Multicriteria assignment method PROAFTN - Semantic Scholar

European Journal of Operational Research 125 (2000) 175±183

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Theory and Methodology

Multicriteria assignment method PROAFTN: Methodology and medical application Nabil Belacel

*

Institute of Statistical and Operational Research and Service de Math ematiques de la Gestion, Universit e Libre de Bruxelles, Bd. Du Triomphe, C.P. 210/01, B-1050 Brussels, Belgium Received 5 October 1998; accepted 26 March 1999

Abstract This paper presents a new fuzzy multicriteria classi®cation method, called PROAFTN, for assigning alternatives to prede®ned categories. This method belongs to the class of supervised learning algorithms and enables to determine the fuzzy indi€erence relations by generalising the indices (concordance and discordance) used in the ELECTRE III method. Then, it assigns the fuzzy belonging degree of the alternatives to the categories. We also present a clinical application of the proposed method in the cytopathological diagnosis of acute leukaemia. Ó 2000 Elsevier Science B.V. All rights reserved. Keywords: Multicriteria decision analysis; Classi®cation; Fuzzy assignment; Medical diagnosis

1. Introduction Classi®cation methods are divided into two groups. The ®rst one includes the automatic classi®cation methods called clustering methods, based on the notion of unsupervised learning, and consists in assembling individuals in restricted classes so that all objects of the same class are less dispersed [5]. The second one comprises the assignment methods, which are based on the notion of supervised learning and employ a set of examples belonging to well-known classes. From these examples, the assignment rules are de®ned *

Tel.: +32 2 650 5505; fax: +32 2 650 5970. E-mail address: [email protected] (N. Belacel).

[2,6,8,13,20]. Our problem is formulated in terms of assigning objects to one or several classes by examining the intrinsic value of each object and by referring to pre-existing rules. Thus, our problem can be considered as a supervised learning problem. The aim of this paper is to develop the multicriteria assignment method based on the preference relational system described by Roy [16] and Vincke [18]. This method employs a comparison between the alternatives through the scores of di€erent criteria. So, it avoids resorting to distance and allows us to use qualitative and/or quantitative criteria. Moreover, it helps us to overcome some diculties encountered when data are expressed in di€erent units. The aims of the present study were: (1) to elaborate a new fuzzy multicriteria

0377-2217/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 9 9 ) 0 0 1 9 2 - 7

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assignment method, which uses the MCDA approach and (2) to apply it to a clinical classi®cation problem. Acute leukaemia is used as an example of this application. The present paper is organised as follows: Section 2 presents a general study of the sorting problems. Section 3 describes the di€erent stages of the PROAFTN method. Section 4 is devoted to the clinical application of the proposed method. 2. The sorting problematic 2.1. De®nitions The decision-maker is sometimes confronted with absolute valuation problems [3]. In this case, the analyst may opt for the sorting problematic. This problematic consists in formulating the decision problem in terms of assigning each object from a set A to one or several categories. This assignment is achieved through the examination of the intrinsic value of the action by referring to preestablished norms. We distinguish two situations of sorting. Ordered categories: They are characterised by a sequence of boundary reference objects. This type of the problematic is known as an ordinal sorting problematic [7,9,19]. The rule for assigning objects to categories is formulated as follows: each object, which is judged between the boundaries' reference objects of a category must be assigned to the category. The evaluation of students is an example, which can be treated by using this problematic. Not ordered categories: They are characterised by one or several central reference objects called prototypes. The prototype is considered as a good representative of its category and is described by the score upon each of the n criteria. This type of the problematic is known as a nominal sorting problematic [11]. The assignment rule, the so-called NSORT rule, is formulated as follows: object `a' is assigned to a category if and only if `a' is indi€erent or roughly equivalent to at least one of the prototypes of this category. As an example, we can cite medical diagnosis, where the actions are represented by di€erent symptoms and the categories are represented by typical symptoms.

2.2. Assignment methodology To resolve the assignment problems, we proceed by the following stages: Stage 1. Modelling of categories: In this stage, the prototypes are conceived using the two following steps: · Step 1. Structuring: The prototypes and their parameters (thresholds, weights, etc.) are established using the available knowledge given by the decision maker. · Step 2. Validation: We use one of the two following techniques in order to validate or adjust the parameters obtained in the ®rst step through the assignment examples known as a training set. · Direct technique: It consists in adjusting the parameters through the training set and with decision-maker intervention. · Indirect technique: It consists in ®tting the parameters without decision-maker intervention. This technique requires less cognitive e€ort than the former technique; it uses an automatic method to determine the optimal parameters, which minimise the assignment errors [10]. Stage 2: Assignment: After conceiving the prototypes, we proceed to assign the new objects to speci®c categories. 3. The developed method In this section, we propose the PROAFTN method, which is able to resolve the multicriteria assignment problems in the nominal sorting problematic. In order to assign the fuzzy belonging degree of action a to each category, PROAFTN determines the fuzzy indi€erence relations by generalising concordance and discordance indices used in the ELECTRE III method [15]. The data and notations used by PROAFTN are: A F

set of objects or actions to assign to di€erent categories (throughout this paper, set A will be ®nite and non-empty) ˆ fg1 ; g2 ; . . . ; gn g set of n criteria or attributes

N. Belacel / European Journal of Operational Research 125 (2000) 175±183

X Bh

B A^

set of k categories or classes such as X ˆ fC 1 ; . . . ; C k g; k P 2 prototype set of hth category, where Bh ˆ fbhi jh ˆ 1; . . . ; k; i ˆ 1; . . . ; Lh g with bhi designating the ith prototype of the hth category Sk set of all prototypes, such as B ˆ hˆ1 Bh (the set B will be ®nite and non-empty) set of actions A and B such as A^ ˆ A [ B

If ‰…aIbh1 † and=or …aIbh2 † and=or . . . and=or …aIbhLh †Š;

then a 2 C h :

The comprehensive indi€erence index is determined by aggregating the partial indi€erence indices. These indices indicate if the action a is indi€erent or not to a prototype bhi according to a criterion gj . The partial indi€erence relation is given as follows: aIj bhi () gj …a† 2 ‰Sj1 …bhi †; Sj2 …bhi †Š:

The score of actions are evaluated on the criteria set F, such as: 8a 2 A, we have g…a† ˆ …g1 …a†; g2 …a†; . . . ; gn …a††; 8bhi 2 Bh , we have g…bhi † ˆ …g1 …bhi †; g2 …bhi †; . . . ; gn …bhi ††; where h varies from 1 to k and i varies from 1 to Lh . We note `the action a is assigned to category C h ' by a 2 C h . The PROAFTN method uses a synthesis outranking approach and processes as follows. 3.1. Computing the fuzzy indi€erence relation The fuzzy indi€erence relation is based on the concordance and non-discordance principle [4,11,12,16,17]. The principle is given as follows: when the action a is judged indi€erent to a prototype bhi according to the majority of criteria majority principle and there is no criterion which uses its veto against the armation `a is indi€erent to bhi ' minority respect principle, the action a is considered overall as indi€erent to a prototype bhi . To calculate the fuzzy indi€erence relations we build the partial indi€erence indices, and then we use the concordance and non-discordance concept to aggregate them. 3.1.1. Partial indi€erence relations In general, the prototype scores are given by intervals, so for each criterion gj , we associate to each prototype bhi the interval ‰Sj1 …bhi †; Sj2 …bhi †Š, with Sj2 …bhi † P Sj1 …bhi †. Therefore, the NSORT rule is formulated like this:

177

…1†

If the score of the action a according to the criterion gj is equal to Sj1 …bhi † or to Sj2 …bhi †, the action a will be indi€erent to prototype bhi according to Eq. (1). However, considering the imperfection and imprecision of the data, we can assess the action a on the criterion gj by the score: gj …a† ˆ Sj1 …bhi † ÿ e1 or gj …a† ˆ Sj2 …bhi † ‡ e2 , where e1 and e2 are two positive real numbers, which take very small values. In this case, the application of the rule (1) leads to transform the indi€erence situation into a non-indi€erence situation between the action a and prototype bhi according to criterion gj , despite the fact that variation is not signi®cant. So, to remedy this inconvenience, we have introduced two discrimination thresholds djÿ …bhi † P 0 and dj‡ …bhi † P 0, which correspond, respectively to two functions of Sj1 …bhi † and Sj2 …bhi †. Formally, three comparative situations between the action a and prototype bhi according to criterion gj are obtained using the two discrimination thresholds: · If Sj1 …bhi † 6 gj …a† 6 Sj2 …bhi †, then a is clearly indifferent to bhi . · If ‰gj …a† 6 Sj1 …bhi † ÿ djÿ …bhi †Š or ‰gj …a† P Sj2 …bhi † ‡ dj‡ …bhi †Š, then a is not indi€erent to bhi . · If ‰Sj1 …bhi † ÿ djÿ …bhi † < gj …a† < Sj1 …bhi †Š or ‰Sj2 …bhi † < gj …a† < Sj2 …bhi † ‡ dj‡ …bhi †Š, then there is a weak indi€erence between a and bhi . Fig. 1 illustrates the various situations created by introducing discrimination thresholds. Based on argumentation of discrimination thresholds, we de®ne the partial indi€erence index Cj …a; bhi †, which represents the degree of validity of the three previous situations and veri®es the following properties: Cj …a; bhi † ˆ 1 () Sj1 …bhi † 6 gj …a† 6 Sj2 …bhi †;

…2†

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Fig. 1. Graphic illustration of the comparison zones.

0 < Cj …a; bhi † 8 1 h < Sj …bi † ÿ djÿ …bhi † < gj …a† < Sj1 …bhi †; < 1 () or : 2 h Sj …bi † < gj …a† < Sj2 …bhi † ‡ dj‡ …bhi †; …3† 8 < gj …a† 6 Sj1 …bhi † ÿ djÿ …bhi †; h Cj …a; bi † ˆ 0 () or : gj …a† P Sj2 …bhi † ‡ dj‡ …bhi †;

…4†

The index Cj …a; bhi † is represented between the values Sj1 …bhi † ÿ djÿ …bhi † and Sj1 …bhi † on one hand, and Sj2 …bhi † and ‰Sj2 …bhi † ‡ dj‡ …bhi †Š on the other hand, through the linear interpolation function (Fig. 2). From Fig. 2, we deduce the value of Cj …a; bhi † as follows: Cj …a; bhi † ˆ minfCjÿ …a; bhi †; Cj‡ …a; bhi †g; where

…5†

Cjÿ …a; bhi † ˆ

djÿ …bhi † ÿ minfSj1 …bhi † ÿ gj …a†; djÿ …bhi †g ; djÿ …bhi † ÿ minfSj1 …bhi † ÿ gj …a†; 0g

Cj‡ …a; bhi † ˆ

dj‡ …bhi † ÿ minfgj …a† ÿ Sj2 …bhi †; dj‡ …bhi †g : dj‡ …bhi † ÿ minfgj …a† ÿ Sj2 …bhi †; 0g

3.1.2. Comprehensive indi€erence relation based on the concordance principle In Section 3.1.1, we determined n fuzzy partial relations Cj : A  B ! ‰0; 1Š; j ˆ 1; . . . ; n. Each relation represents the credibility degree of the following situation: `the action a is indi€erent to a prototype bhi for criterion gj '. These n relations enable to determine a comprehensive credibility degree associated with indi€erence relation I using concordance and non-discordance principles and some parameters such as: veto thresholds and importance weights of criteria [11,12]. The con-

Fig. 2. Graphic representation of the partial indi€erence index.

N. Belacel / European Journal of Operational Research 125 (2000) 175±183

cordance concept is based on the majority principle. We de®ne it again in our context. De®nition 1. A fuzzy binary relation CI de®ned on A^ is said to be a comprehensive concordance relation associated with I, if there is an aggregation n function M, de®ned from ‰0; 1Š to ‰0; 1Š, verifying:

To calculate CI …a; bhi †, we chose an aggregation operator M, which realised the compromise between the partial indi€erence relations and the comprehensive indi€erence relation. Among those, we have a weighted mean aggregation operator, which is given as follows: ! n  X ÿ ÿ h

 ÿ1 h h wj  / Cj a; bi ; …6† CI …a; bi † ˆ / jˆ1

where / is a strictly increasing continuous function de®ned from ‰0; 1Š to ‰0; 1Š such that /…0† ˆ 0 and /…1† ˆ 1. If we take the function / such as /…x† ˆ x; then CI …a; bhi † becomes n  X jˆ1

ÿ
 whj  Cj a; bhi ;

De®nition 2. A criterion gj is discordant with the indi€erence relation between the action a and the prototype bhi , if this criterion is not in concordance with the same indi€erence (i.e. Cj …a; bhi † ˆ 0†, such that: Cj …a; bhi † ˆ 0 () ‰gj …a† P Sj2 …bhi † ‡ dj‡ …bhi †Š or ‰gj …a† 6 Sj1 …bhi † ÿ djÿ …bhi †Š:

M is non-decreasing in each argument; M…0; . . . ; 0† ˆ 0; M…1; . . . ; 1† ˆ 1; 8…a; bhi † 2 A  B we have CI …a; bhi † ˆ M…C1 …a; bhi †; C2 …a; bhi †; . . . ; Cn …a; bhi ††:

CI …a; bhi † ˆ

179

…7†

where whj ; j ˆ 1; . . . ; n and h ˆ 1; . . . ; k; are positive weights adding to one and re¯ecting the intrinsic relative importance attached by a decision maker to a criterion gj of category Ch and independently of other categories. We assume that whj are assessed in the absolute scale [14] and they take into account the following conventions: · whj ˆ 0 means that the criterion gj is not pertinent for the assignment of action a to category Ch . · whj ˆ 1 means that the criterion gj is the only pertinent criterion for the assignment of action a to category Ch . 3.1.3. Calculating the comprehensive indi€erence relation We introduce the partial discordance index Dj …a; bhi †:

By de®nition, we call the right veto threshold h ÿ h v‡ j …bi † (resp. the left veto threshold vj …bi †† for criterion gj the minimum value of the di€erence gj …a† ÿ Sj2 …bhi † (resp. Sj1 …bhi † ÿ gj …a††, which is incompatible with the assertion aIbhi . The veto h ‡ h ‡ h thresholds vÿ j …bi † and vj …bi † such as vj …bi † P ‡ h ÿ h ÿ h dj …bi † and vj …bi † P dj …bi †; j ˆ 1; . . . ; n, are used to de®ne the values from which the action a is considered as very di€erent to prototype bhi for criterion gj . They are characterised by the following condition: 9j 2 1; . . . ; n such as: h ÿ
ÿ h
i gj …a† P Sj2 bhi ‡ m‡ j bi h ÿ
ÿ h
i or gj …a† 6 Sj1 bhi ÿ mÿ j bi ÿ h
) non aIbi : The aim of determining the discordance index Dj …a; bhi † of the criterion gj is to apprehend the fact that such a criterion is more or less discordant with the assertion `a is indi€erent to bhi '. The discordance index is maximum …Dj …a; bhi † ˆ 1† when the criterion gj uses its veto against this assertion aIbhi . It is minimum …Dj …a; bhi † ˆ 0† when the criterion is not in discordance with this indi€erence (i.e. Cj …a; bhi † 62 0†. If the criterion gj is in discordance (i.e. Cj …a; bhi † ˆ 0† with indi€erence and it does not use its veto against this indi€erence, we have: 0 < Dj …a; bhi † < 1, which represents the intermediary zones between the non-discordance and discordance situations (Fig. 3). The discordance index Dj …a; bhi † is represented h 1 h between the values Sj1 …bhi † ÿ vÿ j …bi † and Sj …bi † ÿ ÿ h 2 h ‡ h dj …bi † on one hand and Sj …bi † ‡ dj …bi † and h Sj2 …bhi † ‡ v‡ j …bi † on the other hand, by the linear interpolation function (see Fig. 3). From Fig. 3, the discordance index Dj …a; bhi † is determined as follows:

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Fig. 3. Graphic representation of the partial discordance index.

h ‡ h Dj …a; bhi † ˆ max…Dÿ j …a; bi †; Dj …a; bi ††;

…8†

where h Dÿ j …a; bi † ˆ

gj …a† ÿ maxfgj …a†; Sj1 …bhi † ÿ djÿ …bhi †g ; h djÿ …bhi † ÿ maxfSj1 …bhi † ÿ gj …a†; mÿ j …bi †g

h D‡ j …a; bi †

gj …a† ÿ minfgj …a†; Sj2 …bhi † ‡ dj‡ …bhi †g : ˆ h ÿdj‡ …bhi † ‡ maxfÿSj2 …bhi † ‡ gj …a†; m‡ j …bi †g

jˆ1

ÿ

whj 1 ÿ Dj a; bhi :

I…a; bhi † ˆ

jˆ1

…9†

ÿ


h

!

whj Cj a; bi

n Y ÿ jˆ1

De®nition 3. A fuzzy binary relation DI de®ned on A^ is said to be a comprehensive discordance relation for assertion aIbhi , if there exists an n aggregation function h (that maps ‰0; 1Š to ‰0; 1Š) such that: · h is non-decreasing in each argument; · h…0; . . . ; 0† ˆ 0; · h…x1 ; . . . ; xn † ˆ 1 () 9i 2 1; . . . ; n such that xi ˆ 1; · 8…a; bhi † 2 A  B; DI …a; bhi † ˆ h…D1 …a; bhi †; D2 …a; bhi †; . . . ; Dn …a; bhi ††. To determine a comprehensive discordance index DI , we can use a compromise aggregation operator that has a value 0 if at least one Dj ; j ˆ 1; . . . ; n; is equal to 1 (see [12]). Then n Y ÿ

n X



From the partial discordance indices, we de®ne the comprehensive discordance index in the following way.

DI …a; bhi † ˆ 1 ÿ

From Eqs. (5)±(9) and on the basis of the concordance and non-discordance principles, the comprehensive fuzzy indi€erence relation is given as follows:

h ˆ 1; . . . ; k;

ÿ



h h wj

1 ÿ Dj a; bi

! ;

…10†

i ˆ 1; . . . ; Lh :

For a more detailed analysis of the fuzzy relations based on the concordance and non-discordance principles, see Refs. [4,11,12,15±17]. 3.2. Assigning the action to pre-de®ned categories The categories C h ; h ˆ 1; . . . ; k, are represented by a set Bh of Lh prototypes, Bh ˆ bh1 ; bh2 ; . . . ; bhLh , which are considered as good as representative of their category and are described by the score upon each of n criteria. On the basis of the NSORT rule, to assign the action a to the corresponding category Ch , we process as follows: 1. Computing the fuzzy indi€erence relations: the comprehensive indi€erence relations I…a; bhi †; h ˆ 1; . . . ; k; i ˆ 1; . . . ; Lh ; are calculated on the so-called concordance and non-discordance principles from Eq. (10). 2. Evaluating the fuzzy belonging degree d…a; C h †: the fuzzy belonging degree of an action a to category C h ; h ˆ 1; . . . ; k, is de®ned by a set of

N. Belacel / European Journal of Operational Research 125 (2000) 175±183

prototypes Bh ; h ˆ 1; . . . ; k, and it is measured by the indi€erence degrees between a and its nearest neighbour in Bh according to fuzzy indi€erence relations I(a; bhi †; i ˆ 1; . . . ; Lh , [11]. Thus, the fuzzy belonging degree is given as follows: d…a; C h †

n o ˆ max I…a; bh1 †; I…a; bh2 †; . . . ; I…a; bhLh † : …11†

3. Assigning the action `a' to a speci®c category: once the fuzzy belonging degrees d…a; C h †; h ˆ 1; . . . ; k, have been computed for an action a for categories C h ; h ˆ 1; . . . ; k, a crisp assignment of action a is made a 2 C h () d…a; C h † ˆ maxfd…a; C i †=i 2 f1; . . . ; kgg:

4. Application as an aid to medical diagnosis 4.1. Acute leukaemia diagnosis The PROAFTN method developed in Section 3, was implemented and applied to the cytopathological diagnosis of acute leukaemias (AL). AL are a neoplastic proliferation of haematopoietic cells. They form a very heterogeneous group of diseases of varying pathogenesis, etiology and

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prognosis [21]. Many systems of clinical classi®cation of AL have been suggested in the past, but the most widely used today is that proposed by the French±American±British (FAB) haematologists group [1]. AL were chosen for this application because they are more adapted to our situation. Knowing that FAB classi®cation rules are given by intervals, this facilitates the prototype scores determination. According to the FAB classi®cation, AL are divided into two groups: acute myeloblastic leukaemia (AML) and acute lymphoblastic leukaemia (ALL). Each group is morphologically homogeneous and divided into several subgroups [1]. As a function of origin, stage of proliferation and maturation of blast cells (immature cells), nine categories of AL were used for classi®cation (AML M1, AML M2, AML M3, AML M4, AML M5, AML M6, ALL L1, ALL L2 and ALL L3). Each category was labelled according to its cytopathological group as established previously by haematological diagnosis. The features' value used for AL classi®cation of cases were obtained by examining the stained bone marrow smears from each patient by a haematologist using an optical microscope. Then, they were submitted to the PROAFTN method, which determines for each case the fuzzy belonging degree into each category of AL. A total of 47 features were used including 34 features for morphological criteria, seven features for ALL criteria and six features for cytochemical reaction criteria (see Table 1). The cases set of AL was divided into two subsets. The ®rst

Table 1 Classi®cation criteria used by PROAFTN Groups

Classi®cation criteria

Blast cells Granulocytic lineage

Blasts I; blasts II; blasts with Auer rods; total blasts Myeloblasts; promyeloctes I; promyelocytes II; neutrophilic myelocytes; eosinophilic myelocytes; neutrophilic metamyelocytes; eosinophilic metamyelocytes; neutrophils; eosinophils; basophils; total granulocytic lineage Pronormoblasts; basophilic normoblasts; polychromatophilic normoblasts; acidophilic normoblasts; pycnotic normoblasts; promegaloblasts; basophilic megaloblasts; polychromatophilic megaloblasts; acidophilic megaloblasts; pycnotic megaloblasts; total erythroid lineage Lymphoblasts; undi€erentiated lymphocyte; lymphocytes; atypical lymphocytes Monoblasts; promonocytes; monocytes; total monocytic lineage Sudan black, myeloperoxydase, chloro-acetate esterase, butyrate esterase, periodic acid schi€ and lysozyme concentration. Cell size, nuclear chromatin, nuclear shape, nucleolus, cytoplasm, cytoplasmic basophilia and cytoplasmic vacuolation.

Erythroid lineage Lymphocytic lineage Monocytic lineage Cytochemical Morphological criteria for ALL

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Table 2 The number and the percentage of correct classi®cation and misclassi®cation cases of acute leukaemia subtypes obtained by PROAFTN Acute leukaemia subtypes

Correct classi®cation cases, Number (%)

Misclassi®cation cases, Number (%)

AML M1 (n ˆ 11) AML M2 (n ˆ 11) AML M3 (n ˆ 7) AML M4 (n ˆ 11) AML M5 (n ˆ 9) AML M6 (n ˆ 7) ALL L1 (n ˆ 13) ALL L2 (n ˆ 10) ALL L3 (n ˆ 4)

11 11 4 11 9 7 13 10 4

0 0 3 0 0 0 0 0 0

Total (n ˆ 83)

80 (96.4)

(100) (100) (55.5) (100) (100) (100) (100) (100) (100)

(0) (0) (45.5) (0) (0) (0) (0) (0) (0)

3 (3.6)

one included the training set (108 cases), which is used to conceive the prototypes of the PROAFTN method and their parameters (weights, discrimination thresholds, etc.). The second subset represents the test set (83 cases), which is used to test the method. The cases used in this study were supplied courteously by Prof. J.M. Schei€ (laboratory of haematology, Clinique Universitaires Saint-Luc, Brussels, Belgium).

were diagnosed as AML M1 in two cases and AML M2 in one case by the method. The main advantage, which emerges from this analysis is that the PROAFTN procedure can be combined with computer-assisted microscope analysis of cell images. This may help the haematologist to identify rapidly the subtypes of AL.

4.2. Results and discussion

We have introduced a new multicriteria classi®cation method to help in medical diagnosis. At this stage, more studies still need to be undertaken with this procedure before any conclusions can be drawn. In the future, the PROAFTN method may be extended to more complex situations where the objects are only partly understood and are described by fuzzy subsets of criteria. These results also show that the multicriteria decision aid approach can be successfully used to help medical diagnosis.

We have tested the PROAFTN method presented in this paper on an experimental set of 83 new cases of AL, diagnosed in the haematology laboratory. The results obtained by the PROAFTN method were compared to the clinical results obtained by haematologists. The percentages of correct classi®cation and misclassi®cation are summarised in Table 2. The classi®cation was perfect for AML M1, AML M2, AML M4, AML M5, AML M6, ALL L1, ALL L2 and ALL L3 subtypes and con®rmed the initial cytopathological diagnosis proposed by haematologists. 96.4% of the cases were correctly classi®ed, while only 3.6% were misdiagnosed by the PROAFTN method, hence suggesting that the method yields good results in terms of discrimination between AL subtypes. Note that the errors were observed only with the AML M3 variant subtype. In fact, these misclassi®cation cases can be partially explained by the heterogeneity of AML M3. They

5. Conclusion

Acknowledgements The author is very grateful to Prof. Vincke of the Statistic and Operational Research Institute, Belgium, and to Dr. Boulassel and Prof. Schei€ of Saint-Luc Hospital, Belgium, for providing the data and their invaluable assistance. This research was supported by a grant from the International Department of the Universite Libre de Bruxelles.

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References [1] J.B. Bain, Leukaemia diagnosis, a guide to the F.A.B Classi®cation, Senior lecture in Haematology, St. Mary's Hospital Medical School, London, 1991. [2] G. Balestra, A. Ostanello, Segmentation problems and neural networks, applying multiple criteria aid for decision to environmental management, in: Paruccini (Ed.), Environmental Management, vol. 3, 1994, pp. 247±262. [3] C. Bana e Costa, Les problematiques dans le cadre de l'activite d'aide a la decision, Universite de Paris Dauphine, Document du LAMSADE (8) 1993. [4] D. Bouyssou, On some properties of outranking relations based on concordance±discordance principle, in: A. Goicoechea, L. Duckstein, S. Zionts (Eds.), Multiple Criteria Decision Marking, Springer, Berlin, 1992, pp. 93±106. [5] G. Celeux, E. Diday, G. Govaert et al., Classi®cation automatique de donnees environnement statistique et informatique, Dunod Informatique, France, 1989. [6] E.R. Kandel, J.H. Schwartz, T.M. Jesse, Principles of Neural Science, Elsevier, Amsterdam, 1991. [7] R. Massaglia, A. Ostanello, N-TOMIC: A support system for multicriteria segmentation problem, in: P. Korhonen, A. Lewandowski, J. Wallenius (Eds.), Mathematical Systems, vol. 356, Springer, Berlin, 1991, pp. 167± 174. [8] G.J. McLachlan, Discriminant Analysis and Statistical Pattern Recognition, Wiley, New York, 1992. [9] J. Moscarola, B. Roy, Procedure automatique d'examende dossiers fondee sur une segmentation trichotomique en presence de criteres multiples, RAIRO Recherche Operationnelle, 11 (2) (1977) 145±173.

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[10] V. Moussou, R. Slowinski, Inferring an ELECTRE TRI model from assignment examples, cahier LAMSADE No. 40, Paris Dauphine Universite, France, 1996. [11] P. Perny, Multicriteria ®ltering methods based on concordance and non-discordance principles, Annals of Operations Research 80 (1998) 137±165. [12] P. Perny, B. Roy, The use of fuzzy outranking relations in preference modelling, Fuzzy Sets & Systems 49 (1992) 33±53. [13] J.R. Quinlan, Induction of decision trees, Machine Learning 1 (1986) 81±106. [14] F.S. Roberts, Measurement Theory with Applications to Decision-making, Utility and the Social Sciences, Addison-Wesley, Reading, MA, 1979. [15] B. Roy, ELECTRE III: Un algorithme de classement fonde sur une representation ¯oue des preferences en presence de criteres multiples, Cahiers du CERO 20 (1978) 3±24. [16] B. Roy, Multicriteria Methodology for Decision Aiding, Kluwer Academic Publishers, Dordrecht, 1996. [17] B. Roy, D. Bouyssou, Aide multicritere  a la decision: Methodes et cas, Economica, Paris, 1994. [18] Ph. Vincke, Multicriteria Decision Aid, Wiley, New York, 1992. [19] W. Yu, Aide multicritere  a la decision dans le cadre de la problematique du tri, concepts, methodes et applications, Ph.D. Thesis, Paris-Dauphine Universite, 1992. [20] S.M. Weiss, C.A. Kulikowski, Computer Systems that learn, Classi®cation and Prediction Methods from Statistics, Neural Nets, Machine Learning and Expert Systems, Morgan Kaufmann, Los Altos, CA, 1991. [21] R. Zittoun, Hemopathies malignes, Encyclopedie des Cancers dirigee par Bernard Hoeni, Flammarion medecine sciences, France, 1986.