a decision support system for the budgeting of the belgian health care system

O.R. Applications

A decision support system for the budgeting of theBelgian health care system

Alain Mosmans a,b,*, Jean-Claude Praet c, Christophe Dumont d

a SMG-ISRO, Universit�ee Libre de Bruxelles, CP 210/01, Bld. du Triomphe, 1050 Brussels, Belgiumb Service soins de sant�ee INAMI-RIZIV, Av. de Tervueren 211, 1150 Brussels, Belgiumc Direction du Budget, Hoopital Erasme, Route de Lennik 808, 1070 Brussels, Belgium

d DULBEA, Universit�ee Libre de Bruxelles, CP 140, Av. F.D. Roosevelt 50, 1050 Brussels, Belgium

Received 28 February 2001; accepted 2 May 2001

Abstract

The Belgian Ministry for Social Affairs asked the departments of Applied Economics (DULBEA) and Operational

Research (SMG) of the Universit�ee Libre de Bruxelles to develop methodological tools aimed at helping to understandhealth care budget consumption. The paper will describe the proposed methodology while focusing on the construction

of ‘‘health care channels’’ through causality analysis and on the multicriteria assignment of the insured to them. The

Minister decided to prolong that research by putting the accent on the development of a decision support system

implementing the worked out methodology. This system is meant for the I.N.A.M.I.-R.I.Z.I.V.’s (National Institute of

Insurance against Disease and Handicap) actuaries to provide them with new analysis elements. Along with various

packages, these elements are being encompassed in a decision support system the characteristics of which are presented

in the last section of the paper. � 2002 Published by Elsevier Science B.V.

Keywords: Health services; Decision support system; Regression; Multicriteria decision aid; Fuzzy logic

1. Introduction

The last years’ evolution of the health care ex-penditures in Belgium made it very difficult not toexceed the imposed budgetary norms. Moreover,the growth of these expenditures is not always easy

to explain and therefore prevents from both cor-rectly forecasting the forthcoming budgets andefficiently satisfying the population’s needs. Thislast uncertainty is reinforced by the fact that thecurrent budgetary procedures do not take the de-mand for health care explicitly into account. Thisimportant weakness led the Minister for SocialAffairs, Magda De Galan, to ask our departmentsto develop new analysis tools addressing thatproblem (the title of the corresponding researchproject is ‘‘Conception and implementation ofmethodological tools for the determination of bothglobal and partial budgetary objectives of the

European Journal of Operational Research 139 (2002) 449–460

www.elsevier.com/locate/dsw

*Corresponding author.

E-mail addresses: [email protected] and Alain.

[email protected] (A. Mosmans), [email protected]

(J.-C. Praet), [email protected] (C. Dumont).

http://smg.ulb.ac.be/�amosmans (A. Mosmans).

0377-2217/02/$ - see front matter � 2002 Published by Elsevier Science B.V.

PII: S0377-2217 (01 )00369-1

Health Care Insurance’’). The methodology thatwe have developed is completely original andconsists in structuring the care consumptionthrough multicriteria assignment of insured tosome ‘‘health care channels’’ obtained throughcausality analysis.A ‘‘health care channel’’ is defined as the suc-

cession of medical acts belonging to a specifictreatment of a given pathology: for example a one-week stay in hospital, including various bloodtests, an electrocardiogram, some X-ray examina-tions, general anaesthetic, a precise surgical act,and drugs. A channel is thus described by themedical acts and drugs it ‘‘consumes’’.The paper is organised as follows. Section 2

describes the organisation of the Belgian healthcare system and especially focuses on the currentbudgeting methods and their limitations. All thesuccessive phases of the methodology we proposeare detailed in Section 3. Section 4 presents thegeneral characteristics of the forthcoming decisionsupport system implementing this methodology.Section 5 concludes the paper.

2. Current organisation of the Belgian health care

system

2.1. Actors and Data

2.1.1. ActorsThe Belgian Health Care System is based on a

redistributive justice where nearly everybody iscovered by an insurance and can be characterisedby the freedom of choice of the practitioner.The four main actors of the Belgian Health

Care System are the State, the Ministries for SocialAffairs and Public Health, the Mutualities and theI.N.A.M.I.-R.I.Z.I.V. (National Institute of In-surance against Disease and Handicap). The latteris the manager of the Health Care System for theState. The Mutualities constitute the links betweenthe Institute and the patients.The I.N.A.M.I.-R.I.Z.I.V. is in charge of the

general administration and of the control of theInsurance. Within the Institute, there exist differ-ent commissions of agreement and of conventionaiming at fixing the honoraria reimbursement. The

honoraria are fixed by agreements and are identi-cal for each kind of suppliers.The Minister of Public Health fixes the daily

price of hospital fees from rules identical foreverybody. Nevertheless, this price is different foreach hospital.Belgium counts five mutualities, a public body

and the Fund of the National Society of the Bel-gian Railways. These ones receive from theI.N.A.M.I.-R.I.Z.I.V. monthlies and allowancesfor their administrative expenditures in order tocarry out the payment of the medical costs of theirrespective insured.

2.1.2. DataAbout 8000 medical acts are reimbursed by the

I.N.A.M.I.-R.I.Z.I.V. in a proportion dependingon the social category the patient belongs to. Everysuch act is identified by a 6-position code called NCode. There are also about 200 C codes which aregroups of N codes related to the same medicalsector (anaesthesiology, transplants, dialysis, etc.);in addition, these C codes are available for 16different categories of the insured. Quarterlyaccounting time series (over period 1988–1997,aggregated at the national level) are available forevery N or C code.Another available kind of data is the ‘‘profile

data’’. These ones give for each provider of services(GP, etc.) all the prestations (medical acts) carriedout by him as well as the amount of expenditures,the number of acts, the date of prestation and thequarter of accounting by the mutual funds. Giventhat every insured has two years to return his med-ical attestation to the mutuality, the accounting fora given year spreads out on 8 trimesters. Thetransformation from 8 to 4 quarters representingthe real medical activity on an annual basis will becalibrated from the profile data. Moreover, theprofile data allow one to calculate the average delaybetween the moment of the prestation and the mo-ment of the accounting to the I.N.A.M.I.-R.I.Z.I.V.Besides the data described above which are

available by the Institute, the mutualities havegiven us anonymous individual data consisting ofboth socio-economic characteristics and the de-tailed 1995 chronological care and drug con-sumption for some insured people.

450 A. Mosmans et al. / European Journal of Operational Research 139 (2002) 449–460

2.2. Actual determination of the budgets

In 1998, the global budget of the sickness in-surance in Belgium was close to 490 billions BEF.It amounts on the one hand to about 453 billionscoming from the I.N.A.M.I.-R.I.Z.I.V. and on theother hand to around 35 billions stemming fromthe Ministry of Public Health which reimburse25% of the daily price of hospital fee. In 1996,15.5% of the public expenditures (out of intereston the public debt) was effected to expendituresrelated to health care, that is to say around 6.2% ofthe gross domestic product (GDP).The total expenditure of a given C code is tra-

ditionally expressed as a product of three factors:the average reimbursement cost of this code, theaverage consumption of it by the insured and thenumber of insured people. When forecastingthe total expenditure, every factor among the threementioned above is extrapolated independently onits own time series for the last five years, generallyby means of a linear regression in the least squaressense. Their product gives the total expendituresfor a given group of prestations or insured people.Generally, we write

Dt ¼ P t � Ct � Et; ð1Þwhere Dt is the (forecasting of the) total expendi-ture at time t of a code, P t means the (forecastingof the) average cost at time t of the code, Ct is the(forecasting of the) average consumption per in-

sured at time t of the code, Et represents the(forecasting of the) number of insured at time t.In order to avoid the presence of biased data

and thereby to keep the extrapolation errors in agiven limit, it is advised to work with groups ofcodes and/or insured people that are as homoge-neous as possible. Therefore, a partition (more orless precise depending on the available data) of thecare and/or the insured is carried out in such a waythat the forecasting of the global expenditures attime t is realised through a set of extrapolations fora group e of the insured (active population, retired,etc.) and a group p of prestations, typically one ofthe 20 partial budgets (dentists, general practitio-ners, etc.) composing the global budget.In this case, the formula is given by

Dtp;e ¼ P t

p;e � Ctp;e � Et

p;e: ð2Þ

The forecasting of the global expenditures of thehealth care sector is then calculated by summingup the expenditures forecasting for all e and pgroups (see Fig. 1):

Dt ¼X

p;e

Dtp;e: ð3Þ

2.3. Some drawbacks

Obviously, the actual way of extrapolatingbudgets only implicitly takes into account the ex-planatory factors of health care consumption

Fig. 1. Actual determination of the budgets.

A. Mosmans et al. / European Journal of Operational Research 139 (2002) 449–460 451

through historical data. Moreover, it completelyneglects the complementarity of medical actsarising in the treatment of a given pathology. Fi-nally, it is based on a continuity hypothesis and istherefore unable to predict a change in the currentevolution or to forecast the effect on the chrono-logical series of taking some legal decision.As can be concluded from the above and illus-

trated in Fig. 1, the current forecasting techniquesprivilege the productive aspect of the health caresystem by considering independently the differentarticles and chapters of the health care nomen-clature, and consist for every sector in projectingthe trend of the last five-year expenditures. It istherefore well suited for answering the question‘‘how are the budgets going to evolve if nothingunusual happens?’’ and it gives good results insuch a context. Nevertheless, there are growinguncertainties due to several factors like the generalageing of the population, the use of still more so-phisticated medical technologies, a limited growthof the total expenditures (annual rate of 1.5%),irregular invoicing delays, etc. which require ad-ditional investigation tools in order to preventunexpected overspending.

3. Our methodology

In comparison, the methodology we developedis explicitly based on the complementarity of thedifferent kinds of care occurring in the treatmentof various pathologies, structured as what we call‘‘health care channels’’ (see Section 1). It is muchmore linked to a medical consumption reality (i.e.prestation data). These ‘‘health care channels’’ arethemselves integrated in a multicriteria model,which structures the consumption of the care ac-cording to the characteristics of the insured people.These channels allow numerous partitions of thehealth care expenditures in the budgets of theI.N.A.M.I.-R.I.Z.I.V.Our model is basically made of four steps,

which are detailed in the reminder of this section.The first step (Fig. 2) consists in preparing the datathrough nomenclature homogenisation andmathematical models of expenditure reallocation.The second step (Fig. 3) aims at identifying care

channels by means of causality analysis. The thirdstep (Fig. 6) performs a validation of them on in-dividual data. Finally, the fourth step (Fig. 7) cal-culates the assignment of various categories of theinsured to these channels in a valued (i.e. fuzzy)multicriteria sorting procedure.The initial situation (Fig. 4) shows the ‘‘raw

data’’: on the one hand the set of insured Belgianswithout any distinction of care consumption be-haviour and, on the other hand, the set of all the

Fig. 2. The first step of the model.

Fig. 3. The second step of the model.


health care prestations (N codes) or the accountingC codes, depending on the desired level of detail.Some ‘‘diagnosis’’ codes (in black on Fig. 4), re-lated to the treatment of a pathology, are pro-posed as representative of the associated healthcare channel.The data used by the methodology are of two

different kinds:(1) quarterly accounting time series (over period1988–1997, aggregated at the national level) forevery N or C code expenditure of theI.N.A.M.I.-R.I.Z.I.V.,(2) anonymous individual data obtained fromthe mutualities and consisting of both socio-economic characteristics and the detailed 1995chronological care and drug consumption forsome insured people.Given the fact that the reliability of the cau-

sality analysis (performed in the second step) di-rectly depends on the quality of the informationincluded in the I.N.A.M.I.-R.I.Z.I.V. expendituretime series (1), it is necessary to identify all themodifications since the creation of the currentnomenclature (14/09/1984) and to rebuild homo-geneous time series.As explained before, a health care channel is

characterised by the medical acts it is composed of.Therefore the identification of the channels isbased on the principle that two medical acts be-longing to a same channel are such that the pres-tation of one of the two goes together with theprestation of the other (eventually with a delay)and that the evolutions of their prestation volumes

are consequently correlated. The nature of the in-formation included in the I.N.A.M.I.-R.I.Z.I.V.expenditures time series is of an accounting typeand introduces a bias due to the invoicing delaythat can be different from one code to another and,for a given code, from one quarter to another. It istherefore necessary in the first step (Fig. 2) to re-allocate the accounting data in their correspondingprestation data before calculating the correlation.With this aim in view, several mathematicalmodels of expenditures reallocation have beendeveloped, that can be calibrated by use of theI.N.A.M.I.-R.I.Z.I.V. profile data (described inSection 2.1.2.). They are able to isolate the trendmovements of the treasury. Therefore, we couldefficiently use them in a later budgetary controlmodel.The second step (Fig. 3) consists in the identifi-

cation of the care channels which is determined bya high level of ‘‘similarity’’ between the time seriesof the prestations of the diagnosis code (in bold)and the series of each of the various codes relatedto the resources which are consumed. This proce-dure is based on the causality analysis (describedin Section 3.1) and on a method developed bySekkat [13]. This method is for the first time ap-plied to the health care field and allows to identifythe kind of links existing between each prestationas well as the codes which are the knots of thesystem. From a budgetary point of view, theknowledge of these knots is fundamental inthe context of the mastering of the expenditures.Given the global level of aggregation of the time

series (1), the causality analysis will provide us

Fig. 4. The initial situation.

Fig. 5. The intermediate situation.


with parts (or skeletons) of channels rather thanwith complete and detailed channels. This begin-ning of consumption structure the causality anal-ysis would have brought to the codes of the initialsituation (Fig. 4) will lead us to an intermediatesituation (Fig. 5) where every indicator code (inbold) will act as a filter to identify and select in theindividual data from the mutualities (2) samples ofinsured suffering from the disease associated to thechannel.On the one hand these data will then be used in

a third step (Fig. 6) to validate the pieces ofchannels obtained through the causality analysisand eventually to complete them (see dotted boxesin figure) by the observation of the paths actuallyfollowed in the channel structure by the selectedpatients. The channels completed in such a waywill finally be submitted to the judgement of spe-cialist doctors. On the other hand, the socio-eco-nomic data give a family of assignment criteria andlead data analysis techniques to divide the set ofinsured into several ‘‘homogeneous’’ subgroups.The fourth step (Fig. 7) proposes to determine

how the socio-economic characteristics of the in-sured can explain the way they consume the vari-ous channels. To do that we will define a valuedmulticriteria assignment procedure of the insuredon the channels viewed as non-ordered categories.Learning on the samples how the insured in thesubgroups consume the channels will allow tocalibrate the parameters of the model and to as-sociate to every group and every channel one orseveral representative profiles according to thechosen criteria.

Afterwards, the budgetary model is completedby evaluating the cost of every channel and bylinking it with the different budgets of the healthcare system. The multicriteria model really appearsas a ‘‘control panel’’ giving the opportunity tohighlight the consumers’ profiles for every diseaseand to follow the insured people’s consumptionbehaviours inside the framework of care channels,on the basis of their characteristics. For this rea-son, the proposed budgetary model is closer to theconsumers’ needs and to the reality of medicalconsumption. Thanks to this control panel (seeFig. 8), it becomes possible to split the partial andglobal budgets according to particular diseases orclasses of insured people and find answers toquestions like ‘‘why did some budgets evolve in anunusual way?’’.In the two following sections, we are going to

detail the two central aspects of the methodology,that is to say the determination of the health carechannels through causality analysis and the valuedmulticriteria sorting procedure of the insured tothese channels.

3.1. Segmentation of the consumption through‘‘health care channels’’

This step consists in the identification of thechannels, 1 which is based on the causality analysis

Fig. 6. The third step of the model.

Fig. 7. The fourth step of the model.

1 The channels can be particularised according to the kind of

beneficiaries.


and on a method developed by Sekkat [13].Applied to the chronological series of theInstitute’s expenditures, this method allows todetect the kind of links existing between each timeseries of prestation as well as the codes which arethe knots of the system. From a budgetary point ofview, the knowledge of the knots of a system isfundamental in the context of the mastering of theexpenditures.

3.1.1. Definition of the notion of CausalityLet us suppose two processes Xt; Yt and the two

associated vectors of random variables:

X t ¼ fXt;Xt�1; . . .g ¼ ðXt�i; iP 0Þ;Y t ¼ fYt; Yt�1; . . .g ¼ ðYt�i; iP 0Þ:

Let us define the best linear forecasting of avariable X based on an information I by EðX=IÞand the corresponding error of forecasting notedeðX=IÞ ¼ X � EðX=IÞ.Among various characterisations of the notion

of the causality, we have chosen to use the cau-sality in Granger’s [3] sense defined as follows:

Y causes X at time t iff

EðXt=X t�1Y t�1Þ 6¼ EðXt=X t�1Þand

Y causes instantaneously X at time t iff

EðXt=X t�1Y tÞ 6¼ EðXt=X t�1; Y t�1Þ:We see that this definition can be interpreted interms of forecasting. If there is causality (i.e. whenY causes X ), Xt is better forecasted when thevariable Y is introduced than when it is not. It is

because of this forecasting dimension as well asdirect interpretation of the result in terms of linearregression that we chose Granger’s causality. In-deed, what we need to know is whether or not oneseries helps to forecast another one.

3.1.2. Sekkat’s method adapted to the health sectorWe start with the simplest model where every

code (i.e. every quarterly time series of expendi-tures) is alone. The first step is to determine forevery such code which one among all the otherspossibly causes it. Once these codes are deter-mined, we will choose the one for which the in-formation criterion (SBIC) 2 is the smallest. Thisprocedure is repeated until no more ‘‘possiblecauses’’ exist. After having determined all thecauses, we are able to build the channels. Thisprocedure rests on the causality analysis, whichtries to emphasise the relations of dominancewithin the channels as described by Sekkat [13].These ones are deciding in the context of an ef-ficient budgetary control. From the point of viewof the implementation, this causality analysis willconsist for every nomenclature code j in carryingout the following operations:(1) Transform the series into stationary onesthrough the application of the difference opera-tor ðYt ¼ Zt � Zt�1Þ on the series or by regressingit on a time trend and a constant.(2) Estimation of the following equation:

Fig. 8. The proposed budgetary model.

2 SBIC Schwartz–Bayes Information Criteria.


yjt ¼Xp

i¼1bji � yjt�i þ

Xp0

i¼1bvi � yvt�i þ ej;t; ð4Þ

where Y j is the indicator of the code j,v ¼ 1; 2; 3; . . . ; k.We increase P and P 0 until there is no more

autocorrelation in the residuals.(3) We test globally the nullity of the coefficientsbvi thanks to the likelihood ratio.(4) The variables Y v the coefficients of which arenot globally significant are definitively rejectedas the cause of Y j. For the other variables (theones which are globally significant are calledpossible causes), we will compare the differentmodels and will choose the ones which possessthe smallest SBIC. The chosen model will giveus the ‘‘first cause’’.(5) The next step consists in estimating themodel of point (2) by adding the ‘‘first cause’’.Then, we estimate the following equation:

yjt ¼Xp

i¼1bji � yjt�i þ

Xp

i¼1bvi � yvt�i

þXp

i¼1bhi � yht�i þ ej;t: ð5Þ

We apply again the same procedure but withthe code h (h¼ all the other ‘‘possible causes’’that were not retained as ‘‘first cause’’ at thefirst step). After having kept the new ’’firstcause’’, we go to steps (3) and (4) again.(6) The same procedure is applied in order to de-termine all the other causes. The iterations arestopped when no more ‘‘possible causes’’ exist.The individual data of the mutualities will then

be used to validate the channels obtained throughthe causality analysis and eventually to completethem by the observation of the paths actuallyfollowed by the patients in reality.

3.2. Multicriteria assignment of the insured to some‘‘health care channels’’

The following sections explain why existingmulticriteria sorting methods are not able to ad-dress our problem and propose the general prin-ciples of a well-suited procedure [7,8].

3.2.1. A brief history of existing multicriteriasorting proceduresThe multicriteria sorting problematic consists in

assigning a set of objects (actions), described bytheir profile of performances according to k crite-ria, to some predefined categories characterised byone or several reference points (prototypes), de-scribed according to the same k criteria. The as-signment to a category is based on the type ofrelation existing between the action to be assignedand the category prototype(s).If the categories are ordered like in the case of

resource allocation or students evaluation, eachone is characterised by a lower and upper proto-type (sometimes two families of lower and upperprototypes) and the relation used is a preferencerelation; an action will then be assigned to a cat-egory if it is ‘‘between’’ that category prototypes,meaning if it is preferred to the lower prototypeand if the upper prototype is preferred to it. If thecategories are non-ordered like in the case ofmedical or company diagnosis, each one is char-acterised by a central prototype (sometimes afamily of central prototypes) and the relation usedis an indifference relation; an action will then beassigned to a category if it is ‘‘close’’ to (at leastone of) that category prototype(s), meaning if it isindifferent to it. The following will only concen-trate on a specific kind of multicriteria method, i.e.the outranking methods (see definition in [9,12]),especially Perny and Henriet’s method dedicatedto the non-ordered category case we are facing inour problem.The first methods to appear concerned the case

of ordered categories. In 1977, Roy and Moscarola[6,11] proposed the first multicriteria sortingmethod, a crisp trichotomic method based on theElectre methodology, aiming at sorting actionsamong three predefined ordered categories con-taining, respectively, the ‘‘good’’ actions, the‘‘bad’’ actions and between them a dubious cate-gory of actions for which the assignment is notstraightforward. Two families of prototypes, re-spectively, ‘‘good’’ ones and ‘‘bad’’ ones, separatethose three categories. The assignment of an actionis based on the relation of that action with thosetwo families of prototypes. Yu [14] introduced theElectre Tri method (and the associated software)


in 1992 in order to solve problems with more thanthree categories but where frontiers are composedof a single prototype only.It was only in 1996 that Perny and Henriet

[4,10] proposed the first multicriteria assignmentmethod to non-ordered categories, 3 a fuzzyprocedure still based on the concordance and non-discordance Electre principle. The degree ofassignment lCkðaÞ of action a to category Ck isdefined as the value (between 0 and 1, totally falseand totally true) of the fuzzy indifference relationIða; ykÞ between the action and the category pro-totype yk. Instead of calculating Iða; ykÞ simply as acompromise aggregation (typically a weightedmean) of the n monocriterion similarity indicesIjða; ykÞ measuring to what extent every criterionj ¼ 1; . . . ; n is in favour of the proposition aIyk(read ‘‘a is indifferent to yk’’), the concordance andnon-discordance Electre principle is expressedthrough the following logical equation:

lCkðaÞ ¼ Iða; ykÞ ðCIða; ykÞ and ðnot DIða; ykÞÞÞ; ð6Þ

where the overall concordance index CIða; ykÞmeasures the overall agreement with the proposi-tion a I yk and the overall discordance indexDIða; ykÞ measures the degree to which there existsat least one important criterion in conflict with theproposition aI yk. Eq. (6) consequently expressesthat aI yk holds if, and only if, the coalition ofcriteria in agreement with this proposition isstrong enough and there is no significant coalitiondiscordant with it.As shown in Eqs. (7) and (8), CIða; ykÞ and

DIða; ykÞ, respectively, aggregate the n monocrite-rion similarity indices Ijða; ykÞ measuring to whatextent every criterion j ¼ 1; . . . ; n is in favour ofthe proposition aI yk and the n monocriteriondiscordance indices Djða; ykÞ measuring to whatextent every criterion j ¼ 1; . . . ; n is in strongconflict with the proposition aI yk;x1; . . . ;xn beingthe respective importance coefficients associated tothe n criteria:

CIða; ykÞ ¼ WðI1ða; ykÞ; . . . ; Inða; ykÞ;x1; . . . ;xnÞ;ð7Þ

DIða; ykÞ ¼ vðD1ða; ykÞ; . . . ;Dnða; ykÞ;x1; . . . ;xnÞ:ð8Þ

Remark that W is a compromise aggregationoperator expressing the respect of the majorityand v a disjunctive one, reflecting thereby theright of veto of any strongly conflicting criterionagainst aI yk.Fig. 9 shows the general shapes of the monoc-

riterion similarity Ijða; ykÞ and discordanceDjða; ykÞ indices as a function of gjðaÞ, the score ofaction a according to criterion j.From a practical point of view, the ‘‘and ’’ and

‘‘not’’ logical operators of Eq. (6) are usually in-terpreted using, respectively, a t-norm T ðx; yÞ(fuzzy conjunctive operator generalising the tra-ditional logical ‘‘and’’) and a strict negation NðxÞ(usually NðxÞ ¼ 1� x). Hence, Eq. (6) becomes:

lCkðaÞ ¼ Iða; ykÞ T ðCIða; ykÞ;NðDIða; ykÞÞÞ: ð9Þ

Let us remind that T ðx; yÞ is defined from ½0; 1�2 to[0, 1] such that:• T ðx; 1Þ ¼ x 8x 2 ½0; 1�,• T ðx; yÞ6T ðz; tÞ 8x; y; z; t 2 ½0; 1� such that x6 zand y6 t,

• T ðx; yÞ ¼ T ðy; xÞ 8x; y 2 ½0; 1�,• T ðx; T ðy; zÞÞ ¼ T ðT ðx; yÞ; zÞ 8x; y; z 2 ½0; 1�.Moreover, we can easily deduce that:

T ðx; yÞ6 minðx; yÞ 8x; y 2 ½0; 1�:

3.2.2. A new multicriteria sorting procedure to beapplied to the Belgian health care budgetsIt is clear that our care channels are non-ordered

and not exclusive categories. The only method wecould use to perform the assignment of the insuredto the care channels is therefore Perny andHenriet’smethod. The available socio-economic criteria be-ing only partially explanatory of the consumptionof a channel, two identical insured could be assignedto very different channels: it is thus compulsory towork with an average insured. Moreover, anotherimportant aspect tomention here is that our insuredactions are not punctual. Indeed, the detailed be-

3 Although the method is valid for categories with multiple

prototypes, we will illustrate it on a single prototype example.


haviour of an insured individual is not relevant forthe managers of the I.N.A.M.I.-R.I.Z.I.V. The ac-tions will consequently be associated with groups ofaverage insured associated with a subspace of thespace of criteria. For these two reasons, the crispdegree of assignment becomes fuzzy. In addition,the value of this degree of assignment has a cardinalmeaning: the proportion of a group of insured whosuffer from a pathology or the probability for aninsured chosen randomly in that group to sufferfrom a pathology. It is then easy to link this pro-portion to the budgets by multiplying it by thenumber of insured in the related group and by thecost of the concerned channel.The Electre principles of Perny and Henriet’s

method being incompatible with the cardinal na-ture of our degree of assignment, we are going toadapt our first definition of the degree of assign-ment in order to clarify our new procedure char-acteristics. Instead of defining it as the proportionof the insured who suffer from a disease in a groupcorresponding to given values of the criteria, wewill define it as the proportion of all the insuredwho suffer from the disease and who have an ad-missible profile of values according to all the cri-teria. When defining in the same way themonocriterion similarity indices, i.e. as the pro-portion of all the insured who suffer from the dis-ease and who have an admissible value according tothe criterion, it becomes obvious that the multicri-teria degree of assignment is less than any of themonocriterion similarity indices since it is obtainedas the intersection of the latter. Hence the aggre-gation operator calculating the multicriteria degreeof assignment from the monocriterion similarityindices belongs to the class of conjunctive operatorsi.e. T -norms (it is equivalent to the product in theparticular case of independence of the events as-

sociated to the criteria). This conjunctive aspectexpresses that the criteria are rather to be seen likeconstraints to be satisfied for assigning the action.More precisely, to be assigned to a category, anaction has to satisfy the first ‘‘and’’ the second‘‘and’’ . . . ‘‘and’’ the nth constraint. Moreover, sinceT ð0; xÞ ¼ 0, as soon as one-criterion constraint isnot satisfied ðIjða; ykÞ ¼ 0Þ, the assignment has to berejected ðlCkðaÞ ¼ 0Þ. Eq. (9) can be rewritten in thefollowing way:

lCkðaÞ ¼ Iða; ykÞ ðI1ða; ykÞ\and" . . . \and"Inða; ykÞÞ¼ T n�1ðI1ða; ykÞ; . . . ; Inða; ykÞÞ; ð10Þ

where T n�1 is the kth order t-norm defined by:T n�1ð� � �Þ ¼ T ðT ð� � � T ðIn�1ða; ykÞ; Inða; ykÞÞÞÞ.In (10) the eventual weights xi given to the

criteria are not taken into account due to thelack of definition of weighted t-norms. Let usinvestigate the possibility of defining weighted t-norms. More precisely, considering a ‘‘standard’’t-norm T ðx; yÞ where x and y are given the sameimportance, let us make the assumption that thetruth value x (resp. y) is now affected by aweight xx (resp. xy) through a function f thattransforms x (resp. y) in x0 (resp. y 0) i.e.x0 ¼ f ðxx; xÞ and y0 ¼ f ðxy ; yÞ. The weighted-norm Txðx; yÞ is equal to T ðx0; y0Þ and can thusbe written T ðf ðxx; xÞ; f ðxy ; yÞÞ. The weights xx

and xy are supposed to belong to [0, 1]. Ifxx ¼ 0, x is not taken into account at all and ifxx ¼ 1, the value x is not modified by xx. Wecan deduce the following properties.

f : ½0; 1�2 ! ½0; 1� : ðxx; xÞ ! x0 ¼ f ðxx; xÞ hasto verify:

f ð0; xÞ ¼ 1 8x 2 ½0; 1�; ð11Þ

Fig. 9. Monocriterion similarity and discordance indices.


1 being neutral for the t-norm, not to take x intoaccount means x0 ¼ 1.f ð1; xÞ ¼ x 8x 2 ½0; 1� ð12Þas explained, a weight ¼ 1 does not affect the valueof x. In particular, f ð1; 0Þ ¼ 0.If xx 6xx0 then f ðxx; xÞP f ðxx0 ; xÞ

8x 2 ½0; 1� ð13Þ

if xx ¼ 0, x is not taken into account (see (11))although when xx increases, the value x is moreand more taken into account, i.e. f ðxx; xÞ, de-creases until reaching f ð1; xÞ ¼ x.

If x6 x0 then f ðxx; xÞ6 f ðxx; x0Þ8xx 2 ½0; 1� ð14Þ

the greatest the value of x, the greatest its effect onthe t-norm, for a same weight.

f ðxx; 1Þ ¼ 1 8xx 2 ½0; 1� ð15Þ

follows from conditions (11)–(13): 1 ¼ f ð0; 1ÞP f ðxx; 1ÞP f ð1; 1Þ ¼ 1.Such properties characterising a fuzzy implica-

tion I!ðxx; xÞ, our function f ðxx; xÞ can be inter-preted as such. It follows that T ðf ðxx; xÞ; f ðxy ; yÞÞis thus finally equivalent to T ðI!ðxx; xÞ; I!ðxy ; yÞÞ.Fodor and Roubens [1] mention such operatorswhen T is the minimum. They can be defined forany t-norm and any implication operator, whichmakes a very large class of operators.Ongoing research concerns the determination of

appropriate fuzzy operators able at the same timeto provide a clear logical interpretation of how theassignment rule works and to express the interac-tion between criteria which explains the depen-dence of the events associated to the criteria.Although there is almost no well-establishedmethod to deal with interacting criteria, fuzzy in-tegrals are able to represent a certain kind of in-teraction between criteria in the case of averagingoperators (for more details see [2,5]).

4. A new decision support system

The development of a decision support system(D.S.S.) arises from the necessity to rapidly pro-

pose tools completing the existing ones by usinghuge databases of individual data. These require-ments led us to choose the Delphi 3 environment.This tool is still under development. In connectionwith that, it is worth noticing that the work issupervised by a team of future users from theI.N.A.M.I.-R.I.Z.I.V.As mentioned above, this decision support

system must be able to deal with huge databases.Indeed, at the moment, our databases which onlycontain data related to a few hundreds insuredalready have a size of more than 200 MB withtables of more than 500.000 records. Therefore,the package proposes different database manage-ment tools, which does not necessitate any pro-gramming from the user. The databases aremanaged by Interbase 4.2 that is based on theversioning principle.This software is being developed as a kind of

kernel around which other specific existing soft-ware turn so as to avoid the reprogramming ofcommon statistical or graphic procedures. Moreprecisely, it gives the user the opportunity to easilyexport data to Excel� or SPSS� for more indepthcalculations and to import the corresponding re-sults or regular data updates stemming from theInstitute or the mutualities.The interface is built as a notebook each page of

which is dedicated either to treatments on a kindof data or to the implementation of a step of themethodology exposed in Section 3.

5. Conclusions

By deciding to promote a research on theevaluation of the Belgian health care budgets, theMinistry for Social Affairs has launched the de-velopment of new original tools. These onesshould lead to new ways of thinking and buildingthe budgets, which become more and more in-volved to understand. Indeed, the D.S.S. willprovide the decision makers with several degrees offreedom in order to analyse the reality of con-sumption from a patient’s point of view and to testprospective scenario.The paper has described a methodology using

both global and detailed data allowing to highlight


the neglected complementarity between cares andto identify the explanatory capacity of criteriacharacterising the main social categories of insuredBelgians.From a theoretical point of view, this applica-

tion has led to the development of a new multi-criteria sorting procedure based on fuzzyconjunctive aggregation operators (t-norms)weighted through fuzzy implication operators. Itseems that such a new formulation of multicriteriasorting problems could be applied in a more gen-eral context than the one studied here and couldopen new application fields.Ongoing research concerns the practical imple-

mentation of the method on the problem as wellas, more generally, the elaboration of the linksbetween the different methodological tools and theprogramming of a decision support system for theI.N.A.M.I.-R.I.Z.I.V.

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a decision support system for the budgeting of the belgian health care system

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