The interactive decision support system is meant to provide ... In other words, we eagerly longed for a decision support system. ..... method [24] of Leclercq.
089s7177189 $3.00 + 0.00 Copyright 0 1989 PergamonPress plc
Mafhl Comput.Modding, Vol. 12, No. 10/l I, pp. 131l-1320, 1989 Printedin Great Britain. All rights reserved
AN INTERACTIVE DECISION SUPPORT SYSTEM (IDSS) FOR MULTICRITERIA DECISION AID JACQUES TEGHEM JR,’ ‘Dkpartement
de Mathkmatique, 2BelgoNucleaire
Faculte
C. DELHAYE’ Polytechnique
S.A., 25 rue du Champ
and
de Mom, de Mars,
PIERRE L.
KUNSCH’
9 rue de Houdain, 1050 Brussels,
7000 Mons, Belgium Belgium
Abstract-In the last 15 years, an increasing number of methods have been developed in multiobjective programming and multicriteria analyses. The interactive decision support system is meant to provide assistance in selecting the most appropriate methods, given any particular decision problem. In the first part of the paper, the main features of the methods and problems are listed, so as to allow comparisons and to develop rules for a selection procedure. In the second part, these rules are applied to a set of popular methods, ending up with an interactive decision tree.
1.
INTRODUCTION
During the first three decades of its existence, starting in the forties, operational research (OR) has been based on the concept of optimization. This classical problem statement is convenient because it allows the analyst to work in precise and mathematically rigorous terms. Algorithms have been developed to find, or at least, to come close to the best solutions, it means the optima of monocriterion functions. This approach has been without any doubt very successful and many OR tools have been developed that way. However, some slowing down in the initial success of OR might be partially explained by the fact that reality is never entirely reducible to one-dimensional schemes. This is specially true in decision theory, where unique points of view are the exception rather than the rule. The consciousness of the multiple facets present in most decision problems might explain the boom in activities observed for about two decades in modelling with multiple criteria. A new paradigm [l] then became widely accepted, that conflicts are almost inevitably present in each decisional model. This has the important consequence that the concept of an optimum solution is no longer meaningful. Rather, the aim is to find satisfactory compromises and to explore the multiple directions along which they can favourably develop. This multicriterion paradigm had to defend itself against much resistance and as Roy, one of its pioneers, said [ 11: “ . . This multicriteria paradigm will not eliminate the monocriterion paradigm: it is of different nature and both must complement each other. . .” “... the monocriterion paradigm is not a reduction of the multicriteria paradigm. The latter is first a different way of looking at and/or of constructing reality . .” A considerable literature is nowadays available on multicriteria modelling. Real-life case studies even if all together still too rare in a published form, have imposed it as a fully accepted branch of OR. The number of methods is now becoming quite impressive, and the layman gets into trouble when trying to keep an overview (see review articles in Refs [2,3]). Clear development lines are not always apparent as each method has its own specific features. The root of the present study has been our need to bring some order into this somewhat chaotic collection of methods. Faced with particular applications, we had the strong desire to pick a suitable approach, to clearly recognize its strengths and drawbacks, and to place it in comparison with other methods. In other words, we eagerly longed for a decision support system. Of course, it is out of the question to be exhaustive, as dozens of new methods are developed each year. Rather, our intention has been to find a suitable classification, based on the methodology described by Roy in his recent book [4], and to fix it in a decision theoretical approach.
1312
,
JACQUESTEGHEMJR et al.
In the first stage, in Section 2, we describe the main characteristics relative to specific multicriteria problems and models. That is, the action set, the nature of criteria, the available informaiion and how to use it, the nature of the interactivity with the decision maker (DM) etc. The second stage, in Section 3, deals with the evaluation and comparison of some multicriteria methods chosen as representative of the larger families and classes defined by the characteristics of Section 2. These representative methods are those which, on the one hand, are the most familiar to the authors, and on the other hand, have been largely discussed in the framework of the EURO Working Group on MCDA. This group meets twice a year in a European research institution [3]. We think that this subset of methods is sufficient to deal with most practical examples. The last stage, in Section 4, concludes in a decision tree to select a suitable method.? In the case of computer implementation, the leaves can be filled with the corresponding software version. The final aim is to have a unique interactive decision support system (IDSS) applicable to any multicriteria decision problem. 2.
DESCRIPTION
OF
THE
MULTIPLE
CRITERIA
DECISION
MODELS
In a recent in-depth study, Roy [4] has given a careful analysis of the MCDA methodology. In the present section, we keep only those aspects which in our view are sufficient to earmark each class of problem and to distinguish the different modelling approaches. For more precise concepts, the reader is referred to the original work [4]. 2.1. The Set of Actions It is assumed that the set of actions clear and unique way. Two types of situations can arise:
K pertaining
to the decision
process can be described
in a
(a) K is a continuous set of actions. Possibly, in the case of purely discrete problems, only a large number of solutions is considered. Typical examples belong to mathematical programming (MP) problems in which the K-set is implicitly defined by the imposed constraints. The set of feasible values represents the possible actions in the decision problem. The variables can either be continuous or discrete (integer MP) or both (mixed integer MP). The constraints are either linear or non-linear. Some of the constraint coefficients are possibly not known with complete accuracy and they may have uncertain values. They are represented by fuzzy or random numbers. (b) K is a finite moderate size set of discrete actions; this is the case which is generally meant in the framework of the so-called multicriteria analysis. 2.2. The Set of Criteria The consequences of each individual action have to be assessed. This evaluation is based on a set of criteria meant to represent as completely and precisely as possible, the preferences of the DM (cf. Refs [4,6] for a detailed study of preference modelling). A criterion is defined on the basis of one or several preference scales related to the actual problem. The scales are either continuous or discrete, ordinal or cardinal. Let us consider two actions a, b E K, and let f (a), f (b) be respective evaluations for some criterion J Let us also define three basic binary relations: a Pfb
l
a is strictly
preferred
to b for criterion
preferred
aQ/b to b for criterion f (asymmetric
0 a is weakly
f (asymmetric
relations); relation);
a 1,b
l
a is indifferent
from b for criterion
f (symmetric
and reflexive
relation).
tin a recent paper, Colson and de Bruyn describe a multiple criteria support on microcomputers and discuss the qualities of several implementations of MCDM techniques [5].
1313
An IDSS for MCDA
Using these definitions, various concepts of criteria fare
defined, assumed to be real valued.
2.2.1. Ordinary criterion aP/b
iff f(a) >f(b)
a&b
iff f(a) =f(b).
In this case, there is indifference between two actions a and b for criterion f, iff their evaluations are equal. In all other cases, strict preference applies for the action having the highest evaluation. 2.2.2. Pseudo-criterion
In this case, indifference and preference thresholds qf(. ) and p,( ’ ) are defined. They apply to the difference between the evaluations of two actions a, b, as follows: a Pfb
iff f(a) -f(b)
Qfb 8 a I/b 8
a
%W(b))
P,(f(bN 2fW qf(f@)) af(a)
-f(b)
z== qf(f@))
-f(b)
2 0.
2.2.3. Quasi-criterion This is a particular pseudo-criterion
for which the two thresholds have the same value: @(.)=P/(.)*
The preference orders induced within these three criterion families are: (a) total pre-order {Pf , I,}; (b) pseudo-order
{P,, I,, Qr };
(c) semi-order {Pr , If >. The evaluations of the criteria on the set of actions might be only approximate ones or the issues might be uncertain: a dispersion index will be an indicator for this incomplete knowledge. In MP, the criteria are mathematical functions of the problem variables whose coefficients are defined if necessary as fuzzy or random coefficients. 2.3. Properties Within The Criteria Set Quite commonly, further knowledge is available on how to compare the various criteria: (a) Some hierarchical statements are possible within the set. They may be expressed either by a total order, or by a total pre-order, or by any other defined relation between the criteria. (b) A weight w/can be assigned to each criterionf, as a measure of the importance given by the DM to f, such as:
(c) Generalized criteria like quasi- and pseudo-criteria, can be represented in association with a preference function P,(d), which is defined as an increasing function of the difference between the evaluations of two actions, d =f(a) -f(b). This function takes its value between 0 and 1, and represents the intensity of the preference of a above b. Let us mention the PROMETHEE approach [7], in which six different types of preference functions are suggested; each type requires the use of some parameters. Note, however, that the use of preference functions always means that evaluations are only used as differences f(a) -f(b). The resulting loss of information might be unacceptable-or unrealistic-in some applications.
JACQUESTEGHEMJR et al
1314
2.4. The Four Problem Statements The first comprehensive step in dealing with a decision problem implies the description of the K-set of actions, the definitions of the set of criteria and the evaluation of the consequences of actions for each criterion. The next step aims at defining the global preferences in the multicriteria problem, according to the DM. These global preferences boil down to four basic binary relations comparing all pairs of actions a, b: aPb
aQb aIb aRb
a a a a
is globally is globally and b are and b are
and strictly preferred to b (asymmetric relation); and weakly preferred to b (asymmetric relation); globally indifferent (symmetric and reflexive relation); globally incomparable (irreflexive and symmetric relation).
The task of the analyst is now to choose a method in order to fulfil the DM’s needs with regard to the specific decision problem. Roy [4] has figured out four main types of problem statements: -
The choice-problem statement (P,) to assist the selection of one or several “good action(s)“. The sorting-problem statement (P,) to arrange the various actions in predefined classes. The rank-problem statement (P,) to order the actions according to a global preference structure. There are several types of preference structures, the most common ones are the following: l
l
l
the total pre-order, defined for instance by the preference structure {P, I, R = a}, in which the P and I relations are transitive; the partial pre-order, defined for instance by the preference structure {P, I, R}, in which the P and I relations are transitive and fulfil the following property: (PI)u(IP)cP [cf. 4,6]; the total interval order, defined for instance by the preference structure (P, I, R = @}, in which the P relationship is transitive and fulfils the following property PIPc P [cf. 4,6].
Before starting any problem-solving within one of those approaches, it is absolutely necessary to analyse with great accuracy and detail, all the various consequences of the actions. Sometimes, the main aim of the problem-solving is to perform this analysis. In this case, Roy considers it as an additional fourth approach: -
The descriptitle-problem statement (P,).
These four problem statements are obviously not contradictory, have to be combined.
and in some situations, they
2.5. The Operational Approach (OA) Multiple criteria methods draw from the assets of several OAs. 2.5. I. Aggregation into a unique criterion (OA. 1) A mathematical function U is defined in order to aggregate the multiple criteria f, into one unique criterion F: FW = W
(a>,. . . Jib>,
. . . ,Ma>>.
This is the approach usually adopted in classical decision theory. The utility function U makes it possible to use the monocriterion paradigm. However, its definition is often not straightforward and assumes the validity of independence conditions among the criteria [cf. 41. This technique results in a total pre-order, excluding any incompatibility of actions in the global preference structure: the Pr and If relationships are always transitive. 2.5.2. Outranking methods (OA.2) These methods aim at determining the global relationship (P, Q, I, R) between pairs of actions (a, b) thanks to the use of comparative tests. In their simplest form, these tests are considering subsets of criteria: C(a, b) = {m I f,(a) 2fm(b)l
1315
An IDSS for MCDA is the subset
of criteria
in which a is preferred
to or indifferent
C(b, a) = {m I f,(b)
from b;
~fmG91
is the subset of criteria for which b is preferred to a or indifferent from a. In order to find any global relation between two actions, some rules are agreed upon in advance. They use indicators which lie between certain thresholds for establishing the relation. Another distinction is made between compensatory and non-compensatory logics in the definition of the rules. The logic is non-compensatory whenever the cardinalities I C(a, b) I and (C(b, a)] are directly used in the rules, and the differences in the evaluations of the actions 1f,(a) -f,,(b)/ are not used in defining a global preference function [S]. If the reverse is true, the logic is said to be compensatory. 2.5.3. Interactioe
methods
(OA.3)
Interactive methods give an active role to the DM while a solution is being set up. This necessitates an alternation of stages of calculation and dialogue between the analyst and the DM. The calculation state processes the information received from the DM to select a new current solution to be put forward. The dialogue stage allows the DM to question the current solution and to provide supplementary information to the analyst on how to improve it. The interactive approach never proceeds according to any explicit rules. Rather, preferences become unravelled in the course of the interactive stages, in which judgements develop to finally reach a “psychological convergence” [3] representing the most satisfactory solution. Until now, this approach has been most successfully applied to multicriteria MP problems. This might be due to the ease of producing efficient solutions when using MP techniques. A method like STRANGE [9] takes advantage of this to present a large set of related efficient solutions to the DM rather than just one. This allows a better overview during the dialogue stage and eases the convergence process. Let us note here that distinct approaches may be merged to generate new hybrid methods. As an example, the PREFCALC method [lo] uses an interactive approach of the (OA.3) type to determine progressively a utility function (OA.l) which best reflects the DM’s preference.
3. ANALYSIS
OF
SEVERAL
REPRESENTATIVE
METHODS
A selection of MCDA methods has been made as described in Section 1 on the basis of the personal experience of the authors and of the research being performed in the EURO Working Group. Assuming that they are representative enough to cover the needs of most applications, the analysis is performed as follows, according to Section 2: -
for each method, the types of decision problems that can be addressed, and analysed, according to their main characteristics; for each method, the methodological aspects are described.
are listed
The selection-although partly arbitrary-should give the general outline of an IDSS that will help the analyst to locate suitable methods for each new application. Further methods not considered so far should also be easily located, either within existing or within still vacant branches of the decision tree to be set up in Section 4. We first go through the list of selected methods in the two main categories: (a) MP methods (infinite K-set of actions); (b) multiple criteria analysis (finite K-set). (a) The following -
MP methods
have been considered:
The family of methods grouped around STRANGE, due to the present authors [9, 11,121: the basic STRANGE [9,12] approach working in an uncertain environment and its extension to piecewise linear functions in R(estricted) B(asis)-STRANGE [ 111; further, their respective
1316
JACQUESTEGHEM JR ef al. Table I. MP methods Problem statementa
Method STEM-STRANGE STEM-RBSTRANGE STRANGE RB-STRANGE MOMIX STRANGE-MOMIX Choo-Atkins FLIP PROMETHEE IV
Characteristics of the method
Criteria
Variables
Uncertainty (modelling)
Type of approach
Operational approach
Linear Non-linear Linear Non-linear Linear Liner Fractional Linear Non-hnear
Continuous Continuous Continuous Continuous Mixed Mixed Continuous Continuous Continuous
NO NO Yes (random) Yes (random) NO Yes (random) NO Yes (fuzzy) Nob
P, P, P, PZ PZ P, P, P, P.
0A.3 OA.3 OA.3 0A.3 0A.3 OA.3 OA.3 OA.3 0A.l
No. of compromise solutions at each interactive stage Infinite Infinite Infinite Infinite One One Finite number Infinite
“The constraints are assumed to be always linear. “Extensions are possible.
deterministic versions, called STEM-STRANGE and STEM-RBSTRANGE, which are an improved extension of the STEM method [13]. The MOMIX method, also due to the present authors [ 141, working in MIP, and its combined version with STRANGE, to deal with uncertainties [15]; The method of Choo and Atkins [16]. The FLIP method [12, 171, developed by Slowinski using a fuzzy set approach to deal with uncertainties. The PROMETHEE IV method [7], developed by Brans et al. (b) The following -
methods
pertaining
to multicriteria
analysis
have been selected:
The family of ELECTRE methods, developed by Roy and co-workers: ELECTRE I [18], II [19], III [20], IV [21]. The family of PROMETHEE I, II, III methods [7], developed by Brans et al. The ORESTE method, initially created by Roubens [22], and further developed by Pastijn and Leysen [23]. The PREFCALC method [lo] of Jacquet-Lagreze. The MELCHIOR method [24] of Leclercq. The AHP method [25] of Saaty.
It goes beyond the scope of the present study to describe any one of these approaches; the interested reader is advised to study the references. The results of the analysis of these various methods using the concepts and notations of Section 2 are summarized in Table 1 (MP) and Table 2 (multicriteria analysis). Looking at Table 1, it becomes clear that practically all selected MP methods belong to the P, problem statement and mostly use an interactive type method OA.3. Differences mainly arise at the level of the solved problems and their related characteristics. Table 2 confirms that a large majority of methods used in multicriteria analysis belong to the P, problem statement and mostly use an outranking OA.2. Table 2. Multicriteria analysis Characteristics of the method
Problem statement
Compensatory character
Criterzi
Information of the criteria
ELECTRE I ELECTRE 11 ELECTRE III ELECTRE IV PROMETHEE I PROMETHEE II PROMETHEE III
Quasi Quasi Pseudo Pseudo Any Any Any
Weight Weight Weight None Preference function Preference function Preference function
P, P. P: P: P: P: P.;
OA.2 OA.2 0A.2 0A.2 OA.2 OA.2 0A.2
NO’21 No”’ N&’ NO Yes Yes Yes
Ordinary Ordinary Pseudo Ordinary
Total preorder Weight’4’ Any relation Hierarchical
P. P: P: P:
0A.2 (OA.3)“’ 0A.l (OA.3)“’ 0A.2 0A.2
Yes’5’ Yes NO Yes
ORESTE PREFCALC MELCHIOR Saaty ” “See Sectmn 3.
Approach
Operational approach
Method
Characteristics of result Set of good actions Total pre-order Partial pre-ord&’ Partial pre-order Partial pre-order Total pre-order Partial or total interval preorder Partial pre-order Total pre-order Partial pre-order Total pre-order
An IDSS for MCDA
1317
However, an important discriminating characteristic relates to the compensatory aspects. Moreover, the characteristics of the addressed problems and the type of results are very different according to the method. Some additional remarks can be made on studying the tables: (i) P, is not very frequently used; one example is ELECTRE I. P, is practically never mentioned in the literature (note, however, Moscarola and Roy [26]). (ii) An important-but more technical-characteristic of the methods is not mentioned in the tables. It relates to the determination of the various parameters, which are of two kinds: -
the thresholds to be used in quasi- and pseudo-criteria (Sections 2.2.2 and 2.2.3) or the parameters defining the preference functions [Section 2.3(c)]; the parameters to be used in tests during the determination of the global preference relations (Section 2.5.3).
The determination of these parameters might be more or less difficult; this grade of difficulty has, however, not been used as a discriminating characteristic affecting the comparison between methods; the necessary degree of knowledge is supposed to be available during the in-depth process of understanding and implementing the method. Some more specific remarks apply to Table 2 (indicated by superscript numerals in the table): (1) The information about the operational approach relates to the main approach used in the original method. Some extensions or hybrid forms are still possible as has been mentioned previously. Regarding for instance the interactivity, PREFCALC [lo] gives the DM the ability to discuss its utility function; in ORESTE, an extension by Pastijn and Leysen [23] allows the DM an interactive choice of several parameters. Note also that PROMETHEE II and III use an aggregation of the functions of preference, in order to define a valued outranking relation on the set of actions. It, therefore, uses to a certain extent the OA.l method. (2) The ELECTRE methods are sometimes described as being “partially compensatory” approaches. A veto test measuring the incomparability between two actions is namely based on the differences between the evaluations of these two actions for different criteria [cf. 18, 191. (3) ELECTRE III starts by determining a fuzzy outranking relation on the actions [cf. 201. A partial pre-order then results from this relation. (4) PREFCALC has an alternate approach for the definition of weights associated to the criteria; it is based on a preliminary ranking of some of the actions, which are already very familiar to the DM [cf. lo]. (5) ORESTE may, in a certain sense, be called a compensatory approach, as it explicitly uses the differences between the ranks of pairs of actions based on their evaluations [cf. 221. 4. AN INTERACTIVE
DECISION
SUPPORT
SYSTEM
(IDSS)
The role of the IDSS is to guide the DM through a given set of multicriteria methods, in order to pinpoint a suitable approach for the specific problem at hand. A particular IDSS is, of course, dependent on the sample set of selected methods. The present section describes the project called CHOICE, which has been developed on the set described in Section 3. It can be generalized to include any further method not considered so far. Its philosophy remains valid if applied to a different sample of representative methods. At the start, each individual method has been analysed according to the methodology of Section 2. A dialogue based on a small number of simple questions orients the search procedure through successive stages. The questions are related to the main characteristics evidenced in Section 2. The order in which they are asked, has been chosen so as to reflect the priorities in making choices. Multiple answers are possible at each stage. In this way, the DM moves along the branches of the decision tree, as shown in Fig. 1. Each final node downwards of the tree corresponds to a possible answer.
tuncuon
Utility . ..
(STRANGEI
Continuous linear
I
I
_.-. ..
MOMIX
Mixed variables _
\
I I
1 ELECTRE II 1 1 ELECTRE III 1
Outranking method
(RESTRANGE/
Modelization
\
P a
MIXBrA variables .
/
/
I
I \
\
rnrervar pre-order
Finite
Total
/
I
I
Any relation
\
1%relation
ierarchical
H-7-n
have the same importance.
r-- --
K-set
tree. *In this case all criteria
pre -oraer
decision
pre -order
I (RRSTRANGE~ lMoMLn 1
I
Fig. 1. The CHOICE
STRANGE
Continuous lizear
K-Set
Non------satorv
IELECTRE IV J
Compensatory
1 ORESTE I
-...-..-
An IDSS for MCDA
1319
Before starting to comment on this procedure, let us describe in details the CHOICE decision tree (Fig. 1). The first node at the top splits the two types of K-sets of actions into either continuous (multicriteria MP) or finite (multicriteria analysis). The MP branch is simpler and should be described first. Remember that the main characteristics of this class of methods are related to the studied problem itself. A first question makes it clear if uncertainties are present or not. If they are, one should choose between a random or a fuzzy representation. Further down the tree comes questioning on the operational approach. From the structure of his or her problem. the DM will have to decide between either aggregation (only for PROMETHEE IV) or interactive methods. Finally, the criteria are discussed, either they are linear or not; variables, on the other hand, can be continuous or mixed. The tree structure on the side of multicriteria analysis is more complex. The first question is to know whether some information is available on the criteria or not. If yes, one has to choose between four possible substructures. Note that sometimes, several solutions are possible-for instance, if both weights and preference functions are provided. The selection has then to be made according to priorities. In a more sophisticated approach using an “expert system”, multiple answers with their respective confidence levels must be provided: (1) if weights are given, P, or P:, approaches are possible; the next stages address the operational approach and further down the type of results; (2) if preference functions are used, the final method will depend on which type of result is needed, like in the case of the three PROMETHEE methods; (3) if some kind of relation is introduced between the criteria, the branchings are on the nature of this relation; (4) if a global preference on some subset of actions is given by the DM, PREFCALC [see footnote’4’ for Table 21 comes out as the unique solution. If no information is available on the criteria, the type of logic is questioned, either it is non-compensatory (ELECTRE IV) or compensatory (ORESTE, if all criteria are given equal priority).
5. COMMENTS
AND CONCLUSIONS
The IDSS just described necessitates a few comments: -
-
-
-
As said before, it is specific to the selected set of methods; an extension of the method set is possible; branches with empty final nodes-not drawn so far as no corresponding method is yet available-might then appear as further methods fill the nodes. Each final node in Fig. 1 corresponds to only one method. For larger sets, several related or similar methods would gather within one final node. Some care is required as the exploration of the tree is entirely deterministic. Once a method has come out, it has to be explained thoroughly to the DM. It might then appear that the latter does not approve of it, perhaps because he is unable to give the internal parameters inherent to the selected method (cf. Section 3). In this case, backtracking or going back to the top is advised and the answers to the questions will be adapted. The decision tree in Fig. 1 is constructed along some assumption on the ranking of the questions to be put to the DM. In a more sophisticated and more flexible approach, this assumption can be dropped. Along some characteristics of the problem and some wishes of the DM-including some fuzziness in the answer-the IDSS will then be able to develop its reasoning like an “expert system” to propose one or several suitable methods. Plans do exist to implement such an advanced IDSS named CHOICE on a microcomputer, using a low-priced expert system shell. The final decision nodes could be filled with the existing-or with specially developed-stand-alone versions of the most familiar multicriteria methods.
1320
JACQUESTEGHEM JR et al.
REFERENCES 1. B. Roy, Des criteres multtples 2. 3. 4. 5. 6. 7. 8. 9. 10. 1 I. 12. 13. 14. 15. 16.
en recherchc operationnclle. Pourquoi? Technical Report LAMSADE. Universite de Paris-Dauphine. France (1987). G. W. Evans. Overview of techmques for solving multiobjcctive mathematical programs. Mgmt Sci. 30(11), 12681282 (1984). Ph. Vincke, Analysis of multtcriteria decision aid in Europe. Eur. J. opl Res. 25(2). 160-168 (1986). B. Roy, MPthudologie MuftkritPre D’Aide ci /a Dkckion. Collection Gestion-Edition Economica, Paris (1985). G. Colson and Chr. de Bruyn. Multtplc criteria supports on micro-computers. Eelg. J. Op.7 Res. Stoti.~t. Comput. Sci. 27(l), 29 58 (1987). M. Roubens and Ph. Vmcke. Prefererrc~ Modelling; LNEMS 250. Springer-Verlag, Berlin (1985). J. P. Brans (Ed.), B. Mareschal and Ph. Vincke, PROMETHEE--a new family of outranking methods in multicriteria analysis. Operational Research X4, pp. 477-490. Elsevier, New York (1984). J. C. Vans Snick, On the problem of weights in multiple criteria decision-making (the non-compensatory approach). Eur. J. opl Res. 24(2), 2288234 (19X6). J Teghem Jr. D. Dufrane, M. Thauvoye and P. L. Kunsch. STRANGE-an interactive method for multi-objective linear programmmg under uncertamty. Eur. J. opl RES. 26(l), 65582 (1986). E. Jacquet-Lagreze, PREFCALC evaluation et decision multicritere. Euro-Decision, Paris. Revue Utilis. IBM-PC 3. 38-55 (April 1984). C. Delhaye, Extension de la methode du simplexe a dcs problemes non lintaires. Technical Report. Faculte Polytechnique de Mans. Belgium (1986). R. Slowinski and J. Teghem Jr. Fuzzy vs stochastic approaches for multi-objective linear programming under uncertainty. Nau. Rex. Logist. 35, 673 695 (1988). R. Benayoun. J. de Montgolher, J. Tergny and 0. Larichev. Linear programming with multiple objective functions: Step method (STEM). Math1 Program. 1, 366375 (1971). J. Teghem Jr and P. L. Kunsch. MOMIX--an interactive method for mixed integer linear programming. Submitted for publication. J. Teghem Jr and P. L. Kunsch. An interactive DSS for multi-objective investment planning. In Mathematical Models ,for Decision Support (Edited by G. Mitra), pp. 123 134. Springer-Verlag, Berlin (1988). E. U. Choo and D. R. Atkins. An interactive algorithm for multicriteria programming. Computers Ops Res. 7, 81 ~87 (1980).
17. R. Slowinski. A multicriteria fuzzy linear programming method for water supply system development planning. Fuzz> SetsS-w. 19, 217 235 (1986). 18. B. Roy, Classement et choix en presence de points de vue multiples (la mtthode ELECTRE). Reoue.fk. autom. fnf: Rech. opb. 8, 57-75 (1968). 19. B. Roy and P. Bertier,
20. 21. 22. 23.
La mtthode ELECTRE-une application au media planning. In Operational Research 7?(Edited by M. Ross), pp. 291-302. North-Holland, Amsterdam (1973). 8. Roy, ELECTRE III-un algorithme de classements fond6 sur une representation plane des preferences en presence de criteres multiples. Technical Report SEMA, Paris 81 (1977). J. M. Skalta et al., ELECTRE III et IV-aspects methodologiques et guide d’utihsation. Document du LAMSADE no 25, Universrte de Paris-Dauphine, France (1984). M. Reubens, Preference relations on actions and criteria in multicriteria decision making. Eur. J. opl Res. 10(l). 5ll55 (1982). H. Pastijn and J. Leysen. Constructmg an outranking relation with ORESTE. Math/ Comput. Modelling 12, 1255-1268
(1989). 24. J. P, Leclercq,
Propositions d’extension de la notion de dominance en presence de relations d’ordre sur pseudo-crittres: la methode MELCHIOR. Belg. J. Ops Res. Statist. Comput. Sci. 24(l), 32-16 (1984). 25. T. Saaty. The Analytic Hierarchj Process. McGraw-Hill, New York (1981). 26. Moscarola and B. Roy, Procedure automatique d’examen de dossiers fondee sur une segmentation trichotomique presence de criteres multiples. Rewe .fr. autom. If. Rech. opPr. 11, 145-173 (1977).
les
en
