Innovative Applications of O.R.
Random yield supply chain with a yield dependent secondary market

https://doi.org/10.1016/j.ejor.2010.02.021Get rights and content

Abstract

This paper considers a simple supply chain with one supplier and one retailer where the supplier’s production is subject to random yield and the retailer faces uncertain demand. There exists a secondary market for acquiring or disposing products by the supplier. We study both the centralized and decentralized systems. In the decentralized system, a no risk sharing contract and a risk sharing minimum commitment contract are analyzed. The supply chain with the risk sharing contract is further analyzed with a constant secondary market price and a yield dependent secondary market price. We present both the supplier’s and the retailer’s optimal strategies and provide insights for managers when making decisions under random yield risk and demand uncertainty. We find that the secondary market generally has a positive impact on supply chain performance and the actual effect of random yield risk on the supply chain performance depends on cost parameters and supply chain contract settings. Under certain conditions, reducing yield randomness may weaken the double marginalization effect and improve the chain performance. From the numerical study, we also show that there exists an optimal commitment level for the supply chain.

Introduction

During 2007 Christmas season, Nintendo Wii was one of the most wanted presents in the United States. It was out of stock in every retail store. One of the authors waited in line outside of a local Best Buy for four hours in hopes of buying one. Before the store opened that day, no customer knew exactly how many Wiis that Best Buy store would have in stock, nor did any of the store employees. In the end, about 400 customers showed up by 9:00am and only 150 tickets were issued to guarantee the buying of a Wii, resulting in 250 unhappy customers without counting those who left after seeing the long queue. If you wonder where else you can get a Wii, at the same time, Wii was listed on the front page of Ebay. In fact, Nintendo Wii was the most listed item on ebay.com with the winning bid price ranging from 75% to 150% above the retail price. If you do not want to wait or pay for the high shipping cost, another source to buy Wiis was from local secondary markets like Craigslist where people in vicinity sell or buy wanted products. Therefore, in this Wii market, customers would like to know where they could buy a Wii at a reasonable price and make a delightful Christmas present; store managers sought how they could get more Wiis to meet the customers’ demand while the supply was so uncertain; similar problems are faced by distribution managers, area directors, and the whole supply chain. What we have seen in this case is a supply chain with random yield supply and at the same time, secondary markets such as Ebay and Craigslist influencing the sales. The main question for different supply chain parties is how to act under such uncertainties so as to maximize their own utilities.

Another example can be found in the textbook market. The supply chain of textbooks is complicated with different factors interacting. From the point of view of a bookstore manager, she needs to make ordering decision of a new textbook before the semester starts when the demand is unknown. She also has uncertain supply of used books from the students (bookstores buy-back used books and due to the price difference, these used books are popular). In addition, she has access to different secondary markets like online markets (different book selling web sites) and wholesale markets for used books, etc. So, her decision is affected by the uncertain demand, uncertain supply, and the secondary markets. In practice, almost every market has somewhat similar characteristics with respect to secondary market and supply randomness. This paper studies such supply chains affected by both random yield and secondary market and provides insights for decision makers to deal with the influences.

In general, any production or logistics process is somewhat related to random yield. With equal amount input, the output of the production usually varies. Examples can be found in areas like agriculture and high-tech productions like semi-conductor and LCD, etc. Due to damages occurred during transferring, any transportation process can also be viewed as random yield process. Yano and Lee (1995) give a comprehensive review on problems related to random yield. In this paper, we study a supply chain with random yield production, vertical competition, and a secondary market for salvaging or emergency production. We wish to answer related questions to help managers better deal with both demand and yield uncertainties. We also want to study the effect of a secondary market on the supply chain and try to address the following questions.

  • How does the existence of secondary market affect the decisions and the supply chain performance?

  • How does yield uncertainty and demand uncertainty affect the supply chain?

  • How to coordinate such supply chains with what contracts?

There are mainly three bodies of literature related to our study. First is the literature studying random yield effect. Gerchak et al. (1988) study a periodic review production system under the assumption of random yield and demand uncertainty. It is shown that a critical inventory level unrelated to the yield distribution exists, and the optimal policy is not to order when the initial inventory level is above this critical level and order a positive amount otherwise. However, the optimal policy is not the traditional order-up-to policy. Henig and Gerchak, 1990, Erdem and Ozekici, 2002 further study the problem under different assumptions and derive similar policies. Various extensions have been made in different directions. For example, Gerchak et al., 1994, Gurnani et al., 2000 study the assembly system with random yield. Wang and Gerchak (1996) consider the capacity issue under the context of random yield. Gerchak (1992) addresses binomial and stochastically proportional yields in both back-order and lost sales environments. In terms of decentralized inventory system, more recently, Guler and Bilgic, 2009, Gurnani and Gerchak, 2007 further study the coordination problem within decentralized assembly systems with both random yield and demand uncertainty. He and Zhang (2008) study the effects of random yield in a simple decentralized supply chain and compare different random yield risk sharing contracts. Our paper extends the model in He and Zhang (2008) by considering the effect of secondary market on the supply chain.

The random yield risk sharing in this paper is through the use of minimum delivery quantity commitment contract. There is a large literature on commitment contracts in supply chains, including quantity commitment, price commitment, time commitment, buyer commitment, seller commitment, etc. Bassok and Anupindi (1997) study the total minimum commitment contract in a single supplier, single buyer supply chain. This contract can be classified as buyer’s quantity commitment contract. The optimal policy is characterized by order-up-to policy under finite horizon and single period Newsvendor problem with discounted price and no commitment. Tibben-Lembke (2004) provides the optimal and heuristic solutions to the problem studied by Bassok and Anupindi, 1997, Wu, 2005, Ozer and Wei, 2006, Cohen and Yano, 2006 study the buyer commitment contract under different constraints. Generally, through forecast updating, the buyer commits to the purchasing amount before the seller proceeds. Specifically, Cohen and Yano (2006) consider the supplier delivery commitment constraints. Another type of commitment contract, sequential commitment contract, is studied in Granot and Yin (2007), where the supplier and the retailer sequentially commit on prices and quantities. Under different power structures, the supply chain performance with sequential commitment contract is analyzed. Our paper studies a seller/supplier commitment contract where the supplier is assumed to face both random yield uncertainty and minimum delivery commitment. We illustrate how commitment contract performs in this supply chain and show how to improve the supply chain performance through adjusting the commitment contract. Notice that we assume symmetric information for the commitment contract. Due to the rapid technology innovation, for example, RFID, automated supply chain management becomes possible and seems to be the future way to go. Interested readers may see Piramuthu (2005) for more details and references on this topic.

The third body of related research is the literature on the impact of secondary market on supply chains. Usually, secondary market is referred as a market for trading beyond the main market. The Ebay market and local markets mentioned in the Wii example are considered to be secondary markets whereas the normal channel through retail stores like Best Buy, Circuit City, etc. is treated as the main market. In the textbook market, online used book market and wholesale used book market are also secondary markets. Haksoz and Seshadri (2007) review the literature on the effects of secondary markets on supply chains. Generally, there are three main functions of secondary markets. First, a secondary market can be used for inventory pooling. Lee and Whang (2002) study a reseller secondary market where resellers buy and sell excess inventories. It is shown that the secondary market has two effects, a quantity effect and an allocation effect, on the supply chain. In Lee and Whang (2002), it is shown that the quantity effect (sales by the supplier) is indeterminate, while the allocation effect (supply chain performance) is positive. Kouvelis and Gutierrez (1997) study the style goods market with the secondary market when the two markets have non-overlapping selling season. Through risk pooling (goods transfer between two markets), the supply chain performance is improved. Second, a secondary market is used for disposal or salvage of the left-over products. Arcelus et al. (2008) analyze a supply chain using buy-back contract with the access of secondary market for disposal. The secondary market is shown to be beneficial to both the supplier and the retailer when there is a buy-back contract. Third, a secondary market is used as emergency resource to satisfy the demand. To certain extent, the inventory pooling effect incorporates the next two effects: disposal effect and emergency resource. We distinguish them in order to address different aspects of secondary market. In this paper, the secondary market is used for both salvaging and emergency resourcing purposes. Our results further justify that the existence of secondary market improves the supply chain performance under our setting.

In this paper, we study a supply chain facing yield and demand uncertainty. The coordinated supply chain is first analyzed as the benchmark model for comparisons. In the decentralized supply chain setting, we investigate two cases. One assumes that no random yield risk is shared between the supplier and the retailer. The other assumes that the two parties share random yield risk through a commitment contract. In the risk sharing case, we further study the supply chain with a non-yield dependent and a yield dependent secondary market price. In both cases, the profit functions of the supplier and the retailer are analyzed and proved to be concave in the production and ordering quantities, respectively. The optimal ordering and production decisions are characterized in functional forms. Numerical examples are used to provide more managerial insights and answer questions related to secondary market, random yield, and commitment contract.

This paper contributes to the current literature in following aspects. First, our model incorporates both random yield risk and demand uncertainty in the supply chain. The results help managers better understand the influences and interactions between these two risks. Though there is a literature on this topic, it is rather limited. Our analysis shows that reducting random yield through technology innovation can affect the supply chain performance positively. Second, our model provides insights on the effect of the secondary market on the supply chain, which renders guidance for managers to choose different contracts or actions under different market situations. We find that the existence of the secondary market has a positive effect on the supply chain performance. Third, this paper contributes to the literature related to commitment contract in supply chains. We find there exists an optimal minimum commitment quantity that would best serve the supply chain.

The rest of the paper is organized as follows: Section 2 discusses the centralized model. Section 3 illustrates the decentralized cases under no risk sharing setting. Sections 4 Risk sharing contract with constant secondary market price, 5 Risk sharing contract with yield dependent secondary market price discuss the decentralized supply chain under risk sharing setting with a constant secondary market price and a yield dependent secondary market price, respectively. Section 6 has a numerical analysis for the risk sharing minimum commitment contract with the presence of a secondary market price. Section 7 summarizes the paper and indicates some future research directions.

Section snippets

Centralized model

There is one supplier and one retailer in the supply chain, and both are profit-maximizers. Let Q denote the supplier’s decision on how many to produce, U be a random yield variable with a cumulative probability distribution function F(u) and E[U]=μ,uQ be the yield from the production, and q be the retailer’s order quantity. Note that here the stochastic proportion yield model is used and the input amount Q is independent of the yield distribution. The retailer faces a random demand X with a

No risk sharing contract: decentralized decisions

When the supplier and the retailer are independent decision makers, they negotiate certain contracts specifying money and products transfer. Depending on the negotiating power of the two parties, different risk sharing contract can be realized. We classify the contracts in this situation into two categories, no risk sharing and risk sharing contracts. Note that the risk sharing here means the supply chain members share the random yield risk.

In a no risk sharing situation, the supplier bears the

Risk sharing contract with constant secondary market price

We first study the risk sharing contract with a constant secondary market price. To share a random yield risk, the supplier and the retailer negotiate a minimum commitment contract. Assume that there is a commitment quantity k(0<kq) that the supplier must supply, and if the supplier chooses to deliver less than q, for every unit in the shortage, the supplier has to pay a penalty cost b to the retailer. The penalty for delivering less than k units is considered prohibitively large and the

Risk sharing contract with yield dependent secondary market price

In this section, we study the situation when the secondary market price is yield dependent. In this model, the secondary market price (s(u)=α-βu) is yield dependent and the supplier still has the same three options to choose: delivering k, delivering between k and q; or delivering q. The supplier’s profit function isπ2S=wD-b(q-D)+-s(U)(D-UQ)+-cQ+ϵs(U)(UQ-D)+.Similarly, the retailer’s profit function isπ2R=rmin{D,X}-wD+b(q-D)+.

The following lemma generalizes Lemma 1 with a yield dependent

Numerical example

To further illustrate our results, we perform a numerical study which assumes a constant secondary market price. The parameters are set as follows: c=1,w=2,r=5,s=3,ϵ{0,0.1,0.2},θ{0.25,0.5,0.75},b{0,0.25,0.5,0.75}, and both U and X are uniformly distributed where UU(0.1,1.4) and XU(0,100). Our results are summarized in Table 3, Table 4, Table 5. We are interested in answering the following questions: 1. How does the secondary market affect this supply chain? 2. How does the commitment

Summary and future research

This paper studies a supply chain with one supplier and one retailer when the supplier has random yield and the option of trading in a secondary market. Both the centralized and decentralized systems are studied and the analytical results are provided. In the decentralized system, the risk sharing contract and no risk sharing contract are studied. For both contracts, the supplier and the retailer’s expected profit functions are analyzed and shown to be concave functions in their production

Acknowledgments

The authors would like to thank the three anonymous referees for their constructive comments and suggestions that have greatly improved the quality of the paper.

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