Abstract
A suggested algorithm to solve triangular fuzzy rough integer linear programming (TFRILP) problems with α-level is introduced in this paper in order to find rough value optimal solutions and decision rough integer variables, where all parameters and decision variables in the constraints and the objective function are triangular fuzzy rough numbers. In real-life situations, the parameters of a linear programming problem model may not be defined precisely, because of the current market globalization and some other uncontrollable factors. In order to solve this problem, a proper methodology is adopted to solve the TFRILP problems by the slice-sum method with the branch-and-bound technique, through which two fuzzy integers linear programming (FILP) problems with triangular fuzzy interval coefficients and variables were constructed. One of these problems is an FILP problem, where all of its coefficients are the upper approximation interval and represent rather satisfactory solutions; the other is an FILP problem, where all of its coefficients are the lower approximation interval and represent completely satisfactory solutions. Moreover, α-level at α = 0.5 is adopted to find some other rough value optimal solutions and decision rough integer variables. Integer programming is used, since a lot of the linear programming problems require that the decision variables be integers. In addition, the motivation behind this study is to enable the decision makers to make the right decision considering the proposed solutions, while dealing with the uncertain and imprecise data. A flowchart is also provided to illustrate the problem-solving steps. Finally, two numerical examples are given to clarify the obtained results.
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References
Ammar EE, Khalifa AM (2015) On multi objective linear programming problems with inexact rough interval fuzzy coefficients. Int J Comput Appl 14(5):5742–5758
Ammar E, Zheng Z, Brikaa MG (2019) Rough set approach to non-cooperative continuous differential games. Granul Comput. https://doi.org/10.1007/s41066-019-00179-1
Atteya TEM (2016) Rough multiple objective programming. Eur J Oper Res 248(1):204–210
Bazaraa MS, Jarvis JJ, Sherali HD (2010) Linear programming and network flows. Wiley, New York
Bellman RE, Zadeh LA (1970) Decision making in a fuzzy environment. Manag Sci 17:141–164
Bera S, Roy SK (2018) Fuzzy rough soft set and its application to lattice. Granul Comput. https://doi.org/10.1007/s41066-018-00148-0
Bhaumik A, Roy SK (2019) Intuitionistic interval-valued hesitant fuzzy matrix games with a new aggregation operator for solving management problem. Granul Comput. https://doi.org/10.1007/s41066-019-00191-5
Brikaa MG, Zheng Z, Ammar E (2019) Fuzzy multi-objective programming approach for constrained matrix games with payoffs of fuzzy rough numbers. Symmetry 11(5):7021–7026
Chakraborty D, Roy TK (2019) A fuzzy rough multi-objective, multi-item inventory model with both stock-dependent demand and holding cost rate. Granul Comput 4:71–88
Chen SM, Chen SW (2014) Fuzzy forecasting based on two-factors second-order fuzzy-trend logical relationship groups and the probabilities of trends of fuzzy logical relationships. IEEE Trans Cybern 45(3):391–403
Chen SM, Niou SJ (2011) Fuzzy multiple attributes group decision-making based on fuzzy preference relations. Expert Syst Appl 38(4):3865–3872
Chen SM, Chu HP, Sheu TW (2012) TAIEX forecasting using fuzzy time series and automatically generated weights of multiple factors. IEEE Trans Syst Man Cybern Part A Syst Hum 42(6):1485–1495
Chen SM, Manalu GMT, Pan JS, Liu HC (2013) Fuzzy forecasting based on two-factors second-order fuzzy-trend logical relationship groups and particle swarm optimization techniques. IEEE Trans Cybern 43(3):1102–1117
Chinneck JW, Ramadan K (2000) Linear programming with interval coefficients. J Oper Res Soc 51(2):209–220
Dantzig G, Wolfe P (1961) The decomposition algorithm for linear programming. Econometric 9(4):767–778
Dubois D, Prade H (1990) Rough fuzzy sets and fuzzy rough sets. Int J Gen Syst 17(2):191–209
Garai T, Chakraborty D, Roy TK (2017a) Expected value of exponential fuzzy number and its application to multi-item deterministic inventory model for deteriorating items. J Uncertain Anal Appl 5:1–8
Garai T, Roy TK, Chakraborty D (2017b) Possibility necessity credibility measures on generalized intuitionistic fuzzy number and its applications to multi-product manufacturing system. J Granul Comput 2:1–15
Garai T, Chakraborty D, Roy TK (2019a) Multi-objective, inventory model with both stock-dependent demand rate and holding cost rate under fuzzy random environment. Ann Data Sci 5:1–21
Garai T, Chakraborty D, Roy TK (2019b) Fully fuzzy inventory model with price-dependent demand and time varying holding cost under fuzzy decision variables. J Intell Fuzzy Syst 36(4):3725–3738
Gupta PK, Mohan Man (2006) Problems in operations research. Sultan Chand and Sons, New Delhi
Gutierrez F, Lujan E, Asmat R, Vergara E (2019) Fully fuzzy linear programming model for the berth allocation problem with two quays. Uncertain Manag Fuzzy Rough Sets 377:87–113
Hamazehee A, Yaghoobi MA, Mashinchi M (2014) Linear programming with rough interval coefficients. J Intell Fuzzy Syst 26:1179–1189
Li D-f, Hong F-x (2013) Alfa-cut based linear programming methodology for constrained matrix games with payoffs of trapezoidal fuzzy numbers. Fuzzy Optim Decis Mak 12:191–213
Liu F, Yuan XH (2007) Fuzzy number intuitionistic fuzzy set. Fuzzy Syst Math 21(1):88–91
Midya S, Roy SK (2017) Analysis of interval programming in different environments and its application to fixed-charge transportation problem. Discret Math Algorithms Appl 9(3):1750040
Olga IA, Doina F, Gheorghe P, Codruta OH (2009) “WinQSB” simulation software—a tool for professional development. Sci Direct 1(4):2786–2790
Osman MS, El-Sherbiny MM, Khalifa HA, Farag HH (2016) A Fuzzy technique for solving rough interval multi objective transportation problem. Int J Comput Appl 147(10):49–57
Pamucar D, Cirovic G, Bozanic D (2019) Application of interval valued fuzzy-rough numbers in multi-criteria decision making: the IVFRN-MAIRCA model. Yugosl J Oper Res 29(2):221–247
Pandian P, Natarajan G, Akilbasha A (2016) Fully rough integer interval transportation problems. Int J Pharm Technol 8(2):13866–13876
Pandian P, Natarajan G, Akilbasha A (2018) Fuzzy interval integer transportation problems. Int J Pure Appl Math 119(9):133–142
Pawlak Z (1982) Rough Sets. Int J Comput Inf Sci 11:341–356
Pawlak Z, Skowron A (2007) Rudiment of rough sets. Inf Sci 177:3–27
Roy SK, Mula P (2015) Rough set approach to bi-matrix game. Int J Oper Res 23(2):229–244
Roy SK, Mula P (2016) Solving a matrix game, with rough payoffs using genetic algorithm. Oper Res Int J 16:117–130
Roy SK, Midya S, Vincent FY (2018) Multi-objective fixed-charge transportation problem with random rough variables. Int J Uncertain Fuzziness Knowl Based Syst 26(6):971–996
Roy SK, Midya S, Weber GW (2019) Multi-objective, multi-item fixed-charge solid transportation problem under twofold uncertainty. Neural Comput Appl. https://doi.org/10.1007/s00521-019-04431-2
Shaocheng T (1994) Interval number and fuzzy number linear programming. Fuzzy Sets Syst 66(3):301–306
Taha HT (1997) Operation research-An introduction, 6th edn. Mac Milan Publishin Co, New York
Zadeh LA (1965) Fuzzy sets. Inf Control 8(5):338–353
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Ammar, ES., Emsimir, A. A mathematical model for solving fuzzy integer linear programming problems with fully rough intervals. Granul. Comput. 6, 567–578 (2021). https://doi.org/10.1007/s41066-020-00216-4
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DOI: https://doi.org/10.1007/s41066-020-00216-4