ELECTRE TRI-C with hesitant outranking functions: Application to supplier development

Abstract

ELECTRE TRI, a family of multi-criteria methods used to sort alternatives into preference-ordered categories, defines an outranking function to measure the membership degree of an alternative to a category, whenever imprecise evaluations are available. Recent extensions use Hesitant Fuzzy Sets (HFS) to consider uncertain evaluations. However, deterministic parameters are considered, which avoids the application to cases in which non-fuzzy scores and unstable parameters are available. In this study, an approach is proposed for: 1) modeling hesitant outranking functions originated from unstable parameters provided by several DMs; 2) using the HFS to calculate the ELECTRE TRI-C indices; 3) reducing the DMs cognitive effort when they are asked to provide information. An application to supplier development is presented by using ELECTRE TRI-C. Results are compared by using different HFS aggregation operators and sensitivity analysis shows that a robust conclusion can be obtained. Future lines of research are also suggested.

Keywords

ELECTRE TRI-C hesitant fuzzy sets sensitivity analysis supplier development

1 Introduction

ELECTRE TRI is family of MCDA methods designed to aid the decision analyst in problems where a set of alternatives need to be sorted in preference-ordered categories. Each category is defined a priori by upper and lower reference alternatives (ELECTRE TRI-B, ELECTRE TRI-nB [15]), or central reference alternatives (ELECTRE TRI-C, ELECTRE TRI-nC [3]). Using two rules, each alternative a is successively compared to boundaries (or central reference actions) b in terms of a global outranking relation such that it may be assigned to one or several categories. ELECTRE TRI-C, one these sorting methods, has been applied in different domains [16 , 26] for cases where defining boundaries is difficult or the concept of boundary remains vague for the DM, then central reference actions are used to characterize categories. When restricted to a single criterion, a fuzzy number, called the outranking function, is built to evaluate the credibility that any alternative be assigned to a category represented by its reference action. The outranking function is built according to the scores of the alternative, the reference action and two parameters representing the imprecision on evaluations.

ELECTRE methods have been extended to consider situations in which evaluations of alternatives may be expressed as fuzzy sets, but membership degrees are uncertain. Extensions of ELECTRE methods that consider uncertain fuzzy sets [22, 39], interval-valued fuzzy sets [43], interval-valued intuitionistic fuzzy sets [48], and intuitionistic fuzzy numbers [20] have been proposed. [32] propose neutrosophic logic to model outranking relations based on non-transitive strong and weak dominance relations, and a transitive indifference relation, built by the pairwise comparison of simplified neutrosophic numbers (scores). In this study, however, non-transitive relations are taken into account in order to deal with several types of outranking functions. Recently, Hesitant Fuzzy Sets (HFS) have been introduced for contexts where DMs cannot precisely express a fuzzy value [9 , 46]. HFS provides a useful tool because it allows a membership degree to have different values in the interval [0, 1] [41]. HFSs for quantitative criteria, and hesitant fuzzy linguistic term sets (HFLTS) for qualitative criteria, have received great attention in MCDA research and multiple applications have been reported with extensions for classical methods [11–13 , 51].

HFS extensions for ELECTRE methods have been also introduced. For instance, [18] propose the use of HFS to model the DMs’ judgements regarding the evaluation of alternatives and weights of criteria. HFSs are aggregated to present as input for the ELECTRE TRI method. [9] define an hesitant fuzzy decision matrix to model uncertain information regarding the evaluation of alternatives. A score and a deviation functions are used to compare hesitant evaluations among pair of alternatives on each criterion. These are the base for constructing the ELECTRE II concordance and discordance indices. However, the remaining steps of the outranking-based method are applied as usual. [31] consider multiple hesitant fuzzy sets (MHFS) for cases where the same hesitant value regarding a preference or data is repeatedly proposed by a decision maker. These authors define the ELECTRE III concordance and discordance indices as functions of evaluations of alternatives, expressed as MHFSs. [23] also uses HFS evaluation of alternatives to calculate concordance and discordance indices in ELECTRE I. In all cases, the outranking concordance and discordance indices are calculated as prescribed in ELECTRE methods.

Given that ELECTRE methods are most applied in decision problems where evaluations correspond to scores, ELECTRE HFS extensions build the outranking functions modeling the concordance and discordance indexes as aggregations of hesitant scores, by introducing deterministic preference and indifference thresholds parameters. However, if the focus is placed on the DM’s preferences, instead of concentrating on hesitant scores, then uncertain preferences could be modeled as hesitant outranking functions. In recent years, researchers have proposed this kind of approach. [55], for instance, define hesitant fuzzy preference relations (HFPR), and hesitant multiplicative preference relations (HMPR), as squared matrices where each entry is an hesitant fuzzy element modeling a preference relation between a pair of alternatives, a set of possible values in [0, 1]. [52] extends the analysis of two cases of incomplete HFPRs and propose a method to calculate consistent priority weights of alternatives for group decision-making (GDM) situations. [53] focus on the computation of additive consistency indices for an HFPR and [54] extends the calculation to the case of incomplete HFPRs in order to propose a new procedure for group analytic hierarchy process. Outranking preference relations, however, are less restrictive than HFPRs and some transitivity desired properties could not be satisfied [8, 37].

Uncertain parameters due to one single DM with unstable preferences, or several DMs with divergent preferences, have not be considered in the HFS extensions of outranking-based methods. In order to fill this gap, we concentrate on the following research question: how to use HFS to model uncertain parameters such that the ELECTRE TRI-C’s rationale be preserved. Thus, we focus on situations of imprecise, but deterministic, evaluations and unstable parameters (thresholds). An indirect method to model parameters is proposed, by assuming that several DMs are asked to provide information about the outranking relation functions, and not directly about the parameter values. In this sense, we adopt a framework similar to that proposed by the HFPR literature, eliciting outranking preference relations, but without imposing specific properties to the respective functions.

ELECTRE TRI-C has been chosen because the proposed approach can be applied in sorting processes where each outranking function may be elicited as a canonical trapezoidal fuzzy number (TrFN), which models the preference relation between an alternative and a central reference action. This means to consider uncertain thresholds, which has not be considered in other contributions found in the literature. Thus, main contributions proposed in this study are: 1) modeling hesitant outranking functions to deal with uncertainty originated from unstable parameters provided by several DMs; 2) using HFS to build the outranking indices, when deterministic scores and uncertain thresholds are considered; 3) introducing an approach that reduces the DMs cognitive effort when they are asked to provide preferential information, as compared to other ELECTRE extensions using HFS.

An application to the supplier development problem (SDP) is introduced to show the potential of this approach. SDP is essentially a MCDA problem and traditional or hybrid approaches could be used to solve it [6 , 44].

The article is organized as follows. In Section 2, the basic notation and definitions are presented, and two ELECTRE TRI-C outranking functions are introduced. The model applying HFS within ELECTRE TRI-C is introduced in Section 3. The application to the supplier development problem is presented in Section 4. Section 5 presents the conclusions of this article and makes some suggestions for future lines of research.

2 Notation and basic definitions

2.1 Hesitant fuzzy sets

HFS [41]. Let X be a reference set. A hesitant fuzzy set (HFS) ξ_E on X is defined in terms of a function h_E (x) that returns a subset [0, 1] when it is applied to X, which is expressed as ξ_E = {〈x, h_E (x) 〉 ∣ x ∈ X}.

HFE. h_E (x) = {γ ∣ γ ∈ [0, 1]} is a set of some different values in [0, 1] where γ represents the possible membership degree of the element x to E. h_E (x) is called a hesitant fuzzy element (HFE), a basic unit of HFS.

As an example, let us consider X = {x₁, x₂} to be a set such that h_E (x₁) = {0.3, 0.2} and h_E (x₂) = {0.1, 0.3, 0.4}, then an HFS can be written as follows: ξ_E = {〈x₁, {0.3, 0.2} 〉, 〈x₂, {0.1, 0.3, 0.4} 〉}.

Score function. For an HFE h, a function $s (h) = \frac{1}{l_{h}} \sum_{γ \in h} γ$ is called the score function of h, where l_h is the number of elements in h.

For h₁ and h₂, if s (h₁) > s (h₂) ⇒ h₁ ≻ h₂ (superior), and s (h₁) = s (h₂) ⇒ h₁ ∼ h₂ (indifferent).

2.2 ELECTRE TRI-C

Let A = {a₁, a₂, …, a_n} be a set of actions (alternatives) to be assigned into categories; let C_h, (h = 1, …, H ; H ≥ 2) be a set of ordered categories; let B = {b₁, …, b_H} be the set of fictitious or realistic actions, called the central reference actions, such that each b_h represents the category C_h; and let F = {g₁, g₂, …, g_m} be a coherent family of criteria used to evaluate each action. The vector of performances of an action a ∈ A ∪ B, i.e., (g₁ (a) , g₂ (a) , …, g_m (a)) is called the evaluation profile. Categories are ordered in terms of preferences from the category of actions having the worst profiles, C₀, to the best ones, C_H.

The outranking relation, interpreted as the assertion “a is at least as good as a′”, can be measured on each criterion in terms of the partial concordance index, defined as follows: $c_{j} (a, a^{'}) = {\begin{matrix} 0 & if g_{j} (a^{'}) - g_{j} (a) > p_{j}, \\ 1 & if g_{j} (a^{'}) - g_{j} (a) \leq q_{j}, \\ \frac{g_{j} (a) - g_{j} (a^{'}) + p_{j}}{p_{j} - q_{j}} & otherwise, \end{matrix}$ (1) where p_j, q_j are two imprecision parameters called the direct preference threshold and the direct indifference threshold, respectively [17]. Whenever inverse thresholds $p_{j}^{'}, q_{j}^{'}$ can be defined [37], a partial inverse concordance index, which considers the inverse thresholds, may be defined as ${cinv}_{j} (a, a^{'}) = {\begin{matrix} 0 & if g_{j} (a^{'}) - g_{j} (a) > p_{j}^{'}, \\ 1 & if g_{j} (a^{'}) - g_{j} (a) \leq q_{j}^{'}, \\ \frac{g_{j} (a) - g_{j} (a^{'}) + p_{j}^{'}}{p_{j}^{'} - q_{j}^{'}} & otherwise . \end{matrix}$ (2)

Let b be a reference alternative and define the following fuzzy number: $μ_{j} (a, b) = min {c_{j} (b, a), {cinv}_{j} (a, b)} .$ (3)μ_j (a, b) can be interpreted as fuzzy indifference relation, constructed from the two outranking relations [33]. It is a membership function and for any action a ∈ A, it helps to evaluate the degree of membership of this action to the category represented by b: the more a is indifferent from b, the more credible it is that the a belongs to the category represented by b. Actually, μ_j (a, b) is a Trapezoidal Fuzzy Number (TrFN) [7] that can be used in the sorting process. The following sections introduce the model proposed and the application that illustrates the usage of the TrFN defined, jointly with HFS.

Although the ELECTRE methods consider an effect which measures the power of veto from some criteria against the outranking relation, In this study, a simplified version is considered. Indeed, the discordance effect in a criterion is also measured through membership functions, then the approach proposed is not greatly altered.

3 ELECTRE TRI-C with hesitantoutranking functions

In Section 2.2 it has been shown that μ_j (a, b) is a TrFN that help to measure the degree of membership of a to the category represented by b. Whenever imprecision is observed, the parameters $p_{j}, q_{j}, p_{j}^{'}, q_{j}^{'}$ are used in ELECTRE to represent it. Usually, these parameters are elicited from information provided by a DM, which means that they model her/his preferences [17]. In general, a unique function defining the parameter values is commonly associated to a DM [26], and in most cases a single value is defined. However, in situations where several DMs participate in the decision analysis process, these parameters can be uncertain because different preferences and value systems must be taken into account. Thus, more than a single function is available to define p, q for each criterion, which implies that there are more than one possible value for μ_j (a, b), given the same triplet {j, a, b}.

In cases above, an HFS can be considered such that the respective HFE is the set {μ_j (a, b) ∣ a ∈ A, b ∈ B, g_j ∈ F}. For instance, let us suppose DM1 with parameters $p'_{j}^{1} = 3, q'_{j}^{1} = 1.5, p_{j}^{1} = 2, q_{j}^{1} = 1$ and another DM2 such that $p'_{j}^{2} = 3.4, q'_{j}^{2} = 1.6, p_{j}^{2} = 1.6, q_{j}^{2} = 0.8$ . In Fig. 1, TrFNs for these two DMs are shown, when g_j (b) =10. Note that imprecision exists for the two DMs, but the magnitude of imprecision is different. Before this situation, the analyst could ask each DM why a parameter has a value and what are the reasons for it.

Fig.1

Outranking functions for two DMs.

Note that thresholds in ELECTRE methods have been defined as a mean to take into account the lack of precision in instrumental and procedural measurement, or when the entity to be measured is ill-defined [36]. Thresholds exist to consider that most of time accuracy is illusory. In general, HFS extensions of ELECTRE methods have considered hesitancy in evaluations of alternatives on each criterion, while thresholds remain deterministic. In our approach, however, we will assume that uncertainty does not exist neither in g_j (a) nor in p_j, q_j when a single DM is considered. Thus, hesitancy in μ_j (a, b) appears ex post, due to the variety of DMs’ preferences. We assume that a DM is able to propose a single function or value for a parameter, but different DMs may potentially propose distinct functions or values for the same parameter. Thus, it may be considered that uncertainty exists. In such case, differences may be due to a variety of reasons: value systems, objectives, knowledge, experience. All these shape the DMs’ preferences.

The model proposed to use HFS in ELECTRE TRI-C assumes that the alternatives are already defined and evaluated on a set of criteria, but thresholds parameters are not defined and, consequently, the outranking functions are not built yet. Thus, an HFS provides a suitable means to express uncertain information coming from different DMs within the decision analysis process [9]. Several DMs involved in the decision process must provide information to built the membership functions, one for each central reference action. The outranking functions are used to compute the global credibility indices, that are the input to the assignment rules. From this step, the sorting process is developed as prescribed in ELECTRE TRI-C.

The steps to apply this model are as follows:

Define the MCDA model. Let us consider a problem with a set A of n alternatives, a set F of m criteria, a set of H categories and a group of K DMs. The alternatives must be sorted into H categories, using ELECTRE TRI-C.

Elicit H HFEs. For each a ∈ A, b ∈ B, h ∈ {1, …, H}, elicit $μ_{j}^{k} (a, b_{h})$ from information asked to the k-th DM, k ∈ {1, …, K}, such that $μ_{j}^{k} (a, b_{h}) = min {c_{j}^{k} (b_{h}, a), {cinv}_{j}^{k} (a, b_{h})} .$ (4) Then, define the HFEs ${hdir}_{j} (b_{h}, a) = {c_{j}^{k} (b_{h}, a) ∣ k = 1, \dots, K},$ (5) ${hinv}_{j} (a, b_{h}) = {{cinv}_{j}^{k} (a, b_{h}) ∣ k = 1, \dots, K} .$ (6)

Aggregate HFEs. Compute the HFS functions sdir_j (b_h, a) and sinv_j (a, b_h), aggregating hdir_j (b_h, a) and hinv_j (a, b_h), respectively.

Compute the credibility indices. For each (a, b_h), compute the credibility that b_h outranks a and the credibility that a outranks b_h.

Sort the alternatives. Use the credibility indices to apply the ELECTRE TRI-C assignment rules.

In Fig. 2 the process is shown. Note that the steps in dashed boxes correspond to the regular steps in an MCDA method, which are not modified by the proposed model.

Fig.2

Model for integrating HFS into ELECTRE TRI-C.

A number of HFE aggregation operators have been proposed in the literature. A detailed presentation of different aggregation operators and their properties is provided by [14]. In this study, to illustrate the approach, one of the operators used before to extend ELECTRE methods with HFS is proposed to aggregate information in (5) and (6) [9, 23]: ${sdir}_{j} (b_{h}, a) = \frac{1}{K} \sum_{k = 1}^{K} c_{j}^{k} (b_{h}, a),$ (7) ${sinv}_{j} (a, b_{h}) = \frac{1}{K} \sum_{k = 1}^{K} {cinv}_{j}^{k} (b_{h}, a) .$ (8)

Using (7) and (8), the global credibility indices can be computed. It is observed that the sets Hdir (b_h, a) = {sdir_j (b_h, a) ∣ j = 1, …, n} and Hdinv (b_h, a) = {sinv_j (b_h, a) ∣ j = 1, …, n} satisfy the definition of an HFE since they denote the possible membership degree of the element a to the category represented by b_h. Therefore, (9) and (10) can be defined as weighted sum scores, computed from HFEs in Hdir (b_h, a) and Hdinv (b_h, a). $σ_{D} (b_{h}, a) = \sum_{j = 1}^{m} w_{j} {sdir}_{j} (b_{h}, a),$ (9) $σ_{I} (a, b_{h}) = \sum_{j = 1}^{m} w_{j} {sinv}_{j} (a, b_{h}),$ (10) where $w_{j} \geq 0, \sum_{j = 1}^{n} w_{j} = 1$ is a set of weights that the DMs accept to correctly represent the importance of the criteria. By defining σ_D (b_h, a) and σ_I (a, b_h) (and weights w_j in these), we assume that DMs accept that these figures aggregate their opinions by using rules and priorities defined by a supra DM, a person or group with the authority to establish these rules of aggregation [25].

Given that TrFNs are proposed to model the outranking functions, in this study, a light modification of the standard assignment rules is introduced:

Descending Rule. Let λ ∈ [0.5, 1] be a minimum credibility level. Decrease h from H + 1 until the first t such that σ_I (a, b_t) ≥ λ:

If t = H + 1, assign a to C_H.

If t = 0, assign a to C₁.

For 0 < t < H + 1, if min {σ_I (a, b_t) , σ_D (b_t, a)}> min {σ_I (a, b_t+1) , σ_D (b_t+1, a)} then assign a to C_t; otherwise, assign a to C_t+1.

Ascending Rule. Let λ ∈ [0.5, 1] be a minimum credibility level. Increase h from 0 until the first t such that σ_D (b_t, a) ≥ λ:

If t = 1, assign a to C₁.

If t = H + 1, assign a to C_H.

For 0 < t < H + 1, if min {σ_I (a, b_t) , σ_D (b_t, a)},> min {σ_I (a, b_t-1) , σ_D (b_t-1, a)} then assign a to C_t; otherwise, assign a to C_t-1.

4 Application

In this section, the model proposed above is applied to the supplier development problem (SDP) [21, 24]. SDP is an integral part of a supplier development program, to be implemented by a buyer firm, which includes identifying what suppliers are candidate to improvement activities. An improvement program involves the intensive use of financial and human resources. Thus, SDP is an strategic endeavor where uncertain, imprecise and incomplete information may be available in which regards: demand requirements; capacity, which restricts the delivery time, manufacturing time and costs; criteria, which impacts on their relative importance and evaluation; in the presence of several DMs, each of whom has her/his own value system, knowledge and priorities [10 , 49].

Although different MCDA approaches exist to deal with uncertain and imprecise information [30], we are interested on the outranking-based methods. [50] present a review of MCDA methods applied to supplier selection and SDP. These authors propose that most hybrid approaches involving MCDA methods and fuzzy sets have been reported, including methods as Analytical Hierarchic Process (AHP), TOPSIS, SMART, QFD, Dempster Shafer theory of evidence, multi-objective programming. These methods can be classified as compensatory, which implies that bad performance on some criteria can be offset by good performance on others. ELECTRE methods, instead, are non-compensatory since the preference thresholds and the aggregation rules eliminate alternatives that perform very bad in one or several criteria. This is a desirable property, but at the cost of complexity in computations and a difficult process to elicit the threshold parameters.

MCDA methods based on the outranking preference to sort alternatives, applied to deal with SDP, are rather scarce. [4] propose a supplier segmentation process using a method based on PROMETHEE, called PROMSORT. [38] propose FlowSort, a sorting method based on PROMETHEE, to classify suppliers according to a set of criteria. [42] develop a hybrid method, which uses the PROMETHEE’s preference function. [34] use a revised version of ELECTRE TRI [2] to sort suppliers. [18] propose the use of HFS to model the DMs’ judgements regarding the evaluation of alternatives and weights of criteria, which is used as input to the ELECTRE TRI sortingmethod.

4.1 Numerical example

In this section, an application is presented which is adapted from a previous study by [5], where a model of criteria to evaluate strategic suppliers in Mexican sectors is developed. Table 1 shows the criteria and weights in one of the sectors, as reported by these authors.

Table 1
Family of criteria

Code Criterion Weights of criteria

g ₁ Product price 0.2316

g ₂ Product quality 0.2282

g ₃ Reliability of the supplies 0.1795

g ₄ Technological development 0.0803

g ₅ Post-sales service 0.0932

g ₆ Financial capability 0.1146

g ₇ Risk management process 0.0726

Code	Criterion	Weights of criteria
g ₁	Product price	0.2316
g ₂	Product quality	0.2282
g ₃	Reliability of the supplies	0.1795
g ₄	Technological development	0.0803
g ₅	Post-sales service	0.0932
g ₆	Financial capability	0.1146
g ₇	Risk management process	0.0726

Experts coming from several buying firms were invited to complete a questionnaire in order to provide information about the performance of suppliers in these criteria. The scale [0, 10] was chosen to evaluate alternatives. It is worth noticing that the original evaluations were obtained in a context of poor information. [5] provide a detailed explanation of scales used in the original case study. Table 2 shows the matrix of supplier evaluations.

Table 2

Evaluations of suppliers

Supplier	g ₁	g ₂	g ₃	g ₄	g ₅	g ₆	g ₇
e ₁	10.00	10.00	10.00	10.00	10.00	10.00	10.00
e ₂	10.00	6.55	10.00	8.53	8.86	8.94	8.00
e ₃	6.00	2.55	5.64	8.13	4.56	6.82	2.00
e ₄	4.00	8.55	6.55	3.20	10.00	8.94	8.00
e ₅	6.00	10.00	8.55	4.95	10.00	10.00	8.00
e ₆	10.00	8.55	8.55	8.80	9.14	8.94	10.00
e ₇	8.00	6.55	7.09	8.00	7.14	8.94	8.00
e ₈	8.00	8.00	8.55	8.00	8.00	8.94	8.00
e ₉	8.00	6.00	8.55	7.33	8.00	8.94	6.00
b ₁	3.5	3.5	3.5	3.5	3.5	3.5	3.5
b ₂	6.5	6.5	6.5	6.5	6.5	6.5	6.5
b ₃	8.5	8.5	8.5	8.5	8.5	8.5	8.5

Three categories are considered in this model: C₃, the best ones, which do not need a buyer firm (BF) to assist them to improve performance; C₂, the candidates for development; and C₁, the suppliers that are not worth assisting because the costs involved and the resources required to improve them are too high. Therefore, the evaluations of three reference alternatives -b₁, b₂ and b₃- are defined, and also shown in the Table 2. These are fictitious profiles and are defined to illustrate the process.

Three DMs are involved in the decision process. Thus, it is supposed that the outranking functions in this problem are elicited from information provided by them, in the form of TrFNs, having the following form $μ (a, b) = {\begin{matrix} 0 & g (b) \leq d_{1}, \\ l_{μ} (a, b) & g (b) \in [d_{1}, d_{2}], \\ 1 & g (b) \in [d_{2}, d_{3}], \\ r_{μ} (a, b) & g (b) \in [d_{3}, d_{4}], \\ 0 & g (b) \geq d_{4} . \end{matrix}$ (11) Functions l_μ (a, b) , r_μ (a, b) are called the left side and the right side parts of the fuzzy set, respectively. As shown in Fig. 1, these are linear functions. In Table 3, the μ fuzzy sets for each criterion and central reference alternative are shown, for each DM. To simplify computations, we consider that, given a reference central action, each DM defines the same TrFN on the seven criteria.

Table 3

Categories and reference profiles of criteria

C ₁		C ₂		C ₃
DM	FS	g_j (b₁)	FS	g_j (b₂)	FS	g_j (b₃)
DM1	(0, 0, 4.5, 5)	3.5	(4.5, 7, 7, 7.5)	6.5	(7, 8, 10, 10)	8.5
DM2	(0, 0, 3.5, 6)	3.5	(3.5, 6.5, 6.5, 8)	6.5	(6.5, 7, 10, 10)	8.5
DM3	(0, 0, 3.8, 5.2)	3.5	(4.3, 5, 7.5, 8)	6.5	(6.7, 8.3, 10, 10)	8.5

Note that a TrFN is a canonical fuzzy set [7], but also degenerated TrFNs are included in this example. For instance, (4.5, 7, 7, 7.5) is a triangular fuzzy set, such that a₂ = a₃ in (11). Equally, left and right open fuzzy sets are shown, for instance, (0, 0, 4.5, 5) and (7, 8, 10, 10), where a₁ = a₂ and a₃ = a₄, respectively.

In order to compare results using different aggregation rules, three procedures are considered to determine the global outranking indices:

A score function is used to aggregate HFEs (5) and (6): $\begin{matrix} {sdir}_{j} (b_{h}, a) = \frac{1}{K} \sum_{k = 1}^{K} c_{j}^{k} (b_{h}, a), \\ {sinv}_{j} (a, b_{h}) = \frac{1}{K} \sum_{k = 1}^{K} {cinv}_{j}^{k} (b_{h}, a) . \end{matrix}$

A max function is used to aggregate HFEs (5) and (6): $\begin{matrix} {sdir}_{j} (b_{h}, a) = max_{K} {c_{j}^{k} (b_{h}, a)}, \\ {sinv}_{j} (a, b_{h}) = max_{K} {{cinv}_{j}^{k} (b_{h}, a)} . \end{matrix}$

Instead of aggregating HFEs, the average value of thresholds is used as input to calculate the concordance indices, that is: $p_{j} = \frac{1}{K} \sum_{k = 1}^{K} p_{j}^{k}, q_{j} = \frac{1}{K} \sum_{k = 1}^{K} q_{j}^{k} .$

The first and second aggregating operators are monotonic non-decreasing operators and fulfill boundary conditions [14]. The first property states that given l-long HFEs where membership degrees are increasing ordered, H = {h^σ(1), …, h^σ(l)} and H′ = {h^σ(1)′, …, h^σ(l)′}, then h^σ(1) ≤ h^σ(1)′, …, h^σ(l) ≤ h^σ(l)′ implies s (H) ≤ s (H′). This means that the HFEs scores can be compared. The second property states that, given H = {0} and H′ = {1}, then S (H) =0, S (H′) =1.

The third procedure says that uncertainty is observed whenever a threshold has several values, due to DMs having different preference systems. In this case, the procedure does not consider hesitancy, but the average value is calculated which assumes that each threshold is a stochastic variable.

Let us illustrate the computation process. First, let us consider the category b₁ and e₄, with g₁ (e₄) =4.0 and g₁ (b₁) =3.5, in Table 2. Next, by (11), any TrFN elicited from DM_k has the form $(d_{1}^{k}, d_{2}^{k}, d_{3}^{k}, d_{4}^{k})$ . Taking into account TrFNs in Table 3, this yields $q^{k} = d_{3}^{k} - g (e_{4}); p^{k} = d_{4}^{k} - g (e_{4})$ and $q^{' k} = g (e_{4}) - d_{2}^{k}; p^{' k} = g (e_{4}) - d_{1}$ . Thus, thresholds for these three DMs are as follows: $\begin{matrix} {p'_{1}^{1}, q'_{1}^{1}, q_{1}^{1}, p_{1}^{1}} = {3.5, 3.5, 1.0, 1.5}, \\ {p'_{1}^{2}, q'_{1}^{2}, q_{1}^{2}, p_{1}^{2}} = {3.5, 3.5, 0.0, 2.5}, \\ {p'_{1}^{3}, q'_{1}^{3}, q_{1}^{3}, p_{1}^{3}} = {3.5, 3.5, 0.3, 1.7} . \end{matrix}$ Using the expressions (1) and (2), the following indices can be computed $\begin{matrix} c_{1}^{1} (b_{1}, e_{4}) = 1.0 & {cinv}_{1}^{1} (e_{4}, b_{1}) = 1.0, \\ c_{1}^{2} (b_{1}, e_{4}) = 0.8, & {cinv}_{1}^{2} (e_{4}, b_{1}) = 1.0, \\ c_{1}^{3} (b_{1}, e_{4}) = 0.9, & {cinv}_{1}^{3} (e_{4}, b_{1}) = 1.0 . \end{matrix}$ These indices can be used to compute the score functions for C₁: sdir₁ (b₁, e₄) = (1.0 + 0.8 + 0.9)/3;sinv₁ (e₄, b₁) = (1.0 + 1.0 + 1.0)/3. Considering all the categories, we obtain:

$\begin{matrix} {sdir}_{1} (b_{1}, e_{4}) = 0.9, & {sinv}_{1} (e_{4}, b_{1}) = 1.0, \\ {sdir}_{1} (b_{2}, e_{4}) = 1.0, & {sinv}_{1} (e_{4}, b_{2}) = 0.1, \\ {sdir}_{1} (b_{3}, e_{4}) = 1.0, & {sinv}_{1} (e_{4}, b_{3}) = 0.0 . \end{matrix}$ Using the weights of criteria in Table 1 and evaluations in Table 2, these calculations can be replicated for each criterion which gives the following global concordance indices

$\begin{matrix} σ_{D} (b_{1}, e_{4}) = 0.3 & σ_{I} (e_{4}, b_{1}) = 1.0, \\ σ_{D} (b_{2}, e_{4}) = 0.5, & σ_{I} (e_{4}, b_{2}) = 0.7, \\ σ_{D} (b_{3}, e_{4}) = 1.0, & σ_{I} (e_{4}, b_{3}) = 0.5 . \end{matrix}$ In consequence, if λ = 0.8, the ascending assignment rule assigns e₄ into C₃ while the descending rule assigns e₄ into C₂. In Table 4, assignments are summarized for all the alternatives, all the categories, and the three operators compared in this study.

Table 4

Assignment of suppliers according to three procedures

	Average score		Max score		Average Thresholds
Supplier	Descending	Ascending	Descending	Ascending	Descending	Ascending
e ₁	C ₃	C ₃	C ₃	C ₃	C ₃	C ₃
e ₂	C ₃	C ₃	C ₃	C ₃	C ₃	C ₃
e ₃	C ₂	C ₂	C ₂	C ₂	C ₂	C ₂
e ₄	C ₂	C ₃	C ₂	C ₃	C ₂	C ₃
e ₅	C ₃	C ₃	C ₃	C ₃	C ₃	C ₃
e ₆	C ₃	C ₃	C ₃	C ₃	C ₃	C ₃
e ₇	C ₃	C ₃	C ₃	C ₃	C ₃	C ₃
e ₈	C ₂	C ₂	C ₂	C ₂	C ₃	C ₃
e ₉	C ₂	C ₃	C ₂	C ₃	C ₃	C ₃

Observe that the descending and the ascending rules may assign an alternative to different categories, which has been already observed in other applications of ELECTRE TRI-C method [2]. In all cases, the average score and the max score agree in the assignments. The average threshold procedure, however, assigns the alternatives to a category equal or superior. This procedure could be said optimistic.

4.2 Sensitivity analysis

Sensitivity analysis has an important status in MCDA [28]. Although different definitions have been proposed, we propose to analyze robustness, which can be conveniently referred as the ability of a solution to cope with uncertainties [45]. Different sources of uncertainty interact in a decision problem, some reflecting more or less arbitrary choices of the decision analyst and others concerning external uncertainties. In this sense, it has been proposed that the “robustness concern” needs to be explicitly stated in a problem such that the analysis be driven by a specific aim [1], defining the uncertainty dimensions that are to be dealt with.

In this study, two sources of uncertainty are considered: the unstable thresholds and the weights of criteria. Above, HFS have been proposed to deal with the unstable thresholds. In order to analyze the second source of uncertainty, we are going to consider unknown but stochastic weights. We assume that DMs are not able to provide any information to model a specific form of their probability distributions, then a uniformly probability distribution is assumed for each weight.

Let $W^{l} = {w_{j} ∣ 0 \leq w_{j} \leq 1, \sum_{j = 1}^{7} w_{j} = 1, w_{j} \sim$ U (0, 1) , w_j ≥ ∈}, with ∈ an Archimedean number, be the set of weights generated in a round l ∈ {1, 2, …, L} of a random generator process. In this study, Monte Carlo Simulation is used to generate weights and L = 10000 rounds are run. In Table 5 the results are summarized. The percentage of times that an alternative is assigned into a category, both by the descending and the ascending rules, for each aggregation operator, is shown.

Table 5
Sensitivity analysis results (L = 10000)

Descending Ascending

Operator Alternative C3 C2 C1 C3 C2 C1

Average HFS e1 100 0 0 100 0 0

e2 100 0 0 100 0 0

e3 0 79 21 0 81 19

e4 20 78 2 74 26 0

e5 98 2 0 98 2 0

e6 100 0 0 100 0 0

e7 100 0 0 100 0 0

e8 0 79 21 0 81 19

e9 0 78 2 74 26 0

Max HFS e1 100 0 0 100 0 0

e2 100 0 0 100 0 0

e3 0 83 17 0 85 15

e4 5 73 2 76 23 0

e5 98 2 0 98 2 0

e6 100 0 0 100 0 0

e7 100 0 0 100 0 0

e8 0 3 17 0 85 15

e9 6 72 2 76 23 0

Mean threshold e1 100 0 0 100 0 0

e2 100 0 0 100 0 0

e3 0 82 8 0 84 16

e4 1 77 2 75 25 0

e5 97 3 0 98 2 0

e6 100 0 0 100 0 0

e7 95 5 0 95 5 0

e8 100 0 0 100 0 0

e9 91 9 0 91 9 0

		Descending	Ascending
Average HFS	e1	100	0	0	100	0	0
	e2	100	0	0	100	0	0
	e3	0	79	21	0	81	19
	e4	20	78	2	74	26	0
	e5	98	2	0	98	2	0
	e6	100	0	0	100	0	0
	e7	100	0	0	100	0	0
	e8	0	79	21	0	81	19
	e9	0	78	2	74	26	0
Max HFS	e1	100	0	0	100	0	0
	e2	100	0	0	100	0	0
	e3	0	83	17	0	85	15
	e4	5	73	2	76	23	0
	e5	98	2	0	98	2	0
	e6	100	0	0	100	0	0
	e7	100	0	0	100	0	0
	e8	0	3	17	0	85	15
	e9	6	72	2	76	23	0
Mean threshold	e1	100	0	0	100	0	0
	e2	100	0	0	100	0	0
	e3	0	82	8	0	84	16
	e4	1	77	2	75	25	0
	e5	97	3	0	98	2	0
	e6	100	0	0	100	0	0
	e7	95	5	0	95	5	0
	e8	100	0	0	100	0	0
	e9	91	9	0	91	9	0

The analysis of results shows that in 22 out of 27 cases the descending and the ascending rules agree in the assignments. However, it can be observed that: (1) e₄ and e₉ in the cases of HFS operators may be assigned into C₃ or C₂; (2) these operators give C₂ for e₃ and e₈; (3) according to the mean threshold operator, it can be also verified that e₈ and e₉ are clearly assigned into C₃, and e₃ into C₂.

If results obtained in these three operators are compared each other, a robust assignment can be identified, except in case of e₄ and e₉, where the HFS operators assign them into C₃ or C₂, while the mean operator assigns it into C₃. Finally, the following categories are concluded:

$\begin{matrix} C_{3} & = & {e_{1}, e_{2}, e_{4}, e_{5}, e_{6}, e_{7}, e_{9}}, \\ C_{2} & = & {e_{3}, e_{4}, e_{8}, e_{9}}, \\ C_{1} & = & {} . \end{matrix}$ (12) Note that C₁ is empty. In addition, given the uncertainty related to e₄ and e₉, we put these alternatives into C₃ and C₂. A recommendation may be proposed which consists of a more in depth analysis to acquire additional information about these alternatives.

4.3 Advantages and limitations

Few methods have been proposed in the literature to deal with ELECTRE sorting methods, in cases where thresholds are uncertain and scores are imprecise. SMAA-TRI [40], for instance, may support cases in which thresholds can be modeled as stochastic variables, even if scores are deterministic. Uncertainty included in the application above, due to the DMs’ different preference systems, may be attributed to different objectives, knowledge systems and experiences and not to some condition establishing their ignorance about the “true” value of a threshold. Thus, instability of parameters is not attributed to stochastic uncertainty. Extensions of ELECTRE methods, using HFS, have all considered stable and unique parameters, whatever the group of DMs in the process. One of the main contributions of this study consists of using HFSs to model uncertainty without restrictive assumptions regarding the probability distribution of threshold values. Instead, stochastic uncertainty of criteria weights has been chosen to undertake robustness analysis. This needs to be investigated in a future research such that a “full HFS” approach be proposed.

Extensions to ELECTRE methods use HFS to model evaluations of alternatives. In such case, the number of fuzzy sets to be elicited for each DM is n × m, which even for small problems supposes a lot of effort demanded to the analyst and/or the DMs. Moreover, this also supposes that scores representing evaluations of alternatives can be modeles as HFSs. We have considered a kind of situation in which scores are imprecise, but not uncertain. In addition, due to uncertain thresholds, the outranking functions are hesitant. Thus, the number of fuzzy sets to be elicited is greatly reduced to H × m, because usually H << n. In the example, the extensions of ELECTRE methods need to elicit 63 HFSs, against 21 in our model, for each DM. This advantage can be obtained whenever uncertainty in thresholds is considered. Nevertheless, a limitation could appear if DMs are not available to provide information used to build the outranking functions. If the DMs are able to provide information, the cognitive effort that our approach requires to elicit outranking functions is quite lower than the effort required to elicit uncertain scores proposed in other approaches.

Several types of trapezoidal fuzzy numbers have been considered in this example, which have been supposed to be the result of the respective elicitation processes run with the DMs. In particular, left and right open fuzzy sets have been used. This suggests that the approach proposed in this study, i.e. modeling outranking functions as HFSs, can be easily extended to other ELECTRE methods where direct indifference and preference thresholds are considered.

5 Conclusions

In this study, an approach that considers hesitant ELECTRE outranking functions is proposed. Hesitancy is due to unstable thresholds modeling imprecision in evaluations on each criterion in the family of criteria in the decision problem. Differently from other approaches incorporating hesitant fuzzy sets in ELECTRE methods, deterministic scores are considered. This is a new kind of situation, which has not been considered in HFS extensions of ELECTRE methods. The approach provides an alternative way to consider uncertain parameters in ELECTRE, complementary to other models proposed in the literature.

An application, regarding a supplier development problem, is presented using ELECTRE TRI-C and HFS, to assign a set of nine alternatives into three categories. Sensitivity analysis, considering stochastic weights of criteria, is proposed to elaborate conclusions. A robust conclusion emerged from the results obtained in the analysis process, which shows the feasibility of our approach. We think that this extension and the proposed approach could be applied in other areas in which non-compensatory methods are appropriate.

A future work will consider the use of HFS in other ELECTRE TRI methods. Future work also consists of modeling the discordance effect, which for the sake of simplicity we have not included in this paper. In addition, we evaluate to consider the extension to mixed problems, using both quantitative and qualitative criteria.

Footnotes

Acknowledgment

The authors are grateful for the support from FACEPE (PRONEX) and for the partial support of CNPq, Brazil.

References

Aissi

, Roy

, Trends in Multiple Criteria Decision Analysis, Chapter Robustness in multi-criteria decision aiding, Springer, 2010, pp. 87–121.

Almeida-Dias

, Figueira

, Roy

, Electre Tri-C: A multiple criteria sorting method based on characteristic reference actions, European Journal of Operational Research 204(3) (2010), 565–580.

Almeida-Dias

, Figueira

, Roy

, A multiple criteria sorting method where each category is characterized by several reference actions: The Electre Tri-nC method, European Journal of Operational Research 217(3) (2012), 567–579.

Araz

, Ozkarahan

, Supplier evaluation and management system for strategic sourcing based on a new multicriteria sorting procedure, International Journal of Production Economics 106 (2007), 585–606.

Arroyo-López

P.E.

, Ramos-Rangel

J.A.

, A methodological proposal to define supplier development programs, Ingeniería, Investigación y Tecnología XIX (1) (2018), 25–36. http://www.revistaingenieria.unam.mx/numeros/2018/v19n1–03.pdf.

Badi

, Abdulshahed

, Shetwan

, A case study of supplier selection for a steelmaking company in libya by using the Combinative Distance-based ASsessment (CODAS) model, Decision Making: Applications in Management and Engineering 1(1) (2018), 1–12.

Ban

, Brândas

, Coroianu

, Negrutiu

, Nica

, Approximations of fuzzy numbers by trapezoidal fuzzy numbers preserving the ambiguity and value, Computers and Mathematics with Applications 61 (2011), 1379–1401.

Bouyssou

, Outranking relations: Do they have special properties?, Journal of Multi-Criteria Decision Analysis 5(2) (1996), 99–111.

Chen

, Xu

, Hesitant fuzzy ELECTRE II approach: A new way to handle multi-criteria decision making problems, Information Sciences 292 (2015), 175–197.

10.

De Boer

, Labro

and Morlacchi

, A review of methods supporting supplier selection, European Journal of Purchasing & Supply Management 7(2) (2001), 75–89.

11.

Dong

, Yuan

, Wan

, Extended VIKOR method for multiple criteria decision-making with linguistic hesitant fuzzy information, Computers & Industrial Engineering 112 (2017), 305–319.

12.

Fahmi

, Kahraman

, Bilen

, ELECTRE I method using hesitant linguistic term sets: An application to supplier selection, International Journal of Computational Intelligence Systems 9(1) (2016), 153–167.

13.

Faizi

, Rashid

, Salabun

, Zafar

, Watrobski

, Decision making with uncertainty using hesitant fuzzy sets, International Journal of Fuzzy Systems 20(1) (2018), 93–103.

14.

Farhadinia

, A series of score functions for hesitant fuzzy sets, Information Sciences 277 (2014), 102–110.

15.

Fernández

, Figueira

, Navarro

, Roy

, ELECTRE TRI-nB: A new multiple criteria ordinal classification method, European Journal of Operational Research 263 (2017), 214–224.

16.

Figueira

, Almeida

, Matias

, Roy

, Carvalho

, Plancha

, Electre Tri-C, a multiple criteria decision aiding sorting model applied to assisted reproduction, International Journal of Medical Informatics (80)(2011), 262–273.

17.

Figueira

, Greco

, Roy

, Slowinski

, Handbook of Multicriteria Analysis, Chapter ELECTRE Methods: Main Features and Recent Developments, Springer, Vol. 103, 2010, pp. 51–89.

18.

Galo

, Calache

, Carpinetti

, A group decision approach for supplier categorization based on hesitant fuzzy and ELECTRE TRI, International Journal of Production Economics 202 (2018), 182–196.

19.

Govindan

, Jepsen

, ELECTRE: A comprehensive literature review on methodologies and applications, European Journal of Operational Research 250(1) (2016a), 1–29.

20.

Govindan

, Jepsen

, Supplier risk assessment based on trapezoidal intuitionistic fuzzy numbers and ELECTRE TRI-C: A case illustration involving service suppliers, Journal of the Operational Research Society (67) (2016b), 339–376.

21.

Hartley

, Choi

, Supplier development: Customers as a catalyst of process change, Business Horizons (1996), 37–44.

22.

Hatami-Marbini

, Tavana

, An extension of the ELECTRE I method for group decision-making under a fuzzy environment, Omega(39) (2011), 373–386.

23.

Jin

, ELECTRE method for multiple attributes decision making problem with hesitant fuzzy information, Journal of Intelligent and Fuzzy Systems 29 (2015), 463–468.

24.

Krause

, Handfield

, Scanell

, An empirical investigation of supplier development: Reactive and strategic processes, Journal of Operations Management 17 (1998), 39–58.

25.

Leyva-López

, Fernández-González

, A new method for group decision support based on ELECTRE III methodology, European Journal of Operational Research 148 (2003), 14–27.

26.

Macary

, Almeida

, Figueira

, A multiple criteria decision analysis model based on ELECTRE TRI-C for erosion risk assessment in agricultural areas, Environmental Modeling & Assessment 19(3) (2014), 221–242.

27.

Mahmoudi

, Sadi-Nezhad

, Makui

, Vakili

, An extension on PROMETHEE based on the typical hesitant fuzzy sets to solve multi-attribute decision-making problem, Kybernetes 45(8) (2016), 1213–1231.

28.

Mukhametzyanov

, Pamucar

, A sensitivity analysis in MCDM problems: A statistical approach, Decision Making: Applications in Management and Engineering 1(2) (2018), 51–80.

29.

Onar

, Oztaysi

, Kahraman

, Strategic decision selection using hesitant fuzzy TOPSIS and interval type-2 fuzzy AHP: A case study, International Journal of Computational Intelligence Systems 7(5) (2014), 1002–1021.

30.

Pamucar

, Bozanic

, Lukovac

, Komazec

, Normalized weighted geometric Bonferroni mean operator of interval rough numbers – application in interval rough DEMATEL-COPRAS model, Mechanical Engineering 16(2) (2018), 171–191.

31.

Peng

, Wang

, Yang

L.-J.

, Chen

, An extension of ELECTRE to multi-criteria decision-making problems with multi-hesitant fuzzy sets, Information Sciences 307 (2015), 113–126.

32.

Peng

, Wang

J.-Q.

, Zhang

H.-Y.

, Chen

X.-H.

, An outranking approach for multi-criteria decision-making problems with simplified neutrosophic sets, Applied Soft Computing 25 (2014), 336–346.

33.

Perny

, Roy

, The use of fuzzy outranking relations in preference modelling, Fuzzy Sets and System(49) (1992), 33–53.

34.

Rezaei

, Kadziński

, Vana

, Tavasszy

, Embedding carbon impact assessment in multi-criteria supplier segmentation using ELECTRE TRI-rC, Annals of Operations Research (2017), 1–23. DOI: 10.1007/s10479-017-2454-y.

35.

Rezaei

, Ortt

, Multi-criteria supplier segmentation using a fuzzy preference relations based on AHP, European Journal of Operational Research 225 (2013), 75–84.

36.

Roy

, Main sources of inaccurate determination, uncertainty and imprecision in decision models, Mathematical and Computer Modelling 12(10-11) (1989), 1245–1254.

37.

Roy

, Figueira

, Almeida

, Discriminating thresholds as a tool to cope with imperfect knowledge in multiple criteria decision aiding: Theoretical results and practical issues, CAHIER DU LAMSADE 2012, Université Paris Dauphine. 2012.

38.

Sepúlveda

, Derpich

, Automated reasoning for supplier performance appraisal in supply chains, Procedia Computer Science 31 (2014), 966–975.

39.

Sevkli

, An application of the fuzzy ELECTRE method for supplier selection, International Journal of Production Research 48(12) (2010), 3393–3405.

40.

Tervonen

, Figueira

, A survey on stochastic multicriteria acceptability analysis methods, Journal of Multi-Criteria Decision Analysis 15(1-2) (2008), 1–14.

41.

Torra

, Hesitant fuzzy sets, International Journal of Intelligent Systems 25 (2010), 529–539.

42.

Tsui

C.-W.

, Tzeng

, Wen

U.-P.

, A hybrid MCDM approach for improving the performance of green suppliers in the TFT-LCD industry, International Journal of Production Research 53(21) (2015), 6436–6454.

43.

Vahdani

, Hadipour

, Extension of the ELECTRE method based on interval-valued fuzzy sets, Soft Computing(15) (2011), 569–579.

44.

Veskovic

, Stevic

, Stojic

, Vasiljevic

, Milinkovic

, Evaluation of the railway management model by using a new integrated model DELPHI-SWARA-MABAC, Decision Making: Applications in Management and Engineering 1(2) (2018), 34–50.

45.

Wallenius

, Dyer

J.S.

, Fishburn

P.C.

, Steuer

R.E.

, Zionts

, Deb

, Multiple criteria decision making, multiattribute utility theory: Recent accomplishments and what lies ahead, Management Science 54(7) (2008), 1336–1349.

46.

Wang

, Wang

, Zhang

H.-Y.

, Chen

, Multi-criteria outranking approach with hesitant fuzzy sets, OR Spectrum 36 (2014), 1001–1019.

47.

Wang

J.-Q.

, Wang

, Chen

Q.-H.

, Zhang

H.-Y.

, Chen

X.-H.

, An outranking approach for multi-criteria decision-making with hesitant fuzzy linguistic term sets, Information Sciences 280 (2014), 338–351.

48.

, Chen

, Interval-valued intuitionistic Fuzzy ELECTRE method, Asian International Journal of Science and Technology in Production and Manufacturing Engineering 5(3) (2012), 33–40.

49.

, Blackhurst

, Chidambaram

, A model for inbound supply risk analysis, Computers in Industry 57 (2006), 350–365.

50.

Yildiz

, Yayla

, Multi-criteria decision-making methods for supplier selection: A literature review, South African Journal of Industrial Engineering 26(2) (2015), 158–177.

51.

, Zhang

, Zhong

, Sun

, Extended TODIM for multi-criteria group decision making based on unbalanced hesitant fuzzy linguistic term sets, Computers & Industrial Engineering 114 (2017), 316–328.

52.

Zhang

, Guo

, Fusion of heterogeneous incomplete hesitant preference relations in group decision making, International Journal of Computational Intelligence Systems 9(2) (2016), 245–262.

53.

Zhang

, Kou

, Dong

, Additive consistency analysis and improvement for hesitant fuzzy preference relations, Expert Systems With Applications 98 (2018), 118–128.

54.

Zhang

, Kou

, Yu

, Guo

, On priority weights and consistency for incomplete hesitant fuzzy preference relations, Knowledge-Based Systems 143 (2018), 115–126.

55.

Zhu

, Xu

, Regression methods for hesitant fuzzy preference relations, Technological and Economic Development of Economy 19(sup1) (2013), S214–S227.

		Descending			Ascending
Operator	Alternative	C3	C2	C1	C3	C2	C1
Average HFS	e1	100	0	0	100	0	0
	e2	100	0	0	100	0	0
	e3	0	79	21	0	81	19
	e4	20	78	2	74	26	0
	e5	98	2	0	98	2	0
	e6	100	0	0	100	0	0
	e7	100	0	0	100	0	0
	e8	0	79	21	0	81	19
	e9	0	78	2	74	26	0
Max HFS	e1	100	0	0	100	0	0
	e2	100	0	0	100	0	0
	e3	0	83	17	0	85	15
	e4	5	73	2	76	23	0
	e5	98	2	0	98	2	0
	e6	100	0	0	100	0	0
	e7	100	0	0	100	0	0
	e8	0	3	17	0	85	15
	e9	6	72	2	76	23	0
Mean threshold	e1	100	0	0	100	0	0
	e2	100	0	0	100	0	0
	e3	0	82	8	0	84	16
	e4	1	77	2	75	25	0
	e5	97	3	0	98	2	0
	e6	100	0	0	100	0	0
	e7	95	5	0	95	5	0
	e8	100	0	0	100	0	0
	e9	91	9	0	91	9	0

ELECTRE TRI-C with hesitant outranking functions: Application to supplier development

Abstract

Keywords

1 Introduction

2 Notation and basic definitions

2.1 Hesitant fuzzy sets

2.2 ELECTRE TRI-C

4.1 Numerical example

Table 1 Family of criteria Code Criterion Weights of criteria g 1 Product price 0.2316 g 2 Product quality 0.2282 g 3 Reliability of the supplies 0.1795 g 4 Technological development 0.0803 g 5 Post-sales service 0.0932 g 6 Financial capability 0.1146 g 7 Risk management process 0.0726

5 Conclusions

Footnotes

Acknowledgment

References