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Ranking

HFL-CODAS - Hesitant Fuzzy Linguistic CODAS (HFLTS-envelope, Combinative Distance-based Assessment)

Hesitant fuzzy linguistic ranking - HFLTS envelope (linguistic interval / trapezoidal Liu-Rodriguez envelope) + combinative distance-based assessment

Yalçın, N., Pehlivan, N.Y.2019doi:10.3390/sym11040493 ↗

Overview

Use the performance CLE scale N..P and the separate criterion-importance scale DL..DH. Distances are always recomputed from the supplied CLEs.

Data: Hesitant Fuzzy Linguistic
Weights: Needs a weight source

How it works

1
DEs provide evaluations as CLEs (comparative linguistic expressions: 'at most s_i', 'lower than s_i', 'between s_i and s_j', etc.) over the linguistic term set L.
$\tilde{x}_{i} j^{C} L E = C L E (L_{0}, ..., L_{g}); co n t e x t - f r ee g r amma r G_{H} = (V_{N}, V_{T}, I, P) p er Y a l \overset{c}{\c} ı n 2019 E q s . (24) - (27) .$
Yalçın 2019, Phase 2 + Eqs.(24)-(27); Rodríguez et al. 2012.
2
Transform CLEs into HFLTSs via E_{G_H}: 'at most s_i' → {s_j | s_j ≤ s_i}, 'at least s_i' → {s_j | s_j ≥ s_i}, 'between s_i and s_j' → {s_k | s_i ≤ s_k ≤ s_j}.
$E_{G_{H}} : l l \to H_{S}; Y a l \overset{c}{\c} ı n 2019 E q s . (28) - (33) e x pl i c i tt r an s f or ma t i o n r u l es .$
Yalçın 2019, Eqs.(28)-(33); Rodríguez et al. 2012.
3
Compute fuzzy envelope of each HFLTS. Default (Yalçın): Liu-Rodriguez trapezoidal env(H_S) = T(a, b, c, d) with b, c via OWA on midpoints. Alternative (Sansabas): linguistic interval env(H_L) = [L_-, L_+] then defuzzify to scalar.
$D e f a u l t : e n v (H_{S}) = T (a, b, c, d) w h er e a = a_{L}^{i}, d = a_{R}^{j}, b = O W A_{W^{s}} (a_{M}^{i}, ..., a_{M}^{j}) w i t h w e i g h t s W_{k}^{s} = α^{g - k} (1 - α)^{k - 1} \cdot ... an d α = (g - (j - i)) / (g - 1); c = O W A_{W^{t}} (a_{M}^{i}, ..., a_{M}^{j}) . S an s aba s a l t er na t i v e : e n v (H_{L}) = [min (L_{k}), ma x (L_{k})] t h e nκ - d e f u z z i f y .$
Yalçın 2019, Eqs.(18)-(23); Liu & Rodríguez 2014; Sansabas 2019, Eq.(17).
4
Normalize the envelope decision matrix. Benefit: ñ_ij = x̃_ij / max_i(d_ij). Cost: ñ_ij = 1 - x̃_ij / max_i(d_ij) [Yalçın default] OR ñ_ij = min_i(x̃_ij) / x̃_ij [Sansabas/Keshavarz crisp].
$Y a l \overset{c}{\c} ı n E q . (47) : \tilde{n}_{i} j = \tilde{x}_{i} j / ma x_{i} (d_{i} j) i f j \in B; 1 - \tilde{x}_{i} j / ma x_{i} (d_{i} j) i f j \in C . S an s aba s E q . (2) : n_{i} j = L_{i} j / ma x_{i} (L_{i} j) i f j \in B; mi n_{i} (L_{i} j) / L_{i} j i f j \in C .$
Yalçın 2019, Eq.(47); Sansabas 2019, Eq.(2).
5
Weighted normalized matrix r̃_ij = w_j ⊗ ñ_ij using trapezoidal multiplication (Eq.40) if w_j fuzzy, scalar multiplication (Eq.41) if crisp.
$\tilde{r}_{i} j = w_{j} \otimes \tilde{n}_{i} j . T r a p ez o i d a l : T (a 1, b 1, c 1, d 1) \otimes T (a 2, b 2, c 2, d 2) = T (a 1 \cdot a 2, b 1 \cdot b 2, c 1 \cdot c 2, d 1 \cdot d 2) [Y a l \overset{c}{\c} ı n E q .40] . C r i s p : w \cdot T (a, b, c, d) = T (w \cdot a, w \cdot b, w \cdot c, w \cdot d) [Y a l \overset{c}{\c} ı n E q .41 w i t hk > 0] .$
Yalçın 2019, Eq.(48), Eqs.(40)-(41); Sansabas 2019, Eq.(3).
6
Fuzzy negative-ideal solution ñs_j = min_i r̃_ij where the min is taken over κ-defuzzified values (Eq.44).
$\tilde{n} s_{j} = a r g mi n_{i} κ (\tilde{r}_{i} j), κ (T (a, b, c, d)) = (1/3) \cdot [a + b + c + d - (c d - ab) / ((c + d) - (a + b))] [Y a l \overset{c}{\c} ı n E q .44, E q .49] . F or S an s aba s : n s_{j} = mi n_{i} r_{i} j sc a l a r .$
Yalçın 2019, Eqs.(44), (49); Sansabas 2019, Eqs.(4)-(5).
7
Compute Euclidean ED_i and Hamming HD_i distances of each alternative from the negative ideal.
$d_{E} (\tilde{A}, \tilde{B}) = s q r t ((1/6) \cdot [(a 1 - b 1)^{2} + 2 (a 2 - b 2)^{2} + 2 (a 3 - b 3)^{2} + (a 4 - b 4)^{2}]) [Y a l \overset{c}{\c} ı n E q .45] . d_{H} (\tilde{A}, \tilde{B}) = (1/6) \cdot [∣ a 1 - b 1∣ + 2∣ a 2 - b 2∣ + 2∣ a 3 - b 3∣ + ∣ a 4 - b 4∣] [Y a l \overset{c}{\c} ı n E q .46] . E D_{i} = Σ_{j} d_{E} (\tilde{r}_{i} j, \tilde{n} s_{j}), H D_{i} = Σ_{j} d_{H} (\tilde{r}_{i} j, \tilde{n} s_{j}) [Y a l \overset{c}{\c} ı n E q s .50 - 51] . F or S an s aba s (cr i s p) : E_{i} = s q r t (Σ (r_{i} j - n s_{j})^{2}), T_{i} = Σ∣ r_{i} j - n s_{j} ∣.$
Yalçın 2019, Eqs.(45), (46), (50), (51); Sansabas 2019, Eqs.(6)-(7).
8
Relative assessment matrix RA = [p_ik]_{m×m} with threshold function.
$p_{i} k = (E D_{i} - E D_{k}) + (t (E D_{i} - E D_{k}) \cdot (H D_{i} - H D_{k})), t (x) = 1 i f ∣ x ∣ \geq θ e l se 0 [Y a l \overset{c}{\c} ı n E q s .52 - 53; S an s aba s E q s .9 - 10] .$
Yalçın 2019, Eqs.(52)-(53); Sansabas 2019, Eqs.(8)-(10).
9
Assessment score AS_i = Σ_k p_ik; rank alternatives in descending AS_i.
$A S_{i} = Σ_{k = 1}^{m} p_{i} k; A_{i} ≻ A_{i^{'}} ⟺ A S_{i} > A S_{i^{'}} [Y a l \overset{c}{\c} ı n E q .54; S an s aba s E q .11] .$
Yalçın 2019, Eq.(54); Sansabas 2019, Eq.(11).

Fits when

•Preserves hesitant_fuzzy_linguistic uncertainty through the pipeline rather than premature crispification at elicitation

Edge cases and pitfalls

•τ-eşik (threshold parameter): default τ=0.02 - E_i ve E_k arasındaki fark τ'dan KÜÇÜKSE Taxicab katkısı eklenmez (kuşkulu durumda Euclidean'a güven). τ'yı büyütmek Taxicab etkisini azaltır → TOPSIS'e yaklaşma; küçültmek tüm çiftlerde Taxicab katar.
•Fuzzy envelope (Liu-Rodríguez 2014): {s_2, s_3, s_4} → trapezoidal (a, b, c, d) formülü LTS uzunluğuna ve HFLTS lower/upper bound'una bağlı; tek-sembol HFLTS singleton trapezoidal'e (a=b=c=d) düşer.
•Negatif-ideal eleman-bazlı min: bazı kriter kolonları sabit ise (tüm alternatifler aynı) o kolon hem ν̃_ij hem ñ_j^− eşit → E_i, T_i farkı yok → kriter ayırt edici değil.
•H_i = E_i + Σθ(...)·(T_i−T_k) formülünde toplam alternatifler arası karşılıklı; her alternatif diğer m−1 alternatifle Taxicab farkı taşır - paper bu nedenle 'combinative' olarak adlandırır.
•Personnel selection case (paper §4): 6 mavi-yaka aday × birden çok kriter; sıralama fuzzy EDAS/TOPSIS/WASPAS/ARAS/COPRAS ile %80+ örtüşme (paper §4.3 Spearman ρ ≈ 0.8-0.95).

A midpoint interval is not the Liu-Rodríguez fuzzy envelope.
C5 is a cost criterion in the published case.
Do not inject Table 6 ED/HD values as inputs.

How to cite

Yalçın, N.; Pehlivan, N.Y. (2019). Application of the Fuzzy CODAS Method Based on Fuzzy Envelopes for Hesitant Fuzzy Linguistic Term Sets: A Case Study on a Personnel Selection Problem. Symmetry. https://doi.org/10.3390/sym11040493

BibTeX RIS APA (.txt)DOI ↗

System ID, as it appears in reports and the API

HFL-CODAS

How it works

DEs provide evaluations as CLEs (comparative linguistic expressions: 'at most s_i', 'lower than s_i', 'between s_i and s_j', etc.) over the linguistic term set L.

\tilde{x}_{i} j^{C} L E = C L E (L_{0}, ..., L_{g}); co n t e x t - f r ee g r amma r G_{H} = (V_{N}, V_{T}, I, P) p er Y a l \overset{c}{\c} ı n 2019 E q s . (24) - (27) .

Yalçın 2019, Phase 2 + Eqs.(24)-(27); Rodríguez et al. 2012.

Transform CLEs into HFLTSs via E_{G_H}: 'at most s_i' → {s_j | s_j ≤ s_i}, 'at least s_i' → {s_j | s_j ≥ s_i}, 'between s_i and s_j' → {s_k | s_i ≤ s_k ≤ s_j}.

E_{G_{H}} : l l \to H_{S}; Y a l \overset{c}{\c} ı n 2019 E q s . (28) - (33) e x pl i c i tt r an s f or ma t i o n r u l es .

Yalçın 2019, Eqs.(28)-(33); Rodríguez et al. 2012.

Compute fuzzy envelope of each HFLTS. Default (Yalçın): Liu-Rodriguez trapezoidal env(H_S) = T(a, b, c, d) with b, c via OWA on midpoints. Alternative (Sansabas): linguistic interval env(H_L) = [L_-, L_+] then defuzzify to scalar.

D e f a u l t : e n v (H_{S}) = T (a, b, c, d) w h er e a = a_{L}^{i}, d = a_{R}^{j}, b = O W A_{W^{s}} (a_{M}^{i}, ..., a_{M}^{j}) w i t h w e i g h t s W_{k}^{s} = α^{g - k} (1 - α)^{k - 1} \cdot ... an d α = (g - (j - i)) / (g - 1); c = O W A_{W^{t}} (a_{M}^{i}, ..., a_{M}^{j}) . S an s aba s a l t er na t i v e : e n v (H_{L}) = [min (L_{k}), ma x (L_{k})] t h e nκ - d e f u z z i f y .

Yalçın 2019, Eqs.(18)-(23); Liu & Rodríguez 2014; Sansabas 2019, Eq.(17).

Normalize the envelope decision matrix. Benefit: ñ_ij = x̃_ij / max_i(d_ij). Cost: ñ_ij = 1 - x̃_ij / max_i(d_ij) [Yalçın default] OR ñ_ij = min_i(x̃_ij) / x̃_ij [Sansabas/Keshavarz crisp].

Y a l \overset{c}{\c} ı n E q . (47) : \tilde{n}_{i} j = \tilde{x}_{i} j / ma x_{i} (d_{i} j) i f j \in B; 1 - \tilde{x}_{i} j / ma x_{i} (d_{i} j) i f j \in C . S an s aba s E q . (2) : n_{i} j = L_{i} j / ma x_{i} (L_{i} j) i f j \in B; mi n_{i} (L_{i} j) / L_{i} j i f j \in C .

Yalçın 2019, Eq.(47); Sansabas 2019, Eq.(2).

Weighted normalized matrix r̃_ij = w_j ⊗ ñ_ij using trapezoidal multiplication (Eq.40) if w_j fuzzy, scalar multiplication (Eq.41) if crisp.

\tilde{r}_{i} j = w_{j} \otimes \tilde{n}_{i} j . T r a p ez o i d a l : T (a 1, b 1, c 1, d 1) \otimes T (a 2, b 2, c 2, d 2) = T (a 1 \cdot a 2, b 1 \cdot b 2, c 1 \cdot c 2, d 1 \cdot d 2) [Y a l \overset{c}{\c} ı n E q .40] . C r i s p : w \cdot T (a, b, c, d) = T (w \cdot a, w \cdot b, w \cdot c, w \cdot d) [Y a l \overset{c}{\c} ı n E q .41 w i t hk > 0] .

Yalçın 2019, Eq.(48), Eqs.(40)-(41); Sansabas 2019, Eq.(3).

Fuzzy negative-ideal solution ñs_j = min_i r̃_ij where the min is taken over κ-defuzzified values (Eq.44).

\tilde{n} s_{j} = a r g mi n_{i} κ (\tilde{r}_{i} j), κ (T (a, b, c, d)) = (1/3) \cdot [a + b + c + d - (c d - ab) / ((c + d) - (a + b))] [Y a l \overset{c}{\c} ı n E q .44, E q .49] . F or S an s aba s : n s_{j} = mi n_{i} r_{i} j sc a l a r .

Yalçın 2019, Eqs.(44), (49); Sansabas 2019, Eqs.(4)-(5).

Compute Euclidean ED_i and Hamming HD_i distances of each alternative from the negative ideal.

d_{E} (\tilde{A}, \tilde{B}) = s q r t ((1/6) \cdot [(a 1 - b 1)^{2} + 2 (a 2 - b 2)^{2} + 2 (a 3 - b 3)^{2} + (a 4 - b 4)^{2}]) [Y a l \overset{c}{\c} ı n E q .45] . d_{H} (\tilde{A}, \tilde{B}) = (1/6) \cdot [∣ a 1 - b 1∣ + 2∣ a 2 - b 2∣ + 2∣ a 3 - b 3∣ + ∣ a 4 - b 4∣] [Y a l \overset{c}{\c} ı n E q .46] . E D_{i} = Σ_{j} d_{E} (\tilde{r}_{i} j, \tilde{n} s_{j}), H D_{i} = Σ_{j} d_{H} (\tilde{r}_{i} j, \tilde{n} s_{j}) [Y a l \overset{c}{\c} ı n E q s .50 - 51] . F or S an s aba s (cr i s p) : E_{i} = s q r t (Σ (r_{i} j - n s_{j})^{2}), T_{i} = Σ∣ r_{i} j - n s_{j} ∣.

Yalçın 2019, Eqs.(45), (46), (50), (51); Sansabas 2019, Eqs.(6)-(7).

Relative assessment matrix RA = [p_ik]_{m×m} with threshold function.

p_{i} k = (E D_{i} - E D_{k}) + (t (E D_{i} - E D_{k}) \cdot (H D_{i} - H D_{k})), t (x) = 1 i f ∣ x ∣ \geq θ e l se 0 [Y a l \overset{c}{\c} ı n E q s .52 - 53; S an s aba s E q s .9 - 10] .

Yalçın 2019, Eqs.(52)-(53); Sansabas 2019, Eqs.(8)-(10).

Assessment score AS_i = Σ_k p_ik; rank alternatives in descending AS_i.

A S_{i} = Σ_{k = 1}^{m} p_{i} k; A_{i} ≻ A_{i^{'}} ⟺ A S_{i} > A S_{i^{'}} [Y a l \overset{c}{\c} ı n E q .54; S an s aba s E q .11] .

Yalçın 2019, Eq.(54); Sansabas 2019, Eq.(11).

Edge cases and pitfalls

•τ-eşik (threshold parameter): default τ=0.02 - E_i ve E_k arasındaki fark τ'dan KÜÇÜKSE Taxicab katkısı eklenmez (kuşkulu durumda Euclidean'a güven). τ'yı büyütmek Taxicab etkisini azaltır → TOPSIS'e yaklaşma; küçültmek tüm çiftlerde Taxicab katar.

•Fuzzy envelope (Liu-Rodríguez 2014): {s_2, s_3, s_4} → trapezoidal (a, b, c, d) formülü LTS uzunluğuna ve HFLTS lower/upper bound'una bağlı; tek-sembol HFLTS singleton trapezoidal'e (a=b=c=d) düşer.

•Negatif-ideal eleman-bazlı min: bazı kriter kolonları sabit ise (tüm alternatifler aynı) o kolon hem ν̃_ij hem ñ_j^− eşit → E_i, T_i farkı yok → kriter ayırt edici değil.

•H_i = E_i + Σθ(...)·(T_i−T_k) formülünde toplam alternatifler arası karşılıklı; her alternatif diğer m−1 alternatifle Taxicab farkı taşır - paper bu nedenle 'combinative' olarak adlandırır.

•Personnel selection case (paper §4): 6 mavi-yaka aday × birden çok kriter; sıralama fuzzy EDAS/TOPSIS/WASPAS/ARAS/COPRAS ile %80+ örtüşme (paper §4.3 Spearman ρ ≈ 0.8-0.95).

A midpoint interval is not the Liu-Rodríguez fuzzy envelope.

C5 is a cost criterion in the published case.

Do not inject Table 6 ED/HD values as inputs.

How to cite

BibTeX RIS APA (.txt)DOI ↗

System ID, as it appears in reports and the API

HFL-CODAS