Solving Discrete MultiAttribute problems under risk based on an approximation

SOLVING DISCRETE MULTIATTRIBUTE PROBLEMS UNDER RISK BASED ON AN APPROXIMATION*

SIXTO RIOS-INSUA, ALFONSO MATEOS and ANTONIO JIMENEZt

A b s t r a c t . W e consider the multiattribute decision making problem under risk with imprecise information on the decision maker’s preferences, modelled by means of a vector utility function. W e propose an interactive decision aid ap-proach, which uses an idea of approximation to the utility efficient set and qual-itative comparisons for the decision maker, to overcome the possible difficulty in generate i t . An application to university selection illustrates the procedure.

1. INTRODUCTION

W e consider the multiattribute decision making problem under risk with partial information on the DM’s (decision maker) preferences, in the sense that his/her preferences are modelled by means of a vector utility function, Roberts (1972, 1979) and Rietveld (1980) instead of a scalar one, Fishburn (1976) and Keeney and Raiffa (1993). This vector utility function represents imprecise preferences and can be seen as a way for lack of precision of the true but unknown scalar utility function, which is usually defined on the attributes associated with the lowest-level objectives of a hierarchy, which often exhibit well defined multi-objective problems. In other words, if no independence assumption is accepted for the DM to reach a more structured form in the utility function (additive, multiplicative, ...), he/she has to cope with such a vector function in the decision making process.

In this framework, the utility efficient set, R ´ıos-Insua and Mateos (1997), plays a fundamental role because of its property: from any strategy in this set, it is not possible to feasibly move in order to increase one component without necessarily decreasing at least one of the remainder. However, the generation of such set can be involved, specially in continuous problems. In

Work supported by the Madrid Regional Government project C A M 07T/0027/2000 and the Ministry of Science and Technology projects D P I 2001-3731 and B F M 2002-11282-E.

†Dept. of Artificial Intelligence, School of Computer Science, Madrid Technical Uni­

the case of discrete problems i t is easier, but we note that the set of utility efficient solutions might be far too big in most practical situations. Hence, the generation of the utility efficient set is not usually considered the resolution of the multiattribute problem because this set may have many elements and is not generally totally ordered. Thus, there is a need for more intelligent strategies to generate a set of representative efficient solutions or an approximation that gives a fair representation of the whole set. Although there are several methods to aid DMs to generate the efficient set or a representative subset, see, e.g., Gal (1972), Goicoechea et al. (1982), Yu (1985), Chankong and Haimes (1983), Steuer (1986), Vincke (1992), Gal (1995) and Gal et al. (1999), there is not a definitive solution to this problem, particularly for problems under risk.

I n this paper we consider a utility-based procedure, see e.g., von Neumann and Morgenstern (1947) or Keeney and Raiffa (1993), which uses an approx­ imation concept, Mateos and R´ıos-Insua (1996), which intends, on one hand, to facilitate the generation process of a representative set of the whole utility efficient set and, on the other, its interactive reduction to reach a final strategy. Throughout the paper we shall employ the following notation: For two scalars a and b, a ^ b denotes a > b or a = b. For two vectors x, y e Wn, x ^ y denotes Xi ^ yi for i = 1 , . . . , n, and x > y denotes x ^ y but x = y . Our framework is the multiattribute decision making problem under risk with a finite set Z of outcomes z\ i = 1 , . . . , s, characterized by a number of at­ tributes zh...,zN and a set Vz of simple probability distributions over Z, w i t h elements p,q,..., called strategies or (risky) prospects, where for ex­ ample, p = (pi,zl;P2,z2;...;ps,zs), Pj ^ 0 for all j and Y!j=iVi = 1- We assume partial information i n the sense that we can assess a vector utility function u : Z -»• Rm, where u = ( u i , . . . , um) represents a preference order y (asymmetric and transitive) on Vz, leading to a dominance principle defined by

pyq « • E(u,p) >E(u,q),

where E(u,p) = (E (ullP),..., E (um,p)) is the expected utility vector with s

E (Ui,p) =

for i = 1 , . . . , m.

Throughout the paper, we will assume that the real-valued functions m,

% = 1 , . . . , m, on Z are continuous, monotonous and bounded. This framework

leads to the vector maximum problem over Vz, defined

(1) max{E(u,p):peVz

that this definition extends the one for problems under certainty, i.e., if Vz

consists of sure prospects pz = (1, z), then £(VZ, u) = £(Z, u), where £(Z, u)

would be the efficient set for Z given u.

I t is clear that if £(VZ, u) has a unique element p, it would be the

compro-mise strategy for the D M . However, as mentioned above, this is not the case

for most real problems, as £(VZ, u) may have a lot of strategies and its

gener-ation can be difficult. Thus, we shall consider an approximgener-ation to the utility efficient set that will be easier to generate and that could be interactively reduced in the process of reaching a final strategy.

The paper includes five more sections. In Section 2 we introduce some the-ory and concepts related to the approximation set. I n Section 3, we propose a procedure to reduce the utility efficient set for the case of two utility compo-nents, considering a linear preference structure. In Section 4, we extend the procedure to any number of utility components. In Section 5, we present an application of the procedures and, finally, in Section 6, some conclusions are provided.

2. APPROXIMATION OF THE UTILITY EFFICIENT SET

We consider the decision making problem where the DM’s preferences are

modelled by means of a vector utility function u : Z -»• Rm and the D M

can reveal more information on his preferences through an interactive process, obtaining a more precise vector utility function, as we next shall see.

Let k1 = (k\,..., * 4 ) , . . . , kr = ( f c j , . . . , krm) be the extreme points of the

polyhedral of possible weights or scaling constants, which we shall denote by

the matrix M = ( k1, . . . , kr)*. Let us define the information set IM associated

to M as

IM = { k e l m: k £ atf, with a = (an,..., ar) € Sr\,

w i t h Sr the simplex on W. Now, assume that DM’s preferences satisfy the

conditions to agree w i t h a (scalar) utility function as a linear function of the components in u, see e.g., French (1986) and Stewart (1996). We define

uM = M u = ( k1u , . . . , kru ) (each component denotes the inner product of two

vectors, and we understand that the first one is a row vector and the second one a column one) as the vector utility function associated to the information set

IM- Thus, IM represents the DM’s information on the weights in uM. Hence,

uM is an imprecise vector utility function. When M = M0 (the identity

matrix of order r) there is null information about the scaling constants and

we have the original vector utility function u = uM o. On the other hand, if

k1 = . . . = kr = k, we have the scalar utility function uM = k u , w i t h complete

From the vector function uM, we can also consider the corresponding utility

efficient set, denoted £(VZ, uM) . We have the following

P r o p o s i t i o n 1. Given a vector utility function u : Z -»• Rm and an infor­ mation set IM with M = ( k1, . . . , kr)*, then for all peVz

(2) E(uM,p) = (klE(u,p),..., krE(u,p)) .

The proof follows immediately.

Now we shall provide a practical way to find E(VL,P>) by means of an al­ gorithm. Let (h,..., im) be a permutation of the set of indexes ( 1 , . . . , m) of components in u. We follow the next algorithm to solve the problem P^...^:

Step 1. Let Yi=Vz.

Step 2. From j = 1 to m, solve

(3) ut= max E(iH.,p)

Yj+1 = {p € Yj : E(m^j)) = u*.} Step 3. End.

Then we obtain the set Ym+1, which usually has a single strategy. Thus, to find E(u,pj) means to solve problem Pj,j+i,...,m,i,...,j-i, which provides a solution pi. I n analogous way, we can find u(zj).

Now, we provide the approximation to the utility efficient set for the case of a function with two components u =(ui,u2). Given u defined over Z, the

approximation set to £(Vz,u) is defined as (4)

A(Vz,u) = lpeVz:E(Ul,p) ^E(ui,p 2) a n d E ( u2, p ) ^E(u2,pl)\,

where the strategies p1, p2 are obtained by solving P12 and P2i , respectively.

As a particular case, we have the definition under certainty, Mateos and R´ıos-Insua (1997a, 1997b). Some desirable properties, that fulfils the approximation set (4), are:

1) Let u : Z -»• M2 be a vector utility function and IM an information set; then

(5) £(Vz,uM)CA(Vz,uM).

2) Monotonicity. Given a vector utility function u : Z -»• M2 and two

information sets, /M l and IM2, such that /M 2 C int ( /M l) , then

(6) A(Vz,u M*)cA(Vz,u Mi).

3) Convergence. Let u : Z -»• M2 be a vector utility function and {IMn} a decreasing sequence of information sets such that IMl C i n t ( /M o) and

U M „ } I { q } , when n ->• oo. If p° € ^ ( Pz, uM« ) , for every n, then

The above definition and properties cannot be extended in immediate way to m-dimensional utilities, i.e. m ^ 3, as we show in next example.

Con-sider the set of outcomes Z = {zl ,z2 ,z3 ,zA] and the vector utility function

u : Z ->• R3, such that u ^1) = (1,2,3), u(z2) = (3,2,1), u(z3) = (1,3,1)

and u(z4) = (3,0,3). Assume that Vz has four strategies p1 = ( ^ z1; ±,z3) ,

p2 = ( § , z2; ± , z3) , p3 = (l,zl;l,z2;l,z3) and p4 = ( ± , z3; ±,z4) . The

ap-proximation set is A(Vz,u) = {pl,p2,p:i} , but note that it does not contain

to £(Vz,u) = {p\p2,p\p4}, which is a fundamental property.

To extend (4) to m-dimensional utilities, let us first consider the

two-dimensional vector utility u(-) = (ui (•) ,u2 (•)) and the nondominated set

by p\ defined as

J V V ) ={v e Vz : E(Uj,p) > E(ujrf), j =i, j = 1,2}

(J {p € V

: E(

,

p) =

,p

), Vj }

for i = 1,2. Thus, the approximation set (4) can be rewritten as follows:

(7) A(Vz,u)=M(p1)nM(p2).

I n analogous way, we can stretch out this idea to the general case of a

vec-tor utility function u(-) = (m (•),••• ,um (•)), with m ^ 3, considering the

nondominated set

J V V ) = { P 6 P z : 3 j = 1 , . . . , m, j = i, with S ( u j , p ) > ^ ( u j , ^ ) }

| J { p € Pz : S ( « j , p ) = E(ujrf), Vj = 1 , . . . , m }

for each i = 1 , . . . , m, being p* the solution to problem Pi,i+i,...,m,i,...,i-i. Thus,

we define the approximation set to the utility efficient set £(VZ, u) as

m

(8) . 4 ( ^ , 1 1 ) = p | J V V ) .

Properties 1. to 3. are also verified in this general case.

3. AN ALGORITHM TO REDUCE THE APPROXIMATION SET

I n this section we provide an algorithm designed to decision aid in dis-crete multiattribute decision making problems where a two-dimensional vec-tor utility function has been assessed. The basic idea is as follows: Assume

that we have two utility efficient strategies pi and p>, with expected

utili-ties vectors E ( u , ^ ) = (u[,i4) and E (u,p>) = (u{,u32), respectively.

Sup-pose that u32 > u\ and the D M prefers E (\i,f) than E(u,p>), then the

D M would like to have a better utility for u{. Among the improvement

E(u2,-) and we are maximizing. In this way, and from the assumption on

continuity of the vector utility function, we can improve E (u,p>) by taking

(u{*,ui*) = (u{,ui) + /3a, /3 > 0, such that (y?* ,v?2*) - £ ( u / ) (~ means

indifference). Then, from the monotonicity of the utility, we only have to

con-sider the strategies peA = A(VZ, u) that verifies E (uup) > u{*, i.e., the set

Vz is reduced to VZt = {p e A : E(uhp) > u{*}. Similarly, if u{ > u[, we

take a = (0,1) as the improvement direction, being the most preferred

strat-egy the one that verifies E(u2,p) > u32*. In this case, the set Vz is reduced to

VZt = {peA: E(u2,p) >ui*}.

Now, we show how to obtain the two utility efficient strategies which are nec-essary on each run of the procedure. Let V\ be the set of strategies obtained

in iteration h. The process find E ( u , ^1) and find E (u,p2) is conducted by

solving P12 and P2i. I n each iteration h, we compute p1 and p2 to obtain

the approximation set (7). We compare their expected utilities vectors and,

consequently, a new set V%t is obtained. I n the algorithm, we call pj(pk) to

the best (worst) current strategy between p1 and p2. We consider a set N

and we add to this set the strategies indifferent to pp. On the other hand, to

find {u\\uf) means to calculate {u\\uf) = E (u,pk) +/3a, /3 e R+, such

that (u\\uf) - (u{,u32). We will use the auxiliary variable x to keep the

superscript of the best strategy in the previous iteration, to know when there is a movement to a better strategy.

The algorithm is:

Step 0. Let VZt =Vz,h = 0, j = 1 and N = 0. Step 1. Find E (u,pl) and E(u,p2).

Step 2. LetV% = A(V%t,u).

Step 3. If E (u,pl) = E\u,p2), the most preferred strategies are

N = N U {p € Vhz : E (u,p) = E (u,pl) }

and stop. Otherwise, go to next step.

Step 4. If E ( u ^1) - E (u,p2) , let (uf,uf) = E (u,p2) (note that we can

also take (u\\u\*) = E(u,p1)) and go to step 9. Otherwise, go to next

step.

Step 5. Let x = j.

Step 6. If E (u,pl) y E (u,p2), then j = 1 (the best)

- If z = 1 , t h e n i V = 0

- Find {u\\uf) and go to step 9 Otherwise, go to next step.

- I f x=2, then N = 0

- Find (u\*,u\*) and go to next step.

Step 8.LetN = N(j{peV%:E (u,p) = (u\\u\*) } , determine

Vh+l = \p^Vhz: E(u2,p) > u\*\

and find E ( u ^1) in V^1. Let h = h +1 and go to step 2.

Step 9. Let N = N U {p € V\ : E (u,p) = (uf,uf)} , determine

Vzt1 = \v^Vhz: E(ui,p) > uf\

and find E(u,p2) in V%+1. Leth = h + 1 and go to step 2.

The procedure find {uf ,uf) may be conducted by means of questions to the D M about two appropriate strategies to converge to an indifferent point.

Graphically, this means that the point E(u,pk) would move in the

improve-ment direction until the intersection with the isocurve through E(VL,P>) in a

point (u\*,uf).

Therefore, we have a search-oriented algorithm that will interactively re-duce the set of strategies because of the irrevocable DM’s responses. The method is not very demanding because the D M only has t o answer t o qual-itative questions he is faced. On the other hand, the computational effort is small. Observe that if we do not consider the procedures find (u\*,v%*)

and calculate V^1, we shall have for each iteration an optimization problem

(maximize or minimize), except in the first iteration where we have two, one for each component of the expected utility vector. Note that in the method we take the approximation set to the utility efficient set instead of the whole set

Vz, this may considerably reduce the number of strategies to be investigated.

Furthermore, if we assume that the D M fulfils the assumptions that lead to have linear preference structure, French (1986), then the above algorithm may be rewritten in another simple way. Before providing the algorithm, we shall

explain the process determine the information set IMh+1.

Let IMo = R+, if the D M reveals that E (uMh,pl) >z E (uM\p2), h>0

( £ means more preferred or indifferent to), the information set will be IMh+1

w i t h Mfc+i = ( k ^ + ^ k2^1) ) * where k ^ + ^ k2^1) are the generators of

the polyhedral cone {Ak : A > 0, k e K}, where K is the set of elements k that verify

kE(uM\pl)^kE(uM\p2)

k € Iuh •

I n case that E {uMh,p2) >z E ( u ^ p1) , then elements k must fulfil

kE(uM\p2)SkE(uM\pl)

The algorithm is:

Step 0. Let VZt = Vz, h = 0, j = 1 and N = 0.

Step 1. Find E(uMh,pl) and ^ ( u ^ p2 ) on VhZt.

Step 2. Determine Vz = A{V^t,uMh).

Step 3. If E (uM\pl) = E (u^\p2), stop. We have an ideal point and the

solutions are N = NU {p € Vhz : E(uM*,p) = E(uM\Pl)}.

Step 4. Let x=j.

Step 5. If E {\iMh,pl) ~ £ ( uM^ , p2) , then N = {p2} U N, {uf,uf) = E (uMh,p2) and go to step 10.

Step 6. If E(uM\pl) y E(uM\p2), then j = 1 and k = 2. Otherwise, j = 2 and A; = 1.

Step 7. If j + x, then JV = 0.

Step 8. Let P ^1 = V\ and fc = fc + 1. Step 9. Determine IMh and go to step 1. Step 10. Calculate

^

= (p £ Pz : E(u^

,p) > uf\

z

and find £ (uM f e,p2) on V^1. Let JM h + 1 = / M , , , h = h + 1 and go to

step 2.

Note that this algorithm is used when the true (but unknown) scalar utility function can be written as a linear combination of the components in u, which is a particular case of the former one.

4. THE GENERAL CASE

I n the general case of a vector utility function u(-) w i t h m ^ 3 components, the set of improvement directions will be included in

(9) D = Uau ..., am) € Rm : on ^ 0, £ a* = l ) .

If one strategy pj has their best values in the expected utility vector E (u,p>)

for coordinates h,..., ir, we take as improvement directions the subset

Dj = {a € D : ah = 0 , . . . , air = 0} .

For example, we can take as improvement direction the vector a € Dj such

that (n = l / ( m -r),i^ii,...,ir,or simply, ask D M to suggest improvement

directions in Dj, to conduct the process find (u\*,..., u*j) in the general case.

The procedure find E (u,pj) is led by solving problem Pj,j+i,...,m

,i,...,j-i-Procedure modify Vhz due to (u\*,..., vk^) is analogous to the case m = 2

The best solution in the current iteration will be E u,p* and Y will remain as the set of indifferent solutions to the current best solution.

Now, let I be the set that contains the superscripts of the strategies that

provides the same expected utility than pl. The algorithm is as follows:

Step 0. Let VZt = Vz, h = 0, t = 1, Y = 0 and 1 = 9. Step 1. For all) e {1,... ,m} - I, find E (u,pl) in VhZt. Step 2. Determine V\ = A (V%t, u ) .

Step 3. If all E(u,pl) were equals, we would have the most preferred

strate-gies in

YuLeV^:E (u,p) = E(u,p l) \

and stop. Otherwise, let x = t and go to next step.

Step 4. Calculate the minimum t that fulfils E (u,p*) y E (u,pl), for all I

and let / = 0.

Step 5. If E (u,p*) = E (u,jf), then Y = 0. Otherwise, go to next step. Step 6. From j = 1 to m :

- Find £ ( u , p > ) ;

- If £ (Uy ) = £ (Uy ) , then / = / U { j } ;

- If E(u,p>) ~JB ( u , p * ) , then

F = F u { j ) e ^ : S ( u , p ) = £ (u,p>)}

and modify V^ due to (u{*, ...,u%)=E (u,p>);

- Otherwise, find (u\*,..., u*j) let

F = r u ! f ) G ^ : E(u,p) = (u\*,...,u^)\

and modify V\ due to{u\\ ...,u^);

- End.

Step 7. Let P ^1 =V%t,h = h + 1, and go to step 1.

Clearly, this algorithm has same advantages than the one for two compo-nents. As above, we can also consider a general algorithm for the case of linear preference structure.

5. AN APPLICATION TO UNIVERSITY SELECTION

I n this section we present an example about university selection to illustrate the algorithms in the case of a vector utility function with two components. However, we want to emphasize that the more components in the vector utility function the more useful is the algorithm.

hierarchy tree. Thus, each university is characterized by a four dimensional

attribute vector z = (21, z2, z3, 24), where

21 - registration fee (x104 Spanish ptas),

z2 - distance from university to family residence (x10 km), z3 - distance from university to accommodation (in km), z4 - accommodation cost ( x 5 • 104 ptas).

However, the registration fee as well as the accommodation cost, depend on the Consumer Price Index (CPI) which may increase or remain. The experts assert that the CPI will remain with probability of .7 and will increase w i t h probability .3.

We have grouped zx and 24 (costs) and, on the other hand, z2 and z3

(dis-tance). We want to minimize them all. The student reveals that it is more important to save money in accommodation than in registration, because he knows that registration fee is proportional to education quality. He considers six times more important a reduction in accommodation cost than in regis-tration fee. Moreover, he considers eight times more important to reduce the distance from university to accommodation than to family residence. Such information is modelled by means of a vector utility function, where the

com-ponents are ui(z) = zx + 624 and u'2(z) = z2 + 823. We have to minimize

E (u,p>) over Vz = {p1,... ,p26} , where we identify, for instance, strategy p1

w i t h lottery

/ .7 .3 \ V (10,10,1,25) (11,10,1,27) )

and so on (see Table 1). We can transform both components of u' to have u ( z ) = - u ' ( z ) , and thus a maximization problem. Furthermore, if we nor-malize the weights, the vector utility function will be

gZ3) . U(z) = " ^ 1 - ^ 4 , - ^ 2

Table 1 presents twenty six strategies p, one of them must be selected by the D M . Such table also presents the respective consequences z = (21,22,23,24)

and utility vectors u = (uhu2). Table 2 shows the expected utility vectors

E (u,pj) for each strategy.

I n step 1 we obtain by applying the first algorithm to our selection problem that (see Table 2))

E (u,pu) = (-.399, -.320) and E (u,pl) = (-.842, -.052).

and in step 2, the approximation set

V° = A(V° u) = r>1 r? »5 »12 »13 »14 »15 »16 »1 7 »18 »19 »20 r>2l\

which contains 13 strategies.

Table 1 : The set of strategies with their consequences and utility vectors. pi P1 p2 P3 P4 P5 P& P7 P8 P9 10 p 11 p 12 p 13 p (1) Remains Prob. = .7 (10,10,1,25) ( - . 8 3 0 , - . 0 5 2 )

(11,50,5,15) ( - . 5 0 4 , - . 2 6 1 ) (12,100,3,18) ( - . 6 0 3 , - . 4 9 4 ) (13,120,2,16) ( - . 5 3 9 , - . 5 8 7 ) (14,60,4,14) ( - . 4 7 4 , - . 3 0 5 )

(15,40,6,30) ( - 1 . 0 , - . 2 1 6 ) (16,200,9,25) ( - . 8 3 7 , - 1 . 0 ) (17,154,1,28) ( - . 9 3 7 , - . 7 4 7 ) (18,160,5,26) ( - . 8 7 2 , - . 7 9 1 ) (19,200,8,24) ( - . 8 0 8 , - . 9 9 6 ) (20,145,7,17) ( - . 5 7 9 , - . 7 2 7 ) (21,45,5,11) ( - . 3 8 4 , - . 2 3 6 )

(22,39,4,15) ( - . 5 1 6 , - . 2 0 3 )

pi P P2 P3 P4 P6 P6 P7 PS P9 10 p 11 p 12 p 13 p

E (u,pl) ~ E

17 —— 1 0

Increases Prob. = .3 (11,10,1,27) ( - . 8 6 8 , - . 0 5 2 )

(13,50,5,17) ( - . 5 5 3 , - . 2 6 1 ) (17,100,3,21) ( - . 6 8 4 , - . 4 9 4 ) (15,120,2,17) ( - . 5 3 5 , - . 5 8 7 ) (15,60,4,19) ( - . 6 1 8 , - . 3 0 5 )

(16,40,6,31) ( - 1 . 0 , - . 2 1 6 ) (18,200,9,27) ( - . 8 7 5 , - 1 . 0 ) (20,154,1,30) ( - . 9 7 2 , - . 7 4 7 ) (21,160,5,27) ( - . 8 7 8 , - . 7 9 1 ) (25,200,8,25) ( - . 8 1 9 , - . 9 9 6 ) (21,145,7,25) ( - . 8 1 5 - . 7 2 7 ) (22,45,5,16) ( - . 5 3 1 , - . 2 3 6 )

(25,39,4,20) ( - . 6 6 1 , - . 2 0 3 )

p> 14 p 15 p 16 p 17 p 18 p 19 p 20 p 21 p 22 p 23 p 24 p 25 p 26 p Remains Prob. = .7 (23,64,3,10) ( - . 3 5 3 , - . 3 2 0 )

(24,15,2,12) ( - . 4 2 0 , - . 0 8 0 )

(25,18,1,16) ( - . 5 5 2 , - . 0 9 0 )

(26,24,6,18) ( - . 6 1 8 , - . 1 3 9 )

(27,29,4,19) ( - . 6 5 2 , - . 1 5 5 )

(28,33,5,17) ( - . 5 8 8 , - . 1 7 8 )

(29,34,3,21) ( - . 7 2 0 , - . 1 7 5 )

(30,42,2,23) ( - . 7 8 7 , - . 2 1 0 )

(31,157,1,25) ( - . 8 5 4 , - . 7 6 2 ) (32,199,5,28) ( - . 9 5 3 , - . 9 8 0 ) (33,165,7,26) ( - . 8 8 8 , - . 8 2 3 ) (34,187,9,23) ( - . 7 9 1 , - . 9 3 7 ) (35,145,8,11) ( - . 3 9 9 , - . 7 3 1 )

Table 2: Expected utility vectors.

Eu,p>

( - . 8 4 2 , - . 0 5 2 ) ( - . 5 1 9 , - . 2 6 1 ) ( - . 6 2 8 , - . 4 9 4 ) ( - . 5 4 4 , - . 5 8 7 ) ( - . 5 1 8 , - . 3 0 5 ) ( - 1 . 0 , - . 2 1 6 ) ( - . 8 4 9 , - 1 . 0 ) ( - . 9 4 8 , - . 7 4 7 ) ( - . 8 7 4 , - . 7 9 1 ) ( - . 8 1 1 , - . 9 9 6 ) ( - . 6 5 2 , - . 7 2 7 ) ( - . 4 2 9 , - . 2 3 6 ) ( - . 5 6 0 , - . 2 0 3 )

p> 14 p 15 p 16 p ₁₇ p 18 p 19 p 20 p 21 p ₂₂ p 23 p 24 p 25 p 26 p

(u,pu) (Step 4). Then

, P W V

,

£ u , pJ

( - . 3 9 9 , - . 3 2 0 ) ( - . 4 2 5 , - . 0 8 0 ) ( - . 5 4 7 , - . 0 9 0 ) ( - . 6 3 3 , - . 1 3 9 ) ( - . 6 5 7 , - . 1 5 5 ) ( - . 5 8 2 , - . 1 7 8 ) ( - . 7 2 3 , - . 1 7 5 ) ( - . 7 8 9 , - . 2 1 0 ) ( - . 8 4 5 , - . 7 6 2 ) ( - . 9 6 4 , - . 9 2 6 ) ( - . 9 1 9 , - . 8 2 3 ) ( - . 8 0 3 , - . 9 3 7 ) ( - . 4 0 5 , - . 7 3 1 )

, we obtain

Increases Prob. = .3 (25,64,3,15) ( - . 5 0 2 , - . 3 2 0 )

(25,15,2,13) ( - . 4 3 9 , - . 0 8 0 )

(27,18,1,16) ( - . 5 3 6 , - . 0 9 0 )

(30,24,6,20) ( - . 6 6 6 , - . 1 3 9 )

(30,29,4,20) ( - . 6 6 6 , - . 1 5 5 )

(28,33,5,17) ( - . 5 6 9 , - . 1 7 8 )

(30,34,3,22) ( - . 7 2 9 , - . 1 7 5 )

(31,42,2,24) ( - . 7 9 4 , - . 2 1 0 ) (31,157,1,25) ( - . 8 2 5 , - . 7 6 2 ) (35,199,5,30) ( - . 9 8 8 , - . 9 8 0 ) (35,165,7,30) ( - . 9 8 8 , - . 8 2 3 ) (35,187,9,25) ( - . 8 3 0 , - . 9 3 7 ) (37,145,8,12) ( - . 4 2 0 , - . 7 3 1 )

and (Step 9) E u,p15 = (-.425, -.080). We come back to Step 2 and

we obtain

Vz = A(VZt,u) = {pu,p15}.

The set of strategies has been reduced to only two, one of them must be chosen by the D M .

(2) E {u,pu) y E ( u ^1) (Step 6). Then, we have to determine (uf,uf)

~ E(u,pu). Suppose the D M considers (uf,uf) = (-.690,-.052).

Therefore,

Vl = {p2,p5,pl2,pl3pu,pl5,pm,pu,pl8,pl9,p21}

(Step 9). We go back to Step 2 and obtain

V1z = A(VZt,u) = {pu,p15}.

Observe that we have same solution as above. I n case that u\* were greater than -.425, then the most preferred solution would have been

(3) E ( u ^1) y E (u,pu) (Step 7). Then, we have to determine (u\\u\*)

~ E(\i,pl). Assume that D M considers (u\*,u\*) = ( - . 3 9 9 , - . 1 ) .

Hence, VlZt = {p\pl5,p16} (Step 8). Coming back to step 2, we obtain

Vlz = A(Vl,u) = {p\p15}.

We have arrived to situation analogous to the above two cases. I n the event that u\* were greater than -.080, then the most preferred

solution would be p1.

If the D M assumes the axioms to have a scalar utility function as a linear combination of the components of the vector utility function, we can use the information provided by the D M in steps 6 (or 7), where he revealed that

E {u,pu) is preferred to E ( u ^1) (or E ( u ^1) is preferred to E (u,p1 4)), to

construct the generators of the information set. I n fact, if we have the first

assertion, the weights k = (fci,fc2) of the true DM’s scalar utility function,

have to verify

k E {u,pu) ^ kE (u,?1)

k e /M o

and define a new information set IMl, where

Ml = ^ .377 .623 ) '

Hence, the associated vector utility function is

uM l( z ) = (-.143*1 -.857*4, - . 0 5 4 * ! - . 0 6 9 *2 - . 5 5 4 *3 -.323*4

)-Thus, the approximation set for uM l (•) (step 2) is

On the other hand, if the D M reveals that E u ^1 y E u,pu , the weights

k have to verify

kE ( i V ) ^ kE (u,pu)

k e /M o

i.e., the information set is IMl, where

Ml = ( -3 0 77 -623 ) .

Now, the associated vector utility function is

uM l( z ) = (-.054zi -.069z2 -.554z3 -.323z4, -.125*2 --878z3).

The approximation set for uM l( - ) (Step 2) is

v1z = A'P1zt,uMl) = {p\p15

}-We note that the solutions obtained applying this algorithm are, in this case, the same than those obtained with the first one. However, the number of iterations usually will be smaller, although in this last method it is necessary for the D M to check the respective assumptions, which might be difficult to verify.

6. CONCLUSIONS

In multiattribute decision making under risk, the utility efficient set plays an important role in the solution process. Its generation may be very difficult and, moreover, it may be too extensive for the D M to make an easy choice. We propose an interactive approach to overcome such drawbacks, which uses an idea of approximation to the utility efficient set. Then, we reduce the set of strategies of interest based on information revealed by the D M from comparison of pairs of strategies. The procedure stops when a single strategy is achieved or a reduced enough set of strategies for the D M is obtained. We also consider a second procedure that is a variant of first one, valid for the case in which the DM has a linear utility function obtained as a combination of the components of the vector utility function.

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