Multi-Criteria Decision-Making with Z-Numbers

Z-numbers constitute an ordered pair comprising a fuzzy restriction and a measure of reliability, enabling decision makers to capture both the value of a variable and the confidence in that assessment. They extend classical fuzzy sets by incorporating a built-in credibility measure and have been embedded within multi-criteria decision-making (MCDM) frameworks such as TOPSIS, ELECTRE, TODIM, COPRAS, QUALIFLEX and MABAC. This dual-component structure allows for more faithful modelling of uncertain, imprecise or subjective information, ensuring that both qualitative judgements and their associated trustworthiness inform alternative ranking and selection. Applications span supplier appraisal, financial portfolio management, renewable energy planning, water resource risk evaluation and emergency response. Recent work has further blended Z-numbers with neutrosophic sets and Dempster–Shafer evidence theory to generalise the treatment of indeterminacy and fuse evidence from multiple experts. These advances underscore the global utility of Z-number-based MCDM in policy, engineering and strategic contexts where information reliability varies and complete data are rare.

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Multi-Criteria Decision-Making with Z-Numbers publication trend

The graph below shows the total number of articles in multi-criteria decision-making with z-numbers across all publications each year (not limited to Nature Index journals).

Technical terms

Z-number: An ordered pair of fuzzy sets representing a restriction on a variable and an associated reliability degree.

Neutrosophic Z-number: A generalisation combining neutrosophic sets (truth, indeterminacy, falsity) with reliability measures.

Aggregation operator: A mathematical function that merges multiple information inputs into a single collective value respecting Z-number semantics.

Dempster–Shafer evidence theory: A framework for modelling and fusing uncertain evidence into combined belief assignments.

TODIM: A decision method reflecting decision-makers’ loss-aversion behaviour under uncertainty.

ELECTRE II: An outranking MCDM method that compares alternatives via concordance and discordance indices.

References

1. Interactive TOPSIS Based Group Decision Making Methodology Using Z-Numbers. International Journal of Computational Intelligence Systems (2016).
2. A New Methodology of Multicriteria Decision‐Making in Supplier Selection Based on Z‐Numbers. Mathematical Problems in Engineering (2016).
3. A multi-criteria decision making for renewable energy selection using Z-numbers in uncertain environment. Technological and Economic Development of Economy (2018).
4. A TODIM-PROMETHEE Ⅱ Based Multi-Criteria Group Decision Making Method for Risk Evaluation of Water Resource Carrying Capacity under Probabilistic Linguistic Z-Number Circumstances. Mathematics (2020).
5. A Novel Linguistic Z‐Number QUALIFLEX Method and Its Application to Large Group Emergency Decision Making. Scientific Programming (2020).
6. Z-MABAC Method for the Selection of Third-Party Logistics Suppliers in Fuzzy Environment. IEEE Access (2020).
7. Some aggregation operators of neutrosophic Z-numbers and their multicriteria decision making method. Complex & Intelligent Systems (2020).
8. A generalized TODIM-ELECTRE II method based on linguistic Z-numbers and Dempster–Shafer evidence theory with unknown weight information. Complex & Intelligent Systems (2021).
9. Linguistic Z-number weighted averaging operators and their application to portfolio selection problem. PLOS ONE (2020).