Coordination in a Single-Retailer Two-Supplier Supply Chain under Random Demand and Random Supply with Disruption

Abstract

This paper studies the coordination issue of a supply chain consisting of one retailer and two suppliers, a main supplier and a backup supplier. The main supplier’s yield is subject to disruption and the retailer faces a random demand. We determine the retailer’s optimal ordering policy and the main supplier’s production quantity that maximize expected profit of the centralized supply chain. We also analyze the decentralized scenario, and a combination of overproduction risk sharing and buy-back contracts with a side payment from/to the backup supplier is provided to coordinate the supply chain. Numerical examples are given to gain some qualitative insights.

1. Introduction

This paper studies the coordination issue of a supply chain among one retailer and two suppliers under random demand and supply disruption. Different from the previous research that focuses on the supplier's random yield, we consider the supply disruption, during which the output of the production is zero. We also propose an overproduction risk sharing (OPRS) and buy-back contracts with a side payment for supply chain coordination. Since our work touches upon two areas of research: random yield with disruption and contracts in a supply chain with demand uncertainty, we confine ourselves to reviewing the following related works of these two topics.

In general, any production or logistics process is somewhat related to random yield; that is, with equal amount input, the output of the production usually varies. Due to damage that occurs during transferring, any transportation process can also be viewed as random yield process [1]; for example, in the liquid crystal display manufacturing industry, it is quite common to get a production yield of less than 50%. Then the actual response of the supplier to the retailer's order will be mandated by the supply contract between them; for example, if the cost of tracking partial orders is high or the transportation costs are high, the supply contract may specify delivery in a single shipment with the uncertainty in delivery time [2]; in other situations, the manufacturer may agree to accept partial shipment of the order quantity [3].

Gerchak and Grosfeld-Nir [4] analyzed the tradeoff between setup cost and production cost when making batch production decisions, where both the random yield and random demand follow a general discrete distribution. Granot and Yin [5] investigated the effect of sequential commitment in the decentralized newsvendor model with price-dependent demand, where the supplier and the retailer sequentially commit on prices and quantities. Mukhopadhyay and Ma [6] developed a single-period model to evaluate the optimal procurement and production decisions with uncertain demand and random yield of the used parts under three different cases. Wang [7] investigated the traditional and vendor-managed inventory arrangements in a supply chain and obtained the optimal production/inventory decisions under random yield and uncertain demand for both arrangements. He and Zhang [8] studied a seller/supplier commitment contract with minimum delivery commitment and analyzed the supply chain with the risk sharing contract under a constant secondary market price- and a yield-dependent secondary market price. He and Zhao [9] investigated the ordering policy of the retailer, raw material planning decision of the supplier, and the optimal contracts for a three-level supply chain with random yield and demand. Gurnani et al. [10] considered an assembly problem, where a firm faces random demand for a final product and a general yield distribution of two critical components from the individual suppliers. Using the modified cost function, they determined the combined component ordering quantity and assembly decisions for the firm. In the model of Güler and Bilgiç [11], they studied the coordination of an assembly system for arbitrary number of suppliers with random demand and random yield and established the concavity of expected supply chain profit and proposed two mixed type of contracts to coordinate the chain under forced compliance. Under the wholesale price contract, Keren [12] analyzed a two-tier supply chain, where a distributor facing a deterministic demand procures a product from a producer confronting a random production yield. And an analytical solution to the distributor's ordering decision is derived when the production yield follows the uniform distribution. Li et al. [13] extended and provided new results on the supply chain model with producer's random yield proposed by Keren [12]. They derived analytic solutions of the supply chain decisions under generalized yield distribution and pointed out that the distributor should order more than the demand if and only if his/her marginal profit from selling this product exceeds a threshold.

Supply disruption is not considered by the above-cited literature on random yield. However, in a practical reality, a supplier may be unable to satisfy the production order for a variety of reasons, such as equipment failures, damaged facilities, problems in procuring the necessary raw materials, and so forth. With more and more enterprises starting to realize that supply disruption severely affects their ability to successfully manage their own supply chains, supply disruption management has received increasing attention. Many researchers have devoted much effort to studying this issue. Based on the sample firms, Hendricks and Singhal [14] estimated the short-term effects of supply disruption such as production or shipment delays on shareholder value. They found that glitch announcements decrease shareholder value by 10.82%. Furthermore, Hendricks and Singhal [14] investigated the long-term negative effects of supply disruption on the financial performance of firms based. They found that the average abnormal stock returns of firms that experienced disruptions is nearly 40% and the firms cannot quickly recover from the negative effects of a disruption. Taking into account the disruption frequency and the loss of market share, Pochard [15] analyzed the value and the benefits of dual sourcing. Oke and Gopalakrishnan [16] investigated the types and management of risks in a retail supply chain. Yu et al. [17] studied the sourcing decision alternatives between single sourcing and dual sourcing in a two-stage supply chain, where the demand is price-sensitive and the market scale increases when a supply disruption occurs. Sarkar and Mohapatra [18] considered the risks of supply disruption due to occurrence of super, semisuper, and unique events and determined the optimal size of supply base. Tomlin [19] studied a single-product model in which a firm can source from a cheap but unreliable supplier and/or an expensive but reliable supplier. They established supplier characteristics and firm characteristics in determining the firm's optimal strategy for managing supply disruption risk. Meena et al. [20] developed an analytical model to determine the optimal number of suppliers under different failure probability, capacity, and compensation. Li et al. [21] investigated the sourcing strategy of a retailer and the pricing strategies of two suppliers in a supply chain with supply disruption. For more existing supply disruption models, we refer to the review in Snyder and Shen [22].

Since double marginalization [23] will directly lead to inefficient performance of the supply chain, coordination of activities among the different members in the supply chain is necessary for the whole supply chain's effective management. A great deal of effort has been devoted to researching the supply chain coordination issues. All kinds of popular contracts have been explored in the literature for the supply chain coordination, such as buy-back contracts or returns policies [24–26], revenue sharing contracts [27, 28], wholesale price contracts [27, 29], risk sharing contracts [30], quantity discount policies [31], quantity flexibility contracts [32], sales rebate contracts [33], and so on. Recently, Ding and Chen [34] studied how to fully coordinate a three-level supply chain with a flexible return policies by setting the rules of pricing while postponing the determination of the final contract prices. He et al. [35] studied supply chain contracts and coordination when the downstream retailer faces both effort- and price-dependent stochastic demand. In such a situation, since all traditional contracts fail to coordinate the retailer's action and distort the retailer's marginal incentive to exert system optimal effort and price, none of them can coordinate the supply chain. Hence they explored a variety of other combined contracts and found that only a properly designed returns policy with a sales rebate and a penalty contract is able to achieve channel coordination and lead to a Pareto improving win-win situation for supply chain members. Xiao et al. [36] studied coordination of a two-echelon supply chain with consumer return and random demand, where consumers also face uncertainty in their valuations for products. Granot and Yin [5] investigated the effect of sequential commitment in the decentralized newsvendor model with price-dependent demand, where the supplier and the retailer sequentially commit on prices and quantities. van der Rhee et al. [37] propose a new spanning revenue sharing contract mechanism that the most downstream entity initiates a single contract involving all upstream entities for multiechelon supply chains, and analyzed the new revenue sharing contract in the linear supply chain setting with stochastic demand. Lin et al. [38] proposed an insurance contract under which the supplier shares the risk of overstock and understock with the retailer, and showed that the insurance contract can coordinate the supply chain with a newsvendor-type product. Li et al. [39] explored a generalized supply chain model and showed the double marginalization effect in the supply chain under supply uncertainty. When the demand is deterministic, a wholesale price contract and a shortage penalty contract is developed to coordinate the supply chain. When the demand is random, two equivalent coordination contracts (called accept-all type contracts for the buyer) are proposed to induce the supplier into the systematically optimal production input level.

The works by He and Zhang [40] and Hou et al. [41] are similar to our study here. He and Zhang [40] studied the effects of random yield in a decentralized supply chain and proposed several risk sharing contracts. They also compared the results of proposed contracts and found that random yield might enhance the supply chain performance and decrease the double marginalization effect under certain conditions. But there is no alternative sources available and the yield disruption is not considered in their paper. Hou et al. [41] investigated a buy-back contract between a buyer and a backup supplier when the buyer's main supplier is subject to disruption. They derived the buyer's optimal order quantity and the backup supplier's optimal return price under the recurrent supply uncertainty and the demand uncertainty, respectively. But in their study, the coordination mechanism between the main supplier is not concerned, and they do not investigate the buyer's optimal decision when both recurrent supply uncertainty and the demand uncertainty exist.

The majority of the literature considers that the yield is a stochastic proportion of the planned quantity and does not consider the coordination and the yield disruption in the chain, and most of the existing literature that studies the coordination mechanism is confined to the supply chain consisting of one retailer and one supplier. In this paper, we integrate the works of He and Zhang [40] and Hou et al. [41] and study the retailer's optimal ordering policy and the main supplier's optimal planned production quantity under random demand and random supply with disruption. We investigate the coordination between one retailer and two suppliers—one is a main source and the other is a backup, and show that two contracts with three components: overproduction risk sharing, buy-back, and a payment from/to the backup supplier cannot only coordinate our supply chain but also divide the whole chain's expected profit at any proportion among the three members.

The rest of the paper is organized as follows. In Section 2, we provide a description of the model. In Section 3, we study optimal solution for the integrated supply chain. In Section 4, both the retailer's decision and the main supplier's decision are given in the decentralized scenario. In Section 5, we argue that simple wholesale and overproduction risk sharing contract cannot coordinate the chain. A combination of overproduction risk sharing and buy-back contracts with a side payment from the backup supplier to the main supplier is provided to coordinate the chain. Numerical examples to illustrate further insights are provided in Section 5. Finally, some conclusions and further research topics are given in Section 6.

2. Model Description

We consider a single-period supply chain consisting of one retailer and two supplier with the assumption that the information is symmetrical, where one supplier (called main supplier) is unreliable with cheaper wholesale price, and the other supplier (called backup supplier) is perfectly reliable with more expensive wholesale price. This supply phenomenon is often seen in the off-shoring situation [42].

We use 𝐿 to denote the main supplier's decision on how many products to produce, and assume that the yields of the supplier will be 𝑌𝐿, where 𝑌 is a random yield variable and positive support on [0,1] which satisfies 𝑃{𝑌=0}=𝑝 and 𝑃{0<𝑌⩽1}=∫¹₀𝑔(𝑦)𝑑𝑦=1−𝑝=𝑞, where 𝑔(𝑦) is a nonnegative function (note that the yield is subject to disruption with a probability 𝑝, and 𝑔(𝑦) is not a probability density function unless 𝑝=0, i.e., there is no disruption). We assume that the backup supplier's production has a perfect yield, for example, the backup supplier can convert similar or better products from his inventory to satisfy the order. We also assume that the retailer faces a random demand, 𝑋, with cumulative distribution function, 𝐹(𝑥), probability density function, 𝑓(𝑥), and 𝐸(𝑋)=𝜇_𝑋.

The main supplier's per unit production cost is 𝑐_𝑚, and a marginal cost 𝛼𝑐_𝑚 is incurred in the event of a disruption [21]. The unit understock cost to the retailer is 𝑐_𝑢 (this cost may include loss of reputation). In order to mitigate supply risks, the retailer orders 𝑄 units from the backup supplier at the beginning of the selling seasons. The main supplier's whole price is 𝑤_𝑚 per delivery unit, and the backup supplier charges the retailer a wholesale price 𝑤_𝑏 per delivery unit. The main supplier's production cost of each planned unit is 𝑐_𝑚, and the unit production cost of the backup supplier is 𝑐_𝑏. The retailer sells the products to the end customer at the unit price 𝑠. At the end of the selling season, all unsold products will be salvaged at a value of 𝑣 per unit.

Based on the reliability of the backup supplier, it is reasonable to assume that 𝑐_𝑏>𝑐_𝑚>𝑣. In addition, we assume that 𝑠>𝑤_𝑏>𝑐_𝑏 and 𝑠>𝑤_𝑚>𝑐_𝑚. These inequalities make sure that each member has a positive profit.

3. Centralized Model

To establish a performance benchmark, we first analyze the optimal solution of an integrated supply chain. In the centralized model, the expected integrated supply chain profit is given by Π_𝑐(𝑄,𝐿)=𝑠𝐸min(𝑋,𝑌𝐿+𝑄)+𝑣𝐸(𝑌𝐿+𝑄−𝑋)⁺−𝑐_𝑢𝐸(𝑋−𝑌𝐿−𝑄)⁺−𝑐_𝑏𝑄−𝛿𝑐_𝑚𝐿, (1) where 𝛿=𝛼𝑝+𝑞. The first term in (1) is the expected revenue from sales, the second term is the salvaged value, the third term is the opportunity costs due to the lost sales, and the last two terms are the production costs.

From (1), the integrated supply chain's expected profit function can be rewritten as follows: Π_𝑐(𝑄,𝐿)=𝑝^𝑄₀[(𝑓(𝑥)+𝑠𝑥+𝑣𝑄−𝑥)]𝑑𝑥_∞𝑄𝑠𝑄−𝑐_𝑢(𝑓(𝑥)+𝑥−𝑄)𝑑𝑥_{100𝑦𝐿+𝑄}𝑓(𝑥)+𝑠𝑥+𝑣𝑦𝐿+𝑄−𝑥𝑑𝑥_{∞𝑦𝐿+𝑄}𝑠𝑦𝐿+𝑄−𝑐_𝑢𝑓(𝑥)𝑦𝑥−𝑦𝐿−𝑄𝑑𝑥×𝑔𝑑𝑦−𝑐_𝑏𝑄−𝛿𝑐_𝑚𝐿. (2) The following Proposition 1 states that the objective function in (2) is concave. Hence, the optimal production quantity and order quantity from the backup supplier can be determined easily. All proofs, if not provided in the paper, are in the appendix.

Proposition 1. Π_𝑐(𝑄,𝐿) is jointly concave in 𝑄 and 𝐿.

From Proposition 1, we can characterize the optimal order quantity, 𝑄^𝑐, and the optimal planned production quantity, 𝐿^𝑐, through the first-order conditions. In particular, we have the following Proposition 2.

Proposition 2. (1) If 𝛿𝑐_𝑚<𝑐_𝑏𝜇_𝑌 and ∫¹₀𝐹(𝑦𝐿₀)𝑔(𝑦)d𝑦<(𝑠−𝑐_𝑏+𝑐_𝑢)/(𝑠−𝑣+𝑐_𝑢), then (𝑄^c,𝐿^𝑐) is uniquely solved by 𝑝𝐹𝑄^𝑐+¹₀𝐹𝑦𝐿^𝑐+𝑄^𝑐𝑔𝑦d𝑦=𝑠−𝑐_𝑏+𝑐_𝑢𝑠−𝑣+𝑐_𝑢10,𝑦𝐹𝑦𝐿^𝑐+𝑄^𝑐𝑔𝑦d𝑦=𝑠+𝑐_𝑢𝜇_𝑌−𝛿𝑐_𝑚𝑠−𝑣+𝑐_𝑢, (3)(4) where 𝐿₀ satisfies ∫¹₀𝑦𝐹(𝑦𝐿₀)𝑔(𝑦)d𝑦=(𝜇_𝑌(𝑠+𝑐_𝑢)−𝛿𝑐𝑚)/(𝑠−𝑣+𝑐_𝑢).
(2) If 𝛿𝑐_𝑚<𝜇_𝑌𝑐_𝑏 and ∫¹₀𝐹(𝑦𝐿₀)𝑔(𝑦)d𝑦⩾(𝑠−𝑐_𝑏+𝑐_𝑢)/(𝑠−𝑣+𝑐_𝑢), then (𝑄^𝑐,𝐿^𝑐)=(0,𝐿₀).
(3) If 𝛿𝑐_𝑚⩾𝜇_𝑌𝑐_𝑏, then (𝑄^𝑐,𝐿^𝑐)=(𝐹⁻¹(1−(𝑐_𝑏−𝑣)/(𝑠−𝑣+𝑐_𝑢)),0).

From Proposition 2, we can obtain the maximum expected profit of the integrated supply chain as follows: Π_𝑐𝑄^𝑐,𝐿^𝑐=𝑠−𝑣+𝑐_𝑢𝜋𝑄^𝑐,𝐿^𝑐−𝑐_𝑢𝜇_𝑋, (5) where 𝜋(𝑄^𝑐,𝐿^𝑐)=∫¹₀∫^{𝑦𝐿𝑐+𝑄𝑐}₀𝑥𝑓(𝑥)𝑔(𝑦)𝑑𝑥𝑑𝑦+𝑝∫^𝑄𝑐₀𝑥𝑓(𝑥)𝑑𝑥.

4. Decentralized Model under a Traditional Arrangement

Now we consider the decentralized scenario under the traditional arrangement; that is, the two suppliers and the retailer are independent agents and derive their individual expected profit, respectively. At the beginning of the selling season, the retailer places an order quantity 𝑅 from the main supplier, and orders 𝑄 units of products from the backup supplier in order to mitigate supply risks. It is worth mentioning that the retailer does not charge the main supplier any penalty for the unsatisfied orders caused by the supply risks, but when its delivery quantity is higher than 𝑅, the retailer accepts only the amount equal to 𝑅.

4.1. Main Supplier's Production decision

For given order quantity 𝑅, the expected profit of the main supplier is Π_𝑚(𝐿)=𝑤_𝑚𝐸min(𝑌𝐿,𝑅)+𝑣𝐸(𝑌𝐿−𝑅)⁺−𝛿𝑐_𝑚𝐿. (6) Here, the first term is the supplier's revenue from the retailer's order, the second term is the salvaged value, and the last term is the production costs.

Proposition 3. For given 𝑅, the main supplier's optimal planned production quantity 𝐿^∗(𝑅)=𝜁𝑅, where 𝜁 is independent of 𝑅 and is determined by the following: 𝑤_𝑚−𝑣₀^1/𝜁𝑦𝑔𝑦d𝑦=𝛿𝑐_𝑚−𝑣𝜇_𝑌. (7)

Proof. Taking the first and second derivatives in (6), we obtain 𝑑Π_𝑚=𝑤𝑑𝐿^𝑚−𝑣₀^𝑅/𝐿𝑦𝑔𝑦𝑑𝑦+𝑣𝜇_𝑌−𝛿𝑐_𝑚,𝑑²Π_𝑚𝑑𝐿²=−𝑤_𝑚𝑅−𝑣²𝐿³𝑔𝑅𝐿<0. (8)(9) It follows from (9) that Π_𝑚 is concave. Hence, setting 𝑑Π_𝑚/𝑑𝐿=0, we can derive (7).

4.2. The Retailer's Ordering Policy

Taking into account the response from the main supplier, the retailer's expected profit, Π_𝑟(𝑄,𝑅), is given by Π_𝑟(𝑄,𝑅)=𝑠𝐸min(𝑋,𝑅+𝑄,𝑌𝜁𝑅+𝑄)+𝑣𝐸(𝑌𝜁𝑅+𝑄−𝑋)⁺−𝑐_𝑢𝐸𝑋−𝑅min(1,𝜁𝑌)−𝑄⁺−𝑤_𝑚𝑅𝐸min(1,𝜁𝑌)−𝑤_𝑏𝑄. (10) The first term in (10) is the expected revenue from the sales, the second term is the earnings from salvaging the unsold products, the third term is the understock cost, and the last two terms are the cost of buying products from the two suppliers.

From (10), using the same argument as Proposition 1, we can obtain that Π_𝑟(𝑄,𝑅) is jointly concave in 𝑄 and 𝑅, and the unique (𝑄^𝑑,𝑅^𝑑) that maximizes Π_𝑟(𝑄,𝑅) can be derived as follows.

Proposition 4.  (1) If 𝑤_𝑚<𝑤_𝑏 and 𝐹(𝑅₀)𝐺(1/𝜁)+∫₀^1/𝜁𝐹(𝑦𝜁𝑅₀)𝑔(𝑦)d𝑦<(𝑠−𝑤_𝑏+𝑐_𝑢)/(𝑠−𝑣+𝑐_𝑢), then (𝑄^𝑑,𝑅^𝑑) is uniquely solved by 𝑝𝐹𝑄^𝑑+𝐹𝑄^𝑑+𝑅^𝑑𝐺1𝜁+₀^1/𝜁𝐹𝑄^𝑑𝑔𝑦=+𝑦𝜁𝑅d𝑦𝑠−𝑤_𝑏+𝑐_𝑢𝑠−𝑣+𝑐_𝑢,𝐹𝑄^𝑑+𝑅^𝑑𝐺1𝜁+₀^1/𝜁𝐹𝑄^𝑑𝑦=+𝑦𝜁𝑅𝜁𝑦𝑔𝑑𝑦𝑠−𝑤_𝑚+𝑐_𝑢𝑠−𝑣+𝑐_𝑢𝐺1𝜁+𝜁𝛿𝑐_𝑚−𝑣𝜇_𝑌𝑤_𝑚,−𝑣 (11) where 𝑅₀ satisfying 𝐹(𝑅₀)𝐺(1/𝜁)+∫₀^1/𝜁𝐹(𝑦𝜁𝑅₀)𝜁𝑦𝑔(𝑦)d𝑦=((𝑠−𝑤_𝑚+𝑐_𝑢)/(𝑠−𝑣+𝑐_𝑢))[𝐺(1/𝜁)+(𝜁(𝛿𝑐_𝑚−𝑣𝜇_𝑌)/(𝑤_𝑚−𝑣))] and 𝐺(𝑦)=∫¹_𝑦𝑔(𝑦)d𝑦.
(2) If 𝑤_𝑚<𝑤_𝑏 and 𝐹(𝑅₀)𝐺(1/𝜁)+∫₀^1/𝜁𝐹(𝑦𝜁𝑅₀)𝑔(𝑦)d𝑦⩾(𝑠−𝑤_𝑏+𝑐_𝑢)/(𝑠−𝑣+𝑐_𝑢), then (𝑄^𝑑,𝑅^𝑑)=(0,𝑅₀).
(3) If 𝑤_𝑚⩾𝑤_𝑏, then (𝑄^𝑑,𝑅^𝑑)=(𝐹⁻¹((𝑠−𝑤_𝑏+𝑐_𝑢)/(𝑠−𝑣+𝑐_𝑢)),0).

Proof. Similar to the proof of Proposition 2.

Given the retailer's optimal order quantity 𝑄^𝑑 and 𝑅^𝑑, the retailer's expected profit is Π^𝑑_𝑟=𝑠−𝑣+𝑐_𝑢×𝜋𝑄^𝑑,𝜁𝑅^𝑑−¹_1/𝜁^{𝑦𝜁𝑅𝑑+𝑄𝑑}_{𝑅𝑑+𝑄𝑑}𝑥𝑓(𝑥)𝑔𝑦𝑑𝑥𝑑𝑦−𝑐_𝑢𝜇_𝑋, (12) and the backup supplier's expected profit and the main supplier's expected profit are Π^𝑑_𝑏=(𝑤_𝑏−𝑐_𝑏)𝑄^𝑑 and Π^𝑑_𝑚=(𝑤_𝑚−𝑣)𝐺(1/𝜁)𝑅^𝑑, respectively. Let Π_𝑑(𝑄^𝑑,𝑅^𝑑) denote the total expected profit of the decentralized supply chain, then we have the following proposition.

Proposition 5. Π_𝑑(𝑄^𝑑,𝑅^𝑑)<Π_𝑐(𝑄^𝑐,𝐿^𝑐).

Proposition 5 indicates that the decentralized supply chain's expected profit will be lower than that of an integrated supply chain. This phenomenon is well known as double marginalization [23]. It also shows the importance of supply chain coordination. From the main supplier's first-order condition; if 𝑤_𝑚→𝛿𝑐_𝑚/𝜇_𝑌, we have 𝜁→1 and 𝐿^𝑑(𝑅)→𝑅. Furthermore, if 𝑤_𝑏→𝑐_𝑏, then we have 𝐿^𝑑=𝐿^𝑐 and 𝑄^𝑑=𝑄^𝑐 by the retailer's first-order conditions (11). In other words, the total expected profit of the decentralized supply chain is closer to the integrated system as the two supplier's wholesale price approaches their production cost, respectively.

5. Supply Chain Coordination

5.1. Overproduction Risk Sharing Contract

Risk sharing is one of the most common method of achieving supply chain coordination. This method aligns the objectives of the supply chain members and coordinates their activities to optimize system performance. The OPRS contract ensures the main supplier's risk of producing too many units (compared to the quantity ordered) being shared by the retailer [40]. Under the OPRS contract, the retailer commits to pay for all units produced by the main supplier. But he only pays the wholesale price 𝑤_𝑚 per unit for the order quantity 𝑅, and quantities that exceed this amount are compensated at a discount price 𝑤_𝑑 per unit. In order to guarantee that the main supplier is willing to sell the surplus products to the retailer and prevent the main supplier from producing an unlimited amount, we assume that 𝑣<𝑤_𝑑<𝛿𝑐_𝑚/𝜇_𝑌.

Under the OPRS contract, the main supplier's expected profit is Π_𝑚(𝐿)=𝑤_𝑚𝐸min(𝑌𝐿,𝑅)+𝑤_𝑑𝐸(𝑌𝐿−𝑅)⁺−𝛿𝑐_𝑚𝐿. (13)

Proposition 6. For given 𝑅, the main supplier's optimal planned production quantity 𝐿^𝑑(𝑅)=𝜁_𝑐𝑅, where 𝜁_𝑐 is independent of 𝑅 and satisfies 𝑤_𝑚−𝑤_𝑑^1/𝜁𝑐₀𝑦𝑔𝑦d𝑦=𝛿𝑐_𝑚−𝑤_𝑑𝜇_𝑌. (14)

Proof. Similar to the proof of Proposition 3.

Given the main supplier's best response function 𝐿^𝑑(𝑅)=𝜁_𝑐𝑅, the retailer's expected profit can be written as follows: Π_𝑟(𝑄,𝑅)=𝑠𝐸min(𝑋,𝑌𝐿+𝑄)+𝑣𝐸(𝑌𝐿+𝑄−𝑋)⁺−𝑐_𝑢𝐸[𝑋−𝑌𝐿−𝑄]⁺−𝑤_𝑚𝐸min𝜁_𝑐𝑌,1+𝑤_𝑑𝐸𝜁_𝑐𝑌−1⁺𝑅−𝑤_𝑏𝑄. (15)

Proposition 7. The OPRS contract cannot coordinate the supply chain.

Although the OPRS contract can reduce the main supplier's overproduction risk, it increases the retailer's overstock risk. Hence, the retailer will not choose the order quantity that is the same as the integrated supply chain. But if the suppliers are willing to pay the retailer back for unsold products, then the supply chain coordination may be achieved. In the next subsection, we will construct a combined contract with a side payment to coordinate the supply chain.

5.2. Overproduction Risk Sharing with Buy-Back and a Payment from/to the Backup Supplier

This contract is a mixture of overproduction risk sharing and buy-back contracts. For simplification, it is assumed that the main supplier deliver all of his products at the whole price 𝑤_𝑚 (i.e., w_𝑚=𝑤_𝑑) whether his actual output of the product is more than the retailer's order quantity or not, and he will buy all of the retailer's unsold products back with a return price 𝑤_𝑟 per unit at the end of the selling season; for example, the main supplier can disassemble the unsold products and remanufacture at a lower production cost. We assume that 𝑤_𝑟−𝑣>𝑤_𝑚−𝛿𝑐_𝑚/𝜇_𝑌 to avoid the triviality of the main supplier producing an infinite amount (this inequality implies that the main supplier's compensation for a unsold product is more than the profit that he earns for the sale of it). Note that the main supplier may return the unsold quantities that exceed his output, for example, the unsold products will be 𝑌𝐿+𝑄−𝑋>𝑌𝐿 if 𝑋<𝑄 (in this case, the main supplier also returns some of the backup supplier's products). Hence, in order to compensate for the main supplier's product recycling risk, the backup supplier pays a side payment 𝑇(>0) to him. If 𝑇<0, it can be interpreted that the main supplier gives some reimbursement to the backup supplier for coordination, since he produce more products and earns more profit when the supply chain is coordinated. Hence the parameter 𝑇 is adopted to split the expected profit of the coordinated system between two suppliers, and the parameter 𝑇 can be interpreted as that the two suppliers share their expected profits.

Under the above combined contract, the expected profit for the retailer, the main supplier and the backup supplier are given by Π_𝑟=𝑠𝐸[min(𝑋,𝑌𝐿+𝑄)]+𝑤_𝑟𝐸(𝑌𝐿+𝑄−𝑋)⁺−𝑤_𝑚𝐸(𝑌𝐿)−𝑤_𝑏Π𝑄,_𝑚=𝑤_𝑚𝐸(𝑌𝐿)−𝑤_𝑟−𝑣𝐸(𝑌𝐿+𝑄−𝑋)⁺−𝛿𝑐_𝑚Π𝐿+𝑇,_𝑏=𝑤_𝑏−𝑐_𝑏𝑄−𝑇, (16)(17)(18) respectively.

Proposition 8. For given 𝑄, if 𝑄<𝐹⁻¹((𝑤_𝑚−𝛿𝑐_𝑚/𝜇_𝑌)/(𝑤_𝑟−𝑣)), then the main supplier's optimal planned production quantity 𝐿^𝑑(𝑄) is uniquely solved from 𝑤_𝑟−𝑣¹₀𝐹𝑦𝐿^𝑑(𝑄)+𝑄𝑦𝑔𝑦d𝑦=𝑤_𝑚𝜇_𝑌−𝛿𝑐_𝑚, (19) otherwise if 𝑄⩾𝐹⁻¹((𝑤_𝑚−𝛿𝑐_𝑚/𝜇_𝑌)/(𝑤_𝑟−𝑣)), then 𝐿^𝑑(𝑄)=0.

Proof. Taking the first and second derivatives of (15) with respect to 𝐿, we have 𝑑Π_𝑚𝑑𝐿=𝑤_𝑚𝜇_𝑌−𝛿𝑐_𝑚−𝑤_𝑟−𝑣¹₀𝐹𝑦𝑑𝑦𝐿+𝑄𝑦𝑔𝑑𝑦,₂Π_𝑚𝑑𝐿²=−𝑤_𝑟−𝑣¹₀𝑓𝑦𝐿+𝑄𝑦²𝑔𝑦𝑑𝑦<0. (20)(21) It follows from (20) that Π_𝑚 is concave on 𝐿 for given 𝑄. If 𝑄<𝐹⁻¹((𝑤_𝑚−𝛿𝑐_𝑚/𝜇_𝑌)/(𝑤_𝑟−𝑣)), then lim_𝐿→0(𝑑Π_𝑚/𝑑𝐿)=[𝑤_𝑚−(𝑤_𝑟−𝑣)𝐹(𝑄)]𝜇_𝑌−𝛿𝑐_𝑚>0 and lim_𝐿→∞(𝑑Π_𝑚/𝑑𝐿)=(𝑤_𝑚−𝑤_𝑟+𝑣)𝜇_𝑌−𝛿𝑐_𝑚<0, and hence there exists a unique 𝐿^𝑑(𝑅) that satisfies the first-order condition in (18). Otherwise if 𝑄⩾𝐹⁻¹((𝑤_𝑚−𝛿𝑐_𝑚/𝜇_𝑌)/(𝑤_𝑟−𝑣)), then 𝑑Π_𝑚/𝑑𝐿⩽0 for 𝐿>0 and hence 𝐿^𝑑(𝑄)=0.

Set 𝑤₀=𝑠+𝑐_𝑢−𝑠−𝑣+𝑐_𝑢×𝜋𝑄^𝑑,𝜁𝑅^𝑑𝜋(𝑄^𝑐,𝐿^𝑐)−1𝜋(𝑄^𝑐,𝐿^𝑐)¹_1/𝜁^{𝑦𝜁𝑅𝑑+𝑄𝑑}_𝑅^𝑑_+𝑄^𝑑(𝑥)𝑔𝑦,𝑇𝑥𝑓𝑑𝑥𝑑𝑦_min=𝑤_𝑟−𝑣𝜋𝑄^𝑐,𝐿^𝑐−𝑠−𝑐_𝑏+𝑐_𝑢𝑄^𝑐𝑠−𝑣+𝑐_𝑢−𝑤_𝑚−𝑣𝐺1𝜁𝑅^𝑑,𝑇_max=𝑤_𝑏−𝑐_𝑏𝑄^𝑐−𝑄^𝑑 (22)(23)(24) then we have the following proposition.

Proposition 9. Under the combined contracts, if 𝑤_𝑚=𝛿𝑐_𝑚𝜇_𝑌+𝑠+𝑐_𝑢𝜇_𝑌−𝛿𝑐_𝑚𝑤_𝑟−𝑣𝑠−𝑣+𝑐_𝑢𝜇_𝑌,𝑤_𝑏=𝑐_𝑏+𝑠−𝑐_𝑏+𝑐_𝑢𝑤_𝑟−𝑣𝑠−𝑣+𝑐_𝑢, (25)𝑤_𝑟∈[𝑣,𝑤₀] and 𝑇∈[𝑇_min,𝑇_max], then the supply chain can be coordinated.

Proposition 10. Under the combined contract and contract parameters in (25), an arbitrary allocation of the optimal supply chain profit among the three members can be achieved by varying 𝑤_𝑟 and 𝑇.

Proposition 9 shows that the expected profit of the integrated supply chain can be shared with any specified ratios among three members. A simply effective but not exclusive way to set the profit allocated ratios is to let them equal to the profit ratios between the corresponding members before introducing the contracts, which can certainly ensure that each entity earns more as the contracts increase the total profit of the channel.

6. Numerical Examples

In this section, we provide some numerical examples to gain further insights. We assume that the retailer's demand follows a normal distribution with mean 𝜇_𝑋=1000 and standard deviation 𝜎_𝑋=200. We also specify that 𝑔(𝑦)=𝑞/(𝑏−𝑎), where 0⩽𝑎<𝑏⩽1. Hence, the main supplier's yield has a mean of 𝜇_𝑌=(𝑎+𝑏)𝑞/2 and a standard deviation of 𝜎_𝑌=[(4−3𝑞)(𝑎+𝑏)²−4𝑎𝑏]𝑞/(2√3). The base values of the parameters are set as follows: 𝑝=0.25; 𝑐_𝑚=4, 𝑐_𝑏=11, 𝑐_𝑢=2, 𝑣=3, 𝑠=25, 𝑤_𝑚=16, 𝑤_𝑏=18, 𝛼=0.25, 𝑎=0.4, 𝑏=1, 𝜇_𝑋=100, 𝜎_𝑋=50.

6.1. Effect of the Disruption Probability

Under different values of 𝑝, we plot the optimal solutions in Figure 1 and plot the corresponding optimal profits in Figure 2, for the centralized and decentralized systems, respectively.

Figure 1 

Effect of disruption probability on optimal solutions.

Figure 2 

Effect of disruption probability on optimal profits.

Figure 1 shows that the order quantity from the backup supplier increases and the planned production quantity of the main supplier decreases as 𝑝 increases. Since larger disruption probability results in greater supply risk of the main supplier, the retailer will order more products from the backup supplier who has no production risk for avoiding risk increases, and the main supplier will decrease his production in order to reduce the production risk.

Figure 2 shows that an increase in the disruption probability 𝑝 decreases the profits of both the retailer and the main supplier. But this is not the case for the backup supplier. Since the retailer's order quantity from the backup supplier increases with the disruption probability. For all values of 𝑝, the integrated supply chain's profit is significantly lager than the total expected profit of the three members in the decentralized supply chain.

6.2. Effect of the Random Uncertainty and Demand Uncertainty

Tables 1 and 2 present the profits of the supply chain with and without coordination under different yield and different demand, respectively, where (𝑄^∗,𝐿^∗,𝜁^∗)=(𝑄^𝑑,𝐿^𝑑,𝐿^𝑑/𝑅^𝑑) in the decentralized model and (𝑄^∗,𝐿^∗,𝜁^∗)=(𝑄^𝑐,𝐿^𝑐,1) in the centralized model. Table 1 shows that the profits of three members decrease as the yield uncertainty increases, which gives the main supplier an incentive to decrease the planned production quantity for reducing yield uncertainty. Since the retailer also greatly benefits from the decrease in the yield uncertainty, he has an incentive to help the supplier to reduce the uncertainty through increasing the return price, and to place a larger order from the backup supplier for transferring the main supplier's supply risk. Table 2 indicates that lower demand uncertainty benefits the retailer and the backup supplier, but this is not the case for the main suppliers. Since the retailer will order smaller quantities as the demand variability decreases, it will directly reduce the main supplier's profit. We can also from Table 2 find that 𝜁^∗ is invariant. This is true for the reason that 𝜁^∗ is independent of the demand.

Table 1 

Impact of different yield variabilities on the supply chain.

Table 2 

Impact of different demand variabilities on the supply chain.

From Tables 1 and 2, we find that the retailer's expected profit significantly decreases and as the suppliers' whole price increase without coordination. Supply chain coordination greatly benefits the retailer and the whole supply chain, and this becomes more obvious when the demand uncertainty is more severe. We also see that the coordinated supply chain can achieve Pareto improvement by choosing the appropriate contract parameters 𝑤_𝑟 and 𝑇.

7. Conclusion

In this paper, we consider a supply chain including one retailer and two suppliers under random demand and random yield, where the yield is subject to disruption with a determinate probability. Both the centralized scenario and decentralized scenario are studied, and the retailer's optimal ordering policy and the main supplier's optimal planned production quantity are provided. In the decentralized setting, we propose a combined contract that is a combination of OPRS and buy-back contracts with an additional side payment to component regarding the worse performing supplier. With the combined contract, the appropriate whole price and return price can coordinate the supply and allocate the integrate supply chain's profit with any specified ratios between the retailer and the two suppliers, and the side payment can further split the profit between the main supplier and the backup supplier. Numerical examples show that coordination can greatly benefit the retailer and the whole supply chain, and that the coordinated supply chain can achieve Pareto improvement by choosing the appropriate contract parameters.

One direct extension of this work is to study the proposed model under risk measures such as expected utility objective, mean-variance criterion, and conditional value-at-risk. Another extension is to consider our problems with asymmetric information on various factors such as the random yield and the random demand. We leave these problems for further research.

Appendix

Proof of Proposition 1. From (1), the first-order and second-order partial derivatives with respect to 𝐿 and 𝑄, respectively, are given by 𝜕Π_𝑐𝜕𝑄=−𝑠−𝑣+𝑐_𝑢𝑝𝐹(𝑄)+¹₀𝐹𝑦𝐿+𝑄𝑔𝑦𝑑𝑦+𝑠−𝑐_𝑏+𝑐_𝑢,𝜕Π_𝑐𝜕𝐿=−𝑠−𝑣+𝑐_𝑢¹₀𝑔𝑦+𝑦𝐹𝑦𝐿+𝑄𝑑𝑦𝑠+𝑐_𝑢𝜇_𝑌−𝛿𝑐_𝑚,𝜕²Π_𝑐𝜕𝑄²=−𝑠−𝑣+𝑐_𝑢𝑝𝑓(𝑄)+¹₀𝑓𝑔𝑦𝜕𝑦𝐿+𝑄𝑑𝑦<0,₂Π_𝑐𝜕𝐿²=−𝑠−𝑣+𝑐_𝑢¹₀𝑦²𝑓𝑔𝑦𝜕𝑦𝐿+𝑄𝑑𝑦<0,₂Π_𝑐=𝜕𝜕𝑄𝜕𝐿²Π_𝑐𝜕𝐿𝜕𝑄=−𝑠−𝑣+𝑐_𝑢¹₀𝑦𝑓𝑦𝐿+𝑄𝑔𝑦𝑑𝑦. (A.1)(A.2)(A.3)(A.4)(A.5) Furthermore, it is easy to verify that 𝜕²Π_𝑐𝜕𝑄²𝜕²Π_𝑐𝜕𝜕𝑄𝜕𝐿₂Π_𝑐𝜕𝜕𝐿𝜕𝑄²Π_𝑐𝜕𝐿²=𝜕²Π_𝑐𝜕𝑄²𝜕²Π_𝑐𝜕𝐿²−𝜕²Π_𝑐𝜕𝑄𝜕𝐿²=𝑝𝑠−𝑣+𝑐_𝑢²𝑓(𝑄)¹₀𝑓𝑔𝑦−𝑦𝐿+𝑄𝑑𝑦𝑠−𝑣+𝑐_𝑢²¹₀𝑦²𝑓𝑔𝑦×𝑦𝐿+𝑄𝑑𝑦₁₀𝑓𝑔𝑦−𝑦𝐿+𝑄𝑑𝑦₁₀𝑦𝑓𝑦𝐿+𝑄𝑔𝑦𝑑𝑦²⩾𝑝𝑠−𝑣+𝑐_𝑢²𝑓(𝑄)¹₀𝑓𝑦𝐿+𝑄𝑔𝑦𝑑𝑦>0, (A.6) where the the first inequality follows from the Cauchy-Schwarz inequality. Therefore, the Hessian matrix of Π(𝑄,𝐿) is a negative definite matrix, which implies that Π_𝑐(𝑄,𝐿) is jointly concave in (𝑄,𝐿).

Proof of Proposition 2. (i) From (A.4), it is obvious that 𝜕Π_𝑐/𝜕𝐿 is a decreasing function of 𝐿 for given 𝑄. Let 𝑄₀=𝐹⁻¹(((𝑠+𝑐_𝑢)𝜇_𝑌−𝛿𝑐_𝑚)/(𝑠−𝑣+𝑐_𝑢)𝜇_𝑌), then for given 𝑄∈[0,𝑄₀], since (𝜕Π_𝑐/𝜕𝐿)|_𝐿=0>0 and (𝜕Π_𝑐/𝜕𝐿)|_𝐿→∞=𝑣𝜇_𝑌−𝛿𝑐_𝑚<0, there exists a unique optimal 𝐿^𝑐(𝑄) that satisfies the first-order condition in (4). Substituting 𝐿=𝐿^𝑐(𝑄) into (2), we have 𝜕Π_𝑐𝑄,𝐿^𝑐(𝑄)𝜕𝑄=−𝑠−𝑣+𝑐_𝑢×𝑝𝐹(𝑄)+¹₀𝐹𝑦𝐿^𝑐₁(𝑄)+𝑄𝑔𝑦𝑑𝑦+𝑠−𝑐_𝑏+𝑐_𝑢. (A.7) Taking the first-order derivative with respect to 𝑄 in (A.1), we get ¹₀𝑦𝑦𝑑𝐿^𝑐(𝑄)𝑓𝑑𝑄+1𝑦𝐿^𝑐(𝑄)+𝑄𝑔𝑦𝑑𝑦=0. (A.8) From (A.8), we have 𝑑𝐿^𝑐(𝑄)∫𝑑𝑄=−¹⁰𝑦𝑓𝑦𝐿^𝑐(𝑄)+𝑄𝑔𝑦𝑑𝑦∫¹₀𝑦²𝑓𝑦𝐿^𝑐(𝑄)+𝑄𝑔𝑦𝑑𝑦<−1 (A.9) for all 𝑄∈[0,𝑄₀]. By (A.8)-(A.9), we can conclude that there exists a unique constant 𝜆∈(0,1) such that 𝜆𝑑𝐿^𝑐(𝑄)=𝑑𝑄⩽−1,𝑦⩽𝜆,>−1,𝑦>𝜆. (A.10) Since 0=^𝜆₀𝑦𝑦𝑑𝐿^𝑐(𝑄)𝑓𝑑𝑄+1𝑦𝐿^𝑐(𝑄)𝑔𝑦++𝑄𝑑𝑦_1𝜆𝑦𝑦𝑑𝐿^𝑐(𝑄)𝑓𝑑𝑄+1𝑦𝐿^𝑐(𝑄)+𝑄𝑔𝑦𝑑𝑦⩽𝜆^𝜆₀𝑦𝑑𝐿^𝑐(𝑄)𝑓𝑑𝑄+1𝑦𝐿^𝑐(𝑄)𝑔𝑦++𝑄𝑑𝑦_1𝜆𝑦𝑑𝐿^𝑐(𝑄)𝑓𝑑𝑄+1𝑦𝐿^𝑐(𝑄)𝑔𝑦⩽+𝑄𝑑𝑦₁₀𝑦𝑑𝐿^𝑐(𝑄)𝑓𝑑𝑄+1𝑦𝐿^𝑐(𝑄)+𝑄𝑔𝑦𝑑𝑦, (A.11) hence, we have 𝑑²Π_𝑐𝑄,𝐿^𝑐(𝑄)𝑑𝑄²=−𝑝𝑓(𝑄)−¹₀𝑦𝑑𝐿^𝑐(𝑄)𝑓𝑑𝑄+1𝑦𝐿^𝑐(𝑄)+𝑄𝑔𝑦𝑑𝑦<0, (A.12) which implies that Π_𝑐(𝑄,𝐿^𝑐(𝑄)) is concave in 𝑄.
As a result, if 𝛿𝑐_𝑚<𝑐_𝑏𝜇_𝑌 and ∫¹₀𝐹(𝑦𝐿₀)𝑔(𝑦)𝑑𝑦<(𝑠−𝑐_𝑏+𝑐_𝑢)/(𝑠−𝑣+𝑐_𝑢), then 𝑄₀>𝐹⁻¹((𝑠−𝑐_𝑏+𝑐_𝑢)/(𝑠−𝑣+𝑐_𝑢)), and we have (𝑑Π_𝑐(𝑄,𝐿^𝑐(𝑄))/𝑑𝑄)|_𝑄=𝑄0=𝛿𝑐_𝑚/𝜇_𝑌−𝑐_𝑏<0 and (𝑑Π_𝑐(𝑄,𝐿^𝑐(𝑄))/𝑑𝑄)|_𝑄=0=−(𝑠−𝑣+𝑐_𝑢)∫¹₀𝐹(𝑦𝐿₀)𝑔(𝑦)𝑑𝑦+𝑠−𝑐_𝑏+𝑐_𝑢>0. Hence there exists a unique 𝑄^𝑐∈(0,𝑄₀) such that 𝑑Π_𝑐(𝑄,𝐿^𝑐(𝑄))/𝑑Q=0, and the optimal 𝐿^𝑐(𝑄) is solved from ∫¹₀𝑦𝐹(𝑦𝐿^𝑐(𝑄^𝑐)+𝑄^𝑐)𝑔(𝑦)𝑑𝑦=((𝑠+𝑐_𝑢)𝜇_𝑌−𝛿𝑐_𝑚)/(𝑠−𝑣+𝑐_𝑢). That is, (𝑄^𝑐,𝐿^𝑐) can be uniquely solved by the first-order conditions in (3)-(4).
(ii) If 𝛿𝑐_𝑚<𝑐_𝑏𝜇_𝑌 and ∫¹₀𝐹(𝑦𝐿₀)𝑔(𝑦)𝑑𝑦⩾(𝑠−𝑐_𝑏+𝑐_𝑢)/(𝑠−𝑣+𝑐_𝑢), then 𝑄₀>𝐹⁻¹((𝑠−𝑐_𝑏+𝑐_𝑢)/(𝑠−𝑣+𝑐_𝑢)), (𝑑Π_𝑐(𝑄,𝐿^𝑐(𝑄))/𝑑𝑄)|_𝑄=𝑄0<0 and (𝑑Π_𝑐(𝑄,𝐿^𝑐(𝑄))/𝑑𝑄)|_𝑄=0<0. Hence Π_𝑐(𝑄,𝐿^𝑐(𝑄)) is decreasing on 𝑄∈[0,𝑄₀]. Furthermore, it follows from (A.2) that 𝐿^𝑐(𝑄)=0 for all 𝑄>𝑄₀. When 𝐿^𝑐(𝑄)=0, we can obtain that Π_𝑐(𝑄,0) is decreasing on 𝑄∈(𝑄₀,∞) since 𝑑Π_𝑐(𝑄,0)/𝑑𝑄<0 for all 𝑄>𝑄₀. Hence, we have 𝑄^𝑐=0 and 𝐿^𝑐=𝐿₀.
(iii) If 𝛿𝑐_𝑚⩾𝑐_𝑏𝜇_𝑌, then 𝑄₀⩽𝐹⁻¹((𝑠−𝑐_𝑏+𝑐_𝑢)/(𝑠−𝑣+𝑐_𝑢)) and (𝑑Π_𝑐(𝑄,𝐿^𝑐(𝑄))/𝑑𝑄)|_𝑄=𝑄0⩾0. Hence Π_𝑐(𝑄,𝐿^𝑐(𝑄)) is increasing on 𝑄∈[0,𝑄₀]. When 𝑄>𝑄₀, it is easy to verify that Π_𝑐(𝑄,𝐿^𝑐(𝑄))=Π_𝑐(𝑄,0) has the maximum value at the point 𝑄=𝐹⁻¹((𝑠−𝑐_𝑏+𝑐_𝑢)/(𝑠−𝑣+𝑐_𝑢))(>𝑄₀). Hence, we obtain that Π_𝑐(𝑄,𝐿^𝑐(𝑄)) has the maximum value at 𝑄=𝐹⁻¹((𝑠−𝑐_𝑏+𝑐_𝑢)/(𝑠−𝑣+𝑐_𝑢)) for all 𝑄⩾0, and we have (𝑄^𝑐,𝐿^𝑐)=(𝐹⁻¹((𝑠−𝑐_𝑏+𝑐_𝑢)/(𝑠−𝑣+𝑐_𝑢)),0).

Proof of Proposition 5. From Proposition 3, the total expected profit of the decentralized supply chain can be expressed as follows: Π_𝑑𝑄^𝑑,𝑅^𝑑=𝑝^𝑄𝑑₀(𝑠−𝑣)𝑥+𝑣𝑄^𝑑𝑓(𝑥)+𝑑𝑥_∞𝑄𝑑𝑠+𝑐_𝑢𝑄^𝑑−𝑐_𝑢𝑥𝑓(𝑥)+𝑑𝑥_{01/𝜁𝑦𝜁𝑅𝑑+𝑄𝑑0}(𝑠−𝑣)𝑥+𝑣𝑦𝜁𝑅^𝑑+𝑄^𝑑𝑓(𝑥)𝑑𝑥+^∞_{𝑦𝜁𝑅𝑑+𝑄𝑑}𝑠+𝑐_𝑢𝑦𝜁𝑅^𝑑+𝑄^𝑑−𝑐_𝑢𝑥(𝑥)𝑔𝑦+×𝑓𝑑𝑥𝑑𝑦_{11/𝜁𝑅𝑑+𝑄𝑑0}(𝑠−𝑣)𝑥+𝑣𝑄^𝑑𝑓(𝑥)𝑑𝑥+^∞_{𝑅𝑑+𝑄𝑑}𝑠−𝑣+𝑐_𝑢𝑄^𝑑+𝑅^𝑑+𝑣𝑄^𝑑−𝑐_𝑢𝑥𝑓(𝑥)𝑑𝑥+𝑣𝑦𝜁𝑅^𝑑𝑔𝑦𝑑𝑦−𝑐_𝑏𝑄^𝑑−𝛿𝑐_𝑚𝜁𝑅^𝑑. (A.13) For given 𝑄⩾0, let 𝐾()=𝑅∣𝑄_0𝑅+𝑄)]𝑓(𝑥)+[(𝑠−𝑣𝑥+𝑣𝑄𝑑𝑥_∞𝑅+𝑄𝑠−𝑣+𝑐_𝑢(𝑄+𝑅)+𝑣𝑄−𝑐_𝑢𝑥𝑓(𝑥)𝑑𝑥, (A.14) then 𝑑𝐾(𝑅∣𝑄)=𝑑𝑅𝑠−𝑣+𝑐_𝑢[1−𝐹(𝑄+𝑅)]>0, (A.15) which implies that 𝐾(𝑅∣𝑄) is an increasing function of 𝑅. Hence, we get ¹_1/𝜁𝐾𝑅^𝑑∣𝑄^𝑑+𝑣𝑦𝜁𝑅^𝑑𝑔𝑦<𝑑𝑦_11/𝜁𝐾𝑦𝜁𝑅^𝑑∣𝑄^𝑑+𝑣𝑦𝜁𝑅^𝑑𝑔𝑦=𝑑𝑦_{11/𝜁𝑄𝑑0+𝑦𝜁𝑅}(𝑠−𝑣)𝑥+𝑣𝑦𝜁𝑅^𝑑+𝑄^𝑑𝑓(𝑥)+𝑑𝑥_{∞𝑄𝑑+𝑦𝜁𝑅𝑑}𝑠+𝑐_𝑢𝑄^𝑑+𝑦𝜁𝑅^𝑑−𝑐_𝑢𝑥𝑓(𝑥)𝑦𝑑𝑥×𝑔𝑑𝑦. (A.16) As a result, we have Π_𝑑(𝑄^𝑑,𝑅^𝑑)<Π_𝑐(𝑄^𝑑,𝜁𝑅^𝑑)⩽Π_𝑐(𝑄^𝑐,𝐿^𝑐).

Proof of Proposition 7. Take the first derivatives with 𝑄 and 𝑅 in (15), we have 𝜕Π_𝑟=𝜕𝑄𝑠−𝑤_𝑏+𝑐_𝑢−𝑠−𝑣+𝑐_𝑢×𝑝𝐹(𝑄)+¹₀𝐹𝑦𝜂𝑅+𝑄𝑔𝑦𝑑𝑦,𝜕Π_𝑟=𝜕𝑅𝑠+𝑐_𝑢𝜇_𝑌−𝛿𝑐_𝑚−𝑠−𝑣+𝑐_𝑢¹₀𝐹𝑦𝜂−𝑤𝑦𝜂𝑅+𝑄𝑦𝑔𝑑𝑦_𝑚−𝑤_𝑑𝐺1𝜂. (A.17) By comparing (A.17) with (A.1)-(A.2), we find that 𝑄^𝑑=𝑄^𝑐 and 𝑅^𝑑=𝐿^𝑐/𝜂 if and only if 𝑤_𝑏=𝑐_𝑏 and 𝑤_𝑚=𝑤_𝑑. However, if 𝑤_𝑏=𝑐_𝑏, then the backup supplier's expected profit is equal to zero, and 𝑤_𝑚=𝑤_𝑑 will result in the main supplier producing an infinite amount. Hence, the OPRS contract cannot coordinate the supply chain.

Proof of Proposition 9. From (15), we get 𝑑Π_𝑟=𝑑𝑄𝑠+𝑐_𝑢−𝑤_𝑟𝑝𝐹(𝑄)+¹₀𝐹𝑦𝐿+𝑄𝑔𝑦𝑑𝑦+𝑠−𝑤_𝑏+𝑐_𝑢,𝑑Π_𝑚𝑑𝐿=𝑤_𝑚𝜇_𝑌−𝛿𝑐_𝑚−𝑤_𝑟−𝑣¹₀𝐹𝑦𝐿+𝑄𝑦𝑔𝑦𝑑𝑦. (A.18) If conditions in (21)-(22) are fulfilled, then by comparing (A.18) with (A.1)-(A.2), we obtain 𝑑Π_𝑟/𝑑𝑄=𝜕Π_𝑐/𝜕𝐿 and 𝑑Π_𝑚/𝑑𝐿=𝜕Π_𝑐/𝜕𝐿. Hence, we have 𝑄^𝑑=𝑄^𝑐 and 𝐿^𝑑(𝑄^𝑑)=𝐿^𝑐.
If the supply chain coordination is achieved, then the expected profits for the retailer, the main supplier, and the backup supplier are Π^𝑐_𝑚=(𝑠−𝑤_𝑟+𝑐_𝑢)𝜋(𝑄^𝑐,𝐿^𝑐)−𝑐_𝑢𝜇_𝑋Π^𝑐_𝑚=(𝑤_𝑟−𝑣)[𝜋(𝑄^𝑐,𝐿^𝑐)−(𝑠−𝑐_𝑏+𝑐_𝑢)𝑄^𝑐/(𝑠−𝑣+𝑐_𝑢)]+𝑇 and Π^𝑐_𝑏=(𝑤_𝑏−𝑐_𝑏)𝑄^𝑐−𝑇, respectively. The contract (𝑤_𝑟,𝑇) will not be accepted unless the three member's expected profit is better than the reservation expected profit. Hence, by solving Π^𝑐_𝑖⩾Π^𝑑_𝑖, for 𝑖=𝑟,𝑚,𝑏, we get 𝑤_𝑟⩽𝑤₀ and the feasible range of 𝑇, which is [𝑇_min,𝑇_max].

Proof of Proposition 10. For any given nonnegative numbers 𝜙_𝑟, 𝜙_𝑚, and 𝜙_𝑏 that satisfy 𝜙_𝑟+𝜙_𝑚+𝜙_𝑏=1, if 𝑤_𝑟=𝜙_𝑟𝑣+(𝜙_𝑚+𝜙_𝑏)(𝑠+𝑐_𝑢−𝑐_𝑢𝜇_𝑋/𝜋(𝑄^𝑐,𝐿^𝑐)) and 𝑇=(𝑤_𝑟−𝑣)[(𝑠−𝑐_𝑏+𝑐_𝑢)𝑄^𝑐/(𝑠−𝑣+𝑐_𝑢)−𝜙_𝑏𝜋(𝑄^𝑐,𝐿^𝑐)/(𝜙_𝑚+𝜙_𝑏)], then we have Π^𝑐_𝑟∶Π^𝑐_𝑚∶Π^𝑐_𝑏=𝜙_𝑟∶𝜙_𝑚∶𝜙_𝑏.

Acknowledgments

The authors would like to thank the editor and the anonymous referees for their helpful comments on earlier versions of this paper. This research was supported in part by the National Natural Science Foundation of China (no. 71002106) and the Independent Innovation Fund of Tianjin University (no. 2012XZ091).