A Novel (R,S)-Norm Entropy Measure of Intuitionistic Fuzzy Sets and Its Applications in Multi-Attribute Decision-Making
Abstract
:1. Introduction
2. Preliminaries
- ;
- ;
- ;
- : ;
- : ;
- : .
- (P1)
- if and only if A is a crisp set, i.e., either or for all .
- (P2)
- if and only if for all .
- (P3)
- .
- (P4)
- If , that is, if and for any , then .
3. Proposed ()-Norm Intuitionistic Fuzzy Information Measure
- (P1)
- if and only if A is a crisp set, i.e., or for all .
- (P2)
- if and only if for all .
- (P3)
- if A is crisper than B, i.e., if & , for and & , for for all .
- (P4)
- for all .
- Sharpness: In order to prove (P1), we need to show that if and only if A is a crisp set, i.e., either or for all .Firstly, we assume that for and . Therefore, from Equation (6), we have:Since and , therefore, the above equation is satisfied only if or for all .Conversely, we assume that set is a crisp set i.e., either or 1. Now, for and , we can obtain that:Hence, iff A is a crisp set.
- Maximality: We will find maxima of the function ; for this purpose, we will differentiate Equation (6) with respect to and . We get,In order to check the convexity of the function, we calculate its second order derivatives as follows:To find the maximum/minimum point, we set and , which gives that for all i and hence called the critical point of the function .
- (a)
- When , then at the critical point , we compute that:Therefore, the Hessian matrix of is negative semi-definite, and hence, is a concave function. As the critical point of is and by the concavity, we get that has a relative maximum value at .
- (b)
- When , then at the critical point, we can again easily obtain that:This proves that is a concave function and its global maximum at .
Thus, for all or , the global maximum value of attains at the point , i.e., is maximum if and only if A is the most fuzzy set. - Resolution: In order to prove that our proposed entropy function is monotonically increasing and monotonically decreasing with respect to and , respectively, for convince, let , and , then it is sufficient to prove that for , , the entropy function:Taking the partial derivative of f with respect to x and y respectively, we get:For the extreme point of f, we set and and get .Furthermore, , when such that , , i.e., is increasing with , and is decreasing with respect to x, when . On the other hand, and when and , respectively.Further, since is a concave function on the IFS A, therefore, if , then and , which implies that:Thus, we observe that is more around than . Hence, .Similarly, if , then we get .
- Symmetry: By the definition of , we can easily obtain that .
- Joint entropy:
- Conditional entropy:
- When for all , then the proposed measures reduce to the entropy measure of Joshi and Kumar [32].
- When and , then the proposed measures are reduced by the measure of Taneja [27].
- When and , then the measure is equivalent to the R-norm entropy presented by Boekee and Van der Lubbe [28].
- When , then the proposed measure is the well-known Shannon’s entropy.
- When and , then the proposed measure becomes the measure of Bajaj et al. [37].
- ;
- ;
- .
- Consider:
- Consider:
- This can be deduced from Parts (1) and (2).
4. MADM Problem Based on the Proposed Entropy Measure
4.1. Approach I: When the Attribute Weight Is Completely Unknown
- Step 1:
- Normalize the rating values of the decision-maker, if required, by converting the rating values corresponding to the cost type attribute into the benefit type. For this, the following normalization formula is used:
- Step 2:
- Based on the matrix R, the information entropy of attribute is computed as:
- Step 3:
- Based on the entropy matrix, defined in Equation (16), the degree of divergence of the average intrinsic information provided by the correspondence on the attribute can be defined as where . Here, the value of represents the inherent contrast intensity of attribute , and hence, based on this, the attributes weight is given as:
- Step 4:
- Construct the weighted sum of each alternative by multiplying the score function of each criterion by its assigned weight as:
- Step 5:
- Rank all the alternatives according to the highest value of and, hence, choose the best alternative.
- Step 1:
- Since all the attributes are of the same type, so there is no need for the normalization process.
- Step 2:
- Without loss of generality, we take and and, hence, compute the entropy measurement value for each attribute by using Equation (16). The results corresponding to it are , , and .
- Step 3:
- Based on these entropy values, the weight of each criterion is calculated as , , , .
- Step 4:
- The overall weighted score values of the alternative corresponding to , and obtained by using Equation (18) are , , , and .
- Step 5:
- Since , hence the ranking order of the alternatives is . Thus, the best alternative is .
4.2. Approach II: When the Attribute Weight Is Partially Known
- A weak ranking: ;
- A strict ranking: ; .
- A ranking with multiples: , ;
- An interval form: , ;
- A ranking of differences: , .
- Step 1:
- Similar to Approach I.
- Step 2:
- similar to Approach I.
- Step 3:
- The overall entropy of the alternative for the attribute is given by:By considering the importance of each attribute in terms of weight vector , we formulate a linear programming model to determine the weight vector as follows:After solving this model, we get the optimal weight vector .
- Step 4:
- Construct the weighted sum of each alternative by multiplying the score function of each criterion by its assigned weight as:
- Step 5:
- Rank all the alternative according to the highest value of and, hence, choose the best alternative.
- Step 1:
- All the attributes are te same types, so there is no need for normalization.
- Step 2:
- Without loss of generality, we take and and, hence, compute the entropy measurement value for each attribute by using Equation (20). The results corresponding to it are , , and .
- Step 3:
- Formulate the optimization model by utilizing the information of rating values and the partial information of the weight vector as:Hence, we solve the model with the help of MATLAB software, and we can obtain the weight vector as .
- Step 4:
- The overall weighted score values of the alternative corresponding to , and obtained by using Equation (21) are , , and and .
- Step 5:
- Since , hence the ranking order of the alternatives is . Thus, the best alternative is .
5. Conclusions
Author Contributions
Conflicts of Interest
References
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Entropy Measure | A | ||||
---|---|---|---|---|---|
[21] | 0.0818 | 0.1000 | 0.0980 | 0.0934 | 0.0934 |
[19] | 0.3446 | 0.3740 | 0.1970 | 0.1309 | 0.1094 |
[57] | 0.4156 | 0.4200 | 0.2380 | 0.1546 | 0.1217 |
[58] | 0.3416 | 0.3440 | 0.2610 | 0.1993 | 0.1613 |
[25] | 0.2851 | 0.3050 | 0.1042 | 0.0383 | 0.0161 |
[22] | 0.5995 | 0.5981 | 0.5335 | 0.4631 | 0.4039 |
(proposed measure) | |||||
2.3615 | 2.3589 | 1.8624 | 1.4312 | 1.1246 | |
0.8723 | 0.8783 | 0.6945 | 0.5392 | 0.4323 | |
0.5721 | 0.5769 | 0.4432 | 0.3390 | 0.2725 | |
2.2882 | 2.2858 | 1.8028 | 1.3851 | 1.0890 | |
0.8309 | 0.8368 | 0.6583 | 0.5104 | 0.4103 | |
0.5369 | 0.5415 | 0.4113 | 0.3138 | 0.2538 |
S | R | Ranking Order | |||||
---|---|---|---|---|---|---|---|
1.2 | 0.1 | 0.3268 | 0.3084 | 0.3291 | 0.2429 | 0.1715 | |
0.3 | 0.3241 | 0.3081 | 0.3292 | 0.2374 | 0.1690 | ||
0.5 | 0.3165 | 0.2894 | 0.3337 | 0.2368 | 0.1570 | ||
0.7 | 0.1688 | -0.0988 | 0.4296 | 0.2506 | -0.0879 | ||
0.9 | 0.3589 | 0.3992 | 0.3065 | 0.2328 | 0.2272 | ||
1.5 | 0.1 | 0.3268 | 0.3084 | 0.3291 | 0.2429 | 0.1715 | |
0.3 | 0.3239 | 0.3076 | 0.3293 | 0.2374 | 0.1688 | ||
0.5 | 0.3132 | 0.2811 | 0.3359 | 0.2371 | 0.1515 | ||
0.7 | 0.4139 | 0.5404 | 0.2712 | 0.2272 | 0.3185 | ||
0.9 | 0.3498 | 0.3741 | 0.3125 | 0.2334 | 0.2121 | ||
2.0 | 0.1 | 0.3268 | 0.3084 | 0.3291 | 0.2429 | 0.1715 | |
0.3 | 0.3237 | 0.3071 | 0.3294 | 0.2375 | 0.1684 | ||
0.5 | 0.3072 | 0.2666 | 0.3396 | 0.2381 | 0.1415 | ||
0.7 | 0.3660 | 0.4140 | 0.3022 | 0.2308 | 0.2393 | ||
0.9 | 0.3461 | 0.3631 | 0.3150 | 0.2331 | 0.2062 | ||
2.5 | 0.1 | 0.3268 | 0.3084 | 0.3291 | 0.2429 | 0.1715 | |
0.3 | 0.3235 | 0.3067 | 0.3295 | 0.2376 | 0.1681 | ||
0.5 | 0.3010 | 0.2517 | 0.3436 | 0.2396 | 0.1308 | ||
0.7 | 0.3578 | 0.3920 | 0.3074 | 0.2304 | 0.2261 | ||
0.9 | 0.3449 | 0.3591 | 0.3158 | 0.2322 | 0.2045 | ||
3.0 | 0.1 | 0.3268 | 0.3084 | 0.3291 | 0.2429 | 0.1715 | |
0.3 | 0.3234 | 0.3064 | 0.3296 | 0.2376 | 0.1678 | ||
0.5 | 0.2946 | 0.2368 | 0.3476 | 0.2417 | 0.1199 | ||
0.7 | 0.3545 | 0.3829 | 0.3095 | 0.2298 | 0.2209 | ||
0.9 | 0.3442 | 0.3570 | 0.3161 | 0.2314 | 0.2037 | ||
5.0 | 0.1 | 0.3268 | 0.3084 | 0.3291 | 0.2429 | 0.1715 | |
0.3 | 0.3231 | 0.3058 | 0.3298 | 0.2379 | 0.1674 | ||
0.5 | 0.2701 | 0.1778 | 0.3638 | 0.2520 | 0.0767 | ||
0.7 | 0.3496 | 0.3706 | 0.3123 | 0.2277 | 0.2137 | ||
0.9 | 0.3428 | 0.3532 | 0.3168 | 0.2293 | 0.2020 |
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Garg, H.; Kaur, J. A Novel (R,S)-Norm Entropy Measure of Intuitionistic Fuzzy Sets and Its Applications in Multi-Attribute Decision-Making. Mathematics 2018, 6, 92. https://0-doi-org.brum.beds.ac.uk/10.3390/math6060092
Garg H, Kaur J. A Novel (R,S)-Norm Entropy Measure of Intuitionistic Fuzzy Sets and Its Applications in Multi-Attribute Decision-Making. Mathematics. 2018; 6(6):92. https://0-doi-org.brum.beds.ac.uk/10.3390/math6060092
Chicago/Turabian StyleGarg, Harish, and Jaspreet Kaur. 2018. "A Novel (R,S)-Norm Entropy Measure of Intuitionistic Fuzzy Sets and Its Applications in Multi-Attribute Decision-Making" Mathematics 6, no. 6: 92. https://0-doi-org.brum.beds.ac.uk/10.3390/math6060092