# Multiobjective Interaction Programming Problem with Interaction Constraint for Two Players

Authors: Min Jiang; Zhiqing Meng; Xinsheng Xu; Rui Shen; Gengui Zhou
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

This paper extends an existing cooperative multi-objective interaction programming problem with interaction constraint for two players (or two agents). First, we define ans-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.


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## Body

## 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 ans-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

Letf:Rm×Rn→Rk, g:Rm×Rn→Rk be the multiobjective functions, and let X,Y,H⊂Rm×Rn 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):
(FG)min⁡xf(x,y)s.t.(x,y)∈Xmin⁡yg(x,y)s.t.(x,y)∈Ys.t.(x,y)∈H.(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(2.1)f(x*,y*)≤f(x,y),g(x*,y*)≤g(x,y),∀(x,y)∈Z,
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
(2.2)f(x,y*)≤f(x,y),g(x*,y)≤g(x,y),
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 weightλ=(λ1,λ2,…,λk)>0.Definition 2.2.
For(x*,y*)∈Z and s=(s1,s2,…,sk)>0, if it satisfies
(2.3)fi(x*,y*)-λisi≤fi(x,y),gi(x*,y*)-λisi≤gi(x,y),i=1,2,…,k,∀(x,y)∈Z,
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.
Whenλ=(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(Fi) and (Gi) (i=1,2,…k):(Fi)min⁡fi(x,y)s.t.(x,y)∈Z,(Gi)min⁡gi(x,y)s.t.(x,y)∈Z.
Let (xfi*,yfi*) be the optimal solution to (Fi), and let (xgi*,ygi*) be the optimal solution to (Gi). For any given (x′,y′)∈Z, let si′=(1/λi)max{fi(x′,y′)-fi(xfi*,yfi*),gi(x′,y′)-gi(xgi*,ygi*)} (i=1,2,…k) and s′=(s1′,s2′,⋯,sk′). 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
(2.4)fi(xfi*,yfi*)≤fi(x,y),gi(xgi*,ygi*)≤gi(x,y),
which implies
(2.5)fi(x′,y′)-[fi(x′,y′)-fi(xfi*,yfi*)]≤fi(x,y),gi(x′,y′)-[gi(x,y)-gi(xgi*,ygi*)]≤gi(x,y).
It follows with the assumption that
(2.6)fi(x′,y′)-λisi′≤fi(x,y),gi(x′,y′)-λisi′≤gi(x,y).
Then, by Definition 2.2, the proof completes.By Theorem2.3, 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+⋯sk, 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.
Fori=1,2,…,k, let (xfi*,yfi*) be an optimal solution to (Fi), and let (xgi*,ygi*) be the optimal solution to (Gi). Then, (x*,y*,s*) is an optimal solution to the following problem:
(S)min⁡|s|=s1+s2+⋯sks.t.fi(x,y)-λisi≤fi(xfi*,yfi*),i=1,2,…,k,gi(x,y)-λisi≤gi(xgi*,ygi*),i=1,2,…,k,(x,y)∈Z,si≥0,i=1,2,…,k
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
(2.7)fi(x*,y*)-λisi≤fi(xfi*,yfi*)≤f(x,y),i=1,2,…,k,gi(x*,y*)-λisi≤gi(xgi*,ygi*)<gi(x,y),i=1,2,…,k.
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,
(2.8)fi(x′,y′)-λisi′≤fi(x,y),gi(x′,y′)-λisi′≤gi(x,y).
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.
LetZ be compact, and let f and g be continuous functions. Then there exists an s*-optimal joint solution to problem (FG).Theorem 2.7.
Fori=1,2,…,k, let (xfi*,yfi*) be the optimal solution to (Fi), and let (xgi*,ygi*) be the optimal solution to (Gi). If [(x′,y′,s′),(xi1′,yi1′)(xi2′′,yi2′′)]  (i=1,2,…,k) is an optimal solution to the following problem:(S′)min⁡∑i=1k[si+f(xi′,yi′)+g(xi′′,yi′′)]s.t.fi(x,y)-λisi≤fi(xi1′,yi1′),i=1,2,…,k,gi(x,y)-λisi≤gi(xi′′,yi′′),i=1,2,…,k,(x,y),(xi′,yi′),(xi′′,yi′′)∈Z,si≥0,i=1,2,…,k,
then (x′,y′) is an s*-optimal joint solution with weight vector λ to problem (FG), where
(2.9)si*=si′+1λimax⁡{(fi(xi1′,yi1′)-fi(xfi*,yfi*)),(gi(xi2′′,yi2′′)-gi(xgi*,ygi*))},i=1,2,…,k.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*),(xfi*,yfi*)(xgi*,ygi*)]  (i=1,2,…,k) is feasible to problem (S′), which implies
(2.10)∑i=1k[si′+f(xi1′,yi1′)+g(xi2′′,yi2′′)]≤∑i=1k[si*+f(xfi*,yfi*)+g(xgi*,ygi*)],
that is:
(2.11)∑i=1k[si′+(f(xi1′,yi1′)-f(xfi*,yfi*))+(g(xi2′′,yi2′′)-g(xgi*,ygi*))]≤∑i=1ksi*.
Letting
(2.12)si°=si′+1λimax⁡{(f(xi1′,yi1′)-f(xfi*,yfi*)),(g(xi2′′,yi2′′)-g(xgi*,ygi*))},i=1,2,…,k,
then it follows ∑i=1ksi°≤∑i=1ksi*. On the other hand, from the assumption we have that
(2.13)fi(x′,y′)-λisi′≤fi(xi1′,yi1′),i=1,2,…,k,gi(x′,y′)-λisi′≤gi(xi2′′,yi2′′),i=1,2,…,k,
which implies
(2.14)fi(x′,y′)≤λisi′+fi(xi1′,yi1′)-fi(xfi*,yfi*)+fi(xfi*,yfi*),i=1,2,…,k,gi(x′,y′)≤λisi′+gi(xi2′′,yi2′′)-gi(xgi*,ygi*)+gi(xgi*,ygi*),i=1,2,…,k,
that is:
(2.15)fi(x′,y′)≤si°+fi(xfi*,yfi*),i=1,2,…,k,gi(x′,y′)≤si°+gi(xgi*,ygi*),i=1,2,…,k.
Then, (x′,y′,s°) is a feasible solution to problem (S). Thus, ∑i=1ksi°≥∑i=1ksi* holds and (x′,y′) becomes an s°-optimal joint solution with weight vector λ to problem (FG) with ∑i=1ksi°=∑i=1ksi*. This completes the proof.Remark 2.8.
By Theorem2.7, 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 (Fi) and (Gi)  (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:Rm×Rn→Rk1 and g:Rm×Rn→Rk2. 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):
(FG)min⁡xf(x,y)s.t.(x,y)∈X,min⁡yg(x,y)s.t.(x,y)∈Ys.t.(x,y)∈H.
Hence, we need to define a new s-optimal joint solution to problem (FG). Suppose the weight of fi is pi>0 and the weight of gj is qj>0, i=1,2,…,k1, j=1,2,…,k2. Let
(3.1)p=(p1,p2,…,pk1),q=(q1,q2,…,qk1).Definition 3.1.
For(x*,y*)∈Z, s≥0, if it satisfies
(3.2)fi(x*,y*)-pis≤fi(x,y),gj(x*,y*)-qjs≤gj(x,y)
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(xfi*,yfi*) is the optimal solution to (Fi) and (xgj*,ygj*) is the optimal solution to (Gi). For any (x-,y-)∈Z, letting
(3.3)s-=max⁡{1qj(gj(x-,y-)-gj(xgj*,ygj*))1pi(fi(x-,y-)-fi(xfi*,yfi*)),1qj(gj(x-,y-)-gj(xgj*,ygj*))∣i=1,2,…,k1,j=1,2,…,k21pi(fi(x-,y-)-fi(xfi*,yfi*))1pi},
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
(3.4)fi(xfi*,yfi*)≤fi(x,y),gj(xgj*,ygj*)≤gj(x,y),
which implies
(3.5)fi(x-,y-)-[fi(x-,y-)-fi(xfi*,yfi*)]≤fi(x,y),gj(x-,y-)-[gj(x-,y-)-gj(xgj*,ygj*)]≤gj(x,y),
so
(3.6)fi(x-,y-)-pis-≤fi(x,y),gj(x-,y-)-qjs-≤gj(x,y).
This completes the proof with Definition 3.1.By Theorem2.3, for all(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, ifs*=0,(x*,y*) becomes the optimal joint solution for Player 1 (or Agent 1) and Player 2 (or Agent 2).Theorem 3.4.
Fori=1,…,k1,  j=1,…,k2, let (xfi*,yfi*) be the optimal solution to (Fi), and let (xgj*,ygj*) be the optimal solution to (Gi). Then, (x*,y*,s*) is an optimal solution to the following problem (SW):
(SW)min⁡ss.t.fi(x,y)-pis≤fi(xfi*,yfi*),i=1,2,…,k1,gj(x,y)-qjs≤gj(xgj*,ygj*),j=1,2,…,k2,s≥0,(x,y)∈Z,
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
(3.7)fi(x*,y*)-pis*≤fi(xfi*,yfi*)≤fi(x,y),gj(x*,y*)-qjs*≤gj(xgj*,ygj*)≤gj(x,y),
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
(3.8)fi(x-,y-)-pis-≤fi(x,y),gj(x-,y-)-qjs-≤gj(x,y),∀(x,y)∈Z,
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(Fi) and (Gi) 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 (Fi) and (Gi). 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
(3.9)fi(x,y)<fi(x*,y*),gj(x,y)<gj(x*,y*),i=1,2,…,k1,j=1,2,…,k2.Proof.
It is obviously correct ifs*=0. For s*>0, suppose there exists (x′,y′)∈Z such that
(3.10)fi(x′,y′)<fi(x*,y*),gj(x′,y′)<gj(x*,y*),i=1,2,…,k1,j=1,2,…,k2.
Let δ=min{fi(x*,y*)-fi(x′,y′),gj(x*,y*)-gj(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, piα<1, and qjα<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
(3.11)fi(x′,y′)-pis′=fi(x′,y′)-pis*+piαδ<fi(x′,y′)-pis*+fi(x*,y*)-fi(x′,y′)≤fi(x,y),gj(x′,y′)-qjs′=gj(x′,y′)-qjs*+qjαδ<gj(x′,y′)-qjs*+gj(x*,y*)-gj(x′,y′)≤gj(x,y),
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.
Fori=1,…,k1, j=1,…,k2, let (xfi*,yfi*) be an optimal solution to (Fi), and let (xgj*,ygj*) be an optimal solution to (Gi). 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-):
(S-)min⁡s+∑i=1k11pifi(xi′,yi′)+∑j=1k21qjgj(xj′′,yj′′)s.t.fi(x,y)-pis≤fi(xi′,yi′),i=1,2,…,k1,gj(x,y)-qjs≤gj(xj′′,yj′′),j=1,2,…,k2,s≥0,(x,y),(xi′,yi′),(xj′′,yj′′)∈Z,i=1,2,…,k1,j=1,2,…,k2,
then (x-,y-) is an s*-optimal joint solution with weight vector (p,q) to (FG), where
(3.12)s*=s-+max⁡{1qj1pi(fi(x-i′,y-i′)-fi(xfi*,yfi*)),1qj(gj(x-j′′,y-j′′)-gj(xgj*,ygj*))∣i=1,2,…,k1,j=1,2,…,k21pi}1pi.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*),(xfi*,yfi*),(xgj*,ygj*)]  (i=1,2,…,k1, j=1,2,…,k2) is feasible to (S-). Thus, we have
(3.13)s-+∑i=1k11pif(x-i′,y-i′)+∑j=1k21qjg(x-j′′,y-j′′)≤s*+∑i=1k11pifi(xfi*,yfi*)+∑j=1k21qjgj(xgj*,ygj*).
Lets~=s-+max{(1/pi)(fi(x-i′,y-i′)-fi(xfi*,yfi*)),(1/qj)(gj(x-j′′,y-j′′)-gj(xgj*,ygj*))∣i=1,2,…,k1,j=1,2,…,k2}.
It concludes from (3.13) that s~≤s*. Further, with the assumption, we get
(3.14)fi(x-,y-)-pis-≤fi(x-i′,y-i′),i=1,2,…,k1,gj(x-,y-)-qjs-≤gj(x-j′′,y-′′j),j=1,2,…,k2,
which implies
(3.15)fi(x-,y-)≤pis-+fi(x-i′,y-i′)-fi(xfi*,yfi*)+fi(xfi*,yfi*),i=1,2,…,k1,gj(x-,y-)≤qjs-+gj(x-j′′,y-j′′)-gj(xgj*,ygj*)+gj(xgj*,ygj*),j=1,2,…,k2.
Thus,
(3.16)fi(x-,y-)≤pis~+fi(xfi*,yfi*),i=1,2,…,k1,gj(x-,y-)≤qjs~+gj(xgj*,ygj*),j=1,2,…,k2.
Therefore, (x-,y-,s~) is feasible to (S), which implies s~≥s*, and (x-,y-) is the s~-joint solution with weight vector (p,q) to (FG). Thus, s~=s*, and this completes the proof.In Theorem3.4, as soon as the optimal solutions to (Fi) and (Gi) 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 (Fi) and (Gi), it is unluckily that we cannot get an s*.Example 3.8.
Considering the following IPP:(F(y))min⁡f(x,y)=((x-2)2+y2,x2+y2),(G(x))min⁡g(x,y)=(x2+(y-2)2,x2+y2)s.t.x,y∈R,
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 Section2. 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 Pi  (i=1,2,…,n). Let ci denote the production cost for Pi  (i=1,2,…,n), cit the transportation cost for Pi  (i=1,2,…,n), and wi the price of Pi  (i=1,2,…,n). The manufacturer provides products to the retailer at wimin≤wi≤wimax, where wimin is the minimum of wi and wimax is the maximum of wi. Let qi denote the quantity ordered for Pi  (i=1,2,…,n) with qimin≤qi≤qimax, where qimin is the minimum of qi and qimax is the maximum of qi. Supposing the retailer sells the product Pi  (i=1,2,…,n) at the price of pi, then it is clear that wi≤pi. Let Di=Di(pi), which denotes the market demand for Pi  (i=1,2,…,n). Then we get the following pricing model for the bi-level supply chain:
(Fp)max⁡wf(w,p)=((w1-c1-ct1)D1(p1),(w2-c2-ct2)D2(p2),…,((wn-cn-ctn)Dn(pn)))s.t.wimin⁡≤wi≤wimax⁡,i=1,2,…,n,qimin⁡≤Di(pi)≤qimax⁡,i=1,2,…,n,(Gw)max⁡pg(w,p)=((p1-w1)D1(p1),(p2-w2)D2(p2),…,((pn-wn)Dn(pn)))s.t.wimin⁡≤wi≤wimax⁡,i=1,2,…,n,qimin⁡≤Di(pi)≤qimax⁡,i=1,2,…,n,wi≤pi,i=1,2,…,n,
where p=(p1,p2,…,pn) is the parameter for (Fp), w=(w1,w2,…,wn) is the parameter for (Gw), and wi≤pi  (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, leti=1,2,3, and let all the Pi be independent to each other; the transportation cost cit=5 for all Pi; 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 wi  (i=1,2,3) restricted by 20≤w1≤30, 20≤w2≤35 and 20≤w3≤35; the market demand for Pi: D1(p1)=200-4p1,  D2(p2)=250-5p2 and D3(p3)=300-6p3.We have the pricing model of(FG):
(Fp)max⁡wf(w,p)=((w1-15)(200-4p1),(w2-17)(250-5p2),(w3-18)(300-6p3))s.t.20≤w1≤30,20≤w2≤35,20≤w3≤35,40≤200-4p1≤100,30≤250-5p2≤100,30≤300-6p3≤100,(Gw)max⁡pg(w,p)=((p1-w1)(200-4p1),(p2-w2)(250-5p2),(p3-w3)(300-6p3))s.t.20≤w1≤30,20≤w2≤35,20≤w3≤35,40≤200-4p1≤100,30≤250-5p2≤100,30≤300-6p3≤100,p1≥w1,p2≥w2,p3≥w3.
We give the Stackelberg pricing model:
(4.1)max⁡w,pf(w,p)=((w1-15)(200-4p1),(w2-17)(250-5p2),(w3-18)(300-6p3))s.t.20≤w1≤30,20≤w2≤35,20≤w3≤35,40≤200-4p1≤100,30≤250-5p2≤100,30≤300-6p3≤100.p solves the lower level problem:
(4.2)max⁡pg(w,p)=((p1-w1)(200-4p1),(p2-w2)(250-5p2),(p3-w3)(300-6p3))s.t.20≤w1≤30,20≤w2≤35,20≤w3≤35,40≤200-4p1≤100,30≤250-5p2≤100,30≤300-6p3≤100,p1≥w1,p2≥w2,p3≥w3.
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:
(4.3)max⁡pf1(w,p)+g1(w,p)s.t.20≤w1≤30,20≤w2≤35,20≤w3≤35,40≤200-4p1≤100,30≤250-5p2≤100,30≤300-6p3≤100,p1≥w1,p2≥w2,p3≥w3.Then, we solve this pricing problem of supply chain with the previously mentioned three pricing models, and the numerical results are given in Table1.Table 1
Solutions of the three models.
Profit ofs-joint solution
Profit of Stackelberg solution
Profit of pricing solution
Manufacturer
π
m
1
563.22
612.5
350
π
m
2
657.3
680.625
247.5
π
m
3
755.44
765
192
π
m
1976.36
2058.125
789.5
Retailer
π
r
1
273.22
288.75
875
π
r
2
569.1
344.4375
1113.75
π
r
3
571.9
433.5
1344
π
r
1414.22
1066.6875
3332.75
System
π
3390.58
3124.8125
4122.25
π
m
i
(
i
=
1,2
,
3
) denotes the profit of Pi for the manufacturer.
π
m
=
∑
i
=
1
3
π
m
i denotes the total profit of Pi for the manufacturer.
π
r
i
(
i
=
1,2
,
3
) denotes the profit of Pi for the retailer.
π
r
=
∑
i
=
1
3
π
r
i denotes the total profit of Pi for the retailer.
π
=
π
m
+
π
r denotes the total profit of Pi for the system, the manufacturer, and the retailer as a whole.From Table1, it is found that the profit πm of the manufacturer is about 4 times less than the profit πr of the retailer in the solution of joint pricing model which cannot be accepted by the manufacturer. The profit πm of the manufacturer is about 2 times more than the profit πr 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 πm of the manufacturer and the profit πr 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.



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*Source: 618928-2012-08-02.xml*