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# A new methodology for solving bi-criterion fractional stochastic programming

• Solving a bi-criterion fractional stochastic programming using an existing multi criteria decision making tool demands sufficient efforts and it is time consuming. There are many cases in financial situations that a nonlinear fractional programming, generated as a result of studying fractional stochastic programming, must be solved. Often management is not in needs of an optimal solution for the problem but rather an approximate solution can give him/her a good starting for the decision making or running a new model to find an intermediate or final solution. To this end, this author introduces a new linear approximation technique for solving a fractional stochastic programming (CCP) problem. After introducing the problem, the equivalent deterministic form of the fractional nonlinear programming problem is developed. To solve the problem, a fuzzy goal programming model of the equivalent deterministic form of the fractional stochastic programming is provided and then, the process of defuzzification and linearization of the problem is presented. A sample test problem is solved for presentation purposes. There are some limitations to the proposed approach: (1) solution obtains from this type of modeling is an approximate solution and, (2) preparation of approximation model of the problem may take some times for the beginners.

 Citation: • • Figure 1.  Simulation Annealing Goal Technique

Table 2.  upper and lower bound functions and their optimal solution points on the defined feasible region

 $(x_{1}^{*}, x_{2}^{*})$ 1 $f_{1}(x)=8x_{1}+7x_{2}$ (6.6667, 4.6667) 86 Max $f_{1}(x)$ 2 $f_{2}(x)=1.01x_{1}+2.34x_{2}$ (0, 8) 18.72 Max $f_{2}(x)$ 3 $f_{3}(x)=20x_{1}+12x_{2}$ (6.6667, 4.6667) 189 Max $-f_{3}(x)$ 4 $f_{4}(x)=15.964x_{1}+7.3x_{2}$ (9, 0) 143.6789 Max $-f_{4}(x)$ 5 $f_{5}(x)=5x_{1}+4x_{2}$ (6.6667, 4.6667) 52.0003 Max $f_{5}(x)$ 6 $f_{6}(x)=-4.32x_{1}=2.99x_{2}$ (0, 0) 0 Max $f_{6}(x)$ 7 $f_{7}(x)=10x_{1}+9x_{2}$ (0, 0) 0 Max $-f_{7}(x)$ 8 $f_{8}(x)=3.01x_{1}+4.34x_{2}$ (0, 0) 0 Max $-f_{8}(x)$

Table 3.  final solution obtained for the fractional programming problem

 Problems $(x_{1}^{*}, x_{2}^{*})$ Goal Programming Priority Achievement Level $Z_{1}(x^{*})$ $Z_{2}(x^{*})$ Upper bound (0.593, 0.692) 3.593 0.3643 0.5401 Lower bound (0, 7.173) 2.44 0.3477 0.1261
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