# American Institute of Mathematical Sciences

September  2020, 16(5): 2159-2173. doi: 10.3934/jimo.2019048

## Strategic inventory under suppliers competition

 School of Management and Economics, University of Electronic Science and Technology of China, Chengdu, 611731, China

* Corresponding author: Xingzheng Ai

Received  March 2018 Revised  November 2018 Published  May 2019

This paper investigates the impact of competition and the strategic inventories on the performance of a supply chain comprising two competing suppliers and one retailer. Existing literature has shown that the retailer's optimal strategy in equilibrium is to carry inventories, and the suppliers are unable to prevent this. In contrast, our results show that the suppliers will prevent the retailer from carrying strategic inventories when the degree of competition between suppliers is high, and the retailer's carrying strategic inventory is not necessary to force suppliers to lower the future wholesale price. We also find the substitutable relationship between the effect of strategic inventories and the effect of competition. When the degree of competition increases, the suppliers are worse off but the retailer and the total supply chain are both better off when carrying strategic inventories. The retailer could introduce profit sharing contracts so as to encourage suppliers to support strategic inventories which enhance the entire performance of the supply chain.

Citation: Ganfu Wang, Xingzheng Ai, Chen Zheng, Li Zhong. Strategic inventory under suppliers competition. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2159-2173. doi: 10.3934/jimo.2019048
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##### References:
The relationship between $r$ and $h_{0}$ when $a = 10$
The relationship between $r$ and $I^{i}$ when $a = 10, h = 0.5$
the relationship $h$ and $r$ with $a = 1$
Improvement performance
the relationship between $\beta$ and $r$ with $a = 10, h = 0.2$
The Equilibrium Outcome
 The Dynamic Contract The Commitment Contract $\left( {w_{1}^{\ast } , w_{2}^{\ast } } \right)$ $\left( {\begin{array}{l} \dfrac{2a(3-r)^{2}(3-2r-r^{2})-h(12-24r+9r^{2}+2r^{3}-r^{4})}{102-81r+9r^{2}+8r^{3}-2r^{4}}, \\ \dfrac{(2-r)(2a(3-r)(1-r)(3+r)+h(30-r(9+(3-r)r)))}{102-81r+9r^{2}+8r^{3}-2r^{4}}\end{array}} \right)$ $\left( {\dfrac{a(1-r)}{2-r}, \dfrac{a(1-r)}{2-r}} \right)$ $\left( {p_{1}^{\ast } , p_{2}^{\ast } } \right)$ $\left( {\begin{array}{l} \dfrac{a(156-153r+21r^{2}+16r^{3}-4r^{4})-h(2-r)(6-9r+r^{3})}{2(102-81r+9r^{2}+8r^{3}-2r^{4})}, \\ \dfrac{a(138-r(135-r(23+2r(7-2r)))) +h(2-r)(30-r(9+(3-r)r))}{2(102-81r+9r^{2}+8r^{3}-2r^{4})} \end{array}} \right)$ $\left( {\dfrac{a}{2(2-r)}, \dfrac{a}{2(2-r)}} \right)$ $I^{{\rm \ast }}$ $\dfrac{a(30-39r+8r^{2}+r^{3})-h(2-r)^{2}(30-9r-3r^{2}+r^{3})}{2(1-r^{2})(102-81r+9r^{2}+8r^{3}-2r^{4})}$ 0 $\left( {Q_{1}^{\ast } , Q_{2}^{\ast } } \right)$ $\left( {\begin{array}{l} \dfrac{2a(1-r)(39-r(9+2r))+h(2-r)(r(33+6r-4r^{2})-54)}{2(1-r^{2})(102-81r+9r^{2}+8r^{3}-2r^{4})}, \\\dfrac{(2-r)(2a(3-r)(1-r)(3+r)+h(30-r(9+(3-r)r)))}{2(1-r^{2})(102-81r+9r^{2}+8r^{3}-2r^{4})} \end{array}} \right)$ $\left( {\begin{array}{l} \dfrac{a}{2(2-r)(1+r)}, \\ \dfrac{a}{2(2-r)(1+r)}\end{array}} \right)$ $\pi_{m}^{\ast }$ $\dfrac{h^{2}(2-r)^{2}D-4ahE+4a^{2}K}{2(1-r^{2})(102-81r+9r^{2}+8r^{3}-2r^{4})^{2}}$ $\dfrac{a^{2}(1-r)}{(r+1)(2-r)^{2}}$ $\pi_{b}^{\ast }$ $\dfrac{h^{2}(2-r)^{2}A-2ahB+a^{2}C}{2(1-r^{2})(102-81r+9r^{2}+8r^{3}-2r^{4})^{2}}$ $\dfrac{a^{2}(1-r)}{(r+1)(2-r)^{2}}$
 The Dynamic Contract The Commitment Contract $\left( {w_{1}^{\ast } , w_{2}^{\ast } } \right)$ $\left( {\begin{array}{l} \dfrac{2a(3-r)^{2}(3-2r-r^{2})-h(12-24r+9r^{2}+2r^{3}-r^{4})}{102-81r+9r^{2}+8r^{3}-2r^{4}}, \\ \dfrac{(2-r)(2a(3-r)(1-r)(3+r)+h(30-r(9+(3-r)r)))}{102-81r+9r^{2}+8r^{3}-2r^{4}}\end{array}} \right)$ $\left( {\dfrac{a(1-r)}{2-r}, \dfrac{a(1-r)}{2-r}} \right)$ $\left( {p_{1}^{\ast } , p_{2}^{\ast } } \right)$ $\left( {\begin{array}{l} \dfrac{a(156-153r+21r^{2}+16r^{3}-4r^{4})-h(2-r)(6-9r+r^{3})}{2(102-81r+9r^{2}+8r^{3}-2r^{4})}, \\ \dfrac{a(138-r(135-r(23+2r(7-2r)))) +h(2-r)(30-r(9+(3-r)r))}{2(102-81r+9r^{2}+8r^{3}-2r^{4})} \end{array}} \right)$ $\left( {\dfrac{a}{2(2-r)}, \dfrac{a}{2(2-r)}} \right)$ $I^{{\rm \ast }}$ $\dfrac{a(30-39r+8r^{2}+r^{3})-h(2-r)^{2}(30-9r-3r^{2}+r^{3})}{2(1-r^{2})(102-81r+9r^{2}+8r^{3}-2r^{4})}$ 0 $\left( {Q_{1}^{\ast } , Q_{2}^{\ast } } \right)$ $\left( {\begin{array}{l} \dfrac{2a(1-r)(39-r(9+2r))+h(2-r)(r(33+6r-4r^{2})-54)}{2(1-r^{2})(102-81r+9r^{2}+8r^{3}-2r^{4})}, \\\dfrac{(2-r)(2a(3-r)(1-r)(3+r)+h(30-r(9+(3-r)r)))}{2(1-r^{2})(102-81r+9r^{2}+8r^{3}-2r^{4})} \end{array}} \right)$ $\left( {\begin{array}{l} \dfrac{a}{2(2-r)(1+r)}, \\ \dfrac{a}{2(2-r)(1+r)}\end{array}} \right)$ $\pi_{m}^{\ast }$ $\dfrac{h^{2}(2-r)^{2}D-4ahE+4a^{2}K}{2(1-r^{2})(102-81r+9r^{2}+8r^{3}-2r^{4})^{2}}$ $\dfrac{a^{2}(1-r)}{(r+1)(2-r)^{2}}$ $\pi_{b}^{\ast }$ $\dfrac{h^{2}(2-r)^{2}A-2ahB+a^{2}C}{2(1-r^{2})(102-81r+9r^{2}+8r^{3}-2r^{4})^{2}}$ $\dfrac{a^{2}(1-r)}{(r+1)(2-r)^{2}}$
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