PSI - Issue 72

Ruhit Bardhan et al. / Procedia Structural Integrity 72 (2025) 507–519

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The neutrosophic distance measure is used to determine each alternative separation metrics from the NPIS and NNIS:   1 , n i ij i j S d v v      (22)   1 , n i ij i j S d v v      (23) where , as described in neutrosophic sets, is the normalized Euclidean distance between two SVNVs. Step 6: Calculation of Relative Closeness Coefficient where 0 ≤ ≤ 1 . A higher value of indicates a better alternative. Step 7: Ranking of Alternatives The proximity coefficients are used to rank the FGM options in descending order. The option that has the greatest is considered as the most desirable one. 3.3. Methodology for Criteria Weight Determination The neutrosophic TOPSIS approach heavily relies on the weights of evaluation criteria. We provide a combined strategy for FGM selection in a neutrosophic setting that makes use of the Analytic Hierarchy Process (AHP). The following steps are involved in the neutrosophic AHP process: 1. Construct neutrosophic pairwise comparison matrices based on expert judgments 2. Calculate the neutrosophic weights using the geometric mean method 3. De fuzzify the neutrosophic weights to obtain crisp weights 4. Normalize the weights to ensure ∑ w j n j=1 = 1. This approach allows for incorporating uncertainty in the weight determination process, making it particularly suitable for FGM selection problems where the relative importance of different criteria may not be precisely known. 4. Results and Discussion In this section, we apply the proposed neutrosophic TOPSIS framework to a case study involving the selection of functionally graded materials for high- temperature aerospace applications. We then compare the results with those obtained using classical TOPSIS and fuzzy TOPSIS methods. 4.1. Case Study: FGM Selection for Aerospace Applications The case study focuses on selecting the most suitable FGM for thermal barrier coatings in gas turbine engines. Five candidate FGM systems (alternatives) are considered:  A 1 : Ni-Al2O3 FGM  A 2 : Ti-TiB2 FGM Each alternative’s relative proximity coefficient is determined as follows: i    S i i i c S S   (24)