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September  2022, 18(5): 3187-3214. doi: 10.3934/jimo.2021108

## Competitive strategies in the presence of consumers' expected service and product returns

 1 Coordinated Innovation Center for Computable Modelling in Management Science, Tianjin University of Finance and Economics, Tianjin 300222, China 2 Yango University, Fujian 350015, China 3 School of Mathematical Sciences, Sunway University, Malaysia

* Corresponding author: Shuhua Chang

Received  February 2021 Revised  March 2021 Published  September 2022 Early access  June 2021

Fund Project: This project was supported in part by the Major Research Plan of the National Natural Science Foundation of China (91430108), the National Natural Science Foundation of China (11771322), the National Social Science Foundation of China (19CGL002), Tianjin Social Science Planning Project (TJGLQN20-002), and the Russian Science Foundation (20-61-46017)

This paper investigates the optimal strategies and profits of dual channel with product returns in the presence of customers' expected service. Customers' expected service is related to advertising effort and price. We build a two-stage decision making process to analyze the impact of expected services of customers. In addition, we analyze the parameter sensitivity and compare the competitive equilibrium strategies. The results show that the manufacturer will give a lower wholesale price to the retailer in some case. Furthermore, the dual-channel product returns will discourage advertising effort and the service level of the retailer, but it will enable the manufacturer to provide a higher service level. Thus, for managers, the survey of the expected service of customers is very important for the optimal strategies making, and it should not always blindly exploit the retailer's profit for the manufacturer. Finally, when the physical store allows unconditional return of goods, the service level of the online channel will be more considerate.

Citation: Ting Zhang, Shuhua Chang, Yan Dong, Jingyi Yue, Kok Lay Teo. Competitive strategies in the presence of consumers' expected service and product returns. Journal of Industrial and Management Optimization, 2022, 18 (5) : 3187-3214. doi: 10.3934/jimo.2021108
##### References:

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##### References:
The impact of the expected service
 $\omega$ $S_d$ $S_r$ $A$ $Q_d$ $Q_r$ $\pi_m$ $\pi_r$ $a=b=0$ 6.0174 2.1878 1.5861 2.8323 3.6562 6.7535 67.2808 10.7280 $a\neq0,b\neq0$ 6.1059 2.1259 1.5153 2.4894 2.7451 5.8292 55.5396 8.8085 change $+$ $-$ $-$ $-$ $-$ $-$ $-$ $-$
 $\omega$ $S_d$ $S_r$ $A$ $Q_d$ $Q_r$ $\pi_m$ $\pi_r$ $a=b=0$ 6.0174 2.1878 1.5861 2.8323 3.6562 6.7535 67.2808 10.7280 $a\neq0,b\neq0$ 6.1059 2.1259 1.5153 2.4894 2.7451 5.8292 55.5396 8.8085 change $+$ $-$ $-$ $-$ $-$ $-$ $-$ $-$
Sensitivity analysis
 $A$ $\omega$ $S_d$ $S_r$ $Q_d$ $Q_r$ $\pi_d$ $\pi_m$ $\pi_r$ $\tau$ $\nearrow$ $\searrow\nearrow$ $\nearrow\searrow$ $\nearrow\searrow$ $\nearrow$ $\nearrow$ $\nearrow$ $\nearrow$ $\nearrow\searrow$ $\delta$ $\nearrow$ $\searrow$ $\nearrow$ $\nearrow$ $\nearrow$ $\searrow$ $\nearrow$ $\nearrow$ $\searrow$ $\mu$ $\nearrow$ $\searrow$ $\nearrow$ $\nearrow$ $\nearrow$ $\nearrow\searrow$ $\nearrow$ $\nearrow$ $\nearrow$ $g$ $\searrow$ $\nearrow$ $\searrow$ $\searrow$ $\searrow$ $\searrow\nearrow$ $\searrow$ $\searrow$ $\searrow$ $\gamma$ $\nearrow$ $\searrow\nearrow$ $\nearrow$ $\nearrow$ $\nearrow$ $\nearrow$ $\nearrow$ $\nearrow$ $\nearrow$ $a$ $\searrow$ $\nearrow$ $\searrow$ $\searrow$ $\searrow$ $\searrow$ $\searrow$ $\searrow$ $\searrow$ $r$ $\searrow$ $\nearrow$ $\searrow$ $\searrow$ $\searrow$ $\searrow$ $\searrow$ $\nearrow\searrow$ $\nearrow$ * Some parameters are analyzed only within ranges.
 $A$ $\omega$ $S_d$ $S_r$ $Q_d$ $Q_r$ $\pi_d$ $\pi_m$ $\pi_r$ $\tau$ $\nearrow$ $\searrow\nearrow$ $\nearrow\searrow$ $\nearrow\searrow$ $\nearrow$ $\nearrow$ $\nearrow$ $\nearrow$ $\nearrow\searrow$ $\delta$ $\nearrow$ $\searrow$ $\nearrow$ $\nearrow$ $\nearrow$ $\searrow$ $\nearrow$ $\nearrow$ $\searrow$ $\mu$ $\nearrow$ $\searrow$ $\nearrow$ $\nearrow$ $\nearrow$ $\nearrow\searrow$ $\nearrow$ $\nearrow$ $\nearrow$ $g$ $\searrow$ $\nearrow$ $\searrow$ $\searrow$ $\searrow$ $\searrow\nearrow$ $\searrow$ $\searrow$ $\searrow$ $\gamma$ $\nearrow$ $\searrow\nearrow$ $\nearrow$ $\nearrow$ $\nearrow$ $\nearrow$ $\nearrow$ $\nearrow$ $\nearrow$ $a$ $\searrow$ $\nearrow$ $\searrow$ $\searrow$ $\searrow$ $\searrow$ $\searrow$ $\searrow$ $\searrow$ $r$ $\searrow$ $\nearrow$ $\searrow$ $\searrow$ $\searrow$ $\searrow$ $\searrow$ $\nearrow\searrow$ $\nearrow$ * Some parameters are analyzed only within ranges.
Decision changes influenced by parameters
 $g$ $\gamma$ $\phi$ $\delta$ $\tau$ $\mu$ $a$ $B$ $+\rightarrow -$ $-\rightarrow +$ $-\rightarrow +$ $+\rightarrow -$ $-\rightarrow +$ $+\rightarrow -$ $+\rightarrow -$ $E$ $+\rightarrow -$ $-\rightarrow +$ $-\rightarrow +$ $+\rightarrow -$ $-\rightarrow +$ $-$ $-$ $F$ $-$ $-$ $-\rightarrow +$ $+\rightarrow -$ $-$ $-$ $-$ * See Appendix Table 1 for details.
 $g$ $\gamma$ $\phi$ $\delta$ $\tau$ $\mu$ $a$ $B$ $+\rightarrow -$ $-\rightarrow +$ $-\rightarrow +$ $+\rightarrow -$ $-\rightarrow +$ $+\rightarrow -$ $+\rightarrow -$ $E$ $+\rightarrow -$ $-\rightarrow +$ $-\rightarrow +$ $+\rightarrow -$ $-\rightarrow +$ $-$ $-$ $F$ $-$ $-$ $-\rightarrow +$ $+\rightarrow -$ $-$ $-$ $-$ * See Appendix Table 1 for details.
The impact of the expected service
 $\omega$ $S_d$ $S_r$ $A$ $P$ $\pi_m$ $\pi_r$ $a=b=0$ 7.2135 7.5698 3.0290 5.4090 10.9997 47.9312 1.7613 $a\neq0,b\neq0$ 6.4174 6.7444 2.5324 4.1064 9.5829 38.4156 2.5834 change $-$ $-$ $-$ $-$ $-$ $-$ $+$
 $\omega$ $S_d$ $S_r$ $A$ $P$ $\pi_m$ $\pi_r$ $a=b=0$ 7.2135 7.5698 3.0290 5.4090 10.9997 47.9312 1.7613 $a\neq0,b\neq0$ 6.4174 6.7444 2.5324 4.1064 9.5829 38.4156 2.5834 change $-$ $-$ $-$ $-$ $-$ $-$ $+$
The changes of decision due to the influence of the parameters
 $g$ $\gamma$ $\phi$ $\delta$ $\tau$ $\mu$ $a$ B $<0.69$ $>0.69$ $<0.53$ $>0.53$ $<0.2$ $>0.2$ $<0.2$ $>0.2$ $<0.66$ $>0.66$ $<0.6$ $>0.6$ $<0.018$ $>0.018$ $>0$ $<0$ $<0$ $>0$ $<0$ $>0$ $>0$ $<0$ $<0$ $>0$ $>0$ $<0$ $>0$ $<0$ E $<0.3$ $>0.3$ $<0.65$ $>0.65$ $<0.49$ $>0.49$ $<0.09$ $>0.09$ $<0.85$ $>0.85$ $<0$ $<0$ $>0$ $<0$ $<0$ $>0$ $<0$ $>0$ $>0$ $<0$ $<0$ $>0$ F $<0$ $<0$ $<0.25$ $>0.25$ $<0.17$ $>0.17$ $<0$ $<0$ $<0$ $<0$ $>0$ $>0$ $<0$
 $g$ $\gamma$ $\phi$ $\delta$ $\tau$ $\mu$ $a$ B $<0.69$ $>0.69$ $<0.53$ $>0.53$ $<0.2$ $>0.2$ $<0.2$ $>0.2$ $<0.66$ $>0.66$ $<0.6$ $>0.6$ $<0.018$ $>0.018$ $>0$ $<0$ $<0$ $>0$ $<0$ $>0$ $>0$ $<0$ $<0$ $>0$ $>0$ $<0$ $>0$ $<0$ E $<0.3$ $>0.3$ $<0.65$ $>0.65$ $<0.49$ $>0.49$ $<0.09$ $>0.09$ $<0.85$ $>0.85$ $<0$ $<0$ $>0$ $<0$ $<0$ $>0$ $<0$ $>0$ $>0$ $<0$ $<0$ $>0$ F $<0$ $<0$ $<0.25$ $>0.25$ $<0.17$ $>0.17$ $<0$ $<0$ $<0$ $<0$ $>0$ $>0$ $<0$
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