Multiobjective Interaction Programming Problem with Interaction Constraint for Two Players

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

Let 𝑓∶𝑅^𝑚×𝑅^𝑛→𝑅^𝑘, 𝑔∶𝑅^𝑚×𝑅^𝑛→𝑅^𝑘 be the multiobjective functions, and let 𝑋,𝑌,𝐻⊂𝑅^𝑚×𝑅^𝑛 be nonempty sets where 𝑛, 𝑚, and 𝑘 are positive integers. There exist the following multiobjectives interaction programming problem with interaction constraint (𝑥,𝑦)∈𝐻 for Player 1 (or Agent 1) and Player 2 (or Agent 2): min_𝑥𝑓𝑥,𝑦s.t.𝑥,𝑦∈𝑋min_𝑦g𝑥,𝑦s.t.𝑥,𝑦∈𝑌s.t.𝑥,𝑦∈𝐻. (FG)(𝑥,𝑦) satisfies the following constraint (𝑥,𝑦)∈𝐻. Then, such multiobjectives interaction programming problem with interaction constraint is defined as two-player (or two-agent) multiobjective interaction programming problem (FG), and 𝐻 is called the interaction constraint. Let 𝑍={(𝑥,𝑦)∣(𝑥,𝑦)∈𝑋,(𝑥,𝑦)∈𝑌,(𝑥,𝑦)∈𝐻} be a feasible set to problem (FG).

Definition 2.1. For (𝑥^∗,𝑦^∗)∈𝑍,(i)if it satisfies 𝑓𝑥^∗,𝑦^∗≤𝑓𝑥,𝑦,𝑔𝑥^∗,𝑦^∗≤𝑔𝑥,𝑦,∀𝑥,𝑦∈𝑍, (2.1) then (𝑥^∗,𝑦^∗) is called an optimal joint solution for Player 1 (or Agent 1) and Player 2 (or Agent 2) or to problem (FG);(ii)if there does not exist (𝑥,𝑦^∗)∈𝑍,(𝑥^∗,𝑦)∈𝑍 which satisfies 𝑓𝑥,𝑦^∗≤𝑓𝑥,𝑦,𝑔𝑥^∗,𝑦≤𝑔𝑥,𝑦, (2.2) then (𝑥^∗,𝑦^∗) is called a Nash-equilibrium solution for Player 1 (or Agent 1) and Player 2 (or Agent 2) or to problem (FG).
Obviously, the optimal joint solution is a Nash-equilibrium solution.
Let a given weight 𝜆=(𝜆₁,𝜆₂,…,𝜆_𝑘)>0.

Definition 2.2. For (𝑥^∗,𝑦^∗)∈𝑍 and 𝐬=(𝑠₁,𝑠₂,…,𝑠_𝑘)>0, if it satisfies 𝑓_𝑖𝑥^∗,𝑦^∗−𝜆_𝑖𝑠_𝑖≤𝑓_𝑖𝑥,𝑦,𝑔_𝑖𝑥^∗,𝑦^∗−𝜆_𝑖𝑠_𝑖≤𝑔_𝑖𝑥,𝑦,𝑖=1,2,…,𝑘,∀𝑥,𝑦∈𝑍, (2.3) then (𝑥^∗,𝑦^∗) is called an 𝐬-joint solution with weight vector 𝜆 for Player 1 (or Agent 1) and Player 2 (or Agent 2) or to problem (FG). 𝐬 is called a joint value of problem (FG), and the set of all joint values is denoted by 𝑆.
When 𝜆=(1,1,…,1), an s-joint solution with weight vector 𝜆 is an s-joint solution in [11].

Theorem 2.3. Consider the following single objective programming problems 𝐹_𝑖 and 𝐺_𝑖 (𝑖=1,2,…𝑘)∶min𝑓_𝑖𝑥,𝑦s.t.𝑥,𝑦∈𝑍,min𝑔_𝑖𝑥,𝑦s.t.𝑥,𝑦∈𝑍. 𝐹_𝑖𝐺_𝑖 Let (𝑥^∗_𝑓𝑖,𝑦^∗_𝑓𝑖) be the optimal solution to 𝐹_𝑖, and let (𝑥^∗_𝑔𝑖,𝑦^∗_𝑔𝑖) be the optimal solution to 𝐺_𝑖. For any given (𝑥^′,𝑦^′)∈𝑍, let 𝑠^′_𝑖=(1/𝜆_𝑖)max{𝑓_𝑖(𝑥^′,𝑦^′)−𝑓_𝑖(𝑥^∗_𝑓𝑖,𝑦^∗_𝑓𝑖),𝑔_𝑖(𝑥^′,𝑦^′)−𝑔_𝑖(𝑥^∗_𝑔𝑖,𝑦^∗_𝑔𝑖)} (𝑖=1,2,…𝑘) and 𝐬^′=(𝑠^′₁,𝑠^′₂,⋯,𝑠^′_𝑘). Then (𝑥^′,𝑦^′) is an 𝐬^′-joint solution with weight vector 𝜆 to problem (FG).

Proof. For any (𝑥,𝑦)∈𝑍 and 𝑖=1,2,…𝑘, it concludes from the assumption that 𝑓_𝑖𝑥^∗_𝑓𝑖,𝑦^∗_𝑓𝑖≤𝑓_𝑖𝑥,𝑦,𝑔_𝑖𝑥^∗_𝑔𝑖,𝑦^∗_𝑔𝑖≤𝑔_𝑖𝑥,𝑦, (2.4) which implies 𝑓_𝑖𝑥^′,𝑦^′−𝑓_𝑖𝑥^′,𝑦^′−𝑓_𝑖𝑥^∗_𝑓𝑖,𝑦^∗_𝑓𝑖≤𝑓_𝑖,𝑔𝑥,𝑦_𝑖𝑥^′,𝑦^′−𝑔_𝑖𝑥,𝑦−𝑔_𝑖𝑥^∗_𝑔𝑖,𝑦^∗_𝑔𝑖≤𝑔_𝑖𝑥,𝑦. (2.5) It follows with the assumption that 𝑓_𝑖𝑥^′,𝑦^′−𝜆_𝑖𝑠^′_𝑖≤𝑓_𝑖,𝑔𝑥,𝑦_𝑖𝑥^′,𝑦^′−𝜆_𝑖𝑠^′_𝑖≤𝑔_𝑖𝑥,𝑦. (2.6) Then, by Definition 2.2, the proof completes.

By Theorem 2.3, for any (𝑥,𝑦)∈𝑍, there exists a joint value 𝐬 such that (𝑥,𝑦) is an 𝐬-joint solution with weight vector 𝜆 to problem (FG). The 𝐬-joint solution illustrates the same concession 𝐬 the agents make. Furthermore, we define a joint value |𝐬|=𝑠₁+𝑠₂+⋯𝑠_𝑘, expecting to get a minimum of all the joint values as an optimal solution with weight vector 𝜆 to problem (FG).

Definition 2.4. Let (𝑥^∗,𝑦^∗) be an 𝐬^∗-joint solution with weight vector 𝜆 to problem (FG) with the corresponding joint value |𝐬^∗|, that is, the minimum of all the joint values. Then, (𝑥^∗,𝑦^∗) is called an 𝐬^∗-optimal joint solution with weight vector 𝜆 for Player 1 (or Agent 1) and Player 2 (or Agent 2) or to problem (FG).
Obviously, if |𝐬^∗|=0, then the 𝐬^∗-optimal joint solution with weight vector 𝜆 is the optimal joint solution as per Definition 2.1. In fact, an 𝐬^∗-optimal joint solution with weight vector 𝜆 is an 𝐬^∗-joint solution too, but an 𝐬^∗-joint solution with weight vector 𝜆 is not always an 𝐬^∗-optimal joint solution with weight vector 𝜆.

Theorem 2.5. For 𝑖=1,2,…,𝑘, let (𝑥^∗_𝑓𝑖,𝑦^∗_𝑓𝑖) be an optimal solution to 𝐹_𝑖, and let (𝑥^∗_𝑔𝑖,𝑦^∗_𝑔𝑖) be the optimal solution to 𝐺_𝑖. Then, (𝑥^∗,𝑦^∗,𝐬^∗) is an optimal solution to the following problem: min|𝐬|=𝑠₁+𝑠₂+⋯𝑠_𝑘s.t.𝑓_𝑖𝑥,𝑦−𝜆_𝑖𝑠_𝑖≤𝑓_𝑖𝑥^∗_𝑓𝑖,𝑦^∗_𝑓𝑖𝑔,𝑖=1,2,…,𝑘,_𝑖𝑥,𝑦−𝜆_𝑖𝑠_𝑖≤𝑔_𝑖𝑥^∗_𝑔𝑖,𝑦^∗_𝑔𝑖,𝑖=1,2,…,𝑘,𝑥,𝑦∈𝑍,𝑠_𝑖≥0,𝑖=1,2,…,𝑘 (S) if and only if (𝑥^∗,𝑦^∗) is an 𝐬^∗-optimal joint solution with weight vector 𝜆 to problem (FG).

Proof. For any (𝑥,𝑦)∈𝑍, by the assumption, (𝑥^∗,𝑦^∗,𝐬^∗) is an optimal solution to problem (S) such that 𝑓_𝑖𝑥^∗,𝑦^∗−𝜆_𝑖𝑠_𝑖≤𝑓_𝑖𝑥^∗_𝑓𝑖,𝑦^∗_𝑓𝑖𝑔≤𝑓𝑥,𝑦,𝑖=1,2,…,𝑘,_𝑖𝑥^∗,𝑦^∗−𝜆_𝑖𝑠_𝑖≤𝑔_𝑖𝑥^∗_𝑔𝑖,𝑦^∗_𝑔𝑖<𝑔_𝑖𝑥,𝑦,𝑖=1,2,…,𝑘. (2.7) Then, it concludes from Definition 2.2 that (𝑥^∗,𝑦^∗) is an 𝐬^∗-joint solution to problem (FG). Let (𝑥^′,𝑦^′) be an 𝐬^′-optimal joint solution with weight vector 𝜆 to problem (FG). By Definitions 2.2 and 2.4, we have |𝐬^∗|≥|𝐬^′| and, for all (𝑥,𝑦)∈𝑍, 𝑓_𝑖𝑥^′,𝑦^′−𝜆_𝑖𝑠^′_𝑖≤𝑓_𝑖,𝑔𝑥,𝑦_𝑖𝑥^′,𝑦^′−𝜆_𝑖𝑠^′_𝑖≤𝑔_𝑖𝑥,𝑦. (2.8) Then, (𝑥^′,𝑦^′) is feasible solution to problem (S), and with |𝐬^∗|≤|𝐬^′|, we conclude |𝐬^∗|=|𝐬^′|. That is to say, (𝑥^∗,𝑦^∗) is an 𝐬^∗-optimal joint solution with weight vector 𝜆 to problem (FG).
The converse statement is also true. If (𝑥^∗,𝑦^∗) is an 𝐬^∗-optimal joint solution with weight vector 𝜆 to problem (FG), then it implies (𝑥^∗,𝑦^∗,𝐬^∗) is feasible to problem (S) by Definition 2.2. Suppose (𝑥^′,𝑦^′,𝐬^′) is an optimal solution to (S), from the previous proof, it concludes that (𝑥^′,𝑦^′) is an 𝐬^′-optimal joint solution with weight vector 𝜆 to problem (FG) and |𝐬^∗|=|𝐬^′|. Therefore, (𝑥^∗,𝑦^∗,𝐬^∗) is an optimal solution to problem (S), and the proof completes.

Corollary 2.6. Let 𝑍 be compact, and let 𝑓 and 𝑔 be continuous functions. Then there exists an 𝐬^∗-optimal joint solution to problem (FG).

Theorem 2.7. For 𝑖=1,2,…,𝑘, let (𝑥^∗_𝑓𝑖,𝑦^∗_𝑓𝑖) be the optimal solution to 𝐹_𝑖, and let (𝑥^∗_𝑔𝑖,𝑦^∗_𝑔𝑖) be the optimal solution to 𝐺_𝑖. If [(𝑥^′,𝑦^′,𝐬^′),(𝑥^′_𝑖1,𝑦^′_𝑖1)(𝑥^′′_𝑖2,𝑦^′′_𝑖2)]  (𝑖=1,2,…,𝑘) is an optimal solution to the following problem:min^𝑘_𝑖=1𝑠_𝑖+𝑓𝑥_𝑖^′,𝑦_𝑖^′+𝑔𝑥_𝑖^′′,𝑦_𝑖^′′s.t.𝑓_𝑖𝑥,𝑦−𝜆_𝑖𝑠_𝑖≤𝑓_𝑖𝑥^′_𝑖1,𝑦^′_𝑖1𝑔,𝑖=1,2,…,𝑘,_𝑖𝑥,𝑦−𝜆_𝑖𝑠_𝑖≤𝑔_𝑖𝑥_𝑖^′′,𝑦_𝑖^′′,𝑥,𝑖=1,2,…,𝑘,𝑥,𝑦^′_𝑖,𝑦^′_𝑖,𝑥_𝑖^′′,𝑦_𝑖^′′∈𝑍,𝑠_𝑖≥0,𝑖=1,2,…,𝑘, 𝑆^′ then (𝑥^′,𝑦^′) is an 𝐬^∗-optimal joint solution with weight vector 𝜆 to problem (FG), where 𝑠^∗_𝑖=𝑠^′_𝑖+1𝜆_𝑖max𝑓_𝑖𝑥^′_𝑖1,𝑦^′_𝑖1−𝑓_𝑖𝑥^∗_𝑓𝑖,𝑦^∗_𝑓𝑖,𝑔_𝑖𝑥^′′_𝑖2,𝑦^′′_𝑖2−𝑔_𝑖𝑥^∗_𝑔𝑖,𝑦^∗_𝑔𝑖,𝑖=1,2,…,𝑘. (2.9)

Proof. Supposing (𝑥^∗,𝑦^∗) is an 𝐬^∗-optimal joint solution with weight vector 𝜆 to problem (FG), from Theorem 2.5, we have that (𝑥^∗,𝑦^∗,𝐬^∗) is an optimal solution to problem (S). Then, [(𝑥^∗,𝑦^∗,𝐬^∗),(𝑥^∗_𝑓𝑖,𝑦^∗_𝑓𝑖)(𝑥^∗_𝑔𝑖,𝑦^∗_𝑔𝑖)]  (𝑖=1,2,…,𝑘) is feasible to problem 𝑆^′, which implies ^𝑘_𝑖=1𝑠^′_𝑖+𝑓𝑥^′_𝑖1,𝑦^′_𝑖1+𝑔𝑥^′′_𝑖2,𝑦^′′_𝑖2≤^𝑘_𝑖=1𝑠^∗_𝑖+𝑓𝑥^∗_𝑓𝑖,𝑦^∗_𝑓𝑖+𝑔𝑥^∗_𝑔𝑖,𝑦^∗_𝑔𝑖, (2.10) that is: ^𝑘_𝑖=1𝑠^′_𝑖+𝑓𝑥^′_𝑖1,𝑦^′_𝑖1−𝑓𝑥^∗_𝑓𝑖,𝑦^∗_𝑓𝑖+𝑔𝑥^′′_𝑖2,𝑦^′′_𝑖2−𝑔𝑥^∗_𝑔𝑖,𝑦^∗_𝑔𝑖≤^𝑘_𝑖=1𝑠^∗_𝑖. (2.11) Letting 𝑠^∘_𝑖=𝑠^′_𝑖+1𝜆_𝑖max𝑓𝑥^′_𝑖1,𝑦^′_𝑖1−𝑓𝑥^∗_𝑓𝑖,𝑦^∗_𝑓𝑖,𝑔𝑥^′′_𝑖2,𝑦^′′_𝑖2−𝑔𝑥^∗_𝑔𝑖,𝑦^∗_𝑔𝑖,𝑖=1,2,…,𝑘, (2.12) then it follows ∑^𝑘_𝑖=1𝑠^∘_𝑖≤∑^𝑘_𝑖=1𝑠^∗_𝑖. On the other hand, from the assumption we have that 𝑓_𝑖𝑥^′,𝑦^′−𝜆_𝑖𝑠^′_𝑖≤𝑓_𝑖𝑥^′_𝑖1,𝑦^′_𝑖1𝑔,𝑖=1,2,…,𝑘,_𝑖𝑥^′,𝑦^′−𝜆_𝑖𝑠^′_𝑖≤𝑔_𝑖𝑥^′′_𝑖2,𝑦^′′_𝑖2,𝑖=1,2,…,𝑘, (2.13) which implies 𝑓_𝑖𝑥^′,𝑦^′≤𝜆_𝑖𝑠^′_𝑖+𝑓_𝑖𝑥^′_𝑖1,𝑦^′_𝑖1−𝑓_𝑖𝑥^∗_𝑓𝑖,𝑦^∗_𝑓𝑖+𝑓_𝑖𝑥^∗_𝑓𝑖,𝑦^∗_𝑓𝑖𝑔,𝑖=1,2,…,𝑘,_𝑖𝑥^′,𝑦^′≤𝜆_𝑖𝑠^′_𝑖+𝑔_𝑖𝑥^′′_𝑖2,𝑦^′′_𝑖2−𝑔_𝑖𝑥^∗_𝑔𝑖,𝑦^∗_𝑔𝑖+𝑔_𝑖𝑥^∗_𝑔𝑖,𝑦^∗_𝑔𝑖,𝑖=1,2,…,𝑘, (2.14) that is: 𝑓_𝑖𝑥^′,𝑦^′≤𝑠^∘_𝑖+𝑓_𝑖𝑥^∗_𝑓𝑖,𝑦^∗_𝑓𝑖𝑔,𝑖=1,2,…,𝑘,_𝑖𝑥^′,𝑦^′≤𝑠^∘_𝑖+𝑔_𝑖𝑥^∗_𝑔𝑖,𝑦^∗_𝑔𝑖,𝑖=1,2,…,𝑘. (2.15) Then, (𝑥^′,𝑦^′,𝐬^∘) is a feasible solution to problem (S). Thus, ∑^𝑘_𝑖=1𝑠^∘_𝑖≥∑^𝑘_𝑖=1𝑠^∗_𝑖 holds and (𝑥^′,𝑦^′) becomes an 𝐬^∘-optimal joint solution with weight vector 𝜆 to problem (FG) with ∑^𝑘_𝑖=1𝑠^∘_𝑖=∑^𝑘_𝑖=1𝑠^∗_𝑖. This completes the proof.

Remark 2.8. By Theorem 2.7, we can get an 𝐬^∗-optimal joint solution with weight vector 𝜆 to problem (FG) if the optimal solution to problem 𝑆^′ is obtained. However, we cannot get the optimal joint value 𝐬^∗ unless the optimal solutions to problems 𝐹_𝑖 and 𝐺_𝑖  (𝑖=1,2,…,𝑘) are obtained.

3. s-Optimal Joint Solution with Weight Value

In this section, we discuss another optimal joint solution to (FG), where 𝑓∶𝑅^𝑚×𝑅^𝑛→𝑅^𝑘1 and 𝑔∶𝑅^𝑚×𝑅^𝑛→𝑅^𝑘2. When 𝑘₁≠𝑘₂, the definition of s-optimal joint solution to problem (FG) is not appropriate. Consider the following multiobjectives interaction programming problem with interaction constraint (𝑥,𝑦)∈𝐻 for Player 1 (or Agent 1) and Player 2 (or Agent 2): min_𝑥𝑓𝑥,𝑦s.t.𝑥,𝑦∈𝑋,min_𝑦𝑔𝑥,𝑦s.t.𝑥,𝑦∈𝑌s.t.𝑥,𝑦∈𝐻. (FG) Hence, we need to define a new s-optimal joint solution to problem (FG). Suppose the weight of 𝑓_𝑖 is 𝑝_𝑖>0 and the weight of 𝑔_𝑗 is 𝑞_𝑗>0, 𝑖=1,2,…,𝑘₁, 𝑗=1,2,…,𝑘₂. Let 𝑝=𝑝₁,𝑝₂,…,𝑝_𝑘1,𝑞=𝑞₁,𝑞₂,…,𝑞_𝑘1. (3.1)

Definition 3.1. For (𝑥^∗,𝑦^∗)∈𝑍, 𝐬≥0, if it satisfies 𝑓_𝑖𝑥^∗,𝑦^∗−𝑝_𝑖𝐬≤𝑓_𝑖𝑥,𝑦,𝑔_𝑗𝑥^∗,𝑦^∗−𝑞_𝑗𝐬≤𝑔_𝑗𝑥,𝑦 (3.2) for (𝑥,𝑦)∈𝑍  (𝑖=1,2,…,𝑘₁, 𝑗=1,2,…,𝑘₂), then (𝑥^∗,𝑦^∗) is called an 𝐬-joint solution with weight vector (𝑝,𝑞) for Player 1 (or Agent 1) and Player 2 (or Agent 2) or to problem (FG). 𝐬 is called a joint value with weight vector (𝑝,𝑞). Here, s is real value.

Theorem 3.2. Suppose (𝑥^∗_𝑓𝑖,𝑦^∗_𝑓𝑖) is the optimal solution to 𝐹_𝑖 and (𝑥^∗_𝑔𝑗,𝑦^∗_𝑔𝑗) is the optimal solution to 𝐺_𝑖. For any (𝑥,𝑦)∈𝑍, letting 1𝐬=max𝑝_𝑖𝑓_𝑖𝑥,𝑦−𝑓_𝑖𝑥^∗_𝑓𝑖,𝑦^∗_𝑓𝑖,1𝑞_𝑗𝑔_𝑗𝑥,𝑦−𝑔_𝑗𝑥^∗_𝑔𝑗,𝑦^∗_𝑔𝑗∣𝑖=1,2,…,𝑘₁,𝑗=1,2,…,𝑘₂, (3.3) then (𝑥,𝑦) is the 𝐬-joint solution with weight vector (𝑝,𝑞) to (FG).

Proof. For all  (𝑥,𝑦)∈𝑍, 𝑖=1,2,…,𝑘₁, 𝑗=1,2,…,𝑘₂, it follows from the assumption that 𝑓_𝑖𝑥^∗_𝑓𝑖,𝑦^∗_𝑓𝑖≤𝑓_𝑖𝑥,𝑦,𝑔_𝑗𝑥^∗_𝑔𝑗,𝑦^∗_𝑔𝑗≤𝑔_𝑗𝑥,𝑦, (3.4) which implies 𝑓_𝑖𝑥,𝑦−𝑓_𝑖𝑥,𝑦−𝑓_𝑖𝑥^∗_𝑓𝑖,𝑦^∗_𝑓𝑖≤𝑓_𝑖,𝑔𝑥,𝑦_𝑗𝑥,𝑦−𝑔_𝑗𝑥,𝑦−𝑔_𝑗𝑥^∗_𝑔𝑗,𝑦^∗_𝑔𝑗≤𝑔_𝑗𝑥,𝑦, (3.5) so 𝑓_𝑖𝑥,𝑦−𝑝_𝑖𝐬≤𝑓_𝑖𝑥,𝑦,𝑔_𝑗𝑥,𝑦−𝑞_𝑗𝐬≤𝑔_𝑗𝑥,𝑦. (3.6) This completes the proof with Definition 3.1.

By Theorem 2.3, forall(𝑥,𝑦)∈𝑋, and for all (𝑥,𝑦)∈𝑌, there exists a value 𝐬 such that (𝑥,𝑦) becomes an 𝐬-joint solution with weight vector (𝑝,𝑞) to (FG). 𝐬-joint solution implies the decision makers give the same concession 𝐬 with weight vector (𝑝,𝑞), 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 (𝑥^∗,𝑦^∗) is an 𝐬^∗-joint solution with weight vector (𝑝,𝑞) to (FG) with 𝐬^∗ the minimum of all the joint values. Then, (𝑥^∗,𝑦^∗) is called an 𝐬^∗-optimal joint solution with weight vector (𝑝,𝑞) to (FG).
Obviously, if 𝐬^∗=0,(𝑥^∗,𝑦^∗) becomes the optimal joint solution for Player 1 (or Agent 1) and Player 2 (or Agent 2).

Theorem 3.4. For 𝑖=1,…,𝑘₁, 𝑗=1,…,𝑘₂, let (𝑥^∗_𝑓𝑖,𝑦^∗_𝑓𝑖) be the optimal solution to 𝐹_𝑖, and let (𝑥^∗_𝑔𝑗,𝑦^∗_𝑔𝑗) be the optimal solution to 𝐺_𝑖. Then, (𝑥^∗,𝑦^∗,𝐬^∗) is an optimal solution to the following problem (SW): min𝐬s.t.𝑓_𝑖𝑥,𝑦−𝑝_𝑖𝐬≤𝑓_𝑖𝑥^∗_𝑓𝑖,𝑦^∗_𝑓𝑖,𝑖=1,2,…,𝑘₁,𝑔_𝑗𝑥,𝑦−𝑞_𝑗𝐬≤𝑔_𝑗𝑥^∗_𝑔𝑗,𝑦^∗_𝑔𝑗,𝑗=1,2,…,𝑘₂,𝐬≥0,𝑥,𝑦∈𝑍, (SW) if and only if (𝑥^∗,𝑦^∗) is an 𝐬^∗-optimal joint solution with weight vector (𝑝,𝑞) to (FG).

Proof. Suppose (𝑥^∗,𝑦^∗,𝐬^∗) is the optimal solution to the problem (SW). For any (𝑥,𝑦)∈𝑍, 𝑖=1,2,…,𝑘₁, 𝑗=1,2,…,𝑘₂, it follows with the constraints of (SW) that 𝑓_𝑖𝑥^∗,𝑦^∗−𝑝_𝑖𝐬^∗≤𝑓_𝑖𝑥^∗_𝑓𝑖,𝑦^∗_𝑓𝑖≤𝑓_𝑖𝑥,𝑦,𝑔_𝑗𝑥^∗,𝑦^∗−𝑞_𝑗𝐬^∗≤𝑔_𝑗𝑥^∗_𝑔𝑗,𝑦^∗_𝑔𝑗≤𝑔_𝑗𝑥,𝑦, (3.7) which implies (𝑥^∗,𝑦^∗) is an 𝐬^∗-joint solution with weight vector (𝑝,𝑞) to (FG). Assume (𝑥,𝑦) is the 𝐬-optimal joint solution with weight vector (𝑝,𝑞) to (FG). Then, with Definitions 3.1 and 3.3, it gets 𝐬^∗≥𝐬 and 𝑓_𝑖𝑥,𝑦−𝑝_𝑖𝐬≤𝑓_𝑖𝑥,𝑦,𝑔_𝑗𝑥,𝑦−𝑞_𝑗𝐬≤𝑔_𝑗𝑥,𝑦,∀𝑥,𝑦∈𝑍, (3.8) which implies (𝑥,𝑦) is feasible to (S) and 𝐬≥𝐬^∗. Then, we get 𝐬=𝐬^∗, and (𝑥^∗,𝑦^∗) becomes the 𝐬^∗-optimal joint solution with weight vector (𝑝,𝑞) to (FG).
On the contrary, if (𝑥^∗,𝑦^∗) is an 𝐬^∗-optimal joint solution with weight vector (𝑝,𝑞) to (FG), by Definition 3.1, it gets (𝑥^∗,𝑦^∗,𝐬^∗) is feasible to (SW). Supposing (𝑥,𝑦,𝐬) is the optimal solution to (SW), it is easily obtained that (𝑥,𝑦) is the 𝐬-optimal joint solution with weight vector (𝑝,𝑞) to (FG), which implies 𝐬=𝐬^∗. Thus, (𝑥^∗,𝑦^∗,𝐬^∗) is the optimal solution to (SW), and this completes the proof.

Remark 3.5. If the optimal solutions to 𝐹_𝑖 and 𝐺_𝑖 exist, the 𝐬^∗-optimal joint solution with weight vector (𝑝,𝑞) to (FG) exists, too. By Theorem 3.4, we can obtain a method of solving the problem IPP as follows. First, get the optimal solutions to 𝐹_𝑖 and 𝐺_𝑖. Then, we can get an optimal solution (𝑥^∗,𝑦^∗,𝐬^∗) to (SW), and (𝑥^∗,𝑦^∗) is an 𝐬^∗-optimal joint solution with weight vector (𝑝,𝑞) to (FG).

Theorem 3.6. Suppose (𝑥^∗,𝑦^∗) is an 𝐬^∗-optimal joint solution with weight vector (𝑝,𝑞) to (FG). Then, there does not exist (𝑥,𝑦)∈𝑍 such that 𝑓_𝑖𝑥,𝑦<𝑓_𝑖𝑥^∗,𝑦^∗,𝑔_𝑗𝑥,𝑦<𝑔_𝑗𝑥^∗,𝑦^∗,𝑖=1,2,…,𝑘₁,𝑗=1,2,…,𝑘₂. (3.9)

Proof. It is obviously correct if 𝐬^∗=0. For 𝐬^∗>0, suppose there exists (𝑥^′,𝑦^′)∈𝑍 such that 𝑓_𝑖𝑥^′,𝑦^′<𝑓_𝑖𝑥^∗,𝑦^∗,𝑔_𝑗𝑥^′,𝑦^′<𝑔_𝑗𝑥^∗,𝑦^∗,𝑖=1,2,…,𝑘₁,𝑗=1,2,…,𝑘₂. (3.10) Let 𝛿=min{𝑓_𝑖(𝑥^∗,𝑦^∗)−𝑓_𝑖(𝑥^′,𝑦^′),𝑔_𝑗(𝑥^∗,𝑦^∗)−𝑔_𝑗(𝑥^′,𝑦^′)∣𝑖=1,2,…,𝑘₁,𝑗=1,2,…,𝑘₂} with 𝛿>0. Then, there exists sufficiently small 𝛼>0 such that 𝐬^∗−𝛼𝛿>0, 𝑝_𝑖𝛼<1, and 𝑞_𝑗𝛼<1 for 𝑖=1,2,…,𝑘₁, 𝑗=1,2,…,𝑘₂. Letting 𝐬^′=𝐬^∗−𝛼𝛿, then, we get 𝐬^′<𝐬^∗. It follows with Definition 3.1 that 𝑓_𝑖𝑥^′,𝑦^′−𝑝_𝑖𝐬^′=𝑓_𝑖𝑥^′,𝑦^′−𝑝_𝑖𝐬^∗+𝑝_𝑖𝛼𝛿<𝑓_𝑖𝑥^′,𝑦^′−𝑝_𝑖𝐬^∗+𝑓_𝑖𝑥^∗,𝑦^∗−𝑓_𝑖𝑥^′,𝑦^′≤𝑓_𝑖,𝑔𝑥,𝑦_𝑗𝑥^′,𝑦^′−𝑞_𝑗𝐬^′=𝑔_𝑗𝑥^′,𝑦^′−𝑞_𝑗𝐬^∗+𝑞_𝑗𝛼𝛿<𝑔_𝑗𝑥^′,𝑦^′−𝑞_𝑗𝐬^∗+𝑔_𝑗𝑥^∗,𝑦^∗−𝑔_𝑗𝑥^′,𝑦^′≤𝑔_𝑗𝑥,𝑦, (3.11) for any (𝑥,𝑦)∈𝑍. Thus, (𝑥^′,𝑦^′) is 𝐬^′-optimal joint solution with weight vector (𝑝,𝑞) to (FG) and 𝐬^′>𝐬^∗, which contradicts the assumption of 𝐬^′<𝐬^∗; this completes the proof.

Theorem 3.7. For 𝑖=1,…,𝑘₁, 𝑗=1,…,𝑘₂, let (𝑥^∗_𝑓𝑖,𝑦^∗_𝑓𝑖) be an optimal solution to 𝐹_𝑖, and let (𝑥^∗_𝑔𝑗,𝑦^∗_𝑔𝑗) be an optimal solution to 𝐺_𝑖. If [(𝑥,𝑦,𝐬),(𝑥^′_𝑖,𝑦^′_𝑖),(𝑥_𝑗^′′,𝑦_𝑗^′′)]  (𝑖=1,2,…,𝑘₁, 𝑗=1,2,…,𝑘₂) is the optimal solution of the following problem 𝑆: min𝐬+^𝑘1_𝑖=11𝑝_𝑖𝑓_𝑖𝑥^′_𝑖,𝑦^′_𝑖+^𝑘2_𝑗=11𝑞_𝑗𝑔_𝑗𝑥_𝑗^′′,𝑦_𝑗^′′s.t.𝑓_𝑖𝑥,𝑦−𝑝_𝑖𝐬≤𝑓_𝑖𝑥^′_𝑖,𝑦^′_𝑖,𝑖=1,2,…,𝑘₁,𝑔_𝑗𝑥,𝑦−𝑞_𝑗𝐬≤𝑔_𝑗𝑥_𝑗^′′,𝑦_𝑗^′′,𝑗=1,2,…,𝑘₂,𝐬≥0,𝑥,𝑦,𝑥^′_𝑖,𝑦^′_𝑖,𝑥_𝑗^′′,𝑦_𝑗^′′∈𝑍,𝑖=1,2,…,𝑘₁,𝑗=1,2,…,𝑘₂, 𝑆 then (𝑥,𝑦) is an 𝐬^∗-optimal joint solution with weight vector (𝑝,𝑞) to (FG), where 𝐬^∗=1𝐬+max𝑝_𝑖𝑓_𝑖𝑥^′_𝑖,𝑦^′_𝑖−𝑓_𝑖𝑥^∗_𝑓𝑖,𝑦^∗_𝑓𝑖,1𝑞_𝑗𝑔_𝑗𝑥_𝑗^′′,𝑦_𝑗^′′−𝑔_𝑗𝑥^∗_𝑔𝑗,𝑦^∗_𝑔𝑗∣𝑖=1,2,…,𝑘₁,𝑗=1,2,…,𝑘₂. (3.12)

Proof. Supposing (𝑥^∗,𝑦^∗) is the 𝐬^∗-optimal joint solution with weight vector (𝑝,𝑞) to (FG), it concludes from Theorem 3.4 that (𝑥^∗,𝑦^∗,𝐬^∗) is the optimal solution to (SW). Then, [(𝑥^∗,𝑦^∗,𝐬^∗),(𝑥^∗_𝑓𝑖,𝑦^∗_𝑓𝑖),(𝑥^∗_𝑔𝑗,𝑦^∗_𝑔𝑗)]  (𝑖=1,2,…,𝑘₁, 𝑗=1,2,…,𝑘₂) is feasible to 𝑆. Thus, we have 𝐬+^𝑘1_𝑖=11𝑝_𝑖𝑓𝑥^′_𝑖,𝑦^′_𝑖+^𝑘2_𝑗=11𝑞_𝑗𝑔𝑥_𝑗^′′,𝑦_𝑗^′′≤𝐬^∗+^𝑘1_𝑖=11𝑝_𝑖𝑓_𝑖𝑥^∗_𝑓𝑖,𝑦^∗_𝑓𝑖+^𝑘2_𝑗=11𝑞_𝑗𝑔_𝑗𝑥^∗_𝑔𝑗,𝑦^∗_𝑔𝑗. (3.13)
Let 𝐬̃=𝐬+max{(1/𝑝_𝑖)(𝑓_𝑖(𝑥^′_𝑖,𝑦^′_𝑖)−𝑓_𝑖(𝑥^∗_𝑓𝑖,𝑦^∗_𝑓𝑖)),(1/𝑞_𝑗)(𝑔_𝑗(𝑥_𝑗^′′,𝑦_𝑗^′′)−𝑔_𝑗(𝑥^∗_𝑔𝑗,𝑦^∗_𝑔𝑗))∣𝑖=1,2,…,𝑘₁,𝑗=1,2,…,𝑘₂}.
It concludes from (3.13) that 𝐬̃≤𝐬^∗. Further, with the assumption, we get 𝑓_𝑖𝑥,𝑦−𝑝_𝑖𝐬≤𝑓_𝑖𝑥^′_𝑖,𝑦^′_𝑖,𝑖=1,2,…,𝑘₁,𝑔_𝑗𝑥,𝑦−𝑞_𝑗𝐬≤𝑔_𝑗𝑥_𝑗^′′,𝑦_𝑗^′′,𝑗=1,2,…,𝑘₂, (3.14) which implies 𝑓_𝑖𝑥,𝑦≤𝑝_𝑖𝐬+𝑓_𝑖𝑥^′_𝑖,𝑦^′_𝑖−𝑓_𝑖𝑥^∗_𝑓𝑖,𝑦^∗_𝑓𝑖+𝑓_𝑖𝑥^∗_𝑓𝑖,𝑦^∗_𝑓𝑖,𝑖=1,2,…,𝑘₁,𝑔_𝑗𝑥,𝑦≤𝑞_𝑗𝐬+𝑔_𝑗𝑥_𝑗^′′,𝑦_𝑗^′′−𝑔_𝑗𝑥^∗_𝑔𝑗,𝑦^∗_𝑔𝑗+𝑔_𝑗𝑥^∗_𝑔𝑗,𝑦^∗_𝑔𝑗,𝑗=1,2,…,𝑘₂. (3.15) Thus, 𝑓_𝑖𝑥,𝑦≤𝑝_𝑖𝐬̃+𝑓_𝑖𝑥^∗_𝑓𝑖,𝑦^∗_𝑓𝑖,𝑖=1,2,…,𝑘₁,𝑔_𝑗𝑥,𝑦≤𝑞_𝑗𝐬̃+𝑔_𝑗𝑥^∗_𝑔𝑗,𝑦^∗_𝑔𝑗,𝑗=1,2,…,𝑘₂. (3.16) Therefore, (𝑥,𝑦,𝐬)̃ is feasible to (S), which implies 𝐬̃≥𝐬^∗, and (𝑥,𝑦) is the 𝐬̃-joint solution with weight vector (𝑝,𝑞) to (FG). Thus, 𝐬̃=𝐬^∗, and this completes the proof.

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

Example 3.8. Considering the following IPP: min𝑓𝑥,𝑦=(𝑥−2)²+𝑦²,𝑥²+𝑦²,min𝑔𝑥,𝑦=𝑥²+𝑦−2²,𝑥²+𝑦²𝑠.𝑡.𝑥,𝑦∈𝑅, 𝐹𝑦(𝐺(𝑥)) where 𝐻=𝑅×𝑅. For 𝑝₁=𝑝₂=𝑞₁=𝑞₂=2, by Theorem 3.4, it is easily obtained (1,1) that is the 1-optimal joint solution with weight vector (2,2,2,2) 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 𝑃_𝑖  (𝑖=1,2,…,𝑛). Let 𝑐_𝑖 denote the production cost for 𝑃_𝑖  (𝑖=1,2,…,𝑛), 𝑐_𝑖𝑡 the transportation cost for 𝑃_𝑖  (𝑖=1,2,…,𝑛), and 𝑤_𝑖 the price of 𝑃_𝑖  (𝑖=1,2,…,𝑛). The manufacturer provides products to the retailer at 𝑤_𝑖^min≤𝑤_𝑖≤𝑤_𝑖^max, where 𝑤_𝑖^min is the minimum of 𝑤_𝑖 and 𝑤_𝑖^max is the maximum of 𝑤_𝑖. Let 𝑞_𝑖 denote the quantity ordered for 𝑃_𝑖  (𝑖=1,2,…,𝑛) with 𝑞_𝑖^min≤𝑞_𝑖≤𝑞_𝑖^max, where 𝑞_𝑖^min is the minimum of 𝑞_𝑖 and 𝑞_𝑖^max is the maximum of 𝑞_𝑖. Supposing the retailer sells the product 𝑃_𝑖  (𝑖=1,2,…,𝑛) at the price of 𝑝_𝑖, then it is clear that 𝑤_𝑖≤𝑝_𝑖. Let 𝐷_𝑖=𝐷_𝑖(𝑝_𝑖), which denotes the market demand for 𝑃_𝑖  (𝑖=1,2,…,𝑛). Then we get the following pricing model for the bi-level supply chain: max_𝑤𝑓𝑤,𝑝=𝑤₁−𝑐₁−𝑐_𝑡1𝐷₁𝑝₁,𝑤₂−𝑐₂−𝑐_𝑡2𝐷₂𝑝₂,…,𝑤_𝑛−𝑐_𝑛−𝑐_𝑡𝑛𝐷_𝑛𝑝_𝑛s.t.𝑤_𝑖^min≤𝑤_𝑖≤𝑤_𝑖^max𝑞,𝑖=1,2,…,𝑛,_𝑖min≤𝐷_𝑖𝑝_𝑖≤𝑞_𝑖^max,𝑖=1,2,…,𝑛,max_𝑝𝑔𝑤,𝑝=𝑝₁−𝑤₁𝐷₁𝑝₁,𝑝₂−𝑤₂𝐷₂𝑝₂,…,𝑝_𝑛−𝑤_𝑛𝐷_𝑛𝑝_𝑛s.t.𝑤_𝑖^min≤𝑤_𝑖≤𝑤_𝑖^max𝑞,𝑖=1,2,…,𝑛,_𝑖min≤𝐷_𝑖𝑝_𝑖≤𝑞_𝑖^max𝑤,𝑖=1,2,…,𝑛,_𝑖≤𝑝_𝑖,𝑖=1,2,…,𝑛, 𝐹_𝑝𝐺_𝑤 where 𝑝=(𝑝₁,𝑝₂,…,𝑝_𝑛) is the parameter for 𝐹_𝑝, 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛) is the parameter for 𝐺_𝑤, and 𝑤_𝑖≤𝑝_𝑖  (𝑖=1,2,…,𝑛) 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 𝑖=1,2,3, and let all the 𝑃_𝑖 be independent to each other; the transportation cost 𝑐_𝑖𝑡=5 for all 𝑃_𝑖; the production costs: 𝑐₁=10, 𝑐₂=12 and 𝑐₃=13; the production capacity 40≤𝑞₁≤100,  30≤𝑞₂≤100 and 30≤𝑞₃≤100; the price 𝑤_𝑖  (𝑖=1,2,3) restricted by 20≤𝑤₁≤30, 20≤𝑤₂≤35 and 20≤𝑤₃≤35; the market demand for 𝑃_𝑖: 𝐷₁(𝑝₁)=200−4𝑝₁, 𝐷₂(𝑝₂)=250−5𝑝₂ and 𝐷₃(𝑝₃)=300−6𝑝₃.

We have the pricing model of (FG): max_𝑤𝑓𝑤,𝑝=𝑤₁−15200−4𝑝₁,𝑤₂−17250−5𝑝₂,𝑤₃−18300−6𝑝₃s.t.20≤𝑤₁≤30,20≤𝑤₂≤35,20≤𝑤₃≤35,40≤200−4𝑝₁≤100,30≤250−5𝑝₂≤100,30≤300−6𝑝₃≤100,max_𝑝𝑔𝑤,𝑝=𝑝₁−𝑤₁200−4𝑝₁,𝑝₂−𝑤₂250−5𝑝₂,𝑝₃−𝑤₃300−6𝑝₃s.t.20≤𝑤₁≤30,20≤𝑤₂≤35,20≤𝑤₃≤35,40≤200−4𝑝₁≤100,30≤250−5𝑝₂≤100,30≤300−6𝑝₃𝑝≤100,₁≥𝑤₁,𝑝₂≥𝑤₂,𝑝₃≥𝑤₃. 𝐹_𝑝𝐺_𝑤 We give the Stackelberg pricing model: max_𝑤,𝑝𝑓𝑤,𝑝=𝑤₁−15200−4𝑝₁,𝑤₂−17250−5𝑝₂,𝑤₃−18300−6𝑝₃s.t.20≤𝑤₁≤30,20≤𝑤₂≤35,20≤𝑤₃≤35,40≤200−4𝑝₁≤100,30≤250−5𝑝₂≤100,30≤300−6𝑝₃≤100. (4.1)𝑝 solves the lower level problem: max_𝑝𝑔𝑤,𝑝=𝑝₁−𝑤₁200−4𝑝₁,𝑝₂−𝑤₂250−5𝑝₂,𝑝₃−𝑤₃300−6𝑝₃s.t.20≤𝑤₁≤30,20≤𝑤₂≤35,20≤𝑤₃≤35,40≤200−4𝑝₁≤100,30≤250−5𝑝₂≤100,30≤300−6𝑝₃𝑝≤100,₁≥𝑤₁,𝑝₂≥𝑤₂,𝑝₃≥𝑤₃. (4.2) Let 𝑓₁(𝑤,𝑝)=(𝑤₁−15)(200−4𝑝₁)+(𝑤₂−17)(250−5𝑝₂)+(𝑤₃−18)(300−6𝑝₃) and let 𝑔₁(𝑤,𝑝)=(𝑝₁−𝑤₁)(200−4𝑝₁)+(𝑝₂−𝑤₂)(250−5𝑝₂)+(𝑝₃−𝑤₃)(300−6𝑝₃). We give the joint pricing model: max_𝑝𝑓₁𝑤,𝑝+𝑔₁𝑤,𝑝s.t.20≤𝑤₁≤30,20≤𝑤₂≤35,20≤𝑤₃≤35,40≤200−4𝑝₁≤100,30≤250−5𝑝₂≤100,30≤300−6𝑝₃𝑝≤100,₁≥𝑤₁,𝑝₂≥𝑤₂,𝑝₃≥𝑤₃. (4.3)

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.

Table 1 

Solutions of the three models.

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

5. Conclusion

In this paper, 𝐬-optimal joint solution and 𝐬-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.