Description and analysis of some representative interactive multicriteria procedures

0895-7177/89$3.00+ 0.00 Copyright CI 1989PergamonPressplc

Marhlcompur.hhielhg, Vol. 12,No. IO/II, pp. 1221-1238. 1989 Printed in Great Britain. All rights reserved

DESCRIPTION AND ANALYSIS OF SOME REPRESENTATIVE INTERACTIVE MULTICRITERIA PROCEDURES DANIEL VANDERPOOTEN LAMSADE, UniversitC de Paris-Dauphine, Place du Mar&ha1 De Lattre de Tassigny, 75775 Paris Cedex 16, France PHILIPPE

VINCKE

Universitt Libre de Bruxelles, 210 Campus de la F’laine, Boulevard du Triomphe, 1050 Bruxelles, Belgium Abstract-Interactive multicriteria procedures have met with a very great success for about 15 years. We present here some of them which seem to be representative of the literature. Each description is followed by a detailed analysis of the method. This analysis consists of comments about the underlying approach, technical aspects and practical considerations. A final section provides general comments.

1. INTRODUCTION

A very large number of interactive multicriteria procedures have been proposed to date. Each of them rests on various approaches, assumptions and technical concepts. In order to give an idea P ., or tms diversity, we describe, in a chronoiogicai order, i0 procedures. Some of them piayed an important role in the development of the interactive approach. The others are some of the most representative in the literature. We also included our respective methods as a reflection of our own convictions. Many other interesting procedures are presented in the literature and in other surveys [e.g. l-51. Section 2 introduces some basic definitions and theorems which are of interest for the procedures presented in the paper. Each of the following sections is devoted to a specific procedure. Each consists of a description within a unified framework and some comments with respect to the underlying approach of the method, technical aspects and practical considerations. The final section provides general comments. 2. PROBLEM Mathematically,

STATEMENT,

BASIC

DEFINITIONS

AND

THEOREMS

a multiple criteria problem can be stated as max LG(a), . . . , s,(a)1 = da> s.t.

acA

where: denotes the set of potential (or feasible) alternatives; --I&Y,9 . . . , g,> represents a set ofp@ > 2) real-valued functions called criteria which should satisfy some consistency properties [see 61; -the notation max indicates here that we are looking for the best compromise for a decision maker (DM) according to his preference structure, taking into account each of the p criteria (it is assumed, without loss of generality, that each criterion is to be maximized). -A

Additionally,

we shall consider:

-Z( c Rp) the criterion space with its natural partial preordering defined by > (z >, z’ iffziaz(Cj=l,...,p)); -Z,=g[A]=(ZEZ:Zj=gj(a)O’=l,... ,p) where a E A) the set of potential outcomes, which will be assumed to be compact. 1221

(1)

1222

DANIELVANDERPOOTEN and PHILIPPE VINCKE

In some cases, A consists of a list of exhaustively enumerated alternatives. Each alternative a E A is explicitly identified by a vector of criteria outcomes (z,, . . . , zp) with zj = g,(a) Vj. In other cases, a E A is a vector of decision variables: a = (x, , . . . , x,) = x (x E W). Here, A is implicitly identified by a set of constraints over the decision variables. Problems of this type will be referred to as multiple objective programming (MOP) problems. The general formulation of a MOP problem is

. . . , g,(x)1 s.t. hi(X)< 0 (i = 1,.

(2)

maxk, (x),

. . , m)

+ additional constraints

(integrity constraints. . .).

Here, g,(x) (j = 1,. . . , p) are preferably called objective functions and hi(x) (i = 1, . . . , m) are constraint functions. A well-studied special case of problem (2) is the multiple objective linear programming (MOLP) problem: max[c’x, . . . , tix] s.t.

(3)

Dx
where c’ are n-dimensional coefficient vectors, D is an m x n constraint coefficient matrix and b is an m-dimensional vector. Multicriteria interactive procedures have been frequently devoted to MOP problems making use of the large apparatus of techniques from mathematical programming. However, it is clear that problems involving an explicit list of alternatives can also be considered in an interactive way. Although the techniques may be different from those used for MOP problems, the underlying concepts are quite similar. It is to be noticed that problem (1) can be restated in the criterion space using variables zj which indicate the values of criteria gj: max[z,, . . . , zp] s.t.

(4)

zj = gj(a)

(j=l,...,p)

or more compactly, max[z, , . s.t.

. . , z,] = z

ZEZ,.

This formulation will be used in many cases, and particularly for problems involving an explicit list of alternatives. DeJnition

2.1. z’ E Z, is nondominated

iff there is no

z E

Z, s.t. Zj>, zi Vj and

Zj

and zk > z; for

at ieast one ic. De$nition Definition

Vj. 2.2. z’eZA is weakly nondominated iff there is no ZEZ* s.t. Zj > zi 2.3. a E A is (weakfy eficient iff its corresponding criterion vector is (weakly) nondominated. The set of all (weakly) efficient alternatives is the (weakly) efficient set. Other concepts such as proper nondominance (and efficiency) are omitted here. See, for example, Ref. [7] for more details. Let $(j=l,... ,p) be a solution to the following problem: max

gjta>

s.t.

acA.

(5)

We shall denote 2: = gj(zk) and zr = z$ = g,($). Definition 2.4. The vector z* = (z:, . . . , z;), whose coordinates separately reached in each criterion over A, is called the ideal point.

are the optimal

values

Some representative interactive multicriteria procedures

1223

5; are classically gathered into a table called the puyoflmatrix:

The ideal point is on the diagonal of the payoff matrix. Definition 2.5. The vector n = (n,, . . . , n,), where (j = i,. . . ,pj

i.e. the minimum value in the jth column of the payoff matrix, In case of alternative optima for problem (5), the payoff matrix are not uniquely defined. Consulting the payoff matrix gives an estimate of the ranges of it should be noticed that the coordinates of the nadir point do values of the criteria over the efficient set.

is the nadir point. and consequently the nadir point the potential outcomes. However, not correspond to the minimum

Five classical theorems, which are of interest for the procedures presented in this paper, are now introduced. Proofs and other results can be found, for example, in Refs [8,9]. Theorem 2.1 (weighted sum) -If

z’ is the optimal solution to P max C A1z, lCZA

(6)

j=l

-If

with A1> 0, then z’ is nondominated. z’ is nondominated and Z, is convex then there exist Aj> 0 st. z’ is the optimal solution to problem (6).

Theorem 2.2 (weighted Tchebychev norm) -If

z’ is the optimal solution to min max rsZA j=l.....p

( lLj(Z;O -

Zj>l}

(7)

with IV> 0 and z,?* > zl? then z’ is weakly nondominated. Moreover, if z’ is unique then it is nondominated and if not, at least one of the optimal solutions is nondominated. -If z’ is nondominated then there exist Aj> 0 s.t. z’ is the unique optimal solution to problem (7). Theorem 2.3 (augmented weighted Tchebychev norm) --If z’ is the optimal solution to

where A,> 0, zy* > z; and pi are sufficiently small positive values, then z’ is nondominated. --If z’ is nondominated then there exist Aj> 0 and sufficiently small positive values pj, s.t. z’ is the unique optimal solution to problem (8). Results very similar to Theorems 2.2 and 2.3 may be obtained using functions which are no longer norms (they may take negative values) introduced by Wierzbicki [9] as achievement scalarizing functions. It should be noted that the controlling parameters are no longer weighting

DANIEL VANDERPODTENand PHILIPPE VINCKE

1224

vectors 3, which represent the relative importance of each criterion, but reference points Z which represent aspiration levels. Theorem 2.4

-If

z’ is the optimal solution to

$!2 , =~~~ - zj)> .p{&(Z,

(9)

with ZE Z and iL,> 0 then z’ is weakly nondominated. Moreover, if z’ is unique then it is nondominated and if not, at least one of the optimal solutions is nondominated. -If z’ is nondominated then by solving problem (9) with Z = z’ the minimum is attained at z’ and is equal to zero. Theorem 2.5

---If z’ is the optimal solution to 2::

mix [ ,=l,...,p

{n,(zj - zj>> - i

p?j

j=l

1

(10)

where Z E Z, ij > 0 and pj are sufficiently small positive values, then z’ is nondominated. ---If z’ is nondominated then by solving problem (10) with Z = z’ the minimum is attained at z’ and is equal to zero. We are now in a position to describe 10 representative procedures. In some cases, we slightly modified the original description in order to encompass more general cases (when possible). We also replaced weak characterizations of nondominance by stronger ones (e.g. using Theorem 2.3 instead of Theorem 2.2).

3. THE

STEM

PROCEDURE-BENAYOUN

et

al.

[IO]

3.1. Description The procedure progressively reduces Z, by iteratively adding constraints on the criteria values. A compromise solution zh is computed minimizing a weighted Tchebychev norm over Z\, which represents the reduced set at iteration h. Step 0. Determine the payoff matrix and calculate the following “normalization lj = -%

where aj =

C Ofi

27 - n, max{lz,*l,

coefficients”:

(j=l,...,p).

Injl}

Let Z: = Z, and h = 1. Step 1. Calculate a compromise solution Z* by solving min(a -i, s.t.

p

B

(11)

PjZj)

s
-

2,)

(j=

1,

.* .,P>

ZEZ:

with z** = z,? + E,, where p, and cj are sufficiently small positive values. [It should be observed that problem (11) is an equivalent formulation of problem @).I Step 2. zh is presented to the DM (a) if he is satisfied with zh, STOP; (b) otherwise, ask him to indicate on which criterion [index) k he is ready to make a concession and which maximum amount Ak he accepts to concede.

Some representative

interactive

multicriteria

1225

procedures

Step 3. The set of potential outcomes is reduced: z;+’ = {zeZh,: z,azh,-Ak Let&=O,h=h+l

andgotostep

and z,>zfVj#k}. 1.

3.2. Comments (4 STEM was originally proposed in the MOLP framework. Our presentation suggests that this method is general enough to be adapted to other cases, including problems involving an explicit list of alternatives. (b) In the original MOLP presentation, the authors suggest to use use more sophisticated normalization coefficients: z)* - nj a,=max{lz;I.ln,II

1 llc/ll’

)I&I/ being the Euclidean norm of vector cj. However, it should be pointed out that this second normalization term is scale-dependent (because, for any scalar a, we generally have li*1111 /I I) ,.,I\ IIC~II f II P c- II 1.

The main cc>

drawback of the method is its “irrevocability”: when a concession has been made on a criterion, it is definitely registered in the model. If the DM wishes to change his mind, he is obliged to start the procedure again. Similarly, the constraints zj 2 zr Vj #k being irrevocable, it is not possible to reach a compromise which would have a slightly lower value than -_F(Vj# k) and would be much better than zh on all other criteria. (4 As suggested by the authors, more than one criterion could be relaxed at Step 2. Because of the irrevocability and the fact that at least one S is set to zero at each iteration, the authors point out that the procedure stops after at most p iterations. It should be noticed that if 2, = 0 Vj at iteration h, Zi+ ’ is not necessarily reduced to a unique element. In this case, Z:+’ cannot be explored any further by the procedure. This is another difficulty resulting from irrevocability. (e> It is not always easy, for a DM, to specify Ak, particularly if he knows the importance of this value in the procedure and the fact that it is irrevocable. Moreover, it would be more natural for the DM to specify the criteria to be improved, rather than those to be relaxed. (0 The authors point out that the DM can be helped by a sensitivity analysis, giving lower and upper bounds for the variations on the criteria due to a small change on one of them. The idea of a sensitivity analysis to help the DM is very important in such a method. However, the analysis proposed by the authors (in the MOLP case) can produce bounds which do not correspond to feasible solutions, so that one has to be very careful in this step. (g) Calculation steps are simple. In the MOLP case, problem (11) is a linear program. The construction of the payoff matrix requires p optimizations which can be performed efficiently, noticing that the only change is in the objective function. Only one optimization is needed for any subsequent iteration, hnc (h) STEM- .-is certainlv ___ ______, one ____ of the best known interactive nrncd~lrm~ r - _ _ _ _ _ _ _ I ) -it_ Inlcn _- _ _ _I I the me& TV haye been the first such method proposed in the literature and to have opened a fruitful field of research. More recent methods are probably better adapted to the needs of practitioners but the STEM procedure was a pioneer. Moreover, we believe that most of the indicated drawbacks could be overcome by relaxing the irrevocability assumption.

4. THE

GEOFFRION

et al. METHOD

[l l]

4. I. Description

This method has been proposed to solve MOP problems where A is a convex and compact subset of W. Moreover, it is assumed that the DM wants to maximize a function U[g, (x), . . . , g,(x)] which is not known explicitly and is supposed to be differentiable and concave. Roughly speaking, the DM is guided through Z, in a way similar to the progression performed by classical nonlinear optimization techniques.

DANIEL VANDERPOOTENand

I226

PHILIPPE VINCKE

Step 0. A first compromise solution (x’, z’), with z: = g,(x’) Vj, is arbitrarily chosen, Let h = 1. Step 1. Determine the local marginal rates of substitution (or tradeoffs) 2: between criterion g, and criterion g, (the reference criterion) at the point zh. By definition, we have i_: = (3U(zh)/3g,)/(aU(zh)/dg,), where (8u(zh)/8g,) is the ith partial derivative evaluated at zh. An interactive procedure to indirectly determine these tradeoffs was subsequently proposed by Dyer [12] (see below). Step 2. Let V., U[gg,(x*), . . . ,g,(x”)] be the gradient of U at point zh. Determine an optimal solution yk to max V, U[g, (x”), . . . , g,(x”)l

- _.r

s.t.

y

(12)

yEA.

Noticing that V, u lg, (dj, . . , g,jx”jj = X,;,idU(d~j/iTg,j V,g,(x?) and dividing the objective function in problem (12) by (8u(zh)/ag,) (which is positive), we may replace the objective function by 12: . V&(X”) . Y i where V.,g,(xh), which is the gradient of gi evaluated at x*, can be calculated from the data of the problem. dh = yh - x* gives the locally best direction in which to move away from x*. Step 3. Obtain from the DM a solution to the step-size problem max U[g, (x” + tdh), . . , gp(xh + tdh] s.t.

n
This is achieved through a (graphical) procedure showing the simultaneous evolutions of the p criteria as t increases from 0 to 1. By selecting his most preferred criterion vector, the DM provides the required value t *. Let zh+ ’ be the selected point and x* + ’ its inverse image (x” + ’ = x* + t *d”). Step 4. If zh+ ’ = z* (or if the DM is satisfied with z”+ ‘) then STOP with (xh+ ‘, z*+ ‘) as the final prescription else let h = h + 1 and go to Step 1. 4.1.1. Determination of A:. The interactive procedure proposed in Ref. [12] allows us to assess the tradeoffs %,hthrough a. series of pairwise comparisons. The DM is asked to compare the following two solutions:

Zh=
where A, and Aj should be small perturbations relative to z’; and z/h, but large enough to be significant. If the DM prefers z* over z’* (resp. z’* over z*), Aj is decreased (resp. increased) until indifference is obtained; 1.; is then given by Aj/A,. 4.2. Comments

(4 The Geoffrion et al. method is based on the very strong assumption that there pre-exists an implicit utility function U which has to be optimized and that all the answers of the DM will be consistent with this function. (b) The method is an adaptation of the Franke-Wolfe algorithm to the multi-objective case. This algorithm was chosen because of its robust and rather rapid convergence properties. It is clear that many other mathematical programming algorithms could be rendered interactive in a similar way. (4 Unlike most of the methods based on assumptions about an implicit pre-existing utility function, this procedure does not progressively (explicitly or implicitly) eliminate any solution. Consequently, it can be used as an exploratory search process in a trial and error fashion, if prior assumptions and convergence properties are relinquished.

Some representative

interactive

multicriteria

1221

procedures

(d) The main drawback results from the difficulty in providing the preference information required in Steps 1 and 3. It is now widely recognized that DMs are reluctant to specify tradeoffs. The determination of zh+ ‘(or t *), even when graphical displays are used, becomes difficult when p > 3 or 4. (e) The choice of a reference criterion (see Step 1) may prove difficult. Most of the time, criteria involving monetary consequences will be considered. (f) A lot of questions must be answered at each iteration. The Dyer procedure, which allows an indirect specification of tradeoffs, requires many pairwise comparisons. (g) The proposals presented in Step 3 may correspond to dominated points. (h) Calculation steps are easy. In the MOLP case, problem (12) reduces to a linear program. Moreover, it should be noticed, for an efficient implementation, that the only change at each iteration is in the objective function. (i) The Geoffrion et al. [ 1l] procedure can also be considered as a pioneering work. It gave birth to many methods making use of other underlying optimization procedures and trying to reduce the cognitive strain imposed on the DM [e.g. 131.

5. THE

EVOLVING

TARGET

PROCEDURE-ROY

[14]

5. I. Description

This method iteratively determines a region of interest and a search direction (materialized by a weighting vector) which allow us to generate a proposal by minimizing a weighted Tchebychev norm. This process is conducted in a trial and error fashion. StepO. LetZL=Z,andh=l. Step 1. Calculate z *h the ideal point relative to Z: and let z**” = Zig + c. Step 2. Ask the DM to specify a reference point Zh corresponding to aspiration ;h d z/*h yj. Let 1.: = I/(z:*~ - 2;). Step 3. zalculate a compromise solution zh by solving min(p -;, s.t.

levels s.t.

W,)

I* 2 >,(z,** -z,)

(j=l,...,p)

ZEZ;.

Step 4. z” is presented to the DM (a) if he is satisfied with zh, STOP; (b) otherwise, ask him to specify the criteria which can be relaxed. Let K be the corresponding set of criterion indices. For each k E K, ask which maximum amount Ak he accepts to concede. Step 5. The set of potential outcomes becomes Z ;+‘={ZEZA:ZkazZ:Leth=h-tl

andgotostep

Ak VkEK

and

zjaz, ’

V’j~fl,. . . ,p)\K}.

1.

5.2 Comments

(4 The evolving target procedure can be applied in any case, including problems involving an explicit list of alternatives. (b) The originality of this approach is to refute the assumption of a pre-existing and stable utility function. It is clear that within this framework, no consistency is required from the DM. He is free to change his mind. The main purpose of such a procedure is to support learning of preferences in a trial and error fashion, No mathematical convergence is supported, which is natural in a learning-oriented approach. cc>

1228

(4

63

(0

!g?

DANIEL VANDERPCOTENand PHILIPPEVINCKE

Although not indicated in the above description, the original procedure envisaged possible modifications (reductions, and also enlargements) of the current set of alternatives. This could result from the availability of new alternatives or the disappearance of old ones, and also from a change of preferences which could induce the DM to explore other alternatives. The preference information required in Step 4 is similar to STEM. However, no irrevocability is involved here, which makes the information easier to supply. The preference information required in Step 2 is only used to guide the search within Z:. We believe that such an information, which increases the cognitive strain imposed on the DM, is unnecessary in a learning-oriented perspective. Indeed, it would be possible to use a fixed direction of preferences (such as in STEM) and possibly to allow the DM to specify this information when he really wishes to. Calrnlatinn VY.VY.UC."I.

rtpnr "C'yY

rnncict Y"I.O.UL

(p for the construction

6.

THE

nf the v. Cll" rpcnhltinn IV"V.L.C."I.

nf "1

yn

I, 1 1 nntimi7atinn “yC”‘..LuCa””

nrnhlmnc yL”“‘w”‘Y

at “U”ll esarh itpratinn UC lLIlULl”ll

of the local ideal point and 1 for the generation of the proposal).

ZIONTS

AND

WALLENIUS

METHOD

[15,

161

6.1. Description

This method has been proposed in the context of MOLP. It generates a sequence of improved extreme point solutions using local linear approximations of an implicit utility function which is assumed to be pseudo-concave. Step 0. Let ,4 ’ = {k E W: ,I,> E, XjA, = l} be the initial set of weighting vectors (where t is a sufficiently small positive value). Let h = 1. For any 3,E n h, solve the following linear -_^ __^_. p1ug1 a111. (13)

s.t.

Dx 0.

Step 1

Step 2.

Step 3.

Step 4.

Let X” be the resulting optimal solution and zh its corresponding (nondominated) criterion vector. -For each nonbasic variable xk in the optimal solution (xh, zh), test if the introduction of xk into the basis leads to an efficient extreme point (at worst, this test is realized by solving a linear program-see Ref. [17]); if the test is positive, xk is called an efficient nonbasic variable. -Determine the subset M of efficient nonbasic variables whose introduction into the basis do not lead to solutions previously (implicitly) eliminated in Steps 3(b) or 5(b), and its complement N. Let indicator set L = M. For each X~EL, -determine wJk, which represents the decrease in criterion gj due to some specified increase in xk (these quantities are obtained from classical properties of the Simplex algorithm), -ask the DM if he is ready to accept the tradeoff corresponding to the simultaneous variations wlk, . . . , w,~, . . . , wpk; the possible responses are yes, no and I don’t know. (a) If no yes response is obtained then if L = A4 then let L = N and go to Step 2, else STOP with (xh, z”) as the final prescription (b) otherwise, reduce the set of weighting vectors: n h+ ’ = {A E ,4 h: Ejwlk ij < - t for each yes response and xk E L and 2, wiklj > c for each no response and xk E L j (I don’t know responses are not taken into account.) If Ah+’ = 0 drop the oldest active constraint until Ah+’ # 0. (13). Denote the corresponding solution as For any I E A*+‘, solve problem (Xh+I) z/l+ I).

Some representative

interactive

multicriteria

procedures

1229

Step 5. Ask the DM to indicate which of zh and zh+’ is the most preferred: (a) if zh is the most preferred, STOP with (xh, zh); actually, better solutions which are nonextreme points could be found; (b) if zh+’ is the most preferred, modify /1 h+ ’ by adding the following constraint: X1(2,”+ ’ - z))Aj 2 6. Let h = h + 1 and go to Step 1. 6.2. Comments (a) The Zionts and Wallenius method is restricted to MOLP problems. However, some extensions have been proposed in the multiple objective integer programming (MOIP) case and in the discrete case [e.g. 4, 181. Ih\ \“,

(c)

(d)

(e)

(f) (g)

The ctrnno “‘J ““““b 1 ‘A_ methnrl AaIwC.I”U ic 10 hcxcm-4 “U”I.4 fin “11 the CL... ,,,=I-v

acnlmntinn u”Yu”‘y..v..

that Cl...l .an ..I ~~nll~rlvino Y”..“‘_J..‘~

Inc~ndn-mnraw~ \y’-“-.d WV.._..._,

utility function pre-exists and that the answers of the DM are consistent with it. Because of some difficulties involved in this assumption (e.g. in Step 4), the authors decided, in a somewhat arbitrary way, to discard the oldest informations. The compromise solutions provided by the method are always extreme points. This is a consequence of using Theorem 2.1 to characterize nondominated points. It is clear that good compromise solutions may correspond to nonextreme points. The acceptance of a tradeoff in Step 3 involves a constraint in the set of weighting vectors. However, using, in Step 4, a weighting vector which belongs’to the reduced set may result in a solution which does not correspond to the accepted tradeoff [see 191. More precisely, if an increase on a criterion is proposed and accepted in the tradeoff, it may happen that the next solution shows a decrease on this criterion. Even if this is not really an error, the DM may feel uncomfortable thinking he has no control over the procedure. Questions about tradeoffs are generally considered as difficult. This is why the authors proposed in a later description [ 161to replace tradeoffs by pairwise comparisons between the current solution and each of the adjacent efficient extreme solutions. However, questions about tradeoffs cannot be always avoided (e.g. because the DM may reject an adjacent point but accept the limited corresponding tradeoff). The resulting method is seemingly easier to use. The DM has to answer a lot of questions at each iteration. Many calculations have to be performed. Apart from the linear program, which is used at each iteration to generate the new proposal, most of the calculations are devoted to the identification of the efficient nonbasic variables.

7. THE VINCKE

METHOD

[20]

7.1. Description This method, proposed in the context of MOLP, performs an interactive sensitivity analysis using classical Simplex properties. Step 0. Identical to Step 0 of STEM. Step 1. Identical to Step 1 of STEM, with Z: = Z,. Step 2. zh is presented to the DM (a) if he is satisfied with zh, STOP; (b) otherwise, he is successively asked the following questions: l Which criterion do you want to improve? l Are you ready to accept a concession on a criterion? Which criterion? l Are you ready to relax a constraint? Which constraint? l Do you want to be more severe on a constraint? Which constraint? If the DM is interested in a perturbation, he is presented with the corresponding consequence (see Section 7.1.1 below); if he accepts these consequences, a new compromise zh+ ’ is determined (see Section 7.1.2 below). Let h = h + 1 and go to Step 2.

1230

DANIEL VANDERPOOEN

and

PHILIPPE VINCKE

7.1.1. Sensitivity analysis. The linear program giving the successive compromise solutions can be written as follows:

min p - $ p,cJx + i j-l

s.t.

(i=l,...,m)

p + l+Jx

(j = 1,. . . ,P>

x >, 0, 71---1-

T**=~*IL ‘J

I

I

L,’

,, y,

2nd Y&l_

C L,

>

D,x + t; = bi

zj -

where

Mivj

j=l

52~P UAW

- t; + vj = Ajzp*

c’x = 0 tf 2 0,

(j = 1,. . . ,P) t,’ > 0,

CllffiF;‘=ntlV “U”‘V’WLXC’,

CTV,Sll YlllUll

v, > 0, nnQ;ti,,P y”“‘L’.v

zj unrestricted, W,lllPC .U.LsIU)

An ‘.A,

.arc. orh;trcat;lw lavne unu uL”~rLul~~J u.&Lg” nnr;t;xro y”cilLL”.4

values, t) and tj are slack variables and uj are artificial variables. The perturbations proposed in Step 2 correspond to modifications of z:* or b. Noticing that the basic variables, at the beginning of the procedure, are tf , vj and zj, the consequences on the criteria of the proposed perturbations are simply read, in the Simplex tableau, in columns tf and v, and in lines zj (see shadow prices in classical linear programming). Moreover, it is easy to calculate the minimal value of the perturbation leading to a change of basis, i.e. the range of validity of the sensitivity analysis. 7.1.2. Determination of zh+‘. Considering the informations obtained from the sensitivity analysis, the DM may accept the proposed perturbation and choose its amplitude (if he refuses it, the procedure goes to the following question). If he does, zh+ ’ is determined by updating the last Simplex tableau. This is achieved by just changing the values of the basic variables or possibly by performing one dual Simplex iteration if the DM has chosen the maximum amp!itude of ?he perturbation in its range of vaiidity. 7.2. Comments The Vincke method is restricted to MOLP problems. This method follows a learning-oriented perspective. There is no irrevocability: coming back is always possible. No mathematical convergence is supported, which is natural in a learning-oriented approach. The possibility of modifying the set of alternatives under consideration is explicitly included in the method: the DM may change the constraints during the procedure. (e) This method performs an interactive sensitivity analysis. This amounts to saying that the DM must approximatively know his region of interest. More precisely, the first compromise solution z’ should not be too far from this region. (f) The method necessitates a constant dialogue with the DM. The informations required are mainly qualitative. (8) The calculation steps are particularly simple. After thep linear programs to be solved in order to determine the ideal point and another linear program to compute the first compromise solution, at most one dual Simplex iteration is required at each iteration. 8.

THE

REFERENCE

POINT

APPROACH-WIERZBICKI

[21,

221

8.1.

Description This approach is a general framework rather than a specific method. The DM is iteratively asked ,._ _._^_:r.. ,__I_,r:-- I_.._I_ 1__ __c___-__-A:-r-1 la_“& ^___^ ..:__+:,_, “1 _C+L_,_ ,,:,+, n..,-a,,,,*.ln+,A LLJSptXlly dbplrdLlW1 ICVCIS (“I rC,CKl,LX y”lILLS,. DI;bL sl~~,l”Xll,,ilL,“,ls LI,CZG 1.‘““‘LS 01G ~csLCU~CILGU using an achievement scalarizing function denoted by s. Step 0. Present some preliminary information such as the payoff matrix. Determine a weighting vector 1. Let h = 1. Step 1. Ask the DM to specify his aspiration levels: Zh=@,

. . . ,z;>

@EZ).

Some representative interactive multicriteria procedures

1231

Step 2. Let zh be the optimal solution for

s.t.

ZEZ,

if the DM is satisfied with zh then STOP, else let h = h + 1 and go to Step 1. 8.2. Comments (a) The reference point approach can be applied in any case, including problems involving an explicit list of alternatives. (b) This approach, based on aspiration levels, differs from the utility maximization framework. Intuitively, if the aspiration levels are not attainable, s generates a nondominated point closest to the desired levels. If the aspiration levels are attainable with a surplus, s generates a nondominated point, making the best use of this surplus. This is called by the author a quasi-satisficing framework. (c) In accordance with comment (b), the achievement scalarizing function is chosen in order to ensure that the compromise solutions correspond to nondominated points (see Theorems 2.4 and 2.5). The properties of these functions and some examples are indicated in Ref. [9]. (d) This approach follows a learning-oriented perspective. (e) We admit that aspiration levels can be rather easily specified by a DM at the beginning of the procedure and at some specific iterations when he wishes to reorient his exploration. However, it may be difficult for him to provide new aspiration levels at each iteration. Even if zh and Zh give indications in order to set Zh+‘, the relationship is not clear enough to stimulate a natural specification. (f) The basic procedure can be extended in many ways. Relevant dual information can be presented to the DM, several reference points can be used at each iteration. . . A direct extension, proposed by Wierzbicki, consists of using p additional perturbed reference points: zhj= ih + dh . eI (j = 1,. . . ,P>, where dh is the distance between the reference point ihand its corresponding nondominated point (see Step 2), and ej is thejth unit vector. At the expense of extra calculations using problem (14), these perturbed reference points give a description of the set of nondominated points in the neighborhood of Zh. 9. THE

STEUER

AND CHOO

METHOD

[23]

9.1. Description The procedure presents samples of progressively smaller subsets of nondominated points. These samples consist of P(~pj representative points; generated using an~~auemented ~._~~~_ ...u-~~-~~.-- weighted P----- Tchebychev norm, from which the DM is required to select one as his most preferred. Step 0. Calculate the ideal point z* and let z** = z*+~.Let/i’={rZ~IW~:I,~[0,1],~,~,=1}be the initial set of weighting vectors. Let h = 1. Step 1. -Randomly generate a large number (x 50 x p) of weighting vectors from Ah. -Filter this set to obtain a fixed number (2 x p is proposed) of representative weighting vectors. Step 2. -For each representative weighting vector 1, solve the associated augmented Tchebychev program: min (p -$, s.t.

(15)

Pj,>

p 2 S
(j=i,...,pj

where pj are sufficiently small positive values. -Filter the 2 x P resulting nondominated points to obtain P solutions.

DANIELVANDERPCOTEN and F’H~LIPPE VINCKE

I232

Step 3. Present the P compromise solutions and ask the DM to select his most preferred one. Let zh be the selected point. Step 4. (a) If h = t then STOP with zh as the most preferred solution (where t is a prespecified number of iterations), else (b). (b) -Let Ih be a weighting vector which generates zh by solving problem (1.5). Its components are given by

A:=+$&-J -Determine

(j= l,...,p).

the reduced set of weighting vectors: nh+‘=

where

{nEIWP:~jEII,,U,],CjlLJ= l> if E,; < rh/2

[O,rhl rl I, 1 - I L’]Y “jJ 1

[I

_?h

[flf

-

ifAh-.

II 7 ‘J

rh/2, 1: + rh/2]

in which rh is a prespecified “convergence (0 < r < 1). Let h =h +1 and go to Step 1.

,PL--r

..h i? ,L

I

otherwise,

factor” r raised to the hth power

9.2. Commenfs

(4 The Steuer and Choo method can be applied in any case, including problems involving an explicit list of alternatives.

(b) No assumption is made about any implicit utility function. The cc>

DM may change his mind, but only to a certain extent because of the monotonic reduction of the weighting vectors set performed in Step 4(b). Cd)The filtering procedure used in Steps 1 and 2 is described in Ref. [24]. This is an attractive way to present dispersed, and consequently representative, compromise solutions. (e> Many technical parameters (P, t, r), without any preferential meaning have to be prespecified. The authors propose “rules of thumb” to set these values. u-1 The preference information asked of the DM is qualitative and rather natural (Step 3). It may become difficult when the number of criteria increases. (g) The stopping rule in Step 4(a) is somewhat artificial. It is significant that, in a later version, Steuer [5] suggests letting the DM stop the procedure when he wishes, i.e. even when h < t or h > t. Actually, we believe that this parameter t should be ignored. (h) Many computations have to be performed at each iteration. Apart from the p initial optimizations to compute z*, 2 x P problems of type (15) are to be solved at each iteration (nlus \r.-.L 2 filtering procedures): This drawback directlv --- -- --, results from the desire to ensure dispersion and representativeness of the proposed solutions.

10. THE

KORHONEN

AND

LAAKSO

METHOD

[25]

IO. 1. Description

In this procedure, the DM is iteratively asked to specify aspiration levels from which a curve of nondominated points is derived. This curve is graphically presented to the DM who is required to indicate his most preferred solution. PI_._

31cp

n

v.

D~it3TiiiiX

Ziii ai%iirary

_!.-A _” _._> pomp z- ano

_ .-._!_l-A!.__ a weigniing

_.__*_vtxxor

1

A. Let

ii =

Step 1. Ask the DM to specify his aspiration levels (or reference point): Zh=(Y:,...,Z;)

(ZhE2).

Take d” = Zh - zh-’ as the new reference direction,

1.

Some representative

Step 2. Solve the parametric

interactive

multicriteria

procedures

1233

problem: min s(z, y, A) s.t.

(16)

y=z h-‘+tdh ZEZ.4,

t being increased from zero to infinity. In this step, each point y is projected onto the nondominated frontier of ZA (if s is properly chosen). This results in a curve of nondominated points. Step 3. -Graphically present the curve to the DM displaying a diagram similar to the one used in the Geoffrion et al. [I l] procedure. -Ask the DM to select his most preferred compromise solutions. Let zh be this point. Step 4. (a) If zh = Z” _ ’ then check some optimality conditions [see 251. If they are satisfied STOP with zh as the optimal solution, else a new reference direction d”+’ is identified by the optimality test, let h = h + 1 and go to Step 2. (b) If zh # z*-‘thenleth=h+landgotoStepl. 10.2. Comments

(a> The Korhonen

(b)

and Laakso method can be applied in any case, including problems involving an explicit list of alternatives. Optimality conditions are based on the assumption that the DM’s utility function is pseudo-concave. It should be noticed that this assumption is only used when the conditions are to be checked. This procedure is mainly based on a learning-oriented perspective. However, it also aims at strengthening the DM’s confidence in the final prescription. Any classical achievement scalarizing function with convenient nondominance properties could be chosen as s. Using the function discussed in Theorem 2.5, problem (16) becomes:

cc> (4

min(P-!, s.t.

p 2

Pjz,)

(17)

E,(z;- ’ + td; - z,)

(j=1,...,p)

ZEZ,, t being increased from zero to infinity. Preference informations required from the DM consist of aspiration levels. Possible (e> difficulties in using such informations were discussed in Section 8.2(e). (0 The choice of the weighting vector 3, is not indicated by the authors. It is clear that the nondominated curve resulting from problem (16) is greatly influenced by this choice (in some extreme cases, this curve could reduce to the current point). (g> The calculation steps widely depend on the problem. In the MOLP case, classical parametric programming can be applied. In other cases, several optimizations are to be performed (at each iteration) for specific values of t.

11.

THE

JACQUET-LAGREZE

er

al.

METHOD

[26]

11.1. Description In this method, a global utility function is interactively assessed taking into account alternatives. Then, it is applied to the original set in order to derive a prescription. Step 0. Determine

the payoff matrix,

the ideal point

z* and the nadir

point

n.

a subset of

1234

DANIELVANDERPCOTEN and PHILIPPE VINCKE

Step 1. Generate representative points using the following procedure. -Let i,, = l/(z: - n,) Vj and solve the weighted Tchebychev problem to generate a first point z”. -Considering q interior points z” - r/q(z’ - n) (Y = 1, . , q), solve for each point the p following problems (k = 1, . . ,p):

s.t.

z 3 z” - r/q(zO-

n)

ZEZA.

This results in q x p weakly nondominated points. Step 2. The DM is asked to estimate piecewise-linear marginal utility functions u, for each criterion g,. This is achieved through interactive cycles of: -Direct estimations of u, at some breakpoints using graphical displays. -Indirect estimations based on ordinal regression methods [see 271. In this case, the DM is first asked to rank the alternatives selected in Step 1. The general utility function is given by:

U(z) = Step 3. Determine

the final prescription

c y(q).

by extrapolating

U over Z,4:

max U(z) s.t.

ZEZ,.

11.2. Comments et al. method can be applied in any case, including problems involving (a) The Jacquet-Lagreze an explicit list of alternatives. of a local utility (b) In this method, the interaction is primarily directed towards the construction function (Step 2). Unlike the other interactive procedures, the DM is only presented with one proposal (Step 3) which is to be considered as the final prescription. No irrevocability is imposed while the assessment is not completed, i.e. the DM is free to (c) adjust his utility function. (d) The utility function which is assessed from a subset of alternatives is assumed to remain valid with respect to the original set. This assumption is all the stronger as the DM cannot react against the final prescription, However, if this is accepted, the DM may be interested in being provided with an analytical formulation of his preference structure. It can (e) The generation procedure in Step 1 aims at generating several dispersed alternatives. produce weakly nondominated points. Moreover, some nondominated points (like those whose coordinates are less than the corresponding coordinates of the nadir point) cannot be generated. In order to overcome these technical difficulties and even to ensure a better representation, filtering techniques could be applied. the graphical displays of marginal utility functions are usually (f) As to preference information, well-accepted by DMs. The direct specification of a ranking on a subset of alternatives (Step 2) is uneasy. However, a very interesting feature is that the DM can appreciate the impact of a modification of his ranking on the marginal utility functions and conversely. by the authors, and because of the specificity of this approach, only the (8) As indicated calculations which directly involve the DM are to be considered (Steps 0, 1 and 3 can be performed independently). Calculations in Step 2 are very simple. Each ordinal regression is achieved by solving a linear program. 12. THE

VANDERPOOTEN

METHOD

[28]

12.1. Description This method proposes pairwise comparisons between the current most preferred alternative and another one which represents a potential improvement. The response of the DM is analysed in order to derive a region of interest from which could emerge a new proposal.

Some representative interactive multicriteria procedures

1235

Step 0. -Let z** = z* + L and Iz’ > 0 be an arbitrary weighting vector (e.g. determined as in STEM). -Determine a first compromise solution z” arbitrarily or by solving: min s(z, z**, A’) s.t.

ZEZ,.

Step 1. Considering zh- ‘, ask the DM to indicate criteria which should be improved. J is the corresponding set of criterion indices. (a) If J = 0, STOP with zh- ’ as the final prescription. (b) Otherwise, determine the new region of interest: Zh,={zEZ;-‘:z,>zJh-‘,

Vjo.I>.

Let Zh=z*~+L 3 z *h being the ideal point relative to 2:. Or (if the DM prefers to react alternatively) Step I ‘. Considering zh- ‘, ask the DM to specify aspiration levels Zh.Let Z: = Z;-’ and J = 0. Step 2. Calculate a new compromise solution zh by solving: min s (z, Zh,Iz“) s.t.

ZEZ:.

Step 3. Present the DM with zh-’ and zh and ask him to indicate which is the most preferred. (a) If zh is the most preferred, determine a first approximation of the next region of interest: Zh, = {zEZA: z, 2 z;-‘,

VjeJ).

Construct a new weighting vector (preference direction) Ah+’ determined by I!+’

'I

=

1

z**

(j

= 1,.

. . ,p).

I

(b) If zh-’ 1s the most preferred, ask the DM to indicate which criteria (whose values have decreased) are the most responsible for this judgement. Let K be the corresponding set of criterion indices. A first approximation of the next region of interest is determined by =r, = {z&Z,; z, 2 z:,

‘dieEn”].

Let Zh=zh-’

and Ah+‘-;: . Step 4. Let h = h + 1 and go to Step 1 (or Step 1’). ,A _ 11.L. Comments

(a) The Vanderpooten

method can be applied in any case, including problems involving an explicit list of alternatives. @I This method has been designed in a learning-oriented perspective. However, by temporarily preserving information obtained from the last iteration (Step 3), it also aims at directing this learning through locally consistent proposals. Apart from the restriction indicated in (b), previous informations are omitted in order to cc> allow trial and error explorations and changes of mind. Consequently, as usual with learning-oriented procedures, new proposals may be in contradiction with previous ones (which is not necessarily unreasonable). (4 Although not indicated in the above description, the method allows the DM to specify minimal requirement (reservation) levels if he wishes to avoid subsequent contradictions. It should be noticed that such levels are introduced only when the DM experienced a contradiction, i.e. when his preferences are more structured.

1236

DANIEL VANDERPOOTEN and PHILIPPE VINCKE

function with convenient nondominance properties could be (e) Any classical scalarizing chosen as s. The function used in Theorem 2.5 [see problem (lo)] is suggested by the author. The required preference information is of a qualitative nature (except Step 1’ which gives (f ) an alternative way of providing information). Moreover, questions are designed in order to create a logical dialogue with the DM [Step l(b) and Step 3(b)]. At each iteration, calculation steps consist of the resolution of one optimization problem for (g) the generation of the proposal and possibly of p others for the construction of the local ideal point (Step 1).

13. GENERAL

COMMENTS

Although each procedure shows specific features, a basic distinction is to be made concerning the underlying approach. Two main conceptions, highly related to various perceptions of the decision process and the way to improve it, should be distinguished to classify interactive procedures [29]: -a -a

search-oriented learning-oriented

conception; conception.

Our chronological description highlights an evolution from search-oriented methods to learningoriented procedures. It should also be noticed that the most recent methods aim at including both aspects. We now briefly discuss some critical points in the choice (or the design) of an interactive procedure. A summary of the features of each presented method is also proposed in Table 1. 13.1. The calculation

steps

In many cases, it will be useless to introduce sophisticated calculation steps. Unlike other multicriteria approaches, no definitive aggregation has to be performed. Consequently, the choice of a scalarizing function (weighted sum, distance . . ,) should preferably lead to simple formulas and calculations; indeed, this choice is much less important than the quality of the dialogue with the DM. Moreover, in order to be operational and accepted by the DMs, such procedures must have reasonable computation times. However, the choice of a scalarizing function is not completely arbitrary. Some desirable technical requirements (e.g. nondominance properties) should also guide this choice. Theoretical works which indicate the scalarizing re.e. _._____._____._ ____nronerties r- _r_-_-__ of __ classical __________ ___.~_.______ p function __._______~ L-.p 91~ are helpful in this respect. Table I. Summary Prior assumptions (utility function)

STEM[IO]

No

of the features of each interactive

Applicability

No

multicriteria

Trial and error supported

Restriction of the feasible set

No

Yes

Yes

Convergence

procedure No. of questions to the DM

Few

restriction Geoffrion Roy [l41

ef a[. [I I]

Yes No

MOP NO

Yes

No

Yes

Yes

No

No

Very partially Yes

Yes

Yes

Many Rather few Many

No

No

Many

Yes

NO

No

Few

Partially

Yes

Yes

Few

Yes

No

Yes

Yes

No

No

Yes

No

No

Rather few Rather few Few

restriction Zion& and Wallenius [l5. 161 Vincke [20] Wierzbicki

(2 I, 221

Steuer and Choo [23] Korhonen and Laakso 1251 Jacquet-La&e P< al. (26) Vanderpooten [28]

Yes No No No Ye5 (partly used) No

MOLP (extensible) MOLP No restriction No restriction No restriction No restriction

No

No restriction

Difficulty of the questions Rather easy Difficult Rather easy Rather difficult Easy Rather easy Rather easy Rather easy Rather easy Easy

Computational burden

(MOLPcase) Small Small Rather small Large Very small

haii

Large Rather small Small Rather small

Some representative interactive multicriteria procedures

13.2.

Preference

information

and

1237

the dialogue

Preference information

is used by the method to present the DM with potentially improved compromise solutions. From this viewpoint, more information is considered as better. However, it should be remembered that the main purpose is to support the DM and not the procedure. Consequently, qualitative questions will be preferred to quantitative ones. First, in many cases DMs are unable to provide quantitative informations. Second, it is actually useless to require too much precision in the answers as soon as it is accepted that the DM might change his mind. Furthermore, the DM must be prepared to answer a series of similar questions at each iteration. If the cognitive strain is too high for each question, answers, Such anchoring effects should be avoided, Finally, it is important that the DM understands

he may be tempted to maintain his previous above all in a learning-oriented perspective. the reasons why questions arise. A dialogue

should preferably be constructed so that questions logically follow from DM’s answers. In this case, the procedure will be perceived as exhibiting an intelligent behaviour. 13.3. The support to the DM In relation with the former point, the support to the DM is one of the most important aspects of an interactive method. The role of such a method is not to decide for the DM but to enlighten him on his problem: what is possible, what are the consequences of such a choice, how to improve this aspect. . . The procedure should also be able to bring information to the DM.

First of all, proposals must be presented under a convenient form. Graphical displays are valuable in this respect. Additional informations resulting from, for example, sensitivity analysis, are also to be developed. 13.4. Convergence

of

interactiue procedures

The problem of convergence is essential because it is at the origin of a lot of the restrictions and inconveniences of many methods, as for instance: -the -the -the

irrevocability of the decisions; assumption that the DM’s answers are always consistent with a utility function; consideration of only a subset of feasible solutions (extreme points).

The purpose of an interactive method is essentially to find a “satisfactory compromise solution”. The concept of optimality has no validity as soon as it is accepted that learning of preferences should be supported by an interactive procedure: a solution could be rejected at the beginning and finally accepted because of the evolution of the DM’s preference structure. Consequently, the procedure should not be stopped because of any convergence test but only if the DM is satisfied with a solution or when he has the feeling he has enough informations about his problem. Although mathematical convergence is of no interest, an interactive procedure should also aim at guiding the DM’s search for improved solutions. It should be clear that this improvement only refers to the current state of the DM’s preference structure. Finally, we believe that the future of interactive procedure is in trying to reconcile search and learning.

REFERENCES C. L. Hwang and A. S. M. Masud, Multiple Objective Decision Making-Methods and Applications; LNEMS 164. Springer-Verlag. Berlin (1979). A. Goicoechea, D. R. Hansen and L. Duckstein, Multiobjective Decision Analysis with Engineering and Business Applications. Wiley, New York (1982).

5. 6. 7. 8

D. J. White, A selection of multi-objective interactive programming methods. In Multi-objective Decision Makina (Edited by S. French, R. Hartley, L. C. Thomas and 6. J.-White), pp. 99-126. Academic Press, London (1983). ” J. Teghem Jr and P. L. Kunsch, Interactive methods for multi-objective integer linear programming. In Large-scale Modelling and Interactive Decision Analysis, Proceedings Eisenach, G.D.R., 1985 (Edited by G. Fandel, M. Grauer, A. Kurzhanski and A. P. Wierzbicki); LNEMS 273, pp. 75-87. Springer-Verlag, Berlin (1986). R. E. Steuer, Mulriple Criteria Optimization: Theory, Computation and Application. Wiley, New York (1986). B. Roy, MPthodologie MufricritPre d’dide ri la DPcision. Economica, Paris (1985). Y. Sawaragi, H. Nakayama and T. Tanino, Theory of Multiobjective Optimization. Academic Press, New York (1985). V. Chankong and Y. Y. Haimes, Multiobjective Decision Making: Theory and Methodology. North-Holland, New York (1983).

DANIEL

1238

VANDERFQOTEN and PHILIPPEVINCKE

A. P. Wierzbicki, On the completeness and constructiveness of parametric characterizations to vector optimization problems. OR Spektrum 8, 73-87 (1986). 10. R. Benayoun, J. de Montgolfier, J. Tergny and 0. Larichev, Linear programming with multiple objective functions: STEP method (STEM). Math Program. 1, 366-375 (1971). 11. A. M. Geoffrion, J. S. Dyer and A. Feinberg, An interactive approach for multi-criterion optimization, with an application to the operation of an academic department. Mgmt Sci. 19(4), 357-368 (1972). 12. J. S. Dyer, A time-sharing computer program for the solution of the multiple criteria problem. Mgmt Sci. 19(12), 1379-1383 (1973). 13. S. Sadagopan and A. Ravindran, Interactive algorithms for multiple criteria nonlinear programming problems. Eur. J. opl Res. 25(2), 247-257 (1986). 14. B. Roy, From optimization to multicriteria decision aid: three main operatinal attitudes. In MCDM, Proceedings Jouy-en-Josas, France, 1975 (Edited by H. Thiriez and S. Zionts); LNEMS 130, pp. l-32. Springer-Verlag, Berlin (1976). 15. S. Zionts and J. Wallenius, An interactive programming method for solving the multiple criteria problem. Mgmr Sci. 22(6), 652-663 (1976). 16. S. Zionts and J. Wallenius, An interactive multiple objective linear programming method for a class of underlying nonlinear utility functions. Mgmt Sci. 29(5), 519-529 (1983). 17. S. Zionts and J. Wallenius, Identifying efficient vectors: some theory and computational results. Ops Res. 28(3), 785-793 (1980). 18. S. Zionts, A report on a project on multiple criteria decision making. Working paper No. 663, SUNY Buffalo, N.Y. (1985). 19. S. de Samblanckx, P. Depraetere and H. Muller, Critical considerations concerning the multicriteria analysis by the method of Zionts and Wallenius. Eur. J. opl Res. 10(l), 70-76 (1982). 20. Ph. Vincke, Une mtthode interactive en programmation lineaire a plusieurs fonctions economiques. Revue jr. autom. 9.

Inf. Rech. opt+. 10(6), 5-20 (1976).

21. A. P. Wierzbicki, The use of reference objectives in multiobjective optimization. In MCDM Theory and Application, Proceedings HagenlKiinigswinter, F.R.G., 1979 (Edited by G. Fandel and T. Gal); LNEMS 177. pp. 468486. SpringerVerlag, Berlin (1980). 22. A. P. Wierzbicki, A mathematical basis for satisficing decision making. Math1 Modelling 3, 391-405 (1982). 23. R. E. Steuer and E. U. Choo, An interactive weighted Tchebycheff procedure for multiple objective programming. Math1 Program.

26, 326-344

(1983).

24. R. E. Steuer and F. W. Harris, Intra-set point generation and filtering in decision and criterion space. Computers

Ops

Res. 7, 41-53 (1980).

25. P. Korhonen and J. Laakso, A visual interactive method for solving the multiple criteria problem. Eur. J. opl Res. 24(2), 277-287 (1986). 26. E. Jacquet-Lagreze, R. Meziani and R. Slowinski, MOLP with an interactive assessment of a piecewise utility function. Eur. J. opl Res. 31(3), 350-357 (1987). 27. E. Jacquet-Lagreze and J. Siskos, Assessing a set of additive utility functions for multicriteria decision making, the UTA method. Eur. J. opl Res. 10(2), 151-164 (1982). 28. D. Vanderpooten, A multicriteria interactive procedure supporting a directed learning of preferences. Presented at the EURO IX-TIMS XXVIII Co& Paris (1988). 29. D. Vanderpooten, The interactive approach in MCDA: a technical framework and some basic conceptions. Mathl Comput.

Model&g

12, 1213-1220 (1989).