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May  2020, 16(3): 1187-1202. 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  May 2020 Early access  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 and Management Optimization, 2020, 16 (3) : 1187-1202. doi: 10.3934/jimo.2018199
##### References:

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##### References:
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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