Article Contents
Article Contents

# An EOQ inventory model for deteriorating items with controllable deterioration rate under stock-dependent demand rate and non-linear holding cost

• * Corresponding author: Longzhou Cao

This work is supported by the Major Research Plan of the National Natural Science Foundation of China (91430108), the National Natural Science Foundation of China (11771322), the National Social Science Foundation of China (19BGL190) and the Major Program of Tianjin University of Finance and Economics (ZD1302)

• The main purpose of this paper is to investigate the retailer's strategy in selecting the order-up-to level, the reorder point and the preservation technology investment for deteriorating items, aiming to maximize his total profit per unit time. We formulate the problem into a mathematical model that takes into account stock-dependent demand rate, stock-dependent holding cost. The terminal conditions are relaxed to allow that the reorder point can be one of the following two cases: (1) $N\leq0$, i.e., the reorder point may be negative or zero. When the reorder point is negative, the shortage is allowed and partial backlogged. (2) $N\geq0$, i.e., the reorder point may be without shortage or zero. We prove the existence and uniqueness of the optimal order-up-to level, the reorder point and the preservation technology investment under any given two of them for both the two cases. We then present an algorithm to search for decision variables such that the total profit per unit time is maximized. Finally, numerical examples, comparisons in performance and sensitivity analysis are carried out to examine the results obtained. On the basis of the above results, some useful managerial insights are revealed.

Mathematics Subject Classification: Primary: 90B05; Secondary: 91B06.

 Citation:

• Figure 1.  Graphical representation for the case of $N\leq0$

Figure 2.  Graphical representation for the case of $N\geq0$

Figure 3.  The effect of the deterioration rate on $Q$, $N$, $\xi$, $S$, $U$, $AR$, $\theta_1$, $T$, and $t_1$ for the cases with and without preservation technology investment

Table 1.  Notation

 Decision variables $Q$ the order-up-to level at time $t = 0$. $N$ the reorder point at time $T$, i.e., $I(T) = N$. $\xi$ the preservation technology cost per unit time for reducing the deterioration rate. auxiliary variables $S(Q, \xi, N)$ the order quantity per cycle. $T(Q, \xi, N)$ the inventory cycle length. $t_1(Q, \xi)$ the time when the inventory level comes down to zero. Constant parameters $p$ the purchasing cost per unit item. $s$ the selling price per unit item. $h$ the holding cost per unit per unit time. $I(t)$ the inventory level at time $t$. $D(t)$ the demand rate at time $t$. $\gamma$ the elasticity of holding cost. $\beta$ the demand elasticity. $c_s$ the shortage cost per unit per unit time. $c_l$ the lost sale cost per unit. $\lambda$ the scale parameter of the demand rate. $\delta$ the partial backlogging parameter, $\delta\in [0, 1]$. $\theta$ the deteriorating rate. $K$ the replenishment cost per order. $m(\xi)$ the proportion of the reduced deterioration rate, $0\leq m(\xi)\leq 1$. $AR(Q, \xi, N)$ the total profit per unit time. $TR(Q, \xi, N)$ the total profit per inventory cycle.

Table 2.  Sensitivity analysis for Example 3

 parameter value $AR^*$ $Q^*$ $N^*$ $\xi^*$ $T^*$ $S^*$ $\theta$ 0.06 24.1103 3.0660 0.4157 0 2.1865 2.6503 0.13 21.5417 3.0618 0.4074 2.3451 2.1880 2.6544 0.2 20.1058 3.0618 0.4074 3.7811 2.1880 2.6544 0.27 19.1054 3.0618 0.4074 4.7814 2.1880 2.6544 0.34 18.3370 3.0618 0.4074 5.5498 2.1880 2.6544 $K$ 10 25.4420 2.8487 0.6882 3.6616 1.7274 2.1605 15 22.6078 2.9691 0.5264 3.7246 1.9840 2.4427 20 20.1058 3.0618 0.4074 3.7811 2.1880 2.6544 25 17.8250 3.1382 0.3159 3.8343 2.3590 2.8223 30 15.7056 3.2037 0.2439 3.8858 2.5067 2.9598 $a$ 0.08 16.1844 2.8147 0.1615 0 2.1550 2.6532 0.19 18.3422 3.0251 0.3453 3.8236 2.1963 2.6798 0.3 20.1058 3.0618 0.4074 3.7811 2.1880 2.6544 0.41 21.2412 3.0765 0.4378 3.4653 2.1819 2.6387 0.52 22.0291 3.0843 0.4559 3.1586 2.1777 2.6284 $\lambda$ 0.8 13.8535 2.8817 0.3308 3.6707 2.6957 2.5509 0.9 16.9434 2.9760 0.3708 3.7274 2.4150 2.6052 1 20.1058 3.0618 0.4074 3.7811 2.1880 2.6544 1.1 23.3336 3.1408 0.4413 3.8320 2.0008 2.6995 1.2 26.6212 3.2141 0.4729 3.8804 1.8437 2.7412 $h$ 0.2 22.0652 3.3883 0.5140 4.1190 2.2519 2.8743 0.25 20.9776 3.2045 0.4531 3.9328 2.2166 2.7514 0.3 20.1058 3.0618 0.4074 3.7811 2.1880 2.6544 0.35 19.3801 2.9463 0.3716 3.6531 2.1640 2.5747 0.4 18.7597 2.8498 0.3425 3.5425 2.1432 2.5073 $s$ 45 9.0587 2.8849 0.1786 3.2113 2.4050 2.7063 50 14.4533 2.9806 0.2948 3.5260 2.2877 2.6858 55 20.1058 3.0618 0.4074 3.7811 2.1880 2.6544 60 25.9663 3.1330 0.5129 3.9930 2.1035 2.6201 65 31.9997 3.1966 0.6107 4.1732 2.0311 2.5859 $p$ 10 33.3378 3.2234 0.6074 2.9528 2.0160 2.6160 15 26.5681 3.1444 0.5121 3.4419 2.0954 2.6323 20 20.1058 3.0618 0.4074 3.7811 2.1880 2.6544 25 13.9445 2.9707 0.2945 3.9925 2.2981 2.6762 30 8.1104 2.8650 0.1771 4.0781 2.4301 2.6879 $\gamma$ 3 26.2330 4.5559 0.8063 4.9638 2.5784 3.7496 3.5 22.6169 3.6210 0.5504 4.2757 2.3429 3.0706 4 20.1058 3.0618 0.4074 3.7811 2.1880 2.6544 4.5 18.2543 2.6932 0.3182 3.4071 2.0788 2.3750 5 16.8301 2.4333 0.2582 3.1140 1.9977 2.1751 $\beta$ 0.32 19.2226 2.9215 0.2756 3.5565 2.2976 2.6459 0.36 19.6213 2.9920 0.3412 3.6710 2.2467 2.6508 0.4 20.1058 3.0618 0.4074 3.7811 2.1880 2.6544 0.44 20.6777 3.1314 0.4738 3.8865 2.1236 2.6576 0.48 21.3397 3.2011 0.5399 3.9873 2.0550 2.6612
•  [1] H. K. Alfares and and A. M. Ghaithan, EOQ and EPQ production-inventory models with variable holding cost: state-of-the-art review, Arabian Journal for Science and Engineering, 44 (2019), 1737-1755.  doi: 10.1007/s13369-018-3593-4. [2] R. A. Baker and T. L. Urban, A deterministic inventory system with an inventory-level-dependent demand rate, Journal of the Operational Research Society, 39 (1988), 823-831. [3] X. Cai, J. Chen, Y. Xiao and X. Xu, Optimization and coordination of fresh product supply chains with freshness-keeping effort, Production and Operations Management, 19 (2010), 261-278.  doi: 10.1111/j.1937-5956.2009.01096.x. [4] L. E. Cárdenas-Barrón, A. A. Shaikh, S. Tiwari and G. Treviño-Garza, An EOQ inventory model with nonlinear stock dependent holding cost, nonlinear stock dependent demand and trade credit, Computers & Industrial Engineering, 139 (2020), 105557. [5] C. T. Chang, Inventory models with stock-dependent demand and nonlinear holding costs for deteriorating items, Asia-Pacific Journal of Operational Research, 21 (2004), 435-446.  doi: 10.1142/S0217595904000321. [6] Y. Chen and  Z. Lu,  High Efficient and Accuracy Numerical Methods for Optimal Control Problems, Science Press, Beijing, 2015. [7] C. Y. Dye and T. P. Hsieh, An optimal replenishment policy for deteriorating items with effective investment in preservation technology, European Journal of Operational Research, 218 (2012), 106-112.  doi: 10.1016/j.ejor.2011.10.016. [8] C. Y. Dye, The effect of preservation technology investment on a non-instantaneous deteriorating inventory model, Omega, 41 (2013), 872-880.  doi: 10.1016/j.omega.2012.11.002. [9] C. Y. Dye and C. T. Yang, Optimal dynamic pricing and preservation technology investment for deteriorating products with reference price effects, Omega, 62 (2016), 52-67.  doi: 10.1016/j.omega.2015.08.009. [10] L. Feng, Y. L. Chan, L. E. and C árdenas-Barrón, Pricing and lot-sizing polices for perishable goods when the demand depends on selling price, displayed stocks, and expiration date, International Journal of Production Economics, 185 (2017), 11-20. [11] M. Ferguson, V. Jayaraman and G. C. Souza, Note: An application of the EOQ model with nonlinear holding cost to inventory management of perishables, European Journal of Operational Research, 180 (2007), 485-490.  doi: 10.1016/j.ejor.2006.04.031. [12] B. C. Giri and K. S. Chaudhuri, Deterministic models of perishable inventory with stock-dependent demand rate and nonlinear holding cost, European Journal of Operational Research, 105 (1998), 467-474.  doi: 10.1016/S0377-2217(97)00086-6. [13] M. Goh, EOQ models with general demand and holding cost functions, European Journal of Operational Research, 73 (1994), 50-54.  doi: 10.1016/0377-2217(94)90141-4. [14] P. H. Hsu, H. M. Wee and H. M. Teng, Preservation technology investment for deteriorating inventory, International Journal of Production Economics, 124 (2010), 388-394.  doi: 10.1016/j.ijpe.2009.11.034. [15] H. Huang, Y. He and D. Li, Pricing and inventory decisions in the food supply chain with production disruption and controllable deterioration, Journal of Cleaner Production, 180 (2018), 280-296.  doi: 10.1016/j.jclepro.2018.01.152. [16] Y. P. Lee and C. Y. Dye, An inventory model for deteriorating items under stock-dependent demand and controllable deterioration rate, Computers & Industrial Engineering, 63 (2012), 474-482.  doi: 10.1016/j.cie.2012.04.006. [17] R. I. Levin, C. P. McLaughlin, R. P. Lamone and F. J. Kottas, Production Operations Management: Contemporary Policy for Managing Operating Systems, McGraw-Hill, New York, 1972. [18] G. Li, X. He, J. Zhou and H. Wu, Pricing, replenishment and preservation technology investment decisions for non-instantaneous deteriorating items, Omega, 84 (2019), 114-126.  doi: 10.1016/j.omega.2018.05.001. [19] L. Li, M. Zhang, B. Adhikari and Z. Gao, Recent advances in pressure modification-based preservation technologies applied to fresh fruits and vegetables, Food Reviews International, 33 (2017), 538-559.  doi: 10.1080/87559129.2016.1196492. [20] Q. Li, P. Yu and X. Wu, Shelf life extending packaging, inventory control and grocery retailing, Production and Operations Management, 26 (2017), 1369-1382.  doi: 10.1111/poms.12692. [21] J. J. Liao, K. N. Huang and P. S. Ting, Optimal strategy of deteriorating items with capacity constraints under two-levels of trade credit policy, Applied Mathematics and Computation, 233 (2014), 647-658.  doi: 10.1016/j.amc.2014.01.077. [22] G. Liu, J. Zhang and W. Tang, Joint dynamic pricing and investment strategy for perishable foods with price-quality dependent demand, Annals of Operations Research, 226 (2015), 397-416.  doi: 10.1007/s10479-014-1671-x. [23] Z. Lu and S. Zhang, $L^\infty$-error estimates of rectangular mixed finite element methods for bilinear optimal control problem, Applied Mathematics & Computation, 300 (2017), 79-94.  doi: 10.1016/j.amc.2016.12.006. [24] C. K. Jaggi, V. S. S. Yadavalli, M. Verma and A. Sharma, An EOQ model with allowable shortage under trade credit in different scenario, Applied Mathematics and Computation, 252 (2015), 541-551.  doi: 10.1016/j.amc.2014.12.040. [25] U. Mishra, L. E. Cárdenas-Barrón, S. Tiwari, A. A. Shaikh and G. Treviño-Garza, An inventory model under price and stock dependent demand for controllable deterioration rate with shortages and preservation technology investment, Annals of Operations Research, 254 (2017), 165-190.  doi: 10.1007/s10479-017-2419-1. [26] E. Naddor, Inventory Systems, John Wiley, New York, 1982. [27] B. Pal, S. S. Sana and K. Chaudhuri, A distribution-free newsvendor problem with nonlinear holding cost, International Journal of Systems Science, 46 (2015), 1269-1277.  doi: 10.1080/00207721.2013.815828. [28] V. Pando, J. García-Laguna, L. A. San-José and J. Sicilia, Maximizing profits in an inventory model with both demand rate and holding cost per unit time dependent on the stock level, Computers & Industrial Engineering, 62 (2012), 599-608. [29] V. Pando, L. A. San-José, J. García-Laguna and J. Sicilia, An economic lot-size model with non-linear holding cost hinging on time and quantity, International Journal of Production Economics, 145 (2013), 294-303. [30] V. Pando, L. A. San-José, J. García-Laguna and J. Sicilia, Optimal lot-size policy for deteriorating items with stock-dependent demand considering profit maximization, Computers & Industrial Engineering, 117 (2018), 81-93. [31] V. Pando, L. A. San-José and J. Sicilia, Profitability ratio maximization in an inventory model with stock-dependent demand rate and non-linear holding cost, Applied Mathematical Modelling, 66 (2019), 643-661.  doi: 10.1016/j.apm.2018.10.007. [32] S. Saha, I. Nielsen and I. Moon, Optimal retailer investments in green operations and preservation technology for deteriorating items, Journal of Cleaner Production, 140 (2017), 1514-1527.  doi: 10.1016/j.jclepro.2016.09.229. [33] L. A. San-José, J. Sicilia and J. García-Laguna, Analysis of an EOQ inventory model with partial backordering and non-linear unit holding cost, Omega, 54 (2015), 147-157. [34] L. A. San-José, J. Sicilia, M. González-de-la-Rosa and J. Febles-Acosta, An economic order quantity model with nonlinear holding cost, partial backlogging and ramp-type demand, Engineering Optimization, 50 (2018), 1164-1177.  doi: 10.1080/0305215X.2017.1414205. [35] A. Tefera, T. Seyoum and K. Woldetsadik, Effect of disinfection, packaging, and storage environment on the shelf life of mango, Biosystems Engineering, 96 (2007), 201-212.  doi: 10.1016/j.biosystemseng.2006.10.006. [36] G. B. Thomson, Food date labels and hunger in America, Concordia L. Rev., 2 (2017), 143. [37] T. L. Urban, An inventory model with an inventory-level-dependent demand rate and relaxed terminal conditions, Journal of the Operational Research Society, 43 (1992), 721-724. [38] H. J. Weiss, Economic order quantity models with nonlinear holding costs, European Journal of Operational Research, 9 (1982), 56-60.  doi: 10.1016/0377-2217(82)90010-8. [39] Ö. Yaman and L. Bayoındırlı, Effects of an edible coating and cold storage on shelf-life and quality of cherries, LWT-Food Science and Technology, 35 (2002), 146-150.  doi: 10.1006/fstl.2001.0827. [40] C.-T. Yang, An inventory model with both stock-dependent demand rate and stock-dependent holding cost rate, International Journal of Production Economics, 155 (2014), 214-221.  doi: 10.1016/j.ijpe.2014.01.016. [41] L. Yang, R. Tang and K. Chen, Call, put and bidirectional option contracts in agricultural supply chains with sales effort, Applied Mathematical Modelling, 47 (2017), 1-16.  doi: 10.1016/j.apm.2017.03.002. [42] J. Zhang, G. Liu, Q. Zhang and Z. Bai, Coordinating a supply chain for deteriorating items with a revenue sharing and cooperative investment contract, Omega, 56 (2015), 37-49.  doi: 10.1016/j.omega.2015.03.004. [43] J. Zhang, Q. Wei, Q. Zhang and W. Tang, Pricing, service and preservation technology investments policy for deteriorating items under common resource constraints, Computers & Industrial Engineering, 95 (2016), 1-9.  doi: 10.1016/j.cie.2016.02.014.

Figures(3)

Tables(2)