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Analytics under uncertainty: a novel method for solving linear programming problems with trapezoidal fuzzy variables

  • Fuzzy systems and their mathematics
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Abstract

Linear programming (LP) has long proved its merit as the most flexible and most widely used technique for resource allocation problems in various fields. To solve an LP problem, we have traditionally considered crisp values for the parameters, which are unrealistic in real-world decision-making under uncertainty. The fuzzy set theory has been used to model the imprecise parameter values in LP problems to overcome this shortcoming, resulting in a fuzzy LP (FLP) problem. This paper proposes a new method for solving fuzzy variable linear programming (FVLP) problems in which the decision variables and resource vectors are fuzzy numbers. We show how to use the standard simplex algorithm to solve this problem by converting the fuzzy problem into a crisp one once a linear ranking function is chosen. The novelty of the proposed model resides in that it requires less effort on fuzzy computations as opposed to the existing fuzzy methods. Furthermore, to solve the FVLP problem using the existing methods, fuzzy arithmetic operations and the solution to fuzzy systems of equations are required. By contrast, only arithmetic operations of real numbers and the solution to crisp systems of equations are required to solve the same problem with the method proposed in this study. Finally, a transportation case study in the coal industry is presented to demonstrate the applicability of the proposed algorithm.

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References

  • Abbaszadeh Sori A, Ebrahimnejad A, Motameni H (2020) Elite artificial bees’ colony algorithm to solve robot’s fuzzy constrained routing problem. Comput Intell 36(2):659–681

    Article  Google Scholar 

  • Allahviranloo T, Hosseinzadeh Lotfi F, Kiasary MK, Kiani NA, Alizadeh L (2008) Solving full fuzzy linear programming problem by the ranking function. Appl Math Sci 2(1):19–32

    MathSciNet  MATH  Google Scholar 

  • Aviso KB, Sy CL, Tan RR, Ubando AT (2020) Fuzzy optimization of carbon management networks based on direct and indirect biomass co-firing. Renew Sustain Energy Rev 132:110035

    Article  Google Scholar 

  • Baykasoglu A, Gocken T (2012) A direct solution approach to fuzzy mathematical programs with fuzzy decision variables. Exp Syst Appl 39(2):1972–1978

    Article  Google Scholar 

  • Beed R, Roy A, Sankar S, Bhattacharya D (2020) A hybrid multi-objective tour route optimization algorithm based on particle swarm optimization and artificial bee colony optimization. Comput Intell 36(3):884–909

    Article  Google Scholar 

  • Behera D, Peters K, Edalatpanah SA, Qiu D (2021) New methods for solving imprecisely defined linear programming problem under trapezoidal fuzzy uncertainty. J Inform Optim Sci 4(23):603–662

    Google Scholar 

  • Bhardwaj B, Kumar A (2014) A note on the paper “A Simplified novel technique for solving fully fuzzy linear programming problems.” J Optim Theory Appl 163(2):685–696

    Article  MathSciNet  MATH  Google Scholar 

  • Bilgen B (2010) Supply chain network modeling in a golf club industry via fuzzy linear programming approach. J Intell Fuzzy Syst Appl Eng Technol 21:243–253

    MATH  Google Scholar 

  • Charles V, Yadavalli VSS, Rao MCL, Reddy PRS (2011) Stochastic fractional programming approach to a mean and variance model of a transportation problem. Math Prob Eng 2011:657608

    Article  MathSciNet  MATH  Google Scholar 

  • Chen Z-S, Liu X-L, Chin K-S, Pedrycz W, Tsui K-L, Skibniewski MJ (2021a) Online-review analysis based large-scale group decision-making for determining passenger demands and evaluating passenger satisfaction: case study of high-speed rail system in China. Inform Fusion 69:22–39

    Article  Google Scholar 

  • Chen Z-S, Zhang X, Rodriguez RM, Pedrycz W, Martinez L (2021b) Expertise-based bid evaluation for construction-contractor selection with generalized comparative linguistic ELECTRE III. Autom Const 125:103578

    Article  Google Scholar 

  • Das SK, Mandal T, Edalatpanah SA (2017) A mathematical model for solving fully fuzzy linear programming problem with trapezoidal fuzzy numbers. Appl Intell 46(3):509–519

    Article  Google Scholar 

  • Djordjevic I, Petrovic D, Stojic G (2019) A fuzzy linear programming model for aggregated production planning (APP) in the automotive industry. Comput Ind 110:48–63

    Article  Google Scholar 

  • Dong JY, Wan SP (2018) A new trapezoidal fuzzy linear programming method considering the acceptance degree of fuzzy constraints violated. Knowl Based Syst 148:100–114

    Article  Google Scholar 

  • Dorostkar-Ahmadi N, Shafiei Nikabadi M, Babaie-Kafaki S (2020) Optimization of knowledge transferring costs in designing product portfolio: a fuzzy binary linear programming model. VINE J Inform Knowl Manag Syst (in press)

  • Ebrahimnejad A (2011) Some new results in linear programs with trapezoidal fuzzy numbers: finite convergence of the Ganesan and Veeramani’s method and a fuzzy revised simplex method. Appl Math Model 35(9):4526–4540

    Article  MathSciNet  MATH  Google Scholar 

  • Ebrahimnejad A (2015) A duality approach for solving bounded linear programming problems with fuzzy variables based on ranking functions and its application in bounded transportation problems. Int J Syst Sci 46(11):2048–2060

    Article  MathSciNet  MATH  Google Scholar 

  • Ebrahimnejad A (2019) An effective computational attempt for solving fully fuzzy linear programming using MOLP problem J. Ind Prod Eng 36(2):59–69

    MathSciNet  Google Scholar 

  • Ebrahimnejad A, Nasseri SH, Hosseinzadeh Lotfi F, Soltanifar M (2010) A primal-dual method for linear programming problems with fuzzy variables. Eur J Ind Eng 4(2):189–209

    Article  Google Scholar 

  • Ebrahimnejad A, Tavana M (2014) A novel method for solving linear programming problems with symmetric trapezoidal fuzzy numbers. Appl Math Model 38(17–18):4388–4395

    Article  MathSciNet  MATH  Google Scholar 

  • Ebrahimnejad A, Verdegay JL (2014a) A novel approach for sensitivity analysis in linear programs with trapezoidal fuzzy numbers. J Intell Fuzzy Syst 27(1):173–185

    Article  MathSciNet  MATH  Google Scholar 

  • Ebrahimnejad A, Verdegay JL (2014b) On solving bounded fuzzy variable linear program and its applications. J Intel Fuzzy Syst 27(5):2265–2280

    Article  MathSciNet  MATH  Google Scholar 

  • Ebrahimnejad A, Verdegay JL (2016a) A survey on models and methods for solving fuzzy linear programming problems. In: Kahraman C, Kaymak U, Yazici A (eds) Fuzzy logic in its 50th year. Studies in fuzziness and soft computing, vol 341, pp 327–368. Springer, Cham.

  • Ebrahimnejad A, Verdegay JL (2016b) An efficient computational approach for solving type-2 intuitionistic fuzzy numbers based transportation problems. Int J Comput Intell Syst 9(6):1154–1173

    Article  Google Scholar 

  • Ebrahimnejad A, Verdegay JL (2018a) Fuzzy linear programming. In: Fuzzy sets-based methods and techniques for modern analytics. Studies in fuzziness and soft computing, vol 364. Springer, Cham.

  • Ebrahimnejad A, Verdegay JL (2018b) Fuzzy sets-based methods and techniques for modern analytics. Studies in fuzziness and soft computing, vol 364. Springer, Cham.

  • Ebrahimnejad A, Verdegay JL (2018c) A new approach for solving fully intuitionistic fuzzy transportation problems. Fuzzy Optim Dec Making 17(4):447–474

    Article  MathSciNet  MATH  Google Scholar 

  • Ezzati R, Khorram E, Enayati R (2013) A new algorithm to solve fully fuzzy linear programming problems using the MOLP problem. Appl Math Model 39(12):3183–3193

    Article  MathSciNet  MATH  Google Scholar 

  • Ezzati R, Khorram E, Enayati R (2014) A particular simplex algorithm to solve fuzzy lexicographic multi-objective linear programming problems and their sensitivity analysis on the priority of the fuzzy objective functions. J Intel Fuzzy Syst 26(5):2333–2358

    Article  MathSciNet  MATH  Google Scholar 

  • Ghandi M, Roozbahani A (2020) Risk management of drinking water supply in critical conditions using fuzzy PROMETHEE V technique. Water Resour Manag 34:595–615

    Article  Google Scholar 

  • Ganesan K, Veeramani P (2006) Fuzzy linear programming with trapezoidal fuzzy numbers. Ann Oper Res 143(1):305–315

    Article  MATH  Google Scholar 

  • Ghanbari R, Ghorbani-Moghadam K, Mahdavi-Amiri N, De Baets B (2020) Fuzzy linear programming problems: models and solutions. Methodol Appl 24:10043–10073

    Google Scholar 

  • Gupta P, Mehlawat MM, Aggarwal U, Charles V (2018) An integrated AHP-DEA multi-objective optimization model for sustainable transportation in mining industry. Res Policy, 101180. In Press.

  • Hatami-Marbini A, Agrell PJ, Tavana M, Emrouznejad A (2013) A stepwise fuzzy linear programming model with possibility and necessity relations. J Intell Fuzzy Syst 25(1):81–93

    Article  MathSciNet  MATH  Google Scholar 

  • Hatami-Marbini A, Tavana M (2011) An extension of the linear programming method with fuzzy parameters. Int J Math Oper Res 3(1):44–55

    Article  MathSciNet  MATH  Google Scholar 

  • Hillier FS (2010) Introduction to operations research. McGraw-Hill, New York, NY

    MATH  Google Scholar 

  • Hosseinzadeh Lotfi F, Allahviranloo T, Alimardani Jondabeh M, Alizadeh L (2009) Solving a full fuzzy linear programming using lexicography method and fuzzy approximate solution. Appl Math Model 33(7):3151–3156

    Article  MathSciNet  MATH  Google Scholar 

  • Ilbahar E, Kahraman C, Cebi S (2021) Location selection for waste-to-energy plants by using fuzzy linear programming. Energy 234:121189

    Article  Google Scholar 

  • Karakas E, Koyuncu M, Erol R, Kokangul A (2010) Fuzzy programming for optimal product mix decision based on expanded ABC approach. Int J Prod Res 48:729–744

    Article  MATH  Google Scholar 

  • Kaur J, Kumar A (2013) A new method to find the unique fuzzy optimal value of fuzzy linear programming problems J. Optim Theory Appl 156(2):529–534

    Article  MathSciNet  MATH  Google Scholar 

  • Khan IU, Ahmad T, Maan N (2013) A simplified novel technique for solving fully fuzzy linear programming problems J. Optim Theory Appl 159(2):536–546

    Article  MathSciNet  MATH  Google Scholar 

  • Kumar A, Kaur J (2011) A new method for fuzzy linear programming programs with trapezoidal fuzzy numbers. J Fuzzy Valued Syst Anal 2011:1–12

    Article  MATH  Google Scholar 

  • Kumar A, Kaur J (2014) Fuzzy optimal solution of fully fuzzy linear programming problems using ranking function. J Intell Fuzzy Syst 26(1):337–344

    Article  MathSciNet  MATH  Google Scholar 

  • Kumar A, Kaur J, Singh P (2011) A new method for solving fully fuzzy linear programming problems. Appl Math Model 35(2):817–823

    Article  MathSciNet  MATH  Google Scholar 

  • Kundu P, Majumder S, Kar S, Maiti M (2019) A method to solve linear programming problem with interval type-2 fuzzy parameters. Fuzzy Optim Dec Mak 18:103–130

    Article  MathSciNet  MATH  Google Scholar 

  • Li YP, Huang GH, Guo P, Nie SL (2010) Interval-fuzzy possibilistic mixed integer linear programming for environmental management under uncertainty. Int J Environ Pollut 42:93–102

    Article  Google Scholar 

  • Lin F-T (2008) A genetic algorithm for linear programming with fuzzy constraints. J Inform Sci Eng 24(3):801–817

    MathSciNet  Google Scholar 

  • Mahapatra GS, Mahapatra BS, Roy PK (2016) A new concept for fuzzy variable based nonlinear programming problem with application on system reliability via genetic algorithm approach. Ann Oper Res 247:853–866

    Article  MathSciNet  MATH  Google Scholar 

  • Mahdavi I, Mahdi-Paydar MM, Solimanpur M (2011) Solving a new mathematical model for cellular manufacturing system: a fuzzy goal programming approach. Iran J Oper Res 2:35–47

    MATH  Google Scholar 

  • Mahdavi-Amiri N, Nasseri SH (2007) Duality results and a dual simplex method for linear programming problems with trapezoidal fuzzy variables. Fuzzy Sets Syst 158(17):1961–1978

    Article  MathSciNet  MATH  Google Scholar 

  • Maleki HR (2002) Ranking functions and their applications to fuzzy linear programming. Far East J Math Sci 4(3):283–301

    MathSciNet  MATH  Google Scholar 

  • Maleki HR, Tata M, Mashinchi M (2000) Linear programming with fuzzy variables. Fuzzy Sets Syst 109(1):21–33

    Article  MathSciNet  MATH  Google Scholar 

  • Mansoori A, Effati S, Eshaghnezhad M (2017) An efficient recurrent neural network model for solving fuzzy nonlinear programming problems. Appl Intell 46(2):308–327

    Article  Google Scholar 

  • Mottaghi A, Ezzati R, Khorram E (2015) A new method for solving fuzzy linear programming problems based on the fuzzy linear complementary problem (FLCP). Int J Fuzzy Syst 17(2):236–245

    Article  MathSciNet  Google Scholar 

  • Mula J, Peidro D, Poler R (2010) The effectiveness of a fuzzy mathematical programming approach for supply chain production planning with fuzzy demand. Int J Prod Econ 128(1):136–143

    Article  MATH  Google Scholar 

  • Muthukumar S, Srinivasan R, Vijayan V (2020) An optimal solution of unbalanced octagonal fuzzy transportation problem. Mater Today Proc 1–3 (in press)

  • Najafi HS, Edalatpanah SA, Dutta H (2016) A nonlinear model for fully fuzzy linear programming with fully unrestricted variables and parameters. Alexand Eng J 55(3):2589–2595

    Article  Google Scholar 

  • Osuna-Gómez R, Chalco-Cano Y, Hernández-Jiménez B, Aguirre-Cipe I (2019) Optimality conditions for fuzzy constrained programming problems. Fuzzy Sets Syst 362:35–54

    Article  MathSciNet  MATH  Google Scholar 

  • Pérez-Cañedo B, Concepción-Morales ER (2019) On LR-type fully intuitionistic fuzzy linear programming with inequality constraints: solutions with unique optimal values. Exp Syst Appl 128:246–255

    Article  Google Scholar 

  • Ramík J, Vlach M (2016) Intuitionistic fuzzy linear programming and duality: a level sets approach. Fuzzy Optim Dec Mak 15(4):457–489

    Article  MathSciNet  MATH  Google Scholar 

  • Saati S, Hatami-Marbini A, Tavana M, Hajiahkondi E (2012) A two-fold linear programming model with fuzzy data. Int J Fuzzy Syst Appl 2(3):1–12

    Google Scholar 

  • Saati S, Hatami-Marbini A, Tavana M, Hajiahkondi E (2015) A fuzzy linear programming model with fuzzy parameters and decision variables. Int J Inform Dec Sci 7(4):312–333

    Google Scholar 

  • Sakawa M, Katagiri H, Matsui T (2012) Stackelberg solutions for fuzzy random bilevel linear programming through level sets and probability maximization. Oper Res 12(3):271–286

    MATH  Google Scholar 

  • Sanchis R, Díaz-Madroñero M, López-Jiménez A, Pérez-Sánchez M (2019) Solution approaches for the management of the water resources in irrigation water systems with fuzzy costs. Water 11(12):2432

    Article  Google Scholar 

  • Tanaka H, Asai K (1984) Fuzzy solution in fuzzy linear programming problems. IEEE Trans Syst Man Cybern 14(2):325–328

    Article  MATH  Google Scholar 

  • Wang RC, Liang TF (2005) Applying possibilistic linear programming to aggregate production planning. Int J Prod Econ 98(3):328–341

    Article  Google Scholar 

  • Wang G, Peng J (2019) Fuzzy optimal solution of fuzzy number linear programming problems. Int J Fuzzy Syst 21(3):865–881

    Article  MathSciNet  Google Scholar 

  • Yager RR (1981) A procedure for ordering fuzzy subsets of the unit interval. Inform Sci 24:143–161

    Article  MathSciNet  MATH  Google Scholar 

  • Yang G, Li X, Huo L, Liu Q (2020) A solving approach for fuzzy multi-objective linear fractional programming and application to an agricultural planting structure optimization problem. Chaos Solitons Fractals 141:110352

    Article  MathSciNet  Google Scholar 

  • Yucel A, Fuat Guneri A (2011) A weighted additive fuzzy programming approach for multi-criteria supplier selection. Exp Syst Appl 38(5):6281–6286

    Article  Google Scholar 

  • Zadeh LA (1965) Fuzzy sets. Inform. Control 8(3):338–353

    Article  MathSciNet  MATH  Google Scholar 

  • Zimmermann H-J (2010) Fuzzy set theory. WIREs Comut. Stat 2(3):3170332

    Google Scholar 

  • Zhang G, Wu YH, Remias M, Lu J (2003) Formulation of fuzzy linear programming problems as four-objective constrained optimization problems. Appl Math Comput 139(2–3):383–399

    MathSciNet  MATH  Google Scholar 

  • Zhang Q, Zhou J, Wang K, Pantelous AA (2018) An effective solution approach to fuzzy programming with application to project scheduling. Int J Fuzzy Syst 20(8):2383–2398

    Article  MathSciNet  Google Scholar 

  • Zhang G, Zuo H (2014) Research on multi-objective linear programming with fuzzy coefficients constraints. Int J Mach Learn Cybern 5:403–412

    Article  Google Scholar 

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Acknowledgements

Dr. Madjid Tavana is grateful for the partial support he received from the Czech Science Foundation (GCR19-13946S) for this research. The authors are also thankful to the reviewers for their valuable comments on previous drafts of this paper.

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Authors and Affiliations

  1. Department of Mathematics, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran

    Ali Ebrahimnejad

  2. Business Systems and Analytics Department, Distinguished Chair of Business Analytics, La Salle University, Philadelphia, PA, 19141, USA

    Madjid Tavana

  3. Business Information Systems Department, Faculty of Business Administration and Economics, University of Paderborn, 33098, Paderborn, Germany

    Madjid Tavana

  4. CENTRUM Católica Graduate Business School, Lima, Peru

    Vincent Charles

  5. Pontifical Catholic University of Peru, Lima, Peru

    Vincent Charles

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  1. Ali Ebrahimnejad
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  2. Madjid Tavana
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  3. Vincent Charles
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Contributions

Ali Ebrahimnejad contributed to conceptualization, methodology, formal analysis, validation, data curation, writing, and reviewing. Madjid Tavana contributed to methodology, formal analysis, visualization, writing, reviewing, and Editing. Vincent Charles contributed to methodology, investigation, formal analysis, validation, writing, Reviewing, and editing.

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Correspondence to Madjid Tavana.

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Ebrahimnejad, A., Tavana, M. & Charles, V. Analytics under uncertainty: a novel method for solving linear programming problems with trapezoidal fuzzy variables. Soft Comput 26, 327–347 (2022). https://doi.org/10.1007/s00500-021-06389-7

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