Hesitant Fuzzy Linguistic AHP - one-level synthesis

Overview

Use when criterion comparisons and alternative ratings are expressed as hesitant linguistic term sets on the declared seven-term scale.

Output

preference score, higher is better

Data

Hesitant Fuzzy Linguistic, expert linguistic input required

Size

2+ alternatives, 3-10 criteria works best

Used for

Alternative-fuel vehicle selection (Yavuz et al. 2015), technology or supplier selection with linguistic expert judgements

How it works

1. 1

   Validate the seven-term scale (g = 6) and the reciprocal HFLTS criterion matrix: diagonal s_3, lower_ij + upper_ji = 6, upper_ij + lower_ji = 6.

   $$\mathrm{env}(H_S)=[H_S^{-},H_S^{+}]$$

   Yavuz et al. (2015), HFLTS pairwise-comparison construction

2. 2

   Transform every envelope endpoint s_k to k/6.

   $$s_k\mapsto k/6$$

   Yavuz et al. (2015), HFLTS envelope transformation

3. 3

   Sum the semantic intervals by criterion row to get interval utilities.

   $$V_i=[\sum _j{r}_{ij}^{-},\sum _j{r}_{ij}^{+}]$$

   Yavuz et al. (2015), interval utility vector

4. 4

   Normalise interval utilities to interval criterion weights.

   $$W_i=[V_i^{-}/\sum _k{V}_k^{+},V_i^{+}/\sum _k{V}_k^{-}]$$

   Yavuz et al. (2015), Sengupta and Pal (2000) interval normalisation

5. 5

   Validate the alternative ratings table and transform it to semantic intervals with the same k/6 mapping.

   Yavuz et al. (2015), alternative evaluation on the linguistic scale

6. 6

   Aggregate each alternative with the interval criterion weights.

   $$S_a=[\sum _i{W}_i^{-}{R}_{ai}^{-},\sum _i{W}_i^{+}{R}_{ai}^{+}]$$

   Yavuz et al. (2015), weighted interval score

7. 7

   Compute the pairwise preference degree of interval scores, the preference score (row sum minus 0.5, divided by m - 1) and rank descending.

   $$p(S_a\ge S_b)=\min \{1,\max \{0,(S_a^{+}-S_b^{-})/[(S_a^{+}-S_a^{-})+(S_b^{+}-S_b^{-})]\}\}$$

   Yavuz et al. (2015), Eqs. (14) and (15)

Fits when / Look elsewhere when

Assumptions to verify

• One expert (or an already aggregated judgement) supplies both the criterion comparisons and the alternative ratings
• All criteria are benefit-oriented on the seven-term Yavuz scale
• Envelope reciprocity holds in the criterion pairwise matrix

Edge cases and pitfalls

• •Single-term cells ({s_4}) are stored as the degenerate envelope [4, 4]; the method still runs but the intervals collapse.
• •If every alternative score collapses to a point, the preference degree compares the points directly (1, 0 or 0.5).
• •Exactly equal preference scores keep the supplied alternative order; no secondary key is invented.
• •A cost criterion or a scale other than g = 6 is rejected before any computation.

• The main n×n matrix compares criteria; it is not the alternative-rating matrix.

• This manifest is single-expert. Expert aggregation requires a separately sourced group formulation.

• Interval weights are a secondary result and must not be presented as a crisp weight simplex.

• A cost criterion must first be converted by a separately sourced transformation; this kernel rejects it rather than guessing.

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