Multi item inventory model include lead time with demand dependent production cost and set-up-cost in fuzzy environment

Document Type : Review Paper


Department of Mathematics, General Degree College at Gopiballavpur-II, Jhargram, West Bengal, India.


In this paper, we have developed the multi-item inventory model in the fuzzy environment. Here we considered the demand rate is constant and production cost is dependent on the demand rate. Set-up- cost is dependent on average inventory level as well as demand. Lead time crashing cost is considered the continuous function of leading time. Limitation is considered on storage of space. Due to uncertainty all cost parameters of the proposed model are taken as generalized trapezoidal fuzzy numbers. Therefore this model is very real. The formulated multi objective inventory problem has been solved by various techniques like as Geometric Programming (GP) approach, Fuzzy Programming Technique with Hyperbolic Membership Function (FPTHMF), Fuzzy Nonlinear Programming (FNLP) technique and Fuzzy Additive Goal Programming (FAGP) technique. An example is given to illustrate the model. Sensitivity analysis and graphical representation have been shown to test the parameters of the model.


Main Subjects

[1]     Abou-El-Ata, M. O., & Kotb, K. A. M. (1997). Multi-item EOQ inventory model with varying holding cost under two restrictions: a geometric programming approach. Production planning and control8(6), 608-611.
[2]     Bit, A. K. (2004). Fuzzy programming with hyperbolic membership functions for multiobjective capacitated transportation problem. Opsearch41(2), 106-120.
[3]     Ben-Daya, M. A., & Raouf, A. (1994). Inventory models involving lead time as a decision variable. Journal of the operational research society45(5), 579-582.
[4]     Bortolan, G., & Degani, R. (1985). A review of some methods for ranking fuzzy subsets. Fuzzy sets and systems15(1), 1-19.
[5]     Beightler, C., & Phillips, D. T. (1976). Applied geometric programming. John Wiley & Sons. 
[6]     Biswal, M. P. (1992). Fuzzy programming technique to solve multi-objective geometric programming problems. Fuzzy sets and systems51(1), 67-71.
[7]     Chen, C. K. (2000). Optimal determination of quality level, selling quantity and purchasing price for intermediate firms. Production planning and control11(7), 706-712.
[8]     Chuang, B. R., Ouyang, L. Y., & Chuang, K. W. (2004). A note on periodic review inventory model with controllable setup cost and lead time. Computers and operations research31(4), 549-561.
[9]     Duffin, R. J. (1967). Geometric programming-theory and application (No. 04; QA264, D8.). New York Wiley196278 p.
[10] Duffin, R. J., Peterson, E. L. & Zener, C. (1966). Geometric programming theory and applications. Wiley, New York.
[11] Liang, Y., & Zhou, F. (2011). A two-warehouse inventory model for deteriorating items under conditionally permissible delay in payment. Applied mathematical modelling35(5), 2221-2231.
[12] Das, K., Roy, T. K., & Maiti, M. (2000). Multi-item inventory model with quantity-dependent inventory costs and demand-dependent unit cost under imprecise objective and restrictions: a geometric programming approach. Production planning and control11(8), 781-788.
[13] Das, S. K., & Islam, S. (2019). Multi-objective two echelon supply chain inventory model with lot size and customer demand dependent purchase cost and production rate dependent production cost. Pakistan journal of statistics and operation research, 15(4), 831-847.
[14] Hariga, M., & Ben-Daya, M. (1999). Some stochastic inventory models with deterministic variable lead time. European journal of operational research113(1), 42-51.
[15] Harri, F. (1913). How many parts to make at once factory. Mag. Mannage, (10), 135-136.
[16] Islam, S. (2016). Multi-objective geometric programming problem and its applications. Yugoslav journal of operations research20(2).
[17] Islam, S. (2008). Multi-objective marketing planning inventory model: A geometric programming approach. Applied mathematics and computation205(1), 238-246.
[18] Kotb, K. A., & Fergany, H. A. (2011). Multi-item EOQ model with both demand-dependent unit cost and varying leading time via geometric programming. Applied mathematics2(5), 551-555.
[19] Maiti, M. K. (2008). Fuzzy inventory model with two warehouses under possibility measure on fuzzy goal. European journal of operational research188(3), 746-774.
[20] Ouyang, L. Y., Yeh, N. C., & Wu, K. S. (1996). Mixture inventory model with backorders and lost sales for variable lead time. Journal of the operational research society47(6), 829-832.
[21] Ouyang L., & Wu K. (1998). A min-max distribution free procedure for mixed inventory model with variable lead time. Int J Pro Econ, 56(1), 511-516.
[22] Mandal, N. K., Roy, T. K., & Maiti, M. (2005). Multi-objective fuzzy inventory model with three constraints: a geometric programming approach. Fuzzy sets and systems150(1), 87-106.
[23] Mandal, N. K., Roy, T. K., & Maiti, M. (2006). Inventory model of deteriorated items with a constraint: A geometric programming approach. European journal of operational research173(1), 199-210.
[24] Sarkar, B., Gupta, H., Chaudhuri, K., & Goyal, S. K. (2014). An integrated inventory model with variable lead time, defective units and delay in payments. Applied mathematics and computation237, 650-658.
[25] Sarkar, B., Mandal, B., & Sarkar, S. (2015). Quality improvement and backorder price discount under controllable lead time in an inventory model. Journal of manufacturing systems35, 26-36.
[26] Shaikh, A. A., Bhunia, A. K., Cárdenas-Barrón, L. E., Sahoo, L., & Tiwari, S. (2018). A fuzzy inventory model for a deteriorating item with variable demand, permissible delay in payments and partial backlogging with shortage follows inventory (SFI) policy. International journal of fuzzy systems20(5), 1606-1623.
[27] Zadeth, L. A. (1965). Fuzzy sets. Information and control8, 338-353.
[28] Zimmermann, H. J. (1985). Applications of fuzzy set theory to mathematical programming. Information sciences36(1-2), 29-58.
[29] Garai, T., Chakraborty, D., & Roy, T. K. (2019). Multi-objective inventory model with both stock-dependent demand rate and holding cost rate under fuzzy random environment. Annals of data science6(1), 61-81.
Shaikh, A. A., Panda, G. C., Khan, M. A. A., Mashud, A. H. M., & Biswas, A. (2020). An inventory model for deteriorating items with preservation facility of ramp type demand and trade credit. International journal of mathematics in operational research17(4), 514-551