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doi: 10.3934/naco.2020054

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

 Department of Industrial Engineering, Yazd University, Yazd, Iran

Received  March 2020 Revised  November 2020 Published  November 2020

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: Yahia Zare Mehrjerdi. A new methodology for solving bi-criterion fractional stochastic programming. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2020054
##### References:
 [1] M. Abdel-Baset and I. M. Hezam, An improved flower pollination algorithm for ratio optimization problems, Applied Mathematics and Information Sciences Letters, 3 (2015), 83-91.   Google Scholar [2] H. Arsham and A. B. Kahn, A complete algorithm for linear fractional programs, Computers & Mathematics with Applications, 20 (1990), 11-23.  doi: 10.1016/0898-1221(90)90344-J.  Google Scholar [3] M. S. Bazaraa and C. M. Shetty, Nonlinear Programming, Theory and Algorithms, Wiley, New York, 1979.  Google Scholar [4] E. T. Babaee, A. Mardani, Z. Dashtian, M. Soltani and G. W. Weber, A novel hybrid method using fuzzy decision making and multi-objective programming for sustainable-reliable supplier selection in two-echelon supply chain design, Journal of Ceaner Production, 250 (2019). Google Scholar [5] S. K. Bhatt, Equivalence of various linearization algorithms for linear fractional programming, ZOR $\pm$ Methods and Models of Operations Research, 33 (1989), 39-43.  doi: 10.1007/BF01415516.  Google Scholar [6] S. Bisoi, G. Devi and A. Rath, Neural Networks for nonlinear fractional programming, International Journal Of Scientific and Engineering Research, 2 (2011), 1-5.  doi: 10.1155/2012/807656.  Google Scholar [7] A. Biswas, Fuzzy goal programming approach for quadratic fractional bi-level programming, International Conference on Scientific Computing (CSC 2011), 2011. Google Scholar [8] C. T. Chang, Fractional programming with absolute-value functions: A fuzzy goal programming approach, Applied Mathematics and Computations, 167 (2005), 508-515.  doi: 10.1016/j.amc.2004.07.014.  Google Scholar [9] A. Charnes and W. W. Cooper, An explicit general solution in linear fractional programming, Naval Research Logistics Quarterly, 20 (1973), 449-467.  doi: 10.1002/nav.3800200308.  Google Scholar [10] A. Charnes and W. W. Cooper, Chance-constrained programming, Management Science, 6 (1962), 73-79.  doi: 10.1287/mnsc.6.1.73.  Google Scholar [11] A. Charnes and W. W. Cooper, Management Models and Industrial Applications of Linear Programming, Wiley, New York. 1961.  Google Scholar [12] A. Charnes and W. W. Cooper, Programming with linear fractional functions, Naval Research Logistics Q, 9 (1962), 181-186.  doi: 10.1002/nav.3800090303.  Google Scholar [13] J. Crouzeix and J. A. Ferland, Algorithms for generalized fractional programming, Matehmatical Programming, 52 (1991), 191-200.  doi: 10.1007/BF01582887.  Google Scholar [14] W. Dinkelbach, On non-linear fractional programming, Management Science, 13 (1967), 492-498.  doi: 10.1287/mnsc.13.7.492.  Google Scholar [15] T. B. Farag, A parametric analysis on multicriteria integer fractional decision making problems, Faculty of Science, Helwan university, 2012. Google Scholar [16] Q. Fu, L. Li, M. Li, T. Li, D. Liu and S. Cui, A simulation based linear fractional programming model for adaptable water allocation planning in the main stream of the Songhua river basin, China Water, 10 (2018), 627. Google Scholar [17] J. Gu, X. Zhang and Y. Cao, Land use structure optimization based on uncertainty fractional joint probabilistic chance constraint programming, Stochastic Environment Research Risk Assessment, 34 (2020), 1699-1712. Google Scholar [18] P. Guo, X. Chen, M. Li and J. Li, Fuzzy chance constrained linear fractional programming approach for optimal water allocation, Stoch. Environ. Res. Risk Assess, (2014), 1601–1612. Google Scholar [19] S. N. Gupta, A chance constrained approach to fractional programming with random numerator, Journal of Mathematical Modeling Algorithm, (2009), Article number: 357. doi: 10.1007/s10852-009-9110-8.  Google Scholar [20] M. B. Hasan and S. Acharjee, Sloving LFP by converting it into a single LP, International Journal of Operations Research, 8 (2011), 1-14.   Google Scholar [21] I. M. Hezam and O. A. Raouf, Particle Swarm optimization programming for solving complex variable fractional programming problems, International Journal of Engineering, 2 (2013), 123-130.   Google Scholar [22] S. Kataoka, A stochastic programming model, Econometrica, 31 (1963), 181-196.  doi: 10.2307/1910956.  Google Scholar [23] P. Lara, Multiple objective fractional programming and livestock ration formulation, A case study for dairy cow diets in spain, Agricultural Systems, 41 (1993), 321-334.   Google Scholar [24] P. Lara, Linking production theory and multi-objective fractional programming as a support tool for animal diet formulation, Advances in Multiple Objective and Goal Programming, (2007), 301–309. doi: 10.1007/978-3-642-46854-4_33.  Google Scholar [25] P. Lara and I. Stancu-Minasian, Fractional programming: a tool for the assessment of sustainability, Agricultural Systems, 62 (1999), 131-141.   Google Scholar [26] M. Li, P. Guo and C. F. Ren, Water resources management models based on the two-level linear fractional programming method under uncertainty, Journal of Water Resources Planning And Management, 141 (2015), ID: 05015001. Google Scholar [27] B. Martos, Nonlinear Programming, Theory and Methods, North-Holland, Amsterdam, 1975.  Google Scholar [28] S. Masatoshi and K. Kosuke, Conducted a research on the interactive decision-making for multi-objective linear fractional programming problems with block angular structure involving fuzzy numbers, Fuzzy Sets and Systems, 97 (1998), 19-31.  doi: 10.1016/S0165-0114(96)00352-1.  Google Scholar [29] B. Metev and D. Gueorguieva, A simple method for obtaining weakly efficient points in multi objective linear fractional programming problems, European Journal of Operational Research, 126 (2000), 386-390.  doi: 10.1016/S0377-2217(99)00298-2.  Google Scholar [30] B. Metev, Use of reference points for solving MONLP problems, European Journal of Operational Research, 80 (1995), 193-203.   Google Scholar [31] A. Pal, S. Singh and K. Deep, Solution of fractional programming problems using PSO algorithm, In Advance Computing Conference(IACC), 2013 IEEE $3^{rd}$ International, 1060–1064. Google Scholar [32] B. B. Pal and I. Basu, A goal programming method for solving fractional programming problems via dynamic programming, Optimization, 35 (1995), 145-157.  doi: 10.1080/02331939508844136.  Google Scholar [33] T. Pena, C. Casterodeza and P. Lara, Environmental criteria in pig diet formulation with multiple objective fractional programming, Handbook of Operations Research in Natural Resources, (2007), 53–68. Google Scholar [34] O. M. Saad and K. Abd-Rabo, On the solution of chance-constrained integer linear fractional programs, In The 32nd Annual Conference, ISSR, Cairo University, Egypt, Part (VI), 32 (1997), 134–140. Google Scholar [35] O. M. Saad, On stability of proper efficient solutions in multi objective fractional programming problems under fuzziness, Mathematical and Computer Modelling, 45 (2007), 221-231.  doi: 10.1016/j.mcm.2006.05.008.  Google Scholar [36] O. M. Saad and W. H. Sharif, On the solution of integer linear fractional programs with uncertain data, Institute of Mathematics & Computer Sciences Journal, 12 (2001), 169-173.   Google Scholar [37] A. Sameeullah and S. D. Devi, Palaniappan, B. Genetic algorithm based method to solve linear fractional programming problem, Asian Journal of Information Technology, 7 (2008), 83-86.   Google Scholar [38] R. Steuer, Multiple Criteria Optimization - Theory, Computation, and Application, Wiley, New York, Chichester, 1986.  Google Scholar [39] A. Udhayakumar, V. Charies and V. R. Uthariaraj, Stochastic simulation based genetic approach for solving chance constrained fractional programming problem, International Journal of Operational Research, 9 (2010), 23-38.  doi: 10.1504/IJOR.2010.034359.  Google Scholar [40] H. Wolf, A parametric method for solving the linear fractional programming problems, Operations Research, 33 (1985), 835-841.  doi: 10.1287/opre.33.4.835.  Google Scholar [41] Y. Zare Mehrjerdi and F. Faregh, Using Stochastic Linear Fractional Programming for Waste Management (case study: Yazd city), Sharif Journal of Industrial Engineering and Management, 2017. Google Scholar [42] Y. Zare Mehrjerdi, A linearization technique for solving chance constrained goal programming problems, Fourth Industrial Engineering Conference in Iran, Tehran, Iran, 2006. Google Scholar [43] Y. Zare Mehrjerdi, A Goal Programming Model of the Stochastic Programming, Vehicle Routing Problem, PHD Thesis, Oklahoma State University. 1986. Google Scholar [44] Y. Zare Mehrjerdi, Linear and Nonlinear Goal Programming, Yazd University Publisher, 2019. Google Scholar [45] Y. Zare Mehrjerdi, Planning and Development of Water Resources Systems: A Multiple Objective Approach, Second Conference on Planning and Development, Tehran, Iran, 1993. Google Scholar [46] Y. Zare Mehrjerdi, A decision-making model for flexible manufacturing system, Assembly Automation, 29 (2009), 32-40.   Google Scholar [47] Y. Zare Mehrjerdi, Solving fractional programming problem through fuzzy goal setting and approximation, Applied Soft Computing, 11 (2011), 1735-1742.   Google Scholar [48] C. Zhang, M. Li and P. Guo, Two stage stochastic chance constrained fractional programming model for optimal agricultural cultivation scale in an Arid area, Journal of Irrigation and Drainage Engineering, 143 (2017). Google Scholar [49] C. Zhou, G. Huang and J. Chen, A type-2 fuzzy chance constrained fractional integrated modeling method for energy system management of uncertainties and risks, Energies, 12 (2019), 2472. Google Scholar [50] H. Zhu and G. H. Hung, SLFP: A stochastic linear fractional programming approach for sustainable waste management, Waste Management, 31 (2011), 2612-2619.   Google Scholar

show all references

##### References:
 [1] M. Abdel-Baset and I. M. Hezam, An improved flower pollination algorithm for ratio optimization problems, Applied Mathematics and Information Sciences Letters, 3 (2015), 83-91.   Google Scholar [2] H. Arsham and A. B. Kahn, A complete algorithm for linear fractional programs, Computers & Mathematics with Applications, 20 (1990), 11-23.  doi: 10.1016/0898-1221(90)90344-J.  Google Scholar [3] M. S. Bazaraa and C. M. Shetty, Nonlinear Programming, Theory and Algorithms, Wiley, New York, 1979.  Google Scholar [4] E. T. Babaee, A. Mardani, Z. Dashtian, M. Soltani and G. W. Weber, A novel hybrid method using fuzzy decision making and multi-objective programming for sustainable-reliable supplier selection in two-echelon supply chain design, Journal of Ceaner Production, 250 (2019). Google Scholar [5] S. K. Bhatt, Equivalence of various linearization algorithms for linear fractional programming, ZOR $\pm$ Methods and Models of Operations Research, 33 (1989), 39-43.  doi: 10.1007/BF01415516.  Google Scholar [6] S. Bisoi, G. Devi and A. Rath, Neural Networks for nonlinear fractional programming, International Journal Of Scientific and Engineering Research, 2 (2011), 1-5.  doi: 10.1155/2012/807656.  Google Scholar [7] A. Biswas, Fuzzy goal programming approach for quadratic fractional bi-level programming, International Conference on Scientific Computing (CSC 2011), 2011. Google Scholar [8] C. T. Chang, Fractional programming with absolute-value functions: A fuzzy goal programming approach, Applied Mathematics and Computations, 167 (2005), 508-515.  doi: 10.1016/j.amc.2004.07.014.  Google Scholar [9] A. Charnes and W. W. Cooper, An explicit general solution in linear fractional programming, Naval Research Logistics Quarterly, 20 (1973), 449-467.  doi: 10.1002/nav.3800200308.  Google Scholar [10] A. Charnes and W. W. Cooper, Chance-constrained programming, Management Science, 6 (1962), 73-79.  doi: 10.1287/mnsc.6.1.73.  Google Scholar [11] A. Charnes and W. W. Cooper, Management Models and Industrial Applications of Linear Programming, Wiley, New York. 1961.  Google Scholar [12] A. Charnes and W. W. Cooper, Programming with linear fractional functions, Naval Research Logistics Q, 9 (1962), 181-186.  doi: 10.1002/nav.3800090303.  Google Scholar [13] J. Crouzeix and J. A. Ferland, Algorithms for generalized fractional programming, Matehmatical Programming, 52 (1991), 191-200.  doi: 10.1007/BF01582887.  Google Scholar [14] W. Dinkelbach, On non-linear fractional programming, Management Science, 13 (1967), 492-498.  doi: 10.1287/mnsc.13.7.492.  Google Scholar [15] T. B. Farag, A parametric analysis on multicriteria integer fractional decision making problems, Faculty of Science, Helwan university, 2012. Google Scholar [16] Q. Fu, L. Li, M. Li, T. Li, D. Liu and S. Cui, A simulation based linear fractional programming model for adaptable water allocation planning in the main stream of the Songhua river basin, China Water, 10 (2018), 627. Google Scholar [17] J. Gu, X. Zhang and Y. Cao, Land use structure optimization based on uncertainty fractional joint probabilistic chance constraint programming, Stochastic Environment Research Risk Assessment, 34 (2020), 1699-1712. Google Scholar [18] P. Guo, X. Chen, M. Li and J. Li, Fuzzy chance constrained linear fractional programming approach for optimal water allocation, Stoch. Environ. Res. Risk Assess, (2014), 1601–1612. Google Scholar [19] S. N. Gupta, A chance constrained approach to fractional programming with random numerator, Journal of Mathematical Modeling Algorithm, (2009), Article number: 357. doi: 10.1007/s10852-009-9110-8.  Google Scholar [20] M. B. Hasan and S. Acharjee, Sloving LFP by converting it into a single LP, International Journal of Operations Research, 8 (2011), 1-14.   Google Scholar [21] I. M. Hezam and O. A. Raouf, Particle Swarm optimization programming for solving complex variable fractional programming problems, International Journal of Engineering, 2 (2013), 123-130.   Google Scholar [22] S. Kataoka, A stochastic programming model, Econometrica, 31 (1963), 181-196.  doi: 10.2307/1910956.  Google Scholar [23] P. Lara, Multiple objective fractional programming and livestock ration formulation, A case study for dairy cow diets in spain, Agricultural Systems, 41 (1993), 321-334.   Google Scholar [24] P. Lara, Linking production theory and multi-objective fractional programming as a support tool for animal diet formulation, Advances in Multiple Objective and Goal Programming, (2007), 301–309. doi: 10.1007/978-3-642-46854-4_33.  Google Scholar [25] P. Lara and I. Stancu-Minasian, Fractional programming: a tool for the assessment of sustainability, Agricultural Systems, 62 (1999), 131-141.   Google Scholar [26] M. Li, P. Guo and C. F. Ren, Water resources management models based on the two-level linear fractional programming method under uncertainty, Journal of Water Resources Planning And Management, 141 (2015), ID: 05015001. Google Scholar [27] B. Martos, Nonlinear Programming, Theory and Methods, North-Holland, Amsterdam, 1975.  Google Scholar [28] S. Masatoshi and K. Kosuke, Conducted a research on the interactive decision-making for multi-objective linear fractional programming problems with block angular structure involving fuzzy numbers, Fuzzy Sets and Systems, 97 (1998), 19-31.  doi: 10.1016/S0165-0114(96)00352-1.  Google Scholar [29] B. Metev and D. Gueorguieva, A simple method for obtaining weakly efficient points in multi objective linear fractional programming problems, European Journal of Operational Research, 126 (2000), 386-390.  doi: 10.1016/S0377-2217(99)00298-2.  Google Scholar [30] B. Metev, Use of reference points for solving MONLP problems, European Journal of Operational Research, 80 (1995), 193-203.   Google Scholar [31] A. Pal, S. Singh and K. Deep, Solution of fractional programming problems using PSO algorithm, In Advance Computing Conference(IACC), 2013 IEEE $3^{rd}$ International, 1060–1064. Google Scholar [32] B. B. Pal and I. Basu, A goal programming method for solving fractional programming problems via dynamic programming, Optimization, 35 (1995), 145-157.  doi: 10.1080/02331939508844136.  Google Scholar [33] T. Pena, C. Casterodeza and P. Lara, Environmental criteria in pig diet formulation with multiple objective fractional programming, Handbook of Operations Research in Natural Resources, (2007), 53–68. Google Scholar [34] O. M. Saad and K. Abd-Rabo, On the solution of chance-constrained integer linear fractional programs, In The 32nd Annual Conference, ISSR, Cairo University, Egypt, Part (VI), 32 (1997), 134–140. Google Scholar [35] O. M. Saad, On stability of proper efficient solutions in multi objective fractional programming problems under fuzziness, Mathematical and Computer Modelling, 45 (2007), 221-231.  doi: 10.1016/j.mcm.2006.05.008.  Google Scholar [36] O. M. Saad and W. H. Sharif, On the solution of integer linear fractional programs with uncertain data, Institute of Mathematics & Computer Sciences Journal, 12 (2001), 169-173.   Google Scholar [37] A. Sameeullah and S. D. Devi, Palaniappan, B. Genetic algorithm based method to solve linear fractional programming problem, Asian Journal of Information Technology, 7 (2008), 83-86.   Google Scholar [38] R. Steuer, Multiple Criteria Optimization - Theory, Computation, and Application, Wiley, New York, Chichester, 1986.  Google Scholar [39] A. Udhayakumar, V. Charies and V. R. Uthariaraj, Stochastic simulation based genetic approach for solving chance constrained fractional programming problem, International Journal of Operational Research, 9 (2010), 23-38.  doi: 10.1504/IJOR.2010.034359.  Google Scholar [40] H. Wolf, A parametric method for solving the linear fractional programming problems, Operations Research, 33 (1985), 835-841.  doi: 10.1287/opre.33.4.835.  Google Scholar [41] Y. Zare Mehrjerdi and F. Faregh, Using Stochastic Linear Fractional Programming for Waste Management (case study: Yazd city), Sharif Journal of Industrial Engineering and Management, 2017. Google Scholar [42] Y. Zare Mehrjerdi, A linearization technique for solving chance constrained goal programming problems, Fourth Industrial Engineering Conference in Iran, Tehran, Iran, 2006. Google Scholar [43] Y. Zare Mehrjerdi, A Goal Programming Model of the Stochastic Programming, Vehicle Routing Problem, PHD Thesis, Oklahoma State University. 1986. Google Scholar [44] Y. Zare Mehrjerdi, Linear and Nonlinear Goal Programming, Yazd University Publisher, 2019. Google Scholar [45] Y. Zare Mehrjerdi, Planning and Development of Water Resources Systems: A Multiple Objective Approach, Second Conference on Planning and Development, Tehran, Iran, 1993. Google Scholar [46] Y. Zare Mehrjerdi, A decision-making model for flexible manufacturing system, Assembly Automation, 29 (2009), 32-40.   Google Scholar [47] Y. Zare Mehrjerdi, Solving fractional programming problem through fuzzy goal setting and approximation, Applied Soft Computing, 11 (2011), 1735-1742.   Google Scholar [48] C. Zhang, M. Li and P. Guo, Two stage stochastic chance constrained fractional programming model for optimal agricultural cultivation scale in an Arid area, Journal of Irrigation and Drainage Engineering, 143 (2017). Google Scholar [49] C. Zhou, G. Huang and J. Chen, A type-2 fuzzy chance constrained fractional integrated modeling method for energy system management of uncertainties and risks, Energies, 12 (2019), 2472. Google Scholar [50] H. Zhu and G. H. Hung, SLFP: A stochastic linear fractional programming approach for sustainable waste management, Waste Management, 31 (2011), 2612-2619.   Google Scholar
Simulation Annealing Goal Technique
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 $-f_{8}(x)$
 $(x_{1}^{*}, x_{2}^{*})$ 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 $-f_{8}(x)$
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
 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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