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. [
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
The possibility theory, proposed by Zadeh [
Possibility distribution: Necessity measures:
Then the possibility distribution
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
complementary: transitivity: if
In what follows, we present the different formulas of possibility measures
(i) Yuan and Qu [ Definition Definition Definition
In [
(ii) In [
(iii) In [
(iv) Wei and Tang [
If
(v) Gao [
(vi) Gao [ The first formula [ where The second formula [ Results of integrated formulas (
(vii) According to Chen [
There are some basic concepts related to the interval-valued intuitionistic fuzzy sets (IVIFS) [
if
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) Xu and Wei [
(ii) Wang et al. [
Let
(i) Zhang et al. [ First measure: where This possibility degree satisfies the following properties:
Second measure: where
(ii) Wan and Dong [
(iii) Chen [
Then for two IVIFNs the likelihood
These measures are the same as those of the possibility measures.
For a multicriteria decision making problem, let
Suppose the characteristic information of alternative
Construct the interval-valued intuitionistic fuzzy decision matrix:
Calculate the intuitionistic fuzzy decision matrix
Assign weights to criteria; we use the following standard deviation (IF-SD) formula presented in [
Compute the performance of each alternative:
Compute the likelihood matrix [
Determine the alternatives ranking order, according to the decreasing order of
This section described the data set presented in [
The following interval-valued intuitionistic fuzzy sets (IVIFSs) decision making matrix (
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
The intuitionistic fuzzy decision matrix (
Compute weights
We compute the performance of each alternative using (
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 ( For the possibility measures ( The results presented in Table As for the possibility measure ( The results presented in Table For the possibility measure ( Table For the possibility measure ( Table For possibility measure ( Table For the possibility measure ( Table
Table
Possibility degrees using (
Alternatives |
|
|
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|
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Weights | 0.2543 | 0.2472 | 0.2693 | 0.2292 |
Ranking | 2 | 3 | 1 | 4 |
Possibility degrees using (
Alternatives |
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|
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Weights | 0.2579 | 0.2510 | 0.2204 | 0.2708 |
Ranking | 2 | 3 | 4 | 1 |
Possibility degrees using (
Alternatives |
|
|
|
|
---|---|---|---|---|
Weights | 0.2543 | 0.2472 | 0.2693 | 0.2292 |
Ranking | 2 | 3 | 1 | 4 |
Possibility degrees using (
Alternatives |
|
|
|
|
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Weights | 0.2535 | 0.2531 | 0.2569 | 0.2366 |
Ranking | 2 | 3 | 1 | 4 |
Possibility degrees using (
Alternatives |
|
|
|
|
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Weights | 0.2522 | 0.2481 | 0.2662 | 0.2335 |
Ranking | 2 | 3 | 1 | 4 |
Possibility degrees using (
Alternatives |
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Weights | 0.2565 | 0.2452 | 0.2873 | 0.2110 |
Ranking | 2 | 3 | 1 | 4 |
Possibility degrees using (
Alternatives |
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Weights | 0.2511 | 0.2533 | 0.2557 | 0.2399 |
Ranking | 3 | 2 | 1 | 4 |
Alternatives ranking order for different possibility measures under IFN.
Possibility measure | Ranking | Best alternative |
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We apply possibility measures of IVIFS presented in Section
The possibility measures are applied in two cases. In each case an aggregation operator is also applied to the matrix (
Compute the comprehensive evaluation of each investment (alternative) using the geometric weighted average operator (
Each possibility measure presented in Section For the possibility degree ( Table For possibility measure ( Table For the possibility measure ( The obtained results are presented in Table For the possibility measure ( The obtained results are presented in Table
Table
Ranking IVIFSs alternatives using possibility measure (
Alternatives |
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Weights | 0.2111 | 0.2161 | 0.2540 | 0.3200 |
Ranking | 4 | 3 | 2 | 1 |
Ranking IVIFSs alternatives using possibility measures (
Alternatives |
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Weights | 0.2469 | 0.2401 | 0.2429 | 0.2640 |
Ranking | 2 | 4 | 3 | 1 |
Ranking IVFISs alternatives using possibility measure (
Alternatives |
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Weights | 0.2627 | 0.2654 | 0.2105 | 0.2614 |
Ranking | 2 | 1 | 4 | 3 |
Ranking IVFISs alternatives using possibility measure (
Alternatives |
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Weights | 0.2254 | 0.2448 | 0.2613 | 0.2686 |
Ranking | 4 | 3 | 2 | 1 |
Ranking order of alternatives for each possibility measure using IVIFS.
Possibility measures | Ranking | Best alternative |
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( |
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( |
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( |
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Using the optimal aggregated operator ( For the possibility measure ( The alternatives weight For possibility measure ( We compute the weight For possibility measures ( We compute the weights For possibility measure ( The obtained results are presented in Table
Ranking IVFSs using possibility degree (
Alternatives |
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Weights | 0.2256 | 0.2352 | 0.2779 | 0.2599 |
Ranking | 4 | 3 | 1 | 2 |
Ranking IVFSs using possibility measure (
Alternatives |
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Weights | 0.2369 | 0.2322 | 0.2557 | 0.2629 |
Ranking | 3 | 4 | 2 | 1 |
Ranking IVFSs using possibility degree (
Alternatives |
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Weights | 0.2529 | 0.2596 | 0.2235 | 0.2639 |
Ranking | 3 | 2 | 4 | 1 |
Ranking IVFISs alternatives using possibility measure (
Alternatives |
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Weights | 0.2240 | 0.2392 | 0.2695 | 0.2673 |
Ranking | 4 | 3 | 1 | 2 |
Table
Alternatives ranking order for each possibility measure using IVIFS.
Possibility measures | Ranking | Best alternative |
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( |
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( |
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( |
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( |
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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 are introduced by Atanassov [
If
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.