# American Institute of Mathematical Sciences

• Previous Article
An integrated bi-objective optimization model and improved genetic algorithm for vehicle routing problems with temporal and spatial constraints
• JIMO Home
• This Issue
• Next Article
A smoothing SAA algorithm for a portfolio choice model based on second-order stochastic dominance measures
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  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, 2020, 16 (3) : 1187-1202. doi: 10.3934/jimo.2018199
##### References:

show all references

##### 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
 [1] Chih-Chiang Fang. Bayesian decision making in determining optimal leased term and preventive maintenance scheme for leased facilities. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020127 [2] Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027 [3] J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the Sakawa-Shindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391-406. doi: 10.3934/mcrf.2016008 [4] Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437 [5] Seung-Yeal Ha, Dongnam Ko, Chanho Min, Xiongtao Zhang. Emergent collective behaviors of stochastic kuramoto oscillators. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1059-1081. doi: 10.3934/dcdsb.2019208 [6] María J. Garrido-Atienza, Bohdan Maslowski, Jana  Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088 [7] Min Li. A three term Polak-Ribière-Polyak conjugate gradient method close to the memoryless BFGS quasi-Newton method. Journal of Industrial & Management Optimization, 2020, 16 (1) : 245-260. doi: 10.3934/jimo.2018149 [8] Ardeshir Ahmadi, Hamed Davari-Ardakani. A multistage stochastic programming framework for cardinality constrained portfolio optimization. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 359-377. doi: 10.3934/naco.2017023 [9] Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175 [10] Longxiang Fang, Narayanaswamy Balakrishnan, Wenyu Huang. Stochastic comparisons of parallel systems with scale proportional hazards components equipped with starting devices. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021004 [11] Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213 [12] Shihu Li, Wei Liu, Yingchao Xie. Large deviations for stochastic 3D Leray-$\alpha$ model with fractional dissipation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2491-2509. doi: 10.3934/cpaa.2019113 [13] Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521

2019 Impact Factor: 1.366