Fuzzy similarity metrics and their application to consensus reaching in group decision making

Abstract

This paper presents an account of fuzzy similarity metrics that have been proposed to quantify consensus in Multi Criteria Group Decision Making. Fuzzy similarity metrics are indispensable to determine consensus when experts evaluate alternatives in fuzzy terms, which capture experts’ uncertainty and hesitancy. Furthermore, factors such as the level of expertise or cognitive bias lead to disagreements within the group. The fuzzy similarity metrics described in this article are used to measure the similarity between type 1 and type 2 fuzzy sets, and fuzzy numbers. Consensus can be quantified at three different levels: criteria judgement, alternative judgement, or expert preferences. Promising future work includes the incorporation of social fuzzy measures under the umbrella of multi-agent systems as well as the analysis of fuzzy intuitionistic sets.

Keywords

Consensus measure fuzzy distance aggregation decision making

1. Introduction

Multi Criteria Group Decision Making (MCGDM) is a procedure where a committee of experts evaluate alternatives and select the most suitable option according to a set of criteria for the problem at hand.

The Analytic Hierarchy Process (AHP) constitutes one of the approaches to quantify individual judgements and obtain the expert individual preference i.e expert final ranking of alternatives [1]. In the AHP each expert ranks a finite set of alternatives by pairwise comparison of to what extent alternatives fare in each criterion. This form of evaluation is called preference relations [2]. Other approaches to decision making require that experts employ either linguistic labels [3] or values in a numeric scale to evaluate the alternatives without providing explicit criteria weighing. In this schema evaluators provide their judgements by using a set of utility values [2].

Two options exist in order to obtain a collective ordering of the alternatives under evaluation. They are known as the Aggregation of Individual Judgements (AIJ) and the Aggregation of Individual Preferences (AIP) [4]. In AIJ the individual judgements are grouped into a collective aggregation of judgements from which a final ordering is obtained. In the AIP, each expert solves the decision problem by providing his/her set of preference ordering (SPO), O^e = {a^e (1) , ⋯ , a^e (n)}, where e is an expert and a (·) is a permutation function over n alternatives [2]. As soon as all the SPO’s are generated, the collective ordering is computed.

However, one problem that arises is that experts prefer not to use strict quantitative scales when expressing their judgements. That is to say, experts feel more comfortable providing qualitative or linguistic evaluations because it reflects the uncertainty and impreciseness inherent in human reasoning. The fuzzy set theory developed by Zadeh [5] is adequate to model such vagueness.

To aggravate the inherent uncertainties of the evaluations, diverse sources of disagreement among experts also compromise the group decision. Among them it can be mentioned the cognitive bias of the evaluators or the diversity of specialties in the committee of experts.

Despite having uncertain and biased judgements, it is necessary to measure how similar the expert evaluations are in order to determine whether the solution (the collective ordering of alternatives) is valid. That is to say, it is necessary to quantify consensus among experts.

A strict definition of consensus, term that derives from the Latin consentire, is a general agreement, a shared idea, or the judgement arrived at by most of those concerned (merriam-webster.com).

Saaty and Dong [6] proposed a group consensus index to be applied in AHP-based group decision making, which calculates the separation between individual judgements with the collective judgement. The resultant value is then contrasted with a pre-defined threshold to decide whether the group is in agreement or not. However, experts must give their evaluation in terms of a strict numerical scale.

Seeking unanimous agreement could diverge from pragmatism, making it necessary to consider a softer definition of consensus [7]. Fuzzy logic also serves to model and compute soft consensus because it computes to what extent the evaluators judgements and preferences are acceptable without drifting from the notion of strict consensus.

The open research problem regards measuring consensus among experts in a setting where their evaluations are expressed through membership degrees to fuzzy sets (type 1 and type 2), fuzzy numbers or linguistic labels. Thus, the main topic of the present article is to review and analyze relevant fuzzy similarity metrics in MCGDM suitable to determine consensus.

The article is organized as follows. Section 2 contains formal definitions about MCGDM and the fundamental notions of metric spaces and distance functions. Instances of fuzzy metrics are outlined in Section 3. Next, Section 4 presents an analysis of such metrics and their applicability in MCGDM. Finally, concluding remarks and opportunities to enrich this line of research are given.

2. Preliminaries

This section presents the methodological formulation and the fundamentals regarding MCGDM and fuzzy similarity metrics.

2.1. Multi-Criteria Group Decision Making

To carry MCGDM let:

E be the set of of experts. E : = {e₁, e₂, ⋯ , e_k},

A be the set of alternatives. A : = {a₁, a₂, ⋯ , a_n},

C be the set of evaluation criteria. C : = {c₁, c₂, ⋯ , c_m},

L be the set of utility values. L : = {l₁, l₂, ⋯ , l_p}, and

M_L be the collection of fuzzy subsets of L.

Utility values are divided in three categories:

A numerical Universe of Discourse such that $L = [l_{initial}, l_{final}] \subset R$ ; l_initial ≤ l_final. Every instance l_j ∈ L is fuzzyfied by computing their membership degree μ (l_j) to either type-1 or type-2 fuzzy set.

Fuzzy numbers, either Triangular Fuzzy Numbers (TriFN), Trapezoidal Fuzzy Numbers (TFN), or Generalized Fuzzy Numbers (GFN). The derived membership degree is calculated through different approaches, as it will be explained along this paper.

Linguistic terms. This option is suitable because in some cases experts find it difficult to assign numerical values to judge alternatives. Each linguistic label must be linked with either a fuzzy set or a fuzzy number. An example of how linguistic terms are associated with type-2 fuzzy sets is given in [8].

Set L must comply with the conditions delineated in [3]: Its elements must be ordered l_i ≤ l_j if and only if i ≤ j; there exists a Max operator Max(l_i, l_j) = l_i if l_i > l_j; there exists a Min operator Min(l_i, l_j) = l_i if l_i < l_j.

When experts set their individual judgements by assigning a utility value to each pair alternative-criterion, the following general structure is obtained:

Experts	Alternatives	Criteria
		c1	c2	… cm
e1	a1		expert judgements
e2
⋮
ek
e1	a2		expert judgements
e2
⋮
ek
e1	an		expert judgements
e2
⋮
ek

As soon as expert e_i ∈ E binds criterion c_r ∈ C with a utility value l_j ∈ L, the membership degree μ (l_j) is calculated. Thus, membership degrees of expert e_i judgements of alternative a_a is contained in set MU_ia, as represented in Equation 1.

$\begin{matrix} {MU}_{ia} = {(c_{r}, μ (l_{j})) | c_{r} \in C, μ (l_{j}) \in M}; \\ a = 1, \dots, n; \\ i = 1 \dots k . \end{matrix}$ (1)

To determine whether consensus among experts is reached, it is compulsory to measure the similarity of the elements in set MU_ia with those of set MU_ja, i ≠ j.

Let MU be the set of all membership degrees of the experts’ judgements, and d be a metric on MU that quantifies how close the elements in MU are. This structure represents a metric space.

2.2. Metric spaces

According to [9] a metric space is a pair (X, d), where X is a set and $d : X \times X \to R$ is a function satisfying, for x, y, z ∈ X: $\begin{matrix} d (x, x) = & 0, \\ d (x, y) > & 0 \Leftrightarrow x \neq y, \\ d (x, y) = & d (y, x), \\ d (x, y) + & d (y, z) \geq d (x, z) . \end{matrix}$

If (X, d) is a metric space, then d is called a metric on X. For each (x, y) ∈ X × X, the real number d (x, y) is called the distance between x and y.

In [9] it is exemplified a metric space where X is a set, $d : X \times X \to R$ such as d (x, y) =1 for x ≠ y and d (x, y) =0 for x = y. This particular function d is called the trivial metric on X [10].

2.3. Fuzzy metric spaces

Let X be a set. A mapping μ : X → [0, 1] is called a fuzzy subset of X. The function value μ (x) is called the degree of membership of x in μ. The collection of all fuzzy subsets of X is denoted by F_X.

A definition of fuzzy metric is given in [11], as follows. Let μ_X be the fuzzy metric on F_X, defined by μ_X : F_X × F_X → [0, 1], and $μ_{X} (A, B) = \sup_{x \in X} | A (x) - B (x) |$ . Similarly, it is stated in [12] that a fuzzy metric μ_X on a set X is a pair (μ, *) such that μ is a fuzzy set on X × X × [0, ∞) and * is a continuous t-norm satisfying, for all x, y, z ∈ X and t, s > 0:

$\begin{matrix} μ (x, y, 0) = 0; \\ μ (x, y, t) = 1 \leftrightarrow x = y; \\ μ (x, y, t) * μ (y, z, s) \leq μ (x, z, t + s) . \end{matrix}$

Thus, if μ_X a fuzzy metric on X, (F_X, μ_X) is a fuzzy metric space of X [11].

Earlier work on fuzzy metric spaces includes the formalization of the Hausdorff fuzzy metric presented in [13]. Two common metrics are the Hamming d_H and Euclidean d_E distances, whose general representations for membership degrees are given in Equations 2 and 3, respectively [14, 15].

For any two fuzzy sets A and B with membership functions μ_A (x) , μ_B (x):

$d_{H} (A, B) = \int_{x} ∣ μ_{A} (x) - μ_{B} (x) ∣ dx$ (2)

$d_{E} (A, B) = \sqrt{\int_{x} (μ_{A} (x) - μ_{B} (x))^{2} dx}$ (3)

A metric is established in the Fuzzy C-Means algorithm through the specific function to calculate, simultaneously, the distance between a given element x ∈ X with m cluster prototypes in order to obtain compact clusters [16], that is to say, fuzzy sets formed with elements that display the minimum distance with the cluster prototype. In [17] the conventional Fuzzy C-Means algorithm is modified by replacing the Euclidean distance with Mahalanobis and Minkowski metrics in order to enhance the cluster formation capacity by allowing more accurate detection of arbitrary shapes for high dimensional databases.

2.4. MCGDM as fuzzy metric spaces

In the MCGDM framework, experts employ a set L of utility values to provide their judgements. Then, set L is paired to a collection M_L of fuzzy subsets. In this structure, μ_L is the fuzzy metric on M:

$μ_{L} : M_{L} \times M_{L} \to [0, 1] .$ (4)

Hence, if μ_L is a fuzzy metric on L, a fuzzy metric space on L is (M_L, μ_L); where μ_L represents the fuzzy metric used to calculate the distance between expert evaluations as a necessary condition to measure the level of consensus among experts.

Specific fuzzy similarity metrics are explained next.

3. Fuzzy similarity metrics

In this section, fuzzy similarity metrics are explained. Even though the seminal work on fuzzy metrics has been originally proposed in [15], adaptations and improvements have been carried.

3.1. Similarity between membership degrees

It is reported in [18] a decision support system where evaluators provide crisp assessments, which are then fuzzyfied by membership functions. It is determined to what extent experts agree by obtaining the cardinality of the resultant fuzzy clusters of individual judgements. First, experts set their evaluations on a continuos scale of utility values; then, a multi-agent system groups the evaluations into fuzzy clusters. User-defined α-cut values help find the subset of fuzzy clusters with the highest cardinality by computing how many elements in each cluster have a membership value equal or greater than the required α-cut.

A specific function to measure the similarity between two membership degrees defined by fuzzy sets is given in [19]. Let a and b represent two fuzzy sets that capture experts e₁ and e₂ judgements to an alternative a_a under criteria c_r. The similarity sim (a, b) is expressed by Equation 5. $sim (a, b) = \frac{\int_{x} (\min {μ_{a} (x), μ_{b} (x)})^{2} dx}{\int_{x} (\max {μ_{a} (x), μ_{b} (x)})^{2} dx}$ (5)

Equation 5 satisfies the following properties:

0 ≤ sim (a, b) ≤1.

sim (a, b) =1 ⇔ a = b

sim (a, b) = sim (b, a)

According to the previous properties, it is not evident whether Equation 5 satisfies the triangle inequality.

Another metric applicable for membership degrees of evaluations is described in [20]. Specifically, it consists in a fuzzy concordance index that quantifies the similarity between a pair of fuzzy judgements and a consensus index that computes the similarity among a group of fuzzy evaluations. The authors define a concept called experte_ifuzzy or linguistic estimatel_j of alternative a_a, given criterion c_r, denoted as: $F_{c_{r}}^{e_{i}} (a_{a}) = μ_{ij}^{a_{a}} (c_{r}) .$ (6)

Based on expert fuzzy estimates, the group opinion is calculated through the weighted arithmetic mean: $F {c_{r}}^{G} (a_{a}) = \sum_{e_{i} = 1}^{k} ω_{e_{i}} \otimes F_{c_{r}}^{e_{i}} (a_{a}),$ (7) where e_i ∈ E, ω_{e
_i} is the expert weight, and ⊗ is the fuzzy multiplication operator.

The similarity index between an expert evaluation and the group aggregate is given by the following expression: $\begin{matrix} S_{w} (F_{c_{r}}^{e_{i}} (a_{a}), F {c_{r}}^{G} (a_{a})) = \\ \frac{\int_{E} (\min {F_{c_{r}}^{e_{i}} (a_{a}), F {c_{r}}^{G} (a_{a})}) dE}{\int_{E} (\max {F_{c_{r}}^{e_{i}} (a_{a}), F {c_{r}}^{G} (a_{a})}) dE} \end{matrix}$ (8)

The concordance level is then calculated as the linear aggregation of distances and similarity metrics according to the following function: $\begin{matrix} S_{F}^{e_{i}} (F_{c_{r}}^{e_{i}} (a_{a}), F {c_{r}}^{G} (a_{a})) = \\ β S_{w} (F_{c_{r}}^{e_{i}} (a_{a}), F {c_{r}}^{G} (a_{a})) \\ + (1 - β) (1 - D_{h} (F_{c_{r}}^{e_{i}} (a_{a}), F {c_{r}}^{G} (a_{a}))) \end{matrix}$ (9)

In Equation 9 distance D_h (· , ·) is the normalized Hamming distance between $(F_{c_{r}}^{e_{i}} (a_{a}), F {c_{r}}^{G} (a_{a}))$ , β is a parameter 0 ≤ β ≤ 1, which guarantees that 0 ≤ D_h ≤ 1. Finally, the consensus level among k experts, for each alternative, is computed by using the following expression: $C (a_{a}) = \frac{\sum_{i = 1}^{k} S_{F}^{e_{i}} (F_{c_{r}}^{e_{i}} (a_{a}), F {c_{r}}^{G} (a_{a}))}{k} .$ (10)

It can be inferred from Equation 10 that the higher the value of C (a_a) the higher the consensus among the expert regarding alternative a_a. This form of calculating consensus is valuable because it permits to analyze in which alternative the group clearly disagrees, and then use the information in order to provide feedback to the experts.

3.2. Similarity between fuzzy numbers

In MCGDM it is possible that experts express their judgements through either TriFN, TFN, or GFN.

A TriFN l (∊_Ll, c_l, ∊_Rl), where 0 ≤ ∊_Ll ≤ c_l ≤ ∊_Rl, is defined as follows: $l (x) = {\begin{matrix} (x - ∊_{Ll}) / (c_{l} - ∊_{Ll}) & ∊_{Ll} \leq x \leq c_{l}, \\ (∊_{Rl} - x) / (∊_{Rl} - c_{l}) & c_{l} \leq x \leq ∊_{Rl}, \\ 0 & otherwise . \end{matrix}$ (11)

In Equation 11, ∊_Ll is the left spread, c_l is the center, and ∊_Rl is the right spread, respectively, of TriFN l.

Let l₁, l₂ be two TriFN’s defined by Equation 11. A classic function to measure the distance between two TriFN’s l₁, l₂ is given in [21]: $\begin{matrix} d (l_{1}, l_{2}) = \\ \sqrt{\frac{(∊_{{Ll}_{1}} - ∊_{{Ll}_{2}})^{2} + (c_{l_{1}} - c_{l_{2}})^{2} + (∊_{{Rl}_{1}} - ∊_{{Rl}_{2}})^{2}}{3}} . \end{matrix}$ (12)

Other similarity functions have been proposed to compute the distance between TriFN apart from the one given in Equation 12. Fedrizzi et al. [22, 23] developed a consensus model where experts provide their opinions via TriFN’s. The distance used to determine similarity among judgements, which belongs to the family presented originally in [15], is described next.

Left and right boundaries l_iL and l_iR of TriFN l_i are obtained by introducing an α-level α ∈ [0, 1]: $\begin{matrix} l_{iL} (α) = c_{l_{i}} - ∊_{{Ll}_{i}} + ∊_{{Ll}_{i}} α . \\ l_{iR} (α) = c_{l_{i}} + ∊_{{Rl}_{i}} - ∊_{{Rl}_{i}} α . \end{matrix}$ (13)

Next, integral I_{Ll
_i} is defined as the integral of the squared differences between the left boundaries of TriFN’s l_i, l₂: $I_{L} = \int_{0}^{1} (l_{1 L} (α) - l_{2 L} (α))^{2} .$ (14)

Integral R_{Ll
_i} is defined as the integral of the squared differences between the right boundaries of TriFN’s l_i, l₂:

$I_{R} = \int_{0}^{1} (l_{1 R} (α) - l_{2 R} (α))^{2} .$ (15)

The metric D (l₁, l₂) is defined as:

$D (l_{1}, l_{2}) = (\frac{1}{2} (I_{L} + I_{R}))^{\frac{1}{2}} .$ (16)

Equation 16 complies with all the properties that define a metric.

Experts can also provide their judgements via Trapezoidal Fuzzy Numbers, which is the case presented by Wu et al. [24], who present a similarity measure based on the centroids of the TFN’s. In the following lines the relevant aspects of their proposal are described.

Let a and b two TFN’s that capture experts e₁ and e₂ evaluations of an alternative a_a under criteria c_r. The similarity sim (a, b) is expressed by Equation 17. $sim (a, b) = \frac{1}{1 + E (a, b)} .$ (17)

In Equation 17, the term E (a, b) is: $E (a, b) = \frac{1}{2} (∣ x_{a}^{*} \cdot y_{a}^{*} - x_{b}^{*} \cdot y_{b}^{*} ∣ + ∣ {Ar}_{a} - {Ar}_{b} ∣) .$ (18)

In Equation 18, $x_{a}^{*}, y_{a}^{*}, x_{b}^{*}, y_{b}^{*}$ are the centroids of TFN’s a and b; Ar_a, Ar_b are the areas of TFN’s a and b, respectively.

The similarity metric defined by Equation 17 satisfies the following conditions:

sim (a, b) =1 ⇔ a = b.

sim (a, b) = sim (b, a).

Maturo and Ventre [25] represent the variability in the judgements that decision makers assign to alternatives with fuzzy numbers. The authors propose an algorithm for the achievement of consensus based on suitable fuzzy numbers, on preorder and order relations in sets of fuzzy numbers, and on a procedure to decrease the spreads. In a similar fashion, they propose various metric spaces for representing the movements of decision-makers for reaching consensus and introduce the concept of fuzzy coalition for developing an algorithm for building a feasible fuzzy coalition [26].

Another similarity metric between fuzzy numbers is developed in [27]. The proposed measure is suitable to compare GFN’s and it is based on geometric distance, height and radius of gyration point of the fuzzy number to be compared.

A GFN is defined as follows: $μ_{A_{ω_{1}, ω_{2}}} = {\begin{matrix} \frac{ω_{1} (x - a_{1})}{a_{2} - a_{1}}, & if a_{1} \leq x \leq a_{2}; \\ \frac{ω_{1} (a_{3} - a_{2}) + (ω_{2} - ω_{1}) (x - a_{2})}{a_{3} - a_{2}}, & if a_{2} \leq x \leq a_{3}; \\ \frac{ω_{2} (a_{4} - x)}{a_{4} - a_{3}}, & if a_{3} \leq x \leq a_{4}; \\ 0, & otherwise . \end{matrix}$ (19)

In Equation 19 a₁, a₂, a₃, a₄ are real values; ω₁ is the left heigh of the GFN, and ω₂ is the right height such that (ω₁, ω₂) ∈ [0, 1]. If ω₁ = ω₂ = 1, the GFN is a TFN.

Thus, the similarity measure used to compare two GFN’s A, B, is defined as follows: $\begin{matrix} S (A, B) = \\ (1 - \frac{1}{4} \sum_{i = 1}^{4} | a_{i} - b_{i} |) \times \\ (1 - \frac{1}{2} ([| ω_{1 a} - ω_{1 b} |] + | ω_{2 a} - ω_{2 b} |)) \times \\ \frac{\min (r_{x}^{A}, r_{x}^{B}) + \min (r_{y}^{A}, r_{y}^{B})}{\max (r_{x}^{A}, r_{y}^{B}) + \max (r_{y}^{A}, r_{y}^{B})}, \end{matrix}$ (20) where $r_{x}^{A}, r_{x}^{B}, r_{y}^{A}, r_{y}^{B}$ are the radius of gyration of GFN A, B.

The similarity metric defined in Equation 20 has the following properties:

$\begin{matrix} S (A, B) = 1 \leftrightarrow A = B . \\ S (A, B) = 0 \leftrightarrow A = (0, 0, 0, 0; 1, 1), \\ B = (1, 1, 1, 1; 1, 1) . \\ 0 \leq S (A, B) \leq 1 . \end{matrix}$ (21)

In Equation 21 generalized fuzzy numbers A, B are defined according to Equation 19.

Guha and Chakraborty [28] contribute to the concept of fuzzy similarity in MCGDM for the case when experts provide their judgements by using utility values represented as Generalized Trapezoidal Fuzzy Numbers (GTFN). They refer to the definition and properties of a GTFN A = (a₁, a₂ ; β, γ ; ω), given by Chen in [29].

The fuzzy distance between any two GTFN A₁, A₂, as proposed by Guta and Chakraborty is: $\begin{matrix} d (A_{1}, A_{2}) = (d_{α = ω}^{L}, d_{α = ω}^{R}; θ, σ; ω); \\ θ = d_{α = ω}^{L} - \max (\int_{0}^{ω} d_{α}^{L} d α, 0); \\ σ = | (\int_{0}^{ω} d_{α}^{R} d α) - d_{α = ω}^{R} |; \\ ω = \min (ω_{1}, ω_{2}) . \end{matrix}$ (22)

In Equation 22 $(d_{α}^{L}, d_{α}^{R})$ for α ∈ [0, ω], denote the α-cut representation of the distance measure between two GTFN’s.

The similarity measure proposed by Guha and Chakraborty is: $S (A_{1}, A_{2}) = 1 - d (A_{1}, A_{2}) .$ (23)

3.3. Similarity between type-2 fuzzy evaluations

Even though uncertainties can be captured with type-1 fuzzy logic, as presented previously, in MCGMD the use of type-2 fuzzy logic has also been favored. Consequently, consensus and similarity metrics have been proposed when experts set their preferences in type-2 fuzzy logic models.

Castillo and Melin [30] define an interval type-2 fuzzy set as the one in which the membership grade of every element in the universe of discourse is a crisp set whose domain is the interval [0,1] (see Equation 24).

Generally speaking, a type-2 fuzzy set Ä is defined as follows: $\begin{matrix} Ä = {(x, u), μ_{A} (x, u) | \forall x \in X, \forall u \in J_{X} & \in [0, 1], 0 \leq μ_{A} (x, u) \leq 1 .} \end{matrix}$ (24)

In Equation 24 X is the original universe of discourse, J_X is the interval [0,1].

A type-2 trapezoidal fuzzy number Ä_i is defined as follows: $\begin{array}{l} Ä_{i} = (Ä_{i}^{U}, Ä_{i}^{L}), \\ Ä_{i} = ((a_{i 1}^{U}, a_{i 2}^{U}, a_{i 3}^{U}, a_{i 4}^{U}); H_{1} (Ä_{i}^{U})), \\ (a_{i 1}^{L}, a_{i 2}^{L}, a_{i 3}^{L}, a_{i 4}^{L}); H_{1} (Ä_{i}^{L}), H_{2} (Ä_{i}^{L})) . \end{array}$ (25)

In Equation 25 $Ä_{i}^{U}$ , $Ä_{i}^{L}$ are the upper and lower trapezoidal membership functions of the type-2 trapezoidal fuzzy number, which are type-1 membership functions themselves; $H_{j} (Ä_{i}^{U})$ is the membership degree of the element $a_{i (j + 1)}^{U}$ in the upper trapezoidal membership function $Ä_{i}^{U}$ ; $H_{j} (Ä_{i}^{L})$ is the membership degree of the element $a_{i (j + 1)}^{L}$ in the lower trapezoidal membership function $Ä_{i}^{L}$ .

A consensus model for group decision making under type-2 fuzzy logic is proposed in [31, 32]. According to [30] a type-2 fuzzy set is one which also represents uncertainty about the membership function. Thus, a type-2 fuzzy set is characterized by a fuzzy membership function, that is to say, the membership grade for each element of this set is a fuzzy set in [0,1]. This contrasts with the membership grade in type-1 fuzzy logic which is a crisp value. Zhang et al. [31, 32] rely on type-2 fuzzy trapezoidal numbers to develop their consensus metric. In their model, experts provide judgements through linguistic labels, presented as type-2 interval fuzzy numbers. They propose a modified Hamming metric to measure the separation between type-2 trapezoidal fuzzy numbers: $\begin{matrix} D (Ä_{1}, Ä_{2}) = \frac{d_{1} + d_{2}}{8} . \\ d_{1} = | a_{11}^{U} - a_{21}^{U} | + | a_{12}^{U} \cdot H_{1} (Ä_{1}^{U}) - a_{22}^{U} \cdot H_{1} (Ä_{2}^{U}) | \\ + | a_{13}^{U} \cdot H_{2} (Ä_{1}^{U}) - a_{23}^{U} \cdot H_{2} (Ä_{2}^{U}) | \\ + | a_{14}^{U} - a_{24}^{U} | . \\ d_{2} = | a_{11}^{L} - a_{21}^{L} | + | a_{12}^{L} \cdot H_{1} (Ä_{1}^{L}) - a_{22}^{L} \cdot H_{1} (Ä_{2}^{L}) | \\ + | a_{13}^{L} \cdot H_{2} (Ä_{1}^{L}) - a_{23}^{L} \cdot H_{2} (Ä_{2}^{L}) | \\ + | a_{14}^{L} - a_{24}^{L} | . \end{matrix}$ (26)

4. Discussion

This section presents a discussion regarding the advantages of the fuzzy similarly metrics that have been studied in Section 3.

Firstly, it can stated that the extensions and adaptation of the Euclidean and Hamming distances can be used with a pre-defined threshold in order to determine whether consensus has been achieved. Most of the metrics are devised to calculate the similarity between single evaluations and the collective ordering.

Another important aspect is that experts can provide their preferences by using different utility values, such as linguistic labels or fuzzy numbers, as it has been explained in Section 2. An example of such utility values is given in Table 1, which is adapted from [8]. It contains three utility values sets: a list of linguistic terms, type-1 TFN’s, and and type-2 TFN’s. Moreover, a fourth utility value set is implied if L is a continuous numerical interval, for instance, L = [0, 10]. In this case, the utility values are crisp, and expert judgements must be fuzzyfied.

Table 1
An example of fuzzy representations for linguistic terms

Term Trapezoidal Type 2

fuzzy set TFN

Equation 25

Very low (0,0,1,1.5) ((0,0,0,0.1;1,1),

(0,0,0,0.05;0.9,0.9))

Low (1,1.5,2,2.5) ((0,0.1,0.1,0.3;1,1),

0.05,0.1,0.1,0.2;0.9,0.9))

Medium low (2,2.5,3,3.5) ((0.1,0.3,0.3,0.5;1,1),

0.2,0.3,0.3,0.4;0.9,0.9))

Medium (3,3.5,4,4.5) ((0.3,0.5,0.5,0.7;1,1),

(0.4,0.5,0.5,0.6;0.9,0.9))

Medium high (4,4.5,5,6) ((0.5,0.7,0.7,0.9;1,1),

0.6,0.7,0.7,0.8;0.9,0.9))

High (5,6,8.5,9) ((0.7,0.9,0.9,1;1,1),

(0.8,0.9,0.9,0.95;0.9,0.9))

Very high (8.5,9,10,10) ((0.9,1,1,1;1,1),

(0.95,1,1,1;0.9,0.9))

The values given in Table 1 can be used by experts to judge the relevance of the alternatives as follows. In one approach, expert e₁ can rule that alternative a₁ compliance with criterion c₁ is Very high, while expert e₂ can rule that such compliance is Medium. This is an example of judging alternatives on their merits i.e. not pairwise comparison is involved.

If, on the other hand, expert e₁ rules that alternative a₁ compliance with criterion c₁ is Very high when compared to alternative a₂; and expert e₂ rules that a₁ compliance to criterion c₁ is High compared to alternative a₂, this form of judgement comprises a pairwise comparison between alternatives.

It can be observed from Table 1 that straightforward representations of linguistic terms and direct mappings into the fuzzy domain are the advantages of using type-1 fuzzy sets. Fuzzy numbers extend the notion of crisp scales by providing a way to express uncertain evaluations. More complex representations of uncertainty are provided by type-2 fuzzy sets, which are useful to define a linguistic term i.e. Very high, by spreading the region where crisp values fall. The fuzzy metrics presented in this article are helpful to measure the similarity in these three scenarios.

A particular type of utility values scales is the bipolar rating scales, which is a linearly ordered set with symmetry between elements considered as negative and positive categories. They are amply used in recommender systems and MCGDM. Bipolar scales are [33]:

scales with verbal categories ordered from negative categories on the left to positive categories on the right.

scales with verbal labels assigned to one gradation of the scale.

scales with positive numeric scores or with both negative and positive scores.

The set of utility values can also be multi-granular linguistic terms [34], which are not uniformly and symmetrically distributed. Dong et al. [34] propose a consensus-based GDM model by using 2-tuple linguistic representation based on multi-granular, unbalanced 2-tuple linguistic preference relations.

4.1. Consensus process

In a MCGDM setting, the similarities among evaluations can be calculated on three different levels:

Criterion judgement. This is to determine whether consensus has been reach among the committee of experts regarding the judgement of all the alternatives for each and all the criteria.

Alternative judgement. This is to determine whether consensus has been reached among the committee regarding the compliance of each all the alternatives to each and all criteria.

Expert preferences. This is to determine whether consensus has been reached regarding the order preference given by each evaluator. At this level it must be calculated how separated is an expert judgement from the aggregated group evaluation.

At which level the similarity among evaluations must be measured is indeed a matter of the MCGDM settings. However, consensus reaching is a process that is facilitated when similarity measures are available to provide feedback to the experts involved.

Reaching consensus among experts is a negotiation exercise that is carried iteratively. During this process, some of the experts in the committee are asked either to reconsider their evaluations or to explain their reason that led them to provide their opinions.

In general, the computation of the consensus level in MCGDM is done by measuring the deviation between the individual ranking of alternatives and the collective ranking of alternatives and measuring the deviation between the individual evaluation matrices and the collective evaluation matrix. However, Dong et al. [21] indicates that most of the frameworks used along the consensus reaching process rely on a homogeneous set of criteria used by the experts, which is not always true because in some MCGDM settings each expert is allowed to define the criteria that fit best. Thus, Dong and colleagues devise a process to reach consensus in complex and dynamic environments.

4.2. Kemeni optimal aggregations

A formal approximation to obtaining the best ranking of alternatives in a MCGDM setting is called the rank aggregation problem [35]. Firstly, a finite set of rankings (τ₁, τ₂, ⋯ , τ_k) is obtained (each ranking is associated to an expert); secondly, a distance between any pair of rankings is set, and finally, it must be found the ranking that minimizes the sum of the distances from this current ranking to all the individual rankings. An aggregated ranking τ_aggregated is obtained such that the distance D (τ_aggregated, (τ₁, τ₂ . ⋯ , τ_k)) is minimal.

The aggregation obtained using the Kendal τ distance, called Kemeni optimal aggregations, are the only ones that comply simultaneously with neutrality, consistency and the Condorcet property. However, as stated also in [35], computing the Kemeni optimal aggregation is an NP-Hard problem.

Even though the distance measures explained along this article do not necessarily lead to a Kemeni optimal aggregation, all of them are useful to determine consensus when experts’ opinions convey uncertainty and imprecision, helping to solve how to rank a finite set of alternatives.

4.3. Linguistic consensus

While the hard approach to consensus can be regarded as a Kemeni optimal aggregation, in reality soft consensus is a more pragmatic approach. However, all the similarity metrics outlined in this article lead to a numeric quantification of agreement among experts (crisp consensus), even though their opinions are expressed by using the fuzzy approach. In order to model to what extend linguistic evaluations reflect consensus, Herrera et al. [36] suggest using terms that reflect a fuzzy majority or a widespread agreement among experts, such as most, almost all, much more than a half. This contrasts to the way of expressing consensus in a numeric value.

5. Conclusions

A series of fuzzy similarity metrics have been presented, all of which are intended to measure consensus in a MCGDM.

The fuzzy approach to consensus is necessary because in the interaction among experts, factors such as the degree of preference shown by each expert regarding the set of alternatives, the importance, explicit or implicit, given by each evaluator to every criterion, the relative weight of the evaluator, the cognitive bias of the evaluators, or the diversity of specialties in the committee of experts, provoke that unanimous consensus be not achieved. Hence, by developing fuzzy metric spaces the notion of soft consensus, as opposed to strict consensus, is suitable to deal with the uncertainties mentioned. It has been found that the Euclidean, Hamming and trivial metrics are the functions that have been extended to the fuzzy domain to measure the degree of consensus. These metrics are used to compare membership degrees, fuzzy numbers, or type-2 fuzzy sets. All of them are suitable to determine whether consensus has been reached at three different levels: experts judgements of to what extent alternatives comply to a given criterion, expert preferences or group preference.

A promising line of research regards the incorporation of social fuzzy measures to promote a process of consensus reaching. This is possible by developing social interactions under the multi-agent paradigm, where the notion of consensus has been modeled as the dynamics among intelligent agents [37, 38]. Another line of research refers to visualizing the similarity measures, for which the work presented in [39] can be extended to fuzzy similarities.

Also, given that the similarity metrics provide a numeric quantification of the distance between experts evaluations, and in order to model the concept o majority in fuzzy terms, a promising line of research is then to come up with similarity metrics that map into linguistic expressions.

A current trend where consensus must be reached is the field of opinion dynamics, which investigates the fusion process of the opinion formation in a group of agents, and is a powerful tool for supporting the management of public opinions [40]. However, in real-life situations, firms or administrations are not only interested in the formation of public opinions, but also hope to influence and guide the forming opinions to reach either a consensus or even more a specific consensus point. In [40] it is developed a consensus building process in opinion dynamics, based on the concept leadership, by analyzing the structure of the social network in which all agents can form a consensus. A thorough survey on opinion dynamics is given by Dong et al. [41].

Finally, it is worth highlighting that the original fuzzy set theory has been subject to extensions. Chiefly among them has been the concept of intuitionistic fuzzy sets (IFS) proposed by Atanassov [42]. IFS’s are characterized by both, a membership degree and a non-membership degree, properties that allegedly represent more accurately uncertainty and vagueness. On this approach, similarity measures exist and have been used in a MCGDM environment [43]. This is also a fruitful research to explore.

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Term	Trapezoidal	Type 2
	fuzzy set	TFN
		Equation 25
Very low	(0,0,1,1.5)	((0,0,0,0.1;1,1),
		(0,0,0,0.05;0.9,0.9))
Low	(1,1.5,2,2.5)	((0,0.1,0.1,0.3;1,1),
		0.05,0.1,0.1,0.2;0.9,0.9))
Medium low	(2,2.5,3,3.5)	((0.1,0.3,0.3,0.5;1,1),
		0.2,0.3,0.3,0.4;0.9,0.9))
Medium	(3,3.5,4,4.5)	((0.3,0.5,0.5,0.7;1,1),
		(0.4,0.5,0.5,0.6;0.9,0.9))
Medium high	(4,4.5,5,6)	((0.5,0.7,0.7,0.9;1,1),
		0.6,0.7,0.7,0.8;0.9,0.9))
High	(5,6,8.5,9)	((0.7,0.9,0.9,1;1,1),
		(0.8,0.9,0.9,0.95;0.9,0.9))
Very high	(8.5,9,10,10)	((0.9,1,1,1;1,1),
		(0.95,1,1,1;0.9,0.9))