# A Concession Equilibrium Solution Method without Weighted Aggregation Operators for Multiattribute Group Decision-Making Problems

Authors: Min Jiang; Rui Shen; Zhiqing Meng
Journal: Discrete Dynamics in Nature and Society
 (2019)

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## Abstract

This paper introduces a concession equilibrium solution without weighted aggregation operators to multiattribute group decision-making problems (in short MGDMPs). It is of practical significance for all decision-makers to find an optimal solution to MGDMPs or to sort out all candidate solutions to MGDMPs. It is proved that under certain conditions the optimal concession equilibrium solution does exist, and on this important result the optimal concession equilibrium solution is obtained by solving a single objective optimization problem. Moreover, the optimal concession equilibrium solution is equivalent to the robust optimal solution with the group weight aggregation under the worst weight condition. Finally, it is proved that the concession equilibrium solution is equivalent to a complete order, i.e. all candidate alternatives can be sorted by concession equilibrium solution. By defining the triangular fuzzy numbers of target concession value, the optimal concession equilibrium solution or the order of the alternative solutions can be obtained in the range of objective concession ambiguity. Numerical experiment shows that the solution can balance the evaluations of multiattribute group decision makers. This paper provides a new approach to solving multiattribute group decision-making problems.


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## Body

## 1. Introduction

The multiattribute group decision-making problems (MGDMP) exist in many areas such as social network, supplier selection, competitive business environment, economic analysis, strategic planning, medical diagnosis, venture capital, and etc. Because of the conflict between attributes and decision makers, it is very difficult to solve a MGDMP.Suppose that there arer decision makers, DM1,DM2,…,DMr(r>1) –a group of experts– and the multiattribute evaluation (cost or benefit) function for DMi is fi:Rn→Rm, i=1,2,…,r. Let x be a candidate scheme (solution) and the set of all candidate schemes be X⊂Rn. Each decision maker selects a solution or a ranking of the candidate schemes from X by evaluating fi(x). This paper studies multiattribute group decision-making problem as follows:(MGDMP)(1)P1minf1xP2minf2x⋮Prminfrxs.t.x∈X.When r=1, (MGDMP) becomes a group decision-making problem (GDMP), which is a single-attribute one. So far, almost all the studies on MGDMPs focus on the weighted aggregation methods and fall within the following four main areas:( 1 ) The weighted aggregation method used earlier is mainly based on given weights. For example, Choi (1998) [1] and Kim (1999) [2] et al. gave a method involving multiattribute weights and group members’ weights to find out the best candidate. Wei (2000) [3] et al. gave the compromise weight method with individual preference. This idea of the aggregating OWA operator weights has been widely studied and promoted (see Xu (2009) [4]; Yue (2012) [5]; Zhou, Chen, and Liu (2012) [6]; Liu, Cai, and Martnez (2013) [7]; Liu, Zhang, and Zhang (2014) [8]; Gao, Li, and Liu (2015) [9]; Dong, Xiao, Zhang, and Wang (2016) [10]).( 2 ) Some studies focus on weighted aggregation methods that consider the uncertainty of weights, such as weights being an interval or a probability distribution. For example, Merig, Casanovas, and Yang (2014) [11] studied the uncertain generalized probabilistic weighted averaging (UGPWA) operator. Qi, Liang, and Zhang (2015) [12] presented a method of generalized cross-entropy based group decision-making with unknown experts and attribute weights under interval-valued intuitionistic fuzzy environment.( 3 ) In recent years, some new complex methods using fuzzy theory are applied to the weighted aggregation methods. For example, Sadi-Nezhad and Akhtari (2008) [13] studied a possibilistic programming approach with fuzzy multidimensional analysis preference. Yan and Ma (2015) [14] proposed an approach to uncertain quality function deployment based on fuzzy preference relation and fuzzy majority. More research literatures can be seen in Xu, Chen, Rodrguez, Herrera, and Wang (2016) [15]; Bayrama and Sahin (2016) [16]; Chen and Kuo (2017) [17]; Banaeian, Mobli, Fahimnia, Nielsen, and Omida (2018) [18].( 4 ) The hesitant fuzzy linguistic term set and the linguistic distribution are becoming popular tools to solve MGDMPs. For example, Thuong, Zhang, Li, and Hong (2018) [19] proposed a quantitative hesitant fuzzy judgment description with an embedded assessing attitude to evaluate financial statement quality (FSQ) to overcome the weighting difficulties, and applied a distance-based method to determine the evaluators’ weights and a weighted averaging operator to compute the criteria weights of MGDMPs. Wu, Xu, and Xu (2016) [20] proposed an entropy method that is generalized to a linguistic setting to derive the important weights for the attributes with quite different values, which are considered more important and therefore have higher weights for MGDMPs. Wu and Xu (2018) [21] considered the preferences of the decision-makers using fuzzy preference relations and a novel distance measure over the possibility distribution based hesitant fuzzy elements is defined to compute the various consensus measures. More research literatures can be seen in Wu, Li, Chen, and Dong (2018) [22]; Wu, Dai, Chiclana, Fujita, and Herrera-Viedma (2018) [23]; Li, Rodrguez, Martnez, Dong, and Herrera (2018) [24]; Wu and Xu (2016) [25]; Wu, Jin, and Xu (2018) [26].All the above literatures on weighted aggregation almost all focus on limited number of candidate schemes (solutions) to MGDMPs. But, the weighted aggregation method is a commonly used method in solving MGDMPs. Its fatal weakness is that different weights lead to different ranking of the candidate schemes (or candidates), and it is impossible to prove which weighted aggregation method is the best. In literatures on MGDMPs, the attributes’ weights and the experts’ weights should be determined, e.g. in [5] the weights were determined by using all the schemes, then if there are infinite number of candidate schemes, the method will become ineffective. On the other hand, different attributes’ weights and experts’ weights will lead to different ranking of the final scheme, which would result in an outcome that makes it difficult to determine which ranking is the best. So we propose an s-concession equilibrium solution to MGDMPs to avoid the determination of attributes’ weights and experts’ weights, and it provides an optimum solution to the situation when there are infinite number of schemes for MGDMPs.To solve the infinite-number-of-candidate multi-decision-maker decision-making problems, Meng, Hu and Dang (2005) [27] proposed an s-concession equilibrium solution with single attribute mathematical programming model for the coexistence of competitions and cooperation problems. Meng, Hu, Jiang and Zhou (2007) [28] gave an s-concession equilibrium solution with single attribute interactional programming model for the coexistence of competitions and co-operations problems. Xu, Meng, and Shen (2015) [29] introduced an s-concession equilibrium solution and gave a cooperation model based on CVaR measure for a two-stage supply chain with a single-attribute GDMP. Jiang, Meng and Shen (2018) [30] proposed for the first time the target concession value of s-concession equilibrium solution to the single-attribute GDMPs. But, an s-concession equilibrium solution to MGDMPs with the target concession value has not yet seen in published literatures.Jiang, Meng and Shen (2018) [30] introduced a concept as to the solution to group decision-making problems (GDMPs): s-concession equilibrium solution, which is more adaptive to the situation where the number of candidates is unlimited, and used an example to show how to solve the product ordering and production operation decision-making problem between the retailer and the manufacturer using the s- optimal concession equilibrium solution under the case where the number of alternatives is unlimited. The concept is characterized by that, for each decision maker, each objective attribute gives the corresponding concession value s, and the s-optimal concession equilibrium solution is the minimum concession given. The s-optimal concession equilibrium solution provides a natural criterion for evaluating the merits of the candidates. Obviously, it is different from other existing methods with weighted aggregation operators, because s-optimal concession equilibrium solution is a method that provides a natural criterion for evaluating the merits of the candidates and does not use weighted aggregation operators. According to the definition of s∗-optimal concession equilibrium solution, the s∗-optimal concession equilibrium solution is obviously not dependent on the evaluation function of one DM. On the other hand, for each scheme, the equilibrium value is the same for each decision maker’s goal. Therefore, the s-optimal concession equilibrium solution has its individual rationality.In this paper, based on the idea of concession equilibrium to GDMPs (Jiang, 2018) [30], the optimal concession equilibrium solution to MGDMPs without weighted aggregation operators is defined. Our innovation includes (1) a new s∗- optimal concession equilibrium solution is proposed, where s∗ is a vector, while the concept – s-optimal concession solution defined in [30] cannot solve (MGDMP); (2) the s∗- optimal concession equilibrium solution is a robust solution in all the weighted aggression sets; (3) a new triangular-fuzzy-concession ranking method is proposed based on the s∗- optimal concession equilibrium solution, and the rankings in the numerical experiments show stability under different concession values.


## 2.s∗-Optimal Concession Equilibrium Solution to MGDMP

In this section,s∗-optimal concession equilibrium solution to MGDMPs and its property are discussed. Let ϵi=(ϵ1i,ϵ2i,…,ϵmi)T∈Rm (i=1,2,…,r ) be given, where ϵi is called the target concession value of DMi(i=1,2,…,r). Let ε=(ϵ1,ϵ2,…,ϵr).Definition 1.
Letx∗∈X, s∈Rm, and ϵi≥0(i=1,2,…,r). If there is(2)fix∗-s-ϵi≤fix,∀x∈X,i=1,2,…,r,i.e.,(3)fjix∗-sj-ϵji≤fjix,∀x∈X,i=1,2,…,r,j=1,2,…,m,then x∗ is called s-concession equilibrium solution to (MGDMPs) at the value ε. The s=∑j=1msj is called an equilibrium value of (MGDMP) to x∗. s is called an equilibrium point of (MGDMP) to x∗. The set of all equilibrium values s of all s-concession equilibrium solutions x∗∈X to (MGDMP) is denoted as S. If x∗ is the s∗-concession equilibrium solution to (MGDMP) and s∗ is the minimum of the set S, then x∗ is called the s∗-optimal concession equilibrium solution to (MGDMP) at the target concession value ε. s∗ is called the optimal equilibrium value, and obviously the optimal equilibrium value is unique. s∗ is called the optimal equilibrium point, and the equilibrium point s∗ of x∗ is not always unique.This differs from thes∗-optimal concession equilibrium solution to single attribute group decision-making problem (Jiang, 2018) [30]. Furthermore, to solve the infinite-number-of-candidate MGDMPs and avoid the determination of the attributes’ weights and the experts’ weights, the s-concession equilibrium solution and s∗-concession equilibrium solution are introduced.Obviously, we have the following property.Property 2.
Letx∗ be s∗-optimal concession equilibrium solution to (MGDMP) at the value ε.
(1) Letv∈Rm. If(4)fix∗-v-ϵi≤fix,∀x∈X,i=1,2,…,r,then s∗≤v.
(2) Thenx∗ be 0-optimal concession equilibrium solution to (MGDMP) at the value ε(s∗), where ε(s∗)=(s∗+ϵ1,s∗+ϵ2,…,s∗+ϵr).
(3) IfX has only a finite number of solutions, then the s∗-optimal concession equilibrium solution to (MGDMP) exists.Property2 indicates that, with the given ϵi, s∗ is the minimum concession value among all the candidates, so its corresponding s∗-optimal concession equilibrium is the best solution in all equilibrium values.When the different target concession values, i.e., differentϵ, are given, different s∗-optimal concession equilibrium solutions are obtained, as shown in the following example.Example 3.
Consider the following GDMP.
(MGDMP)(5)P1minf1x,y=x-22+y2,x2+y-22P2minf2x,y=x2+y-22,x-22+y2s.t.x,y∈X=R1×R1.
Ifε1=((0,0)T,(0,0)T), we have (6)f111,1-2-0≤f11x,y,f211,1-2-0≤f21x,y,x,y∈R1×R1,f121,1-2-0≤f12x,y,f221,1-2-0≤f22x,y,x,y∈R1×R1.(x,y)=(1,1) is the (2,2)T-optimal concession equilibrium solution to the problem at the concession value ε1. This solution gives the minimum equilibrium value of each decision-maker’s individual objective.Ifε2=((8,0)T,(0,8)T), we have(7)f110,2-0-8≤f1x,y,f210,2-0-0≤f21x,y,x,y∈R1×R1,f120,2-0-0≤f1x,y,f220,2-0-8≤f22x,y,x,y∈R1×R1.( 0,2 ) is the (0,0)T-optimal concession equilibrium solution to the problem at the concession value ε2.From the above example, it is understood that whenϵ is given, an optimal concession equilibrium solution is obtained. And Lemma 4 gives the conclusion that under certain conditions any feasible solution to (MGDMP) is the concession equilibrium solution.For eachi=1,2,…,r;j=1,2,…,m, we have the following single objective optimization problem:(8)Pjiminfjixs.t.x∈X,Let fji∗ be the optimal target value of (Pji). Assumes that fji∗>-∞,i=1,2,…,r,j=1,2,…,m. If some fji∗=-∞, there is no optimal concession equilibrium solution to (MGDMP).Lemma 4.
Assume that there is an optimal solution to(Pji)(i=1,2,…,r;j=1,2,…,m,). Then for any given x¯∈X, x¯ is an s¯-concession equilibrium solution to (MGDMP), where s¯=(s¯1,s¯2,…,s¯m)T and s¯j=max{fji(x¯)-fji∗-ϵji∣i=1,2,…,r}, j=1,2,…,m.Proof.
For anyx∈X, we have (9)fji∗≤fjix,i=1,2,…,r;j=1,2,…,m.So, we have (10)fjix¯-fjix¯-fji∗-ϵji-ϵji≤fjix,i=1,2,…,r;j=1,2,…,m.So,(11)fix¯-s¯-ϵi≤fix,i=1,2,…,r.Therefore, by Definition 1, the conclusion of the theorem is true.Define the following optimization problem:(12)Smins=s1+s2+⋯+sms.t.fjix-sj-ϵji≤fji∗,i=1,2,…,r;j=1,2,…,m,s∈Rm,x∈X.Theorem 5.
Assume that there is an optimal solution to(Pji)(i=1,2,…,r). Then x∗ is s∗-optimal concession equilibrium solution to (MGDMP) at the value ε if and only if x∗,s∗ is an optimal solution to (S).Proof.
First, assume that ifx∗,s∗ is an optimal solution to (S), then for any x∈X, we have (13)fjix∗-sj∗-ϵji≤fji∗≤fjix,i=1,2,…,r;j=1,2,…,m.So, by Definition 1, we have that x∗ is s∗-concession equilibrium solution to (MGDMP) at the value ε. Let x¯ be s¯-optimal concession equilibrium solution to (MGDMP) at the value ε. By Property 2, we have s∗≥s¯ and (14)fix¯-s¯-ϵi≤fix,∀x∈X,i=1,2,…,r.Therefore, (x¯,s¯) is a feasible solution to (S). So, s¯≥s∗. That is s¯=s∗ and x∗ is s∗-optimal concession equilibrium solution to (MGDMP) at the value ε.Now, assume thatx∗ is an s∗-optimal concession equilibrium solution to (MGDMP) at the value ε. Then by Definition 1 it is known that (x∗,s∗) is a feasible solution to (S). Let (x¯,s¯) be an optimal solution to (S). By the above proof, x¯ is an s¯-optimal concession equilibrium solution to (MGDMP) at the value ε. Therefore, s¯=s∗. So, x∗≡sx¯ and x∗,s∗ is an optimal solution to (S).Theorem5 points out that if x is s-concession equilibrium solution, then (15)sj+ϵji≥0,i=1,2,…,r;j=1,2,…,m.Furthermore, Theorem5 gives that if there exists s∗-optimal concession equilibrium solution, the optimal solution for (Pji) must exist. Then we have the following.Theorem 6.
Assume thatX is a compact set and fji(x)(i=1,2,…,r;j=1,2,…,m) is continuous on X. Then the s∗-optimal concession equilibrium solution to (MGDMP) exists.Proof.
By the assumption, there is an optimal solution to each(Pji). By Lemma 4, we have S≠∅. We prove that S is close. Assume that a sequence sk⊂S converges to s∗. For k=1,2,…, let xk∗∈X be an sk- concession equilibrium solution to (MGDMP). Because X is compact, the sequence xk∗ has a convergent subsequence. Without loss of generality, let xk∗→x∗∈X. By Definition 1, we have (16)fixk∗-sk-ϵi≤fix,∀x∈X,i=1,2,…,r,k=1,2,….Let k→+∞, and then we have (17)fix∗-s∗-ϵi≤fix,∀x∈X,i=1,2,…,r.So, x∗ is the s∗-concession equilibrium solution. Therefore S is close. By Lemma 4, there is a minimum s∗ in S. Given a sufficiently large A>0, define the problem: (18)SAmins=s1+s2+⋯+sms.t.fjix-sj-ϵji≤fji∗,i=1,2,…,r;j=1,2,…,m,A≥sj≥sj∗,x∈X,j=1,2,…,m.It is obvious that the problem (S)A is equivalent to the problem (S), and the feasible set of the problem (S)A is compact too. Therefore, there exists the optimal solution (xA∗,sA∗) to (S)A, then (xA∗,sA∗) is also the optimal solution to the problem (S). By Theorem 5, the conclusion is true.To solve (S),(Pji) must be solved first, which is quite difficult. Therefore, we have the following Theorem 7, where solving a single objective programming problem (S¯) obtains the s∗-optimal concession equilibrium solution to (MGDMP).(19)S¯mins+∑i=1r∑j=1mfjixjis.t.fjix-sj-ϵji≤fjixji,i=1,2,…,r;j=1,2,…,m,s∈Rm,x,xji∈X,i=1,2,…,r;j=1,2,…,m,where s,x,xji,i=1,2,…,r;j=1,2,…,m, are variable of (S¯). We have better results as follows.Theorem 7.
Suppose thatx¯ji∗ is an optimal solution to (Pji)(i=1,2,…,r;j=1,2,…,m). If [(x¯,s¯),(x¯11,…,x¯m1),(x¯12,…,x¯m2),…,(x¯1r,…,x¯mr)] is an optimal solution to (S¯), then x¯ is s∗¯-optimal concession equilibrium solution to (MGDMP) at the value ϵ, where s∗=s¯+s^, s^=(s^1,s^2,…,s^m)T,(20)s^j=max⁡fjix¯ji-fjix¯ji∗∣i=1,2,…,r,j=1,2,…,m.Proof.
Letx¯∗ be an s¯∗-optimal concession equilibrium solution to (MGDMP). By Theorem 5, we have that (x¯∗,s¯∗) is an optimal solution to (S). It is clear that [(x¯∗,s¯∗),(x¯11∗,…,x¯m1∗),(x¯12∗,…,x¯m2∗),…,(x¯1r∗,…,x¯mr∗)] is a feasible solution to (S¯). Therefore, we have(21)s¯+∑i=1r∑j=1mfjix¯ji≤s¯∗+∑i=1r∑j=1mfix¯ji∗.By (20), (21), and fji(x¯ji∗)=fji∗, we have s∗≤s¯∗. On the other hand, according to the theorem hypothesis, we have (22)fjix¯-s¯j-ϵji≤fjix¯ji,i=1,2,…,r;j=1,2,…,m.So,(23)fjix¯-s¯j-ϵji≤fjix¯ji-fjix¯ji∗+fjix¯ji∗,i=1,2,…,r;j=1,2,…,m.Here, (24)fjix¯-ϵji≤sj∗+fjix¯ji∗,i=1,2,…,r;j=1,2,…,m.We have that (x∗,s∗) is a feasible solution to (S). Therefore, s∗≥s¯∗ and x∗ is s∗-optimal concession equilibrium solution to (MGDMP) at the value ε.Based on Theorem7, we have the following corollary.Corollary 8.
LetX=Rn and fji(x)(i=1,2,…,r,j=1,2,…,m) on Rn be a continuous differentiable convex function. Suppose that x¯j∗i is an optimal solution to (Pji)(i=1,2,…,r,j=1,2,…,m). Then x∗ is an s∗-optimal concession equilibrium solution to (MGDMP) at the value ε if and only if there is an incomplete zero of (λj1,λj2,…,λjr)(j=1,2,…,m) to satisfy the following KKT-condition:(25)λj1+λj2+⋯+λjr=1,j=1,2,…,m,∑i=1rλji∇fjix∗=0,j=1,2,…,m,λjifjix∗-sj∗-ϵji-fji∗=0,i=1,2,…,r,j=1,2,…,m,fjix∗-sj∗-ϵji≤fji∗,i=1,2,…,r,j=1,2,…,m,λji≥0,i=1,2,…,r,j=1,2,…,m.As per Corollary8, there is (λj1,λj2,…,λjr)(j=1,2,…,m) which can be seen as the group weighted aggregation of (MGDMP). Next, in the Set (27) consisting of all the aggregations, the s∗-optimal concession equilibrium solution is a robust optimal solution, as shown in Theorem 9.LetDMi’s evaluation be fji(x)-ϵji-fji∗ for x. By using weights 1≥λji≥0, DMi’s evaluation becomes λji(fji(x)-ϵji-fji∗) for x and j=1,2,…,m. With the linear weighting method, the evaluation of all decision-makers for x is defined as(26)Fx,λ≔∑i=1r∑j=1mλjifjix-ϵji-fji∗,where λj1+λj2+⋯+λjr=1,λji≥0,i=1,2,…,r;j=1,2,…,m. Let group weight set(27)Λ=λ=λ11,λ12,…,λ1rT,λ21,λ22,…,λ2rT,…,λm1,λm2,…,λmrT∣∑i=1rλji=1,λji≥0,i=1,2,…,r;j=1,2,…,m.The worst evaluation score of the solution x solves: maxλ∈Λ⁡F(x,λ) for each x∈X. Let λx=argmax⁡{F(x,λ)∣λ∈Λ}. Then, we are to find a minimum score from these worst scores minx∈X⁡F(x,λ(x)), i.e.,(28)S~minx∈X⁡maxλ∈Λ⁡Fx,λ.We prove the following conclusion.Theorem 9.
Supposing that the optimal solution to(Pji)(i=1,2,…,r;j=1,2,…,m) exists, then the problem (S) is equivalent to the problem (S~).Proof.
For a fixedx, the problem maxλ∈Λ⁡F(x,λ) is a linear programming: (29)ΛxmaxFx,λ≔∑i=1r∑j=1mλjifjix-ϵji-fji∗s.t.λj1+λj2+⋯+λjr=1,j=1,2,…,m,λji≥0,i=1,2,…,r;j=1,2,…,m.The dual problem of Λ(x) is (30)Sxmins=s1+s2+⋯+sms.t.fjix-sj-ϵji≤fji∗,i=1,2,…,r;j=1,2,…,m,s∈Rm,x∈X.Based on the strong duality theorem, there exists the optimal solution to the problem Λ(x) and the problem S(x), and the optimal objective values are equal at their optimal solutions. Let λ∗ be an optimal solution to Λ(x) and s∗ an optimal solution to S(x), then s∗=F(x,λ∗)=maxλ∈Λ⁡F(x,λ). Therefore, the conclusion of theorem is true.From the viewpoint of robustness, Theorem9 means the s∗-optimal concession equilibrium solution is the robust solution for the decision-makers under the worst weights in Λ.The evaluation functionfji(x)(i=1,2,…,r;j=1,2,…,m) of all decision-makers should be consistent as far as possible. fji∗(i=1,2,…,r;j=1,2,…,m) is the ideal goal. Obviously, the closer the solution to the ideal goal, the better. A deviation function is defined by (31)△x=1rm∑i=1r∑j=1mfjix-fji∗2,x∈X.According to Theorem 5, if x is the s-concession equilibrium solution, then we have (32)0≤fjix-fji∗≤sj+ϵji,i=1,2,…,r;j=1,2,…,m.That is (33)0≤1rm∑i=1r∑j=1mfjix-fji∗2≤1rm∑i=1r∑j=1msj+ϵji2,≤1ms+1rm∑i=1r∑j=1mϵji,for x∈X. Let ε=∑i=1r∑j=1mϵji. ε is the sum of target concession values of all DMi(i=1,2,…,r). So, we have the following conclusion.Corollary 10.
Assume that there is an optimal solution to(Pji)(i=1,2,…,r,j=1,2,…,m). If x∗ is s∗-optimal concession equilibrium solution to (MGDMP) at the value ε, then minx∈X⁡△(x)≤△x∗≤1/ms∗+1/rmε.Corollary10 means that the optimal concession equilibrium solution x∗ has a minimal upper bound 1/ms∗+1/rmε. △(x∗) can be used to evaluate the closeness between the s∗-optimal concession equilibrium solution and the ideal objective.Example 11.
Consider problem (MGDMP)
(MGDMP)(34)P1minf1x=x,-xP2minf2x=-x,xs.t.x∈X=-1,1.We know that f11∗=f21∗=f12∗=f12∗=-1. When ε=[(0,0)T,(0,0)T], 0 is the (1,1)T-optimal concession equilibrium solution to the problem at the concession value ε. This solution gives the minimum equilibrium value of each decision-maker’s individual objective. Define a weighted function by (35)Fx,λ=λ1f11x+1-λ1f12x+λ2f21x+1-λ2f22x=2λ1-λ2x.By Theorem 9, the optimal solution to minx∈X⁡maxλ1,λ2∈[0,1]⁡F(x,λ) is x∗=0. As a comparison, we are to use the linear weighted method to solve this problem, a very famous method (Kim and Han (1999)) [2] where weighted value λ1,λ2∈[0,1]. When λ1>λ2, an optimal solution to minx∈X⁡F(x,λ) is x∗=-1. When λ1<λ2, an optimal solution to minx∈X⁡F(x,λ) is x∗=1. But, when λ1=λ2, no optimal solution to minx∈X⁡F(x,λ) exists. On other hand, the deviation function △(x) is minimum at x∗=0, but maximum at x∗=1 or x∗=-1. It means that the linear weighted method is invalid or bad in this example. Therefore, no matter how the weight is obtained, the linear weighting method may be invalid.


## 3. Ranking and Fuzzy Target Concession Value of MGDMP

Now, we define the ranking in the setX of s-concession equilibrium solution to (MGDMP) at the value ε. Deviation of equilibrium value s of s-concession equilibrium solution to (MGDMP) at the value ε is defined as (36)σs=1m∑j=1msj-1ms2.σ(s) represents the difference among attribute values.Definition 12.
Letx1,x2∈X, and ε≥0. Let x1 be an s1-concession equilibrium solution to (MGDMP) at the target concession value ε and x2 be an s2-concession equilibrium solution to (MGDMP) at the target concession value ϵ. If s1<s2, we denote x1≺sx2 to indicate that x1 is superior to x2. If s1=s2 and σ(s1)<σ(s2), we denote x1⪯sx2 to indicate that x1 is better than x2. If s1=s2 and σ(s1)=σ(s2), we denote x1≡sx2 to indicate that x1 is equivalent to x2.Obviously, the setX is a serially ordered set about the order ≺s,⪯s or ≡s.Theorem 13.
Letx1,x2∈X, and ε≥0. Let x1 be an s1-concession equilibrium solution to (MGDMP) at the target concession value ε and x2 be an s2-concession equilibrium solution to (MGDMP) at the target concession value ε. If fi(x1)≤fi(x2),i=1,2,…,r, Then x1≺sx2, x1⪯sx2, or x1≡sx2.Proof.
According to assumption, we have(37)fix1-s1-ϵi≤fix,∀x∈X,i=1,2,…,r,(38)fix1-s2-ϵi≤fix2-s2-ϵi≤fix,∀x∈X,i=1,2,…,r,By Definition 1, we have s1≤s2.Theorem13 shows that the concession equilibrium solution must be nondominated for all decision-makers.Example 14.
Consider a MGDMP. Three decision-makers are to rank three candidate solutions through scoring, as shown in Table1. Let ε1=((0,0)T,(0,0)T,(0,0)T) and ε2=((0,0)T,(1,1)T,(2,2)T).Table 1
The score of three decisions and the equilibrium value under different target concession values.
Solution score of DM1 score of DM2 score of DM3 ε 1 ε 2 X f 1 ( x ) f 2 ( x ) f 3 ( x ) s s x 1 (1,1) (2,2) (3,3) (2,2)T ( 0,0 ) T x 2 (2,2) (3,3) (1,1) (2,2)T ( 1,1 ) T x 3 (3,3) (1,1) (2,2) (2,2)T ( 2,2 ) THere, by Definition12, when ε1=((0,0)T,(0,0)T,(0,0)T), we have x1≡sx2≡sx3. But when ε2=((0,0)T,(1,1)T,(2,2)T), we have x1≺sx2≺sx3. Example 14 shows a fact that when the decision-makers do not have the same target concession value ε, the ranking order of s-concession equilibrium solutions is not the same.Theorem 15.
Letε≥0, and let s′=(s′1,s′2,…,s′m)T≥0. If x is an s-concession equilibrium solution to (MGDMP) at the target concession value ε, then x is an s-s′-concession equilibrium solution to (MGDMP) at the target concession value ε′, where ϵ′i=(s′1+ϵ1i,s′2+ϵ2i,…,s′m+ϵmi)T∈Rm (i=1,2,…,r) and ε′=(ϵ′1,ϵ′2,…,ϵ′r). In other word, if the same target concession value ε for each DM is increased, then the ranking orders of X do not change.Theorem 16.
Letx∗∈X, ε1,ε1≥0. Let x∗ be an s1-concession equilibrium solution to (MGDMP) at the target concession value ε1 and an s2-concession equilibrium solution to (MGDMP) at the target concession value ε2. If ε1≤ε2, then s1≥s2.Proof.
According to assumption, we have(39)fix∗-s1-ϵ1i≤fix,∀x∈X,i=1,2,…,r.Then,(40)fix∗-s1-ϵ2i≤fix∗-s1+ϵ2i-ϵ1i-ϵ2i≤fix,∀x∈X,i=1,2,…,r.By Property 2(1), we have |s1|≥|s2|.
Theorem16 means that the bigger the target concession values for all decision-makers, the smaller the equilibrium values. But, it is very difficult to take the target concession value for each decision-maker. The decision-maker can give the approximate range of the target concession value, using fuzzy number. Now, the decision-makers give fuzzy number of the target concession value ϵ which is defined as follows: (41)ϵLi=ϵL1i,ϵL2i,…,ϵLmiT∈Rm,i=1,2,…,r,ϵMi=ϵM1i,ϵM2i,…,ϵMmiT∈Rm,i=1,2,…,r,ϵUi=ϵU1i,ϵU2i,…,ϵUmiT∈Rm,i=1,2,…,r,where (ϵLi,ϵMi,ϵUi) is called a triangular fuzzy target concession value of DMi(i=1,2,…,r). Let εL=(ϵL1,ϵL2,…,ϵLr), εM=(ϵM1,ϵM2,…,ϵMr), and εU=(ϵU1,ϵU2,…,ϵUr).Definition 17.
Givenx∗∈X, εL,εM,εU. If there is (42)fixL∗-sL-ϵLi≤fix,∀x∈X,i=1,2,…,r,andfixM∗-sM-ϵMi≤fix,∀x∈X,i=1,2,…,r,andfixU∗-sU-ϵUi≤fix,∀x∈X,i=1,2,…,r,then xL∗ is called sL-concession equilibrium solution to (MGDMP) at the target concession value εL, xM∗ is called sM-concession equilibrium solution to (MGDMP) at the target concession value εM, and xU∗ is called sU-concession equilibrium solution to (MGDMP) at the target concession value εU.By Theorem16, if εL≤εM≤εU, sL≥sM≥sU. Therefore, sL-optimal concession equilibrium solution, sM-optimal concession equilibrium solution, and sU-optimal concession equilibrium solution to (MGDMP) may not be the same x. We have an example as follows.Example 18.
A company is about to produce a new mobile phone. It needs to give consumers a chance to choose their favorite colors and shapes. So 9 colors and shapes are provided as candidates of the new mobile phone. Five decision-makers were randomly selected to score the nine candidates. The smaller the score, the more satisfied the candidate is. The favoritism to colors and shapes of the new mobile phone is described with the score band0~7: 0~3 mean like the new mobile phone with 0 meaning very much, 1 like, 2 general, and 3 a little, respectively; 4 means indifferent to the new mobile phone; 5~7 means dislike the new mobile phone with 5 a little, 6 dislike, and 7 very much. Hence, (0,0) means like the colors and shapes of the new mobile phone very much. (2,3) means generally like the colors and like a little shapes of the new mobile phone. Then, the triangular fuzzy numbers of target concession value: εL=[(0,0)T,(0,0)T,(0,0)T,(0,0)T,(0,0)T], εM=[(1,2)T,(2,2)T,(2,3)T,(1,2)T,(2,3)T], and εU=[(3,3)T,(3,3)T,(4,4)T,(4,3)T,(4,4)T] describes the degree of preference for color and shape, respectively. Let fi(x)=(f1i(x),f2i(x)),i=1,2,3,4,5, be multiattribute function of 5 decision makers. f1i(x) represents color, and f2i(x) indicates shape in Table 2.Table 2
The scores of 9 candidates and the equilibrium value under triangular fuzzy target concession values.
Solution M1 DM2 DM3 DM4 DM5 ε L ε M ε U X f 1 ( x ) f 2 ( x ) f 3 ( x ) f 4 ( x ) f 5 ( x ) s L sM sU △ ( x ) x 1 (1,5) (2,2) (3,3) (3,4) (5,7) ( 3,5 ) T ( 1,2 ) T (-1,1)T 1.2288 x 2 (2,3) (3,3) (4,2) (4,3) (3,6) ( 3,4 ) T ( 2,1 ) T (-1,0)T 1.1000 x 3 (5,5) (4,2) (3,2) (2,3) (2,2) ( 4,4 ) T ( 3,2 ) T (1,1)T 1.0198 x 4 (4,1) (2,4) (5,3) (6,4) (6,3) ( 5,2 ) T ( 4,0 ) T (1,-1)T 1.2961 x 5 (5,2) (5,5) (3,7) (4,2) (4,2) ( 4,5 ) T ( 3,2 ) T (1,1)T 1.3304 x 6 (3,7) (6,3) (2,5) (1,7) (3,6) ( 4,6 ) T ( 2,4 ) T (1,3)T 1.5067 x 7 (6,6) (2,3) (3,3) (3,5) (2,5) ( 5,5 ) T ( 4,3 ) T (2,2)T 1.2884 x 8 (7,5) (3,3) (4,2) (7,5) (5,3) ( 6,4 ) T ( 5,2 ) T (3,1)T 1.4832 x 9 (3,4) (4,2) (6,4) (6,4) (4,4) ( 5,3 ) T ( 4,1 ) T (1,0)T 1.3454We get the optimal objective valuefji∗: [(1,1),(2,2),(2,2),(1,2),(2,2)]. By Lemma 4, we obtain three concession equilibrium value (sL,sM,sU) for sL-concession equilibrium solution, sM- concession equilibrium solution, and sU-concession equilibrium solution to (MGDMP) respectively, as shown in Table 2.From Table2, we obtain three orders of the 9 candidates as follows:(43)x2⪯sLx4≺sLx3⪯sLx1≡sLx9≺sLx5≺sLx7⪯sLx6≡sLx8,x1≡sMx2≺sMx4≺sMx3≡sMx5⪯sMx9≺sMx6≺sMx7⪯sMx8,x2≺sUx1≡sUx4≺sUx9≺sUx3≡sUx5≺sUx7⪯sUx6≡sUx8.Under the triangular fuzzy numbers of target concession value, the above three ranking orders of the 9 candidates are different. The optimal concession equilibrium solutions to (MGDMP) are different too. The given target concession value can affect the optimal concession equilibrium solution to (MGDMP). Obviously, the consistency given by this example is very poor. When all the candidates have similar consistency, the orders obtained bys-optimal concession equilibrium solution will have better fairness. Of course, it is not difficult to see that if a decision-maker cannot fairly evaluate a program, it directly affects the order of the program.We choose the approach to MGDMP based on determining the weights of experts by using projection method in [5] to rank Example 18. In Table 3 the first line in the first column shows the weights of attributes determined by the experts, the other four lines in the first column show the weights of attributes determined randomly, and the second column shows the final ranking of the nine schemes. The ranking is determined via the values of projections of the approach to MGDMP based on determining the weights of experts by using projection method; i.e., the smaller the projection the better the alternative in [5]. From Table 3, it is found that the ranking results rely on the weights of attributes determined by the experts, and different weights lead to different ranking. In particular in the third line, x8 is ranked as No. 1, while from Table 2 the value of △(x8) is 1.48 so x8 should be ranked around No. 8, which is an obvious deviation from the experts’. However, the difference in our ranking is not so big. Using approach to MGDMP based on determining the weights of experts by using projection method in [5], which relies on the weights determined by the experts, might disturb the final ranking of all the schemes and sometime even brings contradictory rankings. However, our proposal avoids the determination of attributes’ weights and experts’ weights, as the value (f1i(x),f2i(x)) has contained the preference of the attributes and the experts.Table 3
Approach to MGDMP based on determining the weights of experts by using projection method [5].
Weight vector of attributes of five experts Rank all alternatives (0.50,0.50)(0.50,0.50)(0.50,0.50)(0.50,0.50)(0.50,0.50) x 4 ≺ x 8 ≺ x 9 ≺ x 3 ≺ x 5 ≺ x 7 ≺ x 1 ≺ x 2 ≺ x 6 (0.97,0.03)(0.96,0.04)(0.49,0.51)(0.80,0.20)(0.14,0.86) x4≺x2≺x9≺x8≺x6≺x5≺x7≺x3≺x1 (0.42,0.58)(0.92,0.08)(0.79,0.21)(0.96,0.04)(0.66,0.34) x8≺x4≺x6≺x1≺x9≺x2≺x5≺x3≺x7 (0.04,0.96)(0.85,0.15)(0.93,0.07)(0.68,0.32)(0.76,0.24) x1≺x6≺x4≺x8≺x9≺x7≺x3≺x5≺x2 (0.74,0.26)(0.39,0.61)(0.66,0.34)(0.17,0.83)(0.71,0.29) x2≺x7≺x6≺x3≺x4≺x5≺x9≺x8≺x1The merit of our proposal is that there is no need to determine the attributes’ weights and experts’ weights and it is easy to determine the triangular fuzzy numbers of target concession value as it is given through attributes’ value. Another merit of our proposal is that we may find the optimum solution from the infinite number of candidates, as some methods used to solve MGDMPs do not apply to the situations where there are infinite candidates as shown in [5].


## 4. Conclusion

The paper defines a news∗- optimal concession equilibrium solution and proves that when there exist optimal solutions to all the subproblems there exists the s∗- optimal concession equilibrium solution and that it is equivalent to solving a single objective programming problem. Besides, the paper proves that the s∗- optimal concession equilibrium solution is equivalent to the optimal solution with the worst weight using the linear weighted objective method. Finally, we prove that all candidate schemes can be ranked by the concession equilibrium solution. By defining the triangular fuzzy number of target concession value, the ranking order of the schemes or the optimal concession equilibrium solution can be obtained in the range of objective concession ambiguity. The numerical experiments show that the s∗- optimal concession equilibrium solution has stable ranking as compared to that by the weighted aggression method and can balance the preferences of different decision-makers about different attributes.



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