Building fresh product supply chain cooperation in a typical wholesale market

Abstract

In China, the fresh product supply chain is usually connected by a wholesale market. This feature, as well as the product’s perishability, has a great influence on the chain. The market price is exogenous and stochastic because it is actually determined by the uncontrollable supply and demand in the market. The wholesaler reacts to the stochastic price, and based on his experience, he may refuse to sell and may retain fresh products with relatively long lifetime when the price is low. When purchasing at the market, the retailer is incapable of distinguishing the quality change during the delay caused by the wholesaler. However, the difference in quality becomes obvious with faster deterioration in the retail store, and the retailer’s sales are influenced accordingly. The ’market’ works well in pricing the products and matching supply and demand, but it cannot prevent the wholesaler delaying the sales due to the retailer’s incapability. We explain this situation as a result of quality information asymmetry and fix it by coordinating the supply chain; we expect a cooperation mechanism in which the wholesaler cannot improve his profit by delaying the sales even when he has access to the market. The wholesaler’s behavior under market price uncertainty is simplified and described. With the wholesaler’s behavior taken into account, the average quality distribution in the market is approximately estimated. With the approximated description, we construct the model and design the mechanism to coordinate the wholesaler and retailer. A numerical experiment based on an actual market situation is developed and analyzed; the reason why most existing contracts fail is discussed, and managerial suggestions are provided for the practitioners.

Appendices

Appendix A

Lemma

Equations about $P_0$:

$$\begin{aligned} P_0=& \sum _{x=0}^{\overline{t}-n_{\text{s}}}\frac{Pr\{w<g(\theta (0))\}+Pr\{w<g(\theta (1))\}+\cdots +Pr\{w<g(\theta (x+n_s-1))\}}{x+n_{\text{s}}} \nonumber \\&\times\, f(x;n_{\text{s}},1-P_0) \end{aligned}$$

(13)

and

$$\begin{aligned} f(x;n_{\text{s}},1-P_0)= {\left\{ \begin{array}{ll} \left( {\begin{array}{c}x+n_{\text{s}}-1\\ n_{\text{s}}-1\end{array}}\right) P_0^x (1-P_0)^{n_{\text{s}}}&{}x=0,1, \ldots ,\overline{t}-n_{\text{s}}-1\\ 1-\sum _{y=0}^{\overline{t}-n_{\text{s}}-1}f(y;n_{\text{s}},1-P_0)&{}x=\overline{t}-n_{\text{s}} \end{array}\right. } \end{aligned}$$

have a unique solution.

Proof

Assume $\lambda (p)=\sum _{x=0}^{\overline{t}-n_{\text{s}}} \frac{Pr\{w<g(\theta (0))\}+ \cdots +Pr\{w<g(\theta (x+n_{\text{s}}-1))\}}{x+n_{\text{s}}}$ $x+n_{\text{s}}-1\atopwithdelims ()n_{\text{s}}-1$ $p^x(1-p)^{n_{\text{s}}},$ where $p\in [0,1]$. $\frac{Pr\{w<g(\theta (0))\}+\cdots+Pr\{w<g(\theta (x+n_{\text{s}}-1))\} }{x+n_{\text{s}}}$ is the expectation of the probability that a wholesaler does not sell products to the retailer with a total delay of x days. So $\lambda (p)$ is also a conditional probability and we have $\lambda (p)\in [0,1]$. We also know that $\lambda (p)$ is continuous over $p\in [0,1]$ and derivable on $p\in (0,1)$.

Under this space, with Lagrange’s theorem, $d(\lambda (p_1),\lambda (p_2))=|\lambda (p_1)-\lambda (p_2)|=|\lambda ^{\prime }(\xi )||p_2-p_1|,$ where $|\lambda ^\prime (\xi )|=|\lambda (1)-\lambda (0)|=\frac{Pr\{w<g(\theta (0))\}+ \cdots +Pr\{w<g(\theta (n))\}}{n+1}\in (0,1)$ and $p_1,p_2$ can be any value in the interval [0,1]. Thus, $\lambda$ is a contraction mapping on [0, 1]. According to Banach’s fixed point theorem, $P_0=\lambda (P_0)$ has a unique solution and can be solved iteratively with an arbitrary start point in the interval [0, 1].

Appendix B

Proposition

The optimal supply chain profit in the centralized scenario is higher than the optimal profit in the decentralized virtual supply chain.

Proof

After the other virtual wholesaler is considered, the total profit of the virtual chain is

$$\begin{aligned} \Pi ^{\text{d}}=\, & \Pi _{\text{r}}(q_{\text{r}},p_{\text{r}})+\Pi _{\text{s}}(n_{\text{s}})+\Pi _{{\text{s}}^{\prime }}\nonumber \\=\, & p_r\int _{\underline{w}}^{\overline{w}}\int _{\underline{\epsilon }}^{\overline{\epsilon }}\sum _{\theta \in \Theta _y}P_y(\theta )\min \{A\theta p_{\text{r}}^{-k}\epsilon ,q_{\text{r}}\}f_{\epsilon }(x)f_w(y){\mathrm {d}}x{\mathrm {d}}y-E_w\left[wq_{\text{r}}\right]\nonumber \\ +&\quad \sum _{\theta \in \Theta _{\text{m}}}P_{\text{m}}(\theta )\int _{\underline{w}}^{\overline{w}}\left[q_{\text{r}}(y-c_0)-\frac{K}{n_{\text{s}}}\right]f_w(y){\mathrm {d}}y-\frac{c_{\text{h}}n_{\text{s}}q_{\text{r}}}{2}-HC \nonumber \\=\, & p_{\text{r}}\int _{\underline{w}}^{\overline{w}}\int _{\underline{\epsilon }}^{\overline{\epsilon }}\sum _{\theta \in \Theta _y}P_y(\theta )\min \{A\theta p_{\text{r}}^{-k}\epsilon ,q_{\text{r}}\}f_{\epsilon }(x)f_w(y){\mathrm {d}}x{\mathrm {d}}y-q_{\text{r}}c-\frac{K}{n_{\text{s}}}-\frac{c_{\text{h}}n_{\text{s}}q_{\text{r}}}{2}-HC \end{aligned},$$

(14)

while for the centralized scenario, profit $\Pi _c$ is shown in (9).

For any given $(n_{\text{s}},q_{\text{r}},p_{\text{r}})$,

$$\begin{aligned} \Pi ^{\text{c}}-\Pi ^{\text{d}}=\, & {} p_{\text{r}}\int _{\underline{\epsilon }}^{\overline{\epsilon }}\sum _{\theta \in \Theta _{\text{c}}}P_{\text{c}}(\theta )\min \{A\theta p_{\text{r}}^{-k}\epsilon ,q_{\text{r}}\}f_{\epsilon }(x){\mathrm {d}}x\nonumber \\&-\,p_{\text{r}}\int _{\underline{w}}^{\overline{w}}\int _{\underline{\epsilon }}^{\overline{\epsilon }}\sum _{\theta \in \Theta _y}P_y(\theta )\min \{A\theta p_{\text{r}}^{-k}\epsilon ,q_{\text{r}}\}f_{\epsilon }(x)f_w(y){\mathrm {d}}x{\mathrm {d}}y+HC\nonumber \\\ge & {} p_{\text{r}}\int _{\underline{\epsilon }}^{\overline{\epsilon }}\sum _{\theta \in \Theta _{\text{c}}}P_{\text{c}}(\theta )\min \{A\theta p_{\text{r}}^{-k}\epsilon ,q_{\text{r}}\}f_{\epsilon }(x){\mathrm {d}}x\nonumber \\&-p_{\text{r}}\int _{\underline{\epsilon }}^{\overline{\epsilon }}\sum _{\theta \in \Theta _{{\text{m}}}}P_{{\text{m}}}(\theta )\min \{A\theta p_{\text{r}}^{-k}\epsilon ,q_{\text{r}}\}f_{\epsilon }(x){\mathrm {d}}x+HC \nonumber \\ \ge&HC>0. \end{aligned}$$

(15)

We obtain the first inequality by assuming a best case of the decentralized system in which the market price is fixed to its maximal value $\overline{w}$. Then $P_y(\theta )$ becomes $P_{\overline{w}}(\theta )$ and equals $P_{\text{m}}(\theta )$ according to the definition of $P_y(\theta )$.

Then, we have the second inequality by comparing the two probability mass functions $P_{\text{c}}(\theta )$ and $P_{\text{m}}(\theta )$. The result is obvious because we consider the influence of delay in $P_{\text{m}}(\theta )$; therefore, $\Theta _{\text{c}}\subset \Theta _{\text{m}}$ and the quality distribution in the market is approached from the ideal quality distribution by reducing the probability of high quality and increasing the probability of that extended part in $\Theta _{\text{m}}$. This step shows the difference of quality distributions with/without cooperation.

The third inequality $HC>0$ is from our judgment that the virtual wholesaler’s holding cost should be shared by the decentralized system because transactions with the virtual wholesaler exist.

Then, we have $\Pi ^{\text{c}}-\Pi ^{\text{d}}>0$, that is to say, for the optimal setting $(n_{\text{s}}^{\text{d}},q_{\text{r}}^{\text{d}},p_{\text{r}}^{\text{d}})$ in the decentralized scenario, and we also have the same result $\Pi ^{\text{c}}(n_{\text{s}}^{\text{d}},q_{\text{r}}^{\text{d}},p_{\text{r}}^{\text{d}})>\Pi ^{\text{d}}(n_{\text{s}}^{\text{d}},q_{\text{r}}^{\text{d}},p_{\text{r}}^{\text{d}})=\Pi ^{{\text{d}}*}$.

For the centralized scenario, the optimal solution cannot be worse than any other solutions; therefore, it is better for that optimal setting in the decentralized scenario, $\Pi ^{{\text{c}}*}=\Pi ^{\text{c}}(n_{\text{s}}^{\text{c}},q_{\text{r}}^{\text{c}},p_{\text{r}}^{\text{c}})\ge \Pi ^{\text{c}}(n_{\text{s}}^{\text{d}},q_{\text{r}}^{\text{d}},p_{\text{r}}^{\text{d}})$.

Finally, we can obtain the result $\Pi ^{{\text{c}}*}>\Pi ^{{\text{d}}*}$ with the inequalities above.