratic programming (Bector and Chandra, 2005) (iii) convex fractional programming (Gupta and Mehlawat, 2009; Saxena and Jain 2017) under fuzzy environment and established duality results. In this paper the duality results have been established for a linear fractional program under fuzzy environment.