# American Institute of Mathematical Sciences

March  2021, 17(2): 601-631. doi: 10.3934/jimo.2019125

## Competition in a dual-channel supply chain considering duopolistic retailers with different behaviours

 1 Business School, Beijing Technology and Business University, Beijing 100048, China 2 State Grid Beijing Logistic Supply Company, State Grid Beijing Electric Power Company, Beijing 100054, China 3 School of Management, Capital Normal University, Beijing 100048, China

* Corresponding author: Hongxia Sun

Received  October 2018 Revised  May 2019 Published  March 2021 Early access  October 2019

We study competition in a dual-channel supply chain in which a single supplier sells a single product through its own direct channel and through two different duopolistic retailers. The two retailers have three competitive behaviour patterns: Cournot, Collusion and Stackelberg. Three models are respectively constructed for these patterns, and the optimal decisions for the three patterns are obtained. These optimal solutions are compared, and the effects of certain parameters on the optimal solutions are examined for the three patterns by considering two scenarios: a special case and a general case. In the special case, the equilibrium supply chain structures are analysed, and the optimal quantity and profit are compared for the three different competitive behaviours. Furthermore, both parametric and numerical analyses are presented, and some managerial insights are obtained. We find that in the special case, the Stackelberg game allows the supplier to earn the highest profit, the retailer playing the Collusion game makes the supplier earn the lowest profit, and the Stackelberg leader can gain a first-mover advantage as to the follower. In the general case, the supplier can achieve a higher profit by raising the maximum retail price or holding down the self-price sensitivity factor.

Citation: Hongxia Sun, Yao Wan, Yu Li, Linlin Zhang, Zhen Zhou. Competition in a dual-channel supply chain considering duopolistic retailers with different behaviours. Journal of Industrial & Management Optimization, 2021, 17 (2) : 601-631. doi: 10.3934/jimo.2019125
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##### References:
Supply chain structure
Effect of $a_k$ on $\Pi_0$ and ($\Pi_1$-$\Pi_2$) in three patterns
Effect of $\theta_k$ on $q_0$ and $\Pi_0$ in three patterns
Effect of $\theta_k$ on $q_1$ and $\Pi_1$ in three patterns
Effect of $\theta_k$ on $q_2$ and $\Pi_2$ in three patterns
Effect of $\theta_k$ or $\beta$ on ($\Pi_1$-$\Pi_2$) in three patterns
Effect of $\beta$ on $q_k$ or $\Pi_k$ in three patterns
The equilibrium supply chain structure about the three different competitive behaviors
 Structure Wholesale supplier Dual-channel Monopoly retailer Cournot $\delta\leq\frac{2\beta}{2\theta+\beta}$ $\frac{2\beta}{2\theta+\beta}<\delta<\frac{\theta_0}{\beta}$ $\delta\geq\frac{\theta_0}{\beta}$ $q^{co}_0$ - $\frac{2\beta(a-c)-a_0(2\theta+\beta)}{2[2\beta^2-\theta_0(2\theta+\beta)]}$ $\frac{a_0}{2\theta_0}$ $q^{co}_1$ $\frac{a-c}{2(2\theta+\beta)}$ $\frac{V}{2[2\beta^2-\theta_0(2\theta+\beta)]}$ - $q^{co}_2$ $\frac{a-c}{2(2\theta+\beta)}$ $\frac{V}{2[2\beta^2-\theta_0(2\theta+\beta)]}$ - $w^{co}$ $\frac{a+c}{2}$ $\frac{a+c}{2}$ - $p_0^{co}$ - $\frac{a_0}{2}$ $\frac{a_0}{2}$ $p_1^{co}$ $\frac{(a+c)(\theta+\beta)+2a\theta}{2(2\theta+\beta)}$ $\frac{a+c}{2}+\frac{V\theta}{2[2\beta^2-\theta_0(2\theta+\beta)]}$ - $p_2^{co}$ $\frac{(a+c)(\theta+\beta)+2a\theta}{2(2\theta+\beta)}$ $\frac{a+c}{2}+\frac{V\theta}{2[2\beta^2-\theta_0(2\theta+\beta)]}$ - $\Pi_0^{co}$ $\frac{(a-c)^2}{2(2\theta+\beta)}$ $\frac{2(a-c)(a_0\beta+V)-a_0^2(2\theta+\beta)}{4[2\beta^2-\theta_0(2\theta+\beta)]}$ $\frac{a_0^2}{4\theta_0}$ $\Pi_{1}^{co}$ $\frac{\theta(a-c)^2}{4(2\theta+\beta)^2}$ $\frac{\theta V^2}{4[2\beta^2-\theta_0(2\theta+\beta)]^2}$ - $\Pi_{2}^{co}$ $\frac{\theta(a-c)^2}{4(2\theta+\beta)^2}$ $\frac{\theta V^2}{4[2\beta^2-\theta_0(2\theta+\beta)]^2}$ - Collusion $\delta\leq\frac{\beta}{\theta+\beta}$ $\frac{\beta}{\theta+\beta}<\delta<\frac{\theta_0}{\beta}$ $\delta\geq\frac{\theta_0}{\beta}$ $q^{cn}_0$ - $\frac{a_0(\theta+\beta)-\beta(a-c)}{2[\theta_0(\theta+\beta)-\beta^2]}$ $\frac{a_0}{2\theta_0}$ $q^{cn}_1$ $\frac{a-c}{4(\theta+\beta)}$ $\frac{V}{4[\beta^2-\theta_0(\theta+\beta)]}$ - $q^{cn}_2$ $\frac{a-c}{4(\theta+\beta)}$ $\frac{V}{4[\beta^2-\theta_0(\theta+\beta)]}$ - $w^{cn}$ $\frac{a+c}{2}$ $\frac{a+c}{2}$ - $p_0^{cn}$ - $\frac{a_0}{2}$ $\frac{a_0}{2}$ $p_1^{cn}$ $\frac{3a+c}{4}$ $\frac{a+c}{2}+\frac{(\theta+\beta)V}{4[\beta^2-\theta_0(\theta+\beta)]}$ - $p_2^{cn}$ $\frac{3a+c}{4}$ $\frac{a+c}{2}+\frac{(\theta+\beta)V}{4[\beta^2-\theta_0(\theta+\beta)]}$ - $\Pi_0^{cn}$ $\frac{(a-c)^2}{4(\theta+\beta)}$ $\frac{(a-c)(V+a_0\beta)-a_0^2(\theta+\beta)}{4[\beta^2-\theta_0(\theta+\beta)]}$ $\frac{a_0^2}{4\theta_0}$ $\Pi_{1}^{cn}$ $\frac{(a-c)^2}{16(\theta+\beta)}$ $\frac{V^2(\theta+\beta)}{16[\beta^2-\theta_0(\theta+\beta)]^2}$ - $\Pi_{2}^{cn}$ $\frac{(a-c)^2}{16(\theta+\beta)}$ $\frac{V^2(\theta+\beta)}{16[\beta^2-\theta_0(\theta+\beta)]^2}$ - Stackelberg $\delta\leq\frac{\beta U}{T}$ $\frac{\beta U}{T}<\delta<\frac{\theta_0}{\beta}$ $\delta\geq\frac{\theta_0}{\beta}$ $q^{st}_0$ - $\frac{\beta U(a-c)-a_0T}{2(\beta^2 U-\theta_0T)}$ $\frac{a_0}{2\theta_0}$ $q^{st}_1$ $\frac{\theta(a-c)(2\theta-\beta)}{T}$ $\frac{\theta V(2\theta-\beta)}{\beta^2 U-\theta_0T}$ - $q^{st}_2$ $\frac{(a-c)(4\theta^2-\beta^2-2\theta\beta)}{2T}$ $\frac{V(4\theta^2-\beta^2-2\theta\beta)}{2(\beta^2 U-\theta_0T)}$ - $w^{st}$ $\frac{a+c}{2}$ $\frac{a+c}{2}$ - $p_0^{st}$ - $\frac{a_0}{2}$ $\frac{a_0}{2}$ $p_1^{st}$ $a-\frac{(a-c)(4\theta^3+2\theta^2\beta-\beta^3-2\theta\beta^2)}{2T}$ $\frac{a+c}{2}+\frac{V(2\theta^2-\beta^2)(2\theta-\beta)}{2(\beta^2 U-\theta_0T)}$ - $p_2^{st}$ $a-\frac{\theta(a-c)(4\theta^2+2\theta\beta-3\beta^2)}{2T}$ $\frac{a+c}{2}+\frac{\theta V(4\theta^2-\beta^2-2\theta\beta)}{2(\beta^2 U-\theta_0T)}$ - $\Pi_0^{st}$ $\frac{U(a-c)^2}{4T}$ $\frac{U(a-c)(a_0\beta+V)-a_0^2T}{4(\beta^2 U-\theta_0T)}$ $\frac{a_0^2}{4\theta_0}$ $\Pi_{1}^{st}$ $\frac{\theta(a-c)^2(2\theta-\beta)(4\theta^3-2\theta^2\beta+\beta^3-2\theta\beta^2)}{2T^2}$ $\frac{T(2\theta-\beta)^2V^2}{8(\beta^2 U-\theta_0T)^2}$ - $\Pi_{2}^{st}$ $\frac{\theta(a-c)^2(4\theta^2-\beta^2-2\theta\beta)^2}{4T^2}$ $\frac{\theta V^2(4\theta^2-2\theta\beta-\beta^2)^2}{4(\beta^2 U-\theta_0T)^2}$ - where $T = 4\theta(2\theta^2-\beta^2)$, $U = 8\theta^2-4\theta\beta-\beta^2$, $V = a_0\beta-\theta_0(a-c)$
 Structure Wholesale supplier Dual-channel Monopoly retailer Cournot $\delta\leq\frac{2\beta}{2\theta+\beta}$ $\frac{2\beta}{2\theta+\beta}<\delta<\frac{\theta_0}{\beta}$ $\delta\geq\frac{\theta_0}{\beta}$ $q^{co}_0$ - $\frac{2\beta(a-c)-a_0(2\theta+\beta)}{2[2\beta^2-\theta_0(2\theta+\beta)]}$ $\frac{a_0}{2\theta_0}$ $q^{co}_1$ $\frac{a-c}{2(2\theta+\beta)}$ $\frac{V}{2[2\beta^2-\theta_0(2\theta+\beta)]}$ - $q^{co}_2$ $\frac{a-c}{2(2\theta+\beta)}$ $\frac{V}{2[2\beta^2-\theta_0(2\theta+\beta)]}$ - $w^{co}$ $\frac{a+c}{2}$ $\frac{a+c}{2}$ - $p_0^{co}$ - $\frac{a_0}{2}$ $\frac{a_0}{2}$ $p_1^{co}$ $\frac{(a+c)(\theta+\beta)+2a\theta}{2(2\theta+\beta)}$ $\frac{a+c}{2}+\frac{V\theta}{2[2\beta^2-\theta_0(2\theta+\beta)]}$ - $p_2^{co}$ $\frac{(a+c)(\theta+\beta)+2a\theta}{2(2\theta+\beta)}$ $\frac{a+c}{2}+\frac{V\theta}{2[2\beta^2-\theta_0(2\theta+\beta)]}$ - $\Pi_0^{co}$ $\frac{(a-c)^2}{2(2\theta+\beta)}$ $\frac{2(a-c)(a_0\beta+V)-a_0^2(2\theta+\beta)}{4[2\beta^2-\theta_0(2\theta+\beta)]}$ $\frac{a_0^2}{4\theta_0}$ $\Pi_{1}^{co}$ $\frac{\theta(a-c)^2}{4(2\theta+\beta)^2}$ $\frac{\theta V^2}{4[2\beta^2-\theta_0(2\theta+\beta)]^2}$ - $\Pi_{2}^{co}$ $\frac{\theta(a-c)^2}{4(2\theta+\beta)^2}$ $\frac{\theta V^2}{4[2\beta^2-\theta_0(2\theta+\beta)]^2}$ - Collusion $\delta\leq\frac{\beta}{\theta+\beta}$ $\frac{\beta}{\theta+\beta}<\delta<\frac{\theta_0}{\beta}$ $\delta\geq\frac{\theta_0}{\beta}$ $q^{cn}_0$ - $\frac{a_0(\theta+\beta)-\beta(a-c)}{2[\theta_0(\theta+\beta)-\beta^2]}$ $\frac{a_0}{2\theta_0}$ $q^{cn}_1$ $\frac{a-c}{4(\theta+\beta)}$ $\frac{V}{4[\beta^2-\theta_0(\theta+\beta)]}$ - $q^{cn}_2$ $\frac{a-c}{4(\theta+\beta)}$ $\frac{V}{4[\beta^2-\theta_0(\theta+\beta)]}$ - $w^{cn}$ $\frac{a+c}{2}$ $\frac{a+c}{2}$ - $p_0^{cn}$ - $\frac{a_0}{2}$ $\frac{a_0}{2}$ $p_1^{cn}$ $\frac{3a+c}{4}$ $\frac{a+c}{2}+\frac{(\theta+\beta)V}{4[\beta^2-\theta_0(\theta+\beta)]}$ - $p_2^{cn}$ $\frac{3a+c}{4}$ $\frac{a+c}{2}+\frac{(\theta+\beta)V}{4[\beta^2-\theta_0(\theta+\beta)]}$ - $\Pi_0^{cn}$ $\frac{(a-c)^2}{4(\theta+\beta)}$ $\frac{(a-c)(V+a_0\beta)-a_0^2(\theta+\beta)}{4[\beta^2-\theta_0(\theta+\beta)]}$ $\frac{a_0^2}{4\theta_0}$ $\Pi_{1}^{cn}$ $\frac{(a-c)^2}{16(\theta+\beta)}$ $\frac{V^2(\theta+\beta)}{16[\beta^2-\theta_0(\theta+\beta)]^2}$ - $\Pi_{2}^{cn}$ $\frac{(a-c)^2}{16(\theta+\beta)}$ $\frac{V^2(\theta+\beta)}{16[\beta^2-\theta_0(\theta+\beta)]^2}$ - Stackelberg $\delta\leq\frac{\beta U}{T}$ $\frac{\beta U}{T}<\delta<\frac{\theta_0}{\beta}$ $\delta\geq\frac{\theta_0}{\beta}$ $q^{st}_0$ - $\frac{\beta U(a-c)-a_0T}{2(\beta^2 U-\theta_0T)}$ $\frac{a_0}{2\theta_0}$ $q^{st}_1$ $\frac{\theta(a-c)(2\theta-\beta)}{T}$ $\frac{\theta V(2\theta-\beta)}{\beta^2 U-\theta_0T}$ - $q^{st}_2$ $\frac{(a-c)(4\theta^2-\beta^2-2\theta\beta)}{2T}$ $\frac{V(4\theta^2-\beta^2-2\theta\beta)}{2(\beta^2 U-\theta_0T)}$ - $w^{st}$ $\frac{a+c}{2}$ $\frac{a+c}{2}$ - $p_0^{st}$ - $\frac{a_0}{2}$ $\frac{a_0}{2}$ $p_1^{st}$ $a-\frac{(a-c)(4\theta^3+2\theta^2\beta-\beta^3-2\theta\beta^2)}{2T}$ $\frac{a+c}{2}+\frac{V(2\theta^2-\beta^2)(2\theta-\beta)}{2(\beta^2 U-\theta_0T)}$ - $p_2^{st}$ $a-\frac{\theta(a-c)(4\theta^2+2\theta\beta-3\beta^2)}{2T}$ $\frac{a+c}{2}+\frac{\theta V(4\theta^2-\beta^2-2\theta\beta)}{2(\beta^2 U-\theta_0T)}$ - $\Pi_0^{st}$ $\frac{U(a-c)^2}{4T}$ $\frac{U(a-c)(a_0\beta+V)-a_0^2T}{4(\beta^2 U-\theta_0T)}$ $\frac{a_0^2}{4\theta_0}$ $\Pi_{1}^{st}$ $\frac{\theta(a-c)^2(2\theta-\beta)(4\theta^3-2\theta^2\beta+\beta^3-2\theta\beta^2)}{2T^2}$ $\frac{T(2\theta-\beta)^2V^2}{8(\beta^2 U-\theta_0T)^2}$ - $\Pi_{2}^{st}$ $\frac{\theta(a-c)^2(4\theta^2-\beta^2-2\theta\beta)^2}{4T^2}$ $\frac{\theta V^2(4\theta^2-2\theta\beta-\beta^2)^2}{4(\beta^2 U-\theta_0T)^2}$ - where $T = 4\theta(2\theta^2-\beta^2)$, $U = 8\theta^2-4\theta\beta-\beta^2$, $V = a_0\beta-\theta_0(a-c)$
Partial derivatives of optimal quantity with respect to $a_k$ in three patterns
 Cournot $a_0$ $a_1$ $a_2$ $q^{co}_0$ $\frac{-E}{2P}$ $\frac{\beta(2\theta_2-\beta)}{2P}$ $\frac{\beta(2\theta_1-\beta)}{2P}$ $q^{co}_1$ $\frac{-\beta(\beta-2\theta_2)}{2P}$ $\frac{P(2\theta_2D+E)-\beta^2(2\theta_2-\beta)^2D}{2PDE}$ $\frac{-P(\beta D+E)-\beta^2(2\theta_1-\beta)(2\theta_2-\beta)D}{2PDE}$ $q^{co}_2$ $\frac{-\beta(\beta-2\theta_1)}{2P}$ $\frac{-P(\beta D+E)-\beta^2(2\theta_1-\beta)(2\theta_2-\beta)D}{2PDE}$ $\frac{P(2\theta_1D+E)-\beta^2(2\theta_1-\beta)^2D}{2PDE}$ Collusion $a_0$ $a_1$ $a_2$ $q^{cn}_0$ $\frac{\theta_1\theta_2-\beta^2}{Q}$ $\frac{-\beta(\theta_2-\beta)}{2Q}$ $\frac{-\beta(\theta_1-\beta)}{2Q}$ $q^{cn}_1$ $\frac{\beta(\beta-\theta_2)}{2Q}$ $\frac{Q(2\theta_2G+H)+2G\beta^2(\theta_2-\beta)^2}{4QGH}$ $\frac{-Q(2\beta G+H)+2G\beta^2(\theta_1-\beta)(\theta_2-\beta)}{4QGH}$ $q^{cn}_2$ $\frac{\beta(\beta-\theta_1)}{2Q}$ $\frac{-Q(2\beta G+H)+2G\beta^2(\theta_1-\beta)(\theta_2-\beta)}{4QGH}$ $\frac{Q(2\theta_1G+H)+2G\beta^2(\theta_2-\beta)^2}{4QGH}$ Stackelberg $a_0$ $a_1$ $a_2$ $q^{st}_0$ $\frac{2M\theta_2}{R}$ $\frac{-\theta_2\beta(2\theta_2-\beta)}{R}$ $\frac{-\beta S}{2R}$ $q^{st}_1$ $\frac{\theta_2\beta(\beta-2\theta_2)}{R}$ $\frac{R(\theta_2 N+2\theta_2M)+\theta_2\beta^2(2\theta_2-\beta)^2N}{2RMN}$ $\frac{-R(4M\theta_2+\beta N)+\beta^2(2\theta_2-\beta)SN}{4RMN}$ $q^{st}_2$ $\frac{\beta(2\theta_2\beta-E)}{2R}$ $\frac{-R(4M\theta_2+\beta N)+\beta^2(2\theta_2-\beta)SN}{4RMN}$ $\frac{RNE+S^2(\beta^2N-2M\theta_0\theta_2)}{4\theta_2RMN}$ where $P=\beta^2D-\theta_0E$, $Q=\theta_0H-\beta^2G$, $R=4M\theta_0\theta_2-\beta^2N$, $S=E-2\theta_2\beta$.
 Cournot $a_0$ $a_1$ $a_2$ $q^{co}_0$ $\frac{-E}{2P}$ $\frac{\beta(2\theta_2-\beta)}{2P}$ $\frac{\beta(2\theta_1-\beta)}{2P}$ $q^{co}_1$ $\frac{-\beta(\beta-2\theta_2)}{2P}$ $\frac{P(2\theta_2D+E)-\beta^2(2\theta_2-\beta)^2D}{2PDE}$ $\frac{-P(\beta D+E)-\beta^2(2\theta_1-\beta)(2\theta_2-\beta)D}{2PDE}$ $q^{co}_2$ $\frac{-\beta(\beta-2\theta_1)}{2P}$ $\frac{-P(\beta D+E)-\beta^2(2\theta_1-\beta)(2\theta_2-\beta)D}{2PDE}$ $\frac{P(2\theta_1D+E)-\beta^2(2\theta_1-\beta)^2D}{2PDE}$ Collusion $a_0$ $a_1$ $a_2$ $q^{cn}_0$ $\frac{\theta_1\theta_2-\beta^2}{Q}$ $\frac{-\beta(\theta_2-\beta)}{2Q}$ $\frac{-\beta(\theta_1-\beta)}{2Q}$ $q^{cn}_1$ $\frac{\beta(\beta-\theta_2)}{2Q}$ $\frac{Q(2\theta_2G+H)+2G\beta^2(\theta_2-\beta)^2}{4QGH}$ $\frac{-Q(2\beta G+H)+2G\beta^2(\theta_1-\beta)(\theta_2-\beta)}{4QGH}$ $q^{cn}_2$ $\frac{\beta(\beta-\theta_1)}{2Q}$ $\frac{-Q(2\beta G+H)+2G\beta^2(\theta_1-\beta)(\theta_2-\beta)}{4QGH}$ $\frac{Q(2\theta_1G+H)+2G\beta^2(\theta_2-\beta)^2}{4QGH}$ Stackelberg $a_0$ $a_1$ $a_2$ $q^{st}_0$ $\frac{2M\theta_2}{R}$ $\frac{-\theta_2\beta(2\theta_2-\beta)}{R}$ $\frac{-\beta S}{2R}$ $q^{st}_1$ $\frac{\theta_2\beta(\beta-2\theta_2)}{R}$ $\frac{R(\theta_2 N+2\theta_2M)+\theta_2\beta^2(2\theta_2-\beta)^2N}{2RMN}$ $\frac{-R(4M\theta_2+\beta N)+\beta^2(2\theta_2-\beta)SN}{4RMN}$ $q^{st}_2$ $\frac{\beta(2\theta_2\beta-E)}{2R}$ $\frac{-R(4M\theta_2+\beta N)+\beta^2(2\theta_2-\beta)SN}{4RMN}$ $\frac{RNE+S^2(\beta^2N-2M\theta_0\theta_2)}{4\theta_2RMN}$ where $P=\beta^2D-\theta_0E$, $Q=\theta_0H-\beta^2G$, $R=4M\theta_0\theta_2-\beta^2N$, $S=E-2\theta_2\beta$.
The optimal solutions for three different competitive behaviors
 Optimal $q_0$ $q_1$ $q_2$ $w$ $p_0$ $p_1$ $p_2$ $\Pi_0$ $\Pi_1$ $\Pi_2$ Cournot $4.83$ $3.44$ $2.90$ $14.25$ $8.00$ $17.69$ $20.06$ $116.36$ $11.85$ $16.86$ Collusion $5.29$ $2.81$ $2.60$ $14.13$ $8.00$ $18.24$ $20.74$ $108.01$ $11.57$ $17.23$ Stackelberg $4.76$ $3.58$ $2.90$ $14.23$ $8.00$ $17.59$ $20.03$ $117.35$ $12.04$ $16.82$
 Optimal $q_0$ $q_1$ $q_2$ $w$ $p_0$ $p_1$ $p_2$ $\Pi_0$ $\Pi_1$ $\Pi_2$ Cournot $4.83$ $3.44$ $2.90$ $14.25$ $8.00$ $17.69$ $20.06$ $116.36$ $11.85$ $16.86$ Collusion $5.29$ $2.81$ $2.60$ $14.13$ $8.00$ $18.24$ $20.74$ $108.01$ $11.57$ $17.23$ Stackelberg $4.76$ $3.58$ $2.90$ $14.23$ $8.00$ $17.59$ $20.03$ $117.35$ $12.04$ $16.82$
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