Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 812043, 9 pages

http://dx.doi.org/10.1155/2015/812043

## Supply Chain Coordination Contracts under Double Sided Disruptions Simultaneously

^{1}Department of Economic Management, North China Electrical Power University, Baoding 071000, China^{2}Department of Industrial Engineering & Management, College of Engineering, Peking University, Haidian District, Beijing 100871, China

Received 3 June 2015; Revised 18 July 2015; Accepted 27 July 2015

Academic Editor: Young Hae Lee

Copyright © 2015 Huan Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Supply chain coordination models are developed in a two-echelon supply chain with double sided disruptions. In a supply chain system, the supplier may suffer from the product cost disruption and the retailer suffers from the demand disruption simultaneously. The purpose of this study is to design proper supply chain contracts, under which the supply chain with double sided disruption can be coordinated. Firstly, the centralized decision-making models are applied to find the optimal price and quantity under three cases as the baseline. The different cases are divided by the different relationship between the product cost disruption and the demand disruption. Secondly, two different types of contracts are introduced to coordinate the whole supply chain. One is all-unit wholesale quantity discount policy (AQDP) contract, and the other one is capacitated linear pricing policy (CLPP) contract. And it is found out that the gap between the demand disruption and the product cost disruption is the key factor to influence the supply chain coordination. Some numerical examples and sensitivity analysis are given to illustrate the models. The AQDP contracts are listed out under different cases to show how to use it under double sided disruptions.

#### 1. Introduction

Supply chain risk management is becoming an increasingly important area. In the past several years, there has been a shift of focus from creating efficient supply chains to reliable and efficient supply chains. This shift is due to the large-scale negative impacts of supply chain disruptions in global supply chain networks. For example, an earthquake in Taiwan in 1999 damaged the manufacturing facilities of several major semiconductor suppliers and disrupted the flow of components to many PC and laptop manufacturers [1]. During the Chinese Winter storm in early 2010, many power plants in southeastern China contacted coal traders to seek imported coals from foreign countries, as the on-hand coal stocks at many of them were below the critical level [2].

Either the supplier’s or the retailer’s disruptions could not only cut down the efficiency of the whole supply chain but also exert an influence on the supply chain contract [3–5]. Either the supplier or the retailer suffers from disruptions due to economic policies adjustment, transportation delays, or natural disasters. We should be obliged to take the disruption management into consideration. On the one hand, the supplier may experience disruptions, such as [6–10]. On the other hand, the retailer may go through the unexpected changes of market demands, taking, for example, [11–14]. However, the total supply chain may be influenced by the emergency at the same time. Therefore, it is of great significance and urgency to conduct the research on supply chain contracts under double sided disruptions simultaneously. Some researchers have paid some attention to the models including both the supply disruption and the demand disruption, which aimed at the optimal inventory policies [15–17]. The supply disruptions mostly affect the manufacture cost, so normally the supply disruption can be seen as the product cost disruption in [18–21]. So we consider the supplier experiences the product cost disruption. But in this paper, we mean to contribute from the supply chain coordination contract aspect for analysis of the supply chain with double sided disruptions. Specifically, our main contributions are twofold as follows. The first one is, according to the different relationship between the cost disruption and the demand disruption, the centralized decision-making models are divided into three cases. And the optimal price and quantity under three cases are listed out as the baseline. Secondly, two different types of contracts are introduced in order to coordinate the supply chain with cost disruption and demand disruption simultaneously. One is all-unit wholesale quantity discount policy (AQDP) contract, and the other one is capacitated linear pricing policy (CLPP) contract. And it is found out that the gap between the demand disruption and the product cost disruption is the key factor to influence the supply chain coordination.

The rest of this paper is as follows: In Section 2, we review the related literature. Section 3 outlines our basic model under no disruption. In Section 4, we analyze how to develop a scheme facing with double sided disruptions under centralized decision-making. Section 5 shows how the all-unit wholesale quantity discount policy (AQDP) contract and capacitated linear pricing policy (CLPP) contract could coordinate the supply chain when both sides make their own independent decision. In Section 6, we give some numerical examples to elaborate the above theorems. Section 7 will draw the conclusions and future research.

#### 2. Related Literature

Since we study how the supply chain can be coordinated while suffering from double sided disruptions simultaneously, our research is built upon the disruption management literature. Broadly speaking, the impact of disruption management is studied in the supply chain contract framework. On the one hand, demand disruption is a kind of disruption management in supply chains that has been extensively studied in the literature. For instance, Qi et al. [22] design various coordination contract when demand disruption influences the entire supply chain. Xiao et al. [23] examine the coordination scheme when there are several competitive retailers in the supply chain which is suffering from the demand disruption. Cachon and Lariviere [24] develop the allocation reactive plan with the limited capacity of supplier while the retailers meet with the market demand changes. On the other hand, suppliers’ cost disruption is another kind of important disruption management in supply chains and has also received substantial attention (Wang et al. [25]; Parlar and Perry [26]; Parlar and Berkin [27]; Heimann and Waage [28]). Corbett et al. [29] present how contracts should be designed by a retailer when a supplier suffers from cost disruption. They establish a general framework to develop the dilemma of supply chains. Sarkar et al. [6] discuss pricing competition between multiple suppliers, all of whom have a disruption risk. They focus on a single uncertain demand and model the retailer’s products based on the number of suppliers they choose to order from.

Our paper is distinguished from all the aforementioned papers because both sides of the supply chain with their own disruption result in that existing no main principal, whereas all the above papers investigate a unilateral disruption management. There have been some recent papers on the double sided asymmetric information in very different contexts; see, for example, Yao et al. [30], Zhang and Luo [31], Wang et al. [32], and Wang et al. [33]. However, our research cast new light on how to design different styles of supply chain contracts under double sided disruptions simultaneously; this point has no count part in all the sequential screening papers.

Supply chain coordination with contracts is the most closely related research to our work. Cachon [34] introduces six different kinds of contract to coordinate supply chains. Particularly, our paper is related to the vast literature on the wholesale quantity discount contract. Cachon [34] explains the difference between the all-unit and incremental quantity discount contract. Based on that, some papers focus on the discount contract, such as [35–37]. Giri et al. [38] launched researches on a multi-manufacturer-one-retailer model to develop the ordering plan with the discount contract. Therefore, both sides could achieve the maximum joint profit.

#### 3. Baseline Case

Consider a supply chain with one supplier and one retailer [33]. The retailer purchases a kind of product from the supplier and then sells it in the market. The notations that will be used throughout this paper are given below: is the retailer’s demand, is the maximum market scale, is a coefficient of price sensitivity, is the profit of supplier, is the profit of retailer, is the profit of the total supply chain, is the retail price, is the supplier’s unit production cost, is the order quantity, is the unit wholesale price, is the marginal costs associated with the incremental increase in demand, is the marginal costs associated with the incremental decrease in demand, is the demand disruption, is the cost disruption.

For the illustrative purpose, we use an exponential function of the price as follows to describe the retailer’s demand during a supply cycle: [39]. Let , denote the profit of supplier and retailer, respectively. The profit of the supply chain is , supposing that , where .

Suppose that the supplier’s unit quantity cost is ; thus given the retail price , the supply chain profit is . Using simply algebra, we know that when . Substituting this value into the former expression, we can get . Consequently, the retailer’s optimal order quantity equals the supplier’s original quantity plan, as . In the rest of this paper, and denote the demand and cost disruption, respectively. We assume that when supply disruption occurs, the maximum market scale faced by the retail shifts from to .

There is one problem in achieving the maximum supply chain profit , that is, how to share the profit between the two parties. One of the most common approaches is to design appropriate contracts for the supplier. This paper applies two different types. The first one is the wholesale quantity discount policy. Particularly, we study the all-unit wholesale quantity discount policy, denoted by with . If the retailer orders , the unit price is . Otherwise, the unit price becomes . And the latter one is a capacitated linear pricing policy CLPP(). The unit wholesale price is , whereas the retailer is limited to sending for less than the arranging quantity .

#### 4. Centralized Decision-Making

In the disruption model, we assume that there is a central decision-maker who seeks maximizing the total profit. During the disruption captured by the demand disruption and the cost disruption , the realized price is found to be and the supply chain profit can be written aswhere the parameters and are the marginal costs associated with the incremental increase or decrease in demand, respectively, and .

Lemma 1. *Suppose maximizes in (1). Then if ; otherwise .*

Lemma 1 demonstrates that the order quantity will increase with the enlargement of the market scale. From Lemma 1, when , the problem of maximizing reduces to Using the first-order condition and solving for giveWe analyze regarding the constraint . There exist two cases.

*Case 1*. ; then , showing that is maximized at . Therefore, let .

*Case 2*. ; then ; thus is maximized at . Let .

In summary, we know that the quantity will increase if the demand disruption is larger than . Consequently, the quantity plan is dependent on the gap between the demand disruption and the cost disruption. We continue to analyze the corresponding price now.

Nevertheless, the optimal price will increase when the market scale increases. Substituting and into the former given expression , we can obtainNow we can solve the maximum supply chain profit at this price and quantity:

We use the function to denote the maximum profit for the new cost and new market scale. We think out the other case where .

The problem of maximizing reduces to (6)Likewise, we can divide into case () and case (). Leaving out the details, we can obtainSummarizing the above works, we have the following.

Theorem 2. *Given double sided disruptions in both market scale and producing cost , the supply chain profit is maximized for the following values of the price and the quantity :*

*Theorem 2 demonstrates that the initial quantity has some robustness under double sided disrupted market scale and producing cost simultaneously. We find out that case and case can be incorporated to only one case due to the same consequence though they belong to different intervines. Hence, latter part of our paper will substitute case into former case and case . Moreover, we have to use case instead of former case . There are not changes when the gap between disrupted market scale and producing cost is not so large, such as . In this condition, adjusting the price could retrieve any disruption expense, while the adjustment equals which is just related to the demand disruption. Only when either the demand disruption far exceeds the cost disruption () or the other case (), will it be essential to change both the price and the quantity.*

*Figure 1 is used to show the optimal quantity area during centralized decision-making under different cases.*