#### Abstract

This paper extends an existing cooperative multi-objective interaction programming problem with interaction constraint for two players (or two agents). First, we define an **s**-optimal joint solution with weight vector to multi-objective interaction programming problem with interaction constraint for two players and get some properties of it. It is proved that the **s**-optimal joint solution with weight vector to the multi-objective interaction programming problem can be obtained by solving a corresponding mathematical programming problem. Then, we define another **s**-optimal joint solution with weight value to multi-objective interaction programming problem with interaction constraint for two players and get some of its properties. It is proved that the **s**-optimal joint solution with weight vector to multi-objective interaction programming problem can be obtained by solving a corresponding mathematical programming problem. Finally, we build a pricing multi-objective interaction programming model for a bi-level supply chain. Numerical results show that the interaction programming pricing model is better than Stackelberg pricing model and the joint pricing model.

#### 1. Introduction

There exists a kind of interactional and complex decision-making problem characterized with conflicts, incompatibility and complexity among multiagent systems, which has received much attention from researchers. Ever since the 20th century, researchers have studied two-player and multiplayer interaction problems and developed a new field called Game Theory [1], which has been widely applied in economics, engineering, military affairs, computers, and so forth [2, 3]. The game model contains the following main factors: players, strategies, and payoffs. In the game, people focus on finding the optimal strategies that benefit all the players, which are the equilibrium solutions or cooperative solutions to the interactional problems. In recent years, some researchers studied the cooperative games and negotiation games [4], which laid emphasis on the cooperative rules, for example, multiagent model [5]. However, there are still some interactional problems that cannot be solved by the existing game models. In 1999, Meng and Li introduced a definition of interaction decision-making problems [6], which mainly considers the multiagent decision-making problems that involve two persons and more and concerns how decision is made if the decision-making process of every agent is influenced by the other agents. Therefore, the interaction problems turn out to be the interaction decision-making problems. In some cases, the interaction decision-making problems may contain conflicts, so the interaction decision-making problems can be seen as an extension of the game models. The interaction decision-making model is complex and mainly contains the following five factors: decision makers (persons or agents), sets of constraints, decision variables, objective function, and interactional constraints.

Some interaction decision-making problems can be described by nonlinear programming models with parameters, called interaction programming problem (hereinafter called IPP) and studied in [6]. Generally speaking, the game problems can always be described as interaction programming models. However, problems with conflicts and under complex constraints sometimes cannot be described by normal game models, for example, multiagent problems, and cannot be solved. After 1999, researchers made in-depth researches as to the existence and equivalence of the solution to and the method of solving the IPP [7–10]. Ma and Ding studied the relation between interaction programming and multiobjective programming by adopting the converse problem of parametric programming [8]. Meng et al. discussed two new types of IPP that are used to solve the problems with or without conflicts and introduced the definition of its joint optimal solution and the method of solving this model [9, 10]. Jiang et al. discussed the multiobjective interaction programming for two persons [11].

In this paper, first, we introduce a definition of an **s**-optimal joint solution with weight vector to a multiobjective interaction programming problem with two players (or two agents). In fact, Meng et al. have proved the **s**-optimal joint solution is a better solution to interaction programming problems than Nash equilibrium and can be obtained by solving an equivalent mathematical programming problem [7]. Moreover, the **s**-optimal joint solution is obtained under the assumption that all the decision makers make the same concession. For some interaction decision-making problems, there are always multiobjective decisions for decision-makers to make. Therefore, we are to extend the **s**-optimal joint solution of interaction programming problem discussed in [7, 11] to multiobjectives interaction programming problem with two players and study its properties. Then, we introduce a definition of **s**-optimal joint solution with weight value to a multiobjective interaction programming problem with two players (or two agents), which differs from the definition of an **s**-optimal joint solution. Finally, we build an interaction programming pricing model for bilevel supply chain. Numerical results show that the pricing interaction programming model is better than a Stackelberg pricing model or a joint pricing model.

#### 2. **s**-Optimal Joint Solution

**s**

Let , be the multiobjective functions, and let be nonempty sets where , , and are positive integers. There exist the following multiobjectives interaction programming problem with interaction constraint for Player 1 (or Agent 1) and Player 2 (or Agent 2): satisfies the following constraint . Then, such multiobjectives interaction programming problem with interaction constraint is defined as two-player (or two-agent) multiobjective interaction programming problem , and is called the interaction constraint. Let be a feasible set to problem .

*Definition 2.1. *For ,(i)if it satisfies
then is called an optimal joint solution for Player 1 (or Agent 1) and Player 2 (or Agent 2) or to problem ;(ii)if there does not exist which satisfies
then is called a Nash-equilibrium solution for Player 1 (or Agent 1) and Player 2 (or Agent 2) or to problem .

Obviously, the optimal joint solution is a Nash-equilibrium solution.

Let a given weight .

*Definition 2.2. *For and , if it satisfies
then is called an joint solution with weight vector for Player 1 (or Agent 1) and Player 2 (or Agent 2) or to problem . is called a joint value of problem , and the set of all joint values is denoted by .

When , an **s**-joint solution with weight vector is an **s**-joint solution in [11].

Theorem 2.3. *Consider the following single objective programming problems and ()**
Let be the optimal solution to , and let be the optimal solution to . For any given , let () and . Then is an joint solution with weight vector to problem .*

*Proof. *For any and , it concludes from the assumption that
which implies
It follows with the assumption that
Then, by Definition 2.2, the proof completes.

By Theorem 2.3, for any , there exists a joint value such that is an joint solution with weight vector to problem . The joint solution illustrates the same concession the agents make. Furthermore, we define a joint value , expecting to get a minimum of all the joint values as an optimal solution with weight vector to problem .

*Definition 2.4. *Let be an joint solution with weight vector to problem with the corresponding joint value , that is, the minimum of all the joint values. Then, is called an optimal joint solution with weight vector for Player 1 (or Agent 1) and Player 2 (or Agent 2) or to problem .

Obviously, if , then the optimal joint solution with weight vector is the optimal joint solution as per Definition 2.1. In fact, an optimal joint solution with weight vector is an joint solution too, but an joint solution with weight vector is not always an optimal joint solution with weight vector .

Theorem 2.5. *For , let be an optimal solution to , and let be the optimal solution to . Then, is an optimal solution to the following problem:
**
if and only if is an optimal joint solution with weight vector to problem .*

*Proof. *For any , by the assumption, is an optimal solution to problem such that
Then, it concludes from Definition 2.2 that is an joint solution to problem . Let be an optimal joint solution with weight vector to problem . By Definitions 2.2 and 2.4, we have and, for all ,
Then, is feasible solution to problem , and with , we conclude . That is to say, is an optimal joint solution with weight vector to problem .

The converse statement is also true. If is an optimal joint solution with weight vector to problem , then it implies is feasible to problem by Definition 2.2. Suppose is an optimal solution to , from the previous proof, it concludes that is an optimal joint solution with weight vector to problem and . Therefore, is an optimal solution to problem , and the proof completes.

Corollary 2.6. *Let be compact, and let and be continuous functions. Then there exists an optimal joint solution to problem .*

Theorem 2.7. *For , let be the optimal solution to , and let be the optimal solution to . If () is an optimal solution to the following problem:**
then is an optimal joint solution with weight vector to problem , where
*

*Proof. *Supposing is an optimal joint solution with weight vector to problem , from Theorem 2.5, we have that is an optimal solution to problem . Then, () is feasible to problem , which implies
that is:
Letting
then it follows . On the other hand, from the assumption we have that
which implies
that is:
Then, is a feasible solution to problem . Thus, holds and becomes an optimal joint solution with weight vector to problem with . This completes the proof.

*Remark 2.8. *By Theorem 2.7, we can get an optimal joint solution with weight vector to problem if the optimal solution to problem is obtained. However, we cannot get the optimal joint value unless the optimal solutions to problems and () are obtained.

#### 3. s-Optimal Joint Solution with Weight Value

In this section, we discuss another optimal joint solution to , where and . When , the definition of **s**-optimal joint solution to problem is not appropriate. Consider the following multiobjectives interaction programming problem with interaction constraint for Player 1 (or Agent 1) and Player 2 (or Agent 2):
Hence, we need to define a new **s**-optimal joint solution to problem . Suppose the weight of is and the weight of is , , . Let

*Definition 3.1. *For , , if it satisfies
for (, ), then is called an joint solution with weight vector for Player 1 (or Agent 1) and Player 2 (or Agent 2) or to problem . is called a joint value with weight vector . Here, **s** is real value.

Theorem 3.2. *Suppose is the optimal solution to and is the optimal solution to . For any , letting
**
then is the -joint solution with weight vector to .*

*Proof. *For all , , , it follows from the assumption that
which implies
so
This completes the proof with Definition 3.1.

By Theorem 2.3, , and for all , there exists a value such that becomes an -joint solution with weight vector to . -joint solution implies the decision makers give the same concession with weight vector , which is fair for all the decision makers. Thus, it is useful for us to find the minimum of all the joint values.

*Definition 3.3. *Suppose is an joint solution with weight vector to with the minimum of all the joint values. Then, is called an optimal joint solution with weight vector to .

Obviously, if becomes the optimal joint solution for Player 1 (or Agent 1) and Player 2 (or Agent 2).

Theorem 3.4. *For , , let be the optimal solution to , and let be the optimal solution to . Then, is an optimal solution to the following problem :
**
if and only if is an optimal joint solution with weight vector to .*

*Proof. *Suppose is the optimal solution to the problem . For any , , , it follows with the constraints of that
which implies is an joint solution with weight vector to . Assume is the -optimal joint solution with weight vector to . Then, with Definitions 3.1 and 3.3, it gets and
which implies is feasible to and . Then, we get , and becomes the -optimal joint solution with weight vector to .

On the contrary, if is an -optimal joint solution with weight vector to , by Definition 3.1, it gets is feasible to . Supposing is the optimal solution to , it is easily obtained that is the optimal joint solution with weight vector to , which implies . Thus, is the optimal solution to , and this completes the proof.

*Remark 3.5. *If the optimal solutions to and exist, the -optimal joint solution with weight vector to exists, too. By Theorem 3.4, we can obtain a method of solving the problem IPP as follows. First, get the optimal solutions to and . Then, we can get an optimal solution to , and is an optimal joint solution with weight vector to .

Theorem 3.6. *Suppose is an optimal joint solution with weight vector to . Then, there does not exist such that
*

*Proof. *It is obviously correct if . For , suppose there exists such that
Let with . Then, there exists sufficiently small such that , , and for , . Letting , then, we get . It follows with Definition 3.1 that
for any . Thus, is optimal joint solution with weight vector to and , which contradicts the assumption of ; this completes the proof.

Theorem 3.7. *For , , let be an optimal solution to , and let be an optimal solution to . If (, ) is the optimal solution of the following problem :
**
then is an -optimal joint solution with weight vector to , where
*

*Proof. *Supposing is the optimal joint solution with weight vector to , it concludes from Theorem 3.4 that is the optimal solution to . Then, (, ) is feasible to . Thus, we have

Let .

It concludes from (3.13) that . Further, with the assumption, we get
which implies
Thus,
Therefore, is feasible to , which implies , and is the -joint solution with weight vector to . Thus, , and this completes the proof.

In Theorem 3.4, as soon as the optimal solutions to and are found, it is possible to find out an -optimal joint solution with weight vector to by solving the problem . In Theorem 3.7, we can obtain an -optimal joint solution with weight vector to by solving the problem . However, if we do not solve the optimal solutions to and , it is unluckily that we cannot get an .

*Example 3.8. *Considering the following IPP:
where . For , by Theorem 3.4, it is easily obtained that is the optimal joint solution with weight vector to this IPP.

#### 4. A Bilevel Supply Chain with Two Players

Now, we show an example of a bilevel supply chain with two players, which is solved with the method given in Section 2. The bilevel supply chain can be seen as a two-player system where a manufacturer is one agent while a retailer is the other, with the manufacturer providing goods for the retailer to sell. Then, the problem is to decide the prices of the goods at a level such that both the manufacturer and retailer can gain the most. Clearly, this is an IPP with two players. Suppose there is a manufacturer which manufactures products (). Let denote the production cost for (), the transportation cost for (), and the price of (). The manufacturer provides products to the retailer at , where is the minimum of and is the maximum of . Let denote the quantity ordered for () with , where is the minimum of and is the maximum of . Supposing the retailer sells the product () at the price of , then it is clear that . Let , which denotes the market demand for (). Then we get the following pricing model for the bi-level supply chain: where is the parameter for , is the parameter for , and () is the interaction constraint.

As is known to all, Stackelberg pricing model (which is a pricing model constructed as per Stackelberg model) and joint pricing model are widely used in the supply chain pricing decision. Stackelberg pricing model tends to benefit the manufacturer since the price is decided by the manufacturer, while the price is always decided by the retailer in joint pricing model.

Then we compare the results of our model with those of Stackelberg pricing model and joint pricing model. For the previous problem, let , and let all the be independent to each other; the transportation cost for all ; the production costs: , and ; the production capacity , and ; the price () restricted by , and ; the market demand for : , and .

We have the pricing model of : We give the Stackelberg pricing model: solves the lower level problem: Let and let . We give the joint pricing model:

Then, we solve this pricing problem of supply chain with the previously mentioned three pricing models, and the numerical results are given in Table 1.

From Table 1, it is found that the profit of the manufacturer is about 4 times less than the profit of the retailer in the solution of joint pricing model which cannot be accepted by the manufacturer. The profit of the manufacturer is about 2 times more than the profit of the retailer in the solution of Stackelberg model which may not be accepted by the retailer. However, in optimal joint solution to the IPP, the difference between the profit of the manufacturer and the profit of the retailer is much closer, around 40%. Therefore, the -optimal joint solution to provides a better equilibrium solution that can provide maximum profit for both the manufacturer and the retailer. Thus, the optimal joint solution is an acceptable solution for the manufacturer and the retailer.

#### 5. Conclusion

In this paper, -optimal joint solution and optimal joint solution with weight vector to the multiobjective interaction programming problem with two players are discussed, and they are obtained by solving some equivalent mathematical programming problems. Furthermore, the proposed model in this paper can be extended to that of multiagents. The numerical results illustrate that the solution of the multiobjective interaction programming model to the bilevel supply chain is better than those of Stackelberg model and joint pricing model. Moreover, the multiobjective interaction programming may be applied in other fields, such as allocation of multi-jobs in computer networks and allocation of resources in market.

#### Acknowledgments

This paper is supported in part by the National Natural Science Foundation of China under Grant no. 71001089 and the Science Foundation of Binzhou University (BZXYL1009). The authors would like to express their gratitude to anonymous referees’ detailed comments and remarks that help us improve our presentation of this paper considerably.