A new methodology for solving bi-criterion fractional stochastic programming

Yahia Zare Mehrjerdi

doi:10.3934/naco.2020054

Article Contents

2021, Volume 11, Issue 4: 533-554. Doi: 10.3934/naco.2020054

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A new methodology for solving bi-criterion fractional stochastic programming

Yahia Zare Mehrjerdi

Department of Industrial Engineering, Yazd University, Yazd, Iran

Received: March 2020

Revised: November 2020

Early access: November 2020

Published: December 2021

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Abstract

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

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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) $

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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

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