# Coordinating a Three-Level Fresh Agricultural Product Supply Chain considering Option Contract under Spot Price Uncertainty

Authors: Changhua Liao; Qihui Lu
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

We consider a three-level fresh agricultural product supply chain consisting of a producer, a supplier, and a retailer, where the producer plants the initial fresh agricultural products with yield uncertainty, the supplier is the leader of the supply chain and the designer of the contracts, and the retailer sells processed products with random demand and spot price. This paper discusses coordination mechanism of wholesale price contracts or option contracts between the supplier and the retailer and wholesale price contracts between the supplier and the producer. Based on this, we not only explore the conditions for supply chain full coordination and Pareto improvement but also analyze the effect of the retailer’s spot market volatility by comparing two scenarios of the retailer with or without a random spot market. Results show that the wholesale price contract cannot coordinate the supply chain regardless of whether the retailer has a random spot market or not. The supply chain can be simultaneously fully coordinated and achieve Pareto improvement by setting reasonable option contract parameters, but the conditions for coordination are rigorous. When the expected value of reduction from the retailer’s spot market in unit shortage loss cost is greater, the total profit of supply chain will increase. Numerical analysis shows that the option exercise price must be set higher when the retailer has a spot market to achieve Pareto improvement. The loss rate in distribution process and other volatility risks will also affect the coordination of the agricultural supply chain.


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## Body

## 1. Introduction

Fresh agricultural products (herein after FAPs), which are regarded as a special perishable product, are indispensable commodities in daily life and have attracted increasing attention from businesses and scholars. Agriculture supply chain is a complex network, including material suppliers, products producers, distributers, retailers, and other members, and the plating is random and the market demand is random. All these elements will affect the operational efficiency of the supply chain. Our study will consider a three-level fresh agricultural product supply chain. The upstream producer usually faces yield uncertainty caused by uncontrollable factors, such as extreme weather and natural disasters in the planting process [1], and the downstream retailer faces the issue of uncertain demand in the sales process. Furthermore, due to the perishable nature of FAPs and the immaturity of fresh-keeping technology, FAPs have high loss in the distribution process [2]. According to statistics, the annual loss rate of FAPs in China is between 25% and 35% [3]. For the ordering and production decisions of the FAP supply chain, it is particularly important to consider the loss rate at each stage.Thus, under the combined effect of the uncertain output, the loss rate, and the uncertain demand, a serious imbalance would emerge between the supply and demand of FAPs in the entire supply chain, which will ultimately damage the profits of all parties and supply chain. It is of great theoretical and practical significance to study the cooperation mechanism among members of the FAP supply chain for reducing supply and demand imbalance and improving the operation efficiency of the supply chain. We consider using the option contract mechanism to solve agricultural problems because the option contract is an effective tool to hedge the uncertain risks of supply chain and is widely used in practice [4]. For example, China Telecom uses call option contract in its purchases, which accounted for 100 billion RMB annually since 2009 [5]. It also represents 35% of HP’s procurement value in 2011 [6].Furthermore, due to the FAPs occupy an extremely important position in the retail consumption market, there are many enterprises or individuals that plant and sell FAPs. In order to make the research more realistic, we model that when the producer’s output is insufficient to meet the order, he can replenish the products from other cooperative enterprises at a determined price. For the retailer, we consider two scenarios: the retailer does not have a spot market and the retailer has a spot market with uncertain price. Rooted in the background above, we discuss the equilibrium decisions of the producer’s production input and retailer’s order quantity, and we tend to study the following questions: in the three-level FAP supply chain, how to design a reasonable supply chain contract to improve the operation efficiency of the supply chain and make supply chain members better off? What is the effect of the loss rate, random volatility risks, and producer’s wholesale and spot price on the coordination of supply chain? When the retailer has a random spot market, and do the conclusions under no random spot market for the retailer still hold? and what is the impact of retailer’s random spot market?To answer the above questions, this paper constructs a three-level FAP supply chain where the supplier as the leader is the designer of contracts in the transaction between the supplier and the retailer and between the producer and the supplier. We will consider two scenarios: in Scenario 1, there is no random spot market for the retailer; in Scenario 2, the retailer can fulfill his order from a random spot market. In the two scenarios, we first study the centralized decision model as the basis for full coordination. We also investigate the wholesale price contract model and analyze the optimal decisions and expected profits of all parties. Then, we introduce the option contract model and compare the optimal decisions and expected profit of the parties with the previous two models to explore the conditions of full coordination and the region of Pareto improvement to maximize the profit of the supply chain or achieve win-win situations. Finally, we study the influence of the retailer’s random spot market on the decisions and profits of supply chain members by combining the two scenarios and analyze the impact of various factors on supply chain coordination by numerical analysis.The rest of this paper is organized as follows. Section2 reviews related literature and summarizes the research contribution. Section 3 describes the three-level FAP supply chain model, notations, and assumptions. Section 4 analyzes the optimal decision and coordination conditions of the supply chain when there is no random spot market for the retailer. Section 5 analyzes the optimal decision and coordination conditions of the supply chain when the retailer has a random spot market and compares the two scenarios. Section 6 presents managerial insights with some mathematical analyses and several numerical examples. Finally, Section 7 summarizes our research results and future research suggestions. All proofs are presented in Appendix B.


## 2. Literature Review

Previous researches in this field can be summarized into the following three aspects: (1) operation and coordination of the FAP supply chain; (2) option contract in supply chain; (3) spot market.


### 2.1. Operation and Coordination of the FAP Supply Chain

In recent years, a large number of scholars studied operation and coordination issues in the FAP supply chain. He et al. [7] constructed a model to study the impact of adopting blockchain technology on the pricing decisions and profits of the global FAP supply chain members. Wu et al. [8] explored the strategies for adopting blockchain technology in the FAP supply chain consisting of one e-tailer, one 3PLs, and one supplier. To explore whether the E-platform chooses reselling or agency selling mode for fresh food in competition with two kinds of traditional retailers, Liu et al. [9] considered the blockchain-enabled traceability goodwill and product freshness. Feng et al. [10] studied the FAPs supply chain when product freshness and price affect the market demand and showed that cost-sharing and compensation contracts can coordinate this supply chain. Moon et al. [11] studied the investment decisions of the FAPs supply chain and showed that the supply chain can be coordinated by revenue-sharing with investment cost-sharing and achieve a win-win. Qiu et al. [12] showed that the FAP supply chain can be coordinated by the revenue and service cost-sharing contract. Yan et al. [13] proposed the revenue-sharing or wholesale price contracts to coordinate the FAP supply chain, considering strategic consumer behavior. Besides, for a three-level fresh produce supply chain with particularly high FAPs loss rates, Song and He [14] showed a freshness-keeping cost-sharing and revenue-sharing contract can coordinate this supply chain.Ma et al. [15] also analyzed the coordination problem of a three-level FAP supply chain consisting of a supplier, a 3PLs, and a retailer. They showed this supply chain can be coordinated by cost and revenue-sharing contract for the two transaction processes. Most previous studies on the coordination of the FAP supply chain use common contracts, such as revenue-sharing and cost-sharing contracts. Our work differs to this literature stream by considering option contract and analyzing the conditions under which option contract enables the supply chain to achieve full coordination and Pareto improvement, and we consider different loss rates of FAPs in the two transaction processes, which is more realistic.


### 2.2. Option Contract in Supply Chain

The second stream of our research is related to the option contract in the supply chain, which has received increasing attention in recent years. Chen and Wan [16] investigated the multiperiod procurement problem under the call, put, bidirectional option, and wholesale price contracts. Xue et al. [17] studied how the supply disruption risk is shared and mitigated by the option contract and order commitment contract of core company, and Liu et al. [18] proposed a supply chain consisting of a risk-neutral supplier and a risk-averse retailer under the CVaR criteria and studied the option pricing, ordering, and producing problems. In addition to hedging risks, the option contract is used to solve specific practical issues, such as apparel and cotton production with high volatility, carbon emission abatement, and relief material management. Arnab et al. [19] explored how a five-level textile supply chain with demand and supply uncertainty is coordinated by the buyback and option contracts under manufacturer-led or retailer-led scenarios, respectively. Some scholars have introduced the option contract mechanism into relief material supply chain management and found that option contract can enable coordination of a relief supply chain and achievement of Pareto improvement [20–22]. Option contract can also effectively curb carbon emissions [23–25]. For the agricultural supply chain, Yang et al. [3] considered a supplier-retailer agricultural supply chain with option contracts and sales effort and found that a call option contract can reduce the shortage risk, a put option contract can reduce inventory risk, and a bidirectional option contract can reduce the bilateral risk. Recently, Wan et al. [26] gave the option coordination conditions of disrupted FAP supply chain under two supply chain structures and explored the effects of the disruptions and supply chain structure on the option coordination conditions. Unlike existing studies, we introduce an option contract into a three-level FAP supply chain to analyze the producer’s production and the retailer’s order decision and hedge the risks of yield, demand, and retailer’s spot market price uncertainty. The existing literature applied option contract to the two-level supply chain; we apply it to the three-level supply chain to explore whether the same coordination effect is achieved.


### 2.3. Spot Market

Our work is also related to studies on spot market. Luo et al. [27] took the first effort to study the intersection of put option contracts and random yield in the presence of a spot market. Merzifonluoglu [28] helped businesses to achieve the maximum performance from a given production system by optimally selecting customer demands, procurement quantity, spot market purchase, and option contract usage. Wan and Chen [29] studied a finite-horizon replenishment problem with option contracts under spot market. They showed that call, put, and bidirectional option contracts are in combined usage with the spot market which can retain a flexible replenishment strategy, respectively. Besides, Zhao et al. [30] designed an option mechanism to coordinate supply chain and examined the effects of supply competition between the stochastic spot market and the option contract market. Luo et al. [31] explored the supply chain coordination considering yield, demand, and spot price uncertainty under bidirectional option contracts, and Merzifonluoglu [32] analyzed how risk modeling can be used for supply chain management when firm’s primary suppliers are effected by complete disruptions and yield uncertainties and market demand and spot prices are random and possibly correlated. Similarly, we mainly study a three-level FAP supply chain with retailer’s random spot market and further explore the effect of the stochastic spot market by comparing the two scenarios of whether the retailer has a spot market or not. In addition, the producer has a spot market price, but it is an exogenous parameter.


### 2.4. Research Gap

In this subsection, we summarize the research gap between our work and the existing literature and further highlight our contributions. First, this article introduces the optimal decisions of producer and retailer into the three-level FAP supply chain model, analyzes the influence of the loss rates in the two transaction processes on the coordination of the supply chain, and enriches the fresh product supply chain management theory. Second, some related literatures mostly consider option contract to hedge the uncertain risks of the yield and demand in a two-level supply chain. In our article, in addition to the risk of uncertainty in yield and demand, the risk of spot price uncertainty is also considered. We find that option contracts can simultaneously hedge three uncertain risks. In addition, we focus on the analysis of the impact of random spot price and consider that upstream member also has spot market, which makes our model more realistic. Third, some scholars have only studied the optimal equilibrium decision of the three-level supply chain. However, in addition to these, we consider the coordination of the three-level supply chain. We are committed to promoting the balance of supply and demand in the supply chain and optimizing the profits of the supply chain and members, which makes our research more practical significance.


## 2.1. Operation and Coordination of the FAP Supply Chain

In recent years, a large number of scholars studied operation and coordination issues in the FAP supply chain. He et al. [7] constructed a model to study the impact of adopting blockchain technology on the pricing decisions and profits of the global FAP supply chain members. Wu et al. [8] explored the strategies for adopting blockchain technology in the FAP supply chain consisting of one e-tailer, one 3PLs, and one supplier. To explore whether the E-platform chooses reselling or agency selling mode for fresh food in competition with two kinds of traditional retailers, Liu et al. [9] considered the blockchain-enabled traceability goodwill and product freshness. Feng et al. [10] studied the FAPs supply chain when product freshness and price affect the market demand and showed that cost-sharing and compensation contracts can coordinate this supply chain. Moon et al. [11] studied the investment decisions of the FAPs supply chain and showed that the supply chain can be coordinated by revenue-sharing with investment cost-sharing and achieve a win-win. Qiu et al. [12] showed that the FAP supply chain can be coordinated by the revenue and service cost-sharing contract. Yan et al. [13] proposed the revenue-sharing or wholesale price contracts to coordinate the FAP supply chain, considering strategic consumer behavior. Besides, for a three-level fresh produce supply chain with particularly high FAPs loss rates, Song and He [14] showed a freshness-keeping cost-sharing and revenue-sharing contract can coordinate this supply chain.Ma et al. [15] also analyzed the coordination problem of a three-level FAP supply chain consisting of a supplier, a 3PLs, and a retailer. They showed this supply chain can be coordinated by cost and revenue-sharing contract for the two transaction processes. Most previous studies on the coordination of the FAP supply chain use common contracts, such as revenue-sharing and cost-sharing contracts. Our work differs to this literature stream by considering option contract and analyzing the conditions under which option contract enables the supply chain to achieve full coordination and Pareto improvement, and we consider different loss rates of FAPs in the two transaction processes, which is more realistic.


## 2.2. Option Contract in Supply Chain

The second stream of our research is related to the option contract in the supply chain, which has received increasing attention in recent years. Chen and Wan [16] investigated the multiperiod procurement problem under the call, put, bidirectional option, and wholesale price contracts. Xue et al. [17] studied how the supply disruption risk is shared and mitigated by the option contract and order commitment contract of core company, and Liu et al. [18] proposed a supply chain consisting of a risk-neutral supplier and a risk-averse retailer under the CVaR criteria and studied the option pricing, ordering, and producing problems. In addition to hedging risks, the option contract is used to solve specific practical issues, such as apparel and cotton production with high volatility, carbon emission abatement, and relief material management. Arnab et al. [19] explored how a five-level textile supply chain with demand and supply uncertainty is coordinated by the buyback and option contracts under manufacturer-led or retailer-led scenarios, respectively. Some scholars have introduced the option contract mechanism into relief material supply chain management and found that option contract can enable coordination of a relief supply chain and achievement of Pareto improvement [20–22]. Option contract can also effectively curb carbon emissions [23–25]. For the agricultural supply chain, Yang et al. [3] considered a supplier-retailer agricultural supply chain with option contracts and sales effort and found that a call option contract can reduce the shortage risk, a put option contract can reduce inventory risk, and a bidirectional option contract can reduce the bilateral risk. Recently, Wan et al. [26] gave the option coordination conditions of disrupted FAP supply chain under two supply chain structures and explored the effects of the disruptions and supply chain structure on the option coordination conditions. Unlike existing studies, we introduce an option contract into a three-level FAP supply chain to analyze the producer’s production and the retailer’s order decision and hedge the risks of yield, demand, and retailer’s spot market price uncertainty. The existing literature applied option contract to the two-level supply chain; we apply it to the three-level supply chain to explore whether the same coordination effect is achieved.


## 2.3. Spot Market

Our work is also related to studies on spot market. Luo et al. [27] took the first effort to study the intersection of put option contracts and random yield in the presence of a spot market. Merzifonluoglu [28] helped businesses to achieve the maximum performance from a given production system by optimally selecting customer demands, procurement quantity, spot market purchase, and option contract usage. Wan and Chen [29] studied a finite-horizon replenishment problem with option contracts under spot market. They showed that call, put, and bidirectional option contracts are in combined usage with the spot market which can retain a flexible replenishment strategy, respectively. Besides, Zhao et al. [30] designed an option mechanism to coordinate supply chain and examined the effects of supply competition between the stochastic spot market and the option contract market. Luo et al. [31] explored the supply chain coordination considering yield, demand, and spot price uncertainty under bidirectional option contracts, and Merzifonluoglu [32] analyzed how risk modeling can be used for supply chain management when firm’s primary suppliers are effected by complete disruptions and yield uncertainties and market demand and spot prices are random and possibly correlated. Similarly, we mainly study a three-level FAP supply chain with retailer’s random spot market and further explore the effect of the stochastic spot market by comparing the two scenarios of whether the retailer has a spot market or not. In addition, the producer has a spot market price, but it is an exogenous parameter.


## 2.4. Research Gap

In this subsection, we summarize the research gap between our work and the existing literature and further highlight our contributions. First, this article introduces the optimal decisions of producer and retailer into the three-level FAP supply chain model, analyzes the influence of the loss rates in the two transaction processes on the coordination of the supply chain, and enriches the fresh product supply chain management theory. Second, some related literatures mostly consider option contract to hedge the uncertain risks of the yield and demand in a two-level supply chain. In our article, in addition to the risk of uncertainty in yield and demand, the risk of spot price uncertainty is also considered. We find that option contracts can simultaneously hedge three uncertain risks. In addition, we focus on the analysis of the impact of random spot price and consider that upstream member also has spot market, which makes our model more realistic. Third, some scholars have only studied the optimal equilibrium decision of the three-level supply chain. However, in addition to these, we consider the coordination of the three-level supply chain. We are committed to promoting the balance of supply and demand in the supply chain and optimizing the profits of the supply chain and members, which makes our research more practical significance.


## 3. Model Description and Assumptions

This study considers a three-level fresh agricultural product supply chain that includes one producer, one supplier, and one retailer. Due to the particularity of agricultural products, the producer’s output is susceptible to natural disasters and abnormal weather, leading to high uncertainty on yield [33]. We suppose that the producer’s random output factor y is a nonnegative random variable characterized by a cumulative distribution function (CDF) Gy and probability density function (PDF) gy. Given the relatively stable environment of production of finished products, we assume that the supplier does not have loss of processing quantity, and the production yield of the supplier is deterministic. During the selling season, the market demand x is uncertain, which is nonnegative and has a CDF Fx and a PDF fx. In the scenario with random spot market, the retailer can purchase from the spot market with a random price of z, which has a CDF Hz and a PDF hz. We also assume that random variables x, y, and z are independent, and each member of the supply chain is risk-neutral, completely rational, independent, and symmetrical in information.In the distribution periods in the supply chain, the fresh agricultural products face loss in distribution process, such as quantity and quality loss [34]. In this study, we consider the quantity loss and denote the loss rate as βi, i=s,r, 0<β<1. After the producer’s output, in the process from the producer’s loading to the arrival of the supplier, there is a loss caused by natural factors such as loading and unloading, packaging, squeezing, and transportation. The loss rate is denoted as βs. After the supplier’s processing is completed, in the process from loading to the product reaching the retailer, there also exists loss, and βr represents the loss rate. We suppose that the losses in distribution process are borne by the supplier, who is the leading player of the supply chain. In accordance with the reality, the loss rate would decrease from the top to the bottom of the supply chain. We suppose that βr<βs. To simplify the formula, we denote Γt=∫01−βrtxfxdx. We summarize all notations of the paper in Table 1 in Appendix A.Table 1
Summary of basic notation.
Decision variablesInterpretationQQuantity ordered by the retailerqOption quantity ordered by the retailerRProduction input of the producerParametersInterpretationwpWholesale price of the producerwsWholesale price of the supplierpSale price of the retaileroOption priceeOption exercise pricespInitial product price in the producer’s spot market and exogenouscsSupplier’s processing costcpProducer’s production costgrRetailer’s shortage costβrLoss rate in distribution process from the supplier to the retailerβsLoss rate in distribution process from the producer to the supplierxRandom market demand, the mean isμyRandom output factorzRandom spot market price for the retailerΠExpected profitSuperscriptInterpretationsScenario of random spot market for retailer∗Optimal expected profit or decisionSubscriptInterpretationr∖s∖pRetailer\supplier\producer0∖1Wholesale price contract\option contractThe sequence of events in the supply chain is as follows:(i)
The supplier provides wholesale price contracts to the producer and retailer, respectively. In the option contract model, the supplier provides an option contract to the retailer.(ii)
The producer decides a production inputR on the initial FAPs.(iii)
The retailer offers an order quantityQ to the supplier. In the option contract model, he determines the option order quantity q. Then, the supplier provides the same order quantity of initial FAPs to the producer.(iv)
The producer invests in planting. When the producer’s output cannot meet the supplier’s order quantity, the producer will order initial products from his spot market at a price ofsp to fulfill the order [35].(v)
The supplier obtains the initial products from the producer and delivers them to the retailer after processing.(vi)
The retailer obtains finished products from the supplier and sells them to customers at a sale price ofp. The unsatisfied demand will incur shortage cost gr. If there is a spot market, then the retailer can directly purchase from the spot market to meet market demand.We make the following assumptions to avoid uninteresting cases:(1)
Fresh products cannot be returned due to the property of FAPs, and the residual value of the remaining products is small, and we suppose that it is zero.(2)
We assume that the retailer does not have arbitrage behavior when the spot pricez is higher than the option exercise price e, and the retailer will give priority on buying from the supplier when the spot price z is less than the option exercise price e.(3)
Similar to Yang et al. [3] and Luo et al. [31], we suppose that p+gr1−βr>o+e1−βr>1−βrws>cs+wp/1−βs, p+gr−∫0p+grHzdz1−βr>e−∫0eHzdz1−βr+os>1−βrαws, and wp>cp, to ensure that the retailer, supplier, and producer are willing to participate in the supply chain.(4)
We supposesp>cp to avoid the trivial case that the producer never produces.


## 4. Scenario 1: No Random Spot Market for the Retailer

In this section, we study the scenario wherein there is no spot market for the retailer, and the retailer can only order FAPs through the supply chain. To establish a performance benchmark, we first analyze an integrated supply chain.


### 4.1. Centralized Decision Model

In the centralized decision model, we assume that the producer, supplier, and retailer belong to the same virtual economic entity, and it will arrange the overall supply chain ordering strategy and production input level according to the principle of maximizing the overall profit of the supply chain. First, the expected profit of the integrated supply chain is(1)ΠT=Epmin1−βrQT,x−grx−1−βrQT+−csQT−cpRT−spQT1−βs−yRT+=p+gr1−βrQT−p+gr∫01−βrQTFxdx−grμ−csQT−cpRT−spRT∫0QT/RT1−βsGydy,where x+=maxx,0. The first term is the total revenue from sales, and the second term is the shortage cost when ordering products is not enough to meet market demand. The third and fourth terms are costs for processing and planting the initial FAPs, respectively. The fifth term is the cost of buying initial FAPs from the spot market in the case of a low yield.Proposition 1.
ΠT is jointly concave with QT and RT, and optimal QT∗,RT∗ are uniquely solved by(2)QT∗=11−βrF−11−cs+cηp+gr1−βr,∫0ηygydy=cpsp,where η=QT∗/RT∗1−βs, cη=sp/1−βsGη.Thus, the supply chain’s maximum profit is(3)ΠT∗=p+grΓQT∗−grμ.We consider the decentralized supply chain in the following.


### 4.2. Decentralized Model with Wholesale Price Contract

First, in the decentralized model with a wholesale price contract, the retailer’s expected profit function is(4)Πr0=Epmin1−βrQ0,x−ws1−βrQ0−grx−1−βrQ0+=p+gr1−βrQ0−p+gr∫01−βrQ0Fxdx−grμ−ws1−βrQ0.The first term is the retailer’s revenue from sales, the second term is the order cost, and the third term is the shortage cost when ordering products is not enough to meet market demand. Given thatd2Πr0/dQ02=−p+gr1−βr2f1−βrQ0<0, we obtain the uniquely optimal order quantity:(5)Q0∗=11−βrF−11−wsp+gr.Then, the retailer’s maximum expected profit is(6)Πr0∗=p+grΓQ0∗−grμ.Now, we consider the producer’s production planning problem. The producer’s expected profit function is(7)Πp0=EQ0wp1−βs−cpR0−spQ01−βs−yR0+=Q0wp1−βs−cpR0−spR0∫0Q0/R01−βsGydy.The first term is the revenue that the producer receives from selling initial FAPs to the supplier, the second term is the producer’s planting cost, and the third term is the producer’s cost of buying initial FAPs from the spot market in the case of a low yield. Sinced2Πp0/dR02=−spQ02/R031−βs2gQ0/R01−βs<0, the producer’s unique optimal R0∗ satisfies(8)∫0Q0∗/R0∗1−βsygydy=cpsp.According to (2) and (8), η=QT∗/RT∗1−βs=Q0∗/R0∗1−βs. Thus, the producer’s optimal expected profit is(9)Πp0∗=wp−spGηQ0∗1−βs,where cη=sp/1−βsGη. Based on the producer’s optimal expected profit, the producer is profitable and will participate in the transaction only when wp>spGη.Corollary 1.
cη increases with sp.cη is the potential cost of the producer in an economic sense [35]. Corollary 1 shows that an increase in the producer’s spot market price sp will lead to an increase in the potential cost and a decrease in the total profit of the entire supply chain.The above notion implies that the supplier’s optimal expected profit is(10)Πs0∗=Ews1−βrQ0−Q0wp1−βs−csQ0=ws1−βr−wp1−βs−csQ0∗.Proposition 2.
In the decentralized model with wholesale price contract, there areQ0∗<QT∗ and R0∗<RT∗.Proposition2 shows that the wholesale-price-only contract cannot coordinate the three-level FAP supply chain with uncertain output and random demand. Therefore, to maximize the profit of the supply chain or increase the profit of each member of the supply chain, the option contract would be introduced.


### 4.3. Option Contract Model

In the option contract model, the producer and supplier conduct transactions through a wholesale price contract. Meanwhile, the supplier and retailer conduct transactions through an option contract. The option provides the option purchaser with the right to buy a specific quantity of the underlying asset from the option provider at the option exercise price within the future period on the premise of paying option price to the option provider [36]. First, the retailer pays an option fee o to the supplier. When the sales season arrives, the retailer executes the option contract at the exercise price e according to the actual market demand. After the option order quantity is exceeded, the shortage cost gr will be incurred by unsatisfied demand. Thus, the retailer’s expected profit function is(11)Πr1=Ep−emin1−βrq,x−oq−grx−1−βrq+=p+gr−e1−βrq−p+gr−e∫01−βrqFxdx−grμ−oq.The first term is the retailer’s expected sales profit, the second term is the retailer’s option ordering cost, and the third term is the retailer’s shortage cost. Given thatd2Πr1/dq2=−p+gr−e1−βr2f1−βrq<0, we get the unique optimal q∗:(12)q∗=11−βrF−11−op+gr−e1−βr.Then, the retailer’s maximum expected profit is(13)Πr1∗=p+gr−eΓq∗−grμ.Now, we solve the producer’s production planning problem. The producer’s expected profit function is(14)Πp1=Eqwp1−βs−cpR1−spq1−βs−yR1+=qwp1−βs−cpR1−spR1∫0q/R11−βsGydy.The producer’s unique optimalR1∗ can be easily obtained:(15)∫0q∗/R1∗1−βsygydy=cpsp.According to (2) and (15), η=QT∗/RT∗1−βs=q∗/R1∗1−βs. Thus, the producer’s optimal expected profit is(16)Πp1∗=wp−spGηq∗1−βs.In the option contract model, the supplier’s optimal expected profit is(17)Πs1∗=Eoq+emin1−βrq,x−qwp1−βs−csq=p+grop+gr−e−wp1−βs−csq∗+eΓq∗.Proposition 3.
Wheno and e meet the following relationship, the option contract can fully coordinate the supply chain,(18)o=p+gr−ecs+cηp+gr.The producer’s production input is proportional to the retailer’s order quantity, and the production input increases with the order quantity. Accordingly, Proposition3 provides that the profits of the producer and retailer should be distributed through an option contract and encourage the retailer to increase order quantity to ensure that the entire three-level supply chain can be fully coordinated.When the profit of the entire supply chain reaches the maximum, if we want all parties willing to participate in the transaction, we need to ensure that all parties’ profits are greater than those under the wholesale price contract, which is more reasonable. Therefore, we have the following proposition.Proposition 4.
Ife meets the following conditions, then the supply chain can achieve full coordination and Pareto improvement at the same time,(19)1−βrws−cswp/1−βsQ0∗−cη−wp/1−βsq∗Γq∗≤e≤p+grΓq∗−ΓQ0∗Γq∗.Proposition4 shows that the producer’s optimal expected profit function is only related to the order quantity. When the supply chain is fully coordinated, the order quantity increases so that the producer’s profit is always increased. Accordingly, only the profits of the supplier and retailer need to be considered. When the option exercise price e is greater than the threshold, the supplier’s profit can be guaranteed. However, when e is considerably large, the retailer’s cost will increase and the order quantity will decrease; hence, the option exercise price has an upper limit. Only a moderate e can ensure that all three parties benefit and the total profit of the supply chain is maximized.


## 4.1. Centralized Decision Model

In the centralized decision model, we assume that the producer, supplier, and retailer belong to the same virtual economic entity, and it will arrange the overall supply chain ordering strategy and production input level according to the principle of maximizing the overall profit of the supply chain. First, the expected profit of the integrated supply chain is(1)ΠT=Epmin1−βrQT,x−grx−1−βrQT+−csQT−cpRT−spQT1−βs−yRT+=p+gr1−βrQT−p+gr∫01−βrQTFxdx−grμ−csQT−cpRT−spRT∫0QT/RT1−βsGydy,where x+=maxx,0. The first term is the total revenue from sales, and the second term is the shortage cost when ordering products is not enough to meet market demand. The third and fourth terms are costs for processing and planting the initial FAPs, respectively. The fifth term is the cost of buying initial FAPs from the spot market in the case of a low yield.Proposition 1.
ΠT is jointly concave with QT and RT, and optimal QT∗,RT∗ are uniquely solved by(2)QT∗=11−βrF−11−cs+cηp+gr1−βr,∫0ηygydy=cpsp,where η=QT∗/RT∗1−βs, cη=sp/1−βsGη.Thus, the supply chain’s maximum profit is(3)ΠT∗=p+grΓQT∗−grμ.We consider the decentralized supply chain in the following.


## 4.2. Decentralized Model with Wholesale Price Contract

First, in the decentralized model with a wholesale price contract, the retailer’s expected profit function is(4)Πr0=Epmin1−βrQ0,x−ws1−βrQ0−grx−1−βrQ0+=p+gr1−βrQ0−p+gr∫01−βrQ0Fxdx−grμ−ws1−βrQ0.The first term is the retailer’s revenue from sales, the second term is the order cost, and the third term is the shortage cost when ordering products is not enough to meet market demand. Given thatd2Πr0/dQ02=−p+gr1−βr2f1−βrQ0<0, we obtain the uniquely optimal order quantity:(5)Q0∗=11−βrF−11−wsp+gr.Then, the retailer’s maximum expected profit is(6)Πr0∗=p+grΓQ0∗−grμ.Now, we consider the producer’s production planning problem. The producer’s expected profit function is(7)Πp0=EQ0wp1−βs−cpR0−spQ01−βs−yR0+=Q0wp1−βs−cpR0−spR0∫0Q0/R01−βsGydy.The first term is the revenue that the producer receives from selling initial FAPs to the supplier, the second term is the producer’s planting cost, and the third term is the producer’s cost of buying initial FAPs from the spot market in the case of a low yield. Sinced2Πp0/dR02=−spQ02/R031−βs2gQ0/R01−βs<0, the producer’s unique optimal R0∗ satisfies(8)∫0Q0∗/R0∗1−βsygydy=cpsp.According to (2) and (8), η=QT∗/RT∗1−βs=Q0∗/R0∗1−βs. Thus, the producer’s optimal expected profit is(9)Πp0∗=wp−spGηQ0∗1−βs,where cη=sp/1−βsGη. Based on the producer’s optimal expected profit, the producer is profitable and will participate in the transaction only when wp>spGη.Corollary 1.
cη increases with sp.cη is the potential cost of the producer in an economic sense [35]. Corollary 1 shows that an increase in the producer’s spot market price sp will lead to an increase in the potential cost and a decrease in the total profit of the entire supply chain.The above notion implies that the supplier’s optimal expected profit is(10)Πs0∗=Ews1−βrQ0−Q0wp1−βs−csQ0=ws1−βr−wp1−βs−csQ0∗.Proposition 2.
In the decentralized model with wholesale price contract, there areQ0∗<QT∗ and R0∗<RT∗.Proposition2 shows that the wholesale-price-only contract cannot coordinate the three-level FAP supply chain with uncertain output and random demand. Therefore, to maximize the profit of the supply chain or increase the profit of each member of the supply chain, the option contract would be introduced.


## 4.3. Option Contract Model

In the option contract model, the producer and supplier conduct transactions through a wholesale price contract. Meanwhile, the supplier and retailer conduct transactions through an option contract. The option provides the option purchaser with the right to buy a specific quantity of the underlying asset from the option provider at the option exercise price within the future period on the premise of paying option price to the option provider [36]. First, the retailer pays an option fee o to the supplier. When the sales season arrives, the retailer executes the option contract at the exercise price e according to the actual market demand. After the option order quantity is exceeded, the shortage cost gr will be incurred by unsatisfied demand. Thus, the retailer’s expected profit function is(11)Πr1=Ep−emin1−βrq,x−oq−grx−1−βrq+=p+gr−e1−βrq−p+gr−e∫01−βrqFxdx−grμ−oq.The first term is the retailer’s expected sales profit, the second term is the retailer’s option ordering cost, and the third term is the retailer’s shortage cost. Given thatd2Πr1/dq2=−p+gr−e1−βr2f1−βrq<0, we get the unique optimal q∗:(12)q∗=11−βrF−11−op+gr−e1−βr.Then, the retailer’s maximum expected profit is(13)Πr1∗=p+gr−eΓq∗−grμ.Now, we solve the producer’s production planning problem. The producer’s expected profit function is(14)Πp1=Eqwp1−βs−cpR1−spq1−βs−yR1+=qwp1−βs−cpR1−spR1∫0q/R11−βsGydy.The producer’s unique optimalR1∗ can be easily obtained:(15)∫0q∗/R1∗1−βsygydy=cpsp.According to (2) and (15), η=QT∗/RT∗1−βs=q∗/R1∗1−βs. Thus, the producer’s optimal expected profit is(16)Πp1∗=wp−spGηq∗1−βs.In the option contract model, the supplier’s optimal expected profit is(17)Πs1∗=Eoq+emin1−βrq,x−qwp1−βs−csq=p+grop+gr−e−wp1−βs−csq∗+eΓq∗.Proposition 3.
Wheno and e meet the following relationship, the option contract can fully coordinate the supply chain,(18)o=p+gr−ecs+cηp+gr.The producer’s production input is proportional to the retailer’s order quantity, and the production input increases with the order quantity. Accordingly, Proposition3 provides that the profits of the producer and retailer should be distributed through an option contract and encourage the retailer to increase order quantity to ensure that the entire three-level supply chain can be fully coordinated.When the profit of the entire supply chain reaches the maximum, if we want all parties willing to participate in the transaction, we need to ensure that all parties’ profits are greater than those under the wholesale price contract, which is more reasonable. Therefore, we have the following proposition.Proposition 4.
Ife meets the following conditions, then the supply chain can achieve full coordination and Pareto improvement at the same time,(19)1−βrws−cswp/1−βsQ0∗−cη−wp/1−βsq∗Γq∗≤e≤p+grΓq∗−ΓQ0∗Γq∗.Proposition4 shows that the producer’s optimal expected profit function is only related to the order quantity. When the supply chain is fully coordinated, the order quantity increases so that the producer’s profit is always increased. Accordingly, only the profits of the supplier and retailer need to be considered. When the option exercise price e is greater than the threshold, the supplier’s profit can be guaranteed. However, when e is considerably large, the retailer’s cost will increase and the order quantity will decrease; hence, the option exercise price has an upper limit. Only a moderate e can ensure that all three parties benefit and the total profit of the supply chain is maximized.


## 5. Scenario 2: Random Spot Market for the Retailer

In this section, we study the situation where the retailer has a spot market and can not only order FAPs through the supply chain but also directly order finished products from a random spot market to sell to customers. In addition, the retailer’s spot market price is generally volatile 37. Thus, in the following, we consider Scenario 2: random spot market for the retailer. Similar to the previous part, we first study the centralized decision model. Then, we will consider the wholesale price contract and option contract models.


### 5.1. Centralized Decision Model

In the centralized decision model, when the retailer has a random spot market, due to the supply chain can observe the market demandx and the spot market price z at the beginning of the sales period, the supply chain needs to compare the random spot price z with the sum of the sale price p and the shortage cost gr according to the actual market demand. Then, the expected profit of the integrated supply chain is(20)ΠTs=Epx−minp+gr,zx−1−βrQTs+−csQTs−cpRTs−spQTs1−βs−yRTs+=pμ−p+gr−∫0p+grHzdzμ−1−βrQTs+∫01−βrQTsFxdx−csQTs−cpRTs−spRTs∫0QTs/RTs1−βsGydy.Proposition 5.
ΠTs is jointly concave in QTs and RTs, and the maximum QTs∗,RTs∗ is uniquely solved by(21)QTs∗=11−βrF−11−cs+cηp+gr−ν1−βr,∫0ηygydy=cpsp,where ν=∫0p+grHzdz, η=QTs∗/RTs∗1−βs, cη=sp/1−βsGη.According to Luo et al. [31], we define ν to be the expected value of the reduction in unit shortage loss cost when the retailer faces unsatisfied market demand and is affected by the random spot market, which is less than the unit shortage loss cost p+gr of the unsatisfied market demand. Therefore, the maximum supply chain’s profit is(22)ΠTs∗=p+gr−νΓQTs∗+ν−grμ.Corollary 2.
QTs∗<QT∗, RTs∗<RT∗.Corollary2 shows that the competition from the spot market will often reduce the order quantity when there is a spot market for the retailer. This notion is consistent with the reality. This phenomenon occurs because the retailer has more flexibility on the order quantity to adapt to changes in the market demand and price.Corollary 3.
In the centralized decision model,(1)
QTs∗ is decreasing in ν, RTs∗ is also decreasing in ν,(2)
η is increasing in cp and decreasing in sp.According to Corollary3(1), it is more advantageous to order directly from the spot market for the retailer when the expected value of reduction in shortage unit loss cost ν is greater. Consequently, the order quantity from the supply chain will be reduced, and the producer will decrease the production input. According to Corollary 3(2), the ratio of the optimal order quantity to the optimal production input quantity will also increase when the producer’s unit production cost cp increases, indicating that the producer will reduce the production cost by reducing the production input. The ratio η will decrease when the producer’s spot market price sp increases. This notion indicates that the producer will increase the production input to reduce the replenishment quantity, thereby diminishing the replenishment cost.Proposition 6.
In the centralized decision model,(1)
If0<ν<p+gr−cs+cηQTs∗/μ−ΓQTs∗, then ΠTs∗ is increasing in ν; if p+gr−cs+cηQTs∗/μ−ΓQTs∗<ν<p+gr, then ΠTs∗ is decreasing in ν;(2)
Ifp+grΓQT∗−ΓQTs∗/μ−ΓQTs∗<ν<p+gr, then ΠTs∗>ΠT∗; if 0<ν<p+grΓQT∗−ΓQTs∗/μ−ΓQTs∗, then ΠTs∗<ΠT∗.Proposition6(1) indicates that the decrease in revenue caused by the decrease in order quantity is less than the decrease in the shortage loss cost caused by the retailer’s spot market when ν is smaller, thus increasing the supply chain’s total profit. Meanwhile, the decrease in revenue caused by the decrease in order quantity is more than the decrease in the shortage loss cost caused by the retailer’s spot market when ν is larger, thus decreasing the supply chain’s total profit. Proposition 6(2) indicates that the supply chain will increase the amount of replenishment from the retailer’s spot market when ν is larger, which will greatly reduce the cost caused by the shortage, the cost reduction is greater than the reduction in profit caused by the decrease in the order quantity to achieve better total profit. The shortage loss cost is a small amount of reduced when ν is smaller. Hence, the total profit of the supply chain when the retailer operates in the spot market will be reduced due to the decrease in the order quantity.


### 5.2. Decentralized Model with Wholesale Price Contract

In the wholesale price contract model, all parties in the supply chain conduct spot transactions through wholesale contracts. At this time, the wholesale price contract is a forward contract and does not consider the issue of accompanying market. However, the wholesale price of the supplier may be lower due to the entry of the competitive market, that is,αws, where 0<α≤1. Thus, when the retailer has a random spot market, the retailer’s expected profit function is(23)Πr0s=Epx−αws1−βrQ0s−minp+gr,zx−1−βrQ0s+=pμ−αws1−βrQ0s−p+gr−νμ−1−βrQ0s+∫01−βrQ0sFxdx.Given thatd2Πr0s/dQ0s2=−p+gr−ν1−βr2f1−βrQ0s<0, we get the unique optimal Q0s∗(24)Q0s∗=11−βrF−11−αwsp+gr−ν.The retailer’s maximum expected profit is(25)Πr0s∗=p+gr−νΓQ0s∗+ν−grμ.Now, we consider the producer’s production planning problem. The producer’s expected profit function is(26)Πp0s=EQ0swp1−βs−cpR0s−spQ0s1−βs−yR0s+=Q0swp1−βs−cpR0s−spR0s∫0Q0s/R0s1−βsGydy.Therefore, the producer’s optimalR0s∗ uniquely satisfies(27)∫0Q0s∗/R0s∗1−βsygydy=cpsp.According to (2) and (27), η=QTs∗/RTs∗1−βs=Q0s∗/R0s∗1−βs. Thus, the producer’s optimal expected profit is(28)Πp0s∗=wp−spGηQ0s∗1−βs.Correspondingly, the supplier’s optimal expected profit is(29)Πs0s∗=Eαws1−βrQ0s−Q0swp1−βs−csQ0s=αws1−βr−wp1−βs−csQ0s∗.Proposition 7.
Q0s∗<QTs∗ and R0s∗<RTs∗.Corollary 4.
In the wholesale price contract model,Q0s∗ is decreasing in ν, and R0s∗ is also decreasing in ν.Proposition7 provides that the order quantity in the decentralized supply chain Q0s∗ is less than that in the integrated supply chain QTs∗. The producer’s planned production quantity in the decentralized supply chain R0s∗ is also less than that in the integrated supply chain RTs∗. Thus, the wholesale price contract cannot coordinate the supply chain even if the retailer has a random spot market. Corollary 4 shows that it is more advantageous to purchase directly from the retailer’s spot market when ν increases so that the retailer will reduce the fixed order quantity in the supply chain. Accordingly, the producer will decrease the production input.Proposition 8.
In the wholesale price contract model, we find that(1)
Casep+gr−ν/p+gr<α≤1. Q0s∗<Q0∗, R0s∗<R0∗, Πs0s∗<Πs0∗, Πp0s∗<Πp0∗; at the same time, if p+grΓQ0∗−ΓQ0s∗/μ−ΓQ0s∗<ν<p+gr, then Πr0s∗>Πr0∗; If 0<ν<p+grΓQ0∗−ΓQ0s∗/μ−ΓQ0s∗, then Πr0s∗<Πr0∗;(2)
Case0<α<p+gr−ν/p+gr. Q0s∗>Q0∗, R0s∗>R0∗, Πr0s∗>Πr0∗, Πp0s∗>Πp0∗; at the same time, if minp+gr−ν/p+gr,wp/1−βs+csQ0s∗+ws1−βr−wp/1−βs−csQ0∗/ws1−βrQ0s∗<α<p+gr−ν/p+gr, then Πs0s∗>Πs0∗; if 0<α<minp+gr−ν/p+gr,wp/1−βs+csQ0s∗+ws1−βr−wp/1−βs−csQ0∗/ws1−βrQ0s∗, then Πs0s∗<Πs0∗.Proposition8(1) shows when the supplier’s wholesale price ws is reduced by the random spot market to a small extent, the retailer will reduce the order quantity and increase the quantity flexibility to cope with demand and spot price volatility. Correspondingly, the production input quantity and the optimal profit of producer will decrease with the order quantity. At this time, the supplier’s profit will also be reduced due to the reduction in wholesale price and order quantity. For the retailer, as long as the expected value of the reduction in unit loss costs affected by the retailer’s spot market ν is sufficiently large to a certain extent, the shortage loss cost can be greatly reduced, thereby increasing the retailer’s profits. By contrast, if ν is small, then the reduction in the shortage loss cost cannot make up for the loss caused by the decrease in the order quantity, thus diminishing the profit of the retailer. When the supplier’s wholesale price ws is reduced by the random spot market to a large extent, Proposition 8(2) shows the opposite conclusion to that in Proposition 8(1).


### 5.3. Option Contract Model

The retailer can observe the market demandx and the spot market price z at the beginning of the sales period. Accordingly, the retailer must compare the random spot price z with the option exercise price e or the sum of the sale price p and the shortage cost gr according to the actual market demand. Then, the retailer can consider the procurement strategy from the random spot market. Thus, the retailer’s expected profit function is(30)Πr1s=Epx−osqs−mine,zmin1−βrqs,x−minp+gr,zx−1−βrqs+=pμ−osqs−e−ν11−βrqs−∫01−βrqsFxdx−p+gr−νμ−1−βrqs+∫01−βrqsFxdx,where ν1=∫0eHzdz. Similar to ν above, we define ν1 to be the expected value of the reduction in option exercise price when the retailer faces option exercise quantity and is affected by the random spot market, and it is less than the option exercise price e. Given that d2Πr1s/dqs2=−p+gr−e−ν+ν11−βr2f1−βrqs<0, we get the unique optimal qs∗:(31)qs∗=11−βrF−11−osp+gr−e−ν+ν11−βr.Then, the retailer’s optimal expected profit is(32)Πr1s∗=p+gr−e−ν+ν1Γqs∗+ν−grμ.The producer’s expected profit function is(33)Πp1s=Eqswp1−βs−cpR1s−spqs1−βs−yR1s+=qswp1−βs−cpR1s−spR1s∫0qs/R1s1−βsGydy.Easily, the producer’s unique optimalR1s∗ satisfies(34)∫0qs∗/R1s∗1−βsygydy=cpsp.According to (2) and (34), η=QTs∗/RTs∗1−βs=qs∗/R1s∗1−βs. Thus, the producer’s optimal expected profit is(35)Πp1s∗=wp−spGηqs∗1−βs.Then, the supplier’s optimal expected profit is(36)Πs1s∗=Eosqs∗+mine,srmin1−βrqs∗,x−qs∗wp1−βs−csqs∗=p+gr−νosp+gr−e−ν+ν1−wp1−βs−csqs∗+e−ν1Γqs∗.Proposition 9.
Ifos and e satisfy os=p+gr−e−ν+ν1cs+cη/p+gr−ν, then qs∗=QTs∗, R1s∗=RTs∗.Proposition9 shows that the profits between the supplier and the retailer are still reasonably distributed through the option contract when the retailer has a spot market, so that the three-level FAP supply chain can be fully coordinated. It is also necessary to improve the profits of all parties when the retailer has a random spot market. Thus, we get the following proposition.Proposition 10.
Ife meets the following conditions, then the supply chain can achieve full coordination and Pareto improvement at the same time.(37)1−βrαws−cs−wp/1−βsQ0s∗−cη−wp/1−βsqs∗Γqs∗≤e−ν1≤p+gr−νΓqs∗−ΓQ0s∗Γqs∗.Proposition10 illustrates that the profit of producer will always increase regardless of whether the retailer has a spot market or not as long as the supply chain is fully coordinated. When the expected option exercise price is greater, the supplier’s profit will be greater. When the expected option exercise price is smaller, the retailer’s profit will be greater. Therefore, keeping the expected option exercise price e−ν1 at a moderate level can increase the profits of the supplier and the retailer at the same time, thereby enabling the supply chain to achieve a win-win situation.Corollary 5.
If the supply chain is fully coordinated, thenqs∗<q∗, R1s∗<R1∗.Corollary5 shows that when the retailer has a spot market, even if an option contract is used to increase the order quantity flexibility, it will reduce the quantity of option ordered because the retailer wants to reduce the cost of ordering options, thereby replacing the quantity flexibility of option contract through the retailer’s spot market, to deal with the risk of demand volatility.When the supply chain is fully coordinated, the producer’s optimal expected profit will increase with the increase in the quantity of option ordered. Corollary5 shows that the producer’s profit will decrease when the retailer has a spot market so that here we only consider the profit distribution of the supplier and the retailer. As the leader of the supply chain and the designer of the contract, the supplier always wants to maximize his profit and ensure that the retailer participates in the transaction. So, next we will discuss the variation range of the supplier’s profit and obtain the following proposition.Proposition 11.
In the option contract model, when the supply chain is fully coordinated and the supplier can set the option contract satisfyingΠr1s∗=Πr1∗, there are(1)
If0<e<q∗−qs∗wp−spGη−ν11−βsΓqs∗/1−βsΓq∗−Γqs∗, then Πs1s∗>Πs1∗, Πp1s∗<Πp1∗;(2)
Ifq∗−qs∗wp−spGη−ν11−βsΓqs∗/1−βsΓq∗−Γqs∗<e<p+gr, then Πs1s∗<Πs1∗, Πp1s∗<Πp1∗.Proposition11 indicates that when a retailer has a spot market, the producer’s profit always decreases, which means that the supplier’s ordering cost from the producer is reduced, and the supplier’s revenue will also decrease with the decrease in the option order quantity. When the option exercise price e is smaller, the reduction in order costs will be greater than the reduction in revenue, which will increase the supplier’s total profit. However, the reduction in order costs will be less than the reduction in revenue when the option exercise price e is larger, which will decrease the supplier’s total profit. When the option exercise price e is considerably large, the retailer’s profit will decrease, and it will not be able to make the retailer participate in the transaction.Corollary 6.
In the option contract model,qs∗ is decreasing in ν and increasing in ν1, and R1s∗ is also decreasing in ν and increasing in ν1.Corollary6 implies that it is more advantageous to directly purchase from the retailer’s spot market when ν increases so that the retailer will reduce the option order quantity, and the expected option exercise price e−ν1 will decrease when ν1 increases, which can encourage the retailer to increase option order quantity. The expression of η indicates that the production input and option order quantity have the same increase and decrease.Corollary 7.
When the supply chain is fully coordinated,(1)
The option priceo and os are increasing in cs and cη, decreasing in the exercise price e;(2)
The option priceos is increasing in ν1 and decreasing in ν;(3)
Ifν1/e>ν/p+gr, then os>o; if ν1/e<ν/p+gr, then os<o.According to Corollary7(1), the option price will increase when the supplier’s processing cost cs and the producer’s potential cost cη increase. This result shows that the supply chain coordination is actually a process of redistributing supply chain risks among members. When the risks of the producer and the supplier increase, the supplier transfers part of the risk to the retailer through the option contract, and the supply chain maintains coordination. When the option exercise price e increases, the option price will be lowered to encourage the retailer to increase the option order quantity, and when the option exercise price e decreases, the supplier will increase the option price to protect its own profits. According to Corollary 7(2), the expected option exercise price e−ν1 will decrease when ν1 increases; thus, the supplier will increase the option price os to maintain profit. It is more advantageous to purchase directly from the retailer’s spot market when ν increases, hence the supplier will reduce the option price os to encourage the retailer to order more agricultural products. According to Corollary 7(3), the retailer’s profits will increase when the impact of the retailer’s spot market on the option exercise price e is greater than that on the unit shortage loss cost p+gr, and the supplier will increase the option price to maintain its own profits and share the retailer’s revenue. When the impact of the retailer’s spot market on the option exercise price e is less than that on the unit shortage loss cost p+gr, the supplier will encourage the retailer to increase the option order quantity by reducing the option price to ensure that the retailer’s option order quantity reaches the optimal level, so that the supply chain can be fully coordinated.


## 5.1. Centralized Decision Model

In the centralized decision model, when the retailer has a random spot market, due to the supply chain can observe the market demandx and the spot market price z at the beginning of the sales period, the supply chain needs to compare the random spot price z with the sum of the sale price p and the shortage cost gr according to the actual market demand. Then, the expected profit of the integrated supply chain is(20)ΠTs=Epx−minp+gr,zx−1−βrQTs+−csQTs−cpRTs−spQTs1−βs−yRTs+=pμ−p+gr−∫0p+grHzdzμ−1−βrQTs+∫01−βrQTsFxdx−csQTs−cpRTs−spRTs∫0QTs/RTs1−βsGydy.Proposition 5.
ΠTs is jointly concave in QTs and RTs, and the maximum QTs∗,RTs∗ is uniquely solved by(21)QTs∗=11−βrF−11−cs+cηp+gr−ν1−βr,∫0ηygydy=cpsp,where ν=∫0p+grHzdz, η=QTs∗/RTs∗1−βs, cη=sp/1−βsGη.According to Luo et al. [31], we define ν to be the expected value of the reduction in unit shortage loss cost when the retailer faces unsatisfied market demand and is affected by the random spot market, which is less than the unit shortage loss cost p+gr of the unsatisfied market demand. Therefore, the maximum supply chain’s profit is(22)ΠTs∗=p+gr−νΓQTs∗+ν−grμ.Corollary 2.
QTs∗<QT∗, RTs∗<RT∗.Corollary2 shows that the competition from the spot market will often reduce the order quantity when there is a spot market for the retailer. This notion is consistent with the reality. This phenomenon occurs because the retailer has more flexibility on the order quantity to adapt to changes in the market demand and price.Corollary 3.
In the centralized decision model,(1)
QTs∗ is decreasing in ν, RTs∗ is also decreasing in ν,(2)
η is increasing in cp and decreasing in sp.According to Corollary3(1), it is more advantageous to order directly from the spot market for the retailer when the expected value of reduction in shortage unit loss cost ν is greater. Consequently, the order quantity from the supply chain will be reduced, and the producer will decrease the production input. According to Corollary 3(2), the ratio of the optimal order quantity to the optimal production input quantity will also increase when the producer’s unit production cost cp increases, indicating that the producer will reduce the production cost by reducing the production input. The ratio η will decrease when the producer’s spot market price sp increases. This notion indicates that the producer will increase the production input to reduce the replenishment quantity, thereby diminishing the replenishment cost.Proposition 6.
In the centralized decision model,(1)
If0<ν<p+gr−cs+cηQTs∗/μ−ΓQTs∗, then ΠTs∗ is increasing in ν; if p+gr−cs+cηQTs∗/μ−ΓQTs∗<ν<p+gr, then ΠTs∗ is decreasing in ν;(2)
Ifp+grΓQT∗−ΓQTs∗/μ−ΓQTs∗<ν<p+gr, then ΠTs∗>ΠT∗; if 0<ν<p+grΓQT∗−ΓQTs∗/μ−ΓQTs∗, then ΠTs∗<ΠT∗.Proposition6(1) indicates that the decrease in revenue caused by the decrease in order quantity is less than the decrease in the shortage loss cost caused by the retailer’s spot market when ν is smaller, thus increasing the supply chain’s total profit. Meanwhile, the decrease in revenue caused by the decrease in order quantity is more than the decrease in the shortage loss cost caused by the retailer’s spot market when ν is larger, thus decreasing the supply chain’s total profit. Proposition 6(2) indicates that the supply chain will increase the amount of replenishment from the retailer’s spot market when ν is larger, which will greatly reduce the cost caused by the shortage, the cost reduction is greater than the reduction in profit caused by the decrease in the order quantity to achieve better total profit. The shortage loss cost is a small amount of reduced when ν is smaller. Hence, the total profit of the supply chain when the retailer operates in the spot market will be reduced due to the decrease in the order quantity.


## 5.2. Decentralized Model with Wholesale Price Contract

In the wholesale price contract model, all parties in the supply chain conduct spot transactions through wholesale contracts. At this time, the wholesale price contract is a forward contract and does not consider the issue of accompanying market. However, the wholesale price of the supplier may be lower due to the entry of the competitive market, that is,αws, where 0<α≤1. Thus, when the retailer has a random spot market, the retailer’s expected profit function is(23)Πr0s=Epx−αws1−βrQ0s−minp+gr,zx−1−βrQ0s+=pμ−αws1−βrQ0s−p+gr−νμ−1−βrQ0s+∫01−βrQ0sFxdx.Given thatd2Πr0s/dQ0s2=−p+gr−ν1−βr2f1−βrQ0s<0, we get the unique optimal Q0s∗(24)Q0s∗=11−βrF−11−αwsp+gr−ν.The retailer’s maximum expected profit is(25)Πr0s∗=p+gr−νΓQ0s∗+ν−grμ.Now, we consider the producer’s production planning problem. The producer’s expected profit function is(26)Πp0s=EQ0swp1−βs−cpR0s−spQ0s1−βs−yR0s+=Q0swp1−βs−cpR0s−spR0s∫0Q0s/R0s1−βsGydy.Therefore, the producer’s optimalR0s∗ uniquely satisfies(27)∫0Q0s∗/R0s∗1−βsygydy=cpsp.According to (2) and (27), η=QTs∗/RTs∗1−βs=Q0s∗/R0s∗1−βs. Thus, the producer’s optimal expected profit is(28)Πp0s∗=wp−spGηQ0s∗1−βs.Correspondingly, the supplier’s optimal expected profit is(29)Πs0s∗=Eαws1−βrQ0s−Q0swp1−βs−csQ0s=αws1−βr−wp1−βs−csQ0s∗.Proposition 7.
Q0s∗<QTs∗ and R0s∗<RTs∗.Corollary 4.
In the wholesale price contract model,Q0s∗ is decreasing in ν, and R0s∗ is also decreasing in ν.Proposition7 provides that the order quantity in the decentralized supply chain Q0s∗ is less than that in the integrated supply chain QTs∗. The producer’s planned production quantity in the decentralized supply chain R0s∗ is also less than that in the integrated supply chain RTs∗. Thus, the wholesale price contract cannot coordinate the supply chain even if the retailer has a random spot market. Corollary 4 shows that it is more advantageous to purchase directly from the retailer’s spot market when ν increases so that the retailer will reduce the fixed order quantity in the supply chain. Accordingly, the producer will decrease the production input.Proposition 8.
In the wholesale price contract model, we find that(1)
Casep+gr−ν/p+gr<α≤1. Q0s∗<Q0∗, R0s∗<R0∗, Πs0s∗<Πs0∗, Πp0s∗<Πp0∗; at the same time, if p+grΓQ0∗−ΓQ0s∗/μ−ΓQ0s∗<ν<p+gr, then Πr0s∗>Πr0∗; If 0<ν<p+grΓQ0∗−ΓQ0s∗/μ−ΓQ0s∗, then Πr0s∗<Πr0∗;(2)
Case0<α<p+gr−ν/p+gr. Q0s∗>Q0∗, R0s∗>R0∗, Πr0s∗>Πr0∗, Πp0s∗>Πp0∗; at the same time, if minp+gr−ν/p+gr,wp/1−βs+csQ0s∗+ws1−βr−wp/1−βs−csQ0∗/ws1−βrQ0s∗<α<p+gr−ν/p+gr, then Πs0s∗>Πs0∗; if 0<α<minp+gr−ν/p+gr,wp/1−βs+csQ0s∗+ws1−βr−wp/1−βs−csQ0∗/ws1−βrQ0s∗, then Πs0s∗<Πs0∗.Proposition8(1) shows when the supplier’s wholesale price ws is reduced by the random spot market to a small extent, the retailer will reduce the order quantity and increase the quantity flexibility to cope with demand and spot price volatility. Correspondingly, the production input quantity and the optimal profit of producer will decrease with the order quantity. At this time, the supplier’s profit will also be reduced due to the reduction in wholesale price and order quantity. For the retailer, as long as the expected value of the reduction in unit loss costs affected by the retailer’s spot market ν is sufficiently large to a certain extent, the shortage loss cost can be greatly reduced, thereby increasing the retailer’s profits. By contrast, if ν is small, then the reduction in the shortage loss cost cannot make up for the loss caused by the decrease in the order quantity, thus diminishing the profit of the retailer. When the supplier’s wholesale price ws is reduced by the random spot market to a large extent, Proposition 8(2) shows the opposite conclusion to that in Proposition 8(1).


## 5.3. Option Contract Model

The retailer can observe the market demandx and the spot market price z at the beginning of the sales period. Accordingly, the retailer must compare the random spot price z with the option exercise price e or the sum of the sale price p and the shortage cost gr according to the actual market demand. Then, the retailer can consider the procurement strategy from the random spot market. Thus, the retailer’s expected profit function is(30)Πr1s=Epx−osqs−mine,zmin1−βrqs,x−minp+gr,zx−1−βrqs+=pμ−osqs−e−ν11−βrqs−∫01−βrqsFxdx−p+gr−νμ−1−βrqs+∫01−βrqsFxdx,where ν1=∫0eHzdz. Similar to ν above, we define ν1 to be the expected value of the reduction in option exercise price when the retailer faces option exercise quantity and is affected by the random spot market, and it is less than the option exercise price e. Given that d2Πr1s/dqs2=−p+gr−e−ν+ν11−βr2f1−βrqs<0, we get the unique optimal qs∗:(31)qs∗=11−βrF−11−osp+gr−e−ν+ν11−βr.Then, the retailer’s optimal expected profit is(32)Πr1s∗=p+gr−e−ν+ν1Γqs∗+ν−grμ.The producer’s expected profit function is(33)Πp1s=Eqswp1−βs−cpR1s−spqs1−βs−yR1s+=qswp1−βs−cpR1s−spR1s∫0qs/R1s1−βsGydy.Easily, the producer’s unique optimalR1s∗ satisfies(34)∫0qs∗/R1s∗1−βsygydy=cpsp.According to (2) and (34), η=QTs∗/RTs∗1−βs=qs∗/R1s∗1−βs. Thus, the producer’s optimal expected profit is(35)Πp1s∗=wp−spGηqs∗1−βs.Then, the supplier’s optimal expected profit is(36)Πs1s∗=Eosqs∗+mine,srmin1−βrqs∗,x−qs∗wp1−βs−csqs∗=p+gr−νosp+gr−e−ν+ν1−wp1−βs−csqs∗+e−ν1Γqs∗.Proposition 9.
Ifos and e satisfy os=p+gr−e−ν+ν1cs+cη/p+gr−ν, then qs∗=QTs∗, R1s∗=RTs∗.Proposition9 shows that the profits between the supplier and the retailer are still reasonably distributed through the option contract when the retailer has a spot market, so that the three-level FAP supply chain can be fully coordinated. It is also necessary to improve the profits of all parties when the retailer has a random spot market. Thus, we get the following proposition.Proposition 10.
Ife meets the following conditions, then the supply chain can achieve full coordination and Pareto improvement at the same time.(37)1−βrαws−cs−wp/1−βsQ0s∗−cη−wp/1−βsqs∗Γqs∗≤e−ν1≤p+gr−νΓqs∗−ΓQ0s∗Γqs∗.Proposition10 illustrates that the profit of producer will always increase regardless of whether the retailer has a spot market or not as long as the supply chain is fully coordinated. When the expected option exercise price is greater, the supplier’s profit will be greater. When the expected option exercise price is smaller, the retailer’s profit will be greater. Therefore, keeping the expected option exercise price e−ν1 at a moderate level can increase the profits of the supplier and the retailer at the same time, thereby enabling the supply chain to achieve a win-win situation.Corollary 5.
If the supply chain is fully coordinated, thenqs∗<q∗, R1s∗<R1∗.Corollary5 shows that when the retailer has a spot market, even if an option contract is used to increase the order quantity flexibility, it will reduce the quantity of option ordered because the retailer wants to reduce the cost of ordering options, thereby replacing the quantity flexibility of option contract through the retailer’s spot market, to deal with the risk of demand volatility.When the supply chain is fully coordinated, the producer’s optimal expected profit will increase with the increase in the quantity of option ordered. Corollary5 shows that the producer’s profit will decrease when the retailer has a spot market so that here we only consider the profit distribution of the supplier and the retailer. As the leader of the supply chain and the designer of the contract, the supplier always wants to maximize his profit and ensure that the retailer participates in the transaction. So, next we will discuss the variation range of the supplier’s profit and obtain the following proposition.Proposition 11.
In the option contract model, when the supply chain is fully coordinated and the supplier can set the option contract satisfyingΠr1s∗=Πr1∗, there are(1)
If0<e<q∗−qs∗wp−spGη−ν11−βsΓqs∗/1−βsΓq∗−Γqs∗, then Πs1s∗>Πs1∗, Πp1s∗<Πp1∗;(2)
Ifq∗−qs∗wp−spGη−ν11−βsΓqs∗/1−βsΓq∗−Γqs∗<e<p+gr, then Πs1s∗<Πs1∗, Πp1s∗<Πp1∗.Proposition11 indicates that when a retailer has a spot market, the producer’s profit always decreases, which means that the supplier’s ordering cost from the producer is reduced, and the supplier’s revenue will also decrease with the decrease in the option order quantity. When the option exercise price e is smaller, the reduction in order costs will be greater than the reduction in revenue, which will increase the supplier’s total profit. However, the reduction in order costs will be less than the reduction in revenue when the option exercise price e is larger, which will decrease the supplier’s total profit. When the option exercise price e is considerably large, the retailer’s profit will decrease, and it will not be able to make the retailer participate in the transaction.Corollary 6.
In the option contract model,qs∗ is decreasing in ν and increasing in ν1, and R1s∗ is also decreasing in ν and increasing in ν1.Corollary6 implies that it is more advantageous to directly purchase from the retailer’s spot market when ν increases so that the retailer will reduce the option order quantity, and the expected option exercise price e−ν1 will decrease when ν1 increases, which can encourage the retailer to increase option order quantity. The expression of η indicates that the production input and option order quantity have the same increase and decrease.Corollary 7.
When the supply chain is fully coordinated,(1)
The option priceo and os are increasing in cs and cη, decreasing in the exercise price e;(2)
The option priceos is increasing in ν1 and decreasing in ν;(3)
Ifν1/e>ν/p+gr, then os>o; if ν1/e<ν/p+gr, then os<o.According to Corollary7(1), the option price will increase when the supplier’s processing cost cs and the producer’s potential cost cη increase. This result shows that the supply chain coordination is actually a process of redistributing supply chain risks among members. When the risks of the producer and the supplier increase, the supplier transfers part of the risk to the retailer through the option contract, and the supply chain maintains coordination. When the option exercise price e increases, the option price will be lowered to encourage the retailer to increase the option order quantity, and when the option exercise price e decreases, the supplier will increase the option price to protect its own profits. According to Corollary 7(2), the expected option exercise price e−ν1 will decrease when ν1 increases; thus, the supplier will increase the option price os to maintain profit. It is more advantageous to purchase directly from the retailer’s spot market when ν increases, hence the supplier will reduce the option price os to encourage the retailer to order more agricultural products. According to Corollary 7(3), the retailer’s profits will increase when the impact of the retailer’s spot market on the option exercise price e is greater than that on the unit shortage loss cost p+gr, and the supplier will increase the option price to maintain its own profits and share the retailer’s revenue. When the impact of the retailer’s spot market on the option exercise price e is less than that on the unit shortage loss cost p+gr, the supplier will encourage the retailer to increase the option order quantity by reducing the option price to ensure that the retailer’s option order quantity reaches the optimal level, so that the supply chain can be fully coordinated.


## 6. Numerical Analysis

We conduct some numerical experiments to further explore the influence of factors on the full coordination and the Pareto improvement area of the supply chain. We assume that the market demandx follows a normal distribution with mean μ=50 and standard deviation σ=10, the random output factor y of fresh agricultural products follows a normal distribution with mean μ1=1 and standard deviation σ1=0.2, and the retailer’s spot market price z follows a normal distribution with mean μ2=18 and standard deviation σ2=2. The basic parameter values are as follows: p=20, gr=5, βr=0.1, βs=0.2, cs=2, ws=17, sp=8, wp=8.1, α=0.95, cp=6, and e=13. In addition, random volatility risks are ρx=σ/μ, ρy=σ1/μ1, and ρz=σ2/μ2, respectively. “% profit increase” shows the percentage increase in supply chain profit under coordination as compared to supply chain profit without coordination, that is, the full coordination effect of the option contract. This amount is calculated as follows: %profit increase=supply chain profit under coordination/supply chain profit without coordination−1×100% [35]. Thus, we denote κ=Πr1∗+Πs1∗+Πp1∗/Πr0∗+Πs0∗+Πp0∗−1×100% and κs=Πr1s∗+Πs1s∗+Πp1s∗/Πr0s∗+Πs0s∗+Πp0s∗−1×100%.


### 6.1. Impact of Producer’s Wholesale and Spot Market Pricewp,sp

Although this work studies the three-level supply chain, the coordination contract with option is only applied between the supplier and the retailer, which is achieved by the rational distribution of profits between the supplier and the retailer while ignoring the producer. Thus, the influence of producer on the coordination is worth further exploring.Figure1 shows that the full coordination effect monotonically decreases with sp in both scenarios under full coordination with option contract. sp has a greater impact on the full coordination and is more sensitive to the change trend when there is a spot market for the retailer. When the producer’s spot market price is higher, the producer will increase the wholesale price to ensure its own profits, which will reduce the profits of the supplier and retailer. Accordingly, the retailer will reduce the order quantity, thereby decreasing the total profit of the supply chain and the full coordination effect of the option contract.Figure 1
Impact of producer’s spot market price on the effect of full coordination.In Figure2(a), if wp≤wps0, then Πs1∗≥Πs0∗; otherwise, Πs1∗<Πs0∗; if wp≥wpp0, then Πp1∗≥Πp0∗; otherwise, Πp1∗<Πp0∗. In Scenario 1, when e and wp in Region A, both the supplier and the producer get more profit with the option contract than with the wholesale price contract, and we already know the retailer gets more profit in the option contract. Thus, Region A is the Pareto improvement area of Scenario 1. Similarly, Region B is the Pareto improvement area of Scenario 2.Figure 2
Impact of producer’s wholesale and spot market price on Pareto improvement interval.
(a)(b)In Figure2(b), if sp≥sps0, then Πs1∗≥Πs0∗; otherwise, Πs1∗<Πs0∗; if sp≤spr0, then Πr1∗≥Πr0∗; otherwise, Πr1∗<Πr0∗. In Scenario 1, when e and sp in Region C, both the supplier and the retailer get more profit with the option contract than with the wholesale price contract, and we already know the producer gets more profit in the option contract. Hence, Region C is the Pareto improvement area of Scenario 1. Similarly, Region D is the Pareto improvement area of Scenario 2.Figure2(a) shows that the length of the Pareto improvement interval in the two scenarios decreases with the increase in wp. Only the larger e interval can achieve Pareto improvement all the time. The producer’s profit will increase when wp increases, resulting in a decrease in the overall profit of the retailer and supplier. Accordingly, the supplier will choose the originally larger e range to protect its own profits. The length of the Pareto improvement interval and the value of e under Scenario 2 are larger than those under Scenario 1, indicating that the retailer’s spot market has increased the flexibility of Pareto improvement and that the supplier will increase e to share more retailer’s profits. In addition, the supply chain can achieve Pareto improvement when wp∈6.47,8.5 in Scenarios 1 and 2. When wp exceeds 8.5, the profit shared by the producer is considerably large; accordingly, the supplier and retailer will be unable to achieve coordination. wp has a lower limit of 6.47, which confirms wp>spGη. Figure 2(b) shows that since the gap between the producer’s profit under the two contracts is always greater than zero, the producer is not considered (the same applies to all figures below). We know that the length of the Pareto improvement interval will increase with the increase of sp, and e will decrease. The increase in sp will reduce the overall profit of the supply chain; accordingly, the supplier will encourage the retailer to increase the order quantity by reducing e and the flexibility of profit distribution to increase the overall profit of the supply chain. Similarly, the length of the Pareto improvement interval and the value of e under Scenario 2 are larger than those under Scenario 1. The above analysis implies that the coordination of the three-level FAP supply chain is also related to the producer and not entirely determined by the supplier and retailer.


### 6.2. Impact of Random Volatility Risksρx,ρy,ρz

This work studies the producer’s yield and market demand uncertainty. The focus is on the uncertainty of the retailer’s spot market price. Thus, the impact of random factors on the coordination of the supply chain must be analyzed.Figure3(a) shows that the greater the market demand volatility in Scenario 1 or 2, the better the full coordination effect of the option contract on the supply chain, and the impact on Scenario 2 is more obvious. Figure 3(b) shows that the greater the volatility of the random output factor, the poorer the full coordination effect of the option contract on the supply chain. However, the supply chain is kept coordinated, and the coordination effect of Scenario 2 is significantly greater than that of Scenario 1. Figure 3(c) shows that the full coordination effect of the option contract will remain stable when the volatility of retailer’s spot market price is lower. However, the full coordination effect will exponentially intensify when the volatility is larger, indicating that the option contract has a significant control over the risk of price volatility.Figure 3
Impact of random volatility risks on the effect of full coordination.
(a)(b)(c)In Figure4(a), if ρx≤ρxs0, then Πs1∗≥Πs0∗; otherwise, Πs1∗<Πs0∗; if ρx≥ρxr0, then Πr1∗≥Πr0∗; otherwise, Πr1∗<Πr0∗. In Scenario 1, when e and ρx in Region A, both the supplier and the retailer get more profit with the option contract than with the wholesale price contract, and we already know the producer’s profit is not dependent with ρx. Hence, Region A is the Pareto improvement area of Scenario 1. Similarly, Region B is the Pareto improvement area of Scenario 2.Figure 4
Impact of random volatility risks on Pareto improvement interval.
(a)(b)(c)In Figure4(b), if ρy≥ρys0, then Πs1∗≥Πs0∗; otherwise, Πs1∗<Πs0∗; if ρy≤ρyr0, then Πr1∗≥Πr0∗; otherwise, Πr1∗<Πr0∗. In Scenario 1, when e and ρy in Region C, both the supplier and the retailer get more profit with the option contract than with the wholesale price contract, and we already know the producer’s profit is not dependent with ρy. Thus, Region C is the Pareto improvement area of Scenario 1. Similarly, Region D is the Pareto improvement area of Scenario 2.In Figure4(c), if ρz≤ρzs1, then Πs1s∗≥Πs0s∗; otherwise, Πs1s∗<Πs0s∗; if ρz≥ρzr1, then Πr1s∗≥Πr0s∗; otherwise, Πr1s∗<Πr0s∗. In Scenario 1, when e and ρz in Region C, both the supplier and the retailer get more profit with the option contract than with the wholesale price contract, and we already know the producer’s profit is not dependent with ρz. Thus, Region E is the Pareto improvement area of Scenario 2.Figure4(a) shows that regardless of Scenario 1 or 2, when the risk of demand volatility becomes greater, e will increase, and the length of the Pareto improvement interval will also increase. This notion indicates that if the retailer wants the supplier to share the risk of demand volatility, then it must increase the supplier’s income and will increase the supplier’s power to distribute profits. The length of the interval and the value of e in Scenario 2 are larger than those in Scenario 1, showing that the supplier will increase e to ensure its own profits due to the influence of the retailer’s spot market. Figure 4(b) shows that whether it is Scenario 1 or 2, when the volatility risk of random output factor becomes greater, e will decrease, indicating that part of the risk will be transferred to the retailer through the supplier when the producer’s risk increases. To ensure that the retailer’s profit will reduce e, then the supply chain can be kept coordinated. Similarly, in the face of the impact of the retailer’s spot market, the supplier will also increase e to ensure its own profits. Figure 4(c) shows that e will increase with the increase in the retailer’s spot market price volatility risk, indicating that the supplier will give priority to ensuring its own profits by increasing e in the face of uncertainty.In summary, the option contract can obviously cope with the volatility risks of random factors. In particular, the effect is more evident when the retailer has a spot market.


### 6.3. Impact of Loss Rateβr,βs

Given that this work studies the FAP supply chain, the impact of its characteristic loss rate on the coordination of the supply chain must be considered. Figure5 shows that the greater the loss rate, the worse the full coordination effect of the option contract. The full coordination effect for Scenario 2 is better than that for Scenario 1, and the loss rate has a greater impact on Scenario 2. The trend of the lines in the figure shows that the full coordination effect of Scenario 1 will be greater than that of Scenario 2 when the loss rate is larger; however, this is not the reality, because the advancement of fresh-keeping technology will not cause a high loss rate. When βr and βs have the same influencing degree on the full coordination effect, βs is always greater than βr.Figure 5
Impact of loss rate on the effect of full coordination.
(a)(b)In Figure6(a), if βr≥βrs0, then Πs1∗≥Πs0∗; otherwise, Πs1∗<Πs0∗; if βr≤βrr0, then Πr1∗≥Πr0∗; otherwise, Πr1∗<Πr0∗. In Scenario 1, when e and βr in Region A, both the supplier and the retailer get more profit with the option contract than with the wholesale price contract, and we already know the producer’s profit is not dependent with βr. Hence, Region A is the Pareto improvement area of Scenario 1. Similarly, Region B is the Pareto improvement area of Scenario 2. In Figure 6(b), Region C is the Pareto improvement area of Scenario 1, and Region D is the Pareto improvement area of Scenario 2.Figure 6
Impact of loss rate on Pareto improvement interval.
(a)(b)Figure6 illustrates that when the retailer does not have a spot market, the larger the loss rate, the shorter the Pareto improvement interval length, and the smaller the value of e, until there is no Pareto improvement interval at all. This notion indicates that the supplier needs to reduce e to encourage the retailer to increase the order quantity. Accordingly, the increase in supplier profits is greater than that in the loss costs in the distribution process. When the loss rate is considerably large, and e is small, the supplier’s profit will decrease, resulting in the failure to achieve Pareto improvement. When the retailer has a spot market, the length of the Pareto improvement interval first increases and then decreases with the loss rate, and the e value only slightly changes. This result shows that the increase in loss rate is not necessarily negative, and controlling the loss rate at a moderate level can make it easier for the supply chain to achieve Pareto improvement, which differs from Scenario 1 and is better than Scenario 1. Similarly, the supplier’s profit will greatly decrease when the loss rate is considerably large, resulting in the failure to achieve Pareto improvement.The analytical results of the influence of the abovementioned factors demonstrate that the option contract can simultaneously achieve full coordination and Pareto improvement of the supply chain regardless of whether the retailer has a random spot market or not, but the Pareto improvement interval is relatively small, i.e., the conditions for coordination are rigorous. The full coordination effect of the option contract is decreasing in the producer’s spot market price, random output factor volatility risk, and the loss rate, increasing in the demand and the retailer’s spot market price volatility risk. Various factors also have a significant impact on the Pareto improvement of the supply chain, especially the different loss rate of two transaction processes. In addition, when the retailer has a spot market, the full coordination effect of the option contract is more obvious, and the interval ande for Pareto improvement are larger, which proves that the option contract is better than the wholesale price contract.


## 6.1. Impact of Producer’s Wholesale and Spot Market Pricewp,sp

Although this work studies the three-level supply chain, the coordination contract with option is only applied between the supplier and the retailer, which is achieved by the rational distribution of profits between the supplier and the retailer while ignoring the producer. Thus, the influence of producer on the coordination is worth further exploring.Figure1 shows that the full coordination effect monotonically decreases with sp in both scenarios under full coordination with option contract. sp has a greater impact on the full coordination and is more sensitive to the change trend when there is a spot market for the retailer. When the producer’s spot market price is higher, the producer will increase the wholesale price to ensure its own profits, which will reduce the profits of the supplier and retailer. Accordingly, the retailer will reduce the order quantity, thereby decreasing the total profit of the supply chain and the full coordination effect of the option contract.Figure 1
Impact of producer’s spot market price on the effect of full coordination.In Figure2(a), if wp≤wps0, then Πs1∗≥Πs0∗; otherwise, Πs1∗<Πs0∗; if wp≥wpp0, then Πp1∗≥Πp0∗; otherwise, Πp1∗<Πp0∗. In Scenario 1, when e and wp in Region A, both the supplier and the producer get more profit with the option contract than with the wholesale price contract, and we already know the retailer gets more profit in the option contract. Thus, Region A is the Pareto improvement area of Scenario 1. Similarly, Region B is the Pareto improvement area of Scenario 2.Figure 2
Impact of producer’s wholesale and spot market price on Pareto improvement interval.
(a)(b)In Figure2(b), if sp≥sps0, then Πs1∗≥Πs0∗; otherwise, Πs1∗<Πs0∗; if sp≤spr0, then Πr1∗≥Πr0∗; otherwise, Πr1∗<Πr0∗. In Scenario 1, when e and sp in Region C, both the supplier and the retailer get more profit with the option contract than with the wholesale price contract, and we already know the producer gets more profit in the option contract. Hence, Region C is the Pareto improvement area of Scenario 1. Similarly, Region D is the Pareto improvement area of Scenario 2.Figure2(a) shows that the length of the Pareto improvement interval in the two scenarios decreases with the increase in wp. Only the larger e interval can achieve Pareto improvement all the time. The producer’s profit will increase when wp increases, resulting in a decrease in the overall profit of the retailer and supplier. Accordingly, the supplier will choose the originally larger e range to protect its own profits. The length of the Pareto improvement interval and the value of e under Scenario 2 are larger than those under Scenario 1, indicating that the retailer’s spot market has increased the flexibility of Pareto improvement and that the supplier will increase e to share more retailer’s profits. In addition, the supply chain can achieve Pareto improvement when wp∈6.47,8.5 in Scenarios 1 and 2. When wp exceeds 8.5, the profit shared by the producer is considerably large; accordingly, the supplier and retailer will be unable to achieve coordination. wp has a lower limit of 6.47, which confirms wp>spGη. Figure 2(b) shows that since the gap between the producer’s profit under the two contracts is always greater than zero, the producer is not considered (the same applies to all figures below). We know that the length of the Pareto improvement interval will increase with the increase of sp, and e will decrease. The increase in sp will reduce the overall profit of the supply chain; accordingly, the supplier will encourage the retailer to increase the order quantity by reducing e and the flexibility of profit distribution to increase the overall profit of the supply chain. Similarly, the length of the Pareto improvement interval and the value of e under Scenario 2 are larger than those under Scenario 1. The above analysis implies that the coordination of the three-level FAP supply chain is also related to the producer and not entirely determined by the supplier and retailer.


## 6.2. Impact of Random Volatility Risksρx,ρy,ρz

This work studies the producer’s yield and market demand uncertainty. The focus is on the uncertainty of the retailer’s spot market price. Thus, the impact of random factors on the coordination of the supply chain must be analyzed.Figure3(a) shows that the greater the market demand volatility in Scenario 1 or 2, the better the full coordination effect of the option contract on the supply chain, and the impact on Scenario 2 is more obvious. Figure 3(b) shows that the greater the volatility of the random output factor, the poorer the full coordination effect of the option contract on the supply chain. However, the supply chain is kept coordinated, and the coordination effect of Scenario 2 is significantly greater than that of Scenario 1. Figure 3(c) shows that the full coordination effect of the option contract will remain stable when the volatility of retailer’s spot market price is lower. However, the full coordination effect will exponentially intensify when the volatility is larger, indicating that the option contract has a significant control over the risk of price volatility.Figure 3
Impact of random volatility risks on the effect of full coordination.
(a)(b)(c)In Figure4(a), if ρx≤ρxs0, then Πs1∗≥Πs0∗; otherwise, Πs1∗<Πs0∗; if ρx≥ρxr0, then Πr1∗≥Πr0∗; otherwise, Πr1∗<Πr0∗. In Scenario 1, when e and ρx in Region A, both the supplier and the retailer get more profit with the option contract than with the wholesale price contract, and we already know the producer’s profit is not dependent with ρx. Hence, Region A is the Pareto improvement area of Scenario 1. Similarly, Region B is the Pareto improvement area of Scenario 2.Figure 4
Impact of random volatility risks on Pareto improvement interval.
(a)(b)(c)In Figure4(b), if ρy≥ρys0, then Πs1∗≥Πs0∗; otherwise, Πs1∗<Πs0∗; if ρy≤ρyr0, then Πr1∗≥Πr0∗; otherwise, Πr1∗<Πr0∗. In Scenario 1, when e and ρy in Region C, both the supplier and the retailer get more profit with the option contract than with the wholesale price contract, and we already know the producer’s profit is not dependent with ρy. Thus, Region C is the Pareto improvement area of Scenario 1. Similarly, Region D is the Pareto improvement area of Scenario 2.In Figure4(c), if ρz≤ρzs1, then Πs1s∗≥Πs0s∗; otherwise, Πs1s∗<Πs0s∗; if ρz≥ρzr1, then Πr1s∗≥Πr0s∗; otherwise, Πr1s∗<Πr0s∗. In Scenario 1, when e and ρz in Region C, both the supplier and the retailer get more profit with the option contract than with the wholesale price contract, and we already know the producer’s profit is not dependent with ρz. Thus, Region E is the Pareto improvement area of Scenario 2.Figure4(a) shows that regardless of Scenario 1 or 2, when the risk of demand volatility becomes greater, e will increase, and the length of the Pareto improvement interval will also increase. This notion indicates that if the retailer wants the supplier to share the risk of demand volatility, then it must increase the supplier’s income and will increase the supplier’s power to distribute profits. The length of the interval and the value of e in Scenario 2 are larger than those in Scenario 1, showing that the supplier will increase e to ensure its own profits due to the influence of the retailer’s spot market. Figure 4(b) shows that whether it is Scenario 1 or 2, when the volatility risk of random output factor becomes greater, e will decrease, indicating that part of the risk will be transferred to the retailer through the supplier when the producer’s risk increases. To ensure that the retailer’s profit will reduce e, then the supply chain can be kept coordinated. Similarly, in the face of the impact of the retailer’s spot market, the supplier will also increase e to ensure its own profits. Figure 4(c) shows that e will increase with the increase in the retailer’s spot market price volatility risk, indicating that the supplier will give priority to ensuring its own profits by increasing e in the face of uncertainty.In summary, the option contract can obviously cope with the volatility risks of random factors. In particular, the effect is more evident when the retailer has a spot market.


## 6.3. Impact of Loss Rateβr,βs

Given that this work studies the FAP supply chain, the impact of its characteristic loss rate on the coordination of the supply chain must be considered. Figure5 shows that the greater the loss rate, the worse the full coordination effect of the option contract. The full coordination effect for Scenario 2 is better than that for Scenario 1, and the loss rate has a greater impact on Scenario 2. The trend of the lines in the figure shows that the full coordination effect of Scenario 1 will be greater than that of Scenario 2 when the loss rate is larger; however, this is not the reality, because the advancement of fresh-keeping technology will not cause a high loss rate. When βr and βs have the same influencing degree on the full coordination effect, βs is always greater than βr.Figure 5
Impact of loss rate on the effect of full coordination.
(a)(b)In Figure6(a), if βr≥βrs0, then Πs1∗≥Πs0∗; otherwise, Πs1∗<Πs0∗; if βr≤βrr0, then Πr1∗≥Πr0∗; otherwise, Πr1∗<Πr0∗. In Scenario 1, when e and βr in Region A, both the supplier and the retailer get more profit with the option contract than with the wholesale price contract, and we already know the producer’s profit is not dependent with βr. Hence, Region A is the Pareto improvement area of Scenario 1. Similarly, Region B is the Pareto improvement area of Scenario 2. In Figure 6(b), Region C is the Pareto improvement area of Scenario 1, and Region D is the Pareto improvement area of Scenario 2.Figure 6
Impact of loss rate on Pareto improvement interval.
(a)(b)Figure6 illustrates that when the retailer does not have a spot market, the larger the loss rate, the shorter the Pareto improvement interval length, and the smaller the value of e, until there is no Pareto improvement interval at all. This notion indicates that the supplier needs to reduce e to encourage the retailer to increase the order quantity. Accordingly, the increase in supplier profits is greater than that in the loss costs in the distribution process. When the loss rate is considerably large, and e is small, the supplier’s profit will decrease, resulting in the failure to achieve Pareto improvement. When the retailer has a spot market, the length of the Pareto improvement interval first increases and then decreases with the loss rate, and the e value only slightly changes. This result shows that the increase in loss rate is not necessarily negative, and controlling the loss rate at a moderate level can make it easier for the supply chain to achieve Pareto improvement, which differs from Scenario 1 and is better than Scenario 1. Similarly, the supplier’s profit will greatly decrease when the loss rate is considerably large, resulting in the failure to achieve Pareto improvement.The analytical results of the influence of the abovementioned factors demonstrate that the option contract can simultaneously achieve full coordination and Pareto improvement of the supply chain regardless of whether the retailer has a random spot market or not, but the Pareto improvement interval is relatively small, i.e., the conditions for coordination are rigorous. The full coordination effect of the option contract is decreasing in the producer’s spot market price, random output factor volatility risk, and the loss rate, increasing in the demand and the retailer’s spot market price volatility risk. Various factors also have a significant impact on the Pareto improvement of the supply chain, especially the different loss rate of two transaction processes. In addition, when the retailer has a spot market, the full coordination effect of the option contract is more obvious, and the interval ande for Pareto improvement are larger, which proves that the option contract is better than the wholesale price contract.


## 7. Conclusions

To improve the balance of supply and demand and reduce profit losses in the three-level FAP supply chain, all members would jointly increase the profits of each member and the total supply chain by adopting various cooperation mechanisms. This paper quantitatively investigates a three-level FAP supply chain consisting of a producer with yield uncertainty, a leader supplier, and a retailer with demand and spot price uncertainty. We summarize some key findings in the following.First, we characterize the equilibrium decisions on order and production input quantity in the supply chain and find that pure wholesale price contract cannot coordinate the supply chain regardless of whether the retailer has a random spot market or not. Accordingly, we introduce a pure option contract between the supplier and the retailer, while wholesale price contract is still signed between the producer and the supplier. In any scenario, the supplier can design a reasonable option price and exercise price to enable the supply chain to achieve full coordination and Pareto improvement, thereby attaining a win-win situation, but the conditions for coordination are rigorous. Second, the comparison between Scenario 1 and Scenario 2 demonstrates that when the retailer has a spot market, both the order and production input quantity will decrease. However, the total profit of supply chain will increase when the retailer’s spot market has a great effect on the unit shortage loss cost. Third, numerical analytical results show that high producer’s spot market price, random output factor volatility risk, and loss rate will worsen the coordination effect of option contract. However, high demand and retailer’s spot market price volatility risk can enhance the coordination effect. In addition, when the retailer has a spot market, the higher option exercise price can ensure that all parties benefit and achieve Pareto improvement under full coordination, the coordination effect of the option contract is better, and the coordination area is larger.From the research, we can reach some managerial insights. First, as the leader of the supply chain and the designer of the contract, the supplier should provide the option contract with the retailer regardless of whether the retailer has a random spot market, which can not only maximize the overall profit of the supply chain but also benefit all parties under certain conditions. When the retailer has a random spot market, the supplier needs to set the option exercise price to be larger. Moreover, the supply chain cannot achieve Pareto improvement due to the high loss rate, thus the supplier needs to improve transportation and fresh-keeping technology to reduce the loss in the distribution process of FAPs as much as possible. Second, due to the greater the volatility risk of random factors is not necessarily bad, the retailer needs to make rationally use of the random spot market and should tap more ordering channels to improve the competitiveness of the spot market, thereby increasing his profits. Third, the producer not only needs to negotiate the spot channel in advance and make the producer’s spot market price down as much as possible but also it is necessary to reduce the volatility risk of yield, so as to ensure that the producer’s profit constantly rises.Although the findings and managerial insights are important, there are some limitations in this model. First, one major limitation is that the supply chain members are risk-neutral, not risk-averse. The presence of uncertainty will cause the members to be risk-averse in reality, and the optimal utility of supply chain members needs to be obtained through risk measurement criteria (MV, VaR, or CVaR), which ultimately affects the conditions for supply chain coordination. Second, fresh agricultural products will suffer from quality loss during the distribution process, for example, freshness decreases with time. But, we only consider the quantity loss, which will reduce the practicality of this paper. Third, option contracts are not common in reality, especially in the agricultural field, and their operations are not as simple as wholesale price contracts or revenue-sharing contracts, which is another limitation of its practice. Therefore, future research can break through the limitations of this paper and simultaneously consider risk aversion, dynamic quality loss, and other simpler contracts to explore whether it has the same coordination effect as this paper. This will provide more guidance for the management of the three-level FAP supply chain with uncertainties in yield, demand, and spot market price.



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*Source: 2991241-2022-04-30.xml*