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A new methodology for solving bi-criterion fractional stochastic programming

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  • 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.

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  • 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) $
     | Show Table
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    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
     | Show Table
    DownLoad: CSV
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