In this paper, we used the S-curve membership function methodology in a real life problem of mix product selection. This is referred here to as the product mix selection problem (Tabucanon, 1996). This problem occurs in production planning management where a decision maker plays very important role in making a decision in a fuzzy environment. As an analyst, we try to find the best solution for the decision maker to identify a final decision in order to implement it. We have considered this problem because it is very important to decision maker to make a decision when all parameters, such as objective coefficients, technical coefficients and resource variables are uncertain. Therefore, it is inevitable to solve this problem by using fuzzy linear programming approach. A real life industrial problem is selected and a solution is achieved.
As stated by Rubin and Narasimhan (1984) that the heart of the methodology for the FLP problems lies in the construction of membership function for the objection coefficients, technical coefficients, resource variables and decision variables. In this regards many researches used FLP approach in solving its applications problems in their work (Buckley and Feuring, 2000, Chou and Chen, 1996, Guu and Wu, 1999, Inuiguchi and Ramik, 2000, Maleki et al., 2000, Nishizaki and Sakawa, 2001, Parra et al., 1999a, Parra et al., 1999b, Rommelfanger, 1996, Sengupta et al., 2001 and Wang and Wang, 1999).
The transitional step towards fuzzy linear programming models is models that consider some fuzzy values. Some of these models are linear mathematical of objective decision making presented by mainly crisp and some fuzzy values. Many authors studied such models (Cheng, 1999, Chuang, 1986, Lai and Hwang, 1993, Lai and Hwang, 1994, Lai, 1995 and Zimmermann, 1978). Zimmermann offered the solution for the formulation by fuzzy linear programming (Zimmermann, 1978). Lai’s interactive technique contributed to the improvement of flexibility and robustness of objective decision making methodology (Lai, 1995). Bellman and Zadeh (1970) has proposed one of the most powerful tools to deal with problem in fuzzy environment. Fuzzy linear programming has been making tremendous progress after they were given life by Bellman and Zadeh (1970).
Various types of membership functions were used in fuzzy linear programming problem and its application such as a linear membership function (Gasimov and Yenilmez, 2002, Zimmermann, 1976 and Zimmermann, 1978), the linear fuzzy constraints (Dubois & Prade, 1980), a tangent type of a membership function (Leberling, 1981), an interval linear membership function (Hannan, 1981), an exponential membership function (Carlsson & Korhonen, 1986), inverse tangent membership function (Sakawa, 1983), logistic type of membership function (Watada, 1997), concave piecewise linear membership function (Inuiguchi, Ichihashi, & Kume, 1990), piecewise linear membership function (Hu and Fang, 1998 and Hu and Fang, 1999), flexible membership function (Bells, 1999) and dynamics membership function (Buller, 2002). As a tangent type, of a membership function, an exponential membership function, and hyperbolic membership function are non-linear function; a fuzzy mathematical programming defined with a non-linear membership function results in a non-linear programming. Usually a linear membership function is employed in order to avoid non-linearity. Nevertheless, there are some difficulties in selecting the solution of a problem written in a linear membership function (Watada, 1997). Therefore, in this paper a modified S-curve membership function is employed to overcome such deficits, which a linear membership function has. Furthermore, S-curve membership function is more flexible enough to describe the vagueness in the fuzzy parameters for the production planning problems.
Ramik (2002) solved the investment problem using fuzzy linear programming approach and linear membership function. In this research work the S-curve (Kuz’min, 1981) membership function is adopted in a real life industrial production planning of a chocolate manufacturing unit. The unit produces 8 products using 8 raw materials; mixed in various proportions by 9 different processes under 29 constraints. This complex problem has 8 inherent sub-problems. A set of solutions to all these sub-problems are achieved thus establishing the usefulness of the S-curve membership function for decision making in industrial production planning. The paper is organized as follows: next section is about statement of chocolate manufacturing problem with various cases. Section 3 deals with construction of modified S-curve membership function. Section 4 provides the computational result for the fuzzy mix product selection problem. The results of seven cases are presented in the performance measure in table form. The paper ends with conclusion and future research work.
