Abstract
In this article, an improved MULTIMOORA approach is proposed for multi-attribute decision making (MADM) using the fuzzy concept, best–worst method (BWM) and half-quadratic (HQ) theory. The fuzzy concept helps to capture the vague information derived from human judgement at different stages of decision making, while BWM helps to simplify the attribute weighing. The standard MULTIMOORA method uses three utility functions, namely the ratio system (RS) utility function, reference point (RP) utility function and full multiplicative form (FMF) utility function, for evaluating the alternatives and obtaining the ranking orders using each of the utility function scores, which are consolidated using dominance theory. However, the dominance theory in the existing MULTIMOORA method has limitations, like there is no way to ascertain the trust level of the consolidated ranking and the level of consensus among the three ranking orders. Also, there is a need for multiple comparisons during aggregation, difficulty in automation, and the problem of circular reasoning. To overcome the limitations of dominance theory, a new HQ theory-based aggregation procedure has been proposed in this paper, which also has two associated indices, one to ascertain the level of consensus from the three ranking orders from MULTIMOORA and the other to ascertain the trust level or reliability of the final ranking in the aggregated ranking. The new modification is expected to add to the trustworthiness of the MULTIMOORA decision tool. The applicability of the proposed approach has been demonstrated with cases on welding process selection.
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Abbreviations
- ~ (Tilde accent):
-
Fuzzy value
- Ã = (l, m, u),:
-
The fuzzy set’s lower, middle and upper limits are represented by l, m, and u.
- i = 1, 2,..., K :
-
Number of alternatives
- j = 1, 2,... n :
-
Number of criteria
- \(\tilde{V}_{B}\) :
-
Fuzzy best-to-others vector
- \(\tilde{V}_{W}\) :
-
Fuzzy others-to-worst vector
- \(\tilde{w}_{B}\) :
-
Best criterion fuzzy weight
- \(\tilde{w}_{W}\) :
-
Worst criterion fuzzy weight
- \(\tilde{\zeta }\) :
-
Objective function
- :
-
Criteria fuzzy weights
- :
-
Alternative fuzzy weights w.r.t. each criterion
- \(\tilde{w}_{ij}^{a}\) :
-
Fuzzy decision matrix
- \(\tilde{w}_{ij}^{a*}\) :
-
Normalized fuzzy decision matrix
- \(\tilde{Q} = (\tilde{q}_{ij} ) = (l_{ij}^{q} ,m_{ij}^{q} ,u_{ij}^{q} )\) :
-
Fuzzy weighted normalized matrix
- \(\tilde{y}_{i}\) :
-
Fuzzy ratio system utility value
- \(\tilde{r}_{j}\) :
-
Optimum fuzzy reference point
- \(h_{i}\) :
-
Fuzzy reference point utility value
- \(\tilde{z}\) :
-
Fuzzy full multiplicative form utility value
- \(\alpha_{g}\) :
-
Half-quadratic auxiliary
- g = 1,..., G :
-
Number of utility functions
- \(R^{g}\) :
-
Ranking obtained from gth utility function
- \(w_{g}\) :
-
Weight for gth utility function
- \(R^{*}\) :
-
Aggregate ranking of alternative
- \(C(R^{*} )\) :
-
Consensus Index
- \(T(R^{*} )\) :
-
Trust level
- \(N_{\sigma }\) :
-
Probability density function
- σ :
-
Standard deviation
- \(q_{kg}\) :
-
The ratio of probability density function values for the error \((R^{g} - R^{*} )\) to the probability density function values for zero
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RSS contributed to the problem conceptualization, data curation, formal analysis, investigation, methodology, validation, visualization and writing—original draft. VS was involved in the resources, supervision, visualization and writing—review and editing.
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Saluja, R.S., Singh, V. An Improved Fuzzy MULTIMOORA Approach and Its Application in Welding Process Selection. Int. J. Fuzzy Syst. 25, 1707–1726 (2023). https://doi.org/10.1007/s40815-023-01472-7
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DOI: https://doi.org/10.1007/s40815-023-01472-7