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.
Multicriteria decision making methods have been developed widely using fuzzy sets and their generalizations. Park et al.  extended TOPSIS method for decision making problems under interval-valued intuitionistic fuzzy environment. Park et al.  generalized the concepts of correlation coefficient of intuitionistic fuzzy sets into interval-valued intuitionistic fuzzy sets. Ye  proposed weighted correlation coefficients using entropy weights under interval-valued intuitionistic fuzzy environment to rank alternatives. Zhang and Yu  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 [5–8] 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  extended the possibility method of interval-valued numbers defined by [14, 15] to intuitionistic fuzzy sets  and defined a possibility formula to compare two intuitionistic fuzzy numbers (IFNs). In addition, Xu and Da  presented a possibility formula to compare two interval fuzzy numbers and applied possibility measures of interval-valued intuitionistic fuzzy numbers to multicriteria decision making. Gao  presented four possibility measures and proved their equivalence. Liu and Lv  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.
2. Possibility Theory
The possibility theory, proposed by Zadeh , defines a pair of dual set functions: possibility and necessity measures. Therefore a possibility degree quantifies the extent 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 which characterizes the uncertain membership of an element in a (well-defined) known class .(i): the possibility degree that .(ii): the certainty degree (necessity) that and complement of event . Then the possibility distribution can be easily recovered from the possibility measure . (i)Possibility distribution: .(ii)Necessity measures: . If , is certainly true, and if , is not certain ( is still possible).
2.1. Possibility Measures to Compare Intuitionistic Fuzzy Numbers (IFNs) Existing in Literature
The comparison between IFNs (see appendices) can be solved by using the possibility degree formula of the interval values. Thus the possibility measures have to satisfy the following properties .
Let , , and ; then we have the following:(i);(ii)complementary: ;(iii)transitivity: if and , then ;(iv), if and only if ;(v), if and only if . In what follows, we present the different formulas of possibility measures with and being two intuitionistic fuzzy numbers and , .
In [14, 17], the authors have proved the equivalence of the above three formulas. In  these formulas are used to compare two countries. Reference  used possibility degree formula (3) to rank interval rough numbers.
(iii) In  the possibility measure is shown as follows:
(iv) Wei and Tang  generalized possibility measure of interval-valued numbers to intuitionistic fuzzy sets.
If and are different from zero, the possibility measure of is
(vi) Gao  determined the equivalence between formulas (1), (2), (4), and (8) and integrated the following possibility measures:(a)The first formula : where .(b)The second formula :(c)Results of integrated formulas (9) and (10):
(vii) According to Chen , the possibility measure of the event is presented as follows:where and .
3. Interval-Valued Intuitionistic Fuzzy Sets
There are some basic concepts related to the interval-valued intuitionistic fuzzy sets (IVIFS) . Let be a nonempty set of the universe. An IVIFS is defined as , where and denote the intervals of the membership degree and nonmembership degree of the element , satisfying the following:(i),(ii) and for all ,(iii)if and , then is reduced to an IFS.
3.1. 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.
(ii) Wang et al.  defined the optimal aggregated interval-valued intuitionistic fuzzy sets using this formula:
3.2. Possibility Measures to Compare Interval-Valued Intuitionistic Fuzzy Number (IVIFN) Existing in Literature
Let and be two interval-valued intuitionistic fuzzy numbers in , which is the set of all IVIFNs [2, 5, 31, 32]. is the possibility measure of two interval-valued intuitionistic fuzzy numbers. Let us present the existing possibility measures.
(i) Zhang et al.  defined two possibility measures of two interval-valued intuitionistic fuzzy numbers as follows:(a)First measure: where and which represents the performance on the mean value of its membership degree. This possibility degree satisfies the following properties:(1);(2);(3);(4).(b)Second measure: where gives the decision makers’ preference on membership degree or nonmembership degree. When the decision maker is optimal whereas when the decision maker is pessimistic. Then, the below properties are checked:(1);(2) and ;(3) and ;(4).
(iii) Chen  defined a lower likelihood and an upper likelihood on IVIFSs aswhere and , andwhere and .
Then for two IVIFNs the likelihood is defined as follows:
These measures are the same as those of the possibility measures.
4. MCDM Based on Possibility Degree of Interval-Valued Intuitionistic Fuzzy Numbers
For a multicriteria decision making problem, let be the set of alternatives and the set of criteria. is the weight vector of criteria , where and .
Suppose the characteristic information of alternative over criterion is represented by interval-valued intuitionistic fuzzy number , where represents the fuzzy membership degree of the alternative over criterion and represents the fuzzy nonmembership degree of the alternative over criterion . Then the decision matrix is obtained asThe 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  following the steps below.
Step 1. Construct the interval-valued intuitionistic fuzzy decision matrix: .
Step 2. Calculate the intuitionistic fuzzy decision matrix to derive and , and is the transformed IFN decision matrix obtained from using formulas (22):
Step 4. Compute the performance of each alternative:
Step 5. Compute the likelihood matrix . To compare between tow interval fuzzy numbers, we propose to use a possibility measure instead of the formula used in  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  defined as
5. Illustrative Example
5.1. 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 . The fund manager should evaluate each investment considering four criteria: risk , growth , sociopolitical issues , and environmental impacts . The fund manager is satisfied once he provides his assessment of each alternative on each criterion.
Step 1. The following interval-valued intuitionistic fuzzy sets (IVIFSs) decision making matrix (27) presents the relationship between criteria and alternatives of data set as follows: 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 , where the interval 42–48%  reflects that the fund manager has an excellent opportunity to respect the risk criterion (), although the interval 40–50% does not really represent an excellent choice of for risk ().
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 .(a)For the possibility measures (4) and (5), the obtained possibility matrix is The results presented in Table 2 show is the best alternative and ranks first.(b)As for the possibility measure (6), the obtained possibility matrix is The results presented in Table 3 show is the best alternative and ranks first.(c)For the possibility measure (7), the obtained possibility matrix is Table 4 shows that the best alternative is achieving the first rank.(d)For the possibility measure (9), the obtained possibility matrix is Table 5 shows that the best alternative is that ranks first.(e)For possibility measure (10), the obtained possibility matrix is Table 6 shows that the best alternative that ranks first is .(f)For the possibility measure (12), the obtained possibility matrix is Table 7 shows that the best alternative is 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 and the worst alternative . However using formulas (1), (2), (3), (6), (7), (8), (9), (10), and (12) the best alternative is and the worst alternative is . 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 , but they do not produce the same result.
5.2. 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 , , , and and four criteria. We use the IVIF matrix of alternatives (27) and the following criteria’s weight: given in .
5.2.1. 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:
Step 2. Each possibility measure presented in Section 3.2 is applied to the obtained IVIFNs: , , , . (a)For the possibility degree (16), the obtained possibility matrix is Table 9 presents the obtained results and shows that the best alternative is .(b)For possibility measure (15), the obtained possibility degree matrix is Table 10 presents the obtained results and shows that the best alternative is .(c)For the possibility measure (17), the obtained possibility matrix is The obtained results are presented in Table 11 showing that the best alternative is .(d)For the possibility measure (20), the obtained possibility matrix is The obtained results are presented in Table 12 showing that the best alternative is .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 . However (17) provides the best alternative .
5.2.2. Case 2: Application of Optimal Aggregated Interval-Valued Intuitionistic Fuzzy Sets (14)
Using the optimal aggregated operator (14) to IVIF decision matrix, we obtain four interval-valued intuitionistic fuzzy numbers (IVIFNs) representing the alternatives as follows: (a)For the possibility measure (16), the obtained possibility degree matrix is 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 .(b)For possibility measure (15), the obtained possibility degree matrix is 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 which ranks first.(c)For possibility measures (17), the obtained possibility degree matrix is 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 .(d)For possibility measure (20), the obtained possibility matrix is The obtained results are presented in Table 17 showing that the best alternative is .
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 . However (16) and (20) provide the best alternative . We note that the latter is the worst alternative using (17).
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.
Intuitionistic Fuzzy Sets
Intuitionistic fuzzy sets are introduced by Atanassov  who defined a degree of membership , a degree of nonmembership , and a degree of hesitation of an element of an IFS.
denotes an intuitionistic fuzzy number if and are fuzzy numbers with , where denotes the complement of .
If is a discourse universe and a set in , then with the conditions , , , and . Also for each , .
The authors declare that they have no competing interests.
The authors would like to acknowledge the financial support of this work by grants from General Direction of Scientific Research (DGRST), Tunisia, under the ARUB program. They would like to thank Mr. Abdelmajid Dammak for his proofreading and correction of the English of the paper.
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