A Game-theoretical Model of Buy-back Contracts in Assembly

Systems with Uncertain Demand

Zhiqi Lei

Finance major, School of International Education, Wuhan University of Technology, Wuhan, Hubei Province, China

Keywords: Buy-Back Contract, Assembly System, Uncertain Demand, System Coordination.

Abstract: An assembler needs to purchase complete sets of components, each component produced by different

suppliers. Each party in the assembly system shares the risks of demand uncertainty through a buy-back

contract. In this paper, a game-theoretical model is established to analyze how the buy-back contract is

developed under two different mechanisms. The first is one where the assembler sets the buy-back price of

each component, and the second is one where the suppliers set the buy-back prices of their own components.

In both cases, by backward induction, the decision problem is formulated as a constrained optimization

problem. The optimal order quantity is derived. Under the first setting, the system performance is

demonstrated to be increasing in the assembler's share of the profit per unit of product. Under the second

setting, the system performance is demonstrated to be decreasing in the assembler's share of the profit per unit

of product. By comparing the first-order conditions, it is shown that the system performances under the two

settings are equal if the assemblerโs share of profit is larger than the reciprocal of the number of parties in the

system. Finally, numerical examples are provided to illustrate some of the main results.

1 INTRODUCTION

The competition between supply chains has become

the main mode of market competition in the 21st

century. Enterprises in supply chains should

cooperate to maximize the overall benefit. Compared

with the single enterprise, supply chains are usually

faced more demand uncertainties and information

asymmetry, which may lead to the inefficiency of

supply chain. At the same time, each enterprise in the

supply chain pursues the maximization of their own

interests, which may conflict with the overall goal of

the supply chain in the operation process. It has

become a popular research topic to coordinate the

supply chain and improve the profit of the whole

supply chain through well-designed contracts. Supply

chain management is an idea of integration, which

emphasizes the close cooperation of the supply chain

members. However, the supply chain consists of

relatively independent members, whose decision-

making power is decentralized and there are conflicts

of interest among them. The buy-back contract is a

typical coordination mechanism which is widely used

in practice to alleviate the inefficiency of

decentralized supply chain.

In a buy-back contract, suppliers purchase the

product that retailers have not sold out at a specified

buy-back price after the selling season. By

implementing such a contract, the supplier can

provide the retailer a protection, so as to induce the

retailer to increase the order quantity. As the risk of

demand uncertainty is shared by suppliers and

retailers, their revenue and cost are balanced better.

While the retailer can benefit from the buy-back

mechanism, the supplier can also obtain higher profit

from a higher order quantity. Thus, a win-win goal

can be achieved.

In the 1980s, some researchers began to study the

buy-back contract in the supply chain. Paternackde

(Pasternack 1985) studied the coordination of supply

chain where a single supplier and a single retailer sell

a product, focused on the buy-back contract in the

common sales channel, and analyzed the potential

inefficiency of operation due to the influence of

marginal benefit. Under the assumption that the

return price is less than the wholesale price, it is

proved that the total profit of the distribution channel

is similar to that of the vertically integrated supply

chain. His research shows that neither Full Returns

policy nor No Returns policy is effective. A

compromise buy-back contract can promote supply

Lei, Z.

A Game-theoretical Model of Buy-back Contracts in Assembly Systems with Uncertain Demand.

DOI: 10.5220/0011165400003440

In Proceedings of the Inter national Conference on Big Data Economy and Digital Management (BDEDM 2022), pages 137-144

ISBN: 978-989-758-593-7

Copyright

c

๎ 2022 by SCITEPRESS โ Science and Technology Publications, Lda. All rights reserved

137

chain collaboration and improve collaboration

efficiency through Pareto optimization. Emmos and

Gilbert (Emmos 1998) and Donohue (Donohue 2000)

pointed out that it is beneficial for suppliers and

retailers to sign buy-back contracts for their trade.

Gong (Gong 2008) showed that the optimal buy-back

contract between suppliers and retailers could not

always be realized under the condition of information

symmetry. In recent years, buy-back contracts have

attracted extensive attention from the academic

community. The effects and design of buy-back

contracts have been studied from different

perspectives (Das, 2017, Giri, 2014, Lau, 1999,

Padmanabhan, 1995, Webster, 2000, Zhang, 2016,

Hou, 2009, Wang, 2008, Xiao, 2008, Xu, 2008, Xu,

2008).

Assembly system is a common operation mode in

modern manufacturing industry (Wang 2003). With

the popularity of industry subdivision and

outsourcing, the supply of components for assembly

lines is often not controlled by itself. Thus, it is

necessary to coordinate the relationship with

suppliers. Due to the characteristics of the assembly

line, the components of the final product are

complementary. It is necessary to coordinate and

manage multiple suppliers at the same time, which

may be difficult. However, few papers have studied

the buy-back contract in an assembly system.

In the context of uncertain demand, this paper

establishes a Stackelberg game-theoretical model for

the buy-back contract of assembly system, studies the

decisions of all supply chain parties, and analyzes the

influence of various parameters on the buy-back

contract. By comparing the system performance

under two different mechanisms, some managerial

insights about the supply chain structure are

provided. In order to facilitate the presentation and

improve the readability of this paper, the following

section first introduces and analyzes the buy-back

contract model in a simple one-to-one supply chain.

2 BUY-BACK CONTRACT IN

ONE-TO-ONE SYSTEM

Consider the following basic case: the demand for the

final product is uncertain, but the probability

distribution is known. The price of the product is

constant. The product is assembled from a set of

components. In order to produce and sell a product,

the order quantities of all components need to be

determined before the demand is realized.

There is only one supplier and one retailer in the

system, and the demand of the product is the random

variable ๐ท , with probability distribution function

๐น

๏บ

โ

๏ป

, and probability density function ๐

๏บ

โ

๏ป

. The

supplier and retailer use the common demand

distribution. The retailer has to order the product

before the selling season. There is only one chance to

place an order. The unit cost of the product produced

by the supplier is c, the wholesale price of the product

sold to the retailer is w. The price of the product is p.

Only when the profit is positive, the supplier and the

retailer can be willing to participate in the

game. Thus,

๐>๐ค>๐ should be satisfied. After the selling

season, the retailer can sell the leftover product to the

supplier at a price of b, and one unit of product can be

salvaged at a value v by the supplier. Obviously, ๐ฃ<

๐ should hold.

The decision variable is the buy-back price b, then

the supplier decides its optimal supply quantity ๐

๎ฌต

,

and the retailer decides its optimal order quantity ๐

๎ฌถ

.

When the buy-back price b is determined, the optimal

order quantity of the retailer and the optimal supply

quantity of the supplier can be calculated respectively

from the classical newsvendor model (at this time, the

case that the optimal supply quantity is infinite can be

ignored. The detailed analysis is given in the later part

of this paper). Obviously, the actual quantity of the

product is the minimum of the optimal supply quantity

๐

๎ฌต

and the optimal order quantity ๐

๎ฌถ

.

For the supplier, the under-storage cost is ๐คโ๐,

and the over-storage cost is ๐+๐โ๐คโ๐ฃ. Then the

optimal order quantity of the supplier satisfies

๐น

๏บ

๐

๎ฌต

๏ป

=

๎ฏช๎ฌฟ๎ฏ

๎ฏ๎ฌฟ๎ฏฉ

.

(1)

As ๐น

๏บ

โ

๏ป

is an increasing function, both the

supplierโs expected profit and its optimal supply

quantity ๐

๎ฌต

are decreasing in the buy-back price b.

For the retailer, the under-storage cost is

p

w-

,

and the over-storage cost is

wb-

. Then the

retailerโs optimal order quantity satisfies

๐น

๏บ

๐

๎ฌถ

๏ป

=

๎ฏฃ๎ฌฟ๎ฏช

๎ฏฃ๎ฌฟ๎ฏ

.

(2)

As ๐น

๏บ

โ

๏ป

is an increasing function, both the

retailerโs expected profit and its optimal order quantity

๐

๎ฌถ

are increasing in the buy-back price b.

BDEDM 2022 - The International Conference on Big Data Economy and Digital Management

138

2.1 The Retailerโs Decision on the

Buy-back Price

Now, analyze the principles of buy-back pricing from

the perspective of the retailer. Consider the following

setting: The retailer decides the buy-back price of the

product, and its objective is to maximize its own

expected profit.

No matter how the buy-back price is set, the

supplierโs optimal supply quantity and the retailerโs

optimal order quantity cannot be infinite at the same

time, and at least one of them is finite positive. The

actual quantity of the product is the minimum value

between the optimal supply quantity ๐

๎ฌต

and the

optimal order quantity ๐

๎ฌถ

. If ๐

๎ฌถ

<๐

๎ฌต

, it means that

the retailer has made a lower buy-back price, but it is

not conducive to increase the actual quantity of the

product. It can be inferred that the optimal policy of

the retailer must satisfy ๐

๎ฌถ

โฅ๐

๎ฌต

. In addition, an

intuitive conclusion is ๐+๐โฅ๐ค+๐ฃ, because when

๐+๐<๐ค+๐ฃ, the supplierโs optimal quantity is

infinite.

In short, the supplier will become the bottleneck of

both sides, and the final order quantity is decided by

the supplier. Then this process can be summarized as

follows: The buy-back price is set by the retailer, and

the final order quantity is determined by the supplier.

From the above discussion, the following conditions

can be obtained:

๎ฏฃ๎ฌฟ๎ฏช

๎ฏฃ๎ฌฟ๎ฏ

โฅ

๎ฏช๎ฌฟ๎ฏ

๎ฏ๎ฌฟ๎ฏฉ

,๐+๐โฅ๐ค+๐ฃ.

(3)

The final quantity of the product is ๐. As the

conditions of ๐=๐๐๐

๏บ

๐

๎ฌต

,๐

๎ฌถ

๏ป

=๐

๎ฌต

and

๐น

๏บ

๐

๏ป

=

๎ฏช๎ฌฟ๎ฏ

๎ฏ๎ฌฟ๎ฏฉ

holds, the retailerโs profit is

๐

๏บ

๐

๏ป

=๐ธ[โ๐ค๐+๐๐๐๐

๏บ

๐,๐ท

๏ป

+๐

๏บ

๐โ

๐ท

๏ป

๎ฌพ

].

(4)

The above problem can be summarized as a

constrained optimization problem as follows:

๐๐๐ฅ๐

๎ฌต

๏บ

๐

๏ป

=

๏บ

๐โ๐ค

๏ป

๐

โ

๏บ

๐โ๐

๏ป

๎ถฑ๐น

๏บ

๐ฅ

๏ป

๐๐ฅ

๎ฏ

๎ฌด

๐ .๐ก.

๎ต

๐น

๏บ

๐

๎ฌต

๏ป

=

๐คโ๐

๐โ๐ฃ

๐โ๐ค

๐โ๐

โฅ

๐คโ๐

๐โ๐ฃ

,๐+๐โฅ๐ค+๐ฃ

(5)

The solution and discussion of these equations are

carried out in Section 3.

2.2 The Supplierโs Decision on the

Buy-back Price

Now, analyze the principles of buy-back pricing from

the perspective of the supplier. Consider the

following setting: The supplier decides the buy-back

price of the product to maximize its own expected

profit.

The principle is the same as that described in the

previous subsection. When the supplier decides the

buy-back price, the supplier will set the buy-back

price low enough to maximize its own profit. If ๐

๎ฌถ

>

๐

๎ฌต

, the buy-back price is too high, because it is not

conducive to increase the retailerโs order quantity. In

short, the final quantity of the product is decided by

the retailer. Then this process can be summarized as

follows: The buy-back price is determined by the

supplier, and the final quantity of the product is

determined by the retailer.

The actual quantity of the product is the minimum

between the supplierโs optimal supply quantity ๐

๎ฌต

and the retailerโs optimal order quantity ๐

๎ฌถ

. The

retailerโs optimal order quantity is infinite if ๐>๐ค.

From the above discussion, the following conditions

can be obtained:

๎ฏฃ๎ฌฟ๎ฏช

๎ฏฃ๎ฌฟ๎ฏ

โค

๎ฏช๎ฌฟ๎ฏ

๎ฏ๎ฌฟ๎ฏฉ

,๐+๐โค๐ค+๐ฃ.

(6)

The final quantity of the product is

๐. As ๐=๐

๎ฌถ

,

it can be inferred that ๐น

๏บ

๐

๏ป

=

๎ฏฃ๎ฌฟ๎ฏช

๎ฏฃ๎ฌฟ๎ฏ

. The supplierโs

profit is

๐

๎ฌถ

๏บ

๐

๏ป

=๐ธ[

๏บ

๐คโ๐

๏ป

๐โ

๏บ

๐โ

๐ฃ

๏ป๏บ

๐โ๐ท

๏ป

๎ฌพ

].

(7)

The above problem can be summarized as a

constrained optimization problem as follows:

๐๐๐ฅ๐

๎ฌถ

๏บ

๐

๏ป

=

๏บ

๐คโ๐

๏ป

๐

โ

๏บ

๐โ๐ฃ

๏ป

๎ถฑ๐น

๏บ

๐ฅ

๏ป

๐๐ฅ

๎ฏ

๎ฌด

๐ .๐ก.

๎ต

๐น

๏บ

๐

๏ป

=

๐โ๐ค

๐โ๐

๐โ๐ค

๐โ๐

โค

๐คโ๐

๐โ๐ฃ

, ๐+๐โค๐ค+๐ฃ

(8)

The solution and discussion of these equations

will also be carried out in Section 3.

3 BUY-BACK CONTRACT FOR

ASSEMBLY SYSTEM

The assembler has to buy the components before the

actual demand is known. Due to the uncertainty of

A Game-theoretical Model of Buy-back Contracts in Assembly Systems with Uncertain Demand

139

demand, the assembler need to make decisions based

on demand prediction. The output of the assembly

system is limited to each link, and the output capacity

of the system is equal to the weakest link. In addition,

in order to deal with the risk of demand uncertainty,

the supply chain members may sign buy-back

contracts to share the risk. The benefit of this contract

is to reduce the risk downstream of the supply chain,

encourage them to increase their order quantities, and

thereby increase the overall profit of the system. Due

to the complementarity of components, designing the

buy-back contract of an assembly system is relatively

complicated.

The demand for the final product of the assembly

system is a random variable D. The probability

distribution function of D is

๐น

๏บ

โ

๏ป

, and the probability

density function is ๐

๏บ

โ

๏ป

. The unit price of the product

in the market is p. The product consists of n

components. Without loss of generality, suppose that

each supplier produces one of these n components.

For notational convenience, define ๐=

๏ผ

1,2,โฆ,๐

๏ฝ

.

All supply chain parties have to decide the order

quantity or supply quantity before the selling season.

Once the components are in place, the demand is

realized and the product can be assembled in a short

time.

The unit cost of component i is ๐

๎ฏ

. Considering

the assembly process, the production of a final product

also needs to invest ๐

๎ฌด

as the assembly cost.

Obviously,

โ

๐

๎ฏ

<๐

๎ฏก

๎ฏ๎ญ๎ฌด

is required to make sure

that the profit is positive. In fact, ๐

๎ฌด

can be set equal

to zero, and then the market price of the product can

be adjusted to be ๐โ๐

๎ฌด

. The wholesale price of

component i is ๐ค

๎ฏ

, then ๐โ

โ

๐ค

๐

๐

๐=1

is the

assemblerโs profit from one unit of the product. To

ensure that each member does not refuse to participate

in the game, ๐ค

๎ฏ

>๐

๎ฏ

is required. Because the

demand is random and the ordering decisions need to

be made before demand realization, overstocking or

understocking can occur. The unite salvage value of

component i is ๐ฃ

๎ฏ

๏บ

๐ฃ

๎ฏ

<๐

๎ฏ

๏ป

.

The decision variable is the buy-back price ๐

๎ฏ

of

each component. Each component supplier shall

decide its supply quantity ๐

๎ฏ

according to the buy-

back price, and the assembler shall decide the order

quantity ๐

๎ฌด

.

The sum of the corresponding parameters is in

uppercase letters for marking convenience. For

example, define

๐ถโก๎ท ๐

๎ฏ

,๐ตโก๎ท๐

๎ฏ

๎ฏก

๎ฏ๎ญ๎ฌต

,

๎ฏก

๎ฏ๎ญ๎ฌต

(9)

๐โก๎ท ๐ค

๎ฏ

,๐โก๎ท๐ฃ

๎ฏ

๎ฏก

๎ฏ๎ญ๎ฌต

,

๎ฏก

๎ฏ๎ญ๎ฌต

Suppliers and retailers use the same demand

distribution. When the buy-back price ๐

๎ฏ

,๐โ๐ is

determined, the optimal order quantity (supply

quantity) of each supply chain party can be calculated

from the classical newsvendor model. The final

quantity of the product and components should be as

follows: The quantity of each component is the same,

and the actual quantity ๐ of the product is the

minimum among the optimal supply quantity ๐

๎ฏ

๏บ

๐=

1,2,โฆ๐

๏ป

and the optimal order quantity ๐

๎ฌด

. That is

to say, ๐=๐๐๐

๏ผ

๐

๎ฏ

๏ฝ

.

The profit of the system is

๐

๏บ

๐

๏ป

=๐ธ[โ๐ถ๐+๐๐๐๐

๏บ

๐,๐ท

๏ป

+๐

๏บ

๐โ

๐ท

๏ป

๎ฌพ

],

(10)

which can be written as

๐

๏บ

๐

๏ป

=

๏บ

๐โ๐ถ

๏ป

๐โ

๏บ

๐โ๐

๏ป

๎ถฑ๐น

๏บ

๐ฅ

๏ป

๐๐ฅ

๎ฏ

๎ฌด

.

(11)

According to the classical newsvendor model, the

above profit function is concave and has a unique

optimal solution. This property can help compare the

effectiveness of different mechanisms.

3.1 A Contract Model in Which the

Assembler Determines the

Buy-back Price

Now, from the assemblerโs point of view to analyze

the principle of buy-back price formulation. Consider

this situation: the assembler decides the buy-back

price of the product, and the assembler's goal in setting

the buy-back price is to maximize its own profit.

Facing n suppliers, the assembler formulates the

buy-back price ๐

๎ฏ

of each component ๐โ๐. The

optimal strategy of the assembler to formulate the

buy-back price should meet the following conditions:

๐

๎ฌด

โฅ๐

๎ฌต

=๐

๎ฌถ

=โฏ=๐

๎ฏก

;

๐น

๏บ

๐

๎ฌด

๏ป

=

๎ฏฃ๎ฌฟ๎ฏ

๎ฏฃ๎ฌฟ๎ฎป

,๐

๎ฏ

+๐

๎ฏ

โฅ๐ค

๎ฏ

+๐ฃ

๎ฏ

,๐๐๐;

๎ฏฃ๎ฌฟ๎ฏ

๎ฏฃ๎ฌฟ๎ฎป

โฅ

๎ฏ๎ฌฟ๎ฎผ

๎ฎป๎ฌฟ๎ฏ

;

The supplierโs optimal policy must satisfy ๐

๎ฌด

โฅ

๐

๎ฏ

,๐โ๐. If there exists i which makes ๐

๎ฌด

<๐

๎ฏ

,

then it means that the assembler has set a too low buy-

back price, which is not conducive to increase the

order quantity. In addition, when the buy-back price

is determined, the optimal supply quantity of each

component can be obtained immediately according to

the classical newsvendor model. It is useless for one

supplier to have the optimal supply quantity higher

BDEDM 2022 - The International Conference on Big Data Economy and Digital Management

140

than others, which is a waste to the assembler, and the

buy-back price of this component must be increased.

Thus, the optimal supply quantity of each supplier

should be the same. When the sum of the buy-back

prices of each supplier is fixed, the optimal policy is

the one that maximizes the output of the system.

Therefore, the optimal strategy must satisfy ๐

๎ฌด

โฅ

๐

๎ฌต

=๐

๎ฌถ

=โฏ=๐

๎ฏก

. In order to ensure that the

optimal supply quantity of each supplier is limited,

๐

๎ฏ

+๐

๎ฏ

>๐ค

๎ฏ

+๐ฃ

๎ฏ

should hold for all ๐๐๐. The

intuitive explanation is that suppliers will pay cost

when their production exceeds the actual demand.

According to the above analysis, the final quantity

of the product is determined by the supplier. Then the

process of the assembler-as-the-leader buy-back

contract can be summarized as follows: the buy-back

price is determined by the assembler, and then the

optimal order quantity is determined by all of them,

and finally the output of the system is determined by

the suppliers.

For supplier i, the under-storage cost is

๐ค

๎ฏ

+๐

๎ฏ

,

and the over-storage cost is

๐

๎ฏ

+๐

๎ฏ

โ๐ค

๎ฏ

โ๐ฃ

๎ฏ

.

Then the supplierโs optimal order quantity satisfies

๐น

๏บ

๐

๎ฏ

๏ป

=

๎ฏช

๎ณ

๎ฌฟ๎ฏ

๎ณ

๎ฏ

๎ณ

๎ฌฟ๎ฏฉ

๎ณ

.

(12)

It follows that

๐ค

๎ฌต

โ๐

๎ฌต

๐

๎ฌต

โ๐ฃ

๎ฌต

=

๐ค

๎ฌถ

โ๐

๎ฌถ

๐

๎ฌถ

โ๐ฃ

๎ฌถ

=โฏ=

๐ค

๎ฏก

โ๐

๎ฏก

๐

๎ฏก

โ๐ฃ

๎ฏก

=

๐โ๐ถ

๐ตโ

๐

(13)

As ๐น

๏บ

โ

๏ป

is an increasing function, the larger the

buy-back price ๐

๎ฏ

is, the smaller the optimal supply

quantity ๐

๎ฏ

and the supplierโs expected profit are.

For the assembler, the under-storage cost is ๐โ

๐, and the over-storage cost is ๐โ๐ต. Then the

optimal order quantity of the assembler satisfies

๐น

๏บ

๐

๎ฌด

๏ป

=

๐โ๐

๐โ๐ต

.

(14)

As ๐น

๏บ

โ

๏ป

is an increasing function, the larger the

buy-back price B is, the larger the optimal order

quantity ๐

๎ฌด

and the expected profit of the assembler

are.

According to the above discussion, the following

conditions can be obtained:

๐โ๐

๐โ๐ต

โฅ

๐โ๐ถ

๐ตโ๐

,๐

๎ฏ

+๐

๎ฏ

>๐ค

๎ฏ

+๐ฃ

๎ฏ

,๐๐๐.

(15)

The final quantity of the product is ๐=

๐๐๐

๏ผ

๐

๎ฏ

๏ฝ

=๐

๎ฌต

, then ๐น

๏บ

๐

๏ป

=

๎ฏ๎ฌฟ๎ฎผ

๎ฎป๎ฌฟ๎ฏ

.

The profit function of assemblers is

๐

๎ฌด

๏บ

๐

๏ป

=๐ธ

[

โ๐๐+ ๐๐๐๐

๏บ

๐,๐ท

๏ป

+๐ต

๏บ

๐โ๐ท

๏ป

๎ฌพ

]

.

(16)

The expression of the profit function is

๐

๎ฌด

๏บ

๐

๏ป

=

๏บ

๐โW

๏ป

๐

โ

๏บ

๐

โB

๏ป

๎ถฑ๐น

๏บ

๐ฅ

๏ป

๐๐ฅ

๐

0

.

(17)

where ๐น

๏บ

๐

๏ป

=๐๐๐๏

๎ฏฃ๎ฌฟ๎ฏ

๎ฏฃ๎ฌฟ๎ฎป

,

๎ฏ๎ฌฟ๎ฎผ

๎ฎป๎ฌฟ๎ฏ

๏
.

It follows that

๐

๎ฌด

๏บ

๐

๏ป

is continuously

differentiable at the point B=

๏บ

๎ฏฃ๎ฌฟ๎ฏ

๏ป๏บ

๎ฏ๎ฌฟ๎ฎผ

๏ป

๎ฏฃ๎ฌฟ๎ฏ

+๐, which

is equivalent to

๎ฏฃ๎ฌฟ๎ฏ

๎ฏฃ๎ฌฟ๎ฎป

=

๎ฏ๎ฌฟ๎ฎผ

๎ฎป๎ฌฟ๎ฏ

. However, the previous

analysis has shown that the optimal value of B should

satisfy

๎ฏฃ๎ฌฟ๎ฏ

๎ฏฃ๎ฌฟ๎ฎป

โฅ

๎ฏ๎ฌฟ๎ฎผ

๎ฎป๎ฌฟ๎ฏ

. Thus, the problem can be

simplified by narrowing down the feasible region.

The above problem can be summarized as a

constrained optimization problem:

๐๐๐ฅ๐

๎ฌด

๏บ

๐

๏ป

=

๏บ

๐โ๐

๏ป

๐โ

๏บ

๐โ๐ต

๏ป

๎ถฑ๐น

๏บ

๐ฅ

๏ป

๐๐ฅ

๎ฏ

๎ฌด

๐ .๐ก.

โฉ

โจ

โง

๐น

๏บ

๐

๏ป

=

๐โ๐ถ

๐ตโ๐

๐โ๐

๐

โ๐ต

โฅ

๐โ๐ถ

๐ตโ๐

,๐ต+๐ถโฅ๐+

๐

(18)

Here, the inequality constraint ๐ต+๐ถโฅ๐+๐

can be omitted because it can be inferred from the

first equality constraint as follows:

๐โ๐ถ

๐ตโ๐

=๐น

๏บ

๐

๏ป

โค1.

(19)

The derivative of ๐

๎ฌด

๏บ

๐

๏ป

with respect to ๐ is

๐๐

๎ฌด

๏บ

๐

๏ป

๐๐

=๐โ๐ถโ

๏บ

๐โ๐

๏ป

๐น

๏บ

๐

๏ป

โ

๏บ

๐โ๐ถ

๏ป

๐

๏บ

๐

๏ป

.

(20)

Where

๐

๏บ

๐

๏ป

=

๐

๏บ

๐

๏ป

[

๐น

๏บ

๐

๏ป

]

๎ฌถ

๎ถฑ๐น

๏บ

๐ฅ

๏ป

๐๐ฅ

๐

0

.

(21)

If

๎ฏ๎ฐ

๎ฐฌ

๏บ

๎ฏ

๏ป

๎ฏ๎ฏ

is a decreasing function, then the profit

function ๐

๎ฌด

๏บ

๐

๏ป

of the assembler is concave. The

following properties can also be obtained from the

above derivative function.

Theorem 1. If the buy-back price is determined

by the assembler, then

๏ง The output ๐ of the system and the overall

profit ๐

๏บ

๐

๏ป

of the system are not affected by

the number of suppliers. If

,,CWV

are fixed,

the specific parameter of each supplier does not

affect ๐ and ๐

๏บ

๐

๏ป

;

๏ง ๐

๏บ

๐

๏ป

and ๐ are increasing in ๐ค

๎ฏ

and ๐ฃ

๎ฏ

,

decreasing in ๐

๎ฏ

;

A Game-theoretical Model of Buy-back Contracts in Assembly Systems with Uncertain Demand

141

๏ง When the marginal sales profit ๐โ๐ถ of the

product is fixed, the system output ๐ and

profit ๐

๏บ

๐

๏ป

decrease with ๐โ๐ถ, the profit

obtained by the supplier.

The practical implications of these properties will

be discussed in the next section. In addition, it can be

inferred that the inequality condition

๎ฏฃ๎ฌฟ๎ฏ

๎ฏฃ๎ฌฟ๎ฎป

โฅ

๎ฏ๎ฌฟ๎ฎผ

๎ฎป๎ฌฟ๎ฏ

can be ignored in the process of finding the

optimal solution if

๎ฏ๎ฐ

๎ฐฌ

๏บ

๎ฏ

๏ป

๎ฏ๎ฏ

is a decreasing function,

because the stationary point of the objective function

must satisfy this inequality. This can help to simplify

the process of solving the optimization problem.

3.2 A Contract Model in Which the

Suppliers Determine the Buy-back

Prices

Now, analyze the principle of buy-back price from

the supplierโs perspective. Consider this following

setting: The suppliers decide the buy-back prices of

the components to maximize their own profits.

After the n suppliers determine their buy-back

prices respectively, the optimal supply quantity of

each supplier is determined. Then the assembler

determines the order quantity, which is no higher than

each of the supplierโs optimal supply quantity. The

optimal policy for the suppliers to set the buy-back

prices should satisfy the following properties:

๏ง ๐

๎ฏ

โค๐ค

๎ฏ

;๐๐๐;

๏ง ๐

๎ฌด

โค๐

๎ฏ

;๐๐๐;

๏ง

๎ฏฃ๎ฌฟ๎ฏ

๎ฏฃ๎ฌฟ๎ฎป

โค

๎ฏ๎ฌฟ๎ฎผ

๎ฎป๎ฌฟ๎ฏ

,๐น

๏บ

๐

๏ป

=

๎ฏฃ๎ฌฟ๎ฏ

๎ฏฃ๎ฌฟ๎ฎป

.

Each supplier will set a low enough buy-back

price so that the assemblerโs order quantity is no

higher than the supplierโs optimal supply quantity.

That is to say, ๐

๎ฌด

โค๐

๎ฏ

,๐โ๐. If this condition does

not hold, the supplier will decrease the buy-back price

so as to reduce its own risk without affecting its

supply quantity. This can be summarized as the

following conditions:

๐โ๐

๐โ๐ต

โค

๐ค

๐

โ๐

๐

๐

๐

โ๐ฃ

๐

,๐โ๐.

(22)

According to previous analysis, ๐=๐๐๐

๏ผ

๐

๎ฏ

๏ฝ

=

๐

๎ฌด

. The supplierโs profit is

๐

๎ฏ

๏บ

๐

๏ป

=๐ธ

[

๏บ

๐ค

๎ฏ

โ๐

๎ฏ

๏ป

๐โ

๏บ

๐

๎ฏ

โ๐ฃ

๎ฏ

๏ป๏บ

๐โ๐ท

๏ป

๎ฌพ

]

.

(23)

Then the problem can be expressed as the

following constrained optimization problem:

๐๐๐ฅ๐

๎ฌต

๏บ

๐

๏ป

=

๏บ

๐ค

๎ฏ

โ๐

๎ฏ

๏ป

๐โ

๏บ

๐

๎ฏ

โ๐ฃ

๎ฏ

๏ป

๎ฌ

๐น

๏บ

๐ฅ

๏ป

๐๐ฅ

๎ฏ

๎ฌด

(24)

๐ .๐ก.

โฉ

โจ

โง

๐น

๏บ

๐

๏ป

=

๐โ๐

๐โ๐ต

๐โ๐

๐โ๐ต

โค

๐ค

๐

โ๐

๐

๐

๐

โ๐ฃ

๐

,๐

๐

+๐

๐

โฅ๐ค

๐

+๐ฃ

๐

,๐โ๐

The value of ๐ in the above formula depends on

all

๐

๎ฏ

values, i.e., ๐น

๏บ

๐

๏ป

=

๎ฏ๎ฌฟ๎ฎผ

๎ฎป๎ฌฟ๎ฏ

=

๎ฏ๎ฌฟ๎ฎผ

โ

๐

๐

๎ณ

๎ณ๎ฐธ๎ฐญ

๎ฌฟ๎ฏ

, which

makes the problem difficult to solve. The derivative

of the profit function

๐

๎ฏ

๏บ

๐

๏ป

is

๐๐

๎ฏ

๏บ

๐

๏ป

๐๐

=๐ค

๎ฏ

โ๐

๎ฏ

โ

๏บ

๐

๎ฏ

โ๐ฃ

๎ฏ

๏ป

๐น

๏บ

๐

๏ป

โ

๏บ

๐โ๐

๏ป

๐

๏บ

๐

๏ป

.

(25)

From the above equation, it can be concluded that

the output of the system satisfies

๐โ๐ถโ

๏บ

๐โ๐

๏ป

๐น

๏บ

๐

๏ป

โ๐

๏บ

๐โ๐

๏ป

๐

๏บ

๐

๏ป

=0.

(26)

Theorem 2. If the buy-back price is determined

by the suppliers, then

๏ง When the parameters C, W, V are fixed,

()

Qp

and ๐ have nothing to do with the specific

parameters of each supplier;

๏ง ๐

๏บ

๐

๏ป

and ๐ are increasing in ๐ค

๎ฏ

,๐ฃ

๎ฏ

, but

decreasing in

i

c

and the number of suppliers

n;

๏ง When the unit sales profit ๐โ๐ถ is fixed,

๐

๏บ

๐

๏ป

and ๐ decrease with ๐โ๐.

As in the assembler-led case, it can be inferred

that if

๎ฏ๎ฐ

๎ณ

๏บ

๎ฏ

๏ป

๎ฏ๎ฏ

is a decreasing function, the optimal

solution must satisfy the inequality

๎ฏฃ๎ฌฟ๎ฏ

๎ฏฃ๎ฌฟ๎ฎป

โค

๐ค

๐

โ๐

๐

๐

๐

โ๐ฃ

๐

.

Otherwise, it is not optimal. This property can help to

simplify the solution process.

The results in Theorem 2 and Theorem 1 are very

different. In the case that the assembler decides the

buy-back prices, the final quantity of the product has

nothing to do with the number of suppliers, and the

profit proportion of the assembler plays a positive

role in the system performance. In the case that

suppliers decide the buy-back prices, both the number

of suppliers and the profit proportion of the assembler

have negative effects on the system performance.

4 PERFORMANCE ANALYSIS

Sections 2 and 3 discuss two determination

mechanisms of buy-back prices in a decentralized

assembly system and get some results. This section

further compares the system performance (the overall

profit of the system) in different mechanisms.

BDEDM 2022 - The International Conference on Big Data Economy and Digital Management

142

4.1 Performance Analysis of

Decentralized and Centralized

Systems

Assume that the parameters of an assembly system

are given, the profit of the system can be calculated

in both decentralized and centralized cases. An

intuitive conjecture is that the profits of centralized

systems are higher than those of decentralized

systems. Next, some analysis is provided to support

this conjecture.

From the classical newsvendor model, the optimal

output of the centralized system satisfies ๐น

๏บ

๐

๏ป

=

๎ฏฃ๎ฌฟ๎ฎผ

๎ฏฃ๎ฌฟ๎ฏ

. It should also be pointed out that the profit

function of the classical newsvendor model is

concave. Thus, the profit function is increasing on the

left side of the optimal solution, and decreasing on the

right side of the optimal solution.

In the case that the assembler decides the buy-

back price, the constraint condition (18) implies

several intuitive facts. At the critical point

๎ฏฃ๎ฌฟ๎ฏ

๎ฏฃ๎ฌฟ๎ฎป

=

๎ฏ๎ฌฟ๎ฎผ

๎ฎป๎ฌฟ๎ฏ

, the system output satisfies ๐น

๏บ

๐

๏ป

=

๎ฏฃ๎ฌฟ๎ฎผ

๎ฏฃ๎ฌฟ๎ฏ

, which

is the same as that of the centralized system.

However, by substituting

๎ฏฃ๎ฌฟ๎ฏ

๎ฏฃ๎ฌฟ๎ฎป

=

๎ฏ๎ฌฟ๎ฎผ

๎ฎป๎ฌฟ๎ฏ

into the

derivative function, it can be obtained that

๐๐

๎ฌด

๏บ

๐

๏ป

๐๐

=โ

๏บ

๐โ๐ถ

๏ป

๐

๏บ

๐

๏ป

<0.

(27)

The optimality condition of centralized system

does not hold. It is easy to know that the system

output is lower than the case of centralized system.

In the case where the supplier decides the buy-

back prices, the constraint condition (24) implies that

๐น

๏บ

๐

๏ป

=

๐โ๐

๐โ๐ต

โค

๐โ๐ถ

๐ตโ๐

.

(28)

At the critical point

๎ฏฃ๎ฌฟ๎ฏ

๎ฏฃ๎ฌฟ๎ฎป

=

๎ฏ๎ฌฟ๎ฎผ

๎ฎป๎ฌฟ๎ฏ

, ๐น

๏บ

๐

๏ป

=

๎ฏฃ๎ฌฟ๎ฎผ

๎ฏฃ๎ฌฟ๎ฏ

holds but the optimality condition (26) is violated.

According to the concavity of the profit function,

it can be known that the system performance of the

two decentralized system is lower than that of the

centralized system.

4.2 Performance Comparison of the

Two Decentralized System

When the supplier decides the buy-back price of each

component, the optimal output ๐ of the system

solves (26). When the assembler decides the buy-

back price of each component, the optimal output ๐

of the system satisfies

๐โ๐ถโ

๏บ

๐โ๐

๏ป

๐น

๏บ

๐

๏ป

โ

๏บ

๐โ๐ถ

๏ป

๐

๏บ

๐

๏ป

=0.

(29)

As described in the previous theorem, in both

mechanisms, the parameters of the system will affect

the final performance of the system. It can be seen

that (26) and (29) are very similar, except that only

one coefficient is different, i.e., ๐

๏บ

๐โ๐

๏ป

and

๏บ

๐โ๐ถ

๏ป

.

According to Theorems 1 and 2, the output ๐

and the performance ๐

๏บ

๐

๏ป

are the same in the two

decentralized systems only if ๐

๏บ

๐โ๐

๏ป

=๐โ๐ถ

holds. This equality is equivalent to

๏บ

๐+1

๏ป๏บ

๐โ

๐

๏ป

=๐โ๐ถ.

Let ๐ฟ=

๎ฏฃ๎ฌฟ๎ฏ

๎ฏฃ๎ฌฟ๎ฎผ

. If and only if ๐ฟ

๏บ

๐+1

๏ป

=1, the

system performances in the two cases are the same. If

๐ฟ

๏บ

๐+1

๏ป

>1 holds, the system performance is

better in the case that the assembler decides the buy-

back prices. On the contrary, if ๐ฟ

๏บ

๐+1

๏ป

<1 holds,

the system performance is better in the case that the

suppliers decide the buy-back prices. ๐ฟ is a

threshold value of the assembly system, to determine

which mechanism is better for the decentralized

system.

In the case that the assembler decides the buy-

back prices, the output ๐ and expected profit ๐

๏บ

๐

๏ป

of the system are increasing in ๐ฟ, and not affected by

the number of suppliers. In the case that suppliers

decide the buy-back prices, the output ๐ and

expected profit ๐

๏บ

๐

๏ป

of the system are decreasing in

l

, and the number of suppliers.

The parameter ๐ฟ can be interpreted as the

proportion of the unit sales profit owned by the

assembler. The above results can be intuitively

understood as follows: When the assembler has a

strong market position (strong ability to obtain

profits), the whole system will benefit from the

assemblerโs dominant role in the negotiation of buy-

back prices. On the contrary, when the suppliers are

strong, the whole system will benefit from the

suppliersโ dominant role in the negotiation of buy-

back prices.

4.3 Numerical Example

Here is a simple numerical example to illustrate the

results in the previous subsections. Let ๐

๏บ

๐ฅ

๏ป

=2๐ฅ

and ๐น

๏บ

๐ฅ

๏ป

=๐ฅ

๎ฌถ

, ๐ฅโ

[

0,1

]

. The other parameters are

as follow: ๐โ ๐ถ=100, ๐โ๐=50,

๐โ๐=

150. Then ๐โ๐ถ=50,๐ฟ=

๎ฌต

๎ฌถ

.

According to ๐น

๏บ

๐

๏ป

=

๎ฏฃ๎ฌฟ๎ฎผ

๎ฏฃ๎ฌฟ๎ฏ

, the optimal output of

the centralized system is ๐=0.816. In the

decentralized system where the buy-back prices are

A Game-theoretical Model of Buy-back Contracts in Assembly Systems with Uncertain Demand

143

determined by the assembler, the output of the system

satisfies

๐๐

๎ฌด

๏บ

๐

๏ป

๐๐

=

250

3

โ 300๐=0.

(30)

The solution is ๐=0.278. It is obvious that the

system output and system profit are lower than the

centralized system. In the decentralized system where

the supplier sets the buy-back prices, the output of the

system satisfies

100 โ

50

3

๐โ 300๐=0.

(31)

Take ๐=2. The solution is ๐=0.222, which

is lower than 0.278. In fact, ๐ฟ

๏บ

๐+1

๏ป

>1 holds in

the above example. Thus, the system profit is higher

when the assembler decides the buy-back prices. In

addition, (31) can be written as follows:

๐=

1

3

โ

1

18

๐.

(32)

Obviously, ๐ is a decreasing function of n, and

so is the profit of the system. This is consistent with

the theoretical result in the previous section.

5 CONCLUSIONS

The purpose of this paper is to explore the principle

of designing the buy-back contract for the assembly

system. Between the supply chain members,

cooperation and confrontation coexist. The key

feature of the assembly system is that the components

are complementary. In this context, two different

buy-back pricing mechanisms are studied, and the

influence of various parameters on the system

performance is analyzed. By comparison, a critical

condition about the proportion of profit is provided to

identify which mechanism is more beneficial to the

whole system. It is shown that the two mechanisms

will lead to the same system performance only when

the proportion of profit owned to the assembler is

equal to the number of members in the system.

The model in this paper is not without limitation.

In order to facilitate the analysis, only two extreme

cases are considered: The buy-back prices of all

components are determined by either the assembler

or the suppliers. In reality, the buy-back prices may

be set partly by the assembler and partly by the

suppliers. This is a very complicated case which may

be worth further exploration.

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