Multi-attribute group decision making methods with proportional 2-tuple linguistic assessments and weights

The proportional 2-tuple linguistic model provides a tool to deal with linguistic term sets that are not uniformly and symmetrically distributed. This study further develops multi-attribute group decision making methods with linguistic assessments and linguistic weights, based on the proportional 2-tuple linguistic model. Firstly, this study defines some new operations in proportional 2-tuple linguistic model, including weighted average aggregation operator with linguistic weights, ordered weighted average operator with linguistic weights and the distance between proportional linguistic 2-tuples. Then, four multi-attribute group decision making methods are presented. They are the method based on the proportional 2-tuple linguistic aggregation operator, technique for order preference by similarity to ideal solution (TOPSIS) with proportional 2-tuple linguistic information, elimination et choice translating reality (ELECTRE) with proportional 2-tuple linguistic information, preference ranking organization methods for enrichment evaluations (PROMETHEE) with proportional 2-tuple linguistic information. Finally, an example is given to illustrate the effectiveness of the proposed methods.


Introduction
Due to the complexity and uncertainty of decision making environment, some problems cannot be dealt with by precise and exact models.A possible way to solve such problems is the use of linguistic approaches 15,19,20,21,34,36,39,43 .Two different linguistic models are used in decision making, they are the models based on extension principle 9,40,48,49 and the symbolic methods 10,23,24,25,26,47 .The models based on extension principle perform the retranslation step as an approximation process to express the results in the initial term set provoking a lack of accuracy 26 .To avoid such inaccuracy, Herrera and Martínez 24 proposed the 2tuple linguistic model, and the Herrera and Martínez model has been successfully applied in a wide range of applications 2,3,11,12,14,16,33,35,46 .Although the Herrera and Martínez model has no loss of information, it only guarantees accuracy in dealing with uniformly and symmetrically distributed linguistic term sets.And in the real decision-making environment, the linguistic term sets that are not uniformly and symmetrically distributed may be used to express the preferences.In order to deal with this type of linguistic term sets, two different approaches based on linguistic 2-tuples have been presented.
(1) Herrera et al. 22 defined the concept of unbalanced linguistic term set, and proposed an unbalanced linguistic representation model to deal with unbalanced linguistic term set.This model is based on the use of linguistic hierarchy 22 and the 2-tuple linguistic representation model 26 .
(2) Wang and Hao 44,45 developed the proportional 2tuple linguistic representation model, and the Wang and Hao model is based on the concepts of symbolic proportion and the canonical characteristic values (CCVs).By defining the concept of numerical scale, Dong et al. 13 proposed an integration of the Herrera  Multi-attribute group decision making methods linguistic computational model, Massanet et al. 37 proposed a new linguistic computational model based on discrete fuzzy numbers whose support is a subset of consecutive natural numbers.Meanwhile, Rodríguez and Martínez 42 recently presented a comparative study of different symbolic linguistic computing models 21,24,44,47 in decision making.
Several decision making models based on unbalanced linguistic term sets have been presented.For example, Cabrerizo et al. 6,7 presented a consensus-based group decision making with unbalanced linguistic terms.Martínez et al. 32 applied the model with unbalanced linguistic terms to sensory evaluation.Herrera-Viedma and López-Herrera 28 developed a model of information retrieval system with unbalanced fuzzy linguistic information.Herrera-Viedma et al. 27 and Meng and Pei 41 studied the linguistic aggregation operators with unbalanced linguistic information.
However, although the Wang and Hao model can handle linguistic term sets that are not uniformly and symmetrically distributed, there are little studies regarding the decision model based on the Wang and Hao model.The main aim of this paper is to develop multi-attribute group decision making methods with linguistic assessments and linguistic weights, based on the Wang and Hao model (i.e., the proportional 2-tuple linguistic model).Our proposal will provide a novel approach to deal with the multi-attribute group decision making problems, in which decision makers can comfortably express their preferences by linguistic term sets that are not uniformly and symmetrically distributed.Particularly, the weights of the decision makers are also described via a linguistic way.
The remainder of this paper is organized as follows.In Section 2, we introduce the basic knowledge regarding the proportional 2-tuple linguistic model.Then, Section 3 proposes some new operations in proportional 2-tuple linguistic model.In Section 4, four multi-attribute group decision making methods with proportional 2-tuple linguistic assessments and weights are presented.In Section 5, an illustrative example is provided and, finally, concluding remarks are included in Section 6.

New operations in proportional 2-tuple linguistic model
In the real decision situations, there exist problems that need to assess their alternatives by linguistic term sets that are not uniformly and symmetrically distributed 21 .In this case, the proportional 2-tuple can be used.Particularly, the weights of the decision makers (or attributes) may be also needed to be assessed by linguistic information.In order to deal with this type of problems, we present some new operations in proportional 2-tuple linguistic model, that is, weighted average aggregation operator with linguistic weights, ordered weighted average operator with linguistic weights and the distance between proportional linguistic 2-tuples.L l l l where , j l S be a set of proportional 2-tuples to aggregate.w S be an associated proportional 2-tuple weighted vector.Then, the proportional 2-tuple ordered weighted average (OWA) operator with linguistic weights is defined as ...

Note 1:
In this study, we use the proportional 2-tuple weighted average operator to aggregate the linguistic assessment information.And the methods are similar if we use the OW-like operator.In addition, the fusion of the weights with the information provided of the experts

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Copyright: the authors is an important step in the decision making 8 .However, we do not discuss this issue in this study.
be two sets of proportional 2-tuples, where L is defined as .

Approaches to group decision making with linguistic assessments and linguistic weights
The main process of multi-attribute decision making is to find the best alternative(s) from all of the feasible alternatives where all the alternatives can be evaluated according to the attributes.In this section, we develop multi-attribute group decision making methods with linguistic assessments and linguistic weights, based on the proportional 2-tuple linguistic model.First, we introduce several basic concepts which will be used through the rest of this paper.Let 0 1 0 1 { , ,..., }, { , ,..., }, and be the set of alternatives to be evaluated by a set of experts [ , ,..., ] T q W w w w be the weighting vector of decision makers where .

Method based on the proportional 2-tuple linguistic aggregation operator
In the method based on the proportional 2-tuple linguistic aggregation operator, in order to obtain the best alternative(s), the proportional 2-tuple weighted average operator with linguistic weights is employed to aggregate the linguistic evaluation values of all experts using the weights of decision makers, and then the operator is again used to derive the collective overall preference value of each alternative using the weight of each attribute.This method consists of the following steps.
Step 1: Utilizing the proportional 2-tuple weighted average operator with linguistic weights, linguistic evaluation matrices 1 2 , ,..., q x x x and the weighting vector of decision makers 1 2 , ,..., q w w w are aggregated into a group comprehensive evaluation matrix Step 2: Utilizing the weighted average operator with linguistic weights to derive the collective overall preference value i L of the alternative i A .
Step 3: Rank all the alternatives ( 1, 2,..., ) i A i m and select the best one(s) in accordance with .
i L If any alternative has the highest i L value, then it is the most important alternative(s).

TOPSIS with proportional 2-tuple linguistic information
Technique for order preference by similarity to ideal solution (TOPSIS), developed by Hwang and Yoon in 1981 29 is a simple ranking method in conception and application.For the TOPSIS with proportional 2-tuple linguistic information, the optimal alternative(s) is determined by calculating the distances of every alternative from the positive-ideal solution and negative-ideal solution.It is based on the concept that the optimal alternative(s) should have the shortest distance from the positive-ideal solution and on the other side the farthest distance of the negative-ideal solution.The steps are as follows.
Step 1: This step is same as the step 1 in Section 4.1.

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Multi-attribute group decision making methods
Step 2: Determine the positive-ideal solution l and negative-ideal solution l , respectively, the positive-ideal solution 1 1 Step 4: Calculate the relative closeness degree of each alternative to the positive-ideal solution using the following equation Rank the alternatives according to the relative closeness to the ideal solution.If any alternative has the highest i value, then it is the most important alternative(s).

ELECTRE with proportional 2-tuple linguistic information
The methodologies in the elimination et choice translating reality (ELECTRE) 5 family are the most widely used outranking methods for multiple criteria decision analysis 17 .The family of ELECTRE methods includes ELECTRE I, II, III, IV and TRI 38 .In this paper, based on ELECTRE II, we propose the ELECTRE with proportional 2-tuple linguistic information, which leads to a complete ranking of the alternatives by means of three thresholds for concordance conditions and two thresholds for discordance conditions.The steps are as follows.
Step 1: This step is same as the step1 in Section 4.1.
Step 2: Determination of the concordance conditions. (i {1,2,..., }, J n let the following quantities be defined: and kj l represents the performance of alternative k A in terms of criterion j C . (ii) Computing the concordance index.The concordance index for a pair of alternatives i A and k A measures the strength of the hypothesis that alternative i A is at least as good as alternatives k A .The concordance condition for the pair ( , ) ) ) Where a is a number reflecting a minimum acceptable level of concordance, usually set at 0.5 , j h is the weight of the th j criterion, ik I measures the strength of the hypothesis that alternative i A is good at alternative k A .
Step 3: Determination of the discordance conditions.The discordance condition is introduced to handle the criteria for which i A is not preferred to k A ; for those criteria, the decision maker specifies the maximum level of discordance which can be tolerated.Assuming the thresholds of each criterion are 0 * 0 * , ( ) a and ( , ) Step 5: According to the strong preference relationship and weak preference relationship, we can get a forward ranking ( ) i V A and a reverse ranking ( ).
Step 6: Rank all the alternatives ( 1,2,..., ) i A i m and select the best one(s) in accordance with ( ) i Z A .

PROMETHEE with proportional 2-tuple linguistic information
Besides ELECTRE methods, preference ranking organization methods for enrichment evaluations (PROMETHEE) methods are the other most extensively used outranking relation theory techniques 4 .The family of PROMETHEE methods is suitable for ranking and selecting from among a finite set of alternative actions, especially PROMETHEE I and II 1,18 .Based on PROMETHEE II, the PROMETHEE with proportional 2-tuple linguistic information is intended to provide a complete ranking of a finite set of feasible alternatives from the best to the worst, and the basic principle of this method is based on the pair-wise comparison of alternatives along each selected criterion.The steps are as follows.
Step 1: This step is same as the step1 in Section 4.1.
Step2: Determination of performance differences, the performance difference between each pair of alternatives with respect to each criterion is calculated as follows: ( , ) ( ) ( ) where and show the performance of alternatives i A and k A , respectively, with regard to criterion j , and ( , ) d A A denotes the difference between these performances.
where ( , ) j i k P A A denotes the preference of alternative i A over alternative k A with respect to criterion j as a function of ( , ).Copyright: the authors linear preference) of preference functions for each criterion and certain preferential parameters must be specified for each function, for the sake of simplicity, this study uses criterion with linear preference as the preference function for each criterion.The criterion with linear preference ( ) H d is defined as follows: where m is a preferential parameter that may be determined by decision makers.
Step 4: Calculation of the global preference indices, for each pair of alternatives, an aggregated preference index is calculated as follows: where ( , ) A A denotes an overall preference index which reveals the intensity of the preference for i A over k A .
Step 5: Define the leaving flow, entering flow and net flow of the alternatives.
Leaving flow: ( ) ( , ) Entering flow: ( ) ( , ) Then, a complete ranking of the alternatives can be obtained by the net flows, and the alternative with the highest net flow is superior.

Illustrative example
In order to show how the methods with proportional 2-tuple linguistic assessments and linguistic weights work in practice let us consider the following example.
Suppose that a company wants to enhance its IT construction.After preliminary screening, four feasible governance programs 1 2 3 4 , , , A A A A remain for further evaluation.An expert committee of four decision Assume that the decision makers express their preferences by means of linguistic assessments from the ordered linguistic term set of seven labels:

S S VB Very Bad S B Bad S MB Medium Bad S M Medium S MG Medium Good S G Good S VG Very Good
We can express the weights of decision makers (or attributes) by means of linguistic assessments from the ordered linguistic term set of five labels

N S N I S M VI
The initial evaluation matrices k X provided by the decision makers ( 1, 2,3, 4)   k D k are listed as Tables 1-4.Table 5 defines both the CCV and trapezoidal fuzzy numbers in [0,1] of each label in S .Table 6 defines both the CCV and trapezoidal fuzzy numbers in [0,1] of each label in S .The information about the weights of decision makers and attributes are known as follows Utilizing Eq. ( 1) to aggregate linguistic rating values of four experts for each alternative and then we can obtain the group comprehensive evaluation matrix L , as shown in Table 7.
In the following, we shall utilize the proposed four approaches in this paper getting the most desirable alternative(s).

Z A S S Z A S S Z A S S Z A S S
Ranking all the alternatives ( A A A A , and thus the most desirable alternative is 4 A . (2) Ranking the alternatives: TOPSIS with proportional 2-tuple linguistic information According to the group comprehensive evaluation matrix L , define the positive-ideal solution and negative-ideal solution as the positive-ideal solution: Using the group comprehensive evaluation matrix L , we can obtain the preference difference ( , ) 12).After calculating the preference difference ( , ) j i k d A A , the preference function and the threshold of each criterion can be obtained.Table 13 shows the preference function and the threshold of each criterion.
By following the steps described in Section 4.4, one can obtain the multi-attribute preference index ( , )   i k A A (Table 14).
Using Eqs.(10) and (11), the leaving flow ( ) i A , entering flow ( ) i A , and net flow ( ) i A can be derived (Table 15).Therefore, the ranking of four alternatives is 4 3 1 2 A A A A , and obviously 4 A is the best among all alternatives.

Conclusion, limitation and future research
In decision-making problems, the linguistic term sets which are not uniformly and symmetrically distributed may be used to express decision makers opinions.The proportional 2-tuple linguistic model provides a way to deal with this type of linguistic term sets.This study further develops multi-attribute group decision making methods with linguistic assessments and linguistic weights, based on the proportional 2-tuple linguistic model.The main points presented are as follows: (1) Some new operations in proportional 2-tuple linguistic model are defined, they are the weighted average aggregation operator with linguistic weights, ordered weighted average operator with linguistic weights and the distance between proportional linguistic 2-tuples.
(2) Four multi-attribute group decision making methods are presented.They are the method based on the proportional 2-tuple linguistic aggregation operator, TOPSIS with proportional 2-tuple linguistic information, ELECTRE with proportional 2-tuple linguistic information, and PROMETHEE with proportional 2tuple linguistic information.
Meanwhile, the methodologies based on unbalanced linguistic term sets 22,28 can deal with the linguistic term sets which are not uniformly and symmetrically distributed.So, we argue that our study has a limitation: a comparative analysis between our model and the methods based on unbalanced linguistic term sets need to be proposed.In the future, we will pursue this topic.Moreover, we will discuss some possible applications of our model (e.g., new product development 31 and emergency management evaluation 50 ).
and Martínez model and the Wang and Hao model.Moreover, in order to extend the flexibility of the International Journal of Computational Intelligence Systems, Vol. 7, No. 4 (August 2014), 758-770 Co-published by Atlantis Press and Taylor & Francis Copyright: the authors 758 Downloaded by [Swinburne University of Technology] at 07:52 07 January 2015 an associated proportional 2-tuple weighted vector.Then, the proportional 2-tuple weighted average operator with linguistic weights is defined as Then the decision making matrix of k th decision maker can be expressed as to the concordance thresholds and three discordance sets, the strong and weak outranking relations are defined as follows:(i) i A strongly outranks k A if (a)The concordance is high and the discordance is average.(b) The concordance is average and the discordance is low.i.e., Both concordance and discordance are low.(b) Both concordance and discordance are average.i Although decision makers can select many different types (e.g., quasi criterion, level criterion, criterion with kj l ij l Co-published by Atlantis Press and Taylor & Francis

A
represents the outranking character of i A over all training patterns, and ( ) i A represents the outranked character of i A by all training patterns, then the net flow of alternative i A can be calculated as follows

1 C 4 C 5 C
D has been formed to conduct the evaluation and to select the most suitable IT governance program(s) for the company.Five criteria in accordance with the characteristics of IT governance are considered: IT Business applications; IT investment and prioritization.

1
Co-published by Atlantis Press and Taylor & FrancisCopyright: the authors 764Downloaded by [Swinburne University of Technology] at 07:52 07 January 2015 Co-published by Atlantis Press and Taylor & FrancisCopyright: the authors 44finition 244: Let S and S and CCV on S as previously.

Table 5 .
The CCV and trapezoidal fuzzy numbers in [0,1] of each label in S

Table 6 .
The CCV and trapezoidal fuzzy numbers in [0,1] of each label in S

Table 2 .
The linguistic evaluation matrix 2 X provided by the decision maker 2

Table 4 .
The linguistic evaluation matrix 4 X provided by the decision maker 4

Table 9 .
Having calculated the concordance index and the thresholds of five criteria, we get the strong preference relationship and weak preference relationship by following the steps in Section 4.3, that is, 1

Table 7 .
The group comprehensive evaluation matrix L

Table 8 .
The concordance index for each pair of alternatives

Table 9 .
The thresholds of each criterion is presented in Table10.
i Y A

Table 10 .
Direct ranking is obtained, and the reverse ranking is the same as the direct ranking.
i Y A

Table 11 .
Final Ranking

Table 13 .
The preference function and threshold of each criterion

Table 15 .
The leaving, entering and net flows.