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

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 ) = 1 l h ∑ γ ∈ 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 ′ ) = { 0 if g j ( a ′ ) - g j ( a ) > p j , 1 if g j ( a ′ ) - g j ( a ) ≤ q j , g j ( a ) - g j ( a ′ ) + p j p j - q j otherwise ,(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 ′ ) = { 0 if g j ( a ′ ) - g j ( a ) > p j ′ , 1 if g j ( a ′ ) - g j ( a ) ≤ q j ′ , g j ( a ) - g j ( a ′ ) + p j ′ p j ′ - q j ′ otherwise .(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.

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.

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.

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 ) = { 0 g ( b ) ≤ d 1 , l μ ( a , b ) g ( b ) ∈ [ d 1 , d 2 ] , 1 g ( b ) ∈ [ d 2 , d 3 ] , r μ ( a , b ) g ( b ) ∈ [ d 3 , d 4 ] , 0 g ( b ) ≥ d 4 .(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.

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): sdir j ( b h , a ) = 1 K ∑ k = 1 K c j k ( b h , a ) , sinv j ( a , b h ) = 1 K ∑ k = 1 K cinv j k ( b h , a ) .

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: { 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 } . Using the expressions (1) and (2), the following indices can be computed 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 . 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:

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 . 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

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 . 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.

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.

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 ≤ w j ≤ 1 , ∑ j = 1 7 w j = 1 , w j ∼ 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.

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:

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 = { } .(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.

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.