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doi: 10.3934/jimo.2018199

## Optimal ordering policy for inventory mechanism with a stochastic short-term price discount

 1 School of Management Science, Qufu Normal University, Rizhao Shandong, 276800, China 2 School of Management and Economics, Beijing Institute of Technology, Beijing, 100081, China

Received  April 2017 Revised  November 2017 Published  December 2018

Fund Project: This work is supported by the Natural Science Foundation of China (11671228, 71471101) and Shandong Provincial Natural Science Foundation (ZR2015GZ008)

This paper considers an inventory mechanism in which the supplier may provide a short-term price discount to the retailer at a future time with some uncertainty. To maximize the retailer's profit in this setting, we establish an optimal replenishment and stocking strategy model. Based on the retailer's inventory cost-benefit analysis, we present a closed-form solution for the inventory model and provide an optimal ordering policy to the retailer. Numerical experiments and numerical sensitivity are given to provide some high insights to the inventory model.

Citation: Yiju Wang, Wei Xing, Hengxia Gao. Optimal ordering policy for inventory mechanism with a stochastic short-term price discount. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2018199
##### References:
 [1] A. Ardalan, Optimal ordering policies in response to a sale, IIE Transactions, 20 (1988), 292-294. [2] A. Ardalan, Optimal prices and order quantities when temporary price discounts result in increase in demand, Europ. J. Operations Research, 72 (1994), 52-61. [3] F. J. Arcelus, N. H. Shah and G. Srinivasan, Retailer's pricing, credit and inventory policies for deteriorating items in response to temporary price/credit incentives, Inter. J. Production Economics, 81 (2003), 153-162. [4] R. L. Aull-Hyde, A backlog inventory model during restricted sale periods, J. Operational Research Society, 47 (1996), 1192-1200. [5] L. E. Cárdenas-Barrón, N. R. Smith and S. K. Goyal, Optimal order size to take advantage of a one-time discount offer with allowed backorders, Appl. Math. Modelling, 34 (2010), 1642-1652. doi: 10.1016/j.apm.2009.09.013. [6] H. J. Chang, W. F. Lin and J. F. Ho, Closed-form solutions for Wee's and Martin's EOQ models with a temporary price discount, Inter. J. Production Economics, 131 (2011), 528-534. [7] P. Chu, P. S. Chen and T. Niu, Note on supplier-restricted order quantity under temporary price discounts, Math. Methods of Operations Research, 58 (2003), 141-147. doi: 10.1007/s001860200272. [8] R. A. Davis and N. Gaither, Optimal ordering policies under conditions of extended payment privileges, Management Science, 31 (1985), 499-509. [9] J. K. Friend, Stock control with random opportunities for replenishment, Operational Research Quarterly, 11 (1960), 130-136. [10] S. K. Goyal, Economic ordering policy during special discount periods for dynamic inventory problems under certainty, Engineering Costs and Production Economics, 20 (1990), 101-104. [11] W. K. Kevin Hsu and H. F. Yu, EOQ model for imperfective items under a one-time-only discount, Omega, 37 (2009), 1018-1026. [12] M. A. Kindi and B. R. Sarker, Optimal inventory system with two backlog costs in response to a discount offer, Production Planning and Control, 22 (2011), 325-333. [13] B. Lev and H. J. Weiss, Inventory models with cost changes, Operations Research, 38 (1990), 53-63. doi: 10.1287/opre.38.1.53. [14] Z. W. Luo and J. T. Wang, The optimal price discount, order quantity and minimum quantity in newsvendor model with group purchase, J. Industrial Management Optim., 11 (2015), 1-11. doi: 10.3934/jimo.2015.11.1. [15] S. M. Mousavi, V. Hajipour, S. T. A. Niaki and N. Alikar, Optimizing multi-item multi-period inventory control system with discounted cash flow and inflation: two calibrated meta-heuristic algorithms, Appl. Math. Modelling, 37 (2013), 2241-2256. doi: 10.1016/j.apm.2012.05.019. [16] S. M. Mousavi, J. Sadeghi, S. T. A. Niaki, N. Alikar, A. Bahreininejad and H. S. C. Metselaar, Two parameter-tuned meta-heuristics for a discounted inventory control problem in a fuzzy environment, Information Sciences, 276 (2014), 42-62. doi: 10.1016/j.ins.2014.02.046. [17] S. M. Mousavi, J. Sadeghi, S. T. A. Niaki and M. Tavana, A bi-objective inventory optimization model under inflation and discount using tuned Pareto-based algorithms: NSGA-Ⅱ, NRGA, and MOPSO, Appl. Soft Computing, 43 (2016), 57-72. [18] S. H. R. Pasandideh, S. T. A. Niaki and S. M. Mousavi, Two metaheuristics to solve a multi-item multiperiod inventory control problem under storage constraint and discounts, Inter. J. Advanced Manufacturing Technology, 69 (2013), 1671-1684. [19] D. P. Sari, A. Rusdiansyah and L. Huang, Models of joint economic lot-sizing problem with time-based temporary price discounts, Inter. J. Production Economics, 139 (2012), 145-154. [20] B. R. Sarker and M. A. Kindi, Optimal ordering policies in response to a discount offer, Inter. J. Production Economics, 100 (2006), 195-211. [21] Y. Shaposhnik, Y. T. Herer and H. Naseraldin, Optimal ordering for a probabilistic one-time discount, Europ. J. Operations Research, 244 (2015), 803-814. doi: 10.1016/j.ejor.2015.02.020. [22] H. Sun and Y. Wang, Further Discussion on the error bound for generalized LCP over a polyhedral cone, J. Optim. Theory Appl., 159 (2013), 93-107. doi: 10.1007/s10957-013-0290-z. [23] A. A. Taleizadeh, D. W. Pentico, M. Aryanezhad and S. M. Ghoreyshi, An economic order quantity model with partial backordering and a special sale price, Europ. J. Operations Research, 221 (2012), 571-583. doi: 10.1016/j.ejor.2012.03.032. [24] A. A. Taleizadeh, B. Mohammadi, L. E. Cardenas-Barron and H. Samimi, An EOQ model for perishable product with special sale and shortage, Inter. J. Production Economics, 145 (2013), 318-338. [25] R. Tersine and A. Schwarzkopf, Optimal stock replenishment strategies in response to temporary price reductions, J. Business Logistics, 10 (1989), 123-145. [26] Y. J. Wang, X. F. Sun and F. X. Meng, On the conditional and partial trade credit policy with capital constraints: A Stackelberg model, Appl. Math. Modelling, 40 (2016), 1-18. doi: 10.1016/j.apm.2015.04.036. [27] Y. J. Wang, L. Caccetta and G. L. Zhou, Convergence analysis of a block improvement method for polynomial optimization over unit spheres, Numer. Linear Algebra Appl., 22 (2015), 1059-1076. doi: 10.1002/nla.1996. [28] C. T. Yang, L. Y. Ouyang, K. S. Wu and H. F. Yen, Optimal ordering policy in response to a temporary sale price when retailer's warehouse capacity is limited, Europ. J. Industrial Engineering, 6 (2012), 26-49. [29] T. F. Ye and S. H. Ma, Discount-offering and demand-rejection decisions for substitutable products with different profit levels, J. Industrial & Management Optim., 12 (2016), 45-71. doi: 10.3934/jimo.2016.12.45. [30] Y. G. Zhang and X. W. Tang, Retailer's order strategy of delay in payments under cash discount and capital constraints, Systems Engineering, 27 (2009), 30-34. [31] P. H. Zipkin, Foundations of Inventory Management, New York, NY: McGraw-Hill, 2000.

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##### References:
 [1] A. Ardalan, Optimal ordering policies in response to a sale, IIE Transactions, 20 (1988), 292-294. [2] A. Ardalan, Optimal prices and order quantities when temporary price discounts result in increase in demand, Europ. J. Operations Research, 72 (1994), 52-61. [3] F. J. Arcelus, N. H. Shah and G. Srinivasan, Retailer's pricing, credit and inventory policies for deteriorating items in response to temporary price/credit incentives, Inter. J. Production Economics, 81 (2003), 153-162. [4] R. L. Aull-Hyde, A backlog inventory model during restricted sale periods, J. Operational Research Society, 47 (1996), 1192-1200. [5] L. E. Cárdenas-Barrón, N. R. Smith and S. K. Goyal, Optimal order size to take advantage of a one-time discount offer with allowed backorders, Appl. Math. Modelling, 34 (2010), 1642-1652. doi: 10.1016/j.apm.2009.09.013. [6] H. J. Chang, W. F. Lin and J. F. Ho, Closed-form solutions for Wee's and Martin's EOQ models with a temporary price discount, Inter. J. Production Economics, 131 (2011), 528-534. [7] P. Chu, P. S. Chen and T. Niu, Note on supplier-restricted order quantity under temporary price discounts, Math. Methods of Operations Research, 58 (2003), 141-147. doi: 10.1007/s001860200272. [8] R. A. Davis and N. Gaither, Optimal ordering policies under conditions of extended payment privileges, Management Science, 31 (1985), 499-509. [9] J. K. Friend, Stock control with random opportunities for replenishment, Operational Research Quarterly, 11 (1960), 130-136. [10] S. K. Goyal, Economic ordering policy during special discount periods for dynamic inventory problems under certainty, Engineering Costs and Production Economics, 20 (1990), 101-104. [11] W. K. Kevin Hsu and H. F. Yu, EOQ model for imperfective items under a one-time-only discount, Omega, 37 (2009), 1018-1026. [12] M. A. Kindi and B. R. Sarker, Optimal inventory system with two backlog costs in response to a discount offer, Production Planning and Control, 22 (2011), 325-333. [13] B. Lev and H. J. Weiss, Inventory models with cost changes, Operations Research, 38 (1990), 53-63. doi: 10.1287/opre.38.1.53. [14] Z. W. Luo and J. T. Wang, The optimal price discount, order quantity and minimum quantity in newsvendor model with group purchase, J. Industrial Management Optim., 11 (2015), 1-11. doi: 10.3934/jimo.2015.11.1. [15] S. M. Mousavi, V. Hajipour, S. T. A. Niaki and N. Alikar, Optimizing multi-item multi-period inventory control system with discounted cash flow and inflation: two calibrated meta-heuristic algorithms, Appl. Math. Modelling, 37 (2013), 2241-2256. doi: 10.1016/j.apm.2012.05.019. [16] S. M. Mousavi, J. Sadeghi, S. T. A. Niaki, N. Alikar, A. Bahreininejad and H. S. C. Metselaar, Two parameter-tuned meta-heuristics for a discounted inventory control problem in a fuzzy environment, Information Sciences, 276 (2014), 42-62. doi: 10.1016/j.ins.2014.02.046. [17] S. M. Mousavi, J. Sadeghi, S. T. A. Niaki and M. Tavana, A bi-objective inventory optimization model under inflation and discount using tuned Pareto-based algorithms: NSGA-Ⅱ, NRGA, and MOPSO, Appl. Soft Computing, 43 (2016), 57-72. [18] S. H. R. Pasandideh, S. T. A. Niaki and S. M. Mousavi, Two metaheuristics to solve a multi-item multiperiod inventory control problem under storage constraint and discounts, Inter. J. Advanced Manufacturing Technology, 69 (2013), 1671-1684. [19] D. P. Sari, A. Rusdiansyah and L. Huang, Models of joint economic lot-sizing problem with time-based temporary price discounts, Inter. J. Production Economics, 139 (2012), 145-154. [20] B. R. Sarker and M. A. Kindi, Optimal ordering policies in response to a discount offer, Inter. J. Production Economics, 100 (2006), 195-211. [21] Y. Shaposhnik, Y. T. Herer and H. Naseraldin, Optimal ordering for a probabilistic one-time discount, Europ. J. Operations Research, 244 (2015), 803-814. doi: 10.1016/j.ejor.2015.02.020. [22] H. Sun and Y. Wang, Further Discussion on the error bound for generalized LCP over a polyhedral cone, J. Optim. Theory Appl., 159 (2013), 93-107. doi: 10.1007/s10957-013-0290-z. [23] A. A. Taleizadeh, D. W. Pentico, M. Aryanezhad and S. M. Ghoreyshi, An economic order quantity model with partial backordering and a special sale price, Europ. J. Operations Research, 221 (2012), 571-583. doi: 10.1016/j.ejor.2012.03.032. [24] A. A. Taleizadeh, B. Mohammadi, L. E. Cardenas-Barron and H. Samimi, An EOQ model for perishable product with special sale and shortage, Inter. J. Production Economics, 145 (2013), 318-338. [25] R. Tersine and A. Schwarzkopf, Optimal stock replenishment strategies in response to temporary price reductions, J. Business Logistics, 10 (1989), 123-145. [26] Y. J. Wang, X. F. Sun and F. X. Meng, On the conditional and partial trade credit policy with capital constraints: A Stackelberg model, Appl. Math. Modelling, 40 (2016), 1-18. doi: 10.1016/j.apm.2015.04.036. [27] Y. J. Wang, L. Caccetta and G. L. Zhou, Convergence analysis of a block improvement method for polynomial optimization over unit spheres, Numer. Linear Algebra Appl., 22 (2015), 1059-1076. doi: 10.1002/nla.1996. [28] C. T. Yang, L. Y. Ouyang, K. S. Wu and H. F. Yen, Optimal ordering policy in response to a temporary sale price when retailer's warehouse capacity is limited, Europ. J. Industrial Engineering, 6 (2012), 26-49. [29] T. F. Ye and S. H. Ma, Discount-offering and demand-rejection decisions for substitutable products with different profit levels, J. Industrial & Management Optim., 12 (2016), 45-71. doi: 10.3934/jimo.2016.12.45. [30] Y. G. Zhang and X. W. Tang, Retailer's order strategy of delay in payments under cash discount and capital constraints, Systems Engineering, 27 (2009), 30-34. [31] P. H. Zipkin, Foundations of Inventory Management, New York, NY: McGraw-Hill, 2000.
Optimal order policy for Scenario 1
Policy 1 for Scenario 2
Policy 2 for Scenario 2
Policy 3 for Scenario 2
The expected increased profit as a function of parameter $p$
The expected increased profit as a function of parameter $\gamma$
Notation
 Symbol Description Symbol Description $\lambda$ retailer's market demand rate $t_s$ the start time of possible discount $K$ fixed ordering cost $t_e$ the end time of possible discount $c$ retailer's unit purchase price $t_r$ the special ordering time $b$ retailer's unit selling price $q_s~$ remaining inventory at $t_s$ $h$ retailer's inventory holding cost $q_e$ remaining inventory at $t_e$ per unit item per unit time $q_r$ remaining inventory at $t_r$ $p$ probability that the price $Q_0$ order size before $t_s$ discount takes place $Q_d$ special order size $\gamma$ discount rate $*$ indicates the optimal value
 Symbol Description Symbol Description $\lambda$ retailer's market demand rate $t_s$ the start time of possible discount $K$ fixed ordering cost $t_e$ the end time of possible discount $c$ retailer's unit purchase price $t_r$ the special ordering time $b$ retailer's unit selling price $q_s~$ remaining inventory at $t_s$ $h$ retailer's inventory holding cost $q_e$ remaining inventory at $t_e$ per unit item per unit time $q_r$ remaining inventory at $t_r$ $p$ probability that the price $Q_0$ order size before $t_s$ discount takes place $Q_d$ special order size $\gamma$ discount rate $*$ indicates the optimal value
Numerical result for Example 5.1
 Policy $Q_0$ $q_r$ $Q_d$ $E$ $\pi_{ \rm EOQ}$ 130 40 132.22 -3.50 $\pi_s$ 150 0 172.22 21.38 $\pi_e$ 116.67 0 172.22 24.05
 Policy $Q_0$ $q_r$ $Q_d$ $E$ $\pi_{ \rm EOQ}$ 130 40 132.22 -3.50 $\pi_s$ 150 0 172.22 21.38 $\pi_e$ 116.67 0 172.22 24.05
Numerical result for Example 5.2
 Policy $Q_0$ $q_r$ $Q_d$ $E$ $\pi_s$ 150 0 172.22 0.25 $\bar\pi$ 117.86 3.58 168.64 2.54
 Policy $Q_0$ $q_r$ $Q_d$ $E$ $\pi_s$ 150 0 172.22 0.25 $\bar\pi$ 117.86 3.58 168.64 2.54
Impact of parameter $p$ on the retailer's profit
 $p$ $\pi_{ \rm EOQ}$ $\pi_s$ $\pi_e$ $\bar{\pi}$ EOQ ordering policy 0.01 -0.11 -7.02 -4.35 -0.07 0 EOQ 0.05 -0.58 -3.10 -0.43 0.64 0 $\bar{\pi}$ 0.10 -1.16 1.79 4.46 / 0 $\pi_e$ 0.15 -1.74 6.69 9.36 / 0 $\pi_e$ 0.30 -3.50 21.38 24.05 / 0 $\pi_e$ 0.50 -5.83 40.97 43.64 / 0 $\pi_e$ 0.80 -9.32 70.36 73.02 / 0 $\pi_e$ 0.90 -10.49 80.15 82.82 / 0 $\pi_e$ 0.95 -11.07 85.05 87.71 / 0 $\pi_e$
 $p$ $\pi_{ \rm EOQ}$ $\pi_s$ $\pi_e$ $\bar{\pi}$ EOQ ordering policy 0.01 -0.11 -7.02 -4.35 -0.07 0 EOQ 0.05 -0.58 -3.10 -0.43 0.64 0 $\bar{\pi}$ 0.10 -1.16 1.79 4.46 / 0 $\pi_e$ 0.15 -1.74 6.69 9.36 / 0 $\pi_e$ 0.30 -3.50 21.38 24.05 / 0 $\pi_e$ 0.50 -5.83 40.97 43.64 / 0 $\pi_e$ 0.80 -9.32 70.36 73.02 / 0 $\pi_e$ 0.90 -10.49 80.15 82.82 / 0 $\pi_e$ 0.95 -11.07 85.05 87.71 / 0 $\pi_e$
Numerical results for Example 5.3
 Policy $Q_0$ $q_r$ $Q_d$ $E$ $\pi_{ \rm EOQ}$ 130 40 97.76 -22.60 $\pi_s$ 150 0 137.76 -2.91 $\pi_e$ 116.67 0 137.76 -0.24
 Policy $Q_0$ $q_r$ $Q_d$ $E$ $\pi_{ \rm EOQ}$ 130 40 97.76 -22.60 $\pi_s$ 150 0 137.76 -2.91 $\pi_e$ 116.67 0 137.76 -0.24
Impact of parameter $\gamma$ on retailer's ordering policy
 $\gamma$ $\pi_{ \rm EOQ}$ $\pi_s$ $\pi_e$ $\bar{\pi}$ EOQ optimal policy 0.5 280.65 331.45 334.11 / 0 $\pi_e$ 0.6 161.48 205.80 208.47 / 0 $\pi_e$ 0.7 83.57 121.41 124.07 / 0 $\pi_e$ 0.8 31.44 62.80 65.47 / 0 $\pi_e$ 0.9 -3.50 21.38 24.05 / 0 $\pi_e$ 0.95 -16.21 5.43 8.09 / 0 $\pi_e$ 0.98 -22.61 -2.91 -0.24 / 0 EOQ 0.99 -24.55 -5.50 -2.83 / 0 EOQ
 $\gamma$ $\pi_{ \rm EOQ}$ $\pi_s$ $\pi_e$ $\bar{\pi}$ EOQ optimal policy 0.5 280.65 331.45 334.11 / 0 $\pi_e$ 0.6 161.48 205.80 208.47 / 0 $\pi_e$ 0.7 83.57 121.41 124.07 / 0 $\pi_e$ 0.8 31.44 62.80 65.47 / 0 $\pi_e$ 0.9 -3.50 21.38 24.05 / 0 $\pi_e$ 0.95 -16.21 5.43 8.09 / 0 $\pi_e$ 0.98 -22.61 -2.91 -0.24 / 0 EOQ 0.99 -24.55 -5.50 -2.83 / 0 EOQ
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