Duality for a nonlinear fractional programming under fuzzy environment with parabolic concave membership functions

Article history: Received 15 November 2016 Received in revised form 10 January 2017 Accepted 21 January 2017 A particular type of convex fractional programming problem and its dual is studied under fuzzy environment with parabolic concave membership functions. Appropriate duality results are established using aspiration level approach. The use of parabolic concave membership functions to represent the degree of satisfaction of the decision maker makes it unique from the other studies.


*Mathematical
programming finds many applications in the field of management. Optimization of resources in any organization is very much handled by the application of mathematical programming. An important class of mathematical programming problems is fractional programming which deals with situations where a ratio between two mathematical functions is either maximized or minimized.
There are many managerial decision making situations where the uncertainties in working situations is best explained by fuzzy set theory. The concept of fuzzy set theory is introduced by Zadeh (1965), since then a large number of researchers have shown their interest in the application of fuzzy set theory. Bellman and Zadeh (1970) proposed the concept of decision making in fuzzy environment and their concept of fuzzy decision making is used by Tanaka et al. (1984) in mathematical programming. There are many authors have discussed the use of fuzzy set theory in fractional programming e.g., Luhandjula (1984), Dutta et al. (1992), Ravi and Reddy (1998), Gupta and Bhatia (2001), Chakraborty and Gupta (2002), Pop and Stancu-Minasian (2003), and Stancu-Minasian and Pop (2008).
The duality theory plays a very important role in the theory of linear programming so researchers have shown their interest in the concept of duality for a linear program under fuzzy environment as well e.g., Hamacher et al. (1978), Rödder and Zimmermann (1980), Bector and Chandra (2002) and few others. However, only a few studies exploring duality in fractional programming under fuzzy environment are available in literature.  studied duality for a fuzzy multiobjective linear fractional programming problem and developed a parallel algorithm. Wu (2007) developed duality theory in fuzzy optimization problems formulated by the Wolfe's primal and dual pair. Gupta and Mehlawat (2009a) studied duality for a convex fractional programming under fuzzy environment using linear membership functions.
It is important to note that while implementing any fuzzy mathematical programming problem on the basis of aspiration levels the choice of membership function is very important. The chosen membership function should be able to produce desired satisfaction level of the objective of the decision maker. Several membership functions have been employed in fuzzy mathematical programming: (i) linear (Zimmermann, 2001) (ii) piecewise linear (Inuiguchi et al., 1990) (iii) parabolic concave (Saxena and Jain, 2014) (iv) exponential (Gupta and Mehlawat, 2009b;Li and Lee, 1991). In many practical situations, however, a linear membership function is not a suitable representation which is empirically shown by Hersh and Caramazza (1976). A nonlinear membership function can be used to obtain the desired degree of satisfaction of the objective of the decision maker. However it must be noted that the results obtained for a fuzzy environment must conform to the corresponding results for the crisp situation.
In this paper, we attempt to obtain duality results between a particular type of convex fractional programming problem and its dual under fuzzy environment using a nonlinear membership function i.e., parabolic concave membership function. The use of parabolic concave membership functions to obtain the desired satisfaction level of the decision maker's objective makes it unique study in this direction. The duality results obtained under fuzzy environment are also conforming to the corresponding duality results for the crisp situation. The economic interpretation of these results can be understood as explained by Rödder and Zimmermann (1980). The paper is organized into six sections. Section 2 contains notation and prerequisites. In section 3 a pair of fuzzy primal and dual problems for convex fractional programming is presented. In section 4 a modified weak duality theorem and some other related results are proved. In section 5 a numerical example is presented to verify the results established in section 4. A conclusion is presented in the final section 6.

Fractional programming problems
Now consider the fuzzy versions (̃) and (̃) of (P) and (D) respectively, in the sense of Rödder and Zimmermann (1980).
(̃) Find ∈ , ∈ , such that (Eqs. 7 and 8): ( , ) = > 0 (7) Here, "> " and "< " are fuzzy versions of symbols " ≥" and "≤ " respectively, and have the linguistic interpretation "essentially greater than or equal" and "essentially less than or equal" as explained in Rödder and Zimmermann (1980). These indicate that the inequalities are flexible and may be described by a fuzzy set whose membership function represents fulfillment of the decision maker's satisfaction. Also 0 and 0 are aspiration level of the two objectives. We now assume 0 > 0, > 0, ( = 1,2,..., ) as subjectively chosen constants of admissible violations such that 0 is associated with the objective function and ( = 1,2,..., ) is associated with the i-th linear constraint of (P). Now we define parabolic concave membership functions 0 ( ( )): → [0,1] and ( ): → [0,1], ( = 1,2, … , ) for objective function and constraints of the problem (̃) to obtain a degree of satisfaction in the problem: Using the "min" operator to aggregate the overall satisfaction and following Rödder and Zimmermann (1980) with these membership functions, the crisp equivalent of the fuzzy primal convex fractional programming problem (̃) is as follow: We name the pair (EP)-(ED) as the modified primal-dual pair of fuzzy convex fractional programming problems.

Theorem and related results
Now we establish appropriate duality results for the modified primal-dual pair (EP)-(ED) (or equivalently (CP)-(CD)).
Remark 3: Since, (CD) is not a dual to (CP) in the conventional sense, however (CP) and (CD) are the crisp equivalent of the fuzzy pair (̃) and (̃) respectively, therefore, there does not exist any strong duality theorem between them. However, in addition to = 1 and = 1, we also have 0 − 0 = 0, then inequalities (13) and (14) yields ( , ) = ( ) i.e., and ( , ) become optimal solution to the problems (P) and (D) respectively.

Numerical example
In this section we present a simple numerical example to illustrate the construction of the fuzzy primal-dual pair and also to verify the modified weak duality theorem.

Conclusion
In this paper, along the lines of Gupta and Mehlawat (2009a), a pair of primal and dual programs for a convex fractional program under fuzzy environment is presented with parabolic concave membership functions. We consider a conventional primal-dual pair as (P) and (D) and obtain their fuzzified versions (̃) and (̃) using Rödder and Zimmermann (1980) approach. Next, using Saxena and Jain (2014) and Rödder and Zimmermann (1980), the crisp formulations of (̃) and (̃) are obtained as (EP) and (ED) (or equivalently (CP) and (CD)) respectively. A modified weak duality theorem relating feasible solution of (EP) and (ED) is proved and a corollary is also proved relating optimal solutions of (EP) and (ED). The crisp equivalents (EP) and (ED) obtained are nonlinear problems, where non linearity exists in the constraints. Software LINGO (Scharge, 1997) has been used to solve the numerical illustration. Fuzzy decisive set method (Sakawa and Yano, 1985) and the modified subgradient method (Gasimov, 2002) can also be used to solve these problems. Duals of other types can also be examined under fuzzy environment for the fractional programming problem under consideration for similar kinds of results as obtained in this paper. The approach developed here will prove helpful for possible extensions to linear fractional and quadratic fractional programs and to various other nonlinear fractional programming problems under fuzzy environment with parabolic concave membership functions. Other nonlinear membership functions such as hyperbolic, exponential etc. can also be employed, provided it should conform to the corresponding duality results for the crisp situation.