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

This paper introduces a concession equilibrium solution without weighted aggregation operators to multiattribute group decisionmaking problems (in shortMGDMPs). It is of practical significance for all decision-makers to find an optimal solution toMGDMPs 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.


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.
(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 decisionmaking with unknown experts and attribute weights under interval-valued intuitionistic fuzzy environment.
(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 sconcession 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 twostage 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): -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 decisionmaking problem between the retailer and the manufacturer using the 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 , and the -optimal concession equilibrium solution is the minimum concession given.The -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 -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  *optimal concession equilibrium solution, the  * -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 -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  *optimal concession equilibrium solution is proposed, where  * is a vector, while the concept --optimal concession solution defined in [30] cannot solve (MGDMP); (2) the  *optimal concession equilibrium solution is a robust solution in all the weighted aggression sets; (3) a new triangular-fuzzyconcession ranking method is proposed based on the  *optimal concession equilibrium solution, and the rankings in the numerical experiments show stability under different concession values.

s * -Optimal Concession Equilibrium
Solution to MGDMP then  * is called -concession equilibrium solution to (MGDMPs) at the value .The || = ∑  =1   is called an equilibrium value of (MGDMP) to  * . is called an equilibrium point of (MGDMP) to  * .The set of all equilibrium values || of all -concession equilibrium solutions  * ∈  to (MGDMP) is denoted as .If  * is the  * -concession equilibrium solution to (MGDMP) and | * | is the minimum of the set , then  * is called the  * -optimal concession equilibrium solution to (MGDMP) at the target concession value .| * | is called the optimal equilibrium value, and obviously the optimal equilibrium value is unique. * is called the optimal equilibrium point, and the equilibrium point  * of  * is not always unique.This differs from the  * -optimal concession equilibrium solution to single attribute group decision-making problem (Jiang, 2018) [30].Furthermore, to solve the infinite-numberof-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 .Let  * be  * -optimal concession equilibrium solution to (MGDMP) at the value .
(3) If  has only a finite number of solutions, then the  *optimal concession equilibrium solution to (MGDMP) exists.
Property 2 indicates that, with the given   ,  * is the minimum concession value among all the candidates, so its corresponding  * -optimal concession equilibrium is the best solution in all equilibrium values.
When the different target concession values, i.e., different , are given, different  * -optimal concession equilibrium solutions are obtained, as shown in the following example.
From the above example, it is understood that when  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.
So, we have Therefore, by Definition 1, the conclusion of the theorem is true.
Define the following optimization problem: Theorem 5. Assume that there is an optimal solution to (   ) ( = 1, 2, . . ., ).en  * is  * -optimal concession equilibrium solution to (MGDMP) at the value  if and only if ( * ,  * ) is an optimal solution to (S).
Proof.First, assume that if ( * ,  * ) is an optimal solution to (S), then for any  ∈ , we have Now, assume that  * is an  * -optimal concession equilibrium solution to (MGDMP) at the value .Then by Definition 1 it is known that ( * ,  * ) is a feasible solution to (S).Let (, ) be an optimal solution to (S).By the above proof,  is an -optimal concession equilibrium solution to (MGDMP) at the value .Therefore, || = | * |.So,  * ≡   and ( * ,  * ) is an optimal solution to (S).
Theorem 5 points out that if  is -concession equilibrium solution, then Furthermore, Theorem 5 gives that if there exists  *optimal concession equilibrium solution, the optimal solution for (   ) must exist.Then we have the following.
Proof.By the assumption, there is an optimal solution to each (   ).By Lemma 4, we have  ̸ = 0. We prove that  is close.Assume that a sequence {  } ⊂  converges to  * .For  = 1, 2, . .., let  *  ∈  be an   -concession equilibrium solution to (MGDMP).Because  is compact, the sequence { *  } has a convergent subsequence.Without loss of generality, let  *  →  * ∈ .By Definition 1, we have Let  → +∞, and then we have It is obvious that the problem (S)  is equivalent to the problem (S), and the feasible set of the problem (S)  is compact too.Therefore, there exists the optimal solution ( *  ,  *  ) to (S)  , then ( *  ,  *  ) is also the optimal solution to the problem (S).By Theorem 5, the conclusion is true.
To solve (S), (   ) must be solved first, which is quite difficult.Therefore, we have the following Theorem 7, where solving a single objective programming problem () obtains the  * -optimal concession equilibrium solution to (MGDMP).
Based on Theorem 7, we have the following corollary.
Proof.For a fixed , the problem max ∈Λ (, ) is a linear programming: The dual problem of Λ() is Based on the strong duality theorem, there exists the optimal solution to the problem Λ() and the problem (), and the optimal objective values are equal at their optimal solutions.Let  * be an optimal solution to Λ() and  * an optimal solution to (), then | * | = (,  * ) = max ∈Λ (, ).Therefore, the conclusion of theorem is true.
From the viewpoint of robustness, Theorem 9 means the  * -optimal concession equilibrium solution is the robust solution for the decision-makers under the worst weights in Λ.
|| is the sum of target concession values of all   ( = 1, 2, . . ., ).So, we have the following conclusion.(34) We know that When  = [(0,0)  , (0, 0)  ], 0 is the (1, 1)  -optimal concession equilibrium solution to the problem at the concession value .This solution gives the minimum equilibrium value of each decision-maker's individual objective.Define a weighted function by By Theorem 9, the optimal solution to min 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 , an optimal solution to min ∈ (, ) is  * = −1.When  1 <  2 , an optimal solution to min ∈ (, ) is  * = 1.But, when  1 =  2 , no optimal solution to min ∈ (, ) exists.On other hand, the deviation function △() is minimum at  * = 0, but maximum at  * = 1 or  * = −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.

Ranking and Fuzzy Target Concession Value of MGDMP
Now, we define the ranking in the set  of -concession equilibrium solution to (MGDMP) at the value .Deviation of equilibrium value  of -concession equilibrium solution to (MGDMP) at the value  is defined as () represents the difference among attribute values.
Definition .Let  1 ,  2 ∈ , and  ≥ 0. Let  1 be an  1concession equilibrium solution to (MGDMP) at the target concession value  and  2 be an  2 -concession equilibrium solution to (MGDMP) at the target concession value .
Obviously, the set  is a serially ordered set about the order ≺  , ⪯  or ≡  .Theorem 13.Let  1 ,  2 ∈ , and  ≥ 0. Let  1 be an  1 -concession equilibrium solution to (MGDMP) at the target concession value  and  2 be an  2 -concession equilibrium solution to (MGDMP) at the target concession value .If   ( 1 ) ≤   ( 2 ),  = 1, 2, . . ., , en  1 Proof.According to assumption, we have By Definition 1, we have Theorem 13 shows that the concession equilibrium solution must be nondominated for all decision-makers.
From Table 2, we obtain three orders of the 9 candidates as follows: 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 by -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.
3: Approach to MGDMP based on determining the weights of experts by using projection method [5].
The 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].

Conclusion
The paper defines a new  * -optimal concession equilibrium solution and proves that when there exist optimal solutions to all the subproblems there exists the  * -optimal concession equilibrium solution and that it is equivalent to solving a single objective programming problem.Besides, the paper proves that the  * -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  * -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.
Definition .Given  * ∈ ,   ,   ,   .If there is   -concession equilibrium solution to (MGDMP) at the target concession value   ,  *  is called   -concession equilibrium solution to (MGDMP) at the target concession value   , and  *  is called   -concession equilibrium solution to (MGDMP) at the target concession value   .