HFLPR-PRIORITY - Hesitant Fuzzy Linguistic Preference Relation Priority Programming (Ren-Zhu-Xu 2018)

1. Invite decision makers to provide: (1) a criteria HFLPR B^C = (b^C_ij)_{n×n} expressing pairwise preferences among n criteria; (2) for each criterion C_j (j=1,...,n), an alternatives HFLPR B^{C_j}_A = (b^{C_j}_{A,ik})_{m×m} expressing pairwise preferences among m alternatives with respect to criterion C_j. All HFLEs are drawn from LTS S = {s_0,...,s_{2τ}}.
$B = (b_{ij})_{n \times n}, b_{ij} = {b_{ij}^{σ (s)} ∣ s = 1, \dots, l_{b_{ij}}} \subset S$
2. For each HFLPR B, verify: b^σ(s)_ij ⊕ b^σ(s)_ji = s_{2τ} for all i<j and s=1,...,l_{b_ij}; diagonal b_ii = {s_τ}; |b_ij| = |b_ji|. Report E-1 and E-4 violations.
$b_{ij}^{σ (s)} \oplus b_{j i}^{σ (s)} = s_{2 τ}, b_{ii} = {s_{τ}}, ∣ b_{ij} ∣ = ∣ b_{j i} ∣$
3. For a given HFLPR B and priority vector w, define for each pair (i,j) with i<j and each linguistic element index s: F^s_ij(w) = w_i - (w_i + w_j) × I(b^σ(s)_ij) / (2τ), where I(s_k) = k is the index map. F^s_ij = 0 for all s iff w is perfectly consistent with all elements of b_ij.
$F_{ij}^{s} (w) = w_{i} - \frac{w _{i} + w _{j}}{2 τ} \cdot I (b_{ij}^{σ (s)}), I (s_{k}) = k$
4. For each HFLPR, solve the following LP (obtained by transforming Model 6.1 via ξ = ln[t(1-λ)+1], equivalent to minimising the maximum |F^s_ij|): min ξ s.t. ξ ≥ F^s_ij(w) and ξ ≥ -F^s_ij(w) for all i<j and all s; Σw_i=1, w_i≥0. Then λ* = 1 - (e^ξ-1)/t, and the priority vector w* is the optimal solution.
$min ξ s.t. {ξ \geq F_{ij}^{s} (w) ξ \geq - F_{ij}^{s} (w) \forall i < j, \forall s; i = 1 \sum n w_{i} = 1, w_{i} \geq 0$
5. Apply F4 to the criteria HFLPR B^C. The result w^C = (w^C_1, ..., w^C_n)^T gives the weights of the n criteria.
$w^{C} = (w_{1}^{C}, \dots, w_{n}^{C})^{T} \leftarrow Model 6.1 on B^{C}$
6. For each criterion j=1,...,n, apply F4 to the alternatives HFLPR B^{C_j}_A. The result w^{C_j}_A = (w^{C_j}_{A_1}, ..., w^{C_j}_{A_m})^T gives the priority of each alternative with respect to criterion C_j.
$w_{A}^{C_{j}} = (w_{A_{1}}^{C_{j}}, \dots, w_{A_{m}}^{C_{j}})^{T} \leftarrow Model 6.1 on B_{A}^{C_{j}}, j = 1, \dots, n$
7. Aggregate criterion-level priority vectors into a single comprehensive priority for each alternative: w_{A_i} = Σ_j w^C_j × w^{C_j}_{A_i}. Rank alternatives in descending order of w_{A_i}; the best alternative has the highest comprehensive priority.
$w_{A_{i}} = j = 1 \sum n w_{j}^{C} \cdot w_{A_{i}}^{C_{j}}, i = 1, \dots, m$

1. Invite decision makers to provide: (1) a criteria HFLPR B^C = (b^C_ij)_{n×n} expressing pairwise preferences among n criteria; (2) for each criterion C_j (j=1,...,n), an alternatives HFLPR B^{C_j}_A = (b^{C_j}_{A,ik})_{m×m} expressing pairwise preferences among m alternatives with respect to criterion C_j. All HFLEs are drawn from LTS S = {s_0,...,s_{2τ}}.

B = (b_{ij})_{n \times n}, b_{ij} = {b_{ij}^{σ (s)} ∣ s = 1, \dots, l_{b_{ij}}} \subset S

2. For each HFLPR B, verify: b^σ(s)_ij ⊕ b^σ(s)_ji = s_{2τ} for all i<j and s=1,...,l_{b_ij}; diagonal b_ii = {s_τ}; |b_ij| = |b_ji|. Report E-1 and E-4 violations.

b_{ij}^{σ (s)} \oplus b_{j i}^{σ (s)} = s_{2 τ}, b_{ii} = {s_{τ}}, ∣ b_{ij} ∣ = ∣ b_{j i} ∣

3. For a given HFLPR B and priority vector w, define for each pair (i,j) with i<j and each linguistic element index s: F^s_ij(w) = w_i - (w_i + w_j) × I(b^σ(s)_ij) / (2τ), where I(s_k) = k is the index map. F^s_ij = 0 for all s iff w is perfectly consistent with all elements of b_ij.

F_{ij}^{s} (w) = w_{i} - \frac{w _{i} + w _{j}}{2 τ} \cdot I (b_{ij}^{σ (s)}), I (s_{k}) = k

4. For each HFLPR, solve the following LP (obtained by transforming Model 6.1 via ξ = ln[t(1-λ)+1], equivalent to minimising the maximum |F^s_ij|): min ξ s.t. ξ ≥ F^s_ij(w) and ξ ≥ -F^s_ij(w) for all i<j and all s; Σw_i=1, w_i≥0. Then λ* = 1 - (e^ξ-1)/t, and the priority vector w* is the optimal solution.

min ξ s.t. {ξ \geq F_{ij}^{s} (w) ξ \geq - F_{ij}^{s} (w) \forall i < j, \forall s; i = 1 \sum n w_{i} = 1, w_{i} \geq 0

5. Apply F4 to the criteria HFLPR B^C. The result w^C = (w^C_1, ..., w^C_n)^T gives the weights of the n criteria.

w^{C} = (w_{1}^{C}, \dots, w_{n}^{C})^{T} \leftarrow Model 6.1 on B^{C}

6. For each criterion j=1,...,n, apply F4 to the alternatives HFLPR B^{C_j}_A. The result w^{C_j}_A = (w^{C_j}_{A_1}, ..., w^{C_j}_{A_m})^T gives the priority of each alternative with respect to criterion C_j.

w_{A}^{C_{j}} = (w_{A_{1}}^{C_{j}}, \dots, w_{A_{m}}^{C_{j}})^{T} \leftarrow Model 6.1 on B_{A}^{C_{j}}, j = 1, \dots, n

7. Aggregate criterion-level priority vectors into a single comprehensive priority for each alternative: w_{A_i} = Σ_j w^C_j × w^{C_j}_{A_i}. Rank alternatives in descending order of w_{A_i}; the best alternative has the highest comprehensive priority.

w_{A_{i}} = j = 1 \sum n w_{j}^{C} \cdot w_{A_{i}}^{C_{j}}, i = 1, \dots, m