Management of two competitive closed-loop supply chains

Abstract

The two competitive closed-loop supply chains under our study include three members: two manufacturers and one retailer. In this paper, we focus on the management of the wholesale prices, the retail prices and the collecting prices for the two competitive closed-loop supply chains. On the assumption that the return rate of the used-products is an increasing function of the collecting price, we obtain the optimal wholesale prices, the optimal retail prices and the optimal collecting prices based on the following models: Model MMC (two manufacturers for collecting), Model MRC (manufacturer one and retailer for collecting) and Model RRC (retailer for collecting). Furthermore, by comparing the optimal results, we find that the retailer for collecting is the best channel for the two competitive closed-loop supply chains if the two manufacturers would like to transfer all of their cost savings from remanufacturing to the retailer. At the end, we illustrate a numerical example to analyse the impacts of the market share ratio and the substitute ratio of the two products on the optimal results.

Keywords:

• management
• competitive
• closed-loop supply chain
• pricing

1. Introduction

By our market survey of collecting used-products, we find an interesting situation: in order to get more profit, a retailer would try to sell products for two (or more) manufacturers. At the same time, the retailer is responsible for collecting the used-products for the two (or more) manufacturers. Namely, the two (or more) closed-loop supply chains joint in the retailer. The two competitive closed-loop supply chains under our study (CLSC1 and CLSC2) include three members: two manufacturers and one retailer. In CLSC1, manufacturer one makes new product one and sells them to the market via the retailer, and then collects the used-products one after using for some periods. In CLSC2, manufacturer two makes product two and sells them to the market via the retailer, and then collects the used-product two after using for some periods. Product one and product two can be replaced with each other in the consumer market. So, for the two competitive closed-loop supply chains, there are some new optimisation problems. For example, how to decide the wholesale prices, the retail prices and the collecting prices? How to choose the collecting channel? And so on.

For the management of closed-loop supply chain, Gu et al. (2008) study a single closed-loop supply chain that consists of one manufacturer and one retailer. In their study, the wholesale price and the retail price of the new products will be affected by the collecting price of the used-products; the return rate of the used-products is an increasing function of the collecting price; they obtain the optimal collecting price, the optimal wholesale price and the optimal retail price based on the three models (manufacturer for collecting, retailer for collecting and the third party for collecting); and by comparing the optimal pricing and the profits of the models, they find that the manufacturer for collecting is the best channel, and the retailer for collecting is another choice if the manufacturer has decided to transfer all of its cost saving to the retailer.

However, the management of the two competitive closed-loop supply chains is different from the management of the single closed-loop supply chain. In the two competitive closed-loop supply chains, the wholesale price and the retail price of the new products will be not only affected by the collecting price of the used-products but also affected by the ratio of the market share and the substitution ratio of the two products. So, in this paper, we will focus on the optimal results of the wholesale prices, the retail prices and the collecting prices based on the following models of the two competitive closed-loop supply chains: Model MMC (two manufacturers for collecting), Model MRC (manufacturer one and retailer for collecting) and Model RRC (retailer for collecting). Meanwhile, we will present the best collecting channel via comparing the optimal results of the three models. In order to analyse the impacts of the ratio of the market share and the substitution ratio of the two products on the optimal results, we will illustrate a numerical example.

2. Literature review

The literatures related to our study includes three parts: the management of supply chain with competing manufacturers, the management of supply chain with competing retailers and the management of supply chain with chain-to-chain competition.

In the management of supply chain with competing manufacturers, Lippman and McCardle (1997) consider a competitive version of the classical newsboy problem and investigate the impact of competition upon industry inventory. In their study, a splitting rule specifies how initial industry demand is allocated among competing firms and how any excess demand is allocated among firms with remaining inventory. Mahajan and van Ryzin (2001) analyse a model of inventory competition among firms that provide competing, substitutable goods. Parlar (1988) uses game theoretic concepts to analyse the inventory problem with two substitutable products having random demands. Choi (1991) analyses competition between a national brand and a store brand using three game theoretic models: a manufacturer-leader Stackelberg game, a retailer-leader Stackelberg game and a vertical Nash game.

In the management of supply chain with competing retailers, Bernstein and Federgruen (2005) investigate the equilibrium behaviour of decentralised supply chains with a single supplier and competing retailers under demand uncertainty. Savaskan and Wassenhove (2006) focus on the interaction between a manufacturer's reverse channel choice to collect post-consumer goods and the strategic product pricing decisions in the forward channel when retailing is competitive. Ingene and Parry (1995) study the coordination of the supply chain with two retailers competing in price. Iyer (1998) investigates how manufacturers should coordinate the supply chain with one manufacturer and two retailers competing in price and service. Xiao et al. (2005) study the coordination of the supply chain with demand disruption and consider a price–subsidy rate contract to co-ordinate the investments of the competing retailers. Xiao et al. (2007) find that the linear quantity discount scheme can coordinate the supply chain with two competing retailers, and the all-unit quantity discount scheme can coordinate the supply chain if the retailers are identical after the market demand was disrupted. Xiao and Qi (2008) study the coordination of a supply chain with one manufacturer and two competing retailers after the production cost of the manufacturer was disrupted. They also extend the model to the case with both cost and demand disruption.

In the management of supply chain with chain-to-chain competition, McGuire and Staelin (1983) investigate the effect of product substitutability on Nash equilibrium distribution structures in a duopoly where each manufacturer distributes its goods through a single exclusive retailer, which may be either a franchised outlet or a factory store. They find that each manufacturer will distribute its product through a company store for low degrees of substitutability, and manufacturers will be more likely to use a decentralised distribution system for more highly competitive goods. Anderson and Bao (2010) focus on chain-to-chain competition in which different manufacturers sell through exclusive retailers that compete for end-customers. So, there is direct competition between the retailers, while the competition between manufacturers is indirect. Kurata et al. (2007) analyse channel pricing in multiple distribution channels under competition between a national brand and a store brand, where a national brand can be distributed both through a direct channel (e-channel) and an indirect channel (local stores), but a store brand can be distributed only through an indirect channel. Boyaci and Gallego (2004) consider a market with two competing supply chains, each consisting of one wholesaler and one retailer. They assume that the business environment forces supply chains to charge similar prices and to compete strictly on the basis of customer service. They model customer service competition using game theoretical concepts. They consider three competition scenarios between the supply chains. Rezapour and Farahani (2010) develop an equilibrium model to design a centralised supply chain network operating in markets under deterministic price-dependent demands and with a rival chain present. The two chains provide competitive products, either identical or highly substitutable, for some participating retailer markets. Xiao and Yang (2008) consider two competing supply chains facing uncertain demands, where each supply chain consists of one risk-neutral supplier and one risk-averse retailer. Two retailers compete in retail price as well as in service investment. They assume that each retailer has a long-term relationship with his supplier, which is assured by his individual rationality constraint. The products of two suppliers are partially differentiated and each supplier sells products to customers through her retailer.

Our study of the two competitive closed-loop supply chains is based on the above literatures and has the following features: the return rate of the used-products is an increasing function of the collecting price; the competition of the two manufacturers is described by the market share ratio and the substitute ratio of the two products. We will give the optimal decisions of the wholesale prices, the retail prices, the collecting prices and the best channel.

3. Problem description

In this section, we will give the models of the two competitive closed-loop supply chains: Model MMC, Model MRC and Model RRC.

3.1 Model MMC

Figure 1 illustrates the two competitive closed-loop supply chains of Model MMC. In this model, manufacturer one and manufacturer two will collect their own used-products from the end-customers directly.

Figure 1 Model MMC.

In the forward supply chain of CLSC1 (CLSC2) of Model , manufacturer one (manufacturer two) produces his new product with the unit manufacturing cost or with the unit remanufacturing cost , sells the new product to the retailer with the unit wholesale price and the retailer sells the product one (product two) to the end-customers with the unit retail price . Herein, (), () is the unit cost saving from remanufacturing.

In the reverse supply chains of CLSC1 (and CLSC2) of Model , manufacturer one (manufacturer two) will pay the unit collecting price to the end-customers for a used-product, and the unit average operational cost of collecting a used-product is , including inventory cost, transportation cost, etc.

In this model, manufacturer one and manufacturer two will decide the collecting prices and of the used-products and the wholesale prices and of the new products to maximise their profits and , the retailer will decide the retail prices and of the new products to maximise its profit .

3.2 Model MRC

Figure 2 illustrates the two competitive closed-loop supply chains of Model . In this model, manufacturer one will collect used-product one from the end-customers directly while the retailer will engage in collecting the used-product two for manufacturer two. To take the collected used-products back, manufacturer two must pay a unit transfer price to the retailer.

Figure 2 Model MRC.

The forward supply chains of CLSC1 and CLSC2 of Model are similar to the forward supply chains of Model .

In the reverse supply chain of CLSC1 of Model , manufacturer one will pay the unit collecting price to the end-customers for a used-product, and the unit average operational cost of collecting a used-product is . In the reverse supply chain of CLSC2, manufacturer two will take back all of the collected used-products from the retailer with the unit transfer price and the retailer will pay a unit collecting price to the end-customers for a used-product, and the unit average operational cost of collecting a used-product is , including inventory cost, transportation cost, etc.

In this model, manufacturer one will decide the wholesale price and the collecting price to maximise his profit , manufacturer two will decide the wholesale price to maximise his profit , the retailer will decide the collecting price of the used-products and the retail prices and of the new products to maximise his profit .

3.3 Model RRC

Figure 3 illustrates the two competitive closed-loop supply chains of Model . In this model, the retailer will collect the used-products from the end-customers. To take the used-products back, manufacturer one and manufacturer two must pay the unit transfer prices to the retailer.

Figure 3 Model RRC.

The forward supply chains of Model are similar to forward supply chains of Model .

In the reverse supply chain of CLSC1 (CLSC2) of Model , manufacturer one (manufacturer two) will pay a unit transfer price to the retailer, , the retailer will pay the unit collecting prices to the end-customers for a used-product, and the unit average operational cost of collecting a used-product is , including inventory cost, transportation cost, etc.

In this model, manufacturer one and manufacturer two will decide the wholesale prices and of the new products to maximise their profits and , the retailer will decide the collecting prices and of the used-products and the retail prices and of the new products to maximise its profit .

In our study, the following assumptions are postulated: (1) the return rate of the used-products is an increasing function of the collecting price, , (); (2) each manufacturer produces only one kind of product that is manufactured or remanufactured. Namely, there is no difference between the remanufactured products and the manufactured products; (3) the new product demand is a function of retail prices and with and being the positive parameters and , is the ratio of the market share, is the substitute ratio of the two manufacturers' products, , and (4) in each closed-loop supply chain, the manufacturer is the Stackelberg leader and all the members in this closed-loop supply chain share the same information.

4. Optimal results

In this section, we show the optimal results of the wholesale prices, the retail prices and the collecting prices of Model MMC, Model MRC and Model RRC, respectively. And then, we give the best channel by comparing the optimal results.

Theorem 1

In Model , the optimal value of the wholesale prices and , the collecting prices and and the retail prices and are given as follows:

Herein, , .

In Model , the retailer's problem is to decide and ,

Proposition 1

In Model , the retailer's profit is strictly jointly concave in and .

The proof of Proposition 1 is given in Appendix. Proposition 1 indicates that we can find the optimal values of and by using only the first-order conditions. So, the retailer's unique best response can be obtained from the objective function (1):

The derived demand functions of the new products are given by and .

The problems of manufacturer one and manufacturer two can be stated as

Proposition 2

In Model , the profit of manufacturer one is strictly jointly concave in and , the profit of manufacturer two is strictly jointly concave in and .

The proof of Proposition 2 is given in Appendix. Proposition 2 indicates that we can find the optimal values of , , and by using only the first-order conditions.

So, it is easy to get , , and by the manufacturers' first-order condition. Replacing with in Equation (2), can be obtained. Replacing with in Equation (3), can be obtained.

Proposition 3

In Model , the ratio of the market share should satisfy the following condition:

Proposition 1 can be proved by and .

Theorem 2

In Model , the optimal value of the wholesale prices and , the collecting prices and and the retail prices and are presented as follows:

Herein, , .

In Model , the retailer's problem is to decide , and ,

Proposition 4

In Model , the retailer's profit is strictly jointly concave in , and .

The proof of Proposition 4 is given in Appendix. Proposition 4 indicates that we can find the optimal values of , and by using only the first-order conditions. So, the retailer's unique best response can be obtained from objective function (6):

The derived demand functions of the new products are given by

The manufacturer's problem can be stated as

By the same method as Proposition 2, is strictly jointly concave in and and is concave in . We can get , and by the manufacturers' first-order condition. It is easy to get and by substituting and with and in Equations (7) and (8).

Proposition 5

In Model , the ratio of the market share should satisfy the following condition:

Proposition 5 can be obtained from and .

Theorem 3

In Model , the optimal value of the wholesale prices and , the collecting prices and and the retail prices and are given as follows:

Herein, , .

In Model , the retailer's problem is to decide , , and ,

Proposition 6

In Model , the retailer's profit is strictly jointly concave in , , and .

The proof of Proposition 6 is given in Appendix. Proposition 6 indicates that we can find the optimal values of , , and by using only the first-order conditions. So, the retailer's unique best response can be obtained from objective function (12):

The derived demand functions of the new products are and .

The manufacturer's problem can be stated as

The objective function (17) is concave in . The objective function (18) is concave in . We can get and by the manufacturers' first-order condition. It is easy to get and by substituting and with and in Equations (13) and (14).

Proposition 7

In Model , the ratio of the market share should satisfy the following condition: where

Proposition 3 can be obtained from and .

Let and . In terms of the above theorems and propositions, the optimal values of the product demand and the profit of each model are calculated. Based on the optimal results summarised in Table 1, we obtain some propositions as shown below.

Table 1 The optimal results of the two competitive closed-loop supply chains models.

The optimal results	Model	Model	Model
Herein, , , , , ,

Proposition 8

For models , and , when :

i.	If , then ; if , then ;
ii.	If , then ; if , then .

The proof of Proposition 8 can be easily obtained from the optimal values of in Table 1.

Since the return rate is an increasing function of the collecting price and a higher return rate will result in more saving cost from remanufacturing, the manufacturers hope the collecting prices are higher. From Proposition 8, we can get: if the two manufacturers are responsible for collecting the used-products directly, the collecting prices will be higher; if one of the two manufacturers lets the retailer collect the used-products by a contract, in order to get a higher collecting price, he would transfer all of his cost saving from remanufacturing to the retailer; if the two manufacturers contract the retailer for collecting the used-products, they would transfer all of their cost saving from remanufacturing to the retailer for higher collecting prices.

Proposition 9

In models , and , when :

i.	If , then , ; if , then , .
ii.	If , then , ; if , then , .

The proof of Proposition 9 can be easily obtained from the optimal values in Table 1.

Owing to the market demand for product of manufacturer one (or manufacturer two), (or ) is a decreasing function of his retail price and an increasing function of the other one's retail price, and a higher demand will result in more profit, the manufacturers hope their retail prices are lower. From Proposition 9, we can get: if the two manufacturers are responsible for collecting the used-products directly, the retail prices will be lower, and at the same time the wholesale prices are lower; if one of the two manufacturers lets the retailer collect the used-products by a contract, in order to get a lower retail price, he would transfer all of his cost saving from remanufacturing to the retailer; if the two manufacturers contract the retailer for collecting the used-products, they would transfer all of their cost saving from remanufacturing to the retailer for lower retail prices.

Proposition 10

In models , and , when :

i.	If and , then , and .
ii.	If and , then , and .

The proof of Proposition 10 can be obtained from the optimal values in Table 1 and Equations 1 4-6 10 17).

Proposition 10 indicates that all of the members of the two competitive closed-loop supply chains cannot get their highest profits simultaneously if at least one manufacturer is responsible for collecting the used-products directly. Namely, only if the two manufacturers contract the retailer collecting the used-products and transfer all of their cost saving from remanufacturing to the retailer, all of the members of the two competitive closed-loop supply chains can get their highest profits simultaneously.

From Propositions 8–10, we find that the retailer for collecting (Model ) is the best channel for the two competitive closed-loop supply chains. In order to motivate the retailer to collect the used-products, the two manufacturers should transfer all of their cost saving from remanufacturing to the retailer.

Next, we will give a numerical example and analyse the impacts of and on the optimal results.

5. Numerical example and analysis

In this numerical example, the potential market capacity of demand is (unit:piece), the unit manufacturing costs of manufacturer one and manufacturer two and ; the unit remanufacturing costs of manufacturer one and manufacturer two and ; the operational costs of the manufacturers and retailer for collecting are , and ; and the other parameters , and are equal to , and , respectively. Here, as an example, we only show the numerical results when and .

From Table 1, we can find that the collecting prices of used-products in models MMC, MRC, and RRC are not affected by and , so we analyse the impacts of the two parameters on the other optimal results.

5.1 The impacts of α on the optimal results

The numerical results of the optimal wholesale prices, retail prices and profits with different value of parameter and ) are shown in Figure 4, herein, .

Figure 4 The changing of the optimal results with variable . (i) Changing of the wholesale prices; (ii) changing of the retail prices; (iii) changing of the profits.

From Figure 4, we can obtain the following implications.

1.

In models MMC, MRC and RRC, the wholesale prices and the retail prices for the new product one of manufacturer one will become higher while its ratio of the market share () increases, and the wholesale prices and the retail prices for the new product two of manufacturer two will become lower while its ratio of market share (1 − ) decreases (see Figure 4(i) and 4(ii)). In the competitive environment, the higher wholesale prices and retail prices of the new product one may reduce its ratio of the market share, and the lower wholesale prices and the retail prices may enlarge its ratio of the market share. As a result, the market adjustment by two kinds of products' prices can help to balance their ratio of the market share.

2.

In models MMC, MRC and RRC, the profit of manufacturer one will increase while its product's ratio of market share () increases, the profit of manufacturer two will decrease while its product's ratio of market share (1 − ) decreases, and the profit of retailer will decrease first and then increase while increases (see Figure 4 (iii)). Owing to the market adjustment by prices, the profits of the three members will be the same approximately. Moreover, the profit of retailer is the lowest at about half of , it means the retailer hopes that the two manufacturers have different market share.

5.2 The impacts of θ on the optimal results

The numerical results of the optimal wholesale prices, the retail prices and the profit with changing values of parameter are shown in Figure 5, herein, . From Figure 5, we can obtain that the wholesale prices, the retail prices and the profits will increase when the substitute ratio of the two manufacturers' products () increases. It means the retailer prefers a higher substitute ratio of the two products.

Figure 5 The changing of the optimal results with different . (i) Changing of the wholesale prices; (ii) changing of the retail prices; (iii) changing of the profits.

6. Conclusion

In terms of the characteristics of the two competitive closed-loop supply chains, and on the assumption that the return rate of the used-products is an increasing function of the collecting price, this paper focused on the management of the two competitive closed-loop supply chains.

We obtained the optimal wholesale prices, the optimal retail prices and the optimal collecting prices based on three models: Model MMC, Model MRC and Model RRC. By comparing the optimal results, we find that the retailer for collecting is the best channel for the two competitive closed-loop supply chains if the two manufacturers would like to transfer all of their cost saving from remanufacturing to the retailer. Moreover, we analysed the impacts of the market share ratio and the substitute ratio of the two products on the optimal results by a numerical example.

How to make the optimal decisions of the prices and the channel for the two competitive closed-loop supply chains with risk preferences may be the future work.

Acknowledgement

This work was supported by the National Nature Science Foundation under Grant No. 70871089.

Notes

^1. [email protected]

Appendix

Proof of Proposition 1

The second-order partial derivatives of with respect to and are , and . So, we get the Hessian matrix:

Since and (because of ), is strictly jointly concave in and .

Proof of Proposition 2

The second-order partial derivatives of with respect to and are , and . So, we get the Hessian matrix:

Herein, . In Model MMC, manufacturer one is responsible for remanufacturing and collecting the used-product, and condition should be satisfied. Let D ₁ stand for , then we get

Since the market demand is far greater than , and , it is easy to get . Namely, . Because of , is strictly jointly concave in and .

By the same method, is strictly jointly concave in and .

Proof of Proposition 4

Taking the second-order partial derivatives of with respect to , and , respectively, we have the Hessian matrix:

Herein, . Because of , the first principal minor of the above Hessian matrix has a negative value.

For the determinant of the second principal minor,

(because of ). Namely, the determinant of the second principal minor is always positive.

With regard to the third principal minor, because of , , and ,

if equals to and this condition is true by the context that this is the optimal value of .

From the above proofs, the first principal minor of the above Hessian matrix has a negative value, the determinant of the second principal minor is always positive and the third principal minor is negative. Therefore, is strictly jointly concave in , and .

Proof of Proposition 6

Taking the second-order partial derivatives of with respect to , , and , respectively, we have the Hessian matrix:

Herein, , and . Because of , , and , it is easy to prove . It is obvious that the first principal minor of the above Hessian matrix has a negative value.

For the determinant of the second principal minor,

As for the third principal minor because of , , and the market demand ,

With regard to the fourth principal minor, let and . Because of , , , , and ,

Herein, and if equals to and equals to , respectively, and the two conditions are true by the context that they are the optimal value of and .

From the above proofs, the first principal minor of the above Hessian matrix has a negative value, the determinant of the second principal minor is always positive, the third principal minor is negative and the fourth principal minor is positive. Therefore, is strictly jointly concave in , , and .