PSI - Issue 72

512 Ruhit Bardhan et al. / Procedia Structural Integrity 72 (2025) 507–519  Scalability ( C 11 )  Dimensional accuracy ( C 12 ) Economic and Environmental Factors  Material cost ( C 13 )  Production energy consumption ( C 14 )  Environmental impact ( C 15 )  Recyclability ( C 16 ) Each criterion may have different importance depending on the specific application. Moreover, the evaluation of these criteria often involves uncertainties, imprecisions, and indeterminacies due to:  Limited experimental data on gradient properties  Variations in manufacturing processes  Subjective expert judgments  Trade-offs between competing properties These uncertainties make neutrosophic sets particularly suitable for modeling the FGM selection problem. 3.2. Neutrosophic TOPSIS Algorithm for FGM Selection The following phases make up the suggested neutrosophic TOPSIS method for FGM selection: We nowdescribe each step in detail: Step 1: Construction of Neutrosophic Decision Matrix The neutrosophic decision matrix ̃ = [ ̃ ] × is constructed by evaluating each alternative against each criterion using single-valued neutrosophic values (SVNVs): Algorithm 1: Neutrosophic TOPSIS for FGM Selection 1. Input: Set of FGM alternatives A={A 1 ,A 2 , . . . ,A m } , set of criteria C={C 1 ,C 2 , . . . ,C n } , criteria weights W= {w 1 ,w 2 , . . . ,w n }. 2. Output: Ranking of FGM alternatives. 3. Form neutrosophic decision matrix X̃= [ x̃ ij ] m × n , where x̃ ij = ( T ij , I ij ,F ij ) . 4. To get the neutrosophic choice matrix, normalize it. R̃= [ r̃ ij ] m × n 5. Determine the normalized weighted neutrosophic decision matrix. Ṽ= [ ṽ ij ] m × n 6. The neutrosophic negative ideal solution and neutrosophic positive ideal solution should be defined.

7. Determine the NPIS and NNIS separation metrics for every option. 8. Determine the relative closeness coefficient for every option. 9. Rank the alternatives based on the closeness coefficients. 10. Return: Ranked list of FGM alternatives

 

    

X

X

11

1

n

1 (12) where ̃ = ( , , ) is the neutrosophic evaluation of alternative with respect to criterion . Here:  T ij represents the degree to which alternative A i satisfies criterion C j  I ij represents the indeterminacy or uncertainty in the evaluation  F ij represents the degree to which alternative A i does not satisfy criterion C j . The neutrosophic values can be obtained through expert evaluations, experimental data, or a combination of both. For FGM selection, we propose a specialized approach that captures the gradient nature of these materials: m mn X X X    