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

This issue Previous Article A primal-dual interior point method for

$ P_{\ast }\left( \kappa \right) $

-HLCP based on a class of parametric kernel functions Next Article Global and regional constrained controllability for distributed parabolic linear systems: RHUM approach

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

Abstract / Introduction Full Text(HTML) Figure(1) / Table(2) Related Papers Cited by

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.

Keywords:

Citation:

Full Text(HTML)

Figure 1. Simulation Annealing Goal Technique

Download: Full-size image PowerPoint slide

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

DownLoad: CSV

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

Related Papers

Cited by

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.
[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).
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[42]	Y. Zare Mehrjerdi, A linearization technique for solving chance constrained goal programming problems, Fourth Industrial Engineering Conference in Iran, Tehran, Iran, 2006.
[43]	Y. Zare Mehrjerdi, A Goal Programming Model of the Stochastic Programming, Vehicle Routing Problem, PHD Thesis, Oklahoma State University. 1986.
[44]	Y. Zare Mehrjerdi, Linear and Nonlinear Goal Programming, Yazd University Publisher, 2019.
[45]	Y. Zare Mehrjerdi, Planning and Development of Water Resources Systems: A Multiple Objective Approach, Second Conference on Planning and Development, Tehran, Iran, 1993.
[46]	Y. Zare Mehrjerdi, A decision-making model for flexible manufacturing system, Assembly Automation, 29 (2009), 32-40.
[47]	Y. Zare Mehrjerdi, Solving fractional programming problem through fuzzy goal setting and approximation, Applied Soft Computing, 11 (2011), 1735-1742.
[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).
[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.
[50]	H. Zhu and G. H. Hung, SLFP: A stochastic linear fractional programming approach for sustainable waste management, Waste Management, 31 (2011), 2612-2619.