Z-BWM - Z-Number Best-Worst Method

Overview

Z-BWM outputs criterion importance weights (sum=1). Higher weight = more important. Pair with any alternative-ranking method (TOPSIS, MARCOS, WASPAS) that consumes a weight vector. Check CR ≤ 0.10 before trusting weights.

Data

Z-Number

Weights

Derived internally, no weight source needed

How it works

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

Fits when

Edge cases and pitfalls

• •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

• Mistaking the constraint linguistic scale (EI..AI, range ~1..4.5) for the reliability scale (VL..VH, range 0..1) - they are SEPARATE inputs per Z-number cell.

• Skipping Z→TFN transformation and feeding raw linguistic terms into optimization - must apply Eq.(11) first (Table 3 covers all 25 cases).

• Reading CI from Table 5 by the WRONG entry - CI index is the upper bound u_BW of the BEST-to-WORST Z-number's TFN, NOT the linguistic constraint level alone.

• Forgetting Σ R(W̃_j) = 1 constraint uses GMIR-normalized sum, not arithmetic l/m/u sums.

Works with

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