PSI - Issue 72

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

510









    , A A A A x T x I x F x X   (2) where T A (x) ∶ → [0,1] is the truth-membership function, I A (x) ∶ → [0,1] is the indeterminacy-membership function, and F A (x) ∶ → [0,1] is the falsity-membership function, with the condition that 0 ≤ T A (x)+I A (x)+ F A (x) ≤ 3 for all ∈ . For convenience, a SVNV can be denoted as = ( , , ) , where , , ∈ [0,1] and 0 ≤ + + ≤ 3. Let ={( , ( ), ( ), ( ))| ∈ } and = {( , ( ), ( ), ( ))| ∈ } be two SVNSs. The following is a definition of the fundamental operations: 1. Complement: ={( , ( ),1 − ( ), ( ))| ∈ } 2. Intersection: ∩ = {( ,min( ( ), ( )),max( ( ), ( )),max ( ( ), ( )))| ∈ } 3. Union: ∪ = { ( ,max ( ( ), ( ) ) ,min ( ( ), ( ) ) ,min ( ( ), ( ) )) | ∈ } Let A j =(T j , I j ,F j )(j = 1,2,...,n) be a collection of SVNVs. The neutrosophic weighted average (NWA) operator is defined as:         n n n n w w w j j j 1 2 3 n j j j j j 1 1 1 1 NWA A,A ,A,...A wA 1 1 T , I , F j j j j                      (3) where is the weight of with ∈ [0,1] and ∑ = 1 = 1. For two SVNVs =( , , ) and =( , , ) , the normalized Euclidean distance is defined as:         2 2 2 A B A B A B 1 dA,B= T-T +I -I + F -F 3       (4) 2.3. Classical TOPSIS Method A multi criteria decision making process called the TOPSIS ranks options according to how near the ideal answer they are (Lv et al., 2013). The following phases make up the classical TOPSIS procedure: 1. Construct the decision matrix with alternatives A i (i = 1,2,...,m) and criteria C j (j = 1,2,...,n). 2. Use vector normalization to normalize the decision matrix: ij ij x r = (5)   , ,

m 2 ij i=1

x 

1. Determine the normalized weighted decision matrix:

ij j ij v w r  

(6)

where is the weight of criterion with ∑ =1 = 1 . 2. Find the NIS and PIS, or positive and negative ideal solutions, respectively:   1 2 , ,..., n A v v v     

(7)