# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2020133

## Incentive contract design for supplier switching with considering learning effect

 1 School of Management, Tianjin University of Technology, Tianjin, 300384, China 2 College of Management and Economics, Tianjin University, Tianjin, 300072, China

* Corresponding author: Xiaojie Sun

Received  April 2019 Revised  July 2020 Early access  August 2020

Fund Project: This work was supported by the National Natural Science Foundation of China (Grant No. 71971152)

For minimizing purchase cost, a buying firm would switch to suppliers with providing more favorable prices. This paper investigates the optimal switching decision of a buyer that may switch to an entrant supplier with production learning ability (which is regarded as a private information) under a principal-agent framework. The results obtained show that the switching cost and the learning effect have significant impacts on the buyer's switching decision. Only when the fixed component of the switching cost is relatively low, the buyer can be better off from a partial switching strategy; otherwise, the buyer should take an all-or-nothing switching strategy or no switching strategy. As the learning ability of the entrant supplier increases, the buyer prefers to make more switching. Finally, a benefit-sharing contract is proposed to evaluate the performance of the principal-agent contract, and we demonstrate that the principal-agent contract almost completely dominates the benefit-sharing contract.

Citation: Qian Wei, Jianxiong Zhang, Xiaojie Sun. Incentive contract design for supplier switching with considering learning effect. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020133
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##### References:
The optimal switching ratio $r^{*}(x)$ and transfer payment $t^*(x)$
The expected total cost reduction $\hat{R}(\phi)$ with respect to the sharing proportion $\phi$
$\rm{S_2}$'s profit and the total cost reduction difference under the two contracts via $x$
Optimal switching strategies with $I\geq m(0)$
 $\Psi\!=c_0\!-\gamma\!+k\!+I\!-p$ $\Psi\geq 0$ $-m(0)-\gamma\leq\Psi <0$ $\Psi <-m(0)-\gamma$ $r^*(x)$ $\begin{array}{ll}r^*(x)=0,\\ x\in [0,\gamma].\end{array}$ $r^*(x)=\left\{ \begin{array}{ll}0, x\in [0,x_1^*), \\1,x\in [x_1^*,\gamma]. \end{array}\right.$ $\begin{array}{ll}r^*(x)=1,\\x\in [0,\gamma].\end{array}$
 $\Psi\!=c_0\!-\gamma\!+k\!+I\!-p$ $\Psi\geq 0$ $-m(0)-\gamma\leq\Psi <0$ $\Psi <-m(0)-\gamma$ $r^*(x)$ $\begin{array}{ll}r^*(x)=0,\\ x\in [0,\gamma].\end{array}$ $r^*(x)=\left\{ \begin{array}{ll}0, x\in [0,x_1^*), \\1,x\in [x_1^*,\gamma]. \end{array}\right.$ $\begin{array}{ll}r^*(x)=1,\\x\in [0,\gamma].\end{array}$
Optimal switching strategies with $I<m(0)$
 $\Psi\!=c_0\!-\gamma\!+k\!+I\!-p$ $\Psi\geq 0$ $-I-\gamma\leq\Psi<0$ $r^*(x)$ $r^*(x)\!=\!\!0,\ x\in [0,\gamma].$ ${r^*}(x) = \left\{ {\begin{array}{*{20}{l}} {0,\;x\; \in \;[0,x_1^*),}\\ {1,\;x\; \in \;[x_1^*,\gamma ].} \end{array}} \right.$ $\!\!I\!\!-\!\!2\sqrt{Im(0)}\!\!-\!\!\gamma <\!\!\Psi\!\! <-\!\!I\!\!-\!\!\gamma$ $\quad \!\!I\!\!-\!\!2m(0)\!\!-\gamma\!\! <\!\!\Psi\!\!\leq I\!\!-\!\!2\sqrt{Im(0)}\!\!-\!\!\gamma$ $\Psi\leq I-2m(0)-\gamma$ $r^*(x)\!\!=\!\!\left\{ \begin{array}{lll}0,\quad x\in [0,x_3^*),\\ \!\!-\frac{c_0-p+k}{2[m(x)-x]},\\ \quad x\in [x_3^*,x_2^*),\\ 1, \quad x\in [x_2^*,\gamma]. \end{array}\right.$ $r^*(x)\!\!\!=\!\!\!\left\{ \begin{array}{ll} \!\!-\frac{c_0-p+k}{2[m(x)-x]}, \\ \quad \quad \quad x\in [0,x_2^*), \\1,\quad \quad x\in [x_2^*,\gamma]. \end{array}\right.$ $\begin{array}{l} r^*(x)=1,\\ x\in [0,\gamma]. \end{array}$
 $\Psi\!=c_0\!-\gamma\!+k\!+I\!-p$ $\Psi\geq 0$ $-I-\gamma\leq\Psi<0$ $r^*(x)$ $r^*(x)\!=\!\!0,\ x\in [0,\gamma].$ ${r^*}(x) = \left\{ {\begin{array}{*{20}{l}} {0,\;x\; \in \;[0,x_1^*),}\\ {1,\;x\; \in \;[x_1^*,\gamma ].} \end{array}} \right.$ $\!\!I\!\!-\!\!2\sqrt{Im(0)}\!\!-\!\!\gamma <\!\!\Psi\!\! <-\!\!I\!\!-\!\!\gamma$ $\quad \!\!I\!\!-\!\!2m(0)\!\!-\gamma\!\! <\!\!\Psi\!\!\leq I\!\!-\!\!2\sqrt{Im(0)}\!\!-\!\!\gamma$ $\Psi\leq I-2m(0)-\gamma$ $r^*(x)\!\!=\!\!\left\{ \begin{array}{lll}0,\quad x\in [0,x_3^*),\\ \!\!-\frac{c_0-p+k}{2[m(x)-x]},\\ \quad x\in [x_3^*,x_2^*),\\ 1, \quad x\in [x_2^*,\gamma]. \end{array}\right.$ $r^*(x)\!\!\!=\!\!\!\left\{ \begin{array}{ll} \!\!-\frac{c_0-p+k}{2[m(x)-x]}, \\ \quad \quad \quad x\in [0,x_2^*), \\1,\quad \quad x\in [x_2^*,\gamma]. \end{array}\right.$ $\begin{array}{l} r^*(x)=1,\\ x\in [0,\gamma]. \end{array}$
Optimal transfer payments with $I\geq m(0)$
 $\Psi=c_0-\gamma+k+I-p$ $\Psi\geq 0$ $-m(0)-\gamma\leq\Psi <0$ $\Psi <-m(0)-\gamma$ $t^*(x)$ $\begin{array}{lll} t^*(x)=0,\\ x\in [0,\gamma].\end{array}$ $t^*(x)=\left\{ \begin{array}{ll}0, \\ \quad x\in [0,x_1^*),\\ c_0-x_1^*,\\ \quad x\in [x_1^*,\gamma]. \end{array}\right.$ $\begin{array}{lll} t^*(x)\!=\!c_0\!-\!x_1^*,\\ x\in [0,\gamma].\end{array}$
 $\Psi=c_0-\gamma+k+I-p$ $\Psi\geq 0$ $-m(0)-\gamma\leq\Psi <0$ $\Psi <-m(0)-\gamma$ $t^*(x)$ $\begin{array}{lll} t^*(x)=0,\\ x\in [0,\gamma].\end{array}$ $t^*(x)=\left\{ \begin{array}{ll}0, \\ \quad x\in [0,x_1^*),\\ c_0-x_1^*,\\ \quad x\in [x_1^*,\gamma]. \end{array}\right.$ $\begin{array}{lll} t^*(x)\!=\!c_0\!-\!x_1^*,\\ x\in [0,\gamma].\end{array}$
Optimal transfer payments with $I<m(0)$
 $\Psi\!=\!c_0\!-\!\gamma\!+\!k\!+\!I\!-\!p$ $\Psi\!\geq\!0$ $t^*(x)$ $t^*(x)=0,\ x\in [0,\gamma].$ $-I\!-\!\gamma\!\leq\!\Psi\! <\!0$ $I\!-\!2\sqrt{Im(0)}\!-\!\gamma\! <\!\Psi\! <\!-I\!-\!\gamma$ $t^*(x)=\left\{ \begin{array}{ll}0, x\in [0,x_1^*), \\c_0-x_1^*, x\in [x_1^*,\gamma]. \end{array}\right.$ $\!\!\!t^*(x)\!\!=\!\!\left\{ \begin{array}{lll}\!\!0, \\ \quad\quad\quad\quad \quad\quad x\in [0,x_3^*),\\ \!\!c_0 r^*(x)-x{r^2}^*(x)+ \int_{x_3^*}^{x}r^2(v){\rm d}v, \\ \quad\quad \quad\quad \quad\quad x\in [x_3^*,x_2^*), \\\!\!c_0-x_2^*+\int_{x_3^*}^{x_2^*}r^2(v){\rm d}v, \\\quad\quad \quad\quad \quad\quad x\in [x_2^*,\gamma]. \end{array}\right.$ $\!\!I\!-\!\!2m(0)\!\!-\!\!\gamma\! <\!\Psi\!\leq \!I\!-\!\!2\sqrt{Im(0)}\!\!-\!\!\gamma$ $\Psi\leq I-2m(0)-\gamma$ $\!\!\!t^*(x)\!\!\!=\!\!\!\left\{ \begin{array}{lll} \!\!c_0r^*(x)\!\!-\!\!x{r^2}^*(x)\!\!+\!\!\int_{x_3^*}^{x}r^2(v){\rm d}v, \\ \quad x\in [0,x_2^*), \\\!\!c_0\!-\!x_2^*\!+\!\int_{x_3^*}^{x_2^*}r^2(v){\rm d}v, \\ \quad x\in [x_2^*,\gamma]. \end{array}\right.$ $t^*(x)=c_0-x_1^*,\ x\in [0,\gamma].$
 $\Psi\!=\!c_0\!-\!\gamma\!+\!k\!+\!I\!-\!p$ $\Psi\!\geq\!0$ $t^*(x)$ $t^*(x)=0,\ x\in [0,\gamma].$ $-I\!-\!\gamma\!\leq\!\Psi\! <\!0$ $I\!-\!2\sqrt{Im(0)}\!-\!\gamma\! <\!\Psi\! <\!-I\!-\!\gamma$ $t^*(x)=\left\{ \begin{array}{ll}0, x\in [0,x_1^*), \\c_0-x_1^*, x\in [x_1^*,\gamma]. \end{array}\right.$ $\!\!\!t^*(x)\!\!=\!\!\left\{ \begin{array}{lll}\!\!0, \\ \quad\quad\quad\quad \quad\quad x\in [0,x_3^*),\\ \!\!c_0 r^*(x)-x{r^2}^*(x)+ \int_{x_3^*}^{x}r^2(v){\rm d}v, \\ \quad\quad \quad\quad \quad\quad x\in [x_3^*,x_2^*), \\\!\!c_0-x_2^*+\int_{x_3^*}^{x_2^*}r^2(v){\rm d}v, \\\quad\quad \quad\quad \quad\quad x\in [x_2^*,\gamma]. \end{array}\right.$ $\!\!I\!-\!\!2m(0)\!\!-\!\!\gamma\! <\!\Psi\!\leq \!I\!-\!\!2\sqrt{Im(0)}\!\!-\!\!\gamma$ $\Psi\leq I-2m(0)-\gamma$ $\!\!\!t^*(x)\!\!\!=\!\!\!\left\{ \begin{array}{lll} \!\!c_0r^*(x)\!\!-\!\!x{r^2}^*(x)\!\!+\!\!\int_{x_3^*}^{x}r^2(v){\rm d}v, \\ \quad x\in [0,x_2^*), \\\!\!c_0\!-\!x_2^*\!+\!\int_{x_3^*}^{x_2^*}r^2(v){\rm d}v, \\ \quad x\in [x_2^*,\gamma]. \end{array}\right.$ $t^*(x)=c_0-x_1^*,\ x\in [0,\gamma].$
Sensitivity analysis with respect to system parameters $c_0, p, k, I.$
 -80% -60% $-40\%$ $-20\%$ $0$ $20\%$ $40\%$ $60\%$ $80\%$ $\rm{ \rm{E}}(C)$ 1.0200 1.3400 1.6982 1.9775 2.2772 2.5239 2.7191 2.8631 2.9559 $c_0 $$\hat{ \rm{E}}(C) 1.3609 1.6088 1.8492 2.0805 2.3026 2.5406 2.7265 2.8656 2.9559 \Delta \rm{E} 0.3409 0.2688 0.1510 0.1030 0.0254 0.0167 0.0074 0.0025 0 \frac{\Delta \rm{E}}{ \rm{E}(C)} 33% 20% 8.9% 5.2% 1.1% 0.7% 0.3% 0.09% 0% \rm{E}(C) 0.6000 1.2000 1.7375 2.0975 2.2772 2.3000 2.3000 2.3000 2.3000 p$$ \hat{ \rm{E}}(C)$ 0.6000 1.2000 1.7375 2.1060 2.3026 2.4786 2.6235 2.7449 2.8488 $\Delta \rm{E}$ 0 0 0 0.0085 0.0254 0.1786 0.3235 0.4449 0.5488 $\frac{\Delta \rm{E}}{ \rm{E}(C)}$ 0% 0% 0% 0.4% 1.1% 7.7% 14% 19% 24% $\rm{E}(C)$ 2.1340 2.1728 2.2114 2.2498 2.2772 2.3261 2.3639 2.3759 2.4071 $k $$\hat{ \rm{E}}(C) 2.1922 2.2196 2.2469 2.2739 2.3026 2.3340 2.3661 2.3974 2.4278 \Delta \rm{E} 0.0582 0.0468 0.0355 0.0241 0.0254 0.0079 0.0022 0.0215 0.0207 \frac{\Delta \rm{E}}{ \rm{E}(C)} 2.7% 2.2% 1.6% 1.1% 1.1% 0.34% 0.1% 0.9% 0.86% \rm{E}(C) 1.8881 1.9881 2.0880 2.1870 2.2772 2.3600 2.4375 2.5100 2.5775 I$$ \hat{ \rm{E}}(C)$ 2.0793 2.1365 2.1925 2.2473 2.3026 2.3789 2.4507 2.5178 2.5810 $\Delta \rm{E}$ 0.1912 0.1484 0.1045 0.0603 0.0254 0.0189 0.0132 0.0078 0.0035 $\frac{\Delta \rm{E}}{ \rm{E}(C)}$ 10.1% 7.5% 5.0% 2.8% 1.1% 0.8% 0.5% 0.3% 0.1%
 -80% -60% $-40\%$ $-20\%$ $0$ $20\%$ $40\%$ $60\%$ $80\%$ $\rm{ \rm{E}}(C)$ 1.0200 1.3400 1.6982 1.9775 2.2772 2.5239 2.7191 2.8631 2.9559 $c_0 $$\hat{ \rm{E}}(C) 1.3609 1.6088 1.8492 2.0805 2.3026 2.5406 2.7265 2.8656 2.9559 \Delta \rm{E} 0.3409 0.2688 0.1510 0.1030 0.0254 0.0167 0.0074 0.0025 0 \frac{\Delta \rm{E}}{ \rm{E}(C)} 33% 20% 8.9% 5.2% 1.1% 0.7% 0.3% 0.09% 0% \rm{E}(C) 0.6000 1.2000 1.7375 2.0975 2.2772 2.3000 2.3000 2.3000 2.3000 p$$ \hat{ \rm{E}}(C)$ 0.6000 1.2000 1.7375 2.1060 2.3026 2.4786 2.6235 2.7449 2.8488 $\Delta \rm{E}$ 0 0 0 0.0085 0.0254 0.1786 0.3235 0.4449 0.5488 $\frac{\Delta \rm{E}}{ \rm{E}(C)}$ 0% 0% 0% 0.4% 1.1% 7.7% 14% 19% 24% $\rm{E}(C)$ 2.1340 2.1728 2.2114 2.2498 2.2772 2.3261 2.3639 2.3759 2.4071 $k $$\hat{ \rm{E}}(C) 2.1922 2.2196 2.2469 2.2739 2.3026 2.3340 2.3661 2.3974 2.4278 \Delta \rm{E} 0.0582 0.0468 0.0355 0.0241 0.0254 0.0079 0.0022 0.0215 0.0207 \frac{\Delta \rm{E}}{ \rm{E}(C)} 2.7% 2.2% 1.6% 1.1% 1.1% 0.34% 0.1% 0.9% 0.86% \rm{E}(C) 1.8881 1.9881 2.0880 2.1870 2.2772 2.3600 2.4375 2.5100 2.5775 I$$ \hat{ \rm{E}}(C)$ 2.0793 2.1365 2.1925 2.2473 2.3026 2.3789 2.4507 2.5178 2.5810 $\Delta \rm{E}$ 0.1912 0.1484 0.1045 0.0603 0.0254 0.0189 0.0132 0.0078 0.0035 $\frac{\Delta \rm{E}}{ \rm{E}(C)}$ 10.1% 7.5% 5.0% 2.8% 1.1% 0.8% 0.5% 0.3% 0.1%
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