Coordination of Perishable Product Supply Chains with a Joint Contract under Yield and Demand Uncertainty

Abstract

With the complex and changeable environment, the demand and yield in the perishable products supply chain are usually uncertain. This paper studies a joint contract that combines revenue sharing with quantity discount to coordinate the supply chain under demand and yield uncertainty, which consists of one manufacturer and one retailer. The retailer pays the manufacturer a down payment at the beginning, and the manufacturer gives the retailer a quantity discount and shares a proportion of profit from the retailer at last. To make sure that both members in the supply chain want to adopt this contract, we prove the feasibility of the joint contract achieving a win–win situation. In addition, we investigate how the price in the secondary market influence the contract, and the conclusion further proves that supply chain coordination is actually a process of re-sharing risks among all nodes of the supply chain. However, the joint contract in this paper has certain adaptability to such risks. Finally, numerical analysis is given to show the impacts of uncertainties on the profit of the supply chain, the decisions made by the members, and the effectiveness of our joint contract.

1. Introduction

In recent years, people have had more demand for perishable products than before, as their living standard has increased. Perishable products have long been very popular with consumers, such as fruit, flowers, newspapers, electronics, etc. Due to the characteristics of these products and the preferences of target consumer groups, such products have a long production lead time, short life cycle, large demand uncertainty, and low residual value of unsold products at the end of the period. In the post-COVID-19 business environment, the uncertainties of demand and yield become more and more complex and changeable. Since the demand for these perishable products is usually uncertain, we have to forecast the sales volume before ordering. Previous studies highlight the need to focus on the ordering or supply management perspective because it consumes an ample amount of budget in the public and private sector [1]. Several studies addressed different aspects of demand uncertainty to establish supply chain systems to overcome the difficulty of demand forecasting. The common approaches are using influence parameters such as price, sales efforts, quality efforts, and so on [2,3,4]. As the influence parameters and coefficients of the perishable products’ demand are also usually uncertain, we suppose that the demand follows a normal distribution. The yield of the perishable products is influenced by many factors too. The weather may bring agricultural growers yield uncertainty, and yield uncertainty due to uncontrollable weather factors may also affect the retailers downstream of an agricultural supply chain [5]. Supply uncertainty in influenza vaccine supply chains arises primarily from the combination of a long production lead time, a complex and highly uncertain manufacturing process, a short immunization season, and frequent changes in vaccine composition [6]. We consider the yield of these perishable products as a random variable because of the yield feature of the perishable products. The existing literature focuses on different decisions, such as price, order quantity, production quantity, inventory cost, and so on, which are made by one or more members in the supply chain [7,8,9]. At the end of the selling period, the inventory cost for the perishable products is usually higher than their residual value. This paper considers a perishable products supply chain model in a single period under both the demand and yield uncertainty and study order quantity decision by retailers, raw material devoted, and wholesale price decisions by the manufacturer.

With the rapid development, traditional products are updating faster and faster, and the life cycle is shorter and shorter. The increasing demand for perishable products leads to a higher profit; with the demand uncertainty and long lead time, it is harder to manage the perishable products supply chain at the same time [10]. As members of the supply chain are all independent entities, they make decisions from their own interests. If each member of the supply chain just tends to earn their own maximum profit without considering the whole supply chain system’s maximum profit, the optimal order quantity decisions made by the retailer will be less than the quantity, which maximizes the profit of the whole supply chain system, and then may form a kind of typical case about double marginalization [11]. Each member in the supply chain ignores the decisions of other members when making decisions, which will cause distorted demand information to continuously spread upstream, and then form another situation called the bullwhip effect [12]. The notion of sustainability is becoming critically important in the agri-food supply chain. Perishable products such as agri-food businesses must eliminate supply chain inefficiencies and develop a more sustainable ecosystem [13]. The existence of double marginalization and bullwhip effect mentioned above indicates that it is an urgent need to coordinate the benefit distribution of supply chain components for long-term and effective cooperation. Therefore, the research on supply chain coordination has been enriched, using a variety of theoretical methods, including operations research, game theory, mathematical modeling, intelligent optimization, simulation, system dynamics, and so on. The contract coordination mechanism is one of the most effective methods to reduce the impact of double marginalization and the bullwhip effect. Risk is the quantifiable portion of uncertainty [14]. There are two very important objectives of a supply chain contract: the first is to improve the overall benefit of the supply chain to achieve the effect of centralized control under the condition that the benefit of member enterprises is not reduced; the second is risk sharing, the member of the supply chain can jointly bear the risks of the whole supply chain through the design of contracts, which drives the entities to make decisions that benefit to the whole supply chain and themselves [15].

The contributions of this paper are as follows: Firstly, we establish a two-stage perishable products supply chain model based on the newsboy model and optimize the quantity of products ordered and the quantity of raw materials devoted, which maximize the expected profit of the supply chain when it is an entire system. Then, we analyze the decentralized model, which is assumed that there is no coordination in it, the manufacturer and the retailer are independent entities deriving their individual expected profit. The decentralized model is expressed here as a Stackelberg game, where the manufacturer is the leader, and the retailer is the follower, which leads to double marginalization. We find that a single contract cannot coordinate the supply chain when the manufacturer faces diminished sales prospects. Finally, we propose a joint contract combined revenue sharing contract with a quantity discount contract considering the manufacturer is risk neutral and the retailer has limited funds, to coordinate the supply chain and improve the expected profit of each member in the supply chain.

The structure of this paper is as follows. In Section 2, we introduce the related literature. In Section 3, we provide descriptions, parameters, and assumptions of the model. In Section 4, we analyze a centralized model and study the optimal decisions to establish a benchmark. In Section 5, both the manufacturer’s decision and the retailer’s decision are analyzed in the decentralized model. In Section 6, we propose a joint contract that combines a revenue-sharing contract with a quantity discount contract. In Section 7, numerical analysis of the models and the coordination parameters are provided, and some further insights are shown. Finally, some conclusions are summarized in Section 8.

2. Related Literature

2.1. Supply Chain Demand and Yield Uncertainty

The uncertainties of demand and yield in the supply chain play an important role in the economic market. Random demand is widespread, and random yield frequently occurs in the agricultural sector or in the chemical, electronic and mechanical manufacturing industries [16]. In the past decades, the supply chain coordination problem under an uncertain demand situation was studied widely. For instance, Zhao and Wei [17] and Xu and Zhai [18] considered a single-period supply chain and focused on the fuzzies aspect of demand uncertainty. They used a general fuzzy number to describe the estimate for the demand. Zhang and Liu [19] established a three-level green supply chain system in which market demand related to the green degree and research on coordination mechanism under a non-cooperative game. Wang et al. [20] considered a supply chain consisting of two competing suppliers and a retailer, which faced price-dependent stochastic demand. They divided the customers into two types: the price-sensitive type and the brand-loyalty type, and studied the supply coordination issue. Wang et al. [21] built a stylized dual channel model with price competition and demand uncertainty and investigated how price competition and demand uncertainty affected supply chain sustainability. They supposed that the demand was linearly dependent on its channel price, cross-price parameter, and random demand factor. There are also many studies focused on supply chain coordination under uncertain yield and deterministic demand. Inderfurth and Clemens [22] studied a supply chain problem under random production yield and deterministic demand. Wang et al. [23] investigated a two-echelon supply chain with yield uncertainty. They assumed that the manufacturer was overconfident about the yield and examined the influence on the performance of members in the supply chain. Bassok et al. [24] assumed the periodic demands to be known and constant; they considered the problem of setting order quantities for purchasing components subjected to uncertainty in the delivery amounts and modeled this as a random yield problem.

Visibly, the uncertainties of demand and yield are both important parameters that affect the construction of appropriate models, the members’ decisions, and the coordination solutions in supply chain. At present, many scholars extend research on both the uncertain demand and yield in the supply chain. Li et al. [25] proposed a mixed multi-stage stochastic optimization model to solve the production planning problem of a bioethanol plant with multiple raw materials simultaneously, considering uncertain demand and yield. Shi and Wang [5] focused on an agriculture supply chain; the yield was related to the water, and the demand was linked to the market and explained how the randomness of yield and demand influenced the profit of the supply chain members and the performance of the entire supply chain. Xie et al. [26] considered the supply chain where the buyer faced uncertain demand and yield and study that how the yield uncertainty and the relative bargaining power affected the performance of the buy-back contract. Adhikari et al. [27] proposed a textile supply chain model consisting of apparel retailers, apparel manufacturers, a textile firm, a fiber firm, and a cotton firm under simultaneous demand and supply uncertainty. They investigated production uncertainties for all middle-level members and devised a buy-back bidirectional sales rebate penalty contract for coordination. Giri et al. [28] proposed the price-only contract and a new revenue-sharing contract to reduce demand and supply uncertainties in the decentralized model and observed that the revenue-sharing contract can fully coordinate the supply chain.

2.2. The Supply Chain Coordination

In order to coordinate the supply chain, many researchers studied the contracts in different conditions, such as wholesale price contacts [29], revenue sharing contacts [5], buy-back contracts [30], quantity discount contacts [31], sales rebate contracts [32] and so on. Pasternack [33] put forward the concept of supply chain contracts for the first time in 1985. At that time, he proposed two return contract strategies to coordinate the supply chain, returning all products at a partial price or returning some products at a full price. After the supply chain coordination contract was put forward, a lot of research on supply chain contract coordination was explored by researchers. Cachon [34] summarized supply chain contract coordination and defined coordination as Nash equilibrium in supply chain optimization; there is no profit motive for enterprises to deviate from existing actions. Zhong et al. [35] proposed a subsidy mechanism based on revenue-sharing contracts to improve supply chain performance and demonstrated the existence of pareto improvement sets. Hu et al. [36] analyzed a two-stage supply chain and proposed a revenue-sharing contract with an order penalty and rebate contract in which the supplier gave the manufacturer variable rewards linked to the final delivery quantity to coordinate the supply chain. He and Zhao [37] established a multi-echelon supply chain coordination model and found that the commonly used wholesale price contracts cannot coordinate the system. They proposed a returns policy combined with the wholesale price contract used by the manufacturer and the raw-material supplier, which could perfectly coordinate the supply chain. Lee and Rhee [38] modified five common types of contracts, revenue-sharing, buy-backs, quantity flexibility, quantity discount, and two-part tariff for supply chain coordination in the presence of the retailer-run resale market. Gao et al. [39] established a secondary supply chain composed of a terminal manufacturer and a telecom operator. To encourage effective cooperation between both parties, they designed a combined contract of ‘two-way cost sharing and benefit compensation’ to coordinate the supply chain and achieve common development. Ji and Liu [40] studied a three-stage supply chain with perishable production and short shelf life under uncertain yield and demand and used ZRS (zero wholesale price and revenue-sharing-plus-side payment) contract to coordinate the supply chain.

Many contacts have been explored to research this coordination issue; the returns policy and the revenue sharing contract are two of the most commonly used contracts to coordinate the supply chain. However, only revenue-sharing coordination always leads to a situation where the wholesale price is lower than the manufacturer’s cost [5], which means the manufacturer will lose money in the process of ordering. When the manufacturer is slightly hesitant about the immediate prospects for consumer sales, the contract will lose attraction to them.

We list some related literature to compare this paper in

Table 1. Literature comparison.

Literature	Uncertain Factors		Contract Involved
	Yield Uncertainty	Demand Uncertainty	Wholesale Price	Revenue Sharing	Quantity Discount	Penalty Contract
Shi and Wang [5]	√	√	√	√	×	×
Inderfurth and Clemens [40]	√	×	√	×	×	√
Hu et al. [20]	√	√	√	√	×	√
Ji and Liu [33]	√	√	√	√	×	×
Zhang and Liu [25]	×	√	×	×	×	×
Zhang et al. [18]	×	×	×	×	√	×
Adhikari et al. [29]	√	√	×	×	×	√
Giri et al. [32]	√	√	√	√	×	×
This paper	√	√	√	√	√	×

.

For the above discussion, this paper analyzes a joint coordination contracts mechanism of a two-echelon supply chain system. The results of this paper are closely related to the work by Inderfurth and Clemens [22], Shi and Wang [5]. Shi and Wang [5] consider an agricultural supply chain coordination under water-related uncertain yield. However, they established the supply chain without considering the secondary market, shortage cost, and residual value of the products. Inderfurth and Clemens [22] studied a supply chain problem under random production yield and deterministic demand. It studied wholesale price contracts, over-production risk-sharing contracts, and penalty contracts under different conditions. Our work differs from Inderfurth and Clemens [22] in two aspects. On the one hand, the yield and the demand are both uncertain in our study. On the other hand, the coordination contract we studied is different. This paper proposes a joint contract combined revenue sharing contract with a quantity-discount contract considering the manufacturer is risk neutral and the retailer has limited funds.

3. Model Description and Assumptions

This paper considers the issue of coordination for a two-stage perishable products supply chain, which consists of one manufacturer and one retailer. The manufacturer decides Q (the devote quantity of raw materials), the retailer decides L (the quantity of the raw materials devoted), and w (the wholesale price of the product). When the market demand is greater than the supply at the end of the cycle, a shortage cost will occur. Conversely, retailers will convert the superfluous product to salvage value due to the characteristics of perishable products. When the quantity of production ordered by the retailer is more than that produced by the manufacturer, the manufacturer replenishes the goods from the secondary market. The coordination of the supply chain becomes very complicated in the face of the dynamic environment. We suppose the stochastic demand (X) of the supply chain obeys normal distribution, and the rate of production (Y) obeys uniform distribution. Then, we optimize the quantity of product ordered and the quantity of raw materials devoted in the centralized model and the decentralized model, respectively. We propose a joint contract by comparing the optimal decisions in the decentralized and centralized model afterward.

3.1. Parameters

• p: The retail price of the product, a constant determined by the market;
• c: The raw materials cost for the unit product;
• g: The shortage cost for the unit product;
• v: The residual value of the unit product at the end of the selling period;
• s: The unit price in the secondary market;
• X: The stochastic demand when the retail price is p;
• f(x): The probability density function of the stochastic demand;
• F(x): The distribution function of the stochastic demand;
• ${\mu }_1$: The mean value of the stochastic demand, ${\mu }_1=E\left(x\right)={\displaystyle {\int }_0^{\infty }xf\left(x\right)}dx$;
• ${\sigma }_1$: The standard deviation of the stochastic demand;
• Y: The rate of production, which is a random variable;
• g(y): The probability density distribution function of the production rate;
• G(y): The distribution function of the production rate;
• ${\mu }_2$: The mean value of the production rate, ${\mu }_2=E\left(y\right)={\displaystyle {\int }_0^{\infty }yf\left(y\right)}dy$;
• ${\sigma }_2$: The standard deviation of the production rate.

3.2. Decision Variables

• Q: The quantity of the product ordered by the retailer;
• L: The quantity of the raw materials devoted by the manufacturer;
• w: The wholesale price of the product by the manufacturer.

3.3. Assumptions

• The retailer’s demand is stochastic, which obeys normal distribution. The manufacturer’s yield is random, and the production rate obeys uniform distribution $0\le A\le y\le B\le 1$, $E\left(y\right)={\mu }_2=(A+B)/2$, $D\left(y\right)={\sigma }_2={(B-A)}^2/12$ [20].
• p > w > c/y > 0, the wholesale price set by the manufacturer is greater than the manufacturer’s production cost, which ensures that every member of the supply chain can obtain profits.
• The secondary market can be only accessible to the manufacturer, and the retailer can only purchase from the manufacturer. For convenient to discuss, the secondary market can meet all the manufacturers’ needs.
• s > c/y, the price of production replenished from the secondary market is greater than that produced by the manufacturer, which prevents the manufacturer from never producing; w > s, the wholesale price is always greater than the price of restocking from the secondary market in order to ensure that the manufacturer is willing to replenish from the secondary market.

4. Optimal Decisions in the Centralized Model

Firstly, we discuss the optimal decisions in the centralized model to establish a performance benchmark. In the centralized model, Q and L are decision variables that influence the profit of the supply chain a lot; the decisions made by the manufacturer and the retailer are subject to the goal of maximizing the supply chain’s profit. However, w only affects the benefits of each member in the supply chain and has no impact on the profit of the entire supply chain. Therefore, it does not make sense to optimize w in the centralized model. The process of the retailer purchasing goods from the manufacturer is regarded as the transfer of profits within the supply chain. The expected profit function of the supply chain can be expressed by the following equation:

$${\prod }_{sc}\left(Q,L\right)=pE\left[\min \left(Q,x\right)\right]+vE\left[{\left(Q-x\right)}^{+}\right]-sE\left[{\left(Q-yL\right)}^{+}\right]-gE\left[{\left(X-Q\right)}^{+}\right]-cL$$

(1)

where pE[min(Q,x)] is the sales revenue, vE[(Q − x)⁺] is the residual value of unsold products at the end of the selling period, sE[(Q − yL)⁺] is the cost of restocking from the secondary market by the manufacturer to meet the retailer’s order, gE[(X − Q)⁺] is the loss caused by unmet market demand, and cL is the manufacturer’s cost of production. When the quantity of sale (S(Q)) is expressed as $S(Q)=\left[{\displaystyle {\int }_0^{Q}xf\left(x\right)dx+{\displaystyle {\int }_Q^{+\infty }Qf\left(x\right)dx}}\right]$, Equation (1) can be expressed as:

$${\prod }_{sc}\left(Q,L\right)=\left(p-v+g\right)S(Q)-s{\displaystyle {\int }_A^{Q/L}\left(Q-yL\right)}g\left(y\right)dy+vQ-g{\mu }_1-cL$$

(2)

Taking the first and second derivatives of ${\prod }_{sc}\left(Q,L\right)$ respect to Q and L, we have

$$\frac{\partial {\prod }_{sc}\left(Q,L\right)}{\partial Q}=\left(p-g\right)-\left(p-v+g\right)F\left(Q\right)-s{\displaystyle {\int }_A^{Q/L}g\left(y\right)}dy$$

$$\frac{\partial {\prod }_{sc}\left(Q,L\right)}{\partial L}=-c+s{\displaystyle {\int }_A^{Q/L}yg\left(y\right)}dy$$

$$\frac{{\partial }^2{\prod }_{sc}\left(Q,L\right)}{\partial Q^2}=-\left(p-v+g\right)f\left(Q\right)-\frac{s}{L}g\left(\frac{Q}{L}\right)$$

$$\frac{{\partial }^2{\prod }_{sc}\left(Q,L\right)}{\partial L^2}=-s\frac{Q^2}{L^3}g\left(\frac{Q}{L}\right)$$

$$\frac{{\partial }^2{\prod }_{sc}\left(Q,L\right)}{\partial L\partial Q}=\frac{{\partial }^2{\prod }_{sc}\left(Q,L\right)}{\partial Q\partial L}=s\frac{Q}{L^2}g\left(\frac{Q}{L}\right)$$

The Hessian matrix of ${\prod }_{sc}\left(Q,L\right)$ is

$$H\left(Q,M\right)=\left(\begin{array}{cc}\frac{{\partial }^2{\prod }_{sc}\left(Q,L\right)}{\partial Q^2}& \frac{{\partial }^2{\prod }_{sc}\left(Q,L\right)}{\partial Q\partial L}\\ \frac{{\partial }^2{\prod }_{sc}\left(Q,L\right)}{\partial L\partial Q}& \frac{{\partial }^2{\prod }_{sc}\left(Q,L\right)}{\partial L^2}\end{array}\right)=\left(\begin{array}{cc}-\left(p-v+g\right)f\left(Q\right)-\frac{s}{L}g\left(\frac{Q}{L}\right)& s\frac{Q}{L^2}g\left(\frac{Q}{L}\right)\\ s\frac{Q}{L^2}g\left(\frac{Q}{L}\right)& -s\frac{Q^2}{L^3}g\left(\frac{Q}{L}\right)\end{array}\right)$$

We could find

$\left|H_1\left(Q,L\right)\right|<0$, $\left|H_3\left(Q,L\right)\right|<0$, $\left|H_2\left(Q,L\right)\right|=\frac{\left(p-v+g\right)s{Q}^2}{L^3}f\left(Q\right)g\left(\frac{Q}{L}\right)>0$.

It is easy to prove that ${\prod }_{sc}\left(Q,L\right)$ is a concave function, and there is a unique optimal solution ($Q_{sc}^{*}$, $L_{sc}^{*}$) that maximizes the overall profit of the supply chain, which meets the following conditions:

$$\left(p-g\right)-\left(p-v+g\right)F\left(Q\right)-s{\displaystyle {\int }_A^{Q/L}g\left(y\right)}dy=0$$

$$c-s{\displaystyle {\int }_A^{Q/L}y}g\left(y\right)dy=0$$

Then, $Q_{sc}^{*}$ and $L_{sc}^{*}$ are obtained

$$\left\{\begin{array}{l}{Q}_{sc}^{*}=F^{-1}\left(\frac{p+g-s{\displaystyle {\int }_A^{{\epsilon }_{sc}^{*}}g\left(y\right)}dy}{p-v+g}\right)\\ L_{sc}^{*}=\frac{Q_{sc}^{*}}{{\epsilon }_{sc}^{*}}\end{array}\right.$$

(3)

If there is $\epsilon =Q/L$, $H\left(\epsilon \right)={\displaystyle {\int }_A^{\epsilon }yg(y)dy}$ is evidently an increasing function. Therefore, there is a unique ${\epsilon }_{sc}^{*}$ to obtain $H\left(\epsilon \right)=c/s$; we can obtain $L_{sc}^{*}$ and $Q_{sc}^{*}$. Substitute (3) into Equation (2), the maximum total profit can be obtained in the centralized model, which is:

$${\prod }_{sc}^{*}\left(Q,L\right)={\prod }_{sc}^{*}\left(Q_{sc}^{*},L_{sc}^{*}\right)=\left(p-v+g\right){\displaystyle {\int }_0^{Q_{sc}^{*}}xf\left(x\right)}dx-g{\mu }_1$$

(4)

5. Optimal Decisions in the Decentralized Model

In the decentralized model, the manufacturer and the retailer are two independent entities; both the manufacturer and the retailer will try to maximize their own profits, respectively, without considering the group benefits of the supply chain. We formulate the problem as a Stackelberg game. The manufacturer is a leader, and the retailer is a follower. The transaction between the manufacturer and the retailer is only through the wholesale price contract in the decentralized model, in which the manufacturer makes the production and wholesale price decision first, and then the retailer makes the ordering decision. Based on the analysis above, the expected profits of the manufacturer and the retailer as follows:

$${\prod }_m\left(L,w\right)=wQ-sE\left[{\left(Q-yL\right)}^{+}\right]-cL$$

(5)

$${\prod }_r\left(Q\right)=pE\left[\min \left(Q,x\right)\right]-wQ+vE\left[{\left(Q-x\right)}^{+}\right]-gE\left[{\left(x-Q\right)}^{+}\right]$$

(6)

Backward induction is used to solve the question. Firstly, we obtain the retailer’s order quantity response function. Then, we can obtain the manufacturer’s quantity of raw materials devoted and the wholesale price based on the order quantity response function.

5.1. The Decision of Retailer

In Equation (6), pE[min(Q,x)] is the sales revenue, wQ is the cost of wholesaling products from the manufacturer, vE[(Q − x)⁺] is the residual value of unsold products at the end of the selling period, and gE[(X − Q)⁺] is the loss caused by unmet market demand. The Equation (6) is translated as:

$${\prod }_r\left(Q\right)=\left(p-v+g\right)S(Q)-\left(w-v\right)Q-g\mu$$

(7)

The first and second derivatives of ${\prod }_r\left(Q\right)$ are obtained

$$\begin{array}{l}\frac{d{\prod }_r\left(Q\right)}{dQ}=\left(p-w+g\right)-\left(p-v+g\right)F\left(Q\right)\\ \frac{d^2{\prod }_r\left(Q\right)}{d{Q}^2}=-\left(p-v+g\right)f\left(Q\right)<0\end{array}$$

Therefore, Equation (7) has a unique optimal solution:

$$Q_r\left(w\right)=F^{-1}\left(\frac{p-w+g}{p-v+g}\right)$$

(8)

5.2. The Decision of Manufacturer

In Equation (5), wQ is the selling profit of the manufacturer, sE[(Q − yL)⁺] is the cost of restocking from the secondary market by the manufacturer to meet the retailer’s order, and cL is the manufacturer’s cost of production. Equation (5) is translated as:

$${\prod }_m\left(w,L\right)=wQ-s{\displaystyle {\int }_A^{Q/L}\left(Q-yL\right)}g\left(y\right)dy-cL$$

(9)

Equation (8) is substituted into Equation (9), in which w and L are derived as:

$$\frac{\partial {\prod }_m\left(w.L\right)}{\partial L}=s{\displaystyle {\int }_A^{\epsilon }y}g\left(y\right)dy-c$$

$$\frac{\partial {\prod }_m\left(w.L\right)}{\partial w}=Q_r\left(w\right)-\frac{w-sG\left(\epsilon \right)}{\left(p-v+g\right)f\left(Q_r\left(w\right)\right)}$$

If $\epsilon ={\epsilon }_{sc}^{*}$, we can obtain $s{\displaystyle {\int }_A^{Q_r^{*}/L_m^{*}}y}g\left(y\right)dy-c=0$. The parameter of ${\epsilon }_{sc}^{*}$ is a normal number and dependent on c and s. Since $H\left(\epsilon \right)={\displaystyle {\int }_A^{\epsilon }yg(y)dy}$ is an increasing function, there is a unique $L_m^{*}$ that maximizes ${\prod }_m$ no matter how many orders $Q_r^{*}$ the retailer sets.

When $P(x)=xf(x)/\overline{F}(x)$, $P^{\prime }(x)>0$, Lariviere and Porteus [41] term P(x) the generalized failure rate (IGFR), a distribution has an increasing generalized failure rate if P(x) is weakly increasing for all x. The usual distributions of demand all meet the condition of IGFR, such as uniform distribution, normal distribution, exponential distribution, and so on. When the demand distribution F(x) meets the condition of IGFR, the expected profit of the manufacturer is a concave function [42], which proves that there is a unique w that can maximize the profit of the manufacturer. We suppose the unique value is $w_m^{*}$.

Substitute $w_m^{*}$ into Equation (8); we can obtain $Q_r^{*}=F^{-1}\left(\frac{p-w_m^{*}+g}{p-v+g}\right)$. The maximum expected profit of retailer in the decentralized model is:

$${\prod }_r^{*}={\prod }_r^{*}\left(Q_r^{*}\right)=\left(p-v+g\right){\displaystyle {\int }_0^{Q_r^{*}}xf\left(x\right)}dx-g{\mu }_1$$

(10)

$w_m^{*}$ and $L_m^{*}$ are substituted into Equation (9), the maximum expected profit of the manufacturer in the decentralized model is:

$${\prod }_m^{*}={\prod }_m^{*}\left(w_m^{*},L_m^{*}\right)=\left[w_m^{*}-s{\displaystyle {\int }_A^{{\epsilon }_{sc}^{*}}g\left(y\right)}dy\right]Q_r^{*}$$

(11)

5.3. Comparison of Optimal Decisions in the Decentralized and Centralized Model

According to the above analysis, in the centralized model, the order quantity is $Q_{sc}^{*}=F^{-1}\left(\frac{p+g-s{\displaystyle {\int }_A^{{\epsilon }_{sc}^{*}}g\left(y\right)}dy}{p-v+g}\right)$ and which in the decentralized model is $Q_r^{*}=F^{-1}\left(\frac{p-w_m^{*}+g}{p-v+g}\right)$. We can obtain $w_m^{*}>s{\displaystyle {\int }_A^{{\epsilon }_{sc}^{*}}g\left(y\right)}dy$ from Equation (11), so $F^{-1}\left(\frac{p+g-s{\displaystyle {\int }_A^{{\epsilon }_{sc}^{*}}g\left(y\right)}dy}{p-v+g}\right)>F^{-1}\left(\frac{p-w_m^{*}+g}{p-v+g}\right)$, namely $Q_{sc}^{*}>Q_r^{*}$. If there is ${\epsilon }_m^{*}={\epsilon }_{sc}^{*}$, we can obtain $L_{sc}^{*}>L_m^{*}$.

Evidently, in the decentralized model, both the retailer and the manufacturer only consider their own interests when making decisions. The ordering decision made by the retailer is less than the quantity that actually maximizes the supply chain’s profit. At the same time, the production decision made by the manufacturer is less than the quantity that actually maximizes the supply chain’s profit too. The overall profit of the supply chain will decline in the decentralized model, which is well known as double marginalization [22].

The quantity ordered by the retailer in the centralized model is more than that in the decentralized model, indicating that the retailer tends to order fewer products to resist the impact of uncertain market demand without coordination. This decision will reduce the risk that the market demand is lower than expected, resulting in a large backlog of products. Accordingly, the manufacturer tends to devote the quantity of raw materials less than that in the centralized model because of their uncertain yield.

To sum up, the supply chain can achieve a higher level of profit in the centralized model compared with the decentralized model. The manufacturer and retailer are both independent individuals. It is necessary to set a certain coordination contract to maximize their total profit in the decentralized model. This contract will improve the efficiency of the supply chain under yield and demand uncertainty. At the same time, both the manufacturer and the retailer should obtain more profits from the coordination contract than before. Therefore, a coordination contract can motivate the manufacturer and the retailer to abide by the contract and form a good cooperative relationship.

6. Contract Coordination in the Supply Chain

6.1. Joint Contract Based on Credit Payment

The traditional revenue sharing contract generally stipulates that the retailer will give the manufacturer a certain percentage of sales revenue at the end of the sales period, and the manufacturer promises to give the retailer a lower wholesale price to incentivize the retailer to order more goods. As a result, the wholesale price may be lower than the production cost. When manufacturers have low expectations for future sales, they are often reluctant to sell below the production cost. Therefore, this paper establishes a joint contract mechanism for considering long-term cooperation between manufacturers and retailers. The retailer can pay part of the payment to the manufacturer when ordering. Then, the retailer will pay the remaining payment at a quantity discount and share a certain percentage of revenues with the manufacturer in the end.

It is assumed that the joint contract combined revenue sharing contract with a quantity discount contract based on credit payment between manufacturer and retailer is $R\left(F,r,\varphi \right)$. The quantity discount rate for a single period is assumed as r, which is a constant and determined by the manufacturer. If there is $I(Q)=E\left[{\left(Q-x\right)}^{+}\right]$ and $L(Q)=E\left[{\left(x-Q\right)}^{+}\right]$, the profit functions of retailers and manufacturers are expressed as:

$${\prod }_r^{RS}\left(Q\right)=(1-\varphi )\left[pS(Q)+vI(Q)-gL(Q)\right]-(wQ-F)/(1+r)-F$$

(12)

$${\prod }_m^{RS}\left(Q\right)=\varphi \left[pS(Q)+vI(Q)-gL(Q)\right]+(wQ-F)/(1+r)+F-sE\left[{\left(Q-yL\right)}^{+}\right]-cL$$

(13)

The total profit function of the supply chain system remains unchanged; we can obtain ${\prod }_{sc}\left(Q,L\right)={\prod }_r^{RS}\left(Q\right)+{\prod }_m^{RS}\left(L\right)$.

In order to obtain the optimal order quantity of the supply chain under the coordination mechanism, ${\prod }_r^{RS}\left(Q\right)$ is differentiated as:

$$\begin{array}{l}\frac{d{\prod }_r^{RS}\left(Q\right)}{dQ}=\left(1-\varphi \right)\left(p-v+g\right)\left[1-F\left(Q\right)\right]+\left(1-\varphi \right)v-w/(1+r)\\ \frac{d^2{\prod }_r^{RS}\left(Q\right)}{d^{2}Q}=-(1-\varphi )\left(p-v+g\right)f\left(Q\right)<0\end{array}$$

Therefore, there is a unique $Q_r^{RS}$ for $\frac{d{\prod }_r^{RS}\left(Q\right)}{dQ}=0$. The retailer’s optimal decision under the joint contract is obtained as follows:

$$Q_r^{RS}{}^{*}=F^{-1}\left[\frac{\left(1-\varphi \right)\left(p+g\right)-w/(1+r)}{\left(1-\varphi \right)\left(p-v+g\right)}\right]$$

(14)

By comparing Equations (8) and (14), it must be satisfied the following relationships under the joint contract.

$$\frac{\left(1-\varphi \right)\left(p+g\right)-w/(1+r)}{\left(1-\varphi \right)\left(p-v+g\right)}=\frac{p+g-s{\displaystyle {\int }_A^{{\epsilon }_{sc}^{*}}g\left(y\right)}dy}{p-v+g}$$

The $Q_r^{RS}{}^{*}$ is plugged into the equation ${\prod }_m^{RS}\left(L\right)$, we can obtain $\frac{d{\prod }_m^{RS}\left(L\right)}{dL}=s{\displaystyle {\int }_A^{\epsilon }y}g\left(y\right)dy-c=0$.

The equations are deduced as $w=\left(1-\varphi \right)\left(1+r\right)s{\displaystyle {\int }_A^{{\epsilon }_{sc}^{*}}g\left(y\right)}dy$ and $\epsilon ={\epsilon }_{sc}^{*}=Q_{sc}^{*}/L_{sc}^{*}$, then we can obtain

$${\prod }_r^{RS*}=\left(1-\varphi \right){\prod }_{sc}^{*}+R\left(F,r,\varphi \right)$$

$${\prod }_m^{RS*}=\varphi {\prod }_{sc}^{*}-R\left(F,r,\varphi \right)$$

The $R(\varphi ,r)=\left(1-\varphi \right)\left[c-s{\displaystyle {\int }_A^{{\epsilon }_{sc}^{*}}yg\left(y\right)}dy\right]L^{*}-Fr/(1+r)$ is found as a constant; we can obtain ${\prod }_r^{RS*}+{\prod }_m^{RS*}={\prod }_{sc}^{*}$. Therefore, the joint contract can coordinate the supply chain, and the manufacturer and retailer make the optimal decision of Q and L to maximize the profit in the decentralized model through coordination. Nowadays, credit payment is increasingly mature with the development of e-commerce, in which a large number of interest-free loans and small loans are common. The payment method gradually changes from the original full payment to installment payment. Many industries in the post-COVID-19 business environment have become very cautious about their future sales prospects and expectations. The manufacturer is occupied by a large number of inventories, and retailer’s capital turnover is ineffective. Faced with such a dilemma, a joint contract $R\left(F,r,\varphi \right)$ is proposed to solve this problem in this paper. The retailer uses a small amount of funds to obtain a sufficient supply of goods to solve the dilemma of ineffective capital turnover. The retailer pays off the goods and shares a certain proportion of sales profits with manufacturers at the end of sales so as to share risks and profits with manufacturers. When market sales are not expected to be good, the manufacturer can promote the wholesale prices by promoting the ratio of quantity discount to ensure cooperation; on the contrary, the manufacturer can reduce the wholesale prices by promoting the ratio of revenue-sharing to encourage retailers to order more goods.

6.2. Influence of the Secondary Market on Coordination

According to the previous analysis, the relationship between the ratio of manufacturer’s optimal production decision and retailer’s optimal order decision, manufacturer’s production cost, and the price in the secondary market is as follows:

$$\left\{\begin{array}{l}\frac{Q_{sc}^{*}}{L_{sc}^{*}}=\frac{Q_r^{*}}{L_m^{*}}={\epsilon }^{*}\\ {\displaystyle {\int }_A^{{\epsilon }^{*}}y}g\left(y\right)dy=\frac{c}{s}\end{array}\right.$$

Evidently, the ratio of the retailer’s optimal ordering decision Q to the manufacturer’s optimal production decision L is only related to c and s. The price in the secondary market is an external influence factor on the supply chain. In order to study the relationship between ε and s, the first-order derivative functions of ε with respect to s are obtained respectively:

$$\frac{d\epsilon }{ds}=-\frac{c}{s^2\epsilon g\left(\epsilon \right)}$$

We can obtain $\frac{d\epsilon }{ds}<0$, in which the ratio ε of the retailer’s optimal order decision Q and the manufacturer’s optimal production decision L decreases with the increase in secondary market price s. The ordering decision of the retailer is based on the prediction of market demand, and the hypothesis of this paper is that only the manufacturer can enter the secondary market and ensure a sufficient supply of goods for the retailer. Therefore, the ordering decision of the retailer has nothing to do with the price in the secondary market. When the price in the secondary market decreases, the manufacturer will reduce the order quantity of raw materials and reduce a certain amount of production to maximize profit. When the price of the secondary market increases, the manufacturer will increase the order quantity of raw materials to reduce the potential risk of loss of replenishment from the secondary market due to uncertain yield.

The wholesale price under the joint contract is $w=\left(1-\varphi \right)\left(1+r\right)s{\displaystyle {\int }_A^{{\epsilon }_{sc}^{*}}g\left(y\right)}dy$. It has been proved above that when the price in the secondary market increases, the risk of yield uncertainty increases. When the risk of yield uncertainty increases, the manufacturer can promote the proportion of revenue sharing or reduce the quantity discount, and the wholesale price between the manufacturer and the retailer remains unchanged. Then, the external risk of production uncertainty can be partially transferred to the retailer, and finally, the whole supply chain can still achieve system optimization. The conclusion further proves that supply chain coordination is actually a process of re-sharing risks among all nodes of the supply chain. However, the joint contract in this paper has certain adaptability to such risks, and the manufacturer can still maximize the profit of the whole supply chain by adjusting the contract parameter.

7. Numerical Analysis

The impacts of uncertainties on the supply chain’s profit and decisions and the feasibility of the joint contract coordination model are further researched and verified. Model validation with numerical simulation analysis will be conducted below. According to the research results on local Dongtai watermelon, Dongtai watermelon is a popular product among consumers at present. According to the actual production and sales of Dongtai watermelon, the corresponding parameters are given in

Table 2. Numerical assumptions.

Parameters	p	s	c	g	v	μ1	σ1	μ2	σ2
Value	18	12	3	1	2	10,000	10,000	0.5	0.083

.

7.1. The Effect of Uncertainty on the Supply Chain

As we can see from Figure 1, the uncertainty of demand and yield is bad for the optimal profit of the supply chain, not only in the centralized model but also in the decentralized model. In the decentralized model, the manufacturer’s optimal profit decreases with the demand uncertainty increases, and the retailer’s optimal profit is affected by the demand uncertainty a little, but their optimal profits both decrease with the yield uncertainty increases. In the decentralized model with coordination contracts R(F, r, Φ) (F = 10,000; r = 0.4; Φ = 0.71), the optimal profits of the manufacturer, the retailer, and the supply chain are all higher than before, which makes sure that they are willing to cooperate on the basis of the joint contract mentioned above. With the joint contract mechanism, the manufacturer and the retailer work together and reach the supply chain’s maximum profit.

As Figure 2 shows, in the centralized model, the optimal ordering quantity and the optimal raw material devoted quantity both increase with the demand uncertainty increases; on the contrary, they both decrease with the increase in yield uncertainty. We also find that in the decentralized model, with the uncertainty increases, the ordering quantity decision made by the retailer and the raw material devoted decision made by the manufacturer are both affected. The optimal ordering quantity under the decentralized model without coordination contract is less than which under the centralized model, and so does the raw material devoted quantity. It indicates that the profit of the supply chain under a decentralized model without a coordination contract cannot reach the maximum value.

The optimal wholesale price decision is always made by the manufacturer; evidently, the wholesale price cannot coordinate the members in the supply chain. By observing Figure 3, in the decentralized model without coordination, the optimal wholesale price first decreases fast and then decreases slowly with the increase in demand uncertainty, which increases fast and then increases slowly with the increase in yield uncertainty.

7.2. The Profits under Decisions and the Joint Contract

As discussed above, the expected profit function of the supply chain in the centralized model is concave in Q and L. We provide the expected profit surfaces in two conditions. When (σ₁ = 10,000, σ₂ = 0.083333) and (σ₁ = 4000, σ₂ = 0.083333), the expected profits of the supply chain are both smooth surfaces under the decisions of the order quantity (Q) and raw material devoted quantity (L) in the centralized model by observing Figure 4, so there must be a unique (Q, L) to maximize the expected profit of the supply chain. The calculation result shows that when the decisions (Q = 13,015, L = 18,407), the expected profit of the supply chain is 44,505; when the decisions (Q = 11,206, L = 15,848), the expected profit of the supply chain is 69,361, they both can reach the maximum value. We can find from Figure 4 that with the uncertainty of demand decreases, the profit of the supply chain increase.

In the decentralized model, the manufacturer and the retailer are independent individuals. It is significant to design a contract to coordinate their relationship and ensure they made the same decisions as in the centralized model. Assume σ₁ = 10,000, σ₂ = 0.083333; through the coordination of the joint contract R(F, r, Φ) (F = 10,000; r = 0.4; Φ = 0.71), the expected profit of the manufacturer and the retailer under the decisions of the order quantity (Q) and raw material devoted quantity (L) in the decentralized model are smooth surfaces referring to Figure 5, which demonstrates that they both can maximize their own expected profits. When the decisions of the order quantity (Q) and raw material devoted quantity (L) satisfy (Q = 13,015, L = 18,407), which is the supply chain’s optimal decision, the expected profit of both the manufacturer and the retailer reaches maximum simultaneously. It proves that the joint contract we proposed can indeed ensure the manufacturer and the retailer made the same decisions.

7.3. The Effects of Coordination Coefficients on the Supply Chain

Considering the retailer has limited funds, the retailer only pays a deposit at the beginning. Assume the deposit of the retailer F = 10,000. Under the effect of revenue sharing coefficient and quantity discount coefficient, the optimal profits achieved by the manufacturer and the retailer are shown in Figure 6. To establish a benchmark, the optimal profits of the manufacturer and the retailer without a coordination contract are shown within that. It can be found that there is an area where both the manufacturer and the retailer have higher optimal profits after the coordination contract than before.

Assume the quantity discount coefficient and the deposit of the retailer is fixed, r = 0.4, F = 10,000. It can be seen from Figure 7 that with the increase in revenue sharing coefficient, the optimal profit of the manufacturer increases, and the optimal profit of the retailer decreases. It reveals the effect of the revenue sharing coefficient on the optimal profits of the manufacturer and the retailer under the joint contract. For a convenient study, the optimal profits of the manufacturer and the retailer in a decentralized model without coordination are also shown in Figure 7. It is shown above that in the decentralized model with the joint contract; the supply chain can achieve the maximum profit, which is the same as in the centralized model. It is calculated that the manufacturer and the retailer can both achieve higher optimal profits when Φ ∈ (0.570076, 0.875779) than before, which encourages them to adopt the coordination mechanism, and the value of Φ is further determined by their bargaining power.

For further discussion, when r = 0.4, F = 10,000, Φ ∈ (0.570076, 0.875779), the optimal decisions of the wholesale price and related profits under different conditions are collected in

Table 3. The profits and the wholesale price under different conditions.

Profit/Decision	Φ	w	R(cor)	M(cor)	R(cor) + M(cor)
	0.57	5.11	16,276.90	28,228.74	44,505.64
Centralized:	0.59	4.87	15,386.79	29,118.85	44,505.64
SC = 44,505.64	0.61	4.63	14,496.68	30,008.97	44,505.64
	0.63	4.39	13,606.56	30,899.08	44,505.64
	0.65	4.16	12,716.45	31,789.19	44,505.64
	0.67	3.92	11,826.34	32,679.30	44,505.64
Decentralized:	0.69	3.68	10,936.22	33,569.42	44,505.64
R = 2671.38	0.71	3.44	10,046.11	34,459.53	44,505.64
M = 28,228.74	0.73	3.21	9156.00	35,349.64	44,505.64
w = 12.74087	0.75	2.97	8265.89	36,239.76	44,505.64
	0.77	2.73	7375.77	37,129.87	44,505.64
	0.79	2.49	6485.66	38,019.98	44,505.64
	0.81	2.26	5595.55	38,910.09	44,505.64
	0.83	2.02	4705.43	39,800.21	44,505.64
	0.85	1.78	3815.32	40,690.32	44,505.64
	0.87	1.54	2925.21	41,580.43	44,505.64

, which shows that the wholesale price under the decentralized model without coordination is much higher than it in the joint contract. It is because the joint contract through the revenue sharing by the retailer in exchange for a low wholesale price by the manufacturer encourages the retailer to order more products to achieve the optimal ordering quantity of the supply chain. In a coordination contract, as the revenue sharing coefficient decreases, the wholesale price increases, but the profits of all members are more than before. The manufacturer can negotiate with the retailer to obtain a wholesale price higher than the cost when it has quite bad sales prospects.

To study the impacts of the quantity discount coefficient in coordination contract on the optimal profits, we compare the optimal profits of the manufacturer and the retailer under two different revenue sharing coefficients and draw Figure 8. For being convenient to study, their optimal profits in a decentralized model without coordination are also shown in Figure 8. The value of the quantity discount coefficient we chose is related to the revenue sharing coefficient. Figure 8 illustrates that when Φ = 0.71, r ∈ [0, 1] and when Φ = 0.6, r ∈ [0.18, 1], the manufacturer and retailer achieve Pareto improvement. Moreover, when the quantity discount increases, the optimal profit of the manufacturer increases gradually in a narrow range, and the optimal profit of the retailer decreases gradually in a narrow range. It means that the manufacturer can promote the wholesale price by bargaining for a higher quantity discount coefficient, as shown in Figure 9, which does not influence the retailer’s profit too much and makes sure the manufacturer will not lose money at first.

8. Conclusions

This paper investigates the order quantity decision by retailers, raw material devoted quantity, and wholesale price decisions by the manufacturers; and attempts to propose a joint contract for a two-stage perishable products supply chain under both demand and yield uncertainty.

Due to the characteristics of perishable products, we establish a perishable product supply chain model in a single period, the demand is stochastic and normally distributed, and the yield is random and uniformly distributed. When the market demand is greater than the yield at the end of the cycle, a shortage cost will occur. Conversely, retailers will convert superfluous products to salvage value. We first determine the quantity of product ordered and the quantity of raw materials devoted, which maximizes the expected profit of the supply chain when it is an entire system. Then, we analyze the decentralized model, which is formulated as a Stackelberg game, the manufacturer and the retailer are independent entities deriving their individual expected profit, which leads to double marginalization. We find that a single contract cannot coordinate the supply chain when the manufacturer faces diminished sales prospects at the same time. In order to solve this issue, we propose a joint contract that combines revenue sharing with a quantity discount, the retailer pays the manufacturer a down payment at the beginning, and the manufacturer gives the retailer a quantity discount and shares a proportion of profit from the retailer at last. To make sure that both members in the supply chain want to adopt this contract, we prove the feasibility of the joint contract achieving a win–win situation. When the manufacturer is slightly hesitant about the immediate prospects of sales, they can promote the wholesale price by adjusting the coordination coefficient, which makes sure that the manufacturer will not lose money in the process of ordering. To validate the model, we further study the effects of uncertainties in the supply chain’s profit and decisions, the effects of coordination coefficients on the supply chain, and the potential risk of the secondary market. Accordingly, we obtain some managerial implications: (1) The uncertainties of demand and yield harm to the profits of the supply chain members. Because of the risk of such uncertainties, the manufacturer tends to produce fewer products, and the retailer tends to order fewer products. The increase in the price in the secondary market is an external risk for the supply chain, which has a certain impact on the coordination of the supply chain. (2) It is significant to coordinate the relationship of the members in the supply chain, the joint contract we proposed is profitable to all the members, and it is beneficial to maintain the sustainability of the supply chain in the post-COVID-19 business environment.

This research can be extended by considering a dual-channel perishable product supply chain that has two channels for sale; there is more than one manufacturer selling the products, and it will be interesting to analyze the retailer’s decision about the price. Our analysis may be possibly extended to a close-loop perishable products supply chain that takes the unsold products such as electronics for recycling. On the other hand, we can consider more factors related to the demand and yield, such as sales efforts, quality efforts, R&D investment, the weather, and the green degree. It is also meaningful to study the influence of emergencies on the perishable products supply chain coordination with the increase in the frequency and intensity of emergencies in recent years. Our future work will be directed towards these aspects.