American Institute of Mathematical Sciences

doi: 10.3934/jimo.2021087
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Optimal return and rebate mechanism based on demand sensitivity to reference price

 1 School of Management, Management Science and Engineering, Xi'an Jiaotong University, ERC for Process Mining of Manufacturing Services in Shaanxi Province, Xi'an 710049, China 2 Department of Management, Driehaus College of Business, DePaul University, 1 E Jackson Blvd., Chicago, 60604, America

* Corresponding author: Nengmin Wang

Received  October 2020 Revised  March 2021 Early access April 2021

Fund Project: The first author is supported by the Key Project of National Natural Science Foundation of China under Grant 71732006; the National Natural Science Foundation of China under Grants 71572138, 71390331, 71401132, and 71371150

The acceleration of electronic products' upgrade affects consumers' purchase behaviour. How to encourage consumers to return old products in order to upgrade to new products and how to optimize such the closed-loop supply chain are important managerial topics. According to the theory of reference price, the closed-loop supply chain model with fixed rebate and variable rebate is established. The analysis results imply that, when consumers' willingness to return second-hand products depends on manufacturers' rebates and prices of new products, the profit of closed-loop supply chain decreases. In addition, when consumers are sensitive to price difference, enterprises can adopt low profit margin methods to increase new product demand. Furthermore, the profit of the manufacturer is closely related to whether consumers are loss-seeking or loss-averse. Finally, our analysis provides the insights of the relationship between the optimal return and rebate mechanism and the use time of the previous generation of products.

Citation: Junling Han, Nengmin Wang, Zhengwen He, Bin Jiang. Optimal return and rebate mechanism based on demand sensitivity to reference price. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021087
References:

show all references

References:
$\ \pi _{1}^{*}$ when$\ \beta$ varies
$\ \pi _{R_1}^{*}$ when$\ k$ varies
$\ \pi _{M_1}^{*}$ when$\ k$ varies
$\ \pi _{R_1}^{*}$ when$\ k$ varies
$\ \pi _{M_1}^{*}$ when$\ k$ varies
$\ \pi _{1}^{*}$ when$\ k$ varies
$\ \pi _{2}^{*}$ when$\ \beta$ varies
$\ \pi _{R_2}^{*}$ when$\ t$ varies
$\ \pi _{M_2}^{*}$ when$\ t$ varies
$\ \pi _{R_2}^{*}$ when$\ k$ varies
$\ \pi _{M_2}^{*}$ when$\ k$ varies
$\ \pi _{2}^{*}$ when$k$ varies
$\ \pi _{R_2}^{*}$ when$\ k$ varies
$\ \pi _{M_2}^{*}$ when$\ k$ varies
$\ \pi _{2}^{*}$ when$k$ varies
$\ (\pi _{R_1}^{*}-\pi _{R_2}^{*})$ when$\ t$ varies
$\ (\pi _{M_1}^{*}-\pi _{M_2}^{*})$ when$\ t$ varies
$\ \pi _{1}^{*}-\pi _{2}^{*}$ when$\ t$ varies
The description of the symbols
 Notations Description $\ p_0$ The price at which the previous generation of products are sold $\ p$ Sales price of new generation products $\ \omega$ The unit wholesale price of the new product $\ c$ Unit cost of new product $\ T$ Product life cycle $\ t$ Product usage time $\ a$ Market demand scale $\ b$ Price sensitivity coefficient $\ \alpha$ The quantity of used products returned by consumers independent of price and rebate $\ \beta$ Consumers' willingness to return used products $\ \delta$ Discount factor according to product quality $\ k$ Price difference (price difference between the new product and the previous one) sensitivity coefficient ($\ k\geq0$) $\ g$ Cost of the unit of remanufactured product (including remanufacturing, transportation and detection costs) $\ l$ Fixed rebate $\ d$ The demand for new products $\ r_i$ The quantity of previous generation products returned $\ \pi_{M_i}$ Manufacturer's profit $\ \pi_{R_i}$ Retailer's profit $\ \pi_i$ Profits of closed loop supply chains $\ i=1,2$ Represent the corresponding parameters under fixed rebate and variable rebate respectively
 Notations Description $\ p_0$ The price at which the previous generation of products are sold $\ p$ Sales price of new generation products $\ \omega$ The unit wholesale price of the new product $\ c$ Unit cost of new product $\ T$ Product life cycle $\ t$ Product usage time $\ a$ Market demand scale $\ b$ Price sensitivity coefficient $\ \alpha$ The quantity of used products returned by consumers independent of price and rebate $\ \beta$ Consumers' willingness to return used products $\ \delta$ Discount factor according to product quality $\ k$ Price difference (price difference between the new product and the previous one) sensitivity coefficient ($\ k\geq0$) $\ g$ Cost of the unit of remanufactured product (including remanufacturing, transportation and detection costs) $\ l$ Fixed rebate $\ d$ The demand for new products $\ r_i$ The quantity of previous generation products returned $\ \pi_{M_i}$ Manufacturer's profit $\ \pi_{R_i}$ Retailer's profit $\ \pi_i$ Profits of closed loop supply chains $\ i=1,2$ Represent the corresponding parameters under fixed rebate and variable rebate respectively
If consumers are loss-seeking
 $\ \beta$ with fixed rebate mechanism with variable rebate mechanism $\ p _{1}^{*}$ $\ \omega _{1}^{*}$ $\ \pi _{M_1}^{*}$ $\ \pi _{R_1}^{*}$ $\ \pi _{1}^{*}$ $\ p _{2}^{*}$ $\ \omega _{2}^{*}$ $\ \pi _{M_2}^{*}$ $\ \pi _{R_2}^{*}$ $\ \pi _{2}^{*}$ 0 0.989 0.693 0.225 0.092 0.317 0.989 0.693 0.193 0.092 0.285 0.2 0.987 0.688 0.215 0.094 0.309 0.989 0.692 0.191 0.093 0.284 0.4 0.986 0.683 0.205 0.095 0.301 0.988 0.691 0.189 0.093 0.282 0.6 0.982 0.679 0.196 0.097 0.293 0.988 0.690 0.187 0.093 0.280 0.8 0.980 0.674 0.186 0.098 0.285 0.987 0.689 0.185 0.094 0.279 1 0.977 0.669 0.177 0.100 0.277 0.987 0.688 0.183 0.094 0.277
 $\ \beta$ with fixed rebate mechanism with variable rebate mechanism $\ p _{1}^{*}$ $\ \omega _{1}^{*}$ $\ \pi _{M_1}^{*}$ $\ \pi _{R_1}^{*}$ $\ \pi _{1}^{*}$ $\ p _{2}^{*}$ $\ \omega _{2}^{*}$ $\ \pi _{M_2}^{*}$ $\ \pi _{R_2}^{*}$ $\ \pi _{2}^{*}$ 0 0.989 0.693 0.225 0.092 0.317 0.989 0.693 0.193 0.092 0.285 0.2 0.987 0.688 0.215 0.094 0.309 0.989 0.692 0.191 0.093 0.284 0.4 0.986 0.683 0.205 0.095 0.301 0.988 0.691 0.189 0.093 0.282 0.6 0.982 0.679 0.196 0.097 0.293 0.988 0.690 0.187 0.093 0.280 0.8 0.980 0.674 0.186 0.098 0.285 0.987 0.689 0.185 0.094 0.279 1 0.977 0.669 0.177 0.100 0.277 0.987 0.688 0.183 0.094 0.277
If consumers are loss-averse
 $\ \beta$ with fixed rebate mechanism with variable rebate mechanism $\ p _{1}^{*}$ $\ \omega _{1}^{*}$ $\ \pi _{M_1}^{*}$ $\ \pi _{R_1}^{*}$ $\ \pi _{1}^{*}$ $\ p _{2}^{*}$ $\ \omega _{2}^{*}$ $\ \pi _{M_2}^{*}$ $\ \pi _{R_2}^{*}$ $\ \pi _{2}^{*}$ 0 0.989 0.860 0.445 0.018 0.463 0.989 0.860 0.471 0.018 0.489 0.2 0.963 0.807 0.341 0.026 0.367 0.963 0.808 0.369 0.026 0.395 0.4 0.937 0.755 0.240 0.035 0.275 0.937 0.756 0.269 0.035 0.304 0.6 0.911 0.702 0.141 0.046 0.187 0.911 0.704 0.172 0.045 0.217 0.8 0.885 0.650 0.046 0.058 0.104 0.886 0.652 0.078 0.057 0.135 1 0.858 0.598 -0.047 0.071 0.024 0.860 0.600 -0.013 0.071 0.058
 $\ \beta$ with fixed rebate mechanism with variable rebate mechanism $\ p _{1}^{*}$ $\ \omega _{1}^{*}$ $\ \pi _{M_1}^{*}$ $\ \pi _{R_1}^{*}$ $\ \pi _{1}^{*}$ $\ p _{2}^{*}$ $\ \omega _{2}^{*}$ $\ \pi _{M_2}^{*}$ $\ \pi _{R_2}^{*}$ $\ \pi _{2}^{*}$ 0 0.989 0.860 0.445 0.018 0.463 0.989 0.860 0.471 0.018 0.489 0.2 0.963 0.807 0.341 0.026 0.367 0.963 0.808 0.369 0.026 0.395 0.4 0.937 0.755 0.240 0.035 0.275 0.937 0.756 0.269 0.035 0.304 0.6 0.911 0.702 0.141 0.046 0.187 0.911 0.704 0.172 0.045 0.217 0.8 0.885 0.650 0.046 0.058 0.104 0.886 0.652 0.078 0.057 0.135 1 0.858 0.598 -0.047 0.071 0.024 0.860 0.600 -0.013 0.071 0.058
If consumers are loss-seeking
 $\ k$ with fixed rebate mechanism with variable rebate mechanism $\ p _{1}^{*}$ $\ \omega _{1}^{*}$ $\ \pi _{M_1}^{*}$ $\ \pi _{R_1}^{*}$ $\ \pi _{1}^{*}$ $\ p _{2}^{*}$ $\ \omega _{2}^{*}$ $\ \pi _{M_2}^{*}$ $\ \pi _{R_2}^{*}$ $\ \pi _{2}^{*}$ 0 1.088 0.746 0.168 0.081 0.249 1.095 0.761 0.157 0.078 0.235 0.2 1.035 0.714 0.181 0.087 0.268 1.041 0.726 0.169 0.084 0.253 0.4 0.998 0.690 0.194 0.093 0.287 1.003 0.701 0.182 0.090 0.272 0.6 0.970 0.673 0.207 0.099 0.307 0.975 0.682 0.194 0.096 0.290 0.8 0.949 0.659 0.221 0.106 0.327 0.953 0.667 0.208 0.103 0.311 1 0.931 0.648 0.235 0.112 0.347 0.935 0.656 0.221 0.109 0.330
 $\ k$ with fixed rebate mechanism with variable rebate mechanism $\ p _{1}^{*}$ $\ \omega _{1}^{*}$ $\ \pi _{M_1}^{*}$ $\ \pi _{R_1}^{*}$ $\ \pi _{1}^{*}$ $\ p _{2}^{*}$ $\ \omega _{2}^{*}$ $\ \pi _{M_2}^{*}$ $\ \pi _{R_2}^{*}$ $\ \pi _{2}^{*}$ 0 1.088 0.746 0.168 0.081 0.249 1.095 0.761 0.157 0.078 0.235 0.2 1.035 0.714 0.181 0.087 0.268 1.041 0.726 0.169 0.084 0.253 0.4 0.998 0.690 0.194 0.093 0.287 1.003 0.701 0.182 0.090 0.272 0.6 0.970 0.673 0.207 0.099 0.307 0.975 0.682 0.194 0.096 0.290 0.8 0.949 0.659 0.221 0.106 0.327 0.953 0.667 0.208 0.103 0.311 1 0.931 0.648 0.235 0.112 0.347 0.935 0.656 0.221 0.109 0.330
If consumers are loss-averse
 $\ k$ with fixed rebate mechanism with variable rebate mechanism $\ p _{1}^{*}$ $\ \omega _{1}^{*}$ $\ \pi _{M_1}^{*}$ $\ \pi _{R_1}^{*}$ $\ \pi _{1}^{*}$ $\ p _{2}^{*}$ $\ \omega _{2}^{*}$ $\ \pi _{M_2}^{*}$ $\ \pi _{R_2}^{*}$ $\ \pi _{2}^{*}$ 0 1.123 0.818 0.155 0.065 0.220 1.124 0.820 0.186 0.065 0.251 0.2 1.024 0.773 0.173 0.053 0.226 1.024 0.775 0.203 0.052 0.255 0.4 0.952 0.741 0.185 0.044 0.229 0.953 0.743 0.215 0.043 0.259 0.6 0.899 0.717 0.194 0.037 0.231 0.899 0.719 0.225 0.037 0.262 0.8 0.857 0.699 0.202 0.032 0.234 0.858 0.700 0.232 0.032 0.263 1 0.824 0.684 0.208 0.028 0.236 0.825 0.685 0.237 0.027 0.265
 $\ k$ with fixed rebate mechanism with variable rebate mechanism $\ p _{1}^{*}$ $\ \omega _{1}^{*}$ $\ \pi _{M_1}^{*}$ $\ \pi _{R_1}^{*}$ $\ \pi _{1}^{*}$ $\ p _{2}^{*}$ $\ \omega _{2}^{*}$ $\ \pi _{M_2}^{*}$ $\ \pi _{R_2}^{*}$ $\ \pi _{2}^{*}$ 0 1.123 0.818 0.155 0.065 0.220 1.124 0.820 0.186 0.065 0.251 0.2 1.024 0.773 0.173 0.053 0.226 1.024 0.775 0.203 0.052 0.255 0.4 0.952 0.741 0.185 0.044 0.229 0.953 0.743 0.215 0.043 0.259 0.6 0.899 0.717 0.194 0.037 0.231 0.899 0.719 0.225 0.037 0.262 0.8 0.857 0.699 0.202 0.032 0.234 0.858 0.700 0.232 0.032 0.263 1 0.824 0.684 0.208 0.028 0.236 0.825 0.685 0.237 0.027 0.265
If consumers are loss-seeking
 $\ t$ $\ p_{2}^{*}$ $\ \omega_{2}^{*}$ $\ \pi _{M_2}^{*}$ $\ \pi _{R_2}^{*}$ $\ \pi _{2}^{*}$ 0 1.099 0.912 -0.531 0.037 -0.494 0.2 1.075 0.864 -0.308 0.047 -0.261 0.4 1.051 0.817 -0.122 0.058 -0.064 0.6 1.027 0.769 0.026 0.070 0.096 0.8 1.004 0.721 0.136 0.084 0.220 1 0.980 0.674 0.209 0.098 0.307
 $\ t$ $\ p_{2}^{*}$ $\ \omega_{2}^{*}$ $\ \pi _{M_2}^{*}$ $\ \pi _{R_2}^{*}$ $\ \pi _{2}^{*}$ 0 1.099 0.912 -0.531 0.037 -0.494 0.2 1.075 0.864 -0.308 0.047 -0.261 0.4 1.051 0.817 -0.122 0.058 -0.064 0.6 1.027 0.769 0.026 0.070 0.096 0.8 1.004 0.721 0.136 0.084 0.220 1 0.980 0.674 0.209 0.098 0.307
If consumers are loss-averse
 $\ t$ $\ p_{2}^{*}$ $\ \omega_{2}^{*}$ $\ \pi _{M_2}^{*}$ $\ \pi _{R_2}^{*}$ $\ \pi _{2}^{*}$ 0 0.980 0.841 0.080 0.020 0.100 0.2 0.968 0.817 0.127 0.024 0.151 0.4 0.956 0.793 0.165 0.028 0.193 0.6 0.944 0.769 0.194 0.032 0.226 0.8 0.932 0.745 0.213 0.037 0.250 1 0.920 0.721 0.223 0.042 0.265
 $\ t$ $\ p_{2}^{*}$ $\ \omega_{2}^{*}$ $\ \pi _{M_2}^{*}$ $\ \pi _{R_2}^{*}$ $\ \pi _{2}^{*}$ 0 0.980 0.841 0.080 0.020 0.100 0.2 0.968 0.817 0.127 0.024 0.151 0.4 0.956 0.793 0.165 0.028 0.193 0.6 0.944 0.769 0.194 0.032 0.226 0.8 0.932 0.745 0.213 0.037 0.250 1 0.920 0.721 0.223 0.042 0.265
If consumers are loss-seeking
 $\ t$ $\ p_{1}^{*}-p_{2}^{*}$ $\ \omega_{1}^{*}-\omega_{2}^{*}$ $\ \pi _{M_1}^{*}-\pi _{M_2}^{*}$ $\ \pi _{R_1}^{*}-\pi _{R_2}^{*}$ $\ \pi _{1}^{*}-\pi _{2}^{*}$ $t\in [0, 1-\frac{l}{p})$ 0.1 -0.104 -0.207 0.615 0.055 0.670 0.3 -0.080 -0.160 0.410 0.044 0.454 0.5 -0.060 -0.112 0.243 0.032 0.275 $t\in [1-\frac{l}{p}, 1]$ 0.98 0.001 0.002 -0.004 -0.001 -0.005 0.99 0.002 0.005 -0.006 -0.002 -0.008 1 0.004 0.007 -0.009 -0.002 -0.011
 $\ t$ $\ p_{1}^{*}-p_{2}^{*}$ $\ \omega_{1}^{*}-\omega_{2}^{*}$ $\ \pi _{M_1}^{*}-\pi _{M_2}^{*}$ $\ \pi _{R_1}^{*}-\pi _{R_2}^{*}$ $\ \pi _{1}^{*}-\pi _{2}^{*}$ $t\in [0, 1-\frac{l}{p})$ 0.1 -0.104 -0.207 0.615 0.055 0.670 0.3 -0.080 -0.160 0.410 0.044 0.454 0.5 -0.060 -0.112 0.243 0.032 0.275 $t\in [1-\frac{l}{p}, 1]$ 0.98 0.001 0.002 -0.004 -0.001 -0.005 0.99 0.002 0.005 -0.006 -0.002 -0.008 1 0.004 0.007 -0.009 -0.002 -0.011
If consumers are loss-averse
 $\ t$ $\ p_{1}^{*}-p_{2}^{*}$ $\ \omega_{1}^{*}-\omega_{2}^{*}$ {\mbox{$\ \pi _{M_1}^{*}-\pi _{M_2}^{*}$ $\ \pi _{R_1}^{*}-\pi _{R_2}^{*}$ $\ \pi _{1}^{*}-\pi _{2}^{*}$ $t\in [0, 1-\frac{l}{p})$ 0.1 -0.050 -0.100 0.085 0.018 0.103 0.3 -0.038 -0.076 0.043 0.014 0.057 0.5 -0.026 -0.052 0.009 0.010 0.019 $t\in [1-\frac{l}{p}, 1]$ 0.98 0.002 0.005 -0.032 -0.001 -0.033 0.99 0.003 0.006 -0.032 -0.001 -0.033 1 0.004 0.007 -0.033 -0.002 -0.035
 $\ t$ $\ p_{1}^{*}-p_{2}^{*}$ $\ \omega_{1}^{*}-\omega_{2}^{*}$ {\mbox{$\ \pi _{M_1}^{*}-\pi _{M_2}^{*}$ $\ \pi _{R_1}^{*}-\pi _{R_2}^{*}$ $\ \pi _{1}^{*}-\pi _{2}^{*}$ $t\in [0, 1-\frac{l}{p})$ 0.1 -0.050 -0.100 0.085 0.018 0.103 0.3 -0.038 -0.076 0.043 0.014 0.057 0.5 -0.026 -0.052 0.009 0.010 0.019 $t\in [1-\frac{l}{p}, 1]$ 0.98 0.002 0.005 -0.032 -0.001 -0.033 0.99 0.003 0.006 -0.032 -0.001 -0.033 1 0.004 0.007 -0.033 -0.002 -0.035
 [1] Dingzhong Feng, Xiaofeng Zhang, Ye Zhang. Collection decisions and coordination in a closed-loop supply chain under recovery price and service competition. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021117 [2] Zhidan Wu, Xiaohu Qian, Min Huang, Wai-Ki Ching, Hanbin Kuang, Xingwei Wang. Channel leadership and recycling channel in closed-loop supply chain: The case of recycling price by the recycling party. Journal of Industrial & Management Optimization, 2021, 17 (6) : 3247-3268. doi: 10.3934/jimo.2020116 [3] Tinggui Chen, Yanhui Jiang. Research on operating mechanism for creative products supply chain based on game theory. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1103-1112. doi: 10.3934/dcdss.2015.8.1103 [4] Yi Jing, Wenchuan Li. Integrated recycling-integrated production - distribution planning for decentralized closed-loop supply chain. Journal of Industrial & Management Optimization, 2018, 14 (2) : 511-539. doi: 10.3934/jimo.2017058 [5] Wenbin Wang, Peng Zhang, Junfei Ding, Jian Li, Hao Sun, Lingyun He. Closed-loop supply chain network equilibrium model with retailer-collection under legislation. Journal of Industrial & Management Optimization, 2019, 15 (1) : 199-219. doi: 10.3934/jimo.2018039 [6] Reza Lotfi, Yahia Zare Mehrjerdi, Mir Saman Pishvaee, Ahmad Sadeghieh, Gerhard-Wilhelm Weber. A robust optimization model for sustainable and resilient closed-loop supply chain network design considering conditional value at risk. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 221-253. doi: 10.3934/naco.2020023 [7] Benrong Zheng, Xianpei Hong. Effects of take-back legislation on pricing and coordination in a closed-loop supply chain. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021035 [8] Abdolhossein Sadrnia, Amirreza Payandeh Sani, Najme Roghani Langarudi. Sustainable closed-loop supply chain network optimization for construction machinery recovering. Journal of Industrial & Management Optimization, 2021, 17 (5) : 2389-2414. doi: 10.3934/jimo.2020074 [9] Maedeh Agahgolnezhad Gerdrodbari, Fatemeh Harsej, Mahboubeh Sadeghpour, Mohammad Molani Aghdam. A robust multi-objective model for managing the distribution of perishable products within a green closed-loop supply chain. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021107 [10] Guangzhou Yan, Qinyu Song, Yaodong Ni, Xiangfeng Yang. Pricing, carbon emission reduction and recycling decisions in a closed-loop supply chain under uncertain environment. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021181 [11] Masoud Mohammadzadeh, Alireza Arshadi Khamseh, Mohammad Mohammadi. A multi-objective integrated model for closed-loop supply chain configuration and supplier selection considering uncertain demand and different performance levels. Journal of Industrial & Management Optimization, 2017, 13 (2) : 1041-1064. doi: 10.3934/jimo.2016061 [12] Xiao-Xu Chen, Peng Xu, Jiao-Jiao Li, Thomas Walker, Guo-Qiang Yang. Decision-making in a retailer-led closed-loop supply chain involving a third-party logistics provider. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021014 [13] Fatemeh Kangi, Seyed Hamid Reza Pasandideh, Esmaeil Mehdizadeh, Hamed Soleimani. The optimization of a multi-period multi-product closed-loop supply chain network with cross-docking delivery strategy. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021118 [14] Simon Hochgerner. Symmetry actuated closed-loop Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 641-669. doi: 10.3934/jgm.2020030 [15] Xiaohong Chen, Kui Li, Fuqiang Wang, Xihua Li. Optimal production, pricing and government subsidy policies for a closed loop supply chain with uncertain returns. Journal of Industrial & Management Optimization, 2020, 16 (3) : 1389-1414. doi: 10.3934/jimo.2019008 [16] Justine Yasappan, Ángela Jiménez-Casas, Mario Castro. Stabilizing interplay between thermodiffusion and viscoelasticity in a closed-loop thermosyphon. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3267-3299. doi: 10.3934/dcdsb.2015.20.3267 [17] Feimin Zhong, Wei Zeng, Zhongbao Zhou. Mechanism design in a supply chain with ambiguity in private information. Journal of Industrial & Management Optimization, 2020, 16 (1) : 261-287. doi: 10.3934/jimo.2018151 [18] Jianbin Li, Niu Yu, Zhixue Liu, Lianjie Shu. Optimal rebate strategies in a two-echelon supply chain with nonlinear and linear multiplicative demands. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1587-1611. doi: 10.3934/jimo.2016.12.1587 [19] Xiaochen Sun, Fei Hu, Yancong Zhou, Cheng-Chew Lim. Optimal acquisition, inventory and production decisions for a closed-loop manufacturing system with legislation constraint. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1355-1373. doi: 10.3934/jimo.2015.11.1355 [20] Hanxiao Wang, Jingrui Sun, Jiongmin Yong. Weak closed-loop solvability of stochastic linear-quadratic optimal control problems. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2785-2805. doi: 10.3934/dcds.2019117

2020 Impact Factor: 1.801