An extension of PROMETHEE to hierarchicalmulticriteria clustering

J. Rosenfeld∗1 and Y. De Smet†1

1CoDE-SMG Research Unit, Ecole polytechnique de Bruxelles, Universitélibre de Bruxelles, Avenue F.D. Roosevelt 50, 1050 Brussels, Belgium

May 2019

Abstract

Multicriteria clustering can be seen as a hybridization between ranking andsorting problematic. These methods are used to build totally or partially orderedgroups of alternatives based on preference relations. In the context of totallyordered clustering, two hierarchical approaches (top-down and bottom-up) basedon PROMETHEE II have been developed in this paper. These methods relyon the optimization of the clustering structure (by maximizing the intra-clusterhomogeneity and the inter-clusters heterogeneity). A third approach is developedas a hybrid model that merges the information obtained by both previous models.A specific quality index has been introduced to be able to evaluate the method’soutputs and to choose appropriately the desired number of clusters. The threeprocedures have been tested on several dataset (Shanghai Ranking of WorldUniversities, Environmental Performance Index and CPU evaluations) and theresults have been compared with P2Clust.

Keywords: Multiple Criteria Analysis, PROMETHEE, Hierarchical Clustering, Multi-criteria Clustering, Quality Index

1 IntroductionMany engineering decision problems can be modeled as the optimization of a set ofalternatives according to multiple conflicting criteria. In Multiple Criteria Decision Aid(MCDA) one usually distinguishes three main “so-called” problematics [29][7][9]: the∗jean.rosenfeld@ulb.ac.be†yves.de.smet@ulb.ac.be

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selection of a subset among the best alternatives (choice problem), the assignment ofalternatives into predefined classes (sorting problem) or the ranking of the alternativesfrom the best to the worst ones according to a complete or a partial order (rankingproblem).More recently, researchers have started to investigate a new kind of problem: multicriteriaclustering i.e. the identification of groups of alternatives that share similar multicriteriaprofiles. Due to the multicriteria nature of the problem, these clusters are often(completely or partially) ordered. This is referred to as relational clustering (in oppositionto nominal clustering where no order relation exists between the groups).In the competitive society we are living in, there is a large quantity of rankings (ofdifferent kinds). For instance, the Shanghai Ranking of World Universities [26] isdedicated to the assessment of 500 major universities according to six criteria (Alumnias Nobel laureates & Fields Medalists, Staff as Nobel Laureates & Fields Medalists,Highly cited researchers in 21 broad subject categories, Papers published in Nature andScience, Papers indexed in Science Citation Index-expanded and Social Science CitationIndex, Per capita academic performance of an institution). The 100 first institutionsare ranked individually. Then, from 100 to 200 they are listed in two groups of 50; andfrom 200 to 500 they are listed by groups of 100. This constitutes an example where theranking problematic is mixed with sorting. Finally, let us note that the sizes of theseclusters are not based on the structure of the dataset but are rather arbitrarily chosen.On the contrary, clustering algorithms try to detect natural groups structure into theset of alternatives. Multicriteria clustering can be seen as a hybridization between theclustering and the ranking problematic.To the best of our knowledge, the first contribution about multicriteria clustering basedon binary preferences has been proposed by De Smet and Montano [12]. They introduceda specific distance that takes into account the multicriteria preferential informationinduced by the comparison of alternatives. Unfortunately, their algorithm was limitedto the detection of nominal clusters. Later, De Smet and Eppe [11] proposed a naturalextension of the first work to build relations between the groups. It has also beencompleted in the context of valued relations [17]. The transitivity of the cluster relationsor the fact that they were acyclic (even if artificial experiments have shown that suchproblems were seldom) were not guaranteed it this case. Then, an exact algorithm todetect a totally ordered clustering has been developed in [13]. Rocha proposed a methodfor multicriteria clustering in which they distinguish first the clustering approach (whichdoes not integrate the preferences of the decision maker (DM) ) and then a multicriteriatechnique to find the relations between the groups [24]. A formalization of this emergingtopic has been proposed more recently [22].Many MCDA methods have been developed up to now. One may cite Multi-AttributeUtility Theory (MAUT) [15], ELECTRE methods [19], MACBETH [1], PROMETHEE[18][3], etc. In this paper, we will focus on PROMETHEE methods. These have beenapplied to a wide range of application fields such as finance, health care, sport, transport,environmental management, etc [2]. To our point of view, this success is due to (1) theirsimplicity and (2) the existence of user-friendly software such as Visual PROMETHEE,

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Smart Picker [23] or D-Sight [20].Let us stress that the PROMETHEE methodology has already been extended to

clustering. After the initial work of PROMETHEE Cluster [18], a model based on theK-means procedure has been recently proposed. The latter is referred to P2CLUSTand PCLUST [9][25]. In this paper, a complementary contribution is developed witha model based on both an ascending and a descending hierarchical procedure. Thesedeterministic approaches are based on a specific property of the net flow scores. Finally,using the information of both methods, a hybrid approach has been developed based ona quality index characterizing multicriteria ordered clustering. Let one note that DeSmet [10] introduced a first method to divide a cluster in two sub-clusters in the contextof hierarchical multicriteria PROMETHEE clustering a few years ago. In the presentpaper, we extend this first contribution by refining the proof of the property used todivise the clusters (see Section 3). In addition we propose a bottom-up procedure aswell as a hybrid approach. Empirical tests based on several data sets further completethis contribution.

This paper is organized as follows; in section 2 we briefly recall the basics ofPROMETHEE and we demonstrate a new property in section 3. Then, the two newmodels are introduced in section 4 as well as the hybrid approach. A quality indexof multicriteria clustering is introduced in section 5. The models will be tested onillustrative examples and compared to P2CLUST in section 6. Conclusions in section 7will end the paper.

2 The PROMETHEE II rankingThe PROMETHEE outranking methods were developed in the early 80s by Brans andMareschal [4, 6, 28]. They are designed to help DMs to solve multicriteria problemsby using a valued outranking relation. This relation is based on pairwise comparisonsbetween alternatives. We invite the interested reader to [2, 5, 4] for a detailled descriptionof the method.First, let us consider a set of n alternatives A = {a1, a2, ..., an} and a set of q criteriaF = {g1, g2, ..., gq}. Without loss of generality, let us consider that these q criteriahave to be maximized. For each criterion gk, the DM evaluates the preference of analternative ai over an alternative aj by measuring the difference of their evaluations oncriterion gk.

dk(ai, aj) = gk(ai)− gk(aj) (1)

This allows to quantify how alternative ai is better than alternative aj on a givencriterion. However, dk(ai, aj) depends on the units of the considered criterion and doesnot integrate any intra-criterion preference information provided by the DM. Thus,one has to transform these differences into preference degrees using a non-decreasingpreference function Pk : R → [0, 1]. Figure 1 presents an example of such a functionwhere qk is the indifference threshold and pk is the preference threshold. Below the

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indifference threshold, the difference is considered to be so low that the preference issupposed to be equal to 0. Beyond the preference threshold the difference is so importantthat the preference is strict and equal to 1. Between these two thresholds, the preferencefunction evolves linearly. This function represents the unicriterion preference betweenai and aj according to the criterion gk.

Figure 1: Example of a linear preference function

Then, a global preference between each pair of alternatives is computed as follow:

π(ai, aj) =

q∑k=1

ωk · Pk[dk(ai, aj)] (2)

where ωk represents the weight associated to criterion gk. For the sake of simplicity, wewill also use πij to denote π(ai, aj). We assume that ωk > 0 and

∑qk=1 ωk = 1.

Finally, the negative flow score Φ− defines the “average" weaknesses of an alternativeand the positive flow score Φ+ defines its "average" strengths:

Φ+(ai) =1

n− 1

∑aj∈A

π(ai, aj) (3)

Φ−(ai) =1

n− 1

∑aj∈A

π(aj, ai) (4)

The PROMETHEE II score (or net flow score) is computed as follows:

Φ(ai) = Φ+(ai)− Φ−(ai) (5)

The net flow score varies in the range [-1,1]. Based on these values one can rank thealternatives from the worst to the best ones. Obviously, the net flow score depends onthe set of considered alternatives. Therefore the notation ΦA(ai) is sometimes used.

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3 A new property of the net flow scoreThe PROMETHEE II score can easily be interpreted. It is a “balance" between theaverage strengths (positive net flow) and the average weaknesses (negative net flow) ofeach alternative. Some researchers have further analyzed the meaning of such scores.They have proved that net flow scores satisfy some natural properties [4] such asmonotonicity, dominance, independence to non-discriminatory criterion, etc. We willintroduce a complementary argument.Let us consider the following problem: one tries to divide the set of alternatives A intotwo complementary subsets denoted B and B such that B is strongly preferred to B.This can be interpreted to be equivalent to maximize the global preference of B over Band minimize the global preference of B over B. More formally, we want to identify asubset B∗ ⊂ A such that:

B∗ = argmax∑i∈B

∑j∈B

(πij − πji) (6)

The solution of this problem could be found by testing all potential subsets B ⊂ A butthis would be time-consuming for a large number of alternatives. In the following, wedemonstrate it can be done in a more efficient way.Proposition: B∗ is determined by the set of alternatives which have positive net flowscores.Proof :Obviously we have: ∑

i∈B

∑j∈B

(πij − πji) = 0 (7)

Given that A = B ∪B , we have:∑i∈B

∑j∈B

(πij − πji) =∑i∈B

∑j∈A

(πij − πji) (8)

Which amounts to: ∑i∈B

∑j∈B

(πij − πji) = (n− 1)∑i∈B

Φi (9)

As a consequence, the optimal set is strictly composed of alternatives which arecharacterized by positive net flow scores. Therefore, computing the PROMETHEEranking allows solving the previous optimization problem.In non-multicriteria clustering, the divisive scheme of top-down hierarchical clusteringis computationally very demanding because of its combinatorial aspect, making oftenimpossible to investigate all the splitting possibilities at each step [27]. Consequently,bottom-up methods are used in a very large majority of cases. In contrast, the demon-strated property of the net flow score enables to optimize the division step by maximizingthe heterogeneity between the two resulting clusters. This property thus solves one ofthe main issues in classical top-down hierarchical approaches.

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4 Extension of PROMETHEE to hierarchical orderedclustering

In this section, a model for hierarchical ordered clustering is developed. As stated in theintroduction, ordered clustering involves the detection of a relation between each pair ofclusters. One assumes that the first cluster is better than the second one, which is betterthan the third one, etc. The proposed model is based on a deterministic approach (inopposition to P2Clust [9] which is based on a stochastic procedure). Two hierarchicalassignment procedures are considered: the top-down and the bottom-up approaches.

4.1 Top-down approach

In the top-down approach, the initial configuration is constituted by one large clusterincluding all the alternatives. This cluster is then divided in two clusters. Then, one ofthese two clusters is divided in two other parts and the procedure is repeated until onegets the desired number of clusters.It is obvious that the more homogeneous the clusters, the better the clustering. At eachstep the cluster that has to be divided is the least homogeneous one. To estimate thehomogeneity of a cluster Ch, the following index is used:

∆h =1

nh

∑i,j∈Ch

π(i, j) (10)

where nh =(|Ch|

2

)which is the number of pairwise comparisons of elements in Ch.

Obviously, this index has to be minimized; the preferences between alternatives belongingto the same cluster have to be as small as possible. In an ideal case ∆h should be equalto 0. In order to get a partition that is as homogeneous as possible, the cluster Ch

which has the largest ∆h is chosen and divided into two parts according to the propertydemonstrated in section 3. The order of the clustering is determined as follows: thealternatives with a positive net flow score are preferred to the others. The procedure isrepeated until the desired number of clusters, denoted by K, is obtained.To summarize the procedure, the algorithm is described in Algotithm 1.

4.2 Bottom-up approach

In the bottom up procedure, each single alternative forms a cluster in the initialconfiguration. These clusters will then be merged two by two until one gets the desirednumber of clusters. In addition, the order of the individual clusters in the initial step isprovided by the PROMETHEE II ranking.In order to have an ordered clustering, we will merge clusters which are next to each other.To choose which clusters we have to merge, symmetrically to the top down approach,we choose the two clusters which, when merged together, form the most homogeneouscluster. To compute it, we use the index ∆h that we have defined previously. One

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Algorithm 1 Top-down approachInput: A, ω, p, q,Knc = 1while nc < K do

determine h such that Ch = argmaxCl∆l

∀ai ∈ Ch, compute ΦCh(ai)

for l=nc+ 1:h+ 1 by step -1 doCl = Cl−1

end for∀ai ∈ Ch: if ΦCh

(ai) < 0 delete ai∀ai ∈ Ch+1: if ΦCh+1

(ai) ≥ 0 delete ainc = nc+ 1

end while

repeats the procedure until the desired number of clusters is obtained. To summarizethe procedure, the algorithm is described in Algotithm 2.

Algorithm 2 Bottom-up approachInput: A, ω, p, q,Knc = 1while nc > K do

for i=1:nc-1 doBi = CiUCi+1

end fordetermine h such that Bh = argminBl

∆l

Ch = Bh

for l=h+ 1:nc− 1 doCl = Cl+1

end fordelete Cnc

nc = nc+ 1end while

4.3 Example on a small dataset

In order to have a pedagogical representation of the two hierarchical methods, let one usea example with 5 alternatives and 4 criteria as presented in the Table 1. The parameters(weights w, indifference and preference thresholds q and p) are described in the Table 2.In this example, the desired number of clusters K = 3.

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q1 q2 q3 q4a1 5 5 5 4a2 5 4 4 4a3 3 4 1 2a4 1 1 2 2

Table 1: Artificial dataset for the illustrative example

q1 q2 q3 q4w 0.25 0.25 0.25 0.25q 0 0 0 0p 5 5 5 5

Table 2: Parameters for the illustrative example

4.3.1 Top-Down method

Initially, in the top-down method, all the alternatives belong to a single cluster. Thefirst step is to compute the net flow score of each alternative (Table 3) and divide thecluster in two sub-clusters according to the sign of the flow scores. In this example, a1

a1 0.4750a2 0.2875a3 -0.1500a4 -0.2125a5 -0.4000

Table 3: Illustrative example: netflow scores (first step)

and a2 will form the cluster C1 (considered as being the best one). The three otheralternatives will form the cluster C2.The second step consists in choosing which cluster has to be divided. The index ∆h

is then computed for each cluster. The least homogeneous cluster is C2 according to

∆1 0.15∆2 0.20

Table 4: Illustrative example: clusters homogeneity

Table 4. The netflow scores of the alternatives beloging to C2 are then computed andpresented in Table 5. The alternatives a3 and a4 will form the new cluster C2 and a5will form the cluster C3 (according to the sign of the netflow scores). At that time, thereare 3 clusters which corresponds to the desired number of cluster K. The Table 6 givesthe final distribution.

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a3 0.125a4 0.050a5 -0.175

Table 5: Illustrative example: netflow scores (second step)

C1 a1,a2C2 a3,a4C3 a5

Table 6: Illustrative example with top-down approach: final distribution

4.3.2 Bottom-Up method

The first step of the bottom-up method consists in ranking all the alternatives accordingto the net flow scores. The Table 3 gives this ranking. Each alternative will then forma cluster. After, the idea is to test all possibilities for merging neighboring clusters.The Table 7 shows these possibilities and the corresponding index ∆h. Among these

a1-a2 a2-a3 a3-a4 a4-a5∆h 0.15 0.35 0.05 0.25

Table 7: Bottom-up approach: possibilities of merging (step 1)

possibilities, the most homogeneous one will be chosen. In this case, a3 and a4 will bemerged.There are 4 clusters at this step. The same procedure will be applied. All the possibilitiesof merging will be considered in order to choose the most homogeneous one. The Table8 shows all the possibilities of merging for this second step. In this case, the merging of

a1-a2 a2-a3-a4 a3-a4-a5∆h 0.150 0.267 0.200

Table 8: Bottom-up approach: possibilities of merging (step 2)

a1 and a2 (which represent respectively C1 and C2 at that step) would form the mosthomogeneous resulting cluster.The desired number of cluster K = 3 is then reached and the final distribution is thesame as the one with the top-down approach (Table 6).

4.4 Hybrid method

It is obvious that the two previous methods do not offer the warranty to get the samefinal ordered clustering partitions. Therefore, a hybrid model based on the results of thetop-down and the bottom-up approaches has been developed. It pools both information

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to achieve better results in terms of a given quality indicator.Each method gives a vector of n elements, denoted b1 and b2, containing the clusterin which the corresponding alternative belongs. One can pool the information of bothvectors b1 and b2 by defining an adjacency matrix A as follows:

A(i, j) = 1 if

{bm(i) ≤ bm(j)bl(i) < bl(j)

m 6= l

A(i, j) = 0 otherwise

(11)

We have A(i, j) = 1 when an alternative ai is preferred to an alternative aj. Thishappens if it belongs to a better cluster according to both clustering methods. Thenone can build subgroups according to the structure of the graph given by the adjacencymatrix. To form these subgroups, we will use the levels or the ranks of the graph.The clustering results of the top-down and bottom-up approaches, b1 and b2, have somedistinctive features. In fact, there is at least one alternative belonging to each of the Kclusters. It implies there will be at least K subgroups according to the levels or the ranksof the graph. The only way to have exactly K subgroups is to have both hierarchicalmethods giving precisely the same result. However, most of the time, one has more thanK subgroups. It is therefore important to reduce the number of subgroups to remainclose to the desired K clusters. Therefore one will merge some of them.Because of the structure of both clustering methods, the subgroups follow a completeorder. To respect this order, the merging step should concern subgroups which are nextto each other. One will then consider all the merging possibilities for the R subgroupsuntil one gets the desired number of subgroups K. Firstly, one has (R− 1) possibilitiesand then (R− 2), etc. The total number of possibilities is AR−1

K . One will evaluate allthese merging possibilities and select the best one. To do this, one will compare themaccording to a quality index defined for a complete ordered clustering (see Section 5).One applies the following procedure:

• Creation of R ordered subgroups according to the pool of the top-down and thebottom-up hierarchical clustering methods;

• Merging neighboring subgroups until we get the K desired subgroups. Among theAR−1

K merging possibilities, we will keep the best one according to a quality index.

Let us note that in all the hierarchical models we presented, the monotonicity conditionis verified. It means that when an alternative improves its evaluations on at least onecriterion, the cluster in which it belongs has to be at least as good as before:

if

gk(a′i) ≥ gk(ai) ∀k∃s : gs(a

′i) > gs(ai)

ai ∈ Ch, a′i ∈ Cl

⇒ l ≤ h (12)

It implies that if an alternative is the best according to all criteria, it will be in thebest cluster. This arises from the properties of the net flow scores [4].

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5 Quality IndexTo compare the quality of different multicriteria ordered clusterings, one might imagineusing a reference evaluation scheme such as a metric. There exist different qualityindexes in the literature, such as the Dunn Index [14], the Davies-Bouldin Index [21],etc. Generally, these indexes measure both the intra-clusters homogeneity and the inter-clusters heterogeneity. They both should be maximized. Unfortunately, these indexeshave not been extended in the context of multicriteria clustering (where asymmetricrelations are used).It this section we introduce a new quality index for complete multicriteria orderedclustering (based on valued preference relations). The evaluation of the quality ofa multicriteria ordered clustering is done as follows. First, clusters have to be ashomogeneous as possible. In broad terms, the alternatives belonging to the same clusterhave to be as “close" as possible to each other. As it has been introduced (equation 10),∆h is the index of homogeneity of a cluster Ch.

The notion of distance that appears in standard quality indexes is replaced by apreference relationship between the alternatives belonging to the same cluster. Thisindex has to be minimized.Secondly, the set of clusters has to be as heterogeneous as possible. In other words,we might say that the clusters have to be as “far" as possible from each other. Let usdenote δ(h, l) the index of heterogeneity between the clusters Ch and Cl when h < l:

δ(h, l) = π(rh, rl)− π(rl, rh) (13)

where rh and rl represent respectively the mean values of the clusters Ch and Cl. It meansthat, for each criterion, rh will be evaluated by the mean value of all the alternativesbelonging to Ch for that criterion. Taking into account that h < l, the cluster Ch isconsidered to be better than Cl. This index has to be maximized because the alternativesof the best cluster have to be preferred to the other alternatives of the set. On thecontrary, the alternatives of the worst cluster should not be preferred over the others.Here we explicitly take into account the order between the clusters.Finally, to have a global evaluation, we have to pool these indexes into a global qualityindex. For the heterogeneity of the set of clusters, we limit the comparisons to succesiveclusters. Then, we define the quality index D as follows:

D =

∑K−1h=1 δ(h, h+ 1)∑K

h=1 ∆h

(14)

Before using the quality index D to compare different approaches, it was necessary tobe sure it gives us consistent results. This has been done based on naïve dataset.For these tests, the indifference and the preference thresholds of PROMETHEE cor-respond respectively to the third and the seventh decile of the difference between twoalternatives for the considered criterion.

Figure 2 shows one example of a particular dataset tested. It represents a datasetof 900 alternatives evaluated on 2 criteria. Firstly, the dataset is clearly separated in

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Figure 2: Degradation of an artificial dataset

three very homogeneous groups corresponding to the three clusters. These clusters aregradually degraded in the two next graphs.As expected, the quality index decreases when the quality of the clusters is degradedwhich gives an indication of the consistency of the index. Let one note that for the thirdgraph, some clustering distributions with 4 or 5 clusters give approximately the sameresults in terms of quality. It is an expected result considering the dispersion of thedataset.Let one note that this quality index can be used in different ways:

• To compare different methods with the same parameters.

• To compare different parameters for the same method. The number of clusters isa good example. In fact, the DM does not always know how many clusters he hasto form. In that way, he can choose the number of clusters which will give himthe best quality partition (cf Figure 3).

6 Results

6.1 Numerical results

For the hybrid method (4.4), two choices were possible: computing the ranks or thelevels of a graph to form the subgroups. In practice, both choices give almost the same

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results. In fact, experiments on several dataset have shown that less than 3% of thealternatives are assigned to different clusters comparing the two different approaches.In what follows, the levels of a graph will be the only considered method.The different methods will be compared according to the quality index D and thenumber of clusters. In addition, each model will be compared with P2Clust [9].The top-down hierarchical method, the bottom-up hierarchical method, the hybridhierarchical method and the P2Clust method will be called HTD,HBU,HH and P2Crespectively. The models have been tested for three different dataset:

• The first hundred alternatives of the Shanghai Ranking of World Universities(ARWU) according to 6 criteria [26];

• A standard benchmark dataset about CPU evaluation from the UCI repository. Itcontains 209 alternatives and 6 criteria [8];

• The Environmental Performance Index 2014 (EPI). It is composed of 178 alterna-tives and 2 criteria (simplified version) [16];

All methods described will be compared with each other according to the Quality indexD. In order to take into account the sensitivity of the parameters, different valueswill be considered. The values of the indifference and preference thresholds of thePROMETHEE preference function vary as follow:

qk 0 d1 d2 d3

pk d10 d9 d8 d7

Table 9: Different considered values of the PROMETHEE thresholds

where dn corresponds to the nth decile computed on the set of differences of each pairof alternatives according to the considered criterion. For each pair of thresholds, thenumber of clusters K varies from 2 to 6.Table 10 illustrates the comparison for each set of parameters (thresholds and numberof clusters). It means 20 possibilities here (4 pairs of thresholds and 5 different K). Itshows in which proportion a method is better than another according to the Qualityindex D. These tests have been done for each one of the three considered dataset(ARWU, CPU and EPI).

One can see that the hybrid method gives us very interesting results. In fact, inmost cases, it gives better results than the others. Of course this was expected sinceit makes explicit use of the quality index in the procedure. It is also interesting tosee that the results of the bottom-up approach are generally better than the top-downapproach at first sight. This can be explained by the small number of clusters. Infact, the top-down approach has to perform a small number of operations from thelarge single cluster until it has the desired number of clusters K. The bottom-upapproach has to do many operations because it begins with a large number of clusters

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ARWU CPU EPID(HH) ≥ D(HTD) 100% 100% 100%D(HH) ≥ D(HBU) 100% 100% 95%D(HH) ≥ D(P2C) 100% 90% 90%D(HBU) ≥ D(HTD) 100% 90% 80%D(HBU) ≥ D(P2C) 100% 60% 60%D(HTD) ≥ D(P2C) 30% 10% 15%

Table 10: Comparison of the different methods

and merges them until the K clusters are obtained. To the best of our knowledge,P2Clust was the only method for complete ordered clustering using PROMETHEE sofar. Simulations have shown the bottom-up and the hybrid methods give better resultsregarding the quality index than P2Clust in most of the cases. Of course this con-clusion calls from some nuance since P2Clust was not built to optimize this specific index.

On Figure 3 we have a representation of the results for the ARWU dataset according tothe number of clusters K. In this figure, the thresholds qk and pk correspond to d3 andd7 respectively.

Figure 3: Quality of the results for ARWU

We clearly see that the hybrid method has better performance than the others andthat the extreme numbers of clusters do not always give the best results (the evolutionis not monotonous).In this example, the best clustering is given by the hybrid method for 3 clusters (whereD=2.48). Figure 4 represents this clustering in a PCA two-dimensional representationof the ARWU dataset. Let us note that the clustering distribution has been computedusing the original dataset and is represented in this two-dimensional representation.

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Figure 4: Clustering of ARWU with HH, K=3

-40 -20 0 20 40 60 80 100 120 140 160

-50

-40

-30

-20

-10

0

10

20

30

40

We can see that Harvard, which is by far the best university in this dataset, is alonein the best cluster. This illustrates how Harvard is better than the others. A largegroup of 63 universities very close to each other forms the third cluster. The other33 universities form the second cluster (in green). It is important to note that it is atwo-dimensional representation but the clustering is computed taking into account 6criteria. The situation is then more complex than suggested by the figure.

6.2 Convergence and computational time details

It has been shown how the developed models perform in terms of quality. It is nowimportant to know if they do it in a reasonable amount of time. As explained before, thethree hierarchical models are deterministic. Therefore, there is no problem of stabilitywhich represents an advantage to the non-deterministic approach such as P2Clust. Inthe tables [11,12,13] we have the results for the dataset we presented. The tests havebeen executed for different number of clusters K.

ARWU K=2 K=3 K=4 K=5 K=6HTD 0.13 0.15 0.19 0.25 0.26HBU 1.44 1.43 1.39 1.26 1.21HH 1.55 1.96 5.06 7.38 90.63P2Clust 0.06 0.10 0.15 0.24 0.32

Table 11: Comparison of the methods’ convergence for ARWU (seconds)

Without surprise, the hybrid method takes more time than the other hierarchical meth-ods because it needs the results of the bottom-up and the top-down approaches. Wecan see that for the three dataset the amount of time is reasonable. Each hierarchicalmethod leads to compare each pair of alternatives two by two. P2Clust compares each

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CPU K=2 K=3 K=4 K=5 K=6HTD 0.48 1 0.81 1.04 1.06HBU 14.78 13.94 12.88 13.29 12.41HH 15.51 16.57 33.80 311.72 520P2Clust 0.19 0.2 0.43 0.92 1.41

Table 12: Comparison of the methods’ convergence for CPU (seconds)

EPI K=2 K=3 K=4 K=5 K=6HTD 0.13 0.18 0.24 0.26 0.26HBU 2.07 1.93 1.92 1.88 1.84HH 2.22 2.39 4.74 40.60 672.80P2Clust 0.05 0.08 0.12 0.15 0.19

Table 13: Comparison of the methods’ convergence for EPI (seconds)

alternative with central profiles which is less time consuming [9]. It is important to notethat the three hierarchical methods give always a result even if it takes a little moretime. As stressed in [9], the P2Clust model does not always converge to a stable solution.

7 ConclusionsIn this paper, different hierarchical methods for complete multicriteria ordered clusteringbased on PROMETHEE II have been developed. To the best of our knowledge, it isa first contribution regarding deterministic multicriteria clustering method based onPROMETHEE II. The performances of these methods have been tested according todifferent indicators (quality, convergence). The first contribution is a classic hierarchicalmodel developed in two versions: a top-down and a bottom-up methods. Theseapproaches have been developed in order to optimize, at each step, the structureof the clustering by maximizing the intra-cluster homogeneity and the inter-clustersheterogeneity. Then we developed a general quality indicator to characterize the completemulticriteria ordered clustering which helped us to compare different methods outputs.Finally, we proposed a hybrid method which brings together the results of the top-downand the bottom-up approaches. It uses the levels or the ranks of an adjacency matrixwhich gathers information of both hierarchical methods to make ordered subgroups ofalternatives. Then these subgroups are merged until one gets the desired number ofclusters. The best solution among all the possibilities is selected according to the qualityof the distribution.These three methods have been evaluated and they give interesting results. In particularthe hybrid method (which gives most of the time better results than P2Clust) and thebottom-up hierarchical method according to the quality indicator.To conclude, the hierarchical models developed in this paper have led to promising

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results. Moreover, the analysis of the performance of the model on real-world datasetsare encouraging. The comparison of the proposed model with the existing P2Clustmethod has underlined a strong interest in using such an approach to characterizecomplex multicriteria clustering problems.

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