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  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
References:

show all references

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 $4.76$ $3.58$ $2.90$ $14.23$ $8.00$ $17.59$ $20.03$ $117.35$ $12.04$ $16.82$
 [1] David Cantala, Juan Sebastián Pereyra. Endogenous budget constraints in the assignment game. Journal of Dynamics & Games, 2015, 2 (3&4) : 207-225. doi: 10.3934/jdg.2015002 [2] Juliang Zhang, Jian Chen. Information sharing in a make-to-stock supply chain. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1169-1189. doi: 10.3934/jimo.2014.10.1169 [3] Min Li, Jiahua Zhang, Yifan Xu, Wei Wang. Effects of disruption risk on a supply chain with a risk-averse retailer. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021024 [4] 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 [5] 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 [6] Caifang Wang, Tie Zhou. The order of convergence for Landweber Scheme with $\alpha,\beta$-rule. Inverse Problems & Imaging, 2012, 6 (1) : 133-146. doi: 10.3934/ipi.2012.6.133 [7] Alexandre B. Simas, Fábio J. Valentim. $W$-Sobolev spaces: Higher order and regularity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 597-607. doi: 10.3934/cpaa.2015.14.597 [8] Qian Liu. The lower bounds on the second-order nonlinearity of three classes of Boolean functions. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020136 [9] Liqin Qian, Xiwang Cao. Character sums over a non-chain ring and their applications. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020134 [10] Wen-Bin Yang, Yan-Ling Li, Jianhua Wu, Hai-Xia Li. Dynamics of a food chain model with ratio-dependent and modified Leslie-Gower functional responses. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2269-2290. doi: 10.3934/dcdsb.2015.20.2269 [11] A. Aghajani, S. F. Mottaghi. Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (3) : 887-898. doi: 10.3934/cpaa.2018044 [12] 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 [13] Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2411-2428. doi: 10.3934/dcdsb.2020184

2019 Impact Factor: 1.366