Extension card · Hesitant

Dual hesitant fuzzy TOPSIS (Wang, Li, Zhang and Han, 2020)

A dual hesitant fuzzy TOPSIS profile that compares native `(h,g)` membership and nonmembership sets with the source equations.

Base method

TOPSIS →

Philosophy, mechanics, strengths and weaknesses are on the base method card; this card describes only the difference.

Data type (family)

Hesitant →

What this data type is, when to use it, how to write it in a cell: the family's full account is here.

What Changes from the Base Method?

All criteria use the benefit orientation of Eqs. 29–30. A cost transformation has not been verified from this source and is unsupported in this profile. The positive ideal contains the largest membership and smallest nonmembership degrees observed in each criterion; the negative ideal uses the opposite extrema. Each ideal component is a singleton set.

Eqs. 9–12 compute means and volatility. Volatility combines population variance with a cardinality term; it is not merely the range of a set. Eq. 21 compares these features directly without padding sets or applying crisp column normalisation.

When criterion weights are omitted, Eqs. 17 and 27 derive them from pairwise deviations. Supplied nonnegative coefficients summing to one use Eq. 21 directly; this is distinct from the paper's unknown-weight calculation. A zero total deviation cannot define derived weights.

The preferences satisfy 0 ≤ α, β ≤ 1, α + β = 1, and finite λ ≥ 1. Reports must identify the applied parameters and weight source. Eq. 31 gives CC = d⁻ / (d⁺ + d⁻) when the denominator is positive; larger values rank first.

How to Read the Output

Table 2's 16 native cells use α = β = 0.5, λ = 1. Independent Eq. 27 weights are (0.284515766580, 0.109108793809, 0.290843922342, 0.315531517268); Eq. 31 coefficients for A1–A4 are (0.530966001164, 0.451084214060, 0.456836789967, 0.567314452613), giving A4 > A1 > A3 > A2. These reconstruct the equations; the paper's printed numerical example contains unresolved discrepancies.

When to Prefer This over the Base Method

This extension is preferable to base TOPSIS when each cell carries several possible membership degrees and a separate set of several nonmembership degrees. Use a one-sided hesitant family when only membership is hesitant, and base TOPSIS for one crisp value. Because this profile does not verify a cost transformation, studies containing cost criteria require another source-validated profile.

The features can coincide for distinct native sets: h={0.1,0.5,0.6} and h′={0.2,0.3,0.7}, both with g={0.1}, have identical means, variances and cardinalities, hence zero feature distance. The source uses score/accuracy comparison equivalence; this does not establish equality of the original sets. Source fidelity therefore does not imply that every native difference is distinguishable.

Mistakes Specific to This Extension

Deriving membership and nonmembership sets from one another, accepting an empty set, padding sets to equal cardinality, or applying crisp column normalisation changes the source profile. The constraints α+β=1 and λ≥1, together with the weight source, must survive the run. Zero feature distance between distinct sets must not be reported as literal data equality.

Cases

## 1. The source's four-alternative example The Table 2 reconstruction above gives A4, A1, A3, A2 with data-derived weights. It does not alter an equation to conceal discrepancies in the printed numerical example; applied parameters and independently recomputed values remain explicit in the evidence record.

1. The source's four-alternative example

The Table 2 reconstruction above gives A4, A1, A3, A2 with data-derived weights. It does not alter an equation to conceal discrepancies in the printed numerical example; applied parameters and independently recomputed values remain explicit in the evidence record.

2. An assessment with hesitation in support and rejection

When a committee records several possible membership and several possible nonmembership degrees for each alternative, the two sets remain distinct. Valid supplied criterion weights are used directly; otherwise weights are derived from the source equations only when total deviation is positive.

3. What Not to Do

Do not fill an empty rejection set with zero, duplicate values to equalise set lengths, invent a cost transformation, or interpret zero feature distance between different native sets as “the same data.” Each shortcut changes either the input evidence or the source-defined comparison.

Formulae

The computation step by step, with the formula behind each step.

1. Validate native dual hesitant inputs
$D = (a_{ij})_{m \times n} = (⟨ h_{ij}, g_{ij} ⟩)_{m \times n}$
2. Resolve criterion coefficients
$w_{j} = {w_{j}^{supplied}, Y_{j} / \sum_{k = 1}^{n} Y_{k}, if provided otherwise, Y_{j} = i \sum k \sum d_{h} (a_{ij}, a_{k j}), d_{h} (a, b) = \frac{∣Δ G _{m} ∣ + ∣Δ G _{n} ∣ + ∣Δ V _{m} ∣ + ∣Δ V _{n} ∣}{4}$
3. Construct positive and negative ideals
$a_{j}^{+} = ⟨{i max max h_{ij}}, {i min min g_{ij}}⟩, a_{j}^{-} = ⟨{i min min h_{ij}}, {i max max g_{ij}}⟩$
4. Compute mean/volatility distances
$G_{m} (e) = \frac{1}{l ( h )} γ \in h \sum γ, G_{n} (e) = \frac{1}{k ( g )} η \in g \sum η; V_{m} (e) = \frac{1}{2} [\frac{1}{l ( h )} γ \in h \sum (γ - G_{m})^{2} + 1 - \frac{1}{1 + l n l ( h )}], V_{n} (e) = \frac{1}{2} [\frac{1}{k ( g )} η \in g \sum (η - G_{n})^{2} + 1 - \frac{1}{1 + l n k ( g )}]; d_{w p g} (A_{i}, A^{\pm}) = (\frac{1}{2} j = 1 \sum n w_{j} [α ∣ G_{m} (a_{ij}) - G_{m} (a_{j}^{\pm}) ∣^{λ} + α ∣ G_{n} (a_{ij}) - G_{n} (a_{j}^{\pm}) ∣^{λ} + β ∣ V_{m} (a_{ij}) - V_{m} (a_{j}^{\pm}) ∣^{λ} + β ∣ V_{n} (a_{ij}) - V_{n} (a_{j}^{\pm}) ∣^{λ}])^{1/ λ}, α + β = 1, λ \geq 1$
5. Compute closeness and rank alternatives
$C C_{i} = \frac{d _{i}^{-}}{d _{i}^{+} + d _{i}^{-}}, d_{i}^{+} + d_{i}^{-} > 0, rank (i) = 1 + # {k : C C_{k} > C C_{i}}$

Sources

Wang, J., Li, X., Zhang, H., and Han, Z. (2020). Dual hesitant fuzzy TOPSIS profile. Symmetry 12, 191. DOI: 10.3390/sym12020191.

What Changes from the Base Method?

How to Read the Output

When to Prefer This over the Base Method

Mistakes Specific to This Extension

Cases

1. The source's four-alternative example

2. An assessment with hesitation in support and rejection

3. What Not to Do

Formulae

The computation step by step, with the formula behind each step.

1. Validate native dual hesitant inputs

D = (a_{ij})_{m \times n} = (⟨ h_{ij}, g_{ij} ⟩)_{m \times n}

2. Resolve criterion coefficients

w_{j} = {w_{j}^{supplied}, Y_{j} / \sum_{k = 1}^{n} Y_{k}, if provided otherwise, Y_{j} = i \sum k \sum d_{h} (a_{ij}, a_{k j}), d_{h} (a, b) = \frac{∣Δ G _{m} ∣ + ∣Δ G _{n} ∣ + ∣Δ V _{m} ∣ + ∣Δ V _{n} ∣}{4}

3. Construct positive and negative ideals

a_{j}^{+} = ⟨{i max max h_{ij}}, {i min min g_{ij}}⟩, a_{j}^{-} = ⟨{i min min h_{ij}}, {i max max g_{ij}}⟩

4. Compute mean/volatility distances

G_{m} (e) = \frac{1}{l ( h )} γ \in h \sum γ, G_{n} (e) = \frac{1}{k ( g )} η \in g \sum η; V_{m} (e) = \frac{1}{2} [\frac{1}{l ( h )} γ \in h \sum (γ - G_{m})^{2} + 1 - \frac{1}{1 + l n l ( h )}], V_{n} (e) = \frac{1}{2} [\frac{1}{k ( g )} η \in g \sum (η - G_{n})^{2} + 1 - \frac{1}{1 + l n k ( g )}]; d_{w p g} (A_{i}, A^{\pm}) = (\frac{1}{2} j = 1 \sum n w_{j} [α ∣ G_{m} (a_{ij}) - G_{m} (a_{j}^{\pm}) ∣^{λ} + α ∣ G_{n} (a_{ij}) - G_{n} (a_{j}^{\pm}) ∣^{λ} + β ∣ V_{m} (a_{ij}) - V_{m} (a_{j}^{\pm}) ∣^{λ} + β ∣ V_{n} (a_{ij}) - V_{n} (a_{j}^{\pm}) ∣^{λ}])^{1/ λ}, α + β = 1, λ \geq 1

5. Compute closeness and rank alternatives

C C_{i} = \frac{d _{i}^{-}}{d _{i}^{+} + d _{i}^{-}}, d_{i}^{+} + d_{i}^{-} > 0, rank (i) = 1 + # {k : C C_{k} > C C_{i}}