Multi-objective Sustainable Fuzzy Economic Production Quantity (SFEPQ) Model with Demand as Type-2 Fuzzy Number: A Fuzzy Differential Equation Approach

Year 2019, Volume: 48 Issue: 1, 112 - 139, 01.02.2019

B. K. Debnath P. Majumder , U. K. Bera

Abstract

A sustainable fuzzy economic production quantity (SFEPQ) inventory model is formulated by introducing the concept of fuzzy differential equation (FDE) due to dynamic behavior of the production-demand system. Generalized Hukuhara (gH) differentiability proceedure is applied to solve FDE. Since the demand parameter is taken as trapezoidal type-2 fuzzy number, to get corresponding defuzzified values, first critical value (CV)-based reduction method is applied on demand function to transfer into type-1 fuzzy variable which turns to hexagonal fuzzy number in form. After that $\alpha$-cut of a hexagonal fuzzy number is used to find the upper and lower value of demand. To apply the $\alpha$-cut operation on FDE, we divided the interval [0,1] into two sub-intervals [0,0.5] and [0.5,1] and gH-differentiation is applied on this sub-intervals. The objective of this paper is to maximize the profit and simultaneously minimize the carbon emission cost occurring due to the process of inventory management. Finally, the non-linear objective functions are solved by using of multi-objective genetic algorithm and sensitivity analyses on various parameters are also performed in numerically and graphically.

Keywords

Sustainable economic production quantity model, type-2 fuzzy demand, $\alpha$- cut of hexagonal fuzzy number, fuzzy differential equation, generalized Hukuhara differentiation

References

• S.M. Aljazzar, A. Gurtu and M.Y. Jaber, Delay in payments- A strategy to reduce carbon emissions from supply chains, J. Clean. Prod. 170, 636-644, 2018.
• T. Allahviranloo and M. Afsher Kermani, Numerical methods for fuzzy linear partial differential equations under new definition for derivative, Iran. J. Fuzzy Syst. 7 (3), 33-50, 2010.
• D. Battini, A. Persona and F. Sgarbossa, A Sustainable EOQ model: Theoretical formulation and applications, Int. J. Prod. Econ. 149, 145-153, 2014.
• B. Bede, A note on two-point boundary value problems associated with non-linear fuzzy differential equations, Fuzzy Sets and Systems 157, 986-989, 2006.
• B. Bede and S.G. Gal, Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations, Fuzzy Sets and Systems 151, 581-599, 2005.
• S.C. Chang, Fuzzy production inventory for fuzzy product quantity with triangular fuzzy number, Fuzzy Sets and Systems 107, 37-57, 1999.
• Y. Chen and L. Zhang, Some new results about arithmetic of type-2 fuzzy variables, Journal of Uncertain System 5 (3), 227-240, 2011.
• D. Dubois and H. Prade, Operations on fuzzy numbers, Int. J. Syst. Sci. 9 (6), 613-626, 1978.
• R. Ezzati, K. Maleknejad and S. Khezertoo, Convergence, Consistency and stability in fuzzy differential equations, Iran. J. Fuzzy Syst. 12 (3), 95-112, 2015.
• P. Guchhait, M.K. Maiti and M. Maiti, A production inventory model with fuzzy production and demand using fuzzy differential equation: An interval compared genetic algorithm approach, Eng. Appl. Artif. Intell. 26, 766-778, 2013.
• V. Hovelaque and L. Bironneau, The carbon-constrained EOQ model with carbon emission dependent demand, Int. J. Prod. Econ. 164, 285-291, 2015.
• M. Jonas, M. Marland, W. Winiwarter, T. White, Z. Nahorski, R. Bun and S. Nilsson, Benefits of dealing with uncertainty in greenhouse gas inventories: introduction, Climatic Change 103 (1-2), 175-213, 2010b.
• A. Kandel and W.J. Byatt, Fuzzy differential equations, In Proceedings of the International Conference on Cybernetics and Society, Tokyo, 1213-1216, 1978.
• N. Kazemi, S.H. Abdul-Rashid, R.A.R. Ghazila, E. Shekarian and S. Zanoni, Economic order quantity models for items with imperfect quality and emission considerations, Int. J. Syst. Sci.: Oper. & Logist. 5 (2), 99-115, 2018.
• P. Kundu, S. Kar and M. Maiti, Fixed charge transportation problem with type-2 fuzzy variables, Inf. Sci. 255, 170-184, 2014.
• P. Kundu, S. Kar and M. Maiti, Multi-item solid transportation problem with type-2 fuzzy parameters, Appl. Soft Comput. 31, 61-80, 2015.
• H.M. Lee and J.S. Yao, Economic production quantity for fuzzy demand quantity and fuzzy production quantity, European J. Oper. Res. 109, 203-211, 1998.
• D.C. Lin and J.S. Yao, Fuzzy economic production for production inventory, Fuzzy Sets and Systems 111, 465-495, 2000.
• Z.Q. Liu and Y.K. Liu, Type-2 fuzzy variables and their arithmetic, Soft Computing 14, 729-747, 2010.
• P. Majumder, S.P. Mondal, U.K. Bera and M. Maiti, Application of Generalized Hukuhara derivative approach in an economic production quantity model with partial trade credit policy under fuzzy environment, Operations Research Perspective 3, 77- 91, 2016.
• S. Miller, M. Gongora and R. John, Interval type-2 fuzzy modeling and simulated annealing for real world inventory management, International Conference on Hybrid Artificial Intelligence Systems, 231-238, Springer-Verlag Bertin, Heidelberg, 2011.
• M. Mizumoto and K. Tanaka, Fuzzy sets of type-2 under algebraic product and algebraic sum, Fuzzy Sets and Systems 5 (3), 277-280, 1981.
• R. Qin, Y.K. Liu and Z.Q. Liu, Methods of critical value reduction for type-2 fuzzy variables and their applications, J. Comput. Appl. Math. 235, 1454-1481, 2011.
• P. Rajarajeswari, A.S. Sudha and R. Karthika, A New Operation on Hexagonal Fuzzy Number, Int. J. Fuzzy Log. Syst. 3 (3), 15-26, 2013.
• J. Sadeghi, S.T.A. Niaki, M. Malekian and Y. Wang, A Lagrangian relaxation for a fuzzy random EPQ problem with shortages and redundancy allocation: two tuned meta-heuristics, Int. J. Fuzzy Syst. 20 (2), 515-533, 2018.
• S. Sharan, S.P. Tiwary and V.K. Yadav, Interval type-2 fuzzy rough sets and interval type-2 fuzzy closure spaces, Iran. J. Fuzzy Syst. 12 (3), 113-125, 2015.
• E. Shekarian, C.H. Glock, S.M.P. Amiri and K. Schwindl, Optimal manufacturing lot size for a single-stage production system with rework in a fuzzy environment, J. Intell. Fuzzy Syst. 27 (6), 3067-3080, 2014.
• E. Shekarian, M.Y. Jaber, N. Kazemi and E. Ehsani, A fuzzified version of the economic production quantity (EPQ) model with backorders and rework for a single stage system, Eur. J. Ind. Eng. 8 (3), 291-324, 2014.
• E. Shekarian, N. Kazemi, S.H. Abdul-Rashid and E.U. Olugu, Fuzzy inventory models: A comprehensive review, Appl. Soft Comput. 55, 588-621, 2017.
• E. Shekarian, E.U. Olugu, S.H. Abdul-Rashid and E. Bottani, A fuzzy reverse logistics inventory system integrating economic order/production quantity models, Int. J. Fuzzy Syst. 18 (6), 1141-1161, 2016.
• E. Shekarian, E.U. Olugu, S.H. Abdul-Rashid and N. Kazemi, An economic order quantity model considering different holding costs for imperfect quality items subject to fuzziness and learning, J. Intell. Fuzzy Syst. 30 (5), 1985-2997, 2016.
• H.N. Soni, B. Sarkar and M. Joshi, Demand uncertainty and learning in fuzziness in a continuous review inventory model, J. Intell. Fuzzy Syst. 33 (4), 2595-2608, 2017.
• L. Stefanini and B. Bede, Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Anal. 71, 1311-1328, 2009.
• J.R. Stock, S.L. Boyer and T. Harmon, Research opportunities in supply chain management, J. Acad. Mark. Sci. 38 (1), 32-41, 2010.
• Z. Takac, Inclusion and subsethood measure for interval-valued fuzzy sets and for continuous type-2 fuzzy sets, Fuzzy Sets and Systems 224, 106-120, 2013.
• E.J. Villamizar-Roa, V. Angulo-Castilo and Y. Chaleo-Cano, Existence of solutions to fuzzy differential equation with generalized Hukuhara derivative via contractive-like mapping principles, Fuzzy Sets and Systems 265, 24-38, 2015.
• H.C. Wu, The central limit theorems for fuzzy random variables, Inf. Sci. 120, 239- 256, 1999.
• L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning - I, Inf. Sci. 8, 199-249, 1975.
• L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning - II, Inf. Sci. 8, 301-357, 1975.