# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2020167

## Selection and impact of decision mode of encroachment and retail service in a dual-channel supply chain

 School of Mathematics and Physics, Anhui Jianzhu University, Hefei, Anhui 230601, China

* Corresponding author: Jie Min

Received  May 2020 Revised  September 2020 Published  November 2020

Consider a supply chain consisting of one manufacturer and one retailer. The manufacturer may open direct channels through ex-ante or ex-post encroachment, and the retailer can provide consumers with ex-ante or ex-post service. We investigates the effects of encroachment and services on the optimal strategy for two members in three decision modes: MR mode (ex-ante encroachment), MRM mode (ex-post encroachment and ex-post service), and MRMR mode (ex-post encroachment and ex-ante service). The results show that in the MRM mode, both the wholesale and retail prices may become higher with encroachment. Improving the service efficiency may hurt the retailer, and increasing the operating cost for direct channels harms the retailer, while benefits the manufacturer. In addition, only in the MRM mode, the retailer maybe benefits from encroachment under certain conditions. We further study the equilibrium mode and the result shows as follows. The MR mode, widely adopted by the literature on manufacturer encroachment, always is worst for the manufacturer. Only when both the operating cost for direct channels and the service efficiency are low, the equilibrium decision mode is the MRMR mode, otherwise the MRM mode is the equilibrium decision mode.

Citation: Zonghong Cao, Jie Min. Selection and impact of decision mode of encroachment and retail service in a dual-channel supply chain. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020167
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##### References:
The sequence of the events under three decision modes
The effects of $c$ on the optimal outcomes in the MRM mode
Impacts of $\frac{c}{a}$ and $S$ on the profits in the MRM mode
The effects of $c$ on the optimal outcomes in the MRMR mode
The effects of $S$ on the profits in three decision modes
The impact of $\frac{c}{a}$ and $S$ on the equilibrium decision mode
List of notations
 $a$ The maximum potential market size $c$ The unit operating cost for the direct channel $\lambda$ The efficacy of the service level $\eta$ The coefficient for the service cost $S$ The service efficiency, where $S = \frac{\lambda^2}{\eta}$ $p_r$ The retail price for the retail channel, where $p_n = u_n+w_n$ $p_d$ The retail price for the direct channel $\pi_M$/$\pi_R/\pi_C$ Manufacturer's /Retailer's/Whole chain's profit Decision variables $w$ The wholesale price $Q_d$ The sale quantity for the direct channel $Q_r$ The sale quantity for the retail channel $s$ The service level for the retail channel Superscript $MR/RM/RMR$ The optimal outcome in the MR/MRM/MRMR mode $B$ The optimal outcome in the benchmark without encroachment
 $a$ The maximum potential market size $c$ The unit operating cost for the direct channel $\lambda$ The efficacy of the service level $\eta$ The coefficient for the service cost $S$ The service efficiency, where $S = \frac{\lambda^2}{\eta}$ $p_r$ The retail price for the retail channel, where $p_n = u_n+w_n$ $p_d$ The retail price for the direct channel $\pi_M$/$\pi_R/\pi_C$ Manufacturer's /Retailer's/Whole chain's profit Decision variables $w$ The wholesale price $Q_d$ The sale quantity for the direct channel $Q_r$ The sale quantity for the retail channel $s$ The service level for the retail channel Superscript $MR/RM/RMR$ The optimal outcome in the MR/MRM/MRMR mode $B$ The optimal outcome in the benchmark without encroachment
The outcomes in the benchmark without encroachment
 $w^B$ $Q_r^B$ $s^B$ $p_r^B$ $\pi_M^B$ $\pi_R^B$ $\pi_C^B$ $\frac{a}{2}$ $\frac{a}{4-S}$ $\frac{\lambda a}{2\eta(4-S)}$ $\frac{(6-S)a}{2(4-S)}$ $\frac{a^2}{2(4-S)}$ $\frac{a^2}{4(4-S)}$ $\frac{3a^2}{4(4-S)}$
 $w^B$ $Q_r^B$ $s^B$ $p_r^B$ $\pi_M^B$ $\pi_R^B$ $\pi_C^B$ $\frac{a}{2}$ $\frac{a}{4-S}$ $\frac{\lambda a}{2\eta(4-S)}$ $\frac{(6-S)a}{2(4-S)}$ $\frac{a^2}{2(4-S)}$ $\frac{a^2}{4(4-S)}$ $\frac{3a^2}{4(4-S)}$
The optimal outcomes in the MR mode
 $w^{MR}$ $Q_r^{MR}$ $s^{MR}$ $p_r^{MR}$ $\pi_M^{MR}$ $\pi_R^{MR}$ $c<\frac{(2-S)a}{4-S}$ $\frac{a}{2}$ $\frac{c}{2-S}$ $\frac{\lambda c}{2\eta(2-S)}$ $\frac{a}{2}+\frac{c}{2-S}$ $\frac{(a-c)^2}{4}+\frac{c^2}{4-2S}$ $\frac{(4-S)c^2}{4(2-S)^2}$ $c\geq\frac{(2-S)a}{4-S}$ $\frac{a}{2}$ $\frac{a}{4-S}$ $\frac{\lambda a}{2\eta(4-S)}$ $\frac{(6-S)a}{2(4-S)}$ $\frac{a^2}{2(4-S)}$ $\frac{a^2}{4(4-S)}$
 $w^{MR}$ $Q_r^{MR}$ $s^{MR}$ $p_r^{MR}$ $\pi_M^{MR}$ $\pi_R^{MR}$ $c<\frac{(2-S)a}{4-S}$ $\frac{a}{2}$ $\frac{c}{2-S}$ $\frac{\lambda c}{2\eta(2-S)}$ $\frac{a}{2}+\frac{c}{2-S}$ $\frac{(a-c)^2}{4}+\frac{c^2}{4-2S}$ $\frac{(4-S)c^2}{4(2-S)^2}$ $c\geq\frac{(2-S)a}{4-S}$ $\frac{a}{2}$ $\frac{a}{4-S}$ $\frac{\lambda a}{2\eta(4-S)}$ $\frac{(6-S)a}{2(4-S)}$ $\frac{a^2}{2(4-S)}$ $\frac{a^2}{4(4-S)}$
The optimal outcomes in the MRM mode
 $c$ $[0, \frac{(3-2S)a}{5-2S})$ $[\frac{(3-2S)a}{5-2S}, \frac{(3-S+\sqrt{4-S})a}{4-S+\sqrt{4-S}})$ $[\frac{(3-S+\sqrt{4-S})a}{4-S+\sqrt{4-S}}, a)$ $w^{RM}$ $\frac{a}{2}-\frac{c}{2(3-2S)}$ $\frac{(3-S)c-(1-S)a}{2}$ $\frac{a}{2}$ $Q_r^{RM}$ $\frac{2c}{3-2S}$ $a-c$ $\frac{a}{4-S}$ $s^{RM}$ $\frac{\lambda c}{\eta(3-2S)}$ $\frac{\lambda (a-c)}{2\eta}$ $\frac{\lambda a}{2\eta(4-S)}$ $Q_d^{RM}$ $\frac{a}{2}-\frac{(5-2S)c}{2(3-2S)}$ $0$ $0$ $p_r^{RM}$ $\frac{a}{2}+\frac{c}{2(3-2S)}$ $\frac{(2-S)c+Sa}{2}$ $\frac{(6-S)a}{2(4-S)}$ $p_d^{RM}$ $\frac{a}{2}+\frac{(1-2S)c}{2(3-2S)}$ $-$ $-$ $\pi_M^{RM}$ $\frac{(a-c)^2}{4}+\frac{c^2}{3-2S}$ $\frac{(a-c)[(3-S)c-(1-S)a]}{2}$ $\frac{a^2}{2(4-S)}$ $\pi_R^{RM}$ $\frac{(2-S)c^2}{(3-2S)^2}$ $\frac{(2-S)(a-c)^2}{4}$ $\frac{a^2}{4(4-S)}$
 $c$ $[0, \frac{(3-2S)a}{5-2S})$ $[\frac{(3-2S)a}{5-2S}, \frac{(3-S+\sqrt{4-S})a}{4-S+\sqrt{4-S}})$ $[\frac{(3-S+\sqrt{4-S})a}{4-S+\sqrt{4-S}}, a)$ $w^{RM}$ $\frac{a}{2}-\frac{c}{2(3-2S)}$ $\frac{(3-S)c-(1-S)a}{2}$ $\frac{a}{2}$ $Q_r^{RM}$ $\frac{2c}{3-2S}$ $a-c$ $\frac{a}{4-S}$ $s^{RM}$ $\frac{\lambda c}{\eta(3-2S)}$ $\frac{\lambda (a-c)}{2\eta}$ $\frac{\lambda a}{2\eta(4-S)}$ $Q_d^{RM}$ $\frac{a}{2}-\frac{(5-2S)c}{2(3-2S)}$ $0$ $0$ $p_r^{RM}$ $\frac{a}{2}+\frac{c}{2(3-2S)}$ $\frac{(2-S)c+Sa}{2}$ $\frac{(6-S)a}{2(4-S)}$ $p_d^{RM}$ $\frac{a}{2}+\frac{(1-2S)c}{2(3-2S)}$ $-$ $-$ $\pi_M^{RM}$ $\frac{(a-c)^2}{4}+\frac{c^2}{3-2S}$ $\frac{(a-c)[(3-S)c-(1-S)a]}{2}$ $\frac{a^2}{2(4-S)}$ $\pi_R^{RM}$ $\frac{(2-S)c^2}{(3-2S)^2}$ $\frac{(2-S)(a-c)^2}{4}$ $\frac{a^2}{4(4-S)}$
The optimal outcomes in the MRMR mode
 $c$ $[0, \frac{(128-96S+9S^2)a}{256-144S+9S^2})$ $[\frac{(128-96S+9S^2)a}{256-144S+9S^2}, a)$ $w^{RMR}$ $\frac{a}{2}-\frac{9S^2c}{2(128-96S+9S^2)}$ $\frac{a}{2}$ $s^{RMR}$ $\frac{3\lambda (16-3S)c}{\eta(128-96S+9S^2)}$ $\frac{\lambda a}{2\eta(4-S)}$ $Q_d^{RMR}$ $\frac{(128-96S+9S^2)a-(256-144S+9S^2)c}{2(128-96S+9S^2)}$ $0$ $Q_r^{RMR}$ $\frac{4(16-3S)c}{128-96S+9S^2}$ $\frac{a}{4-S}$ $p_r^{RMR}$ $\frac{a}{2}+\frac{(128-24S-9S^2)c}{2(128-96S+9S^2)}$ $\frac{(6-S)a}{2(4-S)}$ $p_d^{RMR}$ $\frac{a}{2}+\frac{(128-120S+9S^2)c}{2(128-96S+9S^2)}$ $-$ $\pi_M^{RMR}$ $\frac{(128-96S+9S^2)(a-c)^2+128c^2 }{4(128-96S+9S^2)}$ $\frac{a^2}{2(4-S)}$ $\pi_R^{RMR}$ $\frac{(16-9S)(16-3S)^2c^2}{(128-96S+9S^2)^2}$ $\frac{a^2}{4(4-S)}$
 $c$ $[0, \frac{(128-96S+9S^2)a}{256-144S+9S^2})$ $[\frac{(128-96S+9S^2)a}{256-144S+9S^2}, a)$ $w^{RMR}$ $\frac{a}{2}-\frac{9S^2c}{2(128-96S+9S^2)}$ $\frac{a}{2}$ $s^{RMR}$ $\frac{3\lambda (16-3S)c}{\eta(128-96S+9S^2)}$ $\frac{\lambda a}{2\eta(4-S)}$ $Q_d^{RMR}$ $\frac{(128-96S+9S^2)a-(256-144S+9S^2)c}{2(128-96S+9S^2)}$ $0$ $Q_r^{RMR}$ $\frac{4(16-3S)c}{128-96S+9S^2}$ $\frac{a}{4-S}$ $p_r^{RMR}$ $\frac{a}{2}+\frac{(128-24S-9S^2)c}{2(128-96S+9S^2)}$ $\frac{(6-S)a}{2(4-S)}$ $p_d^{RMR}$ $\frac{a}{2}+\frac{(128-120S+9S^2)c}{2(128-96S+9S^2)}$ $-$ $\pi_M^{RMR}$ $\frac{(128-96S+9S^2)(a-c)^2+128c^2 }{4(128-96S+9S^2)}$ $\frac{a^2}{2(4-S)}$ $\pi_R^{RMR}$ $\frac{(16-9S)(16-3S)^2c^2}{(128-96S+9S^2)^2}$ $\frac{a^2}{4(4-S)}$
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