Tri-level multi-attribute group decision making based on regret theory in multi-granular linguistic contexts

Abstract

The aim of this paper is to investigate the multi-attribute group decision making (MAGDM) problems with respect to psychological behavior of regret aversion of decision makers (DMs), in which the attribute values are expressed in multi-granular linguistic terms and numerical values. This study proposes a general model based on two-tuple linguistics to deal with multi-granular linguistic information in order to manage information provided by different experts. A practical transformation approach is developed, which makes numerical values expressed in two-tuple linguistics and further converted into values in [0, 1] interval. Compared with the traditional MAGDM model, a tri-level decision making model that includes a top DM, multiple middle level DMs and multiple experts is considered. A novel mathematical programming model is constructed considering all middle level DMs’ preference information and regret aversion. Finally, an illustrative example is given to verify the proposed method. The results and comparative analysis demonstrate its effectiveness and practicability.

Keywords

Tri-level decision making multi-attribute group decision making linguistic two-tuples multi-granular regret theory

1 Introduction

Multi-attribute group decision making (MAGDM) describes a process in which the decision maker (DM) selects the optimal alternative with the attributes expressed in qualitative and quantitative contexts provided by a group of experts. Owing to the uncertainty and complexity of decision problems and the subjective preference and fuzziness of people’s thoughts, experts usually prefer to provide their opinions using linguistic information. Zadeh [1] first introduced linguistic information as a tool for representing imprecise expression. In order to deal with linguistic information without information loss and distortion, Herrera et al. [2] proposed the most widely used two-tuple linguistic representation model applied in MAGDM. In recent years, linguistic multi-attribute group decision making (LMAGDM) methods have been one of the major topics in decision sciences fields [3 –10], and have also extended to handle certain selection problems, including innovative product selection [11], hotel selection [12], online personalized recommendation [13].

In practice, however, experts with different research background tend to evaluate attributes with linguistic term sets in different cardinality. Thus, the consensus of multi-granular linguistic information has attached more and more attention in this field, and different approaches have been proposed to deal with multi-granular LMAGDM problems [14 –19]. Herrera et al. [14] offered an approach for managing linguistic information with the use of the fuzzy set theory. Furthermore, Herrera and Martínez [15] proposed a linguistic model to deal with multi-granular linguistic information assessed in linguistic term sets. In recent years, Dong et al. [16] studied a consensus-based group decision making model to resolve multi-granular unbalanced two-tuple linguistic preference relations. Chen et al. [17] reviewed the fusion process with heterogeneous preference structures including multi-granular linguistic preference relations. Zhang et al. [18] used the projection method to reach consensus with multi-granular hesitant fuzzy linguistic information.

LMAGDM problems deal not only with the fact that experts prefer to provide attribute values in multi-granular linguistic terms, but also the fact that part of the attributes are suited to expression in real numbers [19], triangular fuzzy numbers [20], and interval intuitionistic fuzzy numbers [21] et al. For MAGDM problems, attribute values are expressed in various forms, like linguistic terms, and real numbers and are defined as hybrid LMAGDM. The consistency and transformation of initial information are the major topics in hybrid LMAGDM. Most studies use the fuzzy mathematics method to transform all initial information into fuzzy numbers [22], but the accuracy of the information transformation process and the consensus of group DMs have not been adequately studied.

Owing to the complexity of hybrid LMAGDM problems, the DM has to invite a group of experts to provide comprehensive evaluation information, which is designed to improve the accuracy of information in the transformation process whether or not the linguistic terms are expressed in multi-cardinality or there are and various expression forms of attribute values. The group of experts need to provide their evaluation suggestions independently described as bi-level decision making in which the DM makes the final decisions according to the information provided by the multiple experts in existing studies. The graph of the decision making process is shown in Fig. 1 as follows. Due to the fact that a group of DMs could make evaluation results more consistent with a real situation than a single DM, Bracken et al. [23] developed a multi-level decision making model that differs from the usual bi-level decision making model. With the advantages of reasonability and credibility, the multi-level decision making model aligns MAGDM in accordance with reality, thus, recently, multi-level decision making problems have been studied increasingly [24, 25].

Fig. 1

Bi-level decision making.

With the application of multi-level decision making, psychological behavior of DMs could be affected by subjective factors when they make decisions. DMs possess limited rationality and are usually affected by emotions such as joy, regret, dislike and disappointment. Recently, the most commonly used behavior theories have been the expected utility theory [26], prospect theory [27], and regret theory [28, 29]. Regret theory has a relatively simple structure with fewer parametric hypothesis relative to other behavioral theories and it is consistent with violations of transitivity. However, existing studies of hybrid LMAGDM with the multi-level decision structure have not taken regret aversion of DMs into consideration.

To overcome the aforementioned drawbacks, we proposed a novel approach that transforms numerical attribute values into two-tuple linguistic forms, simplifying the computation process and promising accuracy of the initial evaluation information. We then developed a hybrid LMAGDM model with a multi-level decision structure that includes a top DM, middle level DMs, and experts, taking into consideration the regret aversion of the middle level DMs.

The paper is organized as follows. Section 2 provides a brief introduction to the two-tuple linguistic model and regret theory. In Section 3, the general multi-granular linguistic computation model and the transformed method between numerical attribute values and the linguistic two-tuples are presented. In Section 4, a novel approach for the LMAGDM that considers the regret aversion of DMs based on multi-level decision making is proposed. An illustrative example and comparative analysis with the existing methods are provided in Section 5. Finally, Section 6 concludes the paper.

2 Preliminaries

This section presents basic concepts of the two-tuple linguistic and reviews some formulations about regret theory that are used in the analysis later.

2.1 Two-tuple linguistic

Suppose that S = {s₀, s₁, …, s_g} is a linguistic term set, in which the element s_i represents the ith linguistic term in S. The linguistic term set satisfies the following characteristics [2 –4].

Be ordered: s_i ≤ s_j, if i ≤ j.

Max operator: max (s_i, s_j) = s_i, if s_i ≥ s_j.

Min operator: min (s_i, s_j) = s_i, if s_i ≤ s_j.

Negation operator: neg (s_i) = s_j, such that j = g - i.

Definition 1. [2 –4] Let S = {s₀, s₁, …, s_g} be a linguistic term set, and β ∈ [0, g] be a value supporting the result of a symbolic aggregation operation. The two-tuple linguistic that conveys the equivalent information to β is given by the following function: $Δ : [0, g] \to S \times [- 0.5, 0.5)$ (1) $\begin{matrix} Δ (β) = (s_{i}, α), \\ with {arraycolsep = 3 pt \begin{matrix} s_{i}, & i = round (β) \\ α = β - i, & α \in [- 0.5, 0.5) \end{matrix} \end{matrix}$ (2) where round is the usual round operation.

Definition 2. [2 –4] Let S = {s₀, s₁, …, s_g} be a linguistic term set and (s_i, α) be a two-tuple linguistic. Let β ∈ [0, g] be a value that expresses the information equivalent to the two-tuple linguistic, which can be obtained with the following function: $Δ^{- 1} : S \times [- 0.5, 0.5) \to [0, g]$ (3) $Δ^{- 1} (s_{i}, α_{i}) = α_{i} + i = β$ (4)

2.2 Regret theory

Regret theory was first proposed by Bell [28] and Looms and Sugden [29]. They defined regret as an emotive reflection of not having chosen the optimal alternative and joy as an emotive reflection of having chosen the optimal alternative.

Definition 3. [30] Let a₁, a₂, …, a_n be attribute values. The comprehensive perceived utility value of choosing attribute a₁ is as follows: $u (a_{i}) = v (a_{i}) + r (v (a_{i}) - v (a^{*})), i = 1, 2, \dots, n$ (5) where v (·) represents the utility function, r (·) represents the regret-joy function, and a* represents the attribute value of the optimal alternative.

The utility function v (·) is assumed to be monotonically increasing and decreasingly concave with v′ (·) >0 and v″ (·) <0. In general, power function (v (x) = x^γ) can be used to simulate the utility value of the alternatives, where parameter γ represents the risk aversion coefficient and satisfies 0 < γ < 1 [26]. The graph of utility function v (x) = x^γ with different γ is shown in Fig. 2.

Fig. 2

Utility function.

Regret-joy function r (·) is also monotonically increasing and decreasingly concave with r′ (·) >0 and r″ (·) <0. In general, the exponential function r (Δx) =1 - exp(- δ · (Δx)) can be used to simulate the regret-joy value of choosing an alternative [28]. The graph of the regret-joy function with different δ is shown in Fig. 3.

Fig. 3

Regret-joy function.

3 A general multi-granular linguistic model and transformation

A general method for normalizing attribute values is presented in this section by the general multi-granular linguistic model. The attribute values in the form of numerical numbers can be converted into linguistic terms with different cardinality, which can be further transformed into values in [0, 1] interval.

3.1 A general multi-granular linguistic model

Definition 4. Let S = {S^l|l = 1, 3, 5, ⋯} and $S^{l} = {s_{i}^{l} | i = 0, 1, 2, \dots, l - 1}$ , where S is a set of all possible linguistic term sets and S^l ∈ S is a linguistic term set with odd cardinality. $s_{i}^{l} \in S^{l}$ represents the (i + 1) th linguistic term of S^l. β ∈ [0, 1] is a numerical value that is equivalent to a two-tuple linguistic by the following relations: $\begin{matrix} β \to (s_{i}^{l}, α_{i}^{l}), with \\ {\begin{matrix} s_{i}^{l}, i = round (\frac{β}{1 / - (l - 1)}) = round (β (l - 1)) \\ α_{i}^{l} = \frac{β - i / - (l - 1)}{1 / - (l - 1)} = β (l - 1) - i, α_{i}^{l} \in [- 0.5, 0.5) \end{matrix} \end{matrix}$ (6) where round is the usual round operation. $s_{i}^{l}$ is the considered linguistic term closet to β, $α_{i}^{l}$ is known as symbolic translation.

Remark 1. The interval [0, 1] is viewed as a unit length line segment, which is divided into l - 1 sections, where β is a fixed point on the interval. The way this line is divided is varied. Thus β corresponds to a two-tuple linguistic described by different cardinality. The principle is expressed in Fig. 4.

Fig. 4

Numerical scale of multi-granular linguistic.

Example 1. β = 0.4 can expresses various linguistic hierarchies, including (medium, –0.4) of 5 cardinality, (poor, 0.4) of 7 cardinality, or (poor, 0.2) of 9 cardinality, which are remarked in Fig. 5.

Fig. 5

Numerical scales with β = 0.4.

Proposition 1. The negation operator of a two-tuple linguistic is $Neg (s_{i}^{l}, α_{i}^{l}) = (s_{l - 1 - i}^{l}, - α_{i}^{l})$ .

Proof. Since $Δ^{- 1} (s_{i}^{l}, α_{i}^{l}) = (α_{i}^{l} + i) / - (l - 1)$ and $Δ^{- 1} (s_{l - 1 - i}^{l}, - α_{i}^{l}) = (- α_{i}^{l} + l - 1 - i) / - (l - 1)$ ,then $(α_{i}^{l} + i) / - (l - 1) + (α_{i}^{l} + i) / - (l - 1) + (- α_{i}^{l} + l - 1 - i) / - (l - 1) = 1$ . Thus, $Neg (s_{i}^{l}, α_{i}^{l}) = (s_{l - 1 - i}^{l}, - α_{i}^{l})$ . This completes the proof of Proposition 1.

3.2 A novel method for normalizing numerical attribute values

In hybrid LMAGDM problems, parts of attributes are expressed in numerical values. In practice, however, DMs not only need to know attribute values but also need to understand which level the attribute values are. For example, experts may provide 7 L of oil consumption for a car per 100 km, but DMs do not necessarily know what the numerical value 7 means, whether it represents an oil efficient car or not. In order to handle this problem, we proposed an approach that transformed numerical values into a two-tuple linguistic to grasp the practical meaning of attribute values.

In real life, numerical values and linguistic terms are corresponding equally in some phenomenon, with regards to students’ performance, take 5 cardinality linguistic terms as an example, grades in [50, 60] equals $s_{0}^{5}$ , grades in [60, 70] equals $s_{1}^{5}$ , grades in [70, 80] equals $s_{2}^{5}$ , grades in [80, 90] equals $s_{3}^{5}$ , grades in [90, 100] equals $s_{4}^{5}$ . However, numerical information that linguistic terms correspond to is not always isotonic, for example, the rating of cost cannot be divided meanly. Suppose that cost in [a, a + b] equals $s_{2}^{5}$ , then cost in [a - c, a] equals $s_{3}^{5}$ with c ≤ b, and cost in [a - d, a - c] equals $s_{4}^{5}$ , with d - c ≤ c. Thus, because it is very difficult for cost reduction in practice, for this kind of attributes, we proposed a transformation function to overcome this kind of difficulty.

Let

S^{l_{1}} = {s_{0}^{l_{1}}, s_{1}^{l_{1}}, \dots, s_{l_{1} - 1}^{l_{1}}}

S^{l_{2}} = {s_{0}^{l_{2}}, s_{1}^{l_{2}}, \dots, s_{l_{2} - 1}^{l_{2}}}

be two linguistic term sets. According to industry practices and experts’ experience, the relations between numerical attribute values and linguistic terms are shown as follows, with Table 1 representing benefit attributes and Table 2 representing cost attributes.

Table 1
Corresponding relations between numerical attribute values and linguistic terms

Numerical attribute value	Linguistic term
>b₀	${\tilde{s}}_{0}^{l_{1}}$
[b₀, b₁)	$s_{0}^{l_{1}}$
[b₁, b₂)	$s_{1}^{l_{1}}$
⋮	⋮
[b_i, b_i+1)	$s_{i}^{l_{1}}$
⋮	⋮
[b_l₁-1, b_{l ₁})	$s_{l_{1} - 1}^{l_{1}}$
<b_{l ₁}	${\tilde{s}}_{l_{1} - 1}^{l_{1}}$

Table 2

Corresponding relations between numerical attribute values and linguistic terms

Numerical attribute value	Linguistic term
>c_{l ₂}	${\tilde{s}}_{0}^{l_{2}}$
[c_l₂-1, c_{l ₂})	$s_{0}^{l_{2}}$
[c_l₂-2, c_l₂-1)	$s_{1}^{l_{2}}$
⋮	⋮
[c_l₂-1-j, c_l₂-j)	$s_{j}^{l_{2}}$
⋮	⋮
[c₀, c₁)	$s_{l_{2} - 1}^{l_{2}}$
<c₀	$s_{l_{2} - 1}^{l_{2}}$

where

{\tilde{s}}_{0}^{l_{1}}

{\tilde{s}}_{0}^{l_{2}}

{\tilde{s}}_{l_{1} - 1}^{l_{1}}

{\tilde{s}}_{l_{2} - 1}^{l_{2}}

represent extend linguistic terms of

s_{0}^{l_{1}}

s_{0}^{l_{2}}

s_{l_{1} - 1}^{l_{1}}

s_{l_{2} - 1}^{l_{2}}

, respectively, and express the same level in linguistic evaluation with

s_{0}^{l_{1}}

s_{0}^{l_{2}}

s_{l_{1} - 1}^{l_{1}}

s_{l_{2} - 1}^{l_{2}}

, respectively. Accordingly, attribute values are constrained by minimum and maximum values, for example, attribute values greater than b₀ or less than b_{l
₁} as shown in Table 1 and attribute values greater than c_{l
₂} or less than c₀ as shown in Table 2 are probably not involved in practice. In this paper, only common attribute value range is considered.

In real life, most phenomena approximately obey a certain probability distribution. The most common is normal distribution or uniform distribution, while the latter is convenient for computation. In this paper, suppose that benefit attribute value b_i obeys uniform distribution in the [b_i, b_i+1) interval and cost attribute value c_j obeys uniform distribution in the [c_l₂-1-j, c_l₂-j) interval. The formulations for transforming numerical attribute value to two-tuple linguistic are proposed as follows.

Definition 5. For any benefit attribute value b can convey the equivalent information to two-tuple linguistic $(s_{i}^{l_{1}}, α_{i}^{l_{1}})$ by function ξ₁ as follows: $ξ_{1} (b) = (s_{i}^{l_{1}}, α_{i}^{l_{1}}), with {\begin{matrix} s_{i}^{l_{1}}, b_{i} \leq b < b_{i + 1} \\ α_{i}^{l_{1}} = \frac{b - (b_{i} + b_{i + 1}) / - 2}{b_{i + 1} - b_{i}} \end{matrix}$ (7) where $s_{i}^{l_{1}}$ represents the linguistic term corresponding to benefit attribute value in the [b_i, b_i+1) interval.

For the function ξ₁, it is obvious that an inverse function $ξ_{1}^{- 1}$ exists, which converts a two-tuple linguistic to equivalent benefit attribute value b as follows: $\begin{matrix} ξ_{1}^{- 1} (s_{i}^{l_{1}}, α_{i}^{l_{1}}) \\ = α_{i}^{l_{1}} \times (b_{i + 1} - b_{i}) + b_{i} + b_{i + 1} / - 2 \end{matrix}$ (8)

Definition 6. For any cost attribute value c can convey the equivalent information to two-tuple linguistic $(s_{j}^{l_{2}}, α_{j}^{l_{2}})$ by function ξ₂ as follows: $ξ_{2} (c) = (s_{j}^{l_{2}}, α_{j}^{l_{2}}), with {\begin{matrix} s_{j}^{l_{2}}, c_{l_{2} - 1 - j} \leq c < c_{l_{2} - j} \\ α_{j}^{l_{2}} = \frac{(c_{l - i} + c_{l - 1 - i}) / - 2 - c}{c_{l_{2} - j} - c_{l_{2} - 1 - j}} \end{matrix}$ (9) where $s_{j}^{l_{2}}$ represents the linguistic term corresponding to cost attribute value in the [c_l₂-1-j, c_l₂-j) interval.

For the function ξ₂, it is obvious that an inverse function $ξ_{2}^{- 1}$ exists, which converts a two-tuple linguistic to equivalent cost attribute value c as follows: $\begin{matrix} ξ_{2}^{- 1} (s_{j}^{l_{2}}, α_{j}^{l_{2}}) \\ = (c_{l_{2} - j} + c_{l_{2} - 1 + j}) / - 2 - α_{j}^{l_{2}} \times (c_{l_{2} - j} - c_{l_{2} - 1 - j}) \end{matrix}$ (10)

Based on Equations (7–14), the relations among attribute value, two-tuple linguistic and numerical value are shown in Fig. 6.

Fig. 6

The relationship of transformation.

Fig. 7

Tri-level MAGDM hierarchy and evaluation values.

Example 2. Let $S^{7} = {s_{0}^{7}, s_{1}^{7}, \dots, s_{6}^{7}}$ be a linguistic term set. The corresponding relationships between oil consumption of a car per 100 km and linguistic terms are shown in Table 3. If 8.4 is provided as cost attribute value, then its corresponding two-tuple linguistic is $(s_{2}^{7}, 0.1)$ by Equation (9). On the contrary, there has $ξ^{- 1} (s_{2}^{7}, 0.1) = 8.4$ by Equation (10). Details of the computations are asfollows: $\begin{matrix} ξ_{2} (8.4) = (s_{2}^{7}, 0.1), with {\begin{matrix} s_{2}^{7}, 8 \leq 8.4 < 9 \\ 0.1 = \frac{(8 + 9) / - 2 - 8.4}{9 - 8} \end{matrix} \\ ξ_{2}^{- 1} (s_{2}^{7}, 0.1) = (8 + 9) / - 2 - 0.1 \times (9 - 8) = 8.4 \end{matrix}$

Table 3

Correspondence between oil consumption of a car per 100 km and linguistic term

Oil consumption of a car (L/100 km)	Linguistic term
>12	${\tilde{s}}_{0}^{7}$
[10 –12]	$s_{0}^{7}$
[9–10]	$s_{1}^{7}$
[8–9]	$s_{2}^{7}$
[6 –8]	$s_{3}^{7}$
[5.5–6]	$s_{4}^{7}$
[5–5.5]	$s_{5}^{7}$
[4–5]	$s_{6}^{7}$
<4	${\tilde{s}}_{6}^{7}$

4 A novel LMAGDM method based on the regret theory

In this section, we introduce the tri-level LMAGDM model to deal with the problem of ranking alternatives, taking into consideration the regret aversion of DMs.

Han et al. [31] proposed an uncooperative multi-follower tri-level decision making model which includes leader, middle-level follower, and bottom level follower in order to find an equilibrium solution at the vertical and horizontal levels. In this paper, motivated by this idea, a tri-level LMAGDM hierarchy is presented in Fig. 7, which includes a top DM, multiple middle level DMs and multiple experts. The extended middle level DMs are highlighted by the dashed-line box compared with Fig. 1.

4.1 Description of the LMAGDM problem

Suppose that $X^{q} = (x_{ij}^{q})_{n \times m}$ is the evaluation matrix provided by expert E_q (q = 1, 2, …, k), where $x_{ij}^{q}$ represents attribute value of alternative A_i (i = 1, 2, …, n) with respect to attribute C_j (j = 1, 2, …, m). c_j is the weight of C_j with c_j ∈ [0, 1] and $\sum_{j = 1}^{m} c_{j} = 1$ . D_p (p = 1, 2, …, g) is middle level DM, Middle level DMs make their individual decisions independently according to evaluation matrices. d_p is the weight of D_p with d_p ∈ [0, 1], and $\sum_{p = 1}^{g} d_{p} = 1$ . $e_{q}^{p}$ is the weight of E_q from the perspective of D_p with $e_{q}^{p} \in [0, 1]$ , $e_{q}^{p} \in [e 0.3 e m - q p, {\bar{e}}_{q}^{p}]$ , and $\sum_{q = 1}^{k} e_{q}^{p} = 1 (p = 1, 2, \dots, g)$ . D is the top DM who aggregates all decisions that middle level DMs make and then gives the ultimate decision.

4.2 Tri-level LMAGDM based on regret theory

In the tri-level MAGDM problem, all middle level DMs will be influenced by regret aversion when they make decisions independently according to evaluation matrices, and they have their own preference on experts. In order to resolve this problem, we propose an effective way to quantify their regret aversion with preference information.

Middle level DMs’ regret aversion are derived from two parts: one from choosing A_i referring to X^q instead of X^q*, which is called horizontal regret value. X^q* represents the evaluation matrix that provide the maximum attribute value of A_i, the other from choosing A_i instead of A_i* when referring to X^q, which is called vertical regret value. A_i* represents the optimal alternative in all alternatives. Those two kinds of regret values can be obtained by computation referring to Tables 4 and 5, respectively, as follows.

Table 4
The evaluation values of A_i given by different experts

Table 5

The different evaluation values of alternatives given by E_q

The specific expressions of two kinds of regret values are described as follows: $\begin{matrix} R_{1} (D_{p} \to E_{q} \to A_{i}) = \sum_{j = 1}^{m} R (V_{ij}^{pq} - V_{ij}^{p *}) \\ = \sum_{j = 1}^{m} (1 - exp (- δ ((e_{q}^{p} c_{j} Δ^{- 1} (s_{ij}^{l_{j} q}, α_{ij}^{l_{j} q}))^{γ} \\ - (e_{*}^{j} c_{j} Δ^{- 1} (s_{ij}^{l_{j} *}, α_{ij}^{l_{j} *}))^{γ}))) \end{matrix}$ (11) $\begin{matrix} R_{2} (D_{p} \to E_{q} \to A_{i}) = \sum_{j = 1}^{m} R (V_{ij}^{pq} - V_{* j}^{pq}) \\ = \sum_{j = 1}^{m} (1 - exp (- δ ((e_{q}^{p} c_{j} Δ^{- 1} (s_{ij}^{l_{j} q}, α_{ij}^{l_{j} q}))^{γ} \\ - (e_{q}^{p} c_{j} Δ^{- 1} (s_{* j}^{l_{j} q}, α_{* j}^{l_{j} q}))^{γ}))) \\ i = 1, 2, \dots, n, p = 1, 2, \dots, g, q = 1, 2, \dots, k . \end{matrix}$ (12) where basic assumptions are defined as follows:

R₁ (D_p → E_q → A_i) represents the horizontal regret value. R₂ (D_p → E_q → A_i) represents the vertical regret value. $V_{ij}^{pq}$ represents the utility value of D_p for choosing A_i with respect to C_j referring to X^q. $(s_{ij}^{l_{j} p}, α_{ij}^{l_{j} p})$ is the corresponding attribute value. $V_{* j}^{pq}$ represents the biggest utility value of D_p with respect to C_j of all alternatives referring to X^q. $(s_{* j}^{l_{j} q}, α_{* j}^{l_{j} q})$ is the corresponding attribute value. $e_{*}^{j}$ represents the weight of E* who provided the biggest utility value with respect to C_j of all experts for a certain alternative.

Suppose that λ ∈ [0, 1] represents the psychological preference of middle level DMs, taking into consideration that middle level DMs have different psychological perception effects for two kinds of regret aversion, which are expressed in Equations (11 and 12), respectively. The smaller λ is, the more the middle level DM prefers the horizontal regret aversion. The preference will be maximum when λ = 0 and minimum when λ = 1. The comprehensive regret value is formulated as follows: $\begin{matrix} R (D_{p} \to E_{q} \to A_{i}) = λ R_{1} (D_{p} \to E_{q} \to A_{i}) \\ + (1 - λ) R_{2} (D_{p} \to E_{q} \to A_{i}) \\ i = 1, 2, \dots, n, p = 1, 2, \dots, g, q = 1, 2, \dots, k \end{matrix}$ (13)

To better express regret value, we give the formulations of comprehensive perceived utility value from the perspective of alternatives and middle level DMs, respectively, as follows: $\begin{matrix} U (A_{i}) = V (A_{i}) + R (A_{i}) \\ = \sum_{p = 1}^{g} V (D_{p} \to A_{i}) + \sum_{p = 1}^{g} R (D_{p} \to A_{i}) \\ = \sum_{p = 1}^{g} \sum_{q = 1}^{k} V (D_{p} \to E_{q} \to A_{i}) \\ + \sum_{p = 1}^{g} \sum_{q = 1}^{k} R (D_{p} \to E_{q} \to A_{i}) \\ i = 1, 2, \dots n \end{matrix}$ (14) $\begin{matrix} U (D_{p}) = V (D_{p}) + R (D_{p}) \\ = \sum_{i = 1}^{n} V (D_{p} \to A_{i}) + \sum_{i = 1}^{n} R (D_{p} \to A_{i}) \\ = \sum_{q = 1}^{k} \sum_{i = 1}^{n} V (D_{p} \to E_{q} \to A_{i}) \\ + \sum_{q = 1}^{k} \sum_{i = 1}^{n} R (D_{p} \to E_{q} \to A_{i}) \\ p = 1, 2, \dots, g \end{matrix}$ (15) where basic assumptions are defined as follows:

U (A_i) represents the comprehensive perceived utility value of choosing A_i.

U (D_p) represents the sum of the comprehensive perceived utility value of D_p for choosing each alternative.

V (A_i) represents the utility value of choosing A_i.

V (D_p) represents the sum of the utility value of D_p for choosing each alternative.

R (A_i) represents the regret value of choosing A_i.

R (D_p) represents the sum of the regret value of D_p for choosing a certain alternative instead of the optimal alternative.

V (D_p → A_i) represents the utility value of D_p for choosing A_i.

V (D_p → E_q → A_i) represents the utility value of D_p for choosing A_i referring to X^q.

R (D_p → A_i) represents the regret value of D_p for choosing A_i instead of the optimal alternative.

Based on aforementioned analysis, the comprehensive perceived utility value of D_p is described as follows: $\begin{matrix} U (D_{p}) = \sum_{i = 1}^{n} \sum_{q = 1}^{k} \sum_{j = 1}^{m} (e_{q}^{p} c_{j} Δ^{- 1} (s_{ij}^{l_{j} q}, α_{ij}^{l_{j} q}))^{γ} \\ + λ \sum_{i = 1}^{n} \sum_{q = 1}^{k} \sum_{j = 1}^{m} (1 - exp (- δ ((e_{q}^{p} c_{j} Δ^{- 1} (s_{ij}^{l_{j} q}, α_{ij}^{l_{j} q}))^{γ} \\ - (e_{*}^{j} c_{j} Δ^{- 1} (s_{ij}^{l_{j *}}, α_{ij}^{l_{j *}}))^{γ}))) \\ + (1 - λ) \sum_{i = 1}^{n} \sum_{q = 1}^{k} \sum_{j = 1}^{m} (1 - exp (- δ ((e_{q}^{p} c_{j} Δ^{- 1} (s_{ij}^{l_{j} q}, α_{ij}^{l_{j} q}))^{γ} \\ - (e_{q}^{p} c_{j} Δ^{- 1} (s_{* j}^{l_{j} q}, α_{* j}^{l_{j} q}))^{γ}))) \end{matrix}$ (16)

In summary, for the aforementioned LMAGDM problem based on regret theory, the procedure of choosing the optimal alternative can be described as follows:

Step 1. Transform initial evaluation matrix $X^{q} = (x_{ij}^{q})_{n \times m}$ into two-tuple linguistic evaluation matrix $E^{q} = ((s_{ij}^{l_{j} q}, α_{ij}^{l_{j} q}))_{n \times m}$ using Definitions 5 and 6.

Step 2. Establish a mathematical programming model by maximizing the comprehensive perceived utility value of D_p as follows to obtain the result of $e_{q}^{p}$ . $\begin{array}{l} \max U (D_{p}) \\ s . t . \sum_{q = 1}^{k} e_{q}^{p} = 1 \\ e_{q}^{p} \leq e_{q}^{p} \leq {\bar{e}}_{q}^{p}, p = 1, 2, \dots, g \\ e_{q}^{p} \in [0, 1] \end{array}$ where $e 0.3 e m - q p$ and ${\bar{e}}_{q}^{p}$ represent the lower bound and upper bound of $e_{q}^{p}$ , respectively. The results can be obtained by LINGO11.0, which be denoted as $ω_{e^{p}} = (e_{1}^{p}, e_{2}^{p}, \dots, e_{q}^{p}, \dots, e_{k}^{p}), p = 1, 2, \dots, g$ .

Step 3. Calculate the comprehensive perceived utility value of middle level DMs for choosing each alternative referring to each evaluation matrix by Equations (11–13), which is denoted as $U_{qi}^{p}$ . p = 1, 2, …, g, q = 1, 2, …, k, i = 1, 2, …, n.

Step 4. Establish mathematical programming model as follows to obtain the result of d_p by maximizing the consistency of all middle level DMs.

Distance measures, such as Euclidean distance and Hamming distance, are common tools used widely to measure consistency of different ideas. Euclidean distance is simple and convenient for computing. Thus, we establish a mathematical programming model as follows by minimizing the sum of Euclidean distance D_ed between any two middle level DMs. $\begin{array}{l} \min D_{e d} = \sum_{_{p_{1} \neq p_{2}}^{p_{1}, p_{2} = 1}}^{g} \sum_{i = 1}^{n} {(\sum_{q = 1}^{k} (d_{p_{1}} U (D_{p}_{1} \to E_{q} \to A_{i}) - d_{p_{2}} U (D_{p_{2}} \to E_{q} \to A_{i}))^{2})}^{1 / 2} \\ s . t . \sum_{p = 1}^{g} d_{p} = 1 \\ d_{p} \in [0, 1], p = 1, 2, \dots, g \end{array}$ (M2)

The result can be obtained by LINGO 11.0, which be written as ω_d = (d₁, d₂, …, d_g).

Step 5. Calculate the comprehensive perceived utility value of each alternative with the following: $\begin{matrix} U (A_{i}) & = & \sum_{p = 1}^{g} \sum_{q = 1}^{k} d_{p} e_{q}^{p} U (D_{p} \to E_{q} \to A_{i}) \\ i = 1, 2, \dots, n \end{matrix}$ (17)

Step 6. Rank alternatives according to the value of U (A_i), and select the optimal alternative.

5 Illustrative example

Suppose that a corporation is planning to invest in a car company from four alternatives denoted as A = {A₁, A₂, A₃, A₄}, and considering five attributes that includes: car cost C₁, fuel consumption per 100 kilometers C₂, breaking performance C₃, handling stability C₄, and riding performance C₅. C₁ and C₂ are cost attributes, and C₃, C₄ and C₅ are benefit attributes. The weight of attributes are given as c = 0.25, c₂ = 0.2, c₃ = 0.18, c₄ = 0.2, c₅ = 0.17). The weights of DMs’ preferences are expressed in interval two-tuple linguistic terms, with $p^{1} = {, [s_{2}^{7}, s_{4}^{7}], [s_{1}^{7}, s_{4}^{7}]}$ , $p^{2} = {[s_{2}^{7}, s_{4}^{7}], [s_{3}^{7}, s_{5}^{7}], [s_{1}^{7}, s_{4}^{7}]}$ , $p^{3} = {[s_{3}^{7}, s_{4}^{7}], [s_{3}^{7}, s_{4}^{7}], [s_{2}^{7}, s_{4}^{7}]}$ . The common linguistic termsets are $S^{5} = {s_{0}^{5} =$ $very poor, s_{1}^{5} = poor, s_{2}^{5} = medium,$ $s_{3}^{5} = good, s_{4}^{5} = very good}$ , $S^{7} = {s_{0}^{7} = extremely poor,$ $s_{1}^{7} = very poor,$ $s_{2}^{7} = poor,$ $s_{3}^{7} = medium,$ $s_{4}^{7} = good$ $s_{5}^{7} = very good, s_{6}^{7} = extremely good}$ and $S^{9} = {s_{0}^{9} = extremely poor, s_{1}^{9} = very poor, s_{2}^{9} = poor, s_{3}^{9} = medium,$ $s_{4}^{9} = good,$ $s_{5}^{9} = very good, s_{6}^{9} = good, s_{7}^{9} = very good,$ $s_{8}^{9} = extremely good}$ .

The tri-level LMAGDM hierarchy and evaluation values are provided in Fig. 8. The relations between linguistic terms and C₁ and C₂ are shown in Tables 6 and 7, respectively.

Fig. 8

Tri-level LMAGDM hierarchy and evaluation values.

Table 6

The corresponding relations between attribute C₁, and linguistic term

Attribute C₁ (%)	Linguistic term
>60	${\tilde{S}}_{0}^{7}$
[58–60]	$S_{0}^{7}$
[55–58]	$S_{1}^{7}$
[52–55]	$S_{2}^{7}$
[48–52]	$S_{3}^{7}$
[45–48]	$S_{4}^{7}$
[43–45]	$S_{5}^{7}$
[40–43]	$S_{6}^{7}$
<40	${\tilde{S}}_{6}^{7}$

Table 7

The corresponding relations between attribute C₂, and linguistic term

Attribute C₂ (L/100km)	Linguistic term
>12	${\tilde{S}}_{0}^{7}$
[10 –12]	$S_{0}^{7}$
[9–10]	$S_{1}^{7}$
[8–9]	$S_{2}^{7}$
[6 –8]	$S_{3}^{7}$
[5.5–6]	$S_{4}^{7}$
[5–5.5]	$S_{5}^{7}$
[4–5]	$S_{6}^{7}$
<4	${\tilde{S}}_{6}^{7}$

We described the method proposed in this paper again by this illustrative example used the following steps.

Step 1. Transform the initial evaluation matrix X^q into two-tuple linguistic evaluation matrix E^q using Definitions 5 and 6 as shown in Table 8.

Step 2. Establish mathematical programming model by M1 as follows to obtain the weight of experts. $\begin{array}{l} \max U (D_{p}) = \sum_{i = 1}^{4} \sum_{q = 1}^{3} \sum_{j = 1}^{5} (e_{q}^{p} c_{j} Δ^{- 1} (s_{i j}^{l_{j q}}, α_{i j}^{l j q})) γ \\ + λ \sum_{i = 1}^{4} \sum_{q = 1}^{3} \sum_{j = 1}^{5} (1 - \exp (- δ ((e_{q}^{p} c_{j} Δ^{- 1} (s_{i j}^{l_{j q}}, α_{i j}^{l j q})) γ - (e_{*}^{j} c_{j} Δ^{- 1} (s_{i j}^{l_{j *}}, α_{i j}^{l j *})) γ))) \\ + (1 - λ) \sum_{i = 1}^{4} \sum_{q = 1}^{3} \sum_{j = 1}^{5} (1 - \exp (- δ ((e_{q}^{p} c_{j} Δ^{- 1} (s_{i j}^{l_{j q}}, α_{i j}^{l j q})) γ - (e_{q}^{p} c_{j} Δ^{- 1} (s_{* j}^{l_{j q}}, α_{* j}^{l j q})) γ))) \\ s . t . \sum_{q = 1}^{3} e_{q}^{p} = 1 () \\ {\underline{e}}_{q}^{p} \leq e_{q}^{p} \leq {\bar{e}}_{q}^{p}, p = 1, 2, 3, q = 1, 2, 3 \\ e_{q}^{p} \in [0, 1] \end{array}$ (M3)

Suppose that γ = 0.88, δ = 0.3 [27], and λ = 1 - λ = 0.5 in consideration of DMs’ preference are neutral in general. Using LINGO 11.0 to solve M3 and obtain e¹ = (0.3750, 0.2767, 0.3483), e² = (0.2993, 0.3750, 0.3257) and e³ = (0.3750, 0.3750, 0.2500).

Step 3. Calculate the comprehensive perceived utility value of middle level DMs for choosing each alternative referring to different evaluation matrices using Equations (11–13). The results are shown in Table 9.

Step 4. Establish mathematical programming model using M2 as follows to obtain the weight of middle level DMs. $\begin{array}{l} \min D_{e d} \\ = \sum_{_{p_{1} \neq p_{2}}^{p_{1}, p_{2} = 2}}^{3} \sum_{i = 1}^{4} {(d p_{1} \times U (D p_{1} \to E_{q} \to A_{i}) - d_{p 2} \times U (D_{p_{2}} \to E_{q} \to A_{i}))}^{2})^{1 / 2} \\ s . t . \sum_{p = 1}^{3} d_{p} = 1 \\ d_{1}, d_{2}, d_{3} \in [0, 1] \end{array}$ (M4)

Applying LINGO 11.0 to solve M4, obtaining ω_d = (0.3168, 0.3423, 0.3410).

Step 5. Calculate the comprehensive perceived utility value of each alternative by Equation (21), obtaining U (A₁) =0.2560, U (A₂) =0.2645, U (A₃) =0.2493, U (A₄) =0.2225.

Step 6. Rank alternatives according to the results of comprehensive perceived utility values, resulting in a rank as follows: A₂ ≻ A₁ ≻ A₃ ≻ A₄. Therefore, the corporation will invest in company A₂.

6 Comparative analysis and discussions

This section provides a sensitivity analysis to show the robustness and rationality of the influence of the new parameter on the results. Subsequently, discussions are presented to verify the advantages of the proposed method over existing methods.

6.1 The influence of parameters

The results of the influence on alternatives ranking by resetting parameters γ, δ and λ are shown in Tables 10–12, respectively.

Table 8
Two-tuple linguistic attribute values matrix

Expert	Alternative	Critiria
		C ₁	C ₂	C ₃	C ₄	C ₅
E ₁	A ₁	( $s_{3}^{7}$ , 0.5)	( $s_{3}^{7}$ , 0.1)	( $s_{3}^{7}$ , 0)	( $s_{5}^{9}$ , 0)	( $s_{2}^{5}$ , 0)
	A ₂	( $s_{1}^{7}$ , 0.1667)	( $s_{2}^{7}$ , 0.2)	( $s_{6}^{7}$ , 0)	( $s_{6}^{9}$ , 0)	( $s_{3}^{5}$ , 0)
	A ₃	( $s_{5}^{7}$ , 0.5)	( $s_{5}^{7}$ , 0.5)	( $s_{3}^{7}$ , 0)	( $s_{3}^{9}$ , 0)	( $s_{1}^{5}$ , 0)
	A ₄	( $s_{2}^{7}$ , 0.1667)	( $s_{3}^{7}$ , 0.5)	( $s_{4}^{7}$ , 0)	( $s_{4}^{9}$ , 0)	( $s_{2}^{5}$ , 0)
E ₁	A ₁	( $s_{4}^{7}$ , 0.1667)	( $s_{3}^{7}$ , 0.25)	( $s_{4}^{7}$ , 0)	( $s_{4}^{9}$ , 0)	( $s_{3}^{5}$ , 0)
	A ₂	( $s_{2}^{7}$ , 0.1667)	( $s_{1}^{7}$ , 0)	( $s_{5}^{7}$ , 0)	( $s_{7}^{9}$ , 0)	( $s_{4}^{5}$ , 0)
	A ₃	( $s_{4}^{7}$ , 0.1667)	( $s_{4}^{7}$ , 0.1)	( $s_{2}^{7}$ , 0)	( $s_{4}^{9}$ , 0)	( $s_{2}^{5}$ , 0)
	A ₄	( $s_{3}^{7}$ , 0)	( $s_{3}^{7}$ , 0.2)	( $s_{3}^{7}$ , 0)	( $s_{3}^{9}$ , 0)	( $s_{2}^{5}$ , 0)
E ₁	A ₁	( $s_{4}^{7}$ , 0.5)	( $s_{3}^{7}$ , 0.25)	( $s_{3}^{7}$ , 0)	( $s_{5}^{9}$ , 0)	( $s_{2}^{5}$ , 0)
	A ₂	( $s_{1}^{7}$ , 0.5)	( $s_{2}^{7}$ , 0.1)	( $s_{5}^{7}$ , 0)	( $s_{7}^{9}$ , 0)	( $s_{4}^{5}$ , 0)
	A ₃	( $s_{6}^{7}$ , 0.5)	( $s_{3}^{7}$ , 0.4)	( $s_{3}^{7}$ , 0)	( $s_{3}^{9}$ , 0)	( $s_{2}^{5}$ , 0)
	A ₄	( $s_{3}^{7}$ , 0.5)	( $s_{3}^{7}$ , 0.1)	( $s_{4}^{7}$ , 0)	( $s_{5}^{9}$ , 0)	( $s_{2}^{5}$ , 0)

Table 9

Comprehensive perceived utility value of alternatives

Middle level	Alternative	Criteria
DM	C ₁	C ₂	C ₃
D ₁	A ₁	0.2564	0.2474	0.2573
	A ₂	0.2782	0.2266	0.2741
	A ₃	0.2742	0.1955	0.2649
	A ₄	0.2348	0.1716	0.2555
D ₂	A ₁	0.1937	0.3275	0.2312
	A ₂	0.2184	0.3096	0.2545
	A ₃	0.2176	0.2715	0.2484
	A ₄	0.1849	0.2365	0.2382
	D ₃	A ₁	0.2476	0.3275	0.1725
	A ₂	0.2781	0.3106	0.1925
	A ₃	0.2773	0.2732	0.1887
	A ₄	0.2379	0.2398	0.1835

Table 10

Ranking of alternatives with different γ

γ	U (A₁)	U (A₂)	U (A₃)	U (A₄)	Ranking of alternatives
0.1	3.5620	3.5181	3.5295	3.5147	A₂ ≻ A₁ ≻ A₃ ≻ A₄
0.2	2.5383	2.4880	2.4961	2.4700	A₂ ≻ A₁ ≻ A₃ ≻ A₄
0.3	1.8093	1.7682	1.7687	1.7354	A₂ ≻ A₁ ≻ A₃ ≻ A₄
0.4	1.2901	1.2625	1.2558	1.2190	A₁ ≻ A₂ ≻ A₃ ≻ A₄
0.5	0.9202	0.9054	0.8934	0.8561	A₁ ≻ A₂ ≻ A₃ ≻ A₄
0.6	0.6566	0.6518	0.6370	0.6011	A₁ ≻ A₂ ≻ A₃ ≻ A₄
0.7	0.4688	0.4711	0.4551	0.4219	A₂ ≻ A₁ ≻ A₃ ≻ A₄
0.8	0.3349	0.3417	0.3259	0.2963	A₂ ≻ A₁ ≻ A₃ ≻ A₄
0.9	0.2401	0.2493	0.2345	0.2088	A₂ ≻ A₁ ≻ A₃ ≻ A₄

Based on Tables 10 and 11, it is observed that with the increase of parameter γ and δ, the utility value derived from the alternative itself is increasing, but the comprehensive perceived utility value is reducing, which means the regret aversion is more obvious with the increase of γ. In addition, in Table 10, we could see that the optimal alternative changes from to A₁ to A₂ when γ > 0.6, though, at the same time, the alternatives ranking tend to be stable. Similarly, the larger the δ, the more regret the middle level DMs feel, and alternatives ranking is very robust.

Table 11

Ranking of alternatives with different δ

δ	U (A₁)	U (A₂)	U (A₃)	U (A₄)	Ranking of alternatives
0.05	0.2772	0.2838	0.2701	0.2444	A₂ ≻ A₁ ≻ A₃ ≻ A₄
0.1	0.2729	0.2829	0.2662	0.2405	A₂ ≻ A₁ ≻ A₃ ≻ A₄
0.2	0.2643	0.2724	0.2579	0.2318	A₂ ≻ A₁ ≻ A₃ ≻ A₄
0.3	0.2560	0.2645	0.2493	0.2225	A₂ ≻ A₁ ≻ A₃ ≻ A₄
0.4	0.2483	0.2591	0.2435	0.2164	A₂ ≻ A₁ ≻ A₃ ≻ A₄

Convert Table 12 into Fig. 9 as follows to better explain the influence of parameter λ on the comprehensive perceived utility value of each alternative.

Table 12

Ranking of alternatives with different

λ	U (A₁)	U (A₂)	U (A₃)	U (A₄)	Ranking of alternatives
λ = 0, 1 - λ = 1	0.2499	0.2572	0.2396	0.2063	A₂ ≻ A₁ ≻ A₃ ≻ A₄
λ = 0.1, 1 - λ = 0.9	0.2507	0.2587	0.2420	0.2102	A₂ ≻ A₁ ≻ A₃ ≻ A₄
λ = 0.2, 1 - λ = 0.8	0.2587	0.2676	0.2444	0.2141	A₂ ≻ A₁ ≻ A₃ ≻ A₄
λ = 0.3, 1 - λ = 0.7	0.2606	0.2695	0.2465	0.2182	A₂ ≻ A₁ ≻ A₃ ≻ A₄
λ = 0.4, 1 - λ = 0.6	0.2616	0.2704	0.2510	0.2204	A₂ ≻ A₁ ≻ A₃ ≻ A₄
λ = 0.5, 1 - λ = 0.5	0.2560	0.2645	0.2493	0.2225	A₂ ≻ A₁ ≻ A₃ ≻ A₄
λ = 0.6, 1 - λ = 0.4	0.2645	0.2675	0.2528	0.2275	A₂ ≻ A₁ ≻ A₃ ≻ A₄
λ = 0.7, 1 - λ = 0.3	0.2616	0.2710	0.2570	0.2327	A₂ ≻ A₁ ≻ A₃ ≻ A₄
λ = 0.8, 1 - λ = 0.2	0.2647	0.2768	0.2611	0.2420	A₂ ≻ A₁ ≻ A₃ ≻ A₄
λ = 0.9, 1 - λ = 0.1	0.2684	0.2788	0.2659	0.2439	A₂ ≻ A₁ ≻ A₃ ≻ A₄
λ = 1, 1 - λ = 0	0.2727	0.2839	0.2716	0.2508	A₂ ≻ A₁ ≻ A₃ ≻ A₄

Fig. 9

Comprehensive perceived utility value of alternatives with different λ.

Figure 9 shows that the comprehensive perceived utility value of choosing an alternative is maximized when λ = 1 and is increasing with the increase of λ in general. The reason for this is that compared with other experts’ evaluation matrices this matrix offered bring much higher comprehensive regret-joy value than when compared only with own its own evaluation matrix, befitting the practical aspect of this study. Only when middle level DMs possess all information can they perceive the maximum value of comprehensive regret-joy. The results can be adjusted by the degree of regret aversion the middle level DMs derive from these two regret sources. Hence, the introduction of parameter λ is more reasonable and flexible.

6.2 Comparison with other methods

We reconsider the illustrative example with other two methods. Liu et al. presented an approach based on deviation and TOPSIS in a multi-granular environment [32], which is denoted as Method 1. Zhang et al. set a mathematical programming model to deal with the hybrid multi-criteria decision making problem with incomplete weight information with bi-level decision making model [19], which denoted as Method 2. The comparison results among Method 1, Method 2, and the proposed method are shown in Table 13.

Table 13
Ranking of alternatives with different methods

U (A₁) U (A₂) U (A₃) U (A₄) Ranking of alternatives

Method 1 0.5782 0.6220 0.5816 0.5031 A₂ ≻ A₃ ≻ A₁ ≻ A₄

Method 2 −0.9688 −0.8706 −0.9920 −0.9400 A₂ ≻ A₄ ≻ A₁ ≻ A₃

The proposed method 0.2560 0.2645 0.2493 0.2225 A₂ ≻ A₁ ≻ A₃ ≻ A₄

	U (A₁)	U (A₂)	U (A₃)	U (A₄)	Ranking of alternatives
Method 1	0.5782	0.6220	0.5816	0.5031	A₂ ≻ A₃ ≻ A₁ ≻ A₄
Method 2	−0.9688	−0.8706	−0.9920	−0.9400	A₂ ≻ A₄ ≻ A₁ ≻ A₃
The proposed method	0.2560	0.2645	0.2493	0.2225	A₂ ≻ A₁ ≻ A₃ ≻ A₄

Method 1 transformed all initial attribute values expressed in the form of multi-granular linguistic into linguistic terms with the same 9 cardinalities by a formulation. However, not all attributes are appropriate to expressed in linguistic terms with 9 cardinalities. Furthermore, it is easy to result in loss and distortion of information during the process of transformation of two-tuples into numerical values and computation of the deviation of two two-tuples, thus A₃ ≻ A₁, which is different from A₃ ≺ A₁ as obtained by the proposed method The proposed method avoid this problem and transfer initial multi-granular linguistic information into equivalent numerical value that belongs to the interval directly. This method is simple and convenient for computation. In addition, Method 1 does not take the psychological behavior of middle level DMs into account, thus the comprehensive perceived utility value of alternatives is larger than practice value.

Method 2 normalized linguistic information using triangular fuzzy numbers, however, it still needs to be converted into numerical values for further computation. In addition, the normalization of real number values are dependent on the maximum or minimum evaluation values. According to this method, for example, the evaluation values of attribute C₂ are 7.5, 8.4, 6.2 and 6.8 provided by X³, while the reference value is 6.2 considering C₂ is a cost attribute. Notably, all normalized values may be inaccurate if 6.2 is inaccurate. At the same time, the perceived value derived from the attribute value itself is not considered in Method 2. Furthermore, the traditional bi-level decision model does not fully take into account the psychology of DMs. As a result, A₄ ≻ A₁ ≻ A₃ differs from A₁ ≻ A₃ ≻ A₄ as obtained by the proposed method that avoids this problem.

7 Conclusion

With the increasing complexity of objects and the inherently subjective nature of people, linguistic terms are applied to attribute evaluation. The LMAGDM problem is becoming the major focus of recent studies. The traditional bi-level LMAGDM model may not be well equipped to deal with the difficulty decision making problems. In addition to multiple expert evaluations, evaluations by multiple DMs are also necessary to make decisions more effective and democratic. In this paper, we consider tri-level LMAGDM model that includes a top DM, multiple middle level DMs and multiple experts. It is crucial to assure accuracy of the initial information in the process of transformation and aggregation. Thus, a general computation model based on two-tuple linguistics for dealing with multi-granular linguistic information is proposed, and a practical transformation between numerical attribute values and linguistic terms based on this model is introduced to avoid loss and distortion of information. In addition, the psychological behavior of DMs is an important factor that cannot be ignored because of the bounded rationality of DMs’ perceptions. Ergo, we constructed a novel mathematical programming model based on the tri-level LMAGDM model that considers the regret aversion of all middle level DMs. Finally, an illustrative example is used to verify the effectiveness and practicality of the proposed method.

In future research, we would like to extend this approach to solve LMAGDM problems based on the regret theory under circumstances in which the weight of attributes is also unknown. Moreover, we wish to study the two-tuple linguistic computation model and two-tuple linguistic aggregating operators for better linguistic information fusion.

Footnotes

Acknowledgments

The paper is supported by the National Natural Science Foundation of China (Grant Nos. 71371053; 61773123).

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Tri-level multi-attribute group decision making based on regret theory in multi-granular linguistic contexts

Abstract

Keywords

1 Introduction

2.1 Two-tuple linguistic

3.1 A general multi-granular linguistic model

Table 1 Corresponding relations between numerical attribute values and linguistic terms

4.1 Description of the LMAGDM problem

4.2 Tri-level LMAGDM based on regret theory

Table 4 The evaluation values of A i given by different experts

6.1 The influence of parameters

Table 8 Two-tuple linguistic attribute values matrix

Footnotes

Acknowledgments

References

Table 1
Corresponding relations between numerical attribute values and linguistic terms

Table 4
The evaluation values of A_i given by different experts

Table 8
Two-tuple linguistic attribute values matrix