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A new methodology for solving bi-criterion fractional stochastic programming
Department of Industrial Engineering, Yazd University, Yazd, Iran |
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.
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. |
[3] |
M. S. Bazaraa and C. M. Shetty, Nonlinear Programming, Theory and Algorithms, Wiley, New York, 1979. |
[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. |
[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. |
[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. |
[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. |
[10] |
A. Charnes and W. W. Cooper,
Chance-constrained programming, Management Science, 6 (1962), 73-79.
doi: 10.1287/mnsc.6.1.73. |
[11] |
A. Charnes and W. W. Cooper, Management Models and Industrial Applications of Linear Programming, Wiley, New York. 1961. |
[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. |
[13] |
J. Crouzeix and J. A. Ferland,
Algorithms for generalized fractional programming, Matehmatical Programming, 52 (1991), 191-200.
doi: 10.1007/BF01582887. |
[14] |
W. Dinkelbach,
On non-linear fractional programming, Management Science, 13 (1967), 492-498.
doi: 10.1287/mnsc.13.7.492. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[3] |
M. S. Bazaraa and C. M. Shetty, Nonlinear Programming, Theory and Algorithms, Wiley, New York, 1979. |
[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. |
[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. |
[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. |
[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. |
[10] |
A. Charnes and W. W. Cooper,
Chance-constrained programming, Management Science, 6 (1962), 73-79.
doi: 10.1287/mnsc.6.1.73. |
[11] |
A. Charnes and W. W. Cooper, Management Models and Industrial Applications of Linear Programming, Wiley, New York. 1961. |
[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. |
[13] |
J. Crouzeix and J. A. Ferland,
Algorithms for generalized fractional programming, Matehmatical Programming, 52 (1991), 191-200.
doi: 10.1007/BF01582887. |
[14] |
W. Dinkelbach,
On non-linear fractional programming, Management Science, 13 (1967), 492-498.
doi: 10.1287/mnsc.13.7.492. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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 |

1 | (6.6667, 4.6667) | 86 | Max |
|
2 | (0, 8) | 18.72 | Max |
|
3 | (6.6667, 4.6667) | 189 | Max |
|
4 | (9, 0) | 143.6789 | Max |
|
5 | (6.6667, 4.6667) | 52.0003 | Max |
|
6 | (0, 0) | 0 | Max |
|
7 | (0, 0) | 0 | Max |
|
8 | (0, 0) | 0 | Max |
1 | (6.6667, 4.6667) | 86 | Max |
|
2 | (0, 8) | 18.72 | Max |
|
3 | (6.6667, 4.6667) | 189 | Max |
|
4 | (9, 0) | 143.6789 | Max |
|
5 | (6.6667, 4.6667) | 52.0003 | Max |
|
6 | (0, 0) | 0 | Max |
|
7 | (0, 0) | 0 | Max |
|
8 | (0, 0) | 0 | Max |
Problems | Goal Programming Priority Achievement Level | |||
Upper bound | (0.593, 0.692) | 3.593 | 0.3643 | 0.5401 |
Lower bound | (0, 7.173) | 2.44 | 0.3477 | 0.1261 |
Problems | Goal Programming Priority Achievement Level | |||
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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