An Application of Neutrosophic Set to Relative Importance Assignment in AHP
Abstract
:1. Introduction
- The problem of preferential uncertainty in relative importance assignment for AHP is considered;
- Propose DSVNN as a model of assignment; and
- Illustrate the applications of DSVNN for such a purpose.
2. Preliminaries
2.1. Neutrosophic Set
2.2. Single Valued Neutrosophic Set (SVNS)
2.3. Discrete Single Valued Neutrosophic Number (DSVNN)
2.4. Similarity Measure
- ;
- ;
- if and only if ;
- If , then and ;
- 5.
- , if and ;
2.5. Deneutrosiphication
- Synthesization: It is the transformation fNF applied to convert a neutrosophic set into a fuzzy set . Accordingly, it is a mapping:
- 2.
- Defuzzifization: From the synthesization in the first step, a fuzzy set is obtained, where the membership function is derived using the definition (4). Defuzzification is a process of converting a fuzzy set into a single crisp value. Many methods have been proposed in literature to perform defuzzification. The commonly used defuzzification methods are weighted average method, centroid method, and mean-max method [23]. Following the centroid method [24], the defuzzified value is calculated by the formula
3. Proposed Methodology
- A decision maker would like to assign the relative contribution to the goal of the i-th criterion compared to the j-th one;
- The decision maker is not certain which single scale is suitable for the relative importance among possible scales . Each scale takes just one value from {1,2,3,4,5,6,7,8,9}. The value depends on the level of relative importance. In other words, the scales follow those defined by Saaty, Table 2;
- In addition to the scale assignment, the decision maker has the degrees of truth Tij(rq), indeterminacy Iij(rq), and falsity Fij(rq) for the scale rq;
- The relative importance is then represented by a DSNN ;
- The DSNN undergoes the deneutrosiphication process according to Section 2.5 to obtain the corresponding crisp value cij which builds up a comparison matrix;
- The relative importance cji is obtained from
- The relative importance of the u-th alternative compared to the v-th one with respect to the s-th sub-criterion/criterion contribution can be modeled follows the steps 1–6 above.
- For each comparison between the i-th criterion to the j-th criterion, select possible scales . Each scale takes just one value from {1,2,3,4,5,6,7,8,9} which is the set of relative importance scales following Saaty.
- Assign the degrees of truth Tij(rq), indeterminacy Iij(rq), and falsity Fij(rq) for each scale rq. The relative importance is then represented by a DSNN
- Compute the membership level of the scale rq according to the formula (6), i.e.,
- Compute the final relative importance scale from Equation (8), i.e.,
- The relative importance is obtained from
4. Illustrative Examples
4.1. Assignment with Certainty
4.2. Assignment with Uncertainty
4.3. Determination of Criteria Weights
5. Investment in Equity Market: A Real-World Application
6. Sensitivity Analysis
- When decision makers have absolute indeterminacy about a scale, i.e., indeterminacy degree of 1, their selections of that scale have no effect on final result.
- When decision makers have neither absolute truth nor falsity about a scale, the proposed methodology is recommended for obtaining final relative importance scale.
- The full belongingness happens when the truth degree is equal to 1 and the non-belongingness takes place when the falsity degree is equal to 1.
7. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Relative Importance to Goal | Saaty Scale | Crisp Scale by Score Function (Self-Calculation) | Crisp Scale by Accuracy Function (Self-Calculation) |
---|---|---|---|
Equally important/influential/preferable | 1 | 0.56 | 0.94 |
Slightly important/influential/preferable | 3 | 0.96 | 2.53 |
Strongly important/influential/preferable | 5 | 4.59 | 5.34 |
Very strongly important/influential/preferable | 7 | 7.09 | 7.61 |
Absolutely important/influential/preferable | 9 | 10.13 | 10.13 |
Sporadic values between two close scales | 2 | 0.86 | 1.76 |
4 | 2.78 | 3.98 | |
6 | 4.84 | 6.19 | |
8 | 5.55 | 6.45 |
Scale | Relative Importance to Goal |
---|---|
1 | Equally important/influential/preferable |
3 | Slightly important/influential/preferable |
5 | Strongly important/influential/preferable |
7 | Very strongly important/influential/preferable |
9 | Absolutely important/influential/preferable |
2 4 6 8 | Sporadic values between two close scales |
Possibility | Level | Scale | Truth Degree | Indeterminacy Degree | Falsity Degree |
---|---|---|---|---|---|
Strongly influential | 5 | 0.7 | 0.2 | 0.3 | |
Very strongly influential | 7 | 0.6 | 0.3 | 0.2 |
Possibility | Level | Scale | Truth Degree | Indeterminacy Degree | Falsity Degree |
---|---|---|---|---|---|
Equally influential | 1 | 0.4 | 0.6 | 0.2 | |
Slightly influential | 3 | 0.5 | 0.4 | 0.1 | |
Strongly influential | 5 | 0.4 | 0.5 | 0.2 |
n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|
RI | 0 | 0 | 0.58 | 0.90 | 1.12 | 1.24 | 1.32 | 1.41 | 1.45 |
Criterion | Description |
---|---|
Image | Perceived market image of firm |
Indicators | Key economic indicators |
Earnings | Acceptable level of expected earnings |
Track | Business track record of firm |
Risk | Acceptable level of investment risk |
Account | Reliable accounting information |
Comparison | DSNN Scale | T | I | F | Deneutrosophicated Scale | |
---|---|---|---|---|---|---|
1 to 2 | 2 | 0.5 | 0.2 | 0.4 | 0.55 | 2.61 |
3 | 0.9 | 0.2 | 0.2 | 0.85 | ||
1 to 5 | 1 | 0.3 | 0.1 | 0.6 | 0.35 | 1.70 |
2 | 0.8 | 0.2 | 0.1 | 0.8 | ||
1 to 6 | 3 | 0.5 | 0.2 | 0.5 | 0.5 | 3.62 |
4 | 0.9 | 0.3 | 0.3 | 0.8 | ||
3 to 4 | 1 | 0.8 | 0.2 | 0.3 | 0.75 | 1.44 |
2 | 0.5 | 0.3 | 0.3 | 0.6 | ||
3 to 5 | 3 | 0.3 | 0.1 | 0.6 | 0.35 | 3.70 |
4 | 0.9 | 0.3 | 0.2 | 0.8 | ||
3 to 6 | 3 | 0.5 | 0.3 | 0.4 | 0.55 | 3.59 |
4 | 0.8 | 0.2 | 0.1 | 0.8 | ||
4 to 5 | 5 | 0.9 | 0.2 | 0.4 | 0.75 | 5.40 |
6 | 0.5 | 0.3 | 0.5 | 0.5 | ||
4 to 6 | 3 | 0.6 | 0.2 | 0.4 | 0.6 | 3.50 |
4 | 0.4 | 0.2 | 0.2 | 0.6 | ||
5 to 6 | 1 | 0.5 | 0.2 | 0.4 | 0.55 | 1.58 |
2 | 0.7 | 0.1 | 0.2 | 0.75 |
Truth Degree | Indeterminacy Degree | Falsity Degree | |
---|---|---|---|
1 | 0 | 0 | 1 |
Truth Degree | Indeterminacy Degree | Falsity Degree | |
---|---|---|---|
0 | 0 | 1 | 0 |
0 | 1 | 0 | 0 |
Truth Degree | Indeterminacy Degree | Falsity Degree | |
---|---|---|---|
0.6 | 0 | 0.8 | 0.4 |
0.6 | 0.2 | 0.8 | 0.4 |
0.6 | 0.4 | 0.8 | 0.4 |
0.6 | 0.6 | 0.8 | 0.4 |
0.6 | 0.8 | 0 | 0.4 |
0.6 | 0.8 | 0.2 | 0.4 |
0.6 | 0.8 | 0.4 | 0.4 |
0.6 | 0.8 | 0.6 | 0.4 |
0.6 | 0.8 | 0.8 | 0.4 |
0.6 | 0 | 0.6 | 0.5 |
0.6 | 0.2 | 0.6 | 0.5 |
0.6 | 0.4 | 0.6 | 0.5 |
0.6 | 0.6 | 0 | 0.5 |
0.6 | 0.6 | 0.2 | 0.5 |
0.6 | 0.6 | 0.4 | 0.5 |
0.6 | 0.6 | 0.6 | 0.5 |
0.6 | 0 | 0.4 | 0.6 |
0.6 | 0.2 | 0.4 | 0.6 |
0.6 | 0.4 | 0 | 0.6 |
0.6 | 0.4 | 0.2 | 0.6 |
0.6 | 0.4 | 0.4 | 0.6 |
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Harnpornchai, N.; Wonggattaleekam, W. An Application of Neutrosophic Set to Relative Importance Assignment in AHP. Mathematics 2021, 9, 2636. https://doi.org/10.3390/math9202636
Harnpornchai N, Wonggattaleekam W. An Application of Neutrosophic Set to Relative Importance Assignment in AHP. Mathematics. 2021; 9(20):2636. https://doi.org/10.3390/math9202636
Chicago/Turabian StyleHarnpornchai, Napat, and Wiriyaporn Wonggattaleekam. 2021. "An Application of Neutrosophic Set to Relative Importance Assignment in AHP" Mathematics 9, no. 20: 2636. https://doi.org/10.3390/math9202636
APA StyleHarnpornchai, N., & Wonggattaleekam, W. (2021). An Application of Neutrosophic Set to Relative Importance Assignment in AHP. Mathematics, 9(20), 2636. https://doi.org/10.3390/math9202636