This paper proposes anoptimal fuzzy-based model for obtaining crisp priorities for Fuzzy-AHP comparison matrices. Crisp judgments cannot be given for real-life situations, as most of these include some level of fuzziness and complexity. In these situations, judgments are represented by the set of fuzzy numbers. Most of the fuzzy optimization models derive crisp priorities for judgments represented with Triangular Fuzzy Numbers (TFNs) only. They do not work for other types of Triangular Shaped Fuzzy Numbers (TSFNs) and Trapezoidal Fuzzy Numbers (TrFNs). To overcome this problem, a sum of squared error (SSE) based optimization model is proposed. Unlike some other methods, the proposed model derives crisp weights from all of the above-mentioned fuzzy judgments. A fuzzy number is simulated using the Monte Carlo method. A threshold-based constraint is also applied to minimize the deviation from the initial judgments. Genetic Algorithm (GA) is used to solve the optimization model. We have also conducted casestudiesto show the proposed approach’s advantages over the existingmethods. Results show that the proposed model outperforms other models to minimize SSE and deviation from initial judgments. Thus, the proposed model can be applied in various real time scenarios as it can reduce the SSE value upto 29% compared to the existing studies.
Multi-Criteria decision-making (MCDM) methods are used to solve complex decision-making systems. Rao [
Mikhailov [
All these fuzzy optimization models derive crisp weights for Triangular Fuzzy Numbers (TFNs) only in the literature presented above. Mohtashami [
After introducing some preliminary concepts for the fuzzy judgments in comparison pairwise matrices in Section 2, the following contributions are claimed:
• A novel Sum of Squared Error (SSE) based fuzzy optimization model is proposed. The simulation of the fuzzy number using Monte Carlo’s method and the threshold-based constraint is also presented.
• Further empirical analysis has been conducted based on eight case studiesto compare the proposed SSE-based model with existing models. It has been proved that the proposed model performs better than the existing Mohtashami’s model [
To the best of the author’s knowledge, no other fuzzy optimization model has been proposed till now to derive crisp weights for these types of fuzzy judgments that can surpass the work done by Mohtashami [
In this section, we have discussed how the various fuzzy judgments can be represented.
For modeling many real-time applications, a decision-maker cannot always give crisp judgments. In these situations, judgments are illustrated by fuzzy numbers [
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The main challenge is to devise a model that can derive crisp priorities from all these types of fuzzy judgments. In the next section, the proposed fuzzy optimization model for obtaining crisp priorities is presented.
This section presents a novel fuzzy prioritization method for deriving crisp priorities from fuzzy pairwise comparison matrices. Here, the proposed SSE-based fuzzy optimization model is discussed and the constraint applied to the fuzzy optimization model for minimum deviation from initial judgmentsis also presented. Further, the fuzzy number simulation method is elaborated, and the overall working of the proposed work has been discussed.
In this paper, a new Sum of Squared Error (SSE) based fuzzy optimization model is presented as a nonlinear system. Crisp weights (
The weight ratio (
The value of
To check the effectiveness of this proposed Model, graphs for different types of fuzzy numbers have been drawn considering the frequency (count) of the simulated fuzzy number. If the values of δ and
Each priority vector (crisp weight) should satisfy the constraint defined in
Thus, the crisp weights deviate with ε from the initial judgments far less than the existing models shown in Section 4.
The working of the proposed model is shown in
Then, the crisp weights are derived using the fuzzy optimization model, as shown in
To evaluate the proposed model, eight case studies are solved. As discussed earlier, among the fuzzy optimization models, only Mohtashami’s model [
In this example, a perfectly consistent comparison pairwise matrix with only two elements is considered. In comparison to the second element, the first element is more important and is represented by a TFN
Non-Linear FPP [ |
Linear FPP [ |
Mohtashami [ |
Proposed model | |
---|---|---|---|---|
0.6666 | 0.6666 | 0.6664 | 0.6667 | |
0.3333 | 0.3333 | 0.3336 | 0.3333 | |
ε | 0 | 0 | 0 | 0 |
SSE | 0 | 0 | 0.0002 | 0 |
Consider the comparison matrix of three elements represented by TFNs as
Non-Linear FPP [ |
Linear FPP [ |
Mohtashami [ |
Proposed model | ||
---|---|---|---|---|---|
0.3076 | 0.2738 | 0.3104 | 0.3077 | 0.3103 | |
0.6153 | 0.6492 | 0.6117 | 0.6154 | 0.6115 | |
0.0769 | 0.0769 | 0.0777 | 0.0769 | 0.0782 | |
ε | 0 | 0.031 | 0.036 | 0 | 0.0293 |
SSE | 3.0313 | 4.5624 | 2.8610 | 3.0306 | 2.8328 |
In this example, judgments of the comparison matrices of four elements are given as
Non-Linear FPP [ |
Linear FPP [ |
Mohtashami [ |
Proposed model | |
---|---|---|---|---|
0.5583 | 0.6397 | 0.5262 | 0.4918 | |
0.0519 | 0.0805 | 0.0487 | 0.0584 | |
0.3311 | 0.2221 | 0.3568 | 0.3806 | |
0.0585 | 0.0574 | 0.0680 | 0.0701 | |
ε | 1.8728 | 3.1307 | 1.8049 | 1.7997 |
SSE | 32.2888 | 52.7075 | 34.97 | 24.8729 |
In all the above examples, triangular fuzzy numbers are used. But, the proposed optimization model can work for trapezoidal numbers as well. The followingcases demonstrate the usefulness of the proposed model by taking the judgments in the comparison matrices in the form of TrFNs. As discussed earlier, only Mohtashami’s model [
Consider the comparison matrix of four elements where the judgments are represented in the form of TrFNs as
Linear FPP [ |
Mohtashami [ |
Proposed model | |
---|---|---|---|
0.2423 | 0.3249 | 0.3677 | |
0.0721 | 0.511 | 0.0487 | |
0.6274 | 0.5589 | 0.5147 | |
0.0579 | 0.0649 | 0.0681 | |
ε | 2.84 | 1.99 | 1.6 |
SSE | 45.614 | 35.080 | 34.2698 |
To demonstrate the superiority of the proposed fuzzy optimization model over the other models, another example where judgments are represented in the form of TrFNs is considered. Suppose the judgments of the comparison matrix are as
Linear FPP [ |
Mohtashami [ |
Proposed model | |
---|---|---|---|
0.6544 | 0.5555 | 0.5594 | |
0.0843 | 0.0784 | 0.0932 | |
0.1875 | 0.2923 | 0.2797 | |
0.0728 | 0.0738 | 0.0667 | |
ε | 4.4245 | 3.09 | 3.0 |
SSE | 96.1872 | 71.0065 | 69.7089 |
Till now the fuzzy judgments considered in this manuscript are symmetric like TFNs and TrFNs. But in some situations, a decision-maker makes the judgments of asymmetric nature like TSFNs and TrSFNs. If the values of δ and χ are equal to one, then it demonstrates TFN otherwise it will act as TSFN.
Consider a comparison matrix of four elements in which judgments are represented by
Linear FPP [ |
Mohtashami [ |
Proposed model | |
---|---|---|---|
0.2002 | 0.3341 | 0.3302 | |
0.1194 | 0.0612 | 0.0690 | |
0.6135 | 0.5315 | 0.5283 | |
0.0669 | 0.0732 | 0.0718 | |
ε | 4.1704 | 2.4358 | 2.4001 |
SSE | 73.1194 | 37.5998 | 36.4661 |
To show the proposed model’s effectiveness, another example of comparison matrix with four elements is considered. Judgments are represented as by
Linear FPP [ |
Mohtashami [ |
Proposed model | |
---|---|---|---|
0.0608 | 0.0652 | 0.0681 | |
0.0883 | 0.0514 | 0.0487 | |
0.6263 | 0.5565 | 0.5152 | |
0.2246 | 0.3269 | 0.3680 | |
ε | 3.4 | 1.9846 | 1.6 |
SSE | 57.3787 | 39.4676 | 37.4647 |
In the last example, consider the comparison matrix of four elements as
Linear FPP [ |
Mohtashami [ |
Proposed model | ||
---|---|---|---|---|
0.1705 | 0.2639 | 0.1788 | 0.2270 | |
0.5015 | 0.4339 | 0.3039 | 0.3631 | |
0.0761 | 0.0669 | 0.0483 | 0.0571 | |
0.2518 | 0.2353 | 0.4686 | 0.3530 | |
ε | 4.7623 | 3.4824 | 3.3 | 3.4 |
SSE | 102.4446 | 89.3605 | 90.2756 | 80.5218 |
This paper presents a novel Sum of Squared Error (SSE) based fuzzy optimization model to derive crisp weights for fuzzy pairwise comparison matrices. The proposed optimization model can derive priorities for Triangular Fuzzy Numbers (TFNs), Trapezoidal Fuzzy Numbers (TrFNs), Triangular Shaped Fuzzy (TSFNs), and Trapezoidal Shaped Fuzzy Numbers (TSFNs) unlike most of the optimization models that derive weights in the case of TFNs only. Simulation of the fuzzy number is done by Monte Carlo’s method. The proposed model minimizes the SSE value and deviation from judgments ischecked by applying a threshold-based constraint.Eightexamples are illustrated that include examples of consistent and inconsistent fuzzy judgments represented by TFNs, TrFNs and TSFNs. Results show that the proposed model outperforms other models for all theabove mentioned cases.
Analytic Hierarchy Process (AHP) can be used in many decision-making problems in various fields like healthcare [