Journal of Applied Mathematics

Volume 2016, Article ID 4164295, 13 pages

http://dx.doi.org/10.1155/2016/4164295

## Pricing Strategies in the Remanufacturing Market for the Uncertain Market Size in the Second Period

^{1}College of Economics and Management, Hunan Normal University, Changsha 410081, China^{2}College of Finance and Statistics, Hunan University, Changsha 410082, China

Received 20 December 2015; Accepted 17 March 2016

Academic Editor: Dimitris Fotakis

Copyright © 2016 Liurui Deng and Shenggang Yang. 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

Our main endeavor is to investigate the effect of the uncertain market size in the second period on the pricing strategies in the remanufacturing market. Observing the previous research, we find that the market size in the second period is always supposed to be certain. However, there is substantial empirical and experimental evidence that the market size in reality deviates from this assumption. In fact, though the market size in the first period is definitized, it is difficult to confirm the change of the market scale in the second period, since this change is affected by all kinds of elements, such as the awareness of environmental protection, some consumers’ psychological factors, and the related governments’ policies. Hence, we pay attention to the case in which the change rate of the market scale in the second period is random variable. We suppose that this rate satisfies uniform distribution on . Underlying this assumption, we further provide an insight into the game-playing relationship between original equipment manufacturers (OEMs) and remanufacturers. Moreover, we delicately and subtly incorporate the game theory, stochastic analysis, adversarial risk analysis (ARA), and optimization methods into the pricing strategies in the remanufacturing market. Last but not least, considerable efforts and attempts have been made to subtly test the sensitivity of an optimal solution to the different parameters.

#### 1. Introduction

*(**1) Research Background*. The flow of products in supply chain goes upstream to downstream, such as from the suppliers’ supplier to customers’ customer. Shorter product life cycles and consumers’ consumption behaviors result in faster products flows and subsequently faster generation of waste and depletion of natural resources (see [1]). Varel states that manufacturing generates excess 60% annual nonhazards waste (see [2]). The contradiction between social and economic development and environmental protection is more and more sharp, which makes more and more people and governments pay attention to balancing economic and social development with environmental protection. Both the 1992 Rio Earth Summit and Brundtland Report sharply and frankly declared that coordinating the environmental protection with the social and economic development is one of the most essential challenges in security long-term sustainability (see [3]). Meanwhile, increasing stringent legislation is established to compel enterprises to reduce the influence of products and manufacturing process on the environment. For example, producer responsibility legislation demands that producers should recover used products to reduce landfill. Such pressure, incorporating mounting competition due to global industry activity, challenges companies to strengthen environmental consciousness and give rise to the drive collecting and remanufacturing returned/used products for extending their usable lives in order to reduce waste and conserve natural resource (see [4]).

Remanufacturing is an industrial process in which used products are brought to a “like-new” functional state with warranty to match (see [5, 6]). In other words, remanufacturing means that a product is updated and reprocessed in this process. In more detail, remanufacturing includes disassembly, clearing inspection, reconditioning, and reassembly. Remanufacturing is both more profitable and less harmful to the environment than conventional manufacturing due to the decrements in the raw materials demanded, production processes, and production costs (see [7]), such as the levels of virgin material, landfill, energy, specialised labour, and machine time (see [8, 9]). Giuntini and Gaudette confirm that remanufactured products costs are 40–65% less than new products to produce; thus, remanufactured products’ prices are typically 30–40% lower than new products (see [10]). Globally, remanufacturing can save about 120 trillion BTUs per year of energy, accounting for about 16 million barrels of crude oil and about US$500 million in energy cost. Annual material saved from remanufacturing activities worldwide is 14 million tonnes per year, which is equivalent to fully loaded railway train 1650 miles long (see [11]).

The previously mentioned appealing advantages of remanufacturing motive the remanufacturing industry to burgeon. In 1996, remanufacturing industry reached US$53 billion which consisted of approximately 73,000 remanufacturing companies (see [9]). These sale figures were on a par with the American steel industry. However, the direct employment figure of 48,000 was twice that of the American steel industry and equaled that of the consumer durable industry (see [11]). Global Industry Analysts affirm that the global automotive remanufacturing industry is rapidly growing and is predicted to reach US$104.8 billion by the end of 2015 (see [12]). The boom of the remanufacturing industry gives the related research an incentive and thus the problems about remanufacturing have been stressed in several studies. In the next content of this paper, we will briefly review some related literatures without being completed.

*(2) Literature Review*. Although the pricing strategy in the remanufacturing market is relatively a new term, there are plenty of applausive achievements which abundant researchers manage to obtain [13]. Savaskan et al. discuss the problem of choosing the appropriate reverse channel structure for the collection of used products from customers (see [14]). Jaber and El Saadany create the production, remanufacture, and waste disposal model with lost sales (see [15]). Agrawal et al. experimentally investigate the effect of remanufactured products on the perceived value of new products (see [16]). King et al. provide an insight into the importance of remanufactured products for reducing waste and protecting environment (see [17]). Some researchers focus on the policy influences. Webster and Mitra examine the impact of take-back laws within a manufacturer/remanufacturer competitive framework and develop a general two-period model to investigate questions of interest to policy-makers in government and managers in industry (see [11]). Based on the general research, many investigators begin to explore dynamic models, especially two-period models. Yuksel addresses the design of automobile engines for the remanufacture with quality function deployment (see [18]). Ferrer and Swaminathan characterize the production quant self-selection and explore the effect of various parameters in the Nash equilibrium on the duopoly environment (see [19, 20]). Meanwhile, other researchers are more interested in the pricing strategies for dynamic models. Wun pays attention to the effects of disassemblability and interchangeability on the price competition (see [21, 22]). Besides, he considers price and service between new and remanufactured products (see [23]). Mitra focuses attention on revenue management for remanufactured products and develops a pricing model to maximize the expected revenue from the recovered products (see [24]). In view of service competition, Deng builds a Stackelberg model which contains an original manufacturer, a recycler, and a remanufacturer and further obtains the equilibrium decision of closed-loop supply chain. Based on this strategy model, Deng demonstrates how two different subsidies to utilization rate of raw materials, the service level, and the demand of products impact on the terminal decision (see [25]).

It is noteworthy that the previously mentioned research is based on Stackelberg model. However, beyond the pricing strategies in the remanufacturing market, the classical Stackelberg model believes that since OEMs are the monopoly position, they know the pricing strategy of remanufacturers in the second period. However, the development of remanufacturing industry shakes OEMs monopoly position and makes remanufacturers become stronger and stronger. So, in fact, OEMs do not know the pricing strategy of remanufacturers in the second period. In particular, the rate of recovery changes because of development of remanufacturing industry, improvement of new machines, disassemblability, or other unpredicted reasons. As for OEMs, it is impossible to accurately know the rate of recovery. Thus, OEMs do not know the exact pricing strategies of remanufacturers. Meanwhile, the unit production cost is not fixed but rather changed. Production technology and process improvement cause this cost to change. Similar to OEMs, as for remanufacturers, they have no way to know exactly the unit production cost. The major violation of the traditional Stackelberg model inspires Deng and Ma to build the remanufacturers and the OEMs 1-order adversarial risk analysis (ARA) models, since there exits the game relationship between the OEMs and the remanufacturers in the reality and the above unknown elements by the opponents (see [26]).

The motivation of Deng’s ARA models of OEMs and remanufacturers makes this paper reveal that the pricing strategies in the remanufacturing market for the uncertain market scale in the second period. Differing from Deng’s previous research, in this paper, through observing the previous research, we find that the predominant literatures have another underlying assumption that the market size in the second period is certain. However, there is substantial empirical and experimental evidence that the market size in reality deviates from this assumption. In fact, though the market size of the first period is definitized, it is difficult to confirm the change of the market scale in the second period, since this change is affected by all kinds of elements, such as people’s environmental awareness, some consumers’ psychological factors, the development of remanufacturing industry, and the related governments’ policies.

Noting the above shortcoming of the traditional pricing strategy, we focus on the pricing strategies of OEMs and remanufacturers under uncertain market size in the second period in this paper. The uncertain market scope in the second period brings us the diverse obstacle from the previous work that the optimal pricing strategy becomes a stochastic optimization problem rather than a real one. We tackle this challenge in the approach in which we make great efforts to incorporate stochastic analysis, Stackelberg model, and adversarial risk analysis models into discussing the optimal pricing strategies. Our considerable attempts and efforts have been made to indicate an explicit expression for the optimal pricing strategies. We further provide an insight into the sensitivity of optimal explicit solutions to the different parameters, and , in order to reveal the impact of the various parameters on the optimal strategies in powerfully numerical means.

The rest of this paper is organized as follows. We first describe the remanufacturing market and introduce the related parameters in Section 2. Then, Section 3 presents the pricing strategies of the remanufacturer and the OEM. In this section, we firstly build the pricing strategy model of remanufacturers. These pricing strategies are taken into account in two cases, the high pricing strategy and the low pricing strategy. Based on these pricing strategies of the remanufacturer, we further indicate the pricing strategy models of OEMs. According to the pricing strategies of the remanufacturer, the pricing strategies of the OEM are divided into two different strategies which are the OEM’s responses to the high and low pricing strategies of the remanufacturer. In Section 4, we summarize the concrete strategies studied earlier. Sequently, through testing the sensitivity of the optimal pricing strategies to the different parameters, and , we investigate all kinds of paraments’ influence on the pricing strategies with the numerical method in Section 5. Later, Section 6 summarizes our important theoretical and numerical contributions about the optimal pricing strategy. At last, namely, in Section 7, we propose some interesting and meaningful problems which we will explore in the future.

#### 2. The Description of the Remanufacturing Market and the Related Parameters

Before exploring the model, it is necessary to describe the remanufacturing market and introduce the related paraments.

##### 2.1. The Description of the Remanufacturing Market

We first present the remanufacturing market. Usually, we suppose that there are two kinds of the manufacture, original equipment manufacturers (OEMs) and remanufacturers. There is a game relationship between them. The pricing strategies will directly affect each other’s market share and profit. Meanwhile, OEMs’ design for disassembly influences on the costs of OEMs and remanufacturers. To be specific, high disassemblability makes it easy to repair, clean, and inspect OEMs. Moreover, high disassemblability reduces the recover cost of remanufacturers. However, the design for high disassemblability increases the fixed cost of OEMs. So, in order to save fixed cost and raise the remanufacturers’ cost, OEMs always will choose to design for low disassembly. This measure makes remanufacturers erode the competitiveness and OEMs strengthen competitiveness.

The market is divided into two segments, which are the primary and green segments. Accordingly, the primary market is composed of primary consumers and the green market consists of green consumers (see [27, 28]). Primary consumers believe that the new product has higher value than the remanufactured product. Nevertheless, green consumers are sure that the remanufactured product has the same value as the new product. So, green consumers prefer the remanufactured product to the new product. Because there is no remanufactured product in the first period, the primary and green consumers are not different in this period. However, in the second period, the green consumers are more willing to buy the green product than the new product, since the green product is beneficial for environment.

Noticing that many elements impact on pricing strategies, such as the proportion of green consumers in the whole market, the rate of the used product being available to be remanufactured, and the change of the market size, we set the necessary parameters. In this paper, we consider that parameters are fixed except that the design for disassembly has two levels.

##### 2.2. The Introduction of the Related Parameters

Sequently, we introduce the relevant parameters. The rate of the used product being available to be remanufactured is noted as . Suppose that the market has two periods. The first period consists of consumers, in which green and primary consumers are indifferent, since there is no remanufactured product. Therefore, in this period, the market is the primary market. The second period includes consumers. represents the change rate of the market scale in the second period compared with the market size in the first period. In particular, assume that follows uniform distribution on . In this period, the market is divided into the primary segment and the green segment. The primary segment consists of primary consumers who are willing to pay for green products times as high as primary ones. That is, if we let the price of a new product be accepted by primary consumers in the second period, then the primary consumers prefer to pay or less money for a green product. Otherwise, the primary consumers will not buy the green product. Meanwhile, green consumers think that green products are the same as primary ones. So, the green consumers prefer green products. Assume that the percentage of green consumers in the whole market is and the primary consumers’ proportion is . The quantity of the remanufactured product is restricted by the quantity of the new product in the first period.

We suppose that the utility of consumers in the first period is , where (please see [16]) and is the price of the new product in the first period. The utility of primary and green consumers is, respectively, and , where is the price of the green product in the second period. , , and are, respectively, the demand of the remanufactured product, the demand of the new product in the second period, and the demand of the new product in the first period. Further, suppose that is the proportion in the collected end-life-cycle products which are used by remanufacturers. The levels of disassemblability are high and low . Accordingly, OEM’s different unit production costs are and and the remanufacturer’s different recovery costs are and . and are the fixed costs of OEMs. The remanufactures’ pricing strategies are high and low . The rate of discount is .

#### 3. The Relevant Models

Here, we take Stackelberg model into account. So, firstly, it is needed to provide an insight into the pricing strategies of the remanufacturer which consist of the low and high pricing strategies. Then, relying on the pricing strategy of the remanufacturer, we investigate the pricing strategies of OEMs dealing with ones of the remanufacturer. That is, we, respectively, consider the pricing strategy of OEMs for low pricing strategy of the remanufacturer and the pricing strategy of OEMs for high pricing strategy of the remanufacturer.

##### 3.1. The Model of the Remanufacture

Now, it is needed to think that the remanufacturing problem under restricted condition is as follows:where , .

###### 3.1.1. The Low Pricing Strategy of the Remanufacturer

We firstly think about the low pricing strategy of the remanufacturer, that is, . At this time, since the price of the green products is lower than the price of one expected by the primary consumers, some primary consumers prefer the green product. Hence, in the second period, the remanufacturers compete with the OEMs and the price of the new product affects the profit of remanufacturers. On the other hand, the price of the new product in the first period impacts on the demand of the green product and further impacts on the profit of the remanufacturers. The relationship between the optimal price strategies of the remanufacturers is displayed in the following proposition.

Proposition 1. *Considerwhere , , , , , and are parameters depending on , , , , . And, , , and are the different pricing strategies of the remanufacturers depending on , , , , in the different parameters’ ranges.*

The proof and the description of parameters and pricing strategies are displayed in Appendix A. This proposition displays the low pricing strategy of the remanufacturer.

###### 3.1.2. The High Pricing Strategy of the Remanufacturer

Now, we pay attention to the case and explore the high pricing strategy of the remanufacturer.

Proposition 2. *Let If , we use the high pricing strategy.*(i)*When * *there is the optimal pricing strategy of the remanufacturer as follows:* *if , * *Otherwise, .*(ii)*When * *the optimal solution is that *

*The proof is seen in Appendix B.*

*3.2. The Model of the OEM*

*We think about the OEM’s objective function stated by *

*3.2.1. OEMs’ Pricing Strategy Facing the Remanufacturer’s Low Pricing Strategy*

*At first, we consider what response the OEM should have when the remanufacturer uses the low pricing strategy. The market demand of new products in the second period is At this time, one can get the following result.*

*Proposition 3. When the remanufacturer uses the low pricing strategy, the OEM’s best reaction is as follows:(i)When (ii)When *

*The proof is seen in Appendix C.*

*3.2.2. OEMs’ Pricing Strategy Facing the Remanufacturer’s High Pricing Strategy*

*3.2.2. OEMs’ Pricing Strategy Facing the Remanufacturer’s High Pricing Strategy*

*Next, we think about what response the OEM should have when the remanufacturer uses the high pricing strategy. The market demand of new products in the second period is *

*At this time, one can get the result as follows.*

*Proposition 4. When the remanufacturer uses the high pricing strategy, the OEM’s best response function is as follows:*

*The proof is displayed in Appendix D.*

*4. The Pricing Strategies*

*4. The Pricing Strategies*

*Though we employ the Stackelberg model in the second period in order to present the optimal solution, we are inspired by Deng’s work to notice uncertainty of the market scale in the second period. In more detail, in the second period, the remanufacturer collects the end-life-cycle product produced by OEMs in the first period. We demonstrate that the OEM can observe the pricing strategy of the remanufacturer in the second period as the Stackelberg model believes. Relying on the information about the pricing strategy of the remanufacturer, OEMs can choose the most advantageous pricing strategy for their own. This interactively competitive relationship between OEMs and remanufacturers is a stable basis of the pricing strategies of OEMs and remanufacturers.*

*From remanufacturers’ standpoint, according to the principle of maximizing the expected utility function, they will choose the optimal pricing strategies. Detailedly, the low pricing strategy can impel the remanufacturer to be prone to the benefit from the whole market including the primary and green markets while the high pricing strategy only makes the remanufacturer dominate the green market and obtain the profit from the green market. Therefore, if the low pricing strategy, through penetrating the whole market, can bring the remanufacturer more benefit than the high pricing strategy by which the remanufacturer only stands to gain from the green market, the remanufacturers will prefer the low pricing strategy to the high pricing strategy (Proposition 1). Otherwise, if the high pricing strategy brings more benefit to the remanufacturer than the low pricing strategy, the remanufacturer will choose the high pricing strategy rather than the low pricing strategy (Proposition 2).*

*From OEMs’ standpoint, OEMs also make the optimal choice of their own pricing strategy to get maximal value of their expected utility according to the high and low pricing strategies of the remanufacturer. When the remanufacturer accepts the low pricing strategy, OEMs have to share the primal market with the remanufacture. When the remanufacturer chooses the high pricing strategy, OEMs can monopolize the primary market. Therefore, OEMs should differently react to the different pricing strategy of the remanufacturer. Particularly, if the remanufacturers choose the low strategy, OEMs will choose the pricing strategy of Proposition 3. Otherwise, when the remanufacturers use the high pricing strategy, OEMs will accept the pricing strategy of Proposition 2.*

*5. The Numerical Analysis*

*5. The Numerical Analysis*

*In this section, with a numerical method, we test the sensitivity of the optimal solutions to the different parameters in order to analyze influence of all kinds of paraments in the models on the pricing strategies. Let , , , and .*

*5.1. Testing the Sensitivity of the Optimal Solution to the Parameter *

*5.1. Testing the Sensitivity of the Optimal Solution to the Parameter*

*Firstly, we research the sensitivity of the optimal pricing strategy to the parameter . From Figures 1 and 2, we observe that , , and are equal to the same constant no matter whether there is low or high level of disassemblability, when remanufacturers choose the high pricing strategy. That is, this high pricing is independent of . In the perspective of economics, is the price determined by OEMs in the first period. Since OEMs have to decide the price before the second period, the price is irrelevant to the restriction on the price of the remanufactured products in the second period. Meanwhile, the remanufacturers accept the high pricing strategy; hence, the remanufacturers only dominate the green market and the OEMs monopolize the primary market. Thus, at this time, the profit of remanufacturers reaches maximal value when remanufacturers choose the same price as OEMs (Proposition 2). In other words, the high price strategy’s object is to get interest from the green market in which there are only green consumers not primary consumers. So the price which the primary consumers are willing to pay for the remanufactured product is not related to the high pricing strategy of remanufacturers.*