This study demonstrates an EOQ model with partial backorders over the finite time horizon assuming imprecise demand which is characterized by triangular fuzzy number. Learning effect is considered to reduce the impreciseness of demand as inventory planners get experienced by collecting knowledge from previous cycles. This paper aims to find out the optimal number of replenishments and an optimal fraction of the cycle during the positive inventory to minimize the total annual cost. The optimal policy is derived using analytical approach for crisp and fuzzy model whereas algorithmic procedure is adopted for the fuzzy-learning model. To show the significance of learning effect, numerical analysis is executed and compared results from the crisp, fuzzy and fuzzy-learning case which shows that increasing human learning reduces fuzziness of the demand and approaches to the crisp model.

EOQ model, partial backordering, fuzzy demand, learning in fuzziness, centroid method

[1] F.W. Harris. "How many parts to make at once", Factory, Mag. Manag. Vol.10, Issue.2, pp.135–136. 152, 1913.
[2] C.H. Glock, E.H. Grosse, J.M. Ries Carlo. "The lot sizing problem: A tertiary study", Intern. J. Prod. Econ. Vol.155, pp.39–51, 2014.
[3] D.W. Pentico, M.J. Drake. "A survey of deterministic models for the EOQ and EPQ with partial backordering", Eur. J. Oper. Res. Vol.214, Issue.2, pp.179–198, 2011.
[4] W.J. Fabrycky, J. Banks. Procurement and Inventory Systems, Theory and Analysis, (pp 92-95). Reinhold Publishing Corporation, New York.
[5] A.M. Ali. Inventory problems, in: F.C. Jelen (Ed.), Cost Optim. Eng., McGraw-Hill, New York, pp. 191– 193.
[6] D.C. Montgomery, M.S. Bazaraa, A.K. Keswani. "Inventory models with a mixture of backorders and lost sales", Nav. Res. Logist. Q. Vol.20, Issue.2, pp.255–263, 1973.
[7] C.Y. Dye, H.J. Chang, J.T. Teng. "A deteriorating inventory model with time-varying demand and shortage-dependent partial backlogging", Eur. J. Oper. Res. 2006.
[8] Y.W. Zhou, H.S. Lau, S.L. Yang. "A finite horizon lot-sizing problem with time-varying deterministic demand and waiting-time-dependent partial backlogging", Int. J. Prod. Econ. Vol.91, Issue.2, pp.109–119, 2004.
[9] P.S. You, M.T. Wu. "Optimal ordering and pricing policy for an inventory system with order cancellations", OR Spectr. Vol.29, Issue.4, pp.661–679, 2007.
[10] R. Uthayakumar, K. V. Geetha. "A replenishment policy for non-instantaneous deteriorating inventory system with partial backlogging", Vol.25, Issue.3, pp.313–332, 2009.
[11] H.L. Yang. "A comparison among various partial backlogging inventory lot-size models for deteriorating items on the basis of maximum profit", Int. J. Prod. Econ. Vol.96, Issue.1, pp.119–128, 2005.
[12] H.M. Wee. "Joint pricing and replenishment policy for deteriorating inventory with declining market", Int. J. Prod. Econ. Vol.40, Issue.2–3, pp.163–171, 1995.
[13] L.Y. Ouyang, T.P. Hsieh, C.Y. Dye, H.C. Chang. "An inventory model for deteriorating items with stock-dependent demand under the conditions of inflation and time-value of money", Eng. Econ. Vol.48, Issue.1, pp.52–68, 2003.
[14] H.J. Chang, J.T. Teng, L.Y. Ouyang, C.Y. Dye. "Retailer’s optimal pricing and lot-sizing policies for deteriorating items with partial backlogging", Eur. J. Oper. Res. Vol.168, Issue.1, pp.51–64, 2006.
[15] Y.C. Tsao. "Two-phase pricing and inventory management for deteriorating and fashion goods under trade credit", Math. Methods Oper. Res. Vol.72, Issue.1, pp.107–127, 2010.
[16] A. Zadeh. "Fuzzy S e t s", Inf. Control. Vol.8, pp.338–353, 1965.
[17] K.S. Park. "Fuzzy set theoretic interpretation of economic order quantity", IEEE Trans. Syst. Man Cybern. Vol.17, pp.1082–1084, 1987.
[18] M. Vujosevic, D. Petrovic, R. Petrovic. "EOQ formula when inventory cost is fuzzy", Int. J. Prod. Econ. Vol.45, pp.499–504, 1996.
[19] D. Yadav, S.R. Singh, R. Kumari. "Application of Minimax Distribution Free Procedure and Chebyshev Approach in Mixed Inventory Model Involving Reducible Lead-Time and Setup Cost With Imprecise Demand", Asia-Pacific J. Oper. Res. Vol.30, Issue.04, pp.1350009, 2013.
[20] L.Y. Ouyang, H.C. Chang. "A minimax distribution free procedure for mixed inventory models involving variable lead time with fuzzy lost sales", Int. J. Prod. Econ. Vol.76, Issue.1, pp.1–12, 2002.
[21] H.-C. Chang, J.-S. Yao, L.-Y. Ouyang. "Fuzzy mixture inventory model with variable lead-time based on probabilistic fuzzy set and triangular fuzzy number", Math. Comput. Model. Vol.39, Issue.2–3, pp.287–304, 2004.
[22] H.-C. Chang, J.-S. Yao, L.-Y. Ouyang. "Fuzzy mixture inventory model involving fuzzy random variable lead time demand and fuzzy total demand", Eur. J. Oper. Res. Vol.169, pp.65–80, 2006.
[23] M. Yang, T. Huang, Y. Lin. "Variable lead time fuzzy inventory system under backorder rate with allowed shortage", Int. J. Innov. Comput. Inf. Control. Vol.6, Issue.11, pp.5015–5034, 2010.
[24] L. Wang, Q.L. Fu, Y.R. Zeng. "Continuous review inventory models with a mixture of backorders and lost sales under fuzzy demand and different decision situations", Expert Syst. Appl. Vol.39, Issue.4, pp.4181–4189, 2012.
[25] J.J. Buckley, T. Feuring, Y. Hayashi. "Solving fuzzy problems in operations research: inventory control", Soft Comput. Vol.7, Issue.2, pp.121–129, 2002.
[26] S.K. De, A. Goswami. "An EOQ model with fuzzy inflation rate and fuzzy deterioration rate when a delay in payment is permissible", Int. J. Syst. Sci. Vol.37, Issue.5, pp.323–335, 2006.
[27] K. Maity, M. Maiti. "Possibility and necessity constraints and their defuzzification-A multi-item production-inventory scenario via optimal control theory", Eur. J. Oper. Res. Vol.177, Issue.2, pp.882–896, 2007.
[28] G. Yazgı Tütüncü, O. Aköz, A. Apaydın, D. Petrovic. "Continuous review inventory control in the presence of fuzzy costs", Int. J. Prod. Econ. Vol.113, Issue.2, pp.775–784, 2008.
[29] E. Shekarian, S.H. Abdul-Rashid, E. Bottani, S.K. De, N. Kazemi, S.H. Abdul-Rashid, et al. "Fuzzy inventory models: A comprehensive review", Appl. Soft Comput. J. Vol.55, pp.588–621, 2017.
[30] M. Jaber. Learning Curves: Theory, Models, and Applications.
[31] E.H. Grosse, M. Calzavara, C.H. Glock, F. Sgarbossa. "Incorporating human factors into decision support models for production and logistics: current state of research", IFAC-PapersOnLine. Vol.50, Issue.1, pp.6900–6905, 2017.
[32] C.H. Glock, K. Schwindl, M.Y. Jaber. "An EOQ model with fuzzy demand and learning in fuzziness", Int. J. Serv. Oper. Manag. Vol.12, Issue.1, pp.90–100, 2012.
[33] N. Kazemi, E.U. Olugu, S.H. Abdul-Rashid, R.A.B.R. Ghazilla. "Development of a fuzzy economic order quantity model for imperfect quality items using the learning effect on fuzzy parameters", J. Intell. Fuzzy Syst. Vol.28, Issue.5, pp.2377–2389, 2015.
[34] N. Kazemi, S.H. Abdul-Rashid, E. Shekarian, E. Bottani, R. Montanari. "A fuzzy lot-sizing problem with two-stage composite human learning", Int. J. Prod. Res. Vol.54, Issue.16, pp.5010–5025, 2016.
[35] H.N. Soni, B. Sarkar, M. Joshi. "Demand uncertainty and learning in fuzziness in a continuous review inventory model", J. Intell. Fuzzy Syst. Vol.33, pp.2595–2608, 2017.
[36] A.A. Taleizadeh, M. Nematollahi. "An inventory control problem for deteriorating items with back-ordering and financial considerations", Appl. Math. Model. Vol.38, Issue.1, pp.93–109, 2014.
[37] S.-H. Chen, C.-C. Wang, A. Ramer. "Backorder Fuzzy Inventory Model under Function Principle.", Inf. Sci. (Ny). Vol.95, pp.71–79, 1996.
[38] E.H. Grosse, C.H. Glock, C.H. Glock Carlo. "Production Economics and the Learning Curve: A Meta-Analysis", Intern. J. Prod. Econ. 2015.
[39] T.P. Wright. "Factors Affecting the Cost of Airplanes", J. Aeronaut. Sci. Vol.3, Issue.4, pp.122–128, 1936.