Implicit Utility Function and Pairwise Comparisons

Janusz Majchrzak

Systems Research Institute, Polish Academy of Sciences.

1. INTRODUCTION

A new approach for multicriteria decision making is briefly presented here exploiting the pairwise comparisons of alternatives and assuming the existence of an implicit utility function of the decision maker. We plan t o extend DISCRET along this direction in the future.

The basic problem in the area of the interactive decision support systems is the extraction and utilization of the preferences of decision maker (DM). A rather large number of approaches have been developed during last decade. This chapter reports some basic ideas of a new approach, which seems t o be promising because of its conceptual and methodological simplicity. The presented approach is based on the pairwise comparisons of alternatives and linear approximations of the DM's utility function. Since the approach is a t an early stage of development and its several aspects still have t o be investigated, just some basic ideas and motivations will be presented.

The basic feature of the approach is t h a t t h DM is not forced t o compare pairs of alternatives which are presented t o him but he chooses himself a subset of alternatives t o be evaluated. An underlying quasiconvex DM utility function is assumed.

2. MOTIVATIONS

Let us consider the following multicriteria decision making problem. A decision maker ( a person or an institution) wants t o buy a new car and has some difficulties in choosing from the variety of models available on the market. He is not an expert in cars and he knows just a few models: his old car and those possessed by his friends and rela- tives. So, all he is able t o say about his preferences is a number of statements concerning cars he knows, like for example:

VW Golf is preferred t o Ope1 Kadett, Fiat Uno is preferred t o Peugeot 205, etc.

He refuses t o compare cars he doesn't know or t o supply any other kind of information about them. The reference point approach might be adopted in this case, but what if the DM would not be satisfied with the result?

The task can be formulated as follows. A relatively small number of pairwise com- parisons of alternatives is available. What can be said about the DM's preferences on the basis of this small amount of information and what can be said about the quality of that information ? Note t h a t a statement: "a cheap good car is preferred t o an expensive bad car" is a rather low quality information since, once price and performance have been esta- blished as criteria, this is an obvious statement. The DM should be informed about the quality of t h e alternatives evaluation he had made. Also his inconsistencies should be discovered.

J . Majchrzak - 60 - Implicit Utility Function . .. . 3. BASIC IDEAS

Let

F

be the space of m criteria,

A

=

R T

be the domination cone and let Q C

F

be the set of feasible alternatives. We will assume that there exist an underlying implicit quasiconvex utility function U: -+

R

behind the DM'S preferences. The DM need not recognize it existence; however, we will assume that whenever he decides t h a t alternative

b~ Q is preferred t o alternative a € Q , it is equivalent to U ( b )

>

U ( a ) .

The DM'S utility function U is in general a nonlinear function of criteria. Identifying such function usually requires large amount of d a t a and a significant computational effort.

Therefore, keeping nonlinearity of U in mind, we shall restrict ourselves t o a set of linear approximations of U only.

Suppose that

k

pairs of alternatives were compared by the DM:

b, is preferred to a,, ai,b, E Q

, ⁱ

⁼ 1

,..., k

This set of d a t a may be considered as a set

W

of

k

vectors in the criteria space

F,

^point-

ing from a less preferred alternative a, to a more preferred alternative b,.

Let us also consider the set V of normalized vectors w, E

W

:

Each of the vectors U, represents a direction of improvement in the space of criteria of the function U ( f ) . Hence, the cone C spanned by vectors U, E I/ is the cone of improvement for U (

f )

and can be defined as:

i = k

C =

{

C a , u , : a, E

R + ,

^{u; E} V)

i = l

The cone C* is the corresponding polar cone and can be defined as:

Both cones

C

and C * can be expressed by their generators. The set of cone generators is the minimal subset of vectors belonging to that cone that still span the cone. The genera- tors of cones C and C * will be denoted by c and c*, respectively.

C = { C a , c , a j € _{3 3}

R + ,

C , E

C )

1

C* =

{ C

^{a j c f}

,

^{a j €}

R + ,

cf E

C * )

1

where

C

^and

C*

are corresponding generator sets.

Let us return t o the pairwise comparisons. Since we shall consider linear approxima- tions of the utility function, for the sake of presentation simplicity, assume t h a t U is linear. If the DM has decided that alternative b E Q is preferred to alternative a E

9 ,

then U ( b )

>

U ( a ) . It is clear t h a t < u , u >

> 0,

where u = [a,b], and u is a vector nor- mal to hyperplanes U ( f ) = const. Hence, the vector

u

is contained in cone C*.

From the above analysis it follows that an accurate determination of the vector u normal to the hyperplanes of U will be possible only in the case when the cone C* is spanned by a single vector (namely u ) . In this case the DM'S utility function (or rather actually its linear approximation only) has been obtained and we can easily calculate the

J . Majchrzak - 61 - Implicit Utility F u n c t i o n

. . .

DM's most preferred solution by minimizing

U

over the set Q.

In general, because of obvious reasons, the cone C will be smaller than a halfspace and its polar cone C* will have a nonempty (relative) interior. In such a case, each of the vectors contained in C* may appear t o be the vector u . Fortunately, we can restrict our- selves t o the generators C * of t h e cone C* only. Considering each C; t o be the vector u (minimizing linear function based on CJ ) one can obtain a set of g . E Q being the linear approximation minimizers of DM's utility function. These elements 3 g . define a subset S C Q of nondominated elements of Q in which the DM's most preferred alternative 3 (minimizer of i

U)

is contained.

As it can be seen now, our approach does not pretend t o determine t h e DM's most preferred solution exactly. It will rather tend t o find a domain in which it is contained.

The more information about DM's preferences is contained in alternatives pairwise com- parisons supplied, the smaller this domain will be. Besides, also a good candidate for the most preferred solution may be presented to the DM. It can be obtained a kind of average vector for the cone C*: a sum of c;, a sum of v,, a gravity center of V . etc. The author's favorite method for the candidate selection is the calculation of the minimal (Euclidean)

'.'

norm element from the convex hull spanned by the cone C* generators c;. This technique based on the method of P. Wolfe [ I ] appeared t o be very useful in our approach, serving also for some other purposes. Let us denote the minimal norm element from the convex hull spanned by t h e set

V

of vectors

v

as

The minimizer of the linear function based on vector z will be chosen as the candidate for the DM's most preferred solution.

4. SOME DETAILS

In this chapter, we shall discuss the basic cases t h a t can occur for different sets of pairwise comparisons of alternatives supplied by the DM.

Case 1. Cone C is a halfspace of

F

and llzll = 0.

As it has been already mentioned, in this case the linear approximation of the DM's util- ity f u n c t i 0 n . i ~ defined by the vector u normal t o the halfspace spanned by C . The DM's most preferred solution may be found by the optimization of the linear function based on

u .

Case 2. Cone C is not a halfspace of

F

and llzll = 0.

Since the DM's utility function is assumed t o be quasiconvex, the set

V

of pairwise com- parisons supplied by the DM is inconsistent. Conflicting elements should be selected from the set

V

and presented t o the DM. They are those elements which spann a convex hull containing zero and hence cause llzll = 0. Their selection is automatic during the calcula- tion of the element z.

Case 3. Cone C is contained in a halfspace of

F,

it contains the domination cone

A

and llzll

2

^0.

This is the basic case. After the set of linear functions based on vectors C * optimization, a subset of nondominated elements of set Q will be obtained. This subset is defined by the set of linear approximation optimizers of the utility function. A candidate for the DM's most preferred solution will be found by optimizing over the set Q the linear function based on vector z . Notice t h a t if the number of supplied pairwise comparisons is small (too small t o spann a non-degenerate cone C ) , then generators of the domination cone

A

J . Majchrzak

-

⁶²

-

Implicit Utility Function

. . . .

. can be added t o the set V.

Case 4. Cone

C

is contained in the domination cone

A

and

11~1120.

This is the case of a low quality of information contained in pairwise comparisons of alternatives supplied by the DM (and corresponds to statements like: "a good cheap car is preferred t o an expen- sive bad car"). T h e DM should be informed about this fact and perhaps he will be able t o give some more restrictive statements. If he refuses for some reasons, we cannot proceed along the Case 3 line. However, instead of of considering the supplied information as being of a discriminative type we can treat it as an instructive type information. Each of the vectors uE V can be treated now as an approximation of the DM'S improvement direction or his utility gradient approximation. Hence, we can proceed just like in Case 3, taking the cone

C

instead of

C*

into consideration. Of course the DM should be aware of the new interpretation of the information he has supplied.

Cases 3 and 4 can be distinguished a priori by checking whether

C>A

or

CCA,

respectively.

5. CONCLUDING REMARKS

If the DM is able t o supply a large amount of results on alternatives evaluations, then a technique similar t o one presented in [2] should be used in order to eliminate dom- inated alternatives from further considerations. If it is not the case, the DISCRET pack- age methodology should be applied. Actually, the presented approach is planned t o be included into the DISCRET framework.

Several aspects of the presented approach are still t o be further investigated. The main one is how t o select a small sample of such alternatives t h a t their evaluation by the DM may result in significant improvement of an approximation of DM'S preferences.

REFERENCES

[ I ] P.Wolfe,"Finding the Nearest Point in a Polytope", Mathematical Programming, Vol.11, pp 128-149, 1975.

[2] M.Koksalan, M.H.Karwan, S.Zionts, "An Improved Method for Solving Multiple Criteria Problems Involving Discrete Alternatives", IEEE Transactions on Systems, Man and Cybernetics, Vol. SMC-14, No.1, pp.24-34, 1984.

[3] J.Majchrzak,"Methodological Guide t o the Decision Support System DISCRET for Discrete Alternatives Problems", in this volume.

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