The Maize Supply Chain Coordination Mechanism under Yield Uncertainty Caused by Drought: A Study in the Heilongjiang Province of China

Abstract

Deficient rainfall due to drought is an inevitable factor triggering maize yield uncertainty, thus affecting the performance and sustainability of the supply chain. Therefore, this paper first uses historical data to quantitatively fit the unfavorable effect of rainfall on maize yield affected by drought, taking the maize planting in the Heilongjiang Province of China as an example. Then, using a Stackelberg game, payback revenue sharing (PBRS), and cost revenue sharing (CRS), contract mechanisms are developed to coordinate the maize supply chain incorporating yield uncertainty from drought and demand risk faced by the retailer. We investigate the impact of uncertain maize yield on the supply chain and conduct numerical experiments to verify the analysis. The results reveal that declining rainfall, representing increasing drought severity, negatively affects the profits of the entire supply chain and its members; however, supply chain efficiency is raised under both coordinated contracts. In addition, the yield risk-sharing proportion was higher under a CRS mechanism than a PBRS one, which gives more incentives for the vulnerable supplier to participate in the supply chain. Finally, our analysis indicates that rearranging contract structures and parameters based on classic revenue sharing contracts could realize supply chain coordination.

1. Introduction and Literature Review

As an adverse weather event, drought directly affects crop growth, leading to yield uncertainty. The “Global Report on Food Crises 2018” stated that drought was one of the crucial triggers of food crises affecting 39 million people in 23 countries worldwide [1]. Thus, in response to the impact of drought on agricultural production, most countries have taken measures to reduce yield uncertainty. For example, Australia proposes a “Water Markets” plan that allows individuals and organizations to buy and sell “water rights” [2]; the Chinese government has been committed to promoting and accelerating the construction of water conservancy facilities [3]. These measures can partly alleviate the output decline of crops caused by drought; however, the uncertain yield will further make the producers and retailers in crop supply chains suffer economic losses.

China is a predominantly agricultural country with a vast territory and complex natural conditions. It is one of the countries with the most severe natural disasters in the world [4]. According to the China Statistical Yearbook statistics, from 2010 to 2017 [5], the average annual crop damage area reached 27.196 million hectares, among which approximately 11% of the land had no harvest. In recent years, the crop-affected area caused by drought has been decreasing, which may result from continuous improvement of the field irrigation system. However, drought is still the most severe disaster affecting maize yield [6], and maize growing in the northeastern region, especially in Heilongjiang Province, suffers more severe effects from drought than other regions in China [7].

This paper takes Heilongjiang Province, one of China’s main maize-producing areas, as an example, where drought is the most severe meteorological disaster faced by maize production. The maize growth requires favorable rainfall conditions. However, drought causes insufficient rainfall for maize to grow, resulting in lower output than a planned production quantity. It means that unfavorable weather is accompanied by an output loss for a given input; thus, the maize growers must pay for the production costs against the planned production quantity, whether or not drought occurs. In this context, we designed a two-tier maize supply chain consisting of a supplier and a retailer under the background of contract farming, which is currently a popular form of farm production and sales in China [8]. The supplier grows crops while the retailer buys produce from the supplier and sells them to the market. In this case, the grower as a supplier will transfer the low yield risk due to drought to the demand side through the supply chain in the form of under-producing issues. Meanwhile, the retailer faces market demand risk, resulting in under-ordering issues. The yield and demand risk will hinder the operational efficiency of the supply chain, leading to the supplier and retailer’s profit being below the optimal level. As a result, the cooperative relationship between them has become unstable, which further hampers the sustainable development of the agricultural supply chain [9,10]. In these circumstances, contract mechanisms are introduced to coordinate the chain for improving its performance and constrain the profit distribution between the supplier and the retailer to avoid opportunistic behavior.

Therefore, we aim to explore the adverse impact of drought on maize yield, mainly represented by insufficient rainfall, and its influence on the entire chain’s performance and the members’ profit allocation by contract mechanism. Based on this, two issues need to be resolved. First, we model the relationship between the drought severity represented by rainfall indicators and the maize meteorological yield based on historical data in the Heilongjiang Province of China. Second, the uncertain maize meteorological yield is introduced into the agricultural supply chain, which results in under-produce issues hampering the supply chain coordination; thereby, we design a contract mechanism for coordinating the chain by modifying traditional revenue sharing contracts. For the latter case, centralized and decentralized decision-making models are built for comparative analysis, the optimal solutions of which, respectively, designate the maximum profit under supply chain coordination and the minimum profit under non-coordination; then, two contract mechanisms, i.e., payback revenue sharing (PBRS) and cost revenue sharing (CRS) contracts, which are modified based on revenue sharing contracts, are developed to coordinate the maize supply chain. Consequently, a comparative analysis can be conducted to investigate the profit allocation of different contracts between the supply chain members, thus improving the members’ profits and stimulating the entire supply chain’s sustainable cooperation and development.

With respect to yield risk faced by the supply chain’s upstream members, previous researches focus on the uncertain supply in the manufacturing and transportation processes, where a stochastically proportional yield (SPY) model is commonly used to depict the yield uncertainty [11,12]. For yield uncertainty originating from weather-related factors, most relevant studies also take the form of an SPY, the same as in the manufacturing industry. However, the influences of various disasters on different crops are not precisely the same (e.g., relevant weather indices, frequency of occurrence, the extent of damage, etc.). This paper targets to capture the maize yield law affected by drought leading to insufficient rainfall and thus mitigate its negative effect on yield risk allocation in the maize supply chain, which may be classified under the second level uncertainty of Walker et al. [13]. Hence, differing from these studies, we statistically simulate the relationship between rainfall and maize yield based on the historical data of Heilongjiang Province to quantify the adverse impact of drought on maize output practically. In this context, we define the maize meteorological yield and then incorporate the relationship between rainfall and the maize yield into a contract mechanism to coordinate the maize supply chain, which can be further applied in practice and make more sense. This paper practically simulates the relationship between rainfall and maize output, which brings benefits to exploring the law of the drought’s influence on maize growing. Based on this, two modified revenue sharing contracts are developed, complementing existing research on revenue sharing contracts in the presence of uncertain yield and stochastic demand.

Concerning supply chain coordination, most studies revolve around the contract design based on the Newsboy model [14] and focus on demand and supply uncertainties. In the case of demand uncertainty, studies focus on supply chain coordination using classic contracts, such as revenue sharing and buyback contracts [15,16], or their composites [17,18,19]. However, when facing yield and demand risk, He and Zhao [20] found that classic contracts (i.e., wholesale price, buyback, and revenue sharing contracts) failed to achieve supply chain coordination because of the inability to reallocate yield risk. In contrast, Güler and Keskin [21] found that yield uncertainty did not influence the coordination ability of buyback, revenue sharing, quantity discount, and quantity flexibility contracts, except for wholesale price contracts under the supply chain facing random yield and demand. But the contract structure and parameters differed from the traditional contract arrangements. Previous research implies that although classic contracts cannot coordinate supply chains in the presence of yield and demand risk, redesigning the contract structures and parameters or combining them can resolve the problem. These methods for modifying classic contracts have also been studied in the field of agricultural supply chains, especially in cases where uncertain yields were caused by adverse weather conditions. Zhao and Wu [22] investigated the influence of season and weather factors and used a revenue sharing contract to coordinate a fresh produce supply chain in the case of random yield and demand. To solve the problem that the output and quality of agricultural products are affected by uncontrollable weather and controllable manufacturers’ effort level, Dan et al. [23] proposed a combination contract of risk sharing and surplus buyback. Fu et al. [24] combined a revenue sharing contract with rainfall index insurance and a risk transfer fee to correct the supply side’s distortion from severe weather.

In rural China’s contract farming circumstances, revenue sharing contracts generally apply in agricultural supply chain practices [25,26,27]. However, as pointed out above, classic revenue sharing contracts cannot coordinate the supply chain with both yield and demand uncertainties. Therefore, many studies propose modifying revenue sharing contracts or combining a traditional contract with other contract mechanisms [26,28,29]. With special consideration given to weather’s impact on contract farming output, Ye et al. developed a mechanism combining revenue sharing, production cost sharing, and guaranteed money contracts to coordinate an agricultural supply chain. Anderson and Monjardino designed a double discount contract to coordinate a three-tier supply chain facing yield uncertainty resulting from the level of fertilizer input and random weather factors.

Table 1. Related literature.

Literature	Uncertain Factors		Agricultural Supply Chain	Coordinating Contract
	Yield (SPY)	Demand
Anderson and Monjardino [11]	√ (√)	×	√	Double discount
Xie et al. [12]	√ (√)	√	×	BBRS
Pang et al. [15]	×	√	×	Improved RS
Li et al. [16]	×	√	×	WP, BB, RS
Wang et al. [17]	×	√	×	BB and promotion CS
He et al. [18]	×	√	×	Returns policy and RS
Fakhrzad et al. [19]	×	√	×	Improved BB
He and Zhao [20]	√ (√)	√	×	Composite contracts
Güler and Keski’n [21]	√ (√)	√	×	WP, BB, RS, QD, QF
Zhao and Wu [22]	√ (√)	√	√	RS
Dan et al. [23]	√ (√)	√	√	Risk sharing and surplus BB
Fu et al. [24]	√ (×)	√	√	Insurance + RS + risk transfer fee
Ye et al. [26]	√ (√)	√	√	RS + production CS + guaranteed money
Luo and Chen [28]	√ (√)	√	×	RS and surplus subsidy
Tang and Kouvelis [29]	√ (√)	√	×	PBRS
This paper	√ (×)	√	√	PBRS, CRS

Notations: stochastically proportional yield (SPY); wholesale price (WP) contract; buyback (BB) contract; revenue sharing (RS) contract; cost sharing (CS); quantity discount (QD) contract; quantity flexibility (QF) contract; buyback revenue sharing (BBRS) contract; payback revenue sharing (PBRS) contract; cost revenue sharing (CRS) contract.

summarizes the related literature to this paper as far as we know. In terms of modeling yield risk in the field of supply chain coordination, most literature made use of the SPY model except Fu et al. [24], who did not describe the detailed yield function; all studies are different from this paper. The most relevant to our research is Tang and Kouvelis [29]. They constructed a payback revenue sharing contract, which dealt with the supplier’s under-producing and the retailer’s over-ordering issues due to uncertain yield and demand, respectively. In our analysis, however, the supplier suffers a lower yield than the planned production quantity due to rainfall deficiency from drought. Accordingly, we intend to develop a PBRS contract with different contract structures. The retailer provides a payback price for the actual yield or delivered quantity to guarantee the supplier’s willingness to participate in the chain. Meanwhile, all the retailer’s sales revenue is shared with the supplier to correct under-ordering issues. Furthermore, Tang and Kouvelis pointed out that a cost sharing contract is similar to a revenue sharing contract in the presence of uncertain yield and is essentially equivalent to a payback contract. Considering that the CRS contract is easier to operate in practice than the PBRS one and is a more direct incentive measure for suppliers, this paper also designs a CRS contract in which the retailer shares a portion of the supplier’s cost for all the planned production quantities besides revenue sharing. Then, a comparative analysis can be conducted to investigate the influence of contract structures and parameters on the coordination ability and their influence on profit allocation between supply chain members, thus guiding the practice of maize supply chain coordination.

The rest of this paper is organized as follows. Section 2 analyzes the drought disaster and crop planting situation of Heilongjiang Province in China and quantifies the effect of drought on maize yield. Section 3 introduces the notations, assumptions, and benchmark models in centralized and decentralized systems. Section 4 proposes a payback revenue sharing contract and a cost revenue sharing contract based on the traditional revenue sharing contract. Then, the quantitative effect of drought on maize yield is incorporated into these modified revenue sharing contracts to coordinate the maize supply chain. Section 5 conducts a numerical analysis to verify the results of the above-modified contracts. Finally, Section 6 makes concluding remarks.

2. The Impacts of Drought on Maize Yield

In this subsection, we first solve our first research question, i.e., quantifying the drought’s influence on maize output, i.e., maize meteorological yield. Then, the quantitative relationship between the maize meteorological yield and the drought degree measured by deficient rainfall will be introduced into designing a contract mechanism, including the centralized and decentralized decision-making systems in Section 3 and coordinating contracts, i.e., PBRS and CRS contracts in Section 4 to explore the impact of drought on the performance and profit allocation of the maize supply chain.

In this study, rainfall data are used to depict the degree of drought and then measure their effect on yield. Fang [30], Lobell et al. [31], and Wang et al. [32] proposed a similar method for measuring crop meteorological yield, subtracting trend yield from total crop output. The actual amount of crop harvest measures the entire crop output. The trend yield usually originates from technological progress and is calculated by historical annual crop output quantity data. In addition, Wang et al. [32], taking rice planting in the Heilongjiang Province of China as an example, compared different fitting methods, and it was found that the orthogonal polynomials method had a better fitting effect. Following [30,31,32], in our study, the total maize yield of Heilongjiang Province $Y_T$ is the sum of trend yield and meteorological yield $Y_w$ due to the effect of rainfall, and $\epsilon$ designates the output error affected by other unpredictable factors. The relationship among them can be written as:

$$Y_T=Y_t+Y_w+\epsilon ,$$

(1)

The orthogonal polynomial method [33] is one of the commonly used methods in research similar to ours [32], which has a higher goodness of fit than other methods for calculating maize meteorological yield. Thus, following [32], we adopt an orthogonal polynomial method to obtain trend yield by fitting the time and the maize yield per unit area (100,000 tons per hectare) in Heilongjiang Province from 1998 to 2017. Then, the trend yield function can be expressed as:

$$Y_t=-1.177t^3-44.872t^2-372.623t+5148.824, t=1,2,\cdots \cdots ,20,$$

(2)

Regarding the statistical test results, when the significance level is set as 0.05, $p=0.00<0.05$ means that the null hypothesis is rejected for all variables. The coefficient of determination $R^2=0.929$ signifies that the goodness of fit is good. Figure 1 illustrates the fitting curve of trend yield $Y_t$. Furthermore, we obtain the meteorological yield by subtracting the trend yield from the total output from 1998 to 2017 in Equation (1), the results of which are used to gain Equation (3). The values of the meteorological yield can be plus or minus, indicating a positive or negative effect of rainfall on maize yield in that year, respectively. For example, in our analysis, we take the adverse weather of drought as an example, which corresponds to the negative effect of rainfall deficiency.

The maize growth cycle in Heilongjiang Province is from early May to early October. Hence, the monthly rainfall from May to September is selected, and the sum of which forms an annual total rainfall. Then, the yearly rainfall in the 20 successive years from 1998 to 2017 is calculated to do least-squares fitting with the corresponding meteorological yield by orthogonal polynomials. We assume that if the rain is zero, the total output is zero. Given the significance level and fitting accuracy, we obtain the meteorological yield in Equation (3), which depicts the change in the maize yield (kilogram per hectare, kg/ha) caused by the increase in rainfall at one millimeter (mm). Figure 2 shows the results in the form of a fitting curve.

$$Y_w=-0.031m^2+26.831m-5696.993,$$

(3)

As shown in Figure 2, the maize meteorological yield $Y_w$ is concave in rainfall $m$. When the rain exceeds the favorable range for maize growing, its output is below zero; otherwise, the yield is above zero. Furthermore, rainfall deficiency designates the occurrence of a drought disaster in our analysis, while its corresponding case is a flood disaster not covered by our research.

3. Model Description and Benchmark Models for Contract Mechanism

3.1. Model Description

We consider an agricultural supply chain in which one supplier produces a single crop (i.e., maize) and sells it to a downstream retailer. If the maize yield decreases as the degree of drought increases (reduced rainfall), the retailer faces a random demand, and orders from suppliers face an uncertain output. Therefore, in order to minimize the losses that drought may bring to the two parties in the agricultural supply chain, the total maize yield function (as shown in Equation (1)) is incorporated into the coordinating mechanism. Further, we suppose both players are risk neutral and have common knowledge. For simplicity, the retailer has no inventory costs, goodwill costs, or product salvage value, which will not change the analysis results. Based on Stackelberg game theory [14], the sequence of events is as follows:

• Before the growing season, the supplier publishes a unit wholesale price $w$;
• Given the wholesale price, the retailer decides the order quantity $q$ units;
• Without loss of generality, it is assumed that the supplier takes the retailer’s order quantity as the planned production quantity and one unit of the supplier’s production needs one unit of input; that is, the ratio of input to planned production quantity is 1:1. Thus, $q$ units are also the supplier’s input quantity;
• After harvest, with the impact of rainfall on the yield, the actual maize yield $Q$ units are defined as $Q=q+Y_w$, all of which are delivered to the retailer;
• The market demand $D$ is realized, and the retailer sells the realized maize yield $Q$ at a unit market price $p$.
• The notations are defined as follows:

$x$: stochastic market demand, with probability density function (PDF) $f\left(x\right)$ and cumulative distribution function (CDF) $F\left(x\right)$, which is an increasing function of $x$;

$Q$: the actual yield of the supplier as decided by the rainfall;

$q$: the retailer’s order quantity or the supplier’s planned production quantity; it also equals the supplier’s input quantity;

$p$: retail price of the retailer;

$c$: per-unit production cost of the supplier’s planned production quantity;

$w$: wholesale price of the supplier;

$w_d$: wholesale price in the decentralized system;

$w_1$: wholesale price of the supplier under a payback revenue sharing contract;

$w_2$: wholesale price of the supplier under a cost revenue sharing contract;

${\varphi }_1$: the revenue share earned by the supplier from the retailer’s sales under a payback revenue sharing contract;

${\varphi }_2$: the revenue share made by the supplier from the retailer’s sales under a cost revenue sharing contract;

$w_b$: the retailer gives a payback price for the supplier per unit realized yield under a payback revenue sharing contract;

$\gamma$: the retailer shares a portion of the supplier’s total production cost under a cost revenue sharing contract;

${\Pi }_T^c$: the total expected profit of the centralized supply chain;

${\Pi }_T^d$: the total expected profit of the decentralized supply chain with a wholesale price contract;

${\Pi }_S^d$: the supplier’s expected profit for the supplier under the wholesale price contract;

${\Pi }_R^d$: the retailer’s expected profit for the retailer under the wholesale price contract;

${\Pi }_S^{cr}$: the supplier’s expected profit for the supplier under the CRS contract;

${\Pi }_R^{cr}$: the retailer’s expected profit for the retailer under the CRS contract;

${\Pi }_S^{pr}$: the supplier’s expected profit for the supplier under the PBRS contract;

${\Pi }_R^{pr}$: the retailer’s expected profit for the retailer under the PBRS contract.

To avoid uninteresting trivial, we assume $c<w<p$ to ensure that the players can make profits and thus are willing to stay in the market.

3.2. The Centralized Benchmark Model

We introduce the benchmark model in a centralized setting that aims at maximizing the expected profits of the integrated agricultural supply chain. When the supply chain is coordinated, all members’ total profit equals that in the centralized system. In the benchmark model, the maize supply chain makes decisions as a whole without order quantity and wholesale price. It produces $Q$ units and sells them at a unit price $p$ in the end consumer market. The realized yield $Q$ depends on the input quantity and meteorological yield $Y_w$. Therefore, the total expected profit of the entire agricultural supply chain ${\Pi }_T\left(q\right)$ is:

$$\begin{array}{cc}{\Pi }_T^c\left(q\right)& =E\left[p\min (Q,x)-cq\right]\hfill \\ & =pS\left(q\right)-cq\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em},\hfill \end{array}$$

(4)

where $S\left(q\right)=E\min \left(Q,x\right)=E\min \left(q+Y_w,x\right)={\displaystyle \underset{0}{\overset{q+Y_w}{\int }}xf\left(x\right)dx+{\displaystyle \underset{q+Y_w}{\overset{\infty }{\int }}\left(q+Y_w\right)f\left(x\right)}}dx$ is the expected sales volume. The first term designates the expected sales in the end consumer market. The second term represents the input costs measured against the planned production quantity regardless of whether or not an actual yield is obtained.

Using Equation (4), we can obtain the first-order derivative of $q$ $\frac{d{\Pi }_T^c}{dq}=p\left(1-F\left(q+Y_w\right)\right)-c$. Then, the second-order condition is obtained $\frac{d^2{\Pi }_T^c}{d{q}^2}=-pf\left(q+Y_w\right)<0$, which ensures that the first-order condition equals zero; thus, the following proposition can be put forward.

Proposition 1.

In the centralized system, the expected profit of the integrated supply chain ${\Pi }_T^c$ is concave in the planned production quantity $q$ and maximized by the unique optimal planned production quantity $q_c^{*}$:

$$q_c^{*}=F^{-1}(\frac{p-c}{p})-Y_w,$$

(5)

Substituting Equation (5) into Equation (4), we obtain the optimal profit for the entire supply chain:

$${\Pi }_T^c=P{\displaystyle \underset{0}{\overset{q_c^{*}+Y_w}{\int }}xf\left(x\right)dx}+c{Y}_w,$$

(6)

3.3. The Decentralized Model: A Wholesale Price Contract

In the decentralized supply chain, the supplier and retailer pursue their maximum profit, resulting in a double marginalization problem that makes the integrated chain’s profit lower than in the centralized system. The decentralized model is constructed under a wholesale price contract. The supplier offers the wholesale price $w_d$. Next, the retailer decides on the order quantity $q$ and places an order with the supplier. Then, the supplier determines the planned production quantity $q$ and corresponding input. After harvesting the crop, the supplier delivers the realized yield to the retailer. Finally, the retailer sells the minimum amount between the realized yield $Q$ and the realized demand $x$.

By backward induction, we first solve the retailer’s problem, and the expected profit for the retailer is expressed as:

$$\begin{array}{cc}{\Pi }_R^d\left(q\right)& =E\left[p\min \left(Q,x\right)-w_{d}Q\right]\hfill \\ & =pS\left(q\right)-w_d\left(q+Y_w\right)\hspace{1em},\hfill \end{array}$$

(7)

The first and second terms designate the retailer’s sales revenue and purchase costs. It is easy to know that the second-order condition satisfies $\frac{d^2{\Pi }_T^d}{d{q}^2}=-pf\left(q+Y_w\right)<0$. Hence, the first-order derivative of $q$, $\frac{d{\Pi }_T^d}{dq}=p(1-F(q+Y_w))-w$, equals zero. Then, the following proposition is straightforward.

Proposition 2.

Given the wholesale price $w_d$, the retailer’s expected profit is maximized, and the optimal order quantity is uniquely solved by:

$$q_d^{*}=F^{-1}\left(\frac{p-w_d}{p}\right)-Y_w,$$

(8)

Substituting Equation (8) into Equation (7), we obtain the optimal expected profit of the retailer:

$${\Pi }_R^d=p{\displaystyle \underset{0}{\overset{q_d^{*}+Y_w}{\int }}xf\left(x\right)}dx,$$

(9)

The supplier’s expected profit is the revenue that sells the actual yield to the retailer at a wholesale price $w_d$ after deducting the input costs, as shown in Equation (10).

$$\begin{array}{cc}{\Pi }_S^d& =E\left(w_{d}Q-cq\right)\hfill \\ & =w_d\left(q+Y_w\right)-cq,\hfill \end{array}$$

(10)

Given the retailer’s optimal order quantity $q_d^{*}$, the optimal expected profits of the supplier and the entire supply chain in the decentralized system are decided as:

$${\Pi }_S^d=w_d(q_d^{*}+Y_w)-c{q}_d^{*},$$

(11)

$${\Pi }_T^d=p{\displaystyle \underset{0}{\overset{q_d^{*}+Y_w}{\int }}xf\left(x\right)}dx+w_d\left(q_d^{*}+Y_w\right)-c{q}_d^{*},$$

(12)

Comparing Equation (8) with Equation (5), we can find $q_d^{*}<q_c^{*}$ under the assumption $w_d>c$. It suggests that the optimal expected profit of the integrated supply chain in the decentralized system is lower than in the centralized system. Therefore, to coordinate the agricultural supply chain, new contract mechanisms are needed to give incentives to improve the order quantity and tackle the impact of drought on crop yield.

4. Coordination under the Modified Revenue Sharing Contracts

This section investigates two coordinating mechanisms by modifying the traditional revenue sharing contracts. A cost sharing contract is similar to a payback revenue sharing contract for the yield risk real location. However, the structures and parameters of the two combined contracts are not identical, even if they are modified based on revenue sharing contracts. The purpose is to explore a more favorable coordinating mechanism for guiding practice.

4.1. Coordination under a Payback Revenue Sharing Contract

In this subsection, we consider a payback revenue sharing contract, which is first proposed by Tang and Kouvelis [29]. Their study uses a payback price to correct the supplier’s under-producing problem, and the retailer only shares partial sales revenue with the supplier. However, in our analysis, the supplier suffers a lower yield than the planned production quantity due to rainfall deficiency from drought. Therefore, given the wholesale price $w_d$, the retailer gives the supplier a payback price $w_b$ (lower than the wholesale price $w_d$ or even lower than the production cost $c$) for the realized output $Q$ to cope with lost input costs. In this way, the retailer shares the supplier’s risk of excessive actual production costs, ensuring that the supplier is profitable for growing maize. Meanwhile, in order to correct the retailer’s under-ordering problem, the supplier offers a lower wholesale price $w_1$ to the retailer, and the retailer pays a revenue share ${\varphi }_1$ of his total sales to the supplier. Under the payback revenue sharing contract, coordinating the agricultural supply chain needs to adjust parameters $w_1$, $w_b$, and ${\varphi }_1$. We first deal with the retailer’s problem, and his profit function is expressed as follows:

$$\begin{array}{cc}{\Pi }_R^{pr}& =E\left[(1-{\varphi }_1)p\min \left(Q,x\right)-w_{1}Q-w_{b}Q\right]\\ & =(1-{\varphi }_1)pS\left(q\right)-\left(w_1+w_b\right)\left(q+Y_w\right),\end{array}$$

(13)

The first term designates the retailer’s sales revenue deducted by the share portion paid to the supplier. The second and third terms are the retailer’s purchase costs and the payback subsidy to the supplier’s realized yield. Using Equation (13), we can obtain the first- and second-order conditions as $\frac{d{\Pi }_R^{pr}}{dq}=\left(1-{\varphi }_1\right)p\left(1-F\left(q+Y_w\right)\right)-w_1-w_b$ and $\frac{d^2{\Pi }_R^{pr}}{d{q}^2}=-\left(1-{\varphi }_1\right)pf\left(q+Y_w\right)<0$.

It is straightforward that the optimal order quantity of the retailer can be uniquely solved by the first-order condition, which is $q_1^{*}=F^{-1}\left(\frac{(1-{\varphi }_1)p-\left(w_1+w_b\right)}{(1-{\varphi }_1)p}\right)-Y_w$. Compared to the planned production quantity $q_c^{*}$ in the centralized benchmark model, let $q_1^{*}=q_c^{*}$, and coordinating the agricultural supply chain must satisfy $\frac{p-c}{p}=\frac{(1-{\varphi }_1)p-\left(w_1+w_b\right)}{(1-{\varphi }_1)p}$. Rewriting the above coordinating condition, the relationship between $w_1$, $w_b$, and ${\varphi }_1$ is determined as follows:

$$w_1=(1-{\varphi }_1)c-w_b,$$

(14)

Then, we can come up with the following proposition.

Proposition 3.

${\Pi }_R^{pr}$ is concave in the retailer’s order quantity. The optimal order quantity $q_1^{*}=F^{-1}\left[\frac{(1-{\varphi }_1)p-\left(w_1+w_b\right)}{(1-{\varphi }_1)p}\right]$ maximizes the retailer’s profit; a payback revenue sharing contract can coordinate the agricultural supply chain if the coordinating parameters satisfy Equation (16).

On the supply side, under the payback revenue sharing contract, the supplier’s profit function is expressed as follows:

$$\begin{array}{cc}{\Pi }_S^{pr}& =E\left[{\varphi }_{1}p\min (Q,x)+w_{1}Q+w_{b}Q-cq\right]\\ & ={\varphi }_{1}pS\left(q\right)+\left(w_1+w_b\right)\left(q+Y_w\right)-cq\hspace{1em},\end{array}$$

(15)

Substituting the optimal order quantity $q_1^{*}$ and Equation (14) into Equations (13) and (15), respectively, the maximum profits of the supplier and the retailer can be written as:

$${\Pi }_R^{pr}=\left(1-{\varphi }_1\right)p{\displaystyle \underset{0}{\overset{q_1^{*}+Y_w}{\int }}xf\left(x\right)}dx,$$

(16)

$$\begin{array}{cc}{\Pi }_S^{pr}& ={\varphi }_{1}p{\displaystyle \underset{0}{\overset{q_1^{*}+Y_w}{\int }}xf\left(x\right)}dx+\frac{1}{1-{\varphi }_1}\left(w_1+w_b\right)\left(q_1^{*}+Y_w\right)-c{q}_1^{*}\hfill \\ & ={\varphi }_{1}p{\displaystyle \underset{0}{\overset{q_1^{*}+Y_w}{\int }}xf\left(x\right)}dx+c{Y}_w\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em},\hfill \end{array}$$

(17)

In the decentralized setting, both the contracting parties independently pursue their own optimal expected profit leading to the retailer’s lower order quantity and the supplier’s lower production quantity than those in the centralized system. Proposition 3 shows that if the supplier and the retailer accept the payback revenue sharing contract, the maximum profit of the entire supply chain is equal to the centralized setting. The supplier and the retailer can arbitrarily allocate the optimal profit by adjusting coordination contract parameters. In addition, the supplier and retailer’s profits should be no less than the wholesale price contract. It is evident that if $w_b=0$, the payback revenue sharing contract is the same as a revenue sharing contract.

4.2. Coordination under a Cost Revenue Sharing Contract

In this subsection, we consider a cost revenue sharing contract. Tang and Kouvelis [29] point out that a cost sharing mechanism is equivalent to a payback one, as a buyback contract is equivalent to a revenue sharing one. In our analysis, the supplier suffers a lower yield than the planned production quantity due to rainfall deficiency from drought. We aim to mitigate the economic losses on the maize supplier resulting from drought. Hence, to seek a more beneficial mechanism to guarantee and improve the supplier’s revenue, we will compare the cost revenue sharing contract with the payback revenue sharing contract with a numerical analysis. Under the cost revenue sharing contract, the retailer shares part of the retailer’s production costs depending on the planned output instead of the actual yield in the case of the payback revenue sharing mechanism. Then, the retailer’s profit function is:

$$\begin{array}{cc}{\Pi }_R^{cr}& =E\left[\left(1-{\varphi }_2\right)p\min \left(Q,x\right)-w_{2}Q-\gamma cq\right]\\ & =\left(1-{\varphi }_2\right)pS\left(q\right)-w_2\left(q+Y_w\right)-\gamma cq\hspace{1em},\end{array}$$

(18)

In Equation (18), we obtain the first- and second-order derivatives $\frac{d{\Pi }_R^{cr}}{dq}=\left(1-{\varphi }_2\right)p\left[1-F\left(q+Y_w\right)\right]-w_2-\gamma c$ and $\frac{d^2{\Pi }_R^{cr}}{d{q}^2}=-\left(1-{\varphi }_2\right)pf\left(q+Y_w\right)<0$. The second-order derivative is less than zero, which guarantees that the first-order derivative is equal to zero. Compared to the case in the centralized system, coordinating the chain must satisfy:

$$w_2=\left(1-{\varphi }_2-\gamma \right)c,$$

(19)

Then, it is straightforward to obtain the following proposition.

Proposition 4.

${\Pi }_R^{cr}$ is concave in the planned production quantity $q$, and the retailer achieves maximum profit by the retailer’s optimal $q_2^{*}=F^{-1}\left[\frac{(1-{\varphi }_2)p-\left(w_2+\gamma c\right)}{(1-{\varphi }_2)p}\right]$. The cost revenue sharing contract can coordinate the agricultural supply chain if  $w_2=\left(1-{\varphi }_2-\gamma \right)c$.

The supplier’s profit function is:

$$\begin{array}{cc}{\Pi }_S^{cr}& =E\left[{\varphi }_{2}p\min \left(Q,x\right)+w_{2}Q+\gamma cq-cq\right]\\ & ={\varphi }_{2}pS\left(q\right)+w_2\left(q+Y_w\right)-\left(1-\gamma \right)cq\hspace{1em},\end{array}$$

(20)

Substituting the retailer’s optimal order quantity $q_2^{*}$ and Equation (19) into Equations (18) and (20), we can obtain the maximum profits for the retailer and supplier, respectively:

$${\Pi }_R^{cr}=\left(1-{\varphi }_2\right)p{\displaystyle \underset{0}{\overset{q_2^{*}+Y_w}{\int }}xf\left(x\right)}dx+\gamma c{Y}_w,$$

(21)

$${\Pi }_S^{cr}={\varphi }_{2}p{\displaystyle \underset{0}{\overset{q_2^{*}+Y_w}{\int }}xf\left(x\right)dx}+\left(1-\gamma \right)c{Y}_w,$$

(22)

Equations (16) and (17) show that the supplier mainly bears the yield uncertainty from drought under the payback revenue sharing contract. In contrast, the retailer assumes partial yield risk under the cost revenue sharing contract (see Equations (21) and (22)). Using Propositions 3 and 4, the above two contracts can coordinate the agricultural supply chain in the case of rainfall deficiency due to drought. However, the arbitrary profit allocations between the supplier and retailer are slightly different under the two coordinating mechanisms.

It is worth noting that if $\gamma =0$, the cost revenue contract is the same as a revenue sharing contract, which is a special case of the above coordinating mechanisms. A PBRS mechanism is similar to a CRS one. However, from the perspective of reallocating yield risk, the former emphasizes compensation for the realized order quantity, and the latter underlines cost sharing for the planned production amount.

5. Numerical Analysis

In this section, we make a numerical example to intuitively illustrate the main results of this study and gain further insights. Our paper looks for a favorable mechanism to coordinate the agricultural supply chain with uncertain yield. We take a maize supply chain as an example where under-production results from insufficient rainfall as an adverse effect of drought in Heilongjiang Province. Based on the analysis in Section 3, we assign the rainfall a benchmark value of 307 mm (the total rain from May to September). Thus, the meteorological yield of Heilongjiang Province’s maize output $Y_w$ kilograms per hectare (kg/ha) is negative, representing drought’s adverse effect on maize output. Every year, there is almost no surplus of maize and a slight change in the demand for maize in Heilongjiang Province. Hence, we assume the minimum and maximum maize yield per hectare from 1998 to 2018 in Heilongjiang Province as the lower and upper bound of market demand, respectively. It means that the random demand $D$ kg/ha is uniformly distributed over an interval $\left[3842,6317\right]$ (these data are from the China Statistical Yearbook, 1999–2018). Furthermore, it is assumed that the market price is $p=1.9$ CNY/kg and the planned production costs are $c=0.8$ CNY/kg. The prices and costs are the average values calculated from 2010 to 2018 (these data are from the National Development and Reform Commission of China, 2011–2018 [34]).

In Figure 3, ${\sigma }_D$ designates the standard deviation of the random demand $D$, and $q_c^{*}$ and $q_d^{*}$, respectively, represent the optimal planned production quantity in the centralized and decentralized systems. Figure 3 illustrates that as demand uncertainty increases, the optimal production quantity increases under a centralized system but decreases under a decentralized system. In Figure 4, the rainfall rising on the horizontal axis designates the decline of yield uncertainty. Recall that it is assumed in Section 3.1 that the ratio of input to planned production quantity is 1:1. Thus, $q$ units are both the supplier’s input quantity and planned production quantity, and $Q/q$ represents the supplier’s input–output ratio. As the yield uncertainty increases, the input–output ratio of a decentralized system decreases faster than a centralized system. In the decentralized supply chain, a wholesale price contract is agreed between the supplier and the retailer. Figure 3 and Figure 4 indicate that both the optimal production quantity and the input–output ratio in the decentralized system are lower than the centralized system, signifying that a wholesale price contract cannot coordinate the agricultural supply chain facing uncertain yield and demand. The new coordination mechanisms need to correct the under-producing and under-ordering issues.

Therefore, the coordinating abilities of the modified revenue sharing contracts need to be verified by comparing them with the results of the decentralized model. First, we want to observe the impact of demand uncertainty on the members’ profits and the integrated supply chain. In Figure 5, the larger the value of ${\sigma }_D$ is, the more uncertain the demand is. In this case, the entire supply chain’s profit declines with or without coordination. Further, the profits of the supplier or/and retailer under the payback revenue sharing or the cost revenue sharing contract are higher than under the wholesale price contract, respectively (i.e., ${\Pi }_S^{pr}>{\Pi }_S^d$, ${\Pi }_S^{cr}>{\Pi }_S^d$; ${\Pi }_R^{pr}>{\Pi }_R^d$, ${\Pi }_R^{cr}>{\Pi }_R^d$). Furthermore, the supplier’s profit under the cost revenue sharing contract is slightly higher than under the payback revenue sharing contract (i.e., ${\Pi }_S^{cr}>{\Pi }_S^{pr}$). This is because that under the same wholesale price (i.e., $w_1=w_2$) the retailer shares the costs of the planned production quantity under the former contract rather than the costs of the realized yield under the latter contract.

In contrast, the retailer’s profit under the cost revenue sharing contract is slightly lower than under the payback revenue sharing contract. Referring to He and Zhao (2012), supply chain efficiency is defined as a percentage increase in the entire supply chain profit with coordination compared to without coordination, which is calculated by $\Delta {\Pi }_T^c\%=\frac{{\Pi }_T^c-{\Pi }_T^d}{{\Pi }_T^d}\times 100\%$. Figure 6 indicates that the supply chain efficiency increases as the demand becomes more uncertain, and the entire chain benefits from coordination.

Figure 7 and Figure 8 illustrate how drought rainfall deficiency affects the players’ profits and the integrated supply chain. In Figure 7, whether the supply chain is coordinated or not, the yield uncertainty declines as the rainfall increases, resulting in a rise in the entire chain and the members’ profits, respectively, and vice versa. The supplier and retailer benefit more from the coordination mechanisms (i.e., the PBRS and CRS contracts) than the uncoordinated contract (i.e., the wholesale price contract) as shown in Figure 7 (i.e., ${\Pi }_S^{pr}>{\Pi }_S^d$, ${\Pi }_R^{pr}>{\Pi }_R^d$; ${\Pi }_S^{cr}>{\Pi }_S^d$, ${\Pi }_R^{cr}>{\Pi }_R^d$). In addition, the supplier gains more profit allocation from the PBRS contract than the CRS contract (i.e., ${\Pi }_S^{pr}>{\Pi }_S^{cr}$), while the retailer faces the opposite situation (i.e., ${\Pi }_R^{cr}>{\Pi }_R^{pr}$). In Figure 8, from a converse view, the drop in rainfall means the drought is worsening, corresponding to an increasing supply chain efficiency. It designates that the PBRS and CRS contracts effectively coordinate the agricultural supply chain with yield uncertainty.

6. Conclusions

Extreme weather events (e.g., drought disasters) are inevitable and harm crop harvest. They will further provoke economic loss for an agricultural supply chain in the form of uncertain yield. Therefore, it is essential to consider quantifying the adverse impact of extreme weather events on crop output to achieve supply chain coordination. This paper studied how to coordinate agricultural supply chains where the supplier faced weather-related yield uncertainty (i.e., drought disaster results in deficient rainfall and thus leads to uncertain maize yield) while the retailer confronted random demands. The randomness on the demand side tends to cause a low order quantity issue and is widely studied in the literature.

Similarly, uncertain yields are prone to bring about an under-producing problem, taking the form of a stochastic proportional model in the previous literature. However, different disasters’ influences on various crops are not identical. Hence, in this study, we focused on simulating and modeling the effect of rainfall due to drought on the maize yield with historical data to practically capture the rule of weather-related maize yield. In this case, under a wholesale price contract, crop growers, as a supply side of the chain, transfer the yield risk downstream by producing less than in the centralized system, the same as those retailers who pass demand risk upstream from the chain by ordering a lower quantity than in the centralized system. Therefore, we should design coordinating contracts to correct the supplier and the retailer’s distorted behaviors in the decentralized system to achieve optimal decisions like in the centralized system.

Building upon the above analysis, this study considered two combined coordinating contracts based on a classic revenue sharing contract. As in the previous literature, we used a revenue sharing contract to share the retailer’s demand risk with the supplier. Regarding yield risk, under a PBRS contract, the retailer gives a payback price for the received quantity to share the supplier’s yield risk. In contrast, the retailer shares a portion of the supplier’s planned production costs under a CRS contract rather than the realized quantity. We found that both combined contracts can coordinate the maize supply chain. The prototype of these two contracts is widely observed in the contract farming supply chain with cooperative participation. However, empirical observation finds that when unfavorable weather reduces crop harvest, crop growers are unwilling or unable to take yield risk and tend to withdraw from the chain. In this case, we designed payback price and cost sharing mechanisms to solve yield risk sharing issues and thus investigated the influence of weather-related low output on supply chain coordination. The results indicated that declining rainfall, representing increasing drought severity, negatively affected the members’ profits and the entire supply chain; however, the supply chain efficiency was raised under both coordinating contracts. In addition, the yield risk sharing proportion was higher under a CRS mechanism than a PBRS one, which gave more incentives for the vulnerable supplier to participate in the supply chain.

This study sheds new light on quantifying the yield uncertainty in an agricultural supply chain triggered by extreme weather events. The extension of incorporating the specific impact of meteorological disasters into the supply chain coordination will give more feedback to the practical application. In future work, other classical or combined contracts can be introduced to coordinate the agricultural supply chain considering the adverse effect of different disasters on different crops. Future research may also consider risk-averse suppliers influencing optimal decisions, profit allocation, or contract design. Furthermore, this study found that the maize yield was concave in rainfall. However, we only considered the partial impact of rain on crop yield, which resulted from drought disasters but not flood disasters. Future research can simulate the full effect of rainfall on crop yield and then discuss whether an additive model makes more sense than a multiplicative model (i.e., SPY) in the field of weather-related yield uncertainty. The potential findings will contribute to the literature on coordinating supply chains with random yield and demand. In addition, our study only used historical data to simulate the impact of drought on maize yield, which limits the applicability of the study results. If possible, data collection should be updated, and a more appropriate modeling approach needs to be explored in future research.