Z-BWM: Z-Number Best-Worst Method

Overview

Z-BWM is BWM with a 'how sure are you?' channel. When you say 'Criterion B is Very Important compared to A', BWM accepts that as TFN (5/2,3,7/2). Z-BWM also asks 'how sure?': if 'High' (TFN (0.5,0.7,0.9)), it shrinks your VI-TFN by √α ≈ √0.7 ≈ 0.84. Less-sure judgments contribute less to the optimization. The crisp weight at the end uses GMIR: like a weighted-mean integrating the whole TFN shape (l + 4m + u)/6.

Strengths

• •Captures BOTH vagueness (TFN) AND reliability/confidence (second TFN) of expert judgments simultaneously
• •Demonstrably lower inconsistency (CR=0.034) vs BWM (0.382) and Fuzzy BWM (0.035) on paper's case study
• •Only 2n-3 pairwise comparisons (same as BWM): much fewer than AHP's n(n-1)/2
• •GMIR provides a single defensible defuzzification: no arbitrary choice between centroid/mean-of-maxima

Limitations

• •Fuzzy part retains subjectivity during concept translation (Aboutorab Table 4: acknowledged weakness)
• •Nonlinear constrained optimization requires solver (cannot be solved in closed form)
• •Table 3's 25 precomputed Z→TFN values are paper-specific; alternative scales require re-derivation
• •Single decision-maker: group aggregation requires external mechanism (geometric mean of Z-numbers, voting, etc.)

Edge cases

• •When a_BW = (EI,VL) (i.e., u_BW=1 from Table 5), CI=3: small CI means even tiny ξ* triggers high CR; user should re-elicit
• •When all reliabilities = VH, Z-BWM reduces to Fuzzy BWM (Guo-Zhao 2017) numerically since α=1 → √α=1
• •When n=3, only one OW comparison is non-degenerate; the optimization is barely constrained: encourage n≥4

Common pitfalls

• •Don't confuse constraint scale (EI..AI, range 1..4.5) with reliability scale (VL..VH, range 0..1)
• •Don't skip Z→TFN; the optimizer needs TFNs, not (constraint, reliability) linguistic pairs
• •Don't read CI from Table 5 by the constraint level alone: index is u_BW of the Best-to-Worst TFN
• •Don't compute Σw_j = 1 on arithmetic sums of l/m/u: paper uses GMIR-normalized sum

Worked example

1. 1.Step 1
2. 2.Step 2
3. 3.Step 3
4. 4.Step 4
5. 5.Step 5
6. 6.Step 6
7. 7.Step 7
8. 8.Step 8

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