An extended LINMAP method for MAGDM under linguistic hesitant fuzzy environment

Abstract

The linear programming technique for multidimensional analysis of preference (LINMAP) is a representative decision making method with respect to preference information for given alternatives. In the classical LINMAP method, all of the decision data is known precisely or is given as crisp values. It cannot be used to solve the MAGDM problems under the linguistic hesitant fuzzy environment. In this paper, an extended LINMAP method is proposed to solve the MAGDM problems in which all the attribute values of alternatives and the truth degrees of all pair-wise alternatives’ comparisons are in the form of linguistic hesitant fuzzy sets (LHFSs). In this method, a formula is first presented to calculate the similarity coefficient for the LHFS. On the basis of this, the weight of each expert with respect to each attribute is determined using the support function of the power average (PA) operator. Meanwhile, the collective consistency and inconsistency measurements are introduced to depict the incomplete pair-wise comparison preference relations on alternatives provided by the experts. Then, a linear programming model is constructed to determine the optimal weights of attributes. Furthermore, by calculating the comprehensive ranking values, the ranking of alternatives can be determined. Finally, a numerical example is used to illustrate the use of the proposed method.

Keywords

Linear programming technique for multidimensional analysis of preference (LINMAP)multiple attribute group decision making (MAGDM)linguistic hesitant fuzzy set (LHFS)similarity coefficient

1 Introduction

The linear programming technique for multidimensional analysis of preference (LINMAP) is one of the existing well-known methods for multiple attribute decision making (MADM) or multiple attribute group decision making (MAGDM) problems [1]. The basic idea of the traditional LINMAP method is to define the consistency and inconsistency indices based on pair-wise comparisons of alternatives. Based on the consistency and inconsistency indices, a crisp linear programming model is constructed to derive the ideal solution and the attribute weights. Thus, the best compromise alternative has the shortest distance to the ideal solution from a set of feasible alternatives [2]. In the classical LINMAP method, all of the decision data is known precisely or is given as crisp values [3]. That is to say, the classical LINMAP method is not applicable to solve the MADM or MAGDM problems under complex decision making environments. Some studies on extensions of the LINMAP methods in a variety of different environments have been conducted [2 , 21]. For example, Xia et al. [3], Li and Yang [4], Sadi-Nezhad and Akhtari [5], Sayyaadi and Mehrabipour [6], and Bereketli et al. [7] respectively extend the LINMAP methods for solving the MADM or MAGDM prob-lems under the fuzzy environments. Li [8] and Li et al. [9] respectively extend the LINMAP methods for solving the MADM or MAGDM problems under the intuitionistic fuzzy environments. Li [10], Wang and Li [11], Wang and Liu [12], Chen [13, 14], and Wan and Dong [15] respectively extend the LINMAP methods for solving the MADM or MAGDM problems under the interval-valued intuitionistic fuzzy environments. Chen [16] extends the LINMAP method for solving the MADM problems under the interval type-2 fuzzy environment. Liu et al. [17] extend the LINMAP mehtod for solving the MADM problems under the hesitant fuzzy environment. Zhang and Xu [2], Wan and Li [18, 19], and Li and Wan [20, 21] respectively extend the LINMAP methods for solving the MADM or MAGDM problems under the heterogeneous environments.

In many situations, the use of linguistic information is suitable and straightforward because of the nature of different aspects of the problem [22]. It should be pointed out that there are some challenges for each decision maker (DM)/expert to provide his/her actual preference information by a single linguistic variable within a qualitative decision making environment due to time pressure, lack of knowledge and limited expertise about the problem domain. For providing the DMs/experts a greater flexibility and eliciting linguistic preference, Rodríguez et al. [22] introduce the hesitant fuzzy linguistic term set (HFLTS) that permits a linguistic variable to have several linguistic terms. Since the HFLTS provides many advantages in depicting DMs’/experts’ cognitions and preferences, it has attracted more and more scholars’ attentions, and some studies on this area have been conducted [23 , 28]. Similar to the linguistic variable, the HFLTS also cannot reflect the membership degree of an element to a concrete concept. For overcoming the limitation of the HFLTSs, some researches on extensions of the HFLTSs have been conducted, such as hesitant fuzzy linguistic sets (HFLSs) [29, 30], interval-valued hesitant fuzzy linguistic sets (IVHFLSs) [31], multi-hesitant fuzzy linguistic term sets (MHFLTSs) [32] and linguistic hesitant fuzzy sets (LHFSs) [33, 34]. Especially, Meng et al. [34] introduce a new kind of linguistic variables named LHFSs which can reflect the inconsistency, hesitancy and uncertainty of the DMs/experts. LHFSs are more accurate than HFLTSs and HFLSs when the DM/expert gives evaluation information under fuzzy environment [33]. For example, consider a situation where a linguistic set S = {s₀ : nothing, s₁ : very bad, s₂ : bad, s₃ : medium, s₄ : good, s₅ : very good, s₆ : perfect} is used to evaluatethe quality of a research paper. The expert may hesitant to give the value 0.3 or 0.4 for ‘bad’, the value 0.4 or 0.5 for ‘medium’ and the value 0.7 for ‘good’. The LHFS {(s₂, 0.3, 0.4) , (s₃, 0.4, 0.5) , (s₄, 0.7)} can be used to address this problem, whereas HFLTS and HFLS cannot. In fact, HFLTSs are a special case of LHFSs, with the membership degree being equal to 1 for each linguistic term, and HFLSs are also a special case of LHFSs when DMs/experts use only one linguistic term to assess fuzzy information [33]. However, the existing LINMAP methods are not applicable to solve the MAGDM problems under the linguistic hesitant fuzzy environment. Therefore, it is necessary to extend the LINMAP method for solving such problems.

The purpose of this paper is to propose an extended LINMAP method for solving the MAGDM problems under the linguistic hesitant fuzzy environment, i.e., all the attribute values of alternatives and the truth degrees of all pair-wise alternatives’ comparisons are in the form of LHFSs. First, a formula is presented to calculate the similarity coefficient for the LHFS. With consideration to the unknown weights of multiple experts concerning each attribute, the weight of each expert is determined using the support function of the power average (PA) operator. Then, the collective consistency and inconsistency measurements are introduced to depict the incomplete pair-wise comparison preference relations on alternatives provided by the experts. We also establish a linear programming model to determine the optimal weights of attributes. Furthermore, the ranking of alternatives can be determined by calculating the comprehensive ranking values.

The rest of this paper is arranged as follows.Section 2 gives a brief introduction to the LHFSs and support function of the PA operator. Section 3 presents an extended LINMAP method for solving the MAGDM problem under the linguistic hesitant fuzzy environment. In Section 4, a numerical example is used to illustrate the use of the proposed method. Finally, we summarize and highlight the main contributions of the proposed method in Section 5.

2 Preliminaries

This section introduces the LHFSs and the support function of the PA operator which will be used throughout the paper.

2.1 Linguistic hesitant fuzzy sets (LHFSs)

Hesitant fuzzy set (HFS) [35, 36], which permits the membership of an element to a given set represented by several possible values between 0 and 1, is powerful to determine the membership degree especially when we have several different values on it [37].

Definition 1. [35, 36] Let X be a fixed set, a hesitant fuzzy set (HFS) Z on X is in terms of a function h_Z (x) when applied to X returns a subset of [0, 1], i.e., Z = {〈 x, h_Z (x) 〉 |x ∈ X}, where h_Z (x) is a set of some different values in [0, 1], representing the possible membership degrees of the element x ∈ X to set Z. For the sake of simplicity, Xia and Xu [38, 39] called h_Z (x) a hesitant fuzzy element (HFE).

Similar to the situations of HFSs, a DM/expert may hesitate among several possible linguistic terms to assess linguistic variables within a qualitative decision making environment. Hence, motivated by the idea of HFSs, Rodríguez et al. [22] introduce the HFLTSs, and point out that the envelope of any HFLTS is an uncertain linguistic variable [23].

Definition 2. [22, 24] Let S = {s₀, . . . , s_g} be a linguistic term set. A hesitant fuzzy linguistic term set (HFLTS), H_S, is an ordered finite subset of the consecutive linguistic terms of S.

Let S = {s₀, . . . , s_g} be a linguistic term set, the empty HFLTS and the full HFLTS for a linguistic variable ϑ are respectively defined as follows [22]:

empty HFLTS: H_S (ϑ) = {},

full HFLTS: H_S (ϑ) = S.

Any other HFLTS is formed with at least one linguistic term in S.

Example 1. Let S = {s₀ : nothing, s₁ : very low, s₂ : low, s₃ : medium, s₄ : high, s₅ : very high, s₆ : perfect} be a linguistic term set, two different HFLTSs might be H_S (ϑ) = {s₂ : low, s₃ : medium}, H_S (ϑ) = {s₄ : high, s₅ : very high, s₆ : perfect}.

However, HFLTSs only provide the information about the possible linguistic terms of a linguistic variable, and they cannot reflect the possible membership degree of each linguistic term [34]. To overcome the limitation of the HFLTSs, Meng et al. [34] define a new kind of linguistic variables named LHFSs which not only give the possible linguistic terms of a linguistic variable but also consider the possible membership degrees of each linguistic term.

Definition 3. [34] Let S = {s₀, . . . , s_g} be a linguistic term set; a LHFS in S is a set denoted by LH = {(s_θ(i), lh (s_θ(i))) |s_θ(i) ∈ S}, where lh (s_θ(i)) = {r₁, r₂, . . . , r_{m
_i}} is a set with m_i values in [0,1] denoting the possible membership degrees of the element s_θ(i) ∈ S to the set LH.

The order relationship between any two LHFSs can be defined as follows:

Definition 4. [34] For a LHFS LH, E (LH) = s_e(LH) is called the expectation function of LH with $\begin{matrix} e (LH) & = & \frac{1}{| index (LH) |} (\sum_{θ (i) : θ (i) \in index (LH)} \\ \frac{θ (i)}{| lh (s_{θ (i)}) |} (\sum_{r \in lh (s_{θ (i)})} r)), \end{matrix}$ and D (LH) = s_v(LH) is called its variance function with $\begin{matrix} v (LH) & = & \frac{1}{| index (LH) |} \sum_{θ (i) : θ (i) \in index (LH)} \\ {(\frac{θ (i)}{| lh (s_{θ (i)}) |} (\sum_{r \in lh (s_{θ (i)})} r) - e (LH))}^{2}, \end{matrix}$ where |lh (s_θ(i)) | is the count of real numbers in lh (s_θ(i)), and |index (LH) | is the cardinality of index (LH) = {θ (i) | (s_θ(i), lh (s_θ(i))) ∈ LH, lh (s_θ(i)) ≠ {0}} with s_θ(i) ∈ S. The order relationship for any two LHFSs LH₁ and LH₂ is defined by

If E (LH₁) < E (LH₂), then LH₁ < LH₂.

If E (LH₁) = E (LH₂), then

${\begin{matrix} D (L H_{1}) > D (L H_{2}), & L H_{1} < L H_{2}, \\ D (L H_{1}) = D (L H_{2}), & L H_{1} = L H_{2} . \end{matrix}$

Distance measures are common tools which are widely used in measuring the deviation and closeness degrees of different arguments. Next, we give some extensions of distance measures for LHFSs.

Definition 5. Let S = {s₀, . . . , s_g} be a linguisticterm set, LH₁ = {(s_θ(i), lh₁ (s_θ(i))) |s_θ(i) ∈ S} andLH₂ = {(s_θ(i), lh₂ (s_θ(i))) |s_θ(i) ∈ S} be two LHFSs,where $l h_{1} (s_{θ (i)}) = {r_{1}^{1}, r_{2}^{1}, . . ., r_{m_{i}}^{1}}$ or $l h_{2} (s_{θ (i)}) = {r_{1}^{2}, r_{2}^{2}, . . ., r_{m_{i}}^{2}}$ is a set with m_i values in [0,1] denoting the possible membership degrees of the element s_θ(i) ∈ S to the set LH₁ or LH₂. Then, the distance measure between LH₁ and LH₂ is

$\begin{matrix} d (L H_{1}, L H_{2}) \\ = {\begin{matrix} | ind (E (L H_{1})) - ind (E (L H_{2})) |, \\ if e (L H_{1}) \neq e (L H_{2}) \\ | ind (D (L H_{1})) - ind (D (L H_{2})) |, \\ if e (L H_{1}) = e (L H_{2}) \end{matrix} \end{matrix}$ (1) where E (LH_i) is the expectation function of LH_i, D (LH_i) is the variance function of LH_i, and ind (E (LH_i)) = e (LH_i), ind (D (LH_i)) = v (LH_i), i ∈ {1, 2}.

Property 1.Let LH₁, LH₂ and LH₃ be three LHFSs. The distance measure in Definition 5 satisfies the following conditions:

d (LH₁, LH₂) ≥0 and d (LH₁, LH₂) =0 if and only if LH₁ = LH₂,

d (LH₁, LH₂) = d (LH₂, LH₁)

d (LH₁, LH₂) + d (LH₂, LH₃) ≥ d (LH₁, LH₃).

Proof. 1) By Equation (1), we have d (LH₁, LH₂) ≥0. From Definition 4, if LH₁ = LH₂, then E (LH₁) = E (LH₂) and D (LH₁) = D (LH₂). Furthermore, we can obtain e (LH₁) = e (LH₂) and v (LH₁) = v (LH₂). By Equation (1), we have d (LH₁, LH₂) =0. Also, from Equation (1), if d (LH₁, LH₂) =0, then e (LH₁) = e (LH₂) and v (LH₁) = v (LH₂). Thus, we have E (LH₁) = E (LH₂) and D (LH₁) = D (LH₂). By Definition 4, we have LH₁ = LH₂. In summary, we have that d (LH₁, LH₂) ≥0 and d (LH₁, LH₂) =0 if and only if LH₁ = LH₂.

2) Case 1: e (LH₁) ≠ e (LH₂). By Equation (1), we have that d (LH₁, LH₂) = |ind (E (LH₁)) - ind (E (LH₂)) | = |ind (E (LH₂)) - ind (E (LH₁)) | = d (LH₂, LH₁).

Case 2: e (LH₁) = e (LH₂). By Equation (1), we have that d (LH₁, LH₂) = |ind (D (LH₁)) - ind (D (LH₂)) | = |ind (D (LH₂)) - ind (D (LH₁)) | = d (LH₂, LH₁) .

In summary, we have that

d (LH₁, LH₂) = d (LH₂, LH₁).

Case 4: e (LH₁) = e (LH₃) ≠ e (LH₂). Similar to Case 1, we have that d (LH₁, LH₂) + d (LH₂, LH₃) ≥ d (LH₁, LH₃). Case 5: e (LH₁) = e (LH₂) = e (LH₃). By Equation (1), we have that d (LH₁, LH₂) + d (LH₂, LH₃) = |ind (D (LH₁)) - ind (D (LH₂)) | + |ind (D (LH₂)) - ind (D (LH₃)) | ≥ |ind (D (LH₁)) - ind (D (LH₃)) | = d (LH₁, LH₃).

In summary, we have that

d (LH₁, LH₂) + d (LH₂, LH₃) ≥ d (LH₁, LH₃).

This completes the proof.

2.2 Support function of the PA operator

The PA operator introduced by Yager [40] uses a non-linear weighted average aggregation tool, which can be defined as follows:

Definition 6. [40, 41] A PA operator of dimension n is a mapping PA : Rⁿ → R, according to the following formula: $PA (a_{1}, a_{2}, . . ., a_{n}) = \frac{\sum_{i = 1}^{n} (1 + T (a_{i})) a_{i}}{\sum_{i = 1}^{n} (1 + T (a_{i}))},$ where $T (a_{i}) = \sum_{j = 1, j \neq i}^{n} Sup (a_{i}, a_{j})$ (2) and Sup (a, b) is the support for a from b, which satisfies the following three properties:

Sup (a, b) ∈ [0, 1],

Sup (a, b) = Sup (b, a),

Sup (a, b) ≥ Sup (x, y), if |a - b| ≤ |x - y|.

The PA operator is a non-linear weighted average aggregation operator, and the weight $\frac{\sum_{i = 1}^{n} (1 + T (a_{i})) a_{i}}{\sum_{i = 1}^{n} (1 + T (a_{i}))}$ of the argument a_i depends on all the input arguments a_j (j = 1, 2, . . . , n) and allows the argument values to support each other in the aggregation process. Moreover, the support measure can be regarded as a similarity index and the closer two values are, the more they support each other.

As a crucial part of the PA operator, the support function can be defined as follows:

Definition 7. [40] Let R be the range of the values to be aggregated, then Sup : R × R → [0, 1], such that Sup (a, b) = Sup (b, a), and Sup (a, b) ≥ Sup (x, y) if |a - b| ≤ |x - y|.

Yager [40] proposes two useful forms for the support function as follows:

Sup (a, b) = Ke^-α(a-b)², K ∈ [0, 1], α ∈ [0, + ∞],

Sup (a, b) = K (1 - |a - b|^α), K ∈ [0, 1], α ∈ [0, + ∞].

3 An extended LINMAP method

In this section, we first describe the MAGDM problem under the linguistic hesitant fuzzy environment. Then, a formula is presented to calculate the similarity coefficient for the LHFS. Furthermore, on the basis of the support function of the PA operator, we propose a method for determining the weight of each expert concerning each attribute. Afterwards, we introduce the collective consistency and inconsistency measurements to depict the incomplete pair-wise comparison preference alternatives provided by experts. Based on the collective consistency and inconsistency measurements, a linear programming model is constructed to determine the optimal weights of attributes. The ranking of alternatives can be obtained according to the resulting comprehensive ranking values. At length, a decision procedure is presented.

3.1 Description of the MAGDM problem under the linguistic hesitant fuzzy environment

MAGDM usually refers to the process that multiple experts make evaluations according to their respective knowledge, experience and preference for a set of alternatives over multiple attributes, and then to rank all the alternatives or give evaluation information of each alternative, the decision results from each expert are aggregated to form an overall ranking result for these alternatives [2]. Let T = {1, 2, . . . , t}, M = {1, 2, . . . , m} and N = {1, 2, . . . , n}. Let E = {E₁, E₂, . . . , E_t} be a finite expert set, where E_l denotes the lth expert, l ∈ T; A = {A₁, A₂, . . . , A_m} be a finite alternative set, where A_i denotes the ith alternative, i ∈ M; C = {C₁, C₂, . . . , C_n} be a finite attribute set, where C_k denotes the kth attribute, k ∈ N. Ordinarily, the n attributes can be grouped into the two categories: benefit and cost [42]. The benefit attribute indicates that the larger the attribute value is, the better the attribute will be, while the cost attribute indicates that the smaller the attribute value is, the better the attribute will be. Let N_b and N_c be the subscript sets of benefit and cost attributes, respectively, which satisfy N_b ∪ N_c = N and N_b∩ N_c = ∅. Let $X_{l} = [x_{ik}^{l}]_{m \times n}$ be the decision matrix, where $x_{ik}^{l}$ is the attribute value which is represented by the LHFS given by the expert E_l, i.e., the consequence for alternative A_i with respect to attribute C_k given by the expert E_l.

Thus, a MAGDM problem under the linguistic hesitant fuzzy environment can be concisely expressed in the matrix format as follows: $\begin{matrix} X_{l} = [x_{ik}^{l}]_{m \times n} \\ = [\begin{matrix} x_{11}^{l} & x_{12}^{l} & \dots & x_{1 n}^{l} \\ x_{21}^{l} & x_{22}^{l} & \dots & x_{2 n}^{l} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ x_{m 1}^{l} & x_{m 1}^{l} & \dots & x_{mn}^{l} \end{matrix}], l \in T, \end{matrix}$ where $x_{ik}^{l} = {(s_{θ (t)}, {lh}_{ik}^{l} (s_{θ (t)})) | s_{θ (t)} \in S} \in L H_{S}$ , LH_S denotes all the LHFSs, i ∈ M, k ∈ N, l ∈ T.

In addition to the attribute values, the experts are asked for paired comparisons involving any two alternatives. Assume that the expert E_l gives the preference relations between any two alternatives as follows: $Ω_{l} = {< (i, j), o_{ij}^{l} > | A_{i} ⪰ A_{j}, o_{ij}^{l} \in {LH}_{S}, i, j \in M}$ , l ∈ T, where (i, j) expresses an ordered pair of alternatives A_i and A_j, and $o_{ij}^{l}$ indicates that the expert prefers A_i to A_j (denoted by A_i ⪰ A_j) with the truth degree $o_{ij}^{l} \in L H_{S}$ . The preference relations given by the expert are pair-wise comparisons between any two alternatives, which reflect the opinion of the expert between alternatives. Note the cardinality of Ω_l by |Ω_l|, i.e., the number of alternative pairs in Ω_l, is at most $C_{n}^{2} = n (n - 1) / 2$ . In general, the expert would not be able to specify all the preference relations, so that only some pair-wise comparisons between alternatives are given, i.e., $| Ω_{l} | < C_{n}^{2}$ .

In addition, the weights of multiple experts should be taken into account in decision making process. The weight of each expert is mostly assumed to be known or the same in the existing LINMAP methods. However, the weight of each expert with respect to each attribute is usually unknown a prior and different due to the experts may come from different field or have different knowledge structure. For this, the weight of each expert with respect to each attribute must be determined in our study. Denote the expert weighting vector with respect to each attribute by σ _k = (σ_1k, σ_2k, . . . , σ_tk) ^T (k ∈ N), where σ_lk is the relative weight of the expert E_l with respect to the attribute C_k, satisfying ∑_l∈Tσ_lk = 1 and σ_lk ∈ [0, 1] (l ∈ T, k ∈ N).

Similarly, the weights of attributes are also unknown a prior and must be determined in our study. Denote the attribute weighting vector by w = (w₁, w₂, . . . , w_n) ^T, where w_k is the relative weight of the attribute C_k, such that ∑_k∈Nw_k = 1 and w_k ∈ [0, 1] (k ∈ N). In practical decision problems, the DM may specify some preference relations on the weights of attributes according to his/her knowledge, experience and judgment. With consideration of the situation where the DM’s preference on the attribute weights is incomplete, i.e., the DM only provides partial preference information. Such information of attribute weights is usually incomplete, which can be obtained according to partial preference relations on weights given by the DM and has several different structure forms [43]. Let Λ be the set of all preference information on weights of attributes, which can be constructed by the following five forms [44 , 46], for k, s, v ∈ N, k ≠ s ≠ v:

A weak ranking: {w_k ≥ w_s},

A strict ranking: {w_k - w_s ≥ θ_k},

A ranking with multiples: {w_k ≥ θ_kw_s},

An interval form: {θ_k ≤ w_k ≤ θ_k + ɛ_k},

A ranking of differences: {w_k - w_s ≥ w_k - w_v},

where θ_k and ɛ_k are nonnegative constants.

3.2 Calculation of the similarity coefficient for LHFS

Motivated by the idea of Chen [13, 16], Wan and Dong [15], and Li and Wan [20, 21], this paper simultaneously considers anchored judgments with respect to the positive and negative ideal solutions. On the basis of this, a formula is proposed to calculate the similarity coefficient for the LHFS. And the similarity coefficients can be applied to establish the linear programming model for determining the weights assessment of attributes.

Let S = {s₀, . . . , s_g} be a linguistic term set, LH = {(s_θ(i), lh (s_θ(i))) |s_θ(i) ∈ S} be a LHFS, where lh (s_θ(i)) = {r₁, r₂, . . . , r_{m
_i}} is a set with m_i values in [0,1] denoting the possible membership degrees of the element s_θ(i) ∈ S to the set LH. Then, the positive ideal solution and negative ideal solution of LH, ${LH}^{+}$ and ${LH}^{-}$ , are respectively defined as ${LH}^{+} = {(s_{θ (i)}, \bar{lh (s_{θ (i)})}) | s_{θ (i)} = s_{g} (s_{0})}$ , ${LH}^{-} = {(s_{θ (i)}, \bar{lh (s_{θ (i)})}) | s_{θ (i)} = s_{0} (s_{g})}$ , where $\bar{lh (s_{θ (i)})} = {r_{k_{i}} | r_{k_{i}} = 1, k_{i} = 1, 2, . . ., m_{i}}$ .

Example 2. Let S = {s₀ : nothing, s₁ : very low, s₂ : low, s₃ : medium, s₄ : high, s₅ : very high, s₆ : perfect} be a linguistic term set, a LHFS may be LH = {(s₅, 0.2, 0.6) , (s₆, 0.5)}. Then, the positive ideal solution and negative ideal solution of LH, LH⁺ and LH^-, can be respectively represented by LH⁺ = {(s₆ (s₀) , 1, 1) , (s₆ (s₀) , 1)}, LH^- = {(s₀ (s₆) , 1, 1) , (s₀ (s₆) , 1)}.

Let S = {s₀, . . . , s_g} be a linguistic term set, LH = {(s_θ(i), lh (s_θ(i))) |s_θ(i) ∈ S} be a LHFS, where lh (s_θ(i)) = {r₁, r₂, . . . , r_{m
_i}} is a set with m_i values in [0,1] denoting the possible membership degrees of the element s_θ(i) ∈ S to the set LH, ${LH}^{+} = {(s_{θ (i)}, \bar{lh (s_{θ (i)})}) | s_{θ (i)} = s_{g} (s_{0})}$ and ${LH}^{-} = {(s_{θ (i)}, \bar{lh (s_{θ (i)}})) | s_{θ (i)} = s_{0} (s_{g})}$ be the positive ideal solution and negative ideal solution of LH, respectively, where $\bar{lh (s_{θ (i)})} = {r_{k_{i}} | r_{k_{i}} = 1, k_{i} = 1, 2, . . ., m_{i}}$ . Then using Equation (1), the positive ideal distance measure $d^{+} (LH, {LH}^{+})$ between LH and ${LH}^{+}$ is defined as

$\begin{matrix} d^{+} (LH, {LH}^{+}) & = & d (LH, {LH}^{+}) \\ = & {\begin{matrix} | e (LH) - e (L H^{+}) |, \\ if e (LH) \neq e (L H^{+}) \\ 0, if e (LH) = e (L H^{+}) \end{matrix} \end{matrix}$ (3) and the negative ideal distance measure $d^{-} (LH, {LH}^{-})$ between LH and ${LH}^{-}$ is defined as

$\begin{matrix} d^{-} (LH, {LH}^{-}) & = & d (LH, {LH}^{-}) \\ = & {\begin{matrix} | e (LH) - e (L H^{-}) |, \\ if e (LH) \neq e (L H^{-}) \\ 0, if e (LH) = e (L H^{-}) \end{matrix} \end{matrix}$ (4)

According to the concept of the ideal solution in MADM [47], a similarity coefficient CC (LH) (or simply CC) for any LHFS can be obtained by $CC (LH) = \frac{d^{-} (LH, {LH}^{-})}{d^{-} (LH, {LH}^{-}) + d^{+} (LH, {LH}^{+})}$ (5) where ${LH}^{+}$ and ${LH}^{-}$ are the positive ideal solution and negative ideal solution of LH, respectively, CC (LH) ∈ [0, 1] for any LHFS. Obviously, the greater the similarity coefficient CC (LH) is, the better the LH will be.

3.3 Calculation of the weights of experts

In this section, a method for computing the weight of each expert with respect to each attribute based on the support function of the PA operator is given as follows.

For the expert E_l, the similarity coefficient ${CC}_{ik}^{l}$ of $x_{ik}^{l}$ can be calculated using Equation (5), i.e., ${CC}_{ik}^{l} = \frac{d^{-} (x_{ik}^{l}, x_{ik}^{l -})}{d^{-} (x_{ik}^{l}, x_{ik}^{l -}) + d^{+} (x_{ik}^{l}, x_{ik}^{l +})}$ (6)

where i ∈ M, k ∈ N, l ∈ T, and $\begin{matrix} x_{ik}^{l +} & = & {\begin{matrix} {(s_{θ (t)}, \bar{{lh}_{ik}^{l} (s_{θ (t)})}) | s_{θ (t)} = s_{g}}, \\ if k \in N_{b} \\ {(s_{θ (t)}, \bar{{lh}_{ik}^{l} (s_{θ (t)})}) | s_{θ (t)} = s_{0}}, \\ if k \in N_{c} \end{matrix} \\ x_{ik}^{l -} & = & {\begin{matrix} {(s_{θ (t)}, \bar{{lh}_{ik}^{l} (s_{θ (t)})}) | s_{θ (t)} = s_{0}}, \\ if k \in N_{b} \\ {(s_{θ (t)}, \bar{{lh}_{ik}^{l} (s_{θ (t)})}) | s_{θ (t)} = s_{g}}, \\ if k \in N_{c} \end{matrix} \end{matrix}$

The similarity coefficient matrix ${CC}_{l} = [{CC}_{ik}^{l}]_{m \times n}$ is constructed as follows: $\begin{matrix} {CC}_{l} & = & [{CC}_{ik}^{l}]_{m \times n} \\ = & [\begin{matrix} {CC}_{11}^{l} & {CC}_{12}^{l} & \dots & {CC}_{1 n}^{l} \\ {CC}_{21}^{l} & {CC}_{22}^{l} & \dots & {CC}_{2 n}^{l} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {CC}_{m 1}^{l} & {CC}_{m 1}^{l} & \dots & {CC}_{mn}^{l} \end{matrix}], l \in T . \end{matrix}$

The support degree ${TK}_{ik}^{l}$ of $x_{ik}^{l}$ can be calculated using Equation (2), i.e., ${TK}_{ik}^{l} = \sum_{s \in T, s \neq l} Sup ({CC}_{ik}^{l}, {CC}_{ik}^{s})$ (7) where i ∈ M, k ∈ N, l ∈ T, and $Sup ({CC}_{ik}^{l}, {CC}_{ik}^{s}) = 1 - | {CC}_{ik}^{l} - {CC}_{ik}^{s} |$ , i ∈ M, k ∈ N, l, s ∈ T (l ≠ s), which satisfies the conditions in Definition 6.

The support degree matrix ${TK}_{l} = [{TK}_{ik}^{l}]_{m \times n}$ is constructed as follows: $\begin{matrix} {TK}_{l} & = & [{TK}_{ik}^{l}]_{m \times n} \\ = & [\begin{matrix} {TK}_{11}^{l} & {TK}_{12}^{l} & \dots & {TK}_{1 n}^{l} \\ {TK}_{21}^{l} & {TK}_{22}^{l} & \dots & {TK}_{2 n}^{l} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {TK}_{m 1}^{l} & {TK}_{m 1}^{l} & \dots & {TK}_{mn}^{l} \end{matrix}], l \in T . \end{matrix}$

Then, for the expert E_l, the collective support degree of $x_{ik}^{l}$ , ${CT}_{k}^{l}$ , can be calculated, i.e., ${CT}_{k}^{l} = \sum_{i \in M} {TK}_{ik}^{l}, k \in N, l \in T$ (8)

Hence, the weight of the expert E_l with respect to the attribute C_k, σ_lk, is represented by $σ_{lk} = \frac{{CT}_{k}^{l}}{\sum_{l \in T} {CT}_{k}^{l}}, k \in N, l \in T$ (9)

Obviously, the greater the collective support degree ${CT}_{k}^{l}$ is, the more relative importance of the expert E_l with respect to attribute C_k will be.

3.4 Calculation of the collective consistency and inconsistency measurements

By coupling the weights of experts, the weighted similarity coefficient matrix ${\bar{CC}}_{l} = [{\bar{CC}}_{ik}^{l}]_{m \times n}$ for the expert E_l is constructed as follows: $\begin{matrix} {\bar{CC}}_{l} & = & [{\bar{CC}}_{ik}^{l}]_{m \times n} \\ = & [\begin{matrix} {\bar{CC}}_{11}^{l} & {\bar{CC}}_{12}^{l} & \dots & {\bar{CC}}_{1 n}^{l} \\ {\bar{CC}}_{21}^{l} & {\bar{CC}}_{22}^{l} & \dots & {\bar{CC}}_{2 n}^{l} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {\bar{CC}}_{m 1}^{l} & {\bar{CC}}_{m 1}^{l} & \dots & {\bar{CC}}_{mn}^{l} \end{matrix}], l \in T, \end{matrix}$ where ${\bar{CC}}_{ik}^{l}$ is the weighted similarity coefficient of ${CC}_{ik}^{l}$ , and can be calculated by ${\bar{CC}}_{ik}^{l} = σ_{lk} {CC}_{ik}^{l}, i \in M, k \in N, l \in T$ (10)

Then, for the expert E_l, the comprehensive similarity coefficient ${ICC}_{i}^{l}$ of alternative A_i can be calculated by ${ICC}_{i}^{l} = \sum_{k \in N} w_{k} {\bar{CC}}_{ik}^{l}, i \in M, l \in T$ (11)

It is noted that the attribute vector w = (w₁, w₂, . . . , w_n) ^T is unknown a priori in the LINMAP procedure.

According to Equation (5), the set Ω_l can be transformed into the following set, i.e., $Ω_{l}^{CC} = {< (i, j), CC (o_{ij}^{l}) > | A_{i} ⪰ A_{j}, CC (o_{ij}^{l}) \in [0, 1], i, j \in M}$ , l ∈ T.

For the expert E_l, a large value of ${ICC}_{i}^{l}$ indicates a greater preference for the alternative A_i. For each ordered pair $(i, j) \in Ω_{l}^{CC}$ given by the expert E_l, we expect that the comprehensive similarity coefficients ${ICC}_{i}^{l}$ and ${ICC}_{j}^{l}$ of the alternatives A_i and A_j, which are associated with the attribute weights (i.e., $w = (w_{1}, w_{2}, . . ., w_{n})^{T}$ ), should fulfill the inequality of ${ICC}_{i}^{l} \geq {ICC}_{j}^{l}$ with the truth degree $CC (o_{ij}^{l})$ . An index $({ICC}_{i}^{l} - {ICC}_{j}^{l})^{+}$ is defined to measure the consistency between the ranking order of the alternatives A_i and A_j determined by ${ICC}_{i}^{l}$ and ${ICC}_{j}^{l}$ and the preference relation $(i, j) \in Ω_{l}^{CC}$ given by the expert E_l as follows: $({ICC}_{i}^{l} - {ICC}_{j}^{l})^{+} = {\begin{matrix} 0, if {ICC}_{i}^{l} < {ICC}_{j}^{l} \\ CC (o_{ij}^{l}) ({ICC}_{i}^{l} - {ICC}_{j}^{l}), \\ if {ICC}_{i}^{l} \geq {ICC}_{j}^{l} \end{matrix}$ (12) where $(i, j) \in Ω_{l}^{CC}$ , l ∈ T.

For the expert E_l, if ${ICC}_{i}^{l} < {ICC}_{j}^{l}$ , then the ranking order of alternatives A_i and A_j determined by ${ICC}_{i}^{l}$ and ${ICC}_{j}^{l}$ is inconsistent with the pair $(i, j) \in Ω_{l}^{CC}$ . Thus, $({ICC}_{i}^{l} - {ICC}_{j}^{l})^{+}$ is defined to be 0. While, if ${ICC}_{i}^{l} \geq {ICC}_{j}^{l}$ , then the ranking order of alternatives A_i and A_j determined by ${ICC}_{i}^{l}$ and ${ICC}_{j}^{l}$ is consistent with the pair $(i, j) \in Ω_{l}^{CC}$ with the truth degree $CC (o_{ij}^{l})$ . Thus, $({ICC}_{i}^{l} - {ICC}_{j}^{l})^{+}$ is defined to be $CC (o_{ij}^{l}) ({ICC}_{i}^{l} - {ICC}_{j}^{l})$ . Obviously, Equation (12) can be rewritten as follows:

$\begin{matrix} ({ICC}_{i}^{l} - {ICC}_{j}^{l})^{+} \\ = CC (o_{ij}^{l}) max {{ICC}_{i}^{l} - {ICC}_{j}^{l}, 0} \\ = CC (o_{ij}^{l}) max {\sum_{k \in N} ({\bar{CC}}_{ik}^{l} - {\bar{CC}}_{jk}^{l}) w_{k}, 0} \end{matrix}$ (13) where $(i, j) \in Ω_{l}^{CC}$ , l ∈ T.

Then, a consistency index G_l for the expert E_l is defined as follows:

$\begin{matrix} G_{l} & = & \sum_{(i, j) \in Ω_{l}^{CC}} {({ICC}_{i}^{l} - {ICC}_{j}^{l})}^{+} \\ = & \sum_{(i, j) \in Ω_{l}^{CC}} CC (o_{ij}^{l}) max {{ICC}_{i}^{l} - {ICC}_{j}^{l}, 0} \\ = & \sum_{(i, j) \in Ω_{l}^{CC}} CC (o_{ij}^{l}) max {\sum_{k \in N} ({\bar{CC}}_{ik}^{l} - \\ {\bar{CC}}_{jk}^{l}) w_{k}, 0}, l \in T \end{matrix}$ (14)

Hence, a collective consistency index G is defined as follows:

$\begin{matrix} G & = & \sum_{l \in T} G_{l} = \sum_{l \in T} \sum_{(i, j) \in Ω_{l}^{CC}} {({ICC}_{i}^{l} - {ICC}_{j}^{l})}^{+} \\ = & \sum_{l \in T} \sum_{(i, j) \in Ω_{l}^{CC}} CC (o_{ij}^{l}) max {{ICC}_{i}^{l} - {ICC}_{j}^{l}, 0} \\ = & \sum_{l \in T} \sum_{(i, j) \in Ω_{l}^{CC}} CC (o_{ij}^{l}) max {\sum_{k \in N} ({\bar{CC}}_{ik}^{l} - \\ {\bar{CC}}_{jk}^{l}) w_{k}, 0} \end{matrix}$ (15)

In a similar way, an index $({ICC}_{i}^{l} - {ICC}_{j}^{l})^{-}$ is defined to measure inconsistency between the ranking order of alternatives A_i and A_j determined by ${ICC}_{i}^{l}$ and ${ICC}_{j}^{l}$ and the preference relation $(i, j) \in Ω_{l}^{CC}$ given by the expert E_l as follows: $({ICC}_{i}^{l} - {ICC}_{j}^{l})^{-} = {\begin{matrix} 0, if {ICC}_{i}^{l} \geq {ICC}_{j}^{l} \\ CC (o_{ij}^{l}) ({ICC}_{j}^{l} - {ICC}_{i}^{l}), \\ if {ICC}_{i}^{l} < {ICC}_{j}^{l} \end{matrix}$ (16) where $(i, j) \in Ω_{l}^{CC}$ , l ∈ T.

Similarly, Equation (16) can be rewritten as follows:

$\begin{matrix} ({ICC}_{i}^{l} - {ICC}_{j}^{l})^{-} \\ = CC (o_{ij}^{l}) max {{ICC}_{j}^{l} - {ICC}_{i}^{l}, 0} \\ = CC (o_{ij}^{l}) max {\sum_{k \in N} ({\bar{CC}}_{jk}^{l} - {\bar{CC}}_{ik}^{l}) w_{k}, 0} \end{matrix}$ (17) where $(i, j) \in Ω_{l}^{CC}$ , l ∈ T.

Then, a collective inconsistency index B_l for the expert E_l is defined as follows:

$\begin{matrix} B_{l} & = & \sum_{(i, j) \in Ω_{l}^{CC}} {({ICC}_{i}^{l} - {ICC}_{j}^{l})}^{-} \\ = & \sum_{(i, j) \in Ω_{l}^{CC}} CC (o_{ij}^{l}) max {{ICC}_{j}^{l} - {ICC}_{i}^{l}, 0} \\ = & \sum_{(i, j) \in Ω_{l}^{CC}} CC (o_{ij}^{l}) max {\sum_{k \in N} ({\bar{CC}}_{jk}^{l} - \\ {\bar{CC}}_{ik}^{l}) w_{k}, 0}, l \in T \end{matrix}$ (18)

Hence, a collective inconsistency index B is defined as follows:

$\begin{matrix} B & = & \sum_{l \in T} B_{l} = \sum_{l \in T} \sum_{(i, j) \in Ω_{l}^{CC}} {({ICC}_{i}^{l} - {ICC}_{j}^{l})}^{-} \\ = & \sum_{l \in T} \sum_{(i, j) \in Ω_{l}^{CC}} CC (o_{ij}^{l}) max {{ICC}_{j}^{l} - {ICC}_{i}^{l}, 0} \\ = & \sum_{l \in T} \sum_{(i, j) \in Ω_{l}^{CC}} CC (o_{ij}^{l}) max {\sum_{k \in N} ({\bar{CC}}_{jk}^{l} - \\ {\bar{CC}}_{ik}^{l}) w_{k}, 0} \end{matrix}$ (19)

3.5 Determination of the optimal weights of attributes

According to the above analysis, it is known that the collective consistency and collective inconsistency indices are calculated based on the comprehensive similarity coefficients, and the comprehensive similarity coefficients combine the unknown weights of attributes. To determine the optimal attribute weights, a linear programming model is constructed. On the basis of this, the comprehensive ranking values can be obtained to determine the ranking of alternatives.

The difference between the collective consistency index G and the inconsistency index B can be easily derived from the two indices $({ICC}_{i}^{l} - {ICC}_{j}^{l})^{+}$ and $({ICC}_{i}^{l} - {ICC}_{j}^{l})^{-}$ such that

$\begin{matrix} G - B \\ = \sum_{l \in T} (G_{l} - B_{l}) \\ = \sum_{l \in T} \sum_{(i, j) \in Ω_{l}^{CC}} [{({ICC}_{i}^{l} - {ICC}_{j}^{l})}^{+} - \\ {({ICC}_{i}^{l} - {ICC}_{j}^{l})}^{-}] \\ = \sum_{l \in T} \sum_{(i, j) \in Ω_{l}^{CC}} CC (o_{ij}^{l}) [max {\sum_{k \in N} ({\bar{CC}}_{ik}^{l} - \\ {\bar{CC}}_{jk}^{l}) w_{k}, 0} - max {\sum_{k \in N} ({\bar{CC}}_{jk}^{l} - {\bar{CC}}_{ik}^{l}) w_{k}, 0}] \\ = \sum_{l \in T} \sum_{(i, j) \in Ω_{l}^{CC}} \sum_{k \in N} CC (o_{ij}^{l}) ({\bar{CC}}_{ik}^{l} - {\bar{CC}}_{jk}^{l}) w_{k} \end{matrix}$ (20)

It is unreasonable to request that the DM accepts the consequence that G is smaller than B. Thus, the condition of G - B ≥ 0 must hold. Let o be a nonnegative value given by the DM a priori. Then, the constraint is written as G - B ≥ o . This imposed condition implies that G should be greater than or equal to B, which appears to be a reasonable assumption to make, and the nonnegative value o is viewed as the DM’s lowest acceptable level toward the difference of G - B.

To determine the weighting vector w = (w₁, w₂, . . . , w_n) ^T, the linear programming model may be constructed as follows:

$\begin{matrix} min Z = B \\ s . t . G - B \geq o \\ w \in Λ \\ \sum_{k \in N} w_{k} = 1 \\ w_{k} \in [0, 1], k \in N \end{matrix}$ (21) where Λ is the set of all preference information on weights of attributes, which can be constructed by the five forms in Section 3.1.

The aim of Equation (21) is to minimize the collective inconsistency index B under the condition in which the collective inconsistency index B is smaller than or equals to the collective consistency index G by a nonnegative value o.

Using Equations (19) and (20), Equation (21) can be rewritten as follows:

$\begin{matrix} min Z = \sum_{l \in T} \sum_{(i, j) \in Ω_{l}^{CC}} CC (o_{ij}^{l}) max {\sum_{k \in N} ({\bar{CC}}_{jk}^{l} - \\ {\bar{CC}}_{ik}^{l}) w_{k}, 0} \\ s . t . \sum_{l \in T} \sum_{(i, j) \in Ω_{l}^{CC}} \sum_{k \in N} CC (o_{ij}^{l}) ({\bar{CC}}_{ik}^{l} - \\ {\bar{CC}}_{jk}^{l}) w_{k} \geq o \\ w \in Λ \\ \sum_{k \in N} w_{k} = 1 \\ w_{k} \in [0, 1], k \in N \end{matrix}$ (22)

For each pair (i, j) of $Ω_{l}^{CC}$ , let $y_{ij}^{l} = max {\sum_{k \in N} ({\bar{CC}}_{jk}^{l} - {\bar{CC}}_{ik}^{l}) w_{k}, 0}$ (23) where $(i, j) \in Ω_{l}^{CC}$ , l ∈ T. It follows that $y_{ij}^{l} \geq 0, y_{ij}^{l} \geq \sum_{k \in N} ({\bar{CC}}_{jk}^{l} - {\bar{CC}}_{ik}^{l}) w_{k}$ (24) where $(i, j) \in Ω_{l}^{CC}$ , l ∈ T.

Thus, Equation (22) can be transformed into the equivalent linear programming model as follows:

$\begin{matrix} min Z = \sum_{l \in T} \sum_{(i, j) \in Ω_{l}^{CC}} y_{ij}^{l} \\ s . t . \sum_{l \in T} \sum_{(i, j) \in Ω_{l}^{CC}} \sum_{k \in N} \\ CC (o_{ij}^{l}) ({\bar{CC}}_{ik}^{l} - {\bar{CC}}_{jk}^{l}) w_{k} \geq o \\ y_{ij}^{l} + \sum_{k \in N} ({\bar{CC}}_{ik}^{l} - {\bar{CC}}_{jk}^{l}) w_{k} \geq 0, \\ (i, j) \in Ω_{l}^{CC}, l \in T \\ y_{ij}^{l} \geq 0, (i, j) \in Ω_{l}^{CC}, l \in T \\ w \in Λ, \\ \sum_{k \in N} w_{k} = 1 \\ w_{k} \in [0, 1], k \in N \end{matrix}$ (25)

Equation (25) can be easily solved using the Simplex method to yield the optimal weighting vector $w^{*} = (w_{1}^{*}, w_{2}^{*}, . . ., w_{n}^{*})^{T}$ and the optimal values $y_{ij}^{l *}$ for each pair (i, j) of $Ω_{l}^{CC}$ . The corresponding comprehensive ranking value ${TCC}_{i}^{*}$ of alternative A_i can be obtained by ${TCC}_{i}^{*} = \sum_{l \in T} {ICC}_{i}^{l *} = \sum_{l \in T} \sum_{k \in N} w_{k}^{*} {\bar{CC}}_{ik}^{l}, i \in M$ (26)

Obviously, the greater ${TCC}_{i}^{*}$ is, the better the corresponding alternative A_i will be. The ranking of all alternatives can be obtained according to the comprehensive ranking values. And we choose the desirable alternative with the maximum comprehensive ranking value.

3.6 Decision procedure of the extended LINMAP method

In the above, the extended LINMAP method is proposed, especially the linear programming (i.e., Equation (25)) is constructed to determine the optimal attribute weighting vector. Then, the ranking of all alternatives is generated according to the comprehensive ranking values.

The procedure of the extended LINMAP method for solving the MAGDM problem under the linguistic hesitant fuzzy environment is summarized as follows:

Step 1. Identify all alternatives and attributes to be evaluated. Denote sets of all experts, alternatives and attributes by E = {E₁, E₂, . . . , E_t}, A = {A₁, A₂, . . . , A_m} and C = {C₁, C₂, . . . , C_n}, respectively.

Step 2. Construct the decision matrix X _l for the expert E_l. Then, the expert E_l gives incomplete preference relations on alternatives with the set Ω_l. Further, the DM is asked to provide the preference information on the weights of attributes. All the preference information on the weights of attributes is denoted as Λ.

Step 3. Construct the similarity coefficient matrix CC _l by Equation (6). And construct the support degree matrix TK _l by Equation (7). Then, calculate the collective support degree ${CT}_{k}^{l}$ using Equation (8). Furthermore, determine the expert weighting vector σ _k = (σ_1k, σ_2k, . . . , σ_tk) ^T using Equation (9).

Step 4. Construct the weighted similarity coefficient matrix ${\bar{CC}}_{l}$ by Equation (10). The set Ω_l can be transformed into a set $Ω_{l}^{CC}$ by Equation (5). Furthermore, according to Equations (11–25), the linear programming model can be constructed.

Step 5. Obtain the optimal attribute weighting vector $w^{*} = (w_{1}^{*}, w_{2}^{*}, . . ., w_{n}^{*})^{T}$ through solvingEquation (25).

Step 6. Calculate the comprehensive ranking value ${TCC}_{i}^{*}$ using Equation (26) and determine the ranking of alternatives according to the comprehensive ranking values obtained. The alternative with the largest ${TCC}_{i}^{*}$ value is the best choice.

4 Illustrative example

In this section, an example is used to illustrate the use of the proposed method. A software company NS (i.e., DM) desires to hire a system analysis engineer. After preliminary screening, four candidates (A₁,A₂,A₃,A₄) remain for further evaluation. To conduct the interview and to select the most suitable candidate, a committee consisted of four experts (E₁,E₂,E₃,E₄) has been formed. Here, the three benefit attributes are considered, i.e., past experience (C₁), oral communication skill (C₂), personality (C₃). For evaluating the attributes within a qualitative decision making environment, a linguistic term set is used, i.e., S = {s₀ : nothing, s₁ : very bad, s₂ : bad, s₃ : medium, s₄ : good, s₅ : very good, s₆ : perfect}. The ratings of all the candidates on each attribute are given by each expert as Tables 1–4.

According to the experts’ comprehensions and judgments, all experts give the incomplete preference relations between any two alternatives as follows:

Ω₁ = {< (1, 2) , {(s₅, 0.2, 0.4) , (s₆, 0.7)} > , < (3, 1) , {(s₂, 0.6)} > , < (2, 4) , {(s₆, 0.8)} > , < (4, 3) , {(s₄, 0.8)} >},

Ω₂ = {< (2, 1) , {(s₃, 0.3, 0.4, 0.6)} > , < (4, 3) , {(s₂, 0.2, 0.4) , (s₃, 0.3, 0.7)} > , < (1, 3) , {(s₂, 1)} >},

Ω₃ = {< (1, 2) , {(s₆, 0.4, 0.6, 0.8)} > , < (2, 3) , {(s₁, 0.1, 0.2) , (s₂, 0.3, 0.5) , (s₃, 0.6, 0.8)} >},

Ω₄ = {< (2, 1) , {(s₀, 0.5, 0.7) , (s₁, 0.8)} > , < (4, 1) , {(s₄, 0.7, 0.9) , (s₅, 0.4) , (s₆, 0.1, 0.3)} >}.

The preference information set Λ of attribute importance given by the DM is presented as follows: Λ = { w |w₃ ≥ 2w₂, 0.25 ≤ w₁ ≤ 0.35, 0.2 ≤ w₃ - w₁ ≤ 0.35}.

To solve the problem of engineer selection, the method proposed above is used and the procedure is summarized as follows.

First, the similarity coefficient matrices CC ₁, CC ₂, CC ₃ and CC ₄ are respectively constructed by Equation (6), i.e., $\begin{matrix} {CC}_{1} & = & [\begin{matrix} 0.24 & 0.15 & 0.33 \\ 0.18 & 0.36 & 0.06 \\ 0.33 & 0.40 & 0.22 \\ 0.42 & 0.44 & 0.17 \end{matrix}], \\ {CC}_{2} & = & [\begin{matrix} 0.03 & 0.53 & 0.27 \\ 0.37 & 0.16 & 0.43 \\ 0.60 & 0.26 & 0.14 \\ 0.13 & 0.47 & 0.35 \end{matrix}], \\ {CC}_{3} & = & [\begin{matrix} 0.43 & 0.13 & 0.36 \\ 0.22 & 0.28 & 0.03 \\ 0.41 & 0.26 & 0.36 \\ 0.34 & 0.38 & 0.07 \end{matrix}], \\ {CC}_{4} & = & [\begin{matrix} 0.05 & 1.00 & 0.21 \\ 0.38 & 0.27 & 0.83 \\ 0.63 & 0.33 & 0.45 \\ 0.17 & 0.42 & 0.23 \end{matrix}] . \end{matrix}$

Then, the support degree matrices TK ₁, TK ₂, TK ₃ and TK ₄ are respectively constructed using Equation (7), i.e., $\begin{matrix} {TK}_{1} & = & [\begin{matrix} 2.41 & 1.75 & 2.79 \\ 2.57 & 2.63 & 1.83 \\ 2.35 & 2.65 & 2.55 \\ 2.38 & 2.89 & 2.66 \end{matrix}], \\ {TK}_{2} & = & [\begin{matrix} 2.37 & 1.57 & 2.79 \\ 2.65 & 2.57 & 1.83 \\ 2.51 & 2.79 & 2.39 \\ 2.46 & 2.83 & 2.42 \end{matrix}], \\ {TK}_{3} & = & [\begin{matrix} 2.03 & 1.71 & 2.73 \\ 2.65 & 2.79 & 1.77 \\ 2.51 & 2.79 & 2.55 \\ 2.54 & 2.81 & 2.46 \end{matrix}], \\ {TK}_{4} & = & [\begin{matrix} 2.41 & 0.81 & 2.67 \\ 2.63 & 2.79 & 1.03 \\ 2.45 & 2.79 & 2.37 \\ 2.54 & 2.89 & 2.66 \end{matrix}] . \end{matrix}$

Using Equation (8), the collective support degree ${CT}_{k}^{l}$ is calculated, respectively, i.e., ${CT}_{1}^{1} = 9.71$ , ${CT}_{2}^{1} = 9.92$ , ${CT}_{3}^{1} = 9.83$ ; ${CT}_{1}^{2} = 9.99$ , ${CT}_{2}^{2} = 9.94$ , ${CT}_{3}^{2} = 9.43$ ; ${CT}_{1}^{3} = 9.73$ , ${CT}_{2}^{3} = 10.10$ , ${CT}_{3}^{3} = 9.51$ ; ${CT}_{1}^{4} = 10.03$ , ${CT}_{2}^{4} = 9.28$ , ${CT}_{3}^{4} = 8.73$ .

Furthermore, the expert weighting vector σ _k = (σ_1k, σ_2k, . . . , σ_tk) ^T can be obtained using Equation (9), i.e., $\begin{matrix} σ_{1} & = & (0.2460, 0.2532, 0.2466, 0.2542)^{T}, \\ σ_{2} & = & (0.2528, 0.2533, 0.2574, 0.2365)^{T}, \\ σ_{3} & = & (0.2621, 0.2515, 0.2536, 0.2328)^{T} . \end{matrix}$

Using Equation (5), the set Ω_l can be transformed into the following sets: $\begin{matrix} Ω_{1}^{CC} & = & {< (1, 2), 0.48 >, < (3, 1), 0.2 >, \\ < (2, 4), 0.8 >, < (4, 3), 0.53 >}, \\ Ω_{2}^{CC} & = & {< (2, 1), 0.22 >, < (4, 3), 0.18 >, \\ < (1, 3), 0.33 >}, \\ Ω_{3}^{CC} & = & {< (1, 2), 0.60 >, < (2, 3), 0.17 >}, \\ Ω_{4}^{CC} & = & {< (2, 1), 0.07 >, < (4, 1), 0.36 >} . \end{matrix}$

Using Equation (10), the weighted similarity coefficient matrices ${\bar{CC}}_{1}$ , ${\bar{CC}}_{2}$ , ${\bar{CC}}_{3}$ and ${\bar{CC}}_{4}$ can be respectively constructed, i.e., $\begin{matrix} {\bar{CC}}_{1} & = & [\begin{matrix} 0.06 & 0.04 & 0.09 \\ 0.04 & 0.09 & 0.02 \\ 0.08 & 0.10 & 0.06 \\ 0.10 & 0.11 & 0.04 \end{matrix}], \\ {\bar{CC}}_{2} & = & [\begin{matrix} 0.008 & 0.13 & 0.07 \\ 0.09 & 0.04 & 0.11 \\ 0.15 & 0.07 & 0.04 \\ 0.03 & 0.12 & 0.09 \end{matrix}], \\ {\bar{CC}}_{3} & = & [\begin{matrix} 0.11 & 0.03 & 0.09 \\ 0.05 & 0.07 & 0.008 \\ 0.10 & 0.07 & 0.09 \\ 0.08 & 0.10 & 0.02 \end{matrix}], \\ {\bar{CC}}_{4} & = & [\begin{matrix} 0.01 & 0.24 & 0.05 \\ 0.10 & 0.06 & 0.19 \\ 0.16 & 0.08 & 0.10 \\ 0.04 & 0.10 & 0.05 \end{matrix}] . \end{matrix}$

Taking o = 0.01, the linear programming model can be constructed according to Equations (11–25), i.e., $\begin{matrix} min Z & = & y_{12}^{1} + y_{31}^{1} + y_{24}^{1} + y_{43}^{1} + y_{21}^{2} + y_{43}^{2} \\ + y_{13}^{2} + y_{12}^{3} + y_{23}^{3} + y_{21}^{4} + y_{41}^{4} \\ s . t . & - 0.0296 w_{1} - 1.007 w_{2} + 0.0738 w_{3} \geq 0.01 \\ y_{12}^{1} + 0.02 w_{1} - 0.05 w_{2} + 0.07 w_{3} \geq 0 \\ y_{31}^{1} + 0.02 w_{1} + 0.06 w_{2} - 0.03 w_{3} \geq 0 \\ y_{24}^{1} - 0.06 w_{1} - 0.02 w_{2} - 0.02 w_{3} \geq 0 \\ y_{43}^{1} + 0.02 w_{1} + 0.01 w_{2} - 0.02 w_{3} \geq 0 \\ y_{21}^{2} + 0.082 w_{1} - 0.09 w_{2} + 0.04 w_{3} \geq 0 \\ y_{43}^{2} - 0.12 w_{1} + 0.05 w_{2} + 0.05 w_{3} \geq 0 \\ y_{13}^{2} - 0.142 w_{1} + 0.06 w_{2} + 0.03 w_{3} \geq 0 \\ y_{12}^{3} + 0.06 w_{1} - 0.04 w_{2} + 0.082 w_{3} \geq 0 \\ y_{23}^{3} - 0.05 w_{1} - 0.082 w_{3} \geq 0 \\ y_{21}^{4} + 0.09 w_{1} - 0.18 w_{2} + 0.14 w_{3} \geq 0 \\ y_{41}^{4} + 0.03 w_{1} - 0.14 w_{2} \geq 0 \\ y_{12}^{1}, y_{31}^{1}, y_{24}^{1}, y_{43}^{1}, y_{21}^{2}, y_{43}^{2}, y_{13}^{2}, y_{12}^{3}, y_{23}^{3}, \\ y_{21}^{4}, y_{41}^{4} \geq 0 \\ w_{3} \geq 2 w_{2}, 0.25 \leq w_{1} \leq 0.35, 0.2 \leq w_{3} - \\ w_{1} \leq 0.35 \\ w_{1} + w_{2} + w_{3} = 1 \\ w_{1}, w_{2}, w_{3} \geq 0 \end{matrix}$

The optimal attribute weighting vector $w^{*} = (w_{1}^{*},$ $w_{2}^{*}, . . ., w_{n}^{*})^{T}$ and $y_{ij}^{l *}$ can be obtained by solving the above linear programming model, i.e., w ^* = (0.3103, 0.0294, 0.6603) ^T and $y_{12}^{1 *} = 0$ , $y_{31}^{1 *} = 0.0118$ , $y_{24}^{1 *} = 0.0324$ , $y_{43}^{1 *} = 0.0067$ , $y_{21}^{2 *} = 0$ , $y_{43}^{2 *} = 0.0028$ , $y_{13}^{2 *} = 0.0225$ , $y_{12}^{3 *} = 0.0367$ , $y_{23}^{3 *} = 0.0697$ , $y_{21}^{4 *} = 0$ and $y_{41}^{4 *} = 0$ .

Using Equation (26), the corresponding comprehensive ranking value ${TCC}_{i}^{*}$ can be calculated, i.e., ${TCC}_{1}^{*} = 0.2694$ , ${TCC}_{2}^{*} = 0.3111$ , ${TCC}_{3}^{*} = 0.3529$ , ${TCC}_{4}^{*} = 0.2223$ .

Therefore, the ranking order of the four candidates is generated as following: A₃ ≻ A₂ ≻ A₁ ≻ A₄. Obviously, the best selection is the candidate A₃.

As far as this example is concerned, we do the sensitivity analysis about the nonnegative value o given by the DM a priori.

As o takes the values 0.001, 0.0001, 0.00001, 0.000001, respectively, the corresponding optimal attribute weighting vector and ranking order of the four candidates are depicted in Table 5.

The above sensitivity analysis shows that the best candidate is also A₃. And the ranking order of the four candidates is not sensitive to the parameter o.

5 Conclusions

This paper proposes an extended LINMAP method for the MAGDM problems under the linguistic hesitant fuzzy environment. Using this method, the optimal attribute weights can be determined to calculate the comprehensive ranking values. Then, the ranking of alternatives can be obtained. The contributions of this paper are summarized as follows:

First, an extended LINMAP method is proposed to solve the MAGDM problem under the linguistic hesitant fuzzy environment, i.e., all the attribute values of alternatives and the truth degrees of all pair-wise alternatives’ comparisons are in the form of LHFSs.

Second, the input data, i.e., the ratings of alternatives with respect to each attribute and the truth degrees of all pair-wise alternatives’ comparisons, is allowed to be expressed in the form of LHFSs, and the LHFSs can be able to sufficiently consider the experts’ inconsistency, hesitancy and uncertainty in expressing their linguistic information.

Third, a formula is given to determine the weights of multiple experts concerning each attribute.

In terms of future research, the proposed method can be extended to solve the MAGDM problems under the heterogeneous environment, i.e., all the attribute values of alternatives and the truth degrees of all pair-wise alternatives’ comparisons are in the form of different formats.

Footnotes

Acknowledgements

This research is partly supported by the National Natural Science Foundation of China (Project No. 71271051) and the Fundamental Research Funds for the Central Universities, NEU, China (Project Nos. N130606001 and N140607001).

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