RETRACTED ARTICLE: Three-tier supply chain on temperature control for fresh agricultural products using new differential game model under two decision-making situations

Abstract

Freshness makes the supply chain of fresh agricultural products a concern of consumers. This paper aims to point out how temperature control can improve the freshness of fresh agricultural products and the profits of each member and the whole supply chain. This paper constructs the freshness dynamic game model based on temperature control input and the demand function of fresh agricultural products through the differential game method. It also discusses the impact of consumer preference and discount rate on the profits of fresh agricultural products operating enterprises under centralized decision-making and decentralized decision-making. The changes in the profits of the fresh agricultural products supply chain under the coordination mechanism of “mutual cost-sharing with fixed compensation contract”. The results show that under decentralized decision-making when the profit of the fresh produce retailer is higher than the profit of the supplier or third-party logistics service provider, the retailer is willing to bear a certain proportion of the cost. Moreover, consumer preference can help the company to promote the three-level supply chain of fresh agricultural products to realize a virtuous cycle of “increasing investment in temperature control-prolonging the preservation period of fresh agricultural products-strengthening consumer preference-increasing market demand-improving the members and overall profits of the supply chain”. in addition, under certain conditions, the coordination mechanism of “mutual cost-sharing contract with fixed subsidies” can optimize the temperature control investment level and consumer preference of retailers, suppliers and 3PL, and improve the overall supply chain.

Appendices

Appendix 1

Proof: The optimal profit value function of the fresh produce supply chain system at time t is: ${J}_{FASC1}^{*}\left({T}_{s1},{T}_{r1},{T}_{o1}\right)={e}^{-\rho t}{V}_{1}(G,E)$. Among them, ${V}_{1}(G,E)$ for any $G>0$, and $E>0$(${V}_{1G},{V}_{1E}$ and the existence of the first derivative) satisfy the HJB equation:

$$\begin{aligned}\rho {V}_{1}&=\underset{{T}_{s1},{T}_{r1},{T}_{o1}}{\mathrm{max}}\{p\left(a-bp\right)\left(\chi E+\theta G\right)-\frac{{\eta }^{1}}{2}{T}_{s1}^{2}-\frac{{\eta }^{2}}{2}{T}_{r1}^{2}-\frac{{\eta }^{2}}{2}{T}_{o1}^{2}+{V}_{1G}^{{\prime}}\\&\left({\lambda }_{s}{T}_{s1}+{\lambda }_{r}{T}_{r1}+{\lambda }_{o}{T}_{o1}-\delta G\right)+{V}_{1E}^{{\prime}}(\alpha G-\beta E)\}\end{aligned}$$

(26)

Invest ${T}_{s1}$ yuan in the temperature control of fresh agricultural products suppliers on the right side of the above formula. Retailer temperature control investment ${T}_{r1}$. The 3PL service provider invested ${T}_{o1}$ in temperature control. Find the first partial derivative and make it equal to zero. Come to:

$$\left\{\begin{array}{c}{T}_{s1}=\frac{{V}_{1G}^{{\prime}}{\lambda }_{s}}{{\eta }^{1}}\\ {T}_{r1}=\frac{{V}_{1G}^{{\prime}}{\lambda }_{r}}{{\eta }^{2}}\\ {T}_{o1}=\frac{{V}_{1G}^{{\prime}}{\lambda }_{o}}{{\eta }^{3}}\end{array}\right.$$

(27)

Substitute Eq. (27) into the HJB equation to sort out, and set ${V}_{1}={d}_{1}G+{d}_{2}E+{d}_{3}$, of which ${d}_{1},{d}_{2},{d}_{3}$ are constants. Then ${V}_{1G}^{{\prime}}={d}_{1},{V}_{1E}^{{\prime}}={d}_{2}$, substituting ${V}_{1G}^{{\prime}}={d}_{1},{V}_{1E}^{{\prime}}={d}_{2}$ into the HJB formula, you can get:

$$\left\{\begin{array}{c}{d}_{1}=\frac{p(a-bp)}{\rho +\delta }\left(\theta +\frac{\chi \alpha }{\rho +\beta }\right),{d}_{2}\frac{p(a-bp)\chi }{\rho +\beta }\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\\ {d}_{3}=\left(\frac{{p}^{2}{(a-bp)}^{2}}{2\rho {(\rho +\delta )}^{2}}\right)\left(\frac{{\lambda }_{s}^{2}}{{\eta }^{1}}+\frac{{\lambda }_{r}^{2}}{{\eta }^{2}}+\frac{{\lambda }_{s}^{2}}{{\eta }^{3}}\right){\left(\theta +\frac{\chi \alpha }{\rho +\beta }\right)}^{2}\end{array}\right.$$

(28)

By substituting ${V}_{1G}^{{\prime}}={d}_{1}=\frac{p(a-bp)}{\rho +\delta }\left(\theta +\frac{\chi \alpha }{\rho +\beta }\right)$ into Eq. (27), the optimal input level of temperature control of fresh agricultural products suppliers, retailers and 3PL service providers can be obtained.

Proposition 2 is obtained.

Appendix 2

Proof: The optimal function of the long-term profits of the suppliers, retailers, and 3PL of the fresh produce supply chain at time t is: ${J}_{FASC2S}^{*}\left({T}_{s2}\right)={e}^{-\rho t}{V}_{2S}(G,E)$, ${J}_{FASC2R}^{*}\left({T}_{r2}\right)={e}^{-\rho t}{V}_{2R}(G,E)$, ${J}_{FASC2O}^{*}\left({T}_{o2}\right)={e}^{-\rho t}{V}_{2O}(G,E)$. Among them, ${V}_{2S}(G,E)$, ${V}_{2R}(G,E)$, ${V}_{2O}(G,E)$ for any $G>0$, and $E>0$ satisfy the HJB equation:

$$\left\{\begin{array}{c}\rho {V}_{2S}\left(G,E\right)=\underset{{T}_{s2}}{\mathrm{max}}\left\{w\left(a-bp\right)\left(\theta G+\chi E\right)-\frac{{\eta }^{1}}{2}{T}_{s2}^{2}+{{V}_{2SG}^{\prime}}\left({\lambda }_{s}{T}_{s2}+{\lambda }_{r}{T}_{r2}+{\lambda }_{o}{T}_{o2}-\delta G\right)+{{V}_{2SE}^{\prime}}(\alpha G-\beta E)\right\}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,\\ \rho {V}_{2R}\left(G,E\right)=\underset{{T}_{r2}}{\mathrm{max}}\left\{(p-w-\xi )\left(a-bp\right)\left(\theta G+\chi E\right)-\frac{{\eta }^{2}}{2}{T}_{r2}^{2}+{{V}_{2RG}^{\prime}}\left({\lambda }_{s}{T}_{s2}+{\lambda }_{r}{T}_{r2}+{\lambda }_{o}{T}_{o2}-\delta G\right)+{{V}_{2RE}^{\prime}}(\alpha G-\beta E)\right\}\\ \rho {V}_{2O}\left(G,E\right)=\underset{{T}_{o2}}{\mathrm{max}}\left\{\xi \left(a-bp\right)\left(\theta G+\chi E\right)-\frac{{\eta }^{2}}{2}{T}_{o2}^{2}+{{V}_{2OG}^{\prime}}\left({\lambda }_{s}{T}_{s2}+{\lambda }_{r}{T}_{r2}+{\lambda }_{o}{T}_{o2}-\delta G\right)+{{V}_{2OE}^{\prime}}(\alpha G-\beta E)\right\}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,\end{array}\right.$$

(29)

Find the first-order partial derivative of ${T}_{s2},{T}_{r2},{T}_{o2}$ of fresh agricultural products suppliers, retailers and 3PL on the right side of Eq. (29), and make it equal to zero, and get:

$$\left\{\begin{array}{c}{T}_{s2}=\frac{{\lambda }_{s}{V}_{2SG}^{\prime}}{{\eta }^{1}}\\ {T}_{r2}=\frac{{\lambda }_{r}{V}_{2RG}^{\prime}}{{\eta }^{2}}\\ {T}_{o2}=\frac{{\lambda }_{o}{V}_{1OG}^{\prime}}{{\eta }^{3}}\end{array}\right.$$

(30)

By substituting ${T}_{s2},{T}_{r2},{T}_{o2}$ into Eq. (31), we can get the following results:

$$\left\{\begin{array}{c}\rho {V}_{2S}=\left(w\left(a-bp\right)\theta -{{V}_{2SG}^{\prime}}\delta +{{V}_{2SE}^{\prime}}\alpha \right)G+\left(w\left(a-bp\right)\chi -{{V}_{2SE}^{\prime}}\beta \right)E+\frac{{V}_{2SG}^{\prime{2}}{\lambda }_{s}^{2}}{2{\eta }^{1}}+\frac{{{V}_{2SG}^{\prime}}{{V}_{2RG}^{\prime}}{\lambda }_{r}^{2}}{{\eta }^{2}}+\frac{{{V}_{2SG}^{\prime}}{{V}_{2OG}^{\prime}}{\lambda }_{o}^{2}}{{\eta }^{3}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\\ \rho {V}_{2R}=\left(\left(p-w-\xi \right)(a-bp)\theta -{{V}_{2RG}^{\prime}}\delta +{{V}_{2RE}^{\prime}}\alpha \right)G+\left((p-w-\xi )\left(a-bp\right)\chi -{{V}_{2RE}^{\prime}}\beta \right)E+\frac{{{V}_{2RG}^{\prime}}{{V}_{2SG}^{\prime}}{\lambda }_{s}^{2}}{{\eta }^{1}}+\frac{{{V}_{2RG}^{\prime{2}}}{\lambda }_{r}^{2}}{2{\eta }^{2}}+\frac{{{V}_{2RG}^{\prime}}{{V}_{2OG}^{\prime}}{\lambda }_{o}^{2}}{{\eta }^{3}}\\ \rho {V}_{2O}=\left(\xi (a-bp)\theta -{{V}_{2OG}^{\prime}}\delta +{{V}_{2OE}^{\prime}}\alpha \right)G+\left(\xi \left(a-bp\right)\chi -{{V}_{2RE}^{\prime}}\beta \right)E+\frac{{{V}_{2OG}^{\prime}}{{V}_{2SG}^{\prime}}{\lambda }_{s}^{2}}{{\eta }^{1}}+\frac{{{V}_{2OG}^{\prime}}{{V}_{2RG}^{\prime}}{\lambda }_{r}^{2}}{{\eta }^{2}}+\frac{{V}_{2OG}^{\prime{2}}{\lambda }_{o}^{2}}{2{\eta }^{3}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,\;\;\;\;\end{array}\right.$$

(31)

Therefore, let ${V}_{2S}\left(G,E\right)={k}_{1}G+{k}_{2}E+{k}_{3}$, ${V}_{2O}\left(G,E\right)={f}_{1}G+{f}_{2}E+{f}_{3}$, ${k}_{1},{k}_{2},{k}_{3},{l}_{1},{l}_{2},{l}_{3},{f}_{1},{f}_{2},{f}_{3}$ of which are all constants, then ${{V}_{2SG}^{\prime}}={k}_{1}$, ${{V}_{2SE}^{\prime}}={k}_{2}$, ${{V}_{2RG}^{\prime}}={l}_{1}$, ${{V}_{2RE}^{\prime}}={l}_{2}$, ${{V}_{2OG}^{\prime}}={f}_{1}$, ${{V}_{2OE}^{\prime}}={f}_{2}$ are substituted into the Eq. (31), which can be obtained by the method of undetermined coefficients:

$$\left\{\begin{array}{c}{k}_{1}=\frac{w\left(a-bp\right)}{\rho +\delta }\left(\theta +\frac{\chi \alpha }{\rho +\beta }\right),{k}_{2}=\frac{w\left(a-bp\right)\chi }{\rho +\beta }\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\\ {k}_{3}=\left(\frac{w{\left(a-bp\right)}^{2}}{\rho {\left(\rho +\delta \right)}^{2}}\right)\left(\frac{w{\lambda }_{s}^{2}}{2{\eta }^{1}}+\frac{\left(p-w-\xi \right){\lambda }_{r}^{2}}{{\eta }^{2}}+\frac{\xi {\lambda }_{o}^{2}}{{\eta }^{3}}\right){\left(\theta +\frac{\chi \alpha }{\rho +\beta }\right)}^{2}\end{array}\right.$$

(32)

$$\left\{\begin{array}{c}{l}_{1}=\frac{\left(p-w-\xi \right)\left(a-bp\right)}{\rho +\delta }\left(\theta +\frac{\chi \alpha }{\rho +\beta }\right),{l}_{2}=\frac{\left(p-w-\xi \right)\left(a-bp\right)\chi }{\rho +\beta }\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\\,\\ {l}_{3}=\left(\frac{\left(p-w-\xi \right){\left(a-bp\right)}^{2}}{\rho {\left(\rho +\delta \right)}^{2}}\right)\left(\frac{w{\lambda }_{s}^{2}}{{\eta }^{1}}+\frac{\left(p-w-\xi \right){\lambda }_{r}^{2}}{2{\eta }^{2}}+\frac{\xi {\lambda }_{o}^{2}}{{\eta }^{3}}\right){\left(\theta +\frac{\chi \alpha }{\rho +\beta }\right)}^{2}\end{array}\right.$$

(33)

$$\left\{\begin{array}{c}{f}_{1}=\frac{\xi \left(a-bp\right)}{\rho +\delta }\left(\theta +\frac{\chi \alpha }{\rho +\beta }\right),{f}_{2}=\frac{\xi \left(a-bp\right)\chi }{\rho +\beta }\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\\ {f}_{3}=\left(\frac{\xi {\left(a-bp\right)}^{2}}{\rho {\left(\rho +\delta \right)}^{2}}\right)\left(\frac{w{\lambda }_{s}^{2}}{{\eta }^{1}}+\frac{\left(p-w-\xi \right){\lambda }_{r}^{2}}{{\eta }^{2}}+\frac{\xi {\lambda }_{o}^{2}}{2{\eta }^{3}}\right){\left(\theta +\frac{\chi \alpha }{\rho +\beta }\right)}^{2}\end{array}\right.$$

(34)

However, the values of ${k}_{1},{l}_{1},{f}_{1}$ are substituted into Eq. (30). The optimal temperature control input levels (${T}_{s2},{T}_{r2},{T}_{o2}$) of retailers, suppliers and 3PL service providers 3PL in the three-level supply chain of fresh agricultural products under decentralized decision-making based on wholesale price contract can be obtained. The certificate is completed.

Proposition 4 can be obtained.