An Exhaustive Study of Possibility Measures of Interval-Valued Intuitionistic Fuzzy Sets and Application to Multicriteria Decision Making

This work is interested in showing the importance of possibility theory in multicriteria decision making (MCDM). Thus, we apply some possibility measures from literature to the MCDM method using interval-valued intuitionistic fuzzy sets (IVIFSs). These measures are applied to a decision matrix after being transformed with aggregation operators. The results are compared between each other and concluding remarks are drawn.


Introduction
Multicriteria decision making methods have been developed widely using fuzzy sets and their generalizations.Park et al. [1] extended TOPSIS method for decision making problems under interval-valued intuitionistic fuzzy environment.Park et al. [2] generalized the concepts of correlation coefficient of intuitionistic fuzzy sets into interval-valued intuitionistic fuzzy sets.Ye [3] proposed weighted correlation coefficients using entropy weights under interval-valued intuitionistic fuzzy environment to rank alternatives.Zhang and Yu [4] extended TOPSIS method using cross entropy and generalized an MCMD approach with interval-valued intuitionistic fuzzy sets.The possibility theory has also been applied in many research topics.To rank alternatives, a comparison between the obtained matrix and aggregated IVIFS is mandatory.This is applied by an accuracy function in [5][6][7][8] or a possibility measure.To apply possibility measures to a decision matrix of IVIFS, an aggregation is needed.Some aggregation methods under interval-valued intuitionistic fuzzy information are given in [7,9,10].In the same way, Xu in [11,12] developed some aggregation operators: intuitionistic fuzzy weighted averaging operator, intuitionistic fuzzy ordered weighted averaging (IFOWA), intuitionistic fuzzy weighted averaging (IFWA) operator, and intuitionistic fuzzy hybrid aggregation (IFHA) operator.Wei and Tang [13] extended the possibility method of intervalvalued numbers defined by [14,15] to intuitionistic fuzzy sets [16] and defined a possibility formula to compare two intuitionistic fuzzy numbers (IFNs).In addition, Xu and Da [14] presented a possibility formula to compare two interval fuzzy numbers and applied possibility measures of intervalvalued intuitionistic fuzzy numbers to multicriteria decision making.Gao [17] presented four possibility measures and proved their equivalence.Liu and Lv [18] used possibility measures for the ranking of interval fuzzy numbers.
Our aim is to present and compare several possibility measures under intuitionistic fuzzy and interval-valued intuitionistic fuzzy environment.The remaining of this paper is organized as follows: in Section 2, the possibility theory and measures of IFS are detailed.In Section 3, some preliminaries about IVIFS and the possibility measures are introduced.In Section 4, an IVIF MCDM method is adopted.In Section 5, aggregation operators and possibility measures are applied and their results are compared.In Section 6, the conclusion is drawn.

Possibility Theory
The possibility theory, proposed by Zadeh [19], defines a pair of dual set functions: possibility and necessity measures.Therefore a possibility degree ∏() quantifies the extent 2 Advances in Fuzzy Systems an event  is plausible, and the necessity degree () quantifies the certainty of .The model of imperfect data in the possibility theory is the possibility distribution  () ∈ [0, 1] which characterizes the uncertain membership of an element  in a (well-defined) known class .
Then the possibility distribution  can be easily recovered from the possibility measure ∏.
(ii) In [4,24,25], the authors defined the possibility measures and called them likelihood measures as follows: where   =  + −  − and   =  + −  − .(iii) In [15] the possibility measure is shown as follows: (iv) Wei and Tang [13] generalized possibility measure of interval-valued numbers to intuitionistic fuzzy sets.

Interval-Valued Intuitionistic Fuzzy Sets
There are some basic concepts related to the interval-valued intuitionistic fuzzy sets (IVIFS) [29].Let  = { 1 ,  2 , . . .,   } be a nonempty set of the universe.An IVIFS Ã is defined as and []  Ã(  ), ]  Ã(  )] denote the intervals of the membership degree and nonmembership degree of the element   ∈ Ã, satisfying the following:

Aggregation Operators Existing in Literature.
The aggregation operators are necessary to reduce the IVIFS values; thus, we can compare them using an accuracy function or a possibility measure.In the following, we present two existing aggregation operators.
(i) Zhang et al. [7] defined two possibility measures of two interval-valued intuitionistic fuzzy numbers as follows: (a) First measure: where and  ∈ [0, 1] which represents the performance on the mean value of its membership degree.
This possibility degree satisfies the following properties: (1) (b) Second measure: where  ∈ [0, 1] gives the decision makers' preference on membership degree or nonmembership degree.
When  ≥ 0.5 the decision maker is optimal whereas when  < 0.5 the decision maker is pessimistic.Then, the below properties are checked:

Advances in Fuzzy Systems
(ii) Wan and Dong [33] defined possibility measure by the following formula: where ) can be calculated using ( 4).
(iii) Chen [28] defined a lower likelihood  − and an upper likelihood  + on IVIFSs as where and where Then for two IVIFNs the likelihood  4 (α 1 ≥ α2 ) is defined as follows: These measures are the same as those of the possibility measures.

MCDM Based on Possibility Degree of Interval-Valued Intuitionistic Fuzzy Numbers
For a multicriteria decision making problem, let  = { 1 ,  ) .( 21) The ranking of the alternatives in the multicriteria decision making can be solved using the possibility measure of interval-valued intuitionistic fuzzy numbers.We chose to adopt a modified version of the method described in [4] following the steps below.
Step 4. Compute the performance of each alternative: Advances in Fuzzy Systems 5 Step 5. Compute the likelihood matrix [25].To compare between tow interval fuzzy numbers, we propose to use a possibility measure instead of the formula used in [4] to obtain a possibility matrix.Therefore, each possibility measure presented in Section 2.1 is applied, and all the achieved results are compared in Section 5.
Step 6. Determine the alternatives ranking order, according to the decreasing order of   [25] defined as

Application of Possibility Measure of IFS in Decision
Making Problem.This section described the data set presented in [4,9] to evaluate the four potential investment opportunities  = {1, 2, 3, 4}.The fund manager should evaluate each investment considering four criteria: risk (1), growth (2), sociopolitical issues (3), and environmental impacts (4).The fund manager is satisfied once he provides his assessment of each alternative on each criterion.
Each element of this matrix is presented with IVIFS, giving the fund manager's satisfaction or dissatisfaction degree with an alternative.The element represented for the first alternative [0.42, 0.48], [0.4,0.5], where the interval 42-48% [4] reflects that the fund manager has an excellent opportunity to respect the risk criterion (1), although the interval 40-50% does not really represent an excellent choice of 1 for risk (1).
Step 2. The intuitionistic fuzzy decision matrix (28) is obtained using ( 22 ) . ( Step 3. Compute weights  of the criteria based on (28) and using (23): Step 4. We compute the performance of each alternative using (25), to obtain the interval fuzzy number: ( Step 5.In this step, we apply each possibility measure and determine the achieved results.These are then compared to define the differences between them. Using the possibility measures ( 1), ( 2), (3), and ( 8), we achieved the results presented in Table 1.The best alternative to be ranked first is 3. ) .
The results presented in The results presented in Table 3 show 3 is the best alternative and ranks first.(c) For the possibility measure (7) Table 5 shows that the best alternative is 3 that ranks first.(e) For possibility measure (10) 2), (3), and (8) 3 > 1 > 2 > 4 3 (4) and ( 5) Table 6 shows that the best alternative that ranks first is 2.
Table 7 shows that the best alternative is 3 that clearly ranks first.
Table 8 presents a comparison of the obtained results applying different possibility measures under intuitionistic fuzzy sets and shows the alternatives ranking results.We remark that the possibility measures (4) and ( 5) gave the same best alternative 4 and the worst alternative 3.However using formulas (1), ( 2), ( 3), ( 6), ( 7), ( 8), ( 9), (10), and (12) the best alternative is 3 and the worst alternative is 4.These results show that the measures (4) and ( 2) are different, although they are demonstrated to be equivalent (the operators lead to value 1) in [17], but they do not produce the same result.

Application of Possibility Measures of IVIFS in Decision
Making Problem.We apply possibility measures of IVIFS presented in Section 3.2 to rank IVIFS data sets described in Section 5.It is worth reminding that there are four alternatives 1, 2, 3, and 4 and four criteria.We use the IVIF matrix of alternatives (27) and the following criteria's weight:   = [0.13,0.17, 0.39, 0.31] given in [9].

Case 1: Application of Interval-Valued Intuitionistic Fuzzy
Weighted Geometric (IVIFWG) Operator (13).The possibility measures are applied in two cases.In each case an aggregation operator is also applied to the matrix (27).
Step 1. Compute the comprehensive evaluation of each investment (alternative) using the geometric weighted average operator (13) to aggregate the evaluation of each alternative.Thus, we transform the IVIFS decision matrix to IVIFs for each alternative presented as follows:  ) .
The obtained results are presented in Table 11 showing that the best alternative is 2.
(d) For the possibility measure (20) The obtained results are presented in Table 12 showing that the best alternative is 4.
Table 13 presents all the obtained results applying different possibility methods using the interval-valued intuitionistic fuzzy sets and shows the alternatives ranking results.We remark that the possibility formulas (15), (16), and (20) provide the same best alternative 4.However (17) provides the best alternative 2.
The alternatives weight   is computed using (26) and then ranked in a decreasing order.The results are displayed in Table 14 showing that the best alternative that ranks first is  3 .
We compute the weight   of the alternative using (26) and we rank   in a decreasing order.The results are shown in Table 15 revealing that the best alternative is 4 which ranks first.
(45) We compute the weights   of the alternatives using (26) and we rank   in a decreasing order.The results are displayed in Table 16 showing that the best alternative that rank first is 4. ) . (46) The obtained results are presented in Table 17 showing that the best alternative is 3.
Table 18 presents the results of all applied possibility measures using the interval-valued intuitionistic fuzzy sets and shows the alternatives ranking results.We remark that the possibility formulas ( 15) and ( 17) provide the same best alternative 4.However ( 16) and ( 20) provide the best alternative 3.We note that the latter is the worst alternative using (17).

Conclusion
In this study, we presented different formulas of possibility measures.The formulas exist in literature with IFN and IVIFN.We also presented an MCDM method from the literature.We gave an illustrative examples for applications of different possibility measures and compared their results.First we used an MCDM matrix with intuitionistic fuzzy numbers and then an MCDM matrix with IVIFNs.The values of the latter are aggregated with an aggregation operator in two cases.In each case a different aggregation operator was used.Thus, the appropriate possibility measures are applied.
The results show that the ranked alternatives can be different for each possibility measure, even though some of these measures have already been demonstrated to be equivalent in the literature.

Table 8 :
Alternatives ranking order for different possibility measures under IFN.

Table 13 :
Ranking order of alternatives for each possibility measure using IVIFS.

Table 18 :
Alternatives ranking order for each possibility measure using IVIFS.