The Use of Reference Objectives in Multiobjective Optimization - Theoretical Implications and Practical Experience

NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR

THE USE OF REFERENCE OBJECTIVES IN MULTIOBJECTIVE OPTIMIZATION

--

THEORETICAL IMPLICATIONS AND PRACTICAL EXPERIENCE

A.P. Wierzbicki

Working P a p e r f i are interim reports on work of the International Institute for Applied Systems Analysis and have received only limited review. Views or opinions expressed herein do not necessarily repre- sent those of the Institute or of its National Member Organizations.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS A-2361 Laxenburg, Austria

The paper presents a survey of known results and some.new developments in the use of re.ference objectives --that is, any reasonable or desirable point in the objective space--instead of weighting coefficients in multiobjective optimization. The main conclusions are as follows:

-- Any point in the objective space--no matter whether it. is attainable or not, ideal or not--can be used instead of weight- ing coefficients to derive scalarizing functions which have

minima at Pareto points only. Moreover, entire basic theory of multiobjective optimization--necessary and sufficient con- ditions of optimality and existence of Pareto-optimal solutions, etc. --can be developed with the help of reference objectives instead of weighting coefficients or utility functions.

-- Reference objectives are very practical means for solving

a number of problems such as Pareto-optimality testing, scanning

the set of Pareto-optimal solutions, computer-man interactive

solving of multiobjective problems, group assessment of solu-

tions of multiobjective optimization or cooperative game prob-

lems, or solving dynamic multiobjective optimization problems.

THE USE OF RGFERGNCE OBJECTIVES IN MULTIOBJECTIVE OPTIMIZATION

--

THEORETICAL IMPLICATIONS AND PRACTICAL EXPERIENCE

A.P. Wierzbicki

1

.

INTRODUCTION

This paper is aimed at a revaluation of some basic assump- tions in multicriteria optimization and decision-making from a pragmatical point of view, addressing the question why the known, highly developed methods and techniques in multicriteria analysis are not fully operational in applications. It is assumed that the reader is well acquainted with the state-of-the-art in multi- criteria analysis as represented, for example, by [2,3,7,9,12]

and that he has also encountered some of the vexing problems in the applications of this highly developed theory. The basic question in applications of multicriteria analysis is, in fact, only one though it may take various forms:

--

What is more valuable?

-

the perfection of a compromise based on a model which is never perfect, or the time of a top-rank decision maker? If confronted with a multi- tude of questions "would you prefer this alternative to the other one?", would not the decision maker simply send the analyst back to where he belongs?

. -- - -

--

Does a decision maker think in terms of trade-offs and weighting coefficients or is he rather concerned with aspiration levels and values?

--

Has a decision maker consistent preferences, which under known assumptions could be revealed in the form of a utility function, or does he simply want to attain certain goals?

--

^IS it easier in applications to determine marginal rates of substitution between various objectives, or to choose reasonable ranges or scales for those objectives?

--

^{I s} a compromise i n a g r o u p o f d e c i s i o n makers a t t a i n e d t h r o u g h a b a l a n c e o f t h e i r p r e f e r e n c e s o r r a t h e r by a n a g r e e m e n t o n g o a l s ?

--

Is n o t t h e t e r m " a d e c i s i o n maker" a n a b s t r a c t i o n , con- v e n i e n t f o r t h e a n a l y s t ? O r do w e r a t h e r d e a l w i t h d e c i s i o n making o r g a n i z a t i o n s a s u s u a l , where t h e t o p - l e v e l d e c i s i o n s a r e b a s e d on a c a r e f u l a n d e x t r e m e l y w e l l p r e p a r e d , b u t h i g h l y i n t u i t i v e a p p r a i s a l o f a few c r u c i a l i n d i c e s ?

Many s i m i l a r q u e s t i o n s c a n b e s t a t e d and a l l t h e s e q u e s t i o n s have p u z z l e d many r e s e a r c h e r s . I n f a c t , some r e c e n t p a p e r s

--

see Ackoff 1979, [ I ] - - g o a s f a r a s q u e s t i o n i n g e n t i r e l y t h e p r a c t i c a l v a l u e o f d e c i s i o n a n a l y s i s a n d o p t i m i z a t i o n . Some a u t h o r s p r e f e r a r e t r e a t t o p u r e l y h e u r i s t i c a l ~ r o c e d u r e s f o r d e c i s i o n making t o p s y c h o l o g i c a l , " s o f t - s c i e n c e " a p p r o a c h e s . ,Though h a v i n g much r e s p e c t f o r c a r e f u l , l o g i c a l a n a l y s i s of a

p r o b l e m , f o r d e e p i n t u i t i o n a n d p s y c h o l o g y , I a m n o t e n t i r e l y c o n v i n c e d . I would r a t h e r a d d r e s s a n o t h e r q u e s t i o n , which i n a s e n s e summarizes a l l t h e a b o v e d o u b t s :

\

--

^{What i s} wrong w i t h t h e b a s i c t o o l s o f m u l t i c r i t e r i a a n a l y s i s ? S h o u l d w e n o t r e e x a m i n e i t s b a s i c axioms?

H i s t o r i c a l r e f l e c t i o n c a n h e l p u s i n r e a c h i n g t h i s g o a l . When, i n 1896, P a r e t o [ 1 7 ] h a s f o r m u l a t e d t h e f o u n d a t i o n s o f m u l t i c r i t e r i a o p t i m i z a t i o n a n d u s e d w e i g h t i n g c o e f f i c i e n t s t o

t h i s e n d , h e opened a n e n t i r e f i e l d o f r e s e a r c h . W e i g h t i n g co- e f f i c i e n t s p l a y t h e r e f o r e a c e n t r a l r o l e i n t h e c o n t e m p o r a r y

p a r a d i g m o f m u l t i c r i t e r i a a n a l y s i s - - a l l n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s o f m u l t i o b j e c t i v e o p t i m a l i t y , a l l e q u i l i b r i a and t r a d e - o f f s , a l l u t i l i t y m a x i m i z a t i o n i s b a s i c a l l y r e l a t e d t o w e i g h t i n g c o e f f i c i e n t s . When t h e f o u n d a t i o n s o f t h e g e n e r a l economic

e q u i l i b r i u m t h e o r y w e r e f o r m u l a t e d , a consumer was assumed t o maximize a u t i l i t y f u n c t i o n r e p r e s e n t i n g h i s p r e f e r e n c e o r d e r i n g o f commodity b u n d l e s - - w h a t , i n t h e e q u i l i b r i u m , d i r e c t l y c o r r e - s p o n d s t o P a r e t o w e i g h t i n g c o e f f i c i e n t s f o r m i n g a l i n e a r a p p r o x i - m a t i o n o f t h e u t i l i t y f u n c t i o n . T h i s was a m o s t s a t i s f a c t o r y d e v e l o p m e n t o f economic t h e o r y and s t i l l i s a c o n t e m p o r a r y p a r t o f i t s b a s i c paradigm. I t h a s a l s o found c o n f i r m a t i o n i n empir- i c a l s t u d i e s o f t h e f r e e m a r k e t - - a s f a r a s a n y m a r k e t i s f u l l y f r e e

--

and r e s u l t e d i n f u r t h e r d e e p t h e o r e t i c a l s t u d i e s p r o v i d i n g

for an axiomatic basis of preference orderings and utility theory at a high mathematical level (see e .g. , Debreu 1 959 [5]

)

.

But here is a place for reflection: while a nameless agent on a free market may be well described by his utility function, no individual thinks in terms of preferences of commodity bundles when buying in a supermarket. When I am going to do some shopping, I know that I have to buy, for example, a quantity of milk, sugar, bread, and a shirt for my son; if I have enough money, I might also buy a toy for him and a tool for my gardening. In fact, I am thinking in terms of goals; if they are attainable, I might want to improve them. Moreover, my way of thinking does not change very much when I have to make decisions as a science manager.

However, further extensive studies [2,9,12] on decision making with multiple objectives were related strongly to pref- erences and utility theory. Identification methods for indi- vidual and group preferences as well as utility functions have been developed; statistical approaches have been considered to take into account uncertainty and risks; and even interactive procedures devised to involve a decision maker more directly into decision analysis have been based on learning about his preferences. Moreover, most of the applied studies in multi- objective optimization and decision making are implicitly or explicitly formulated in terms of weighting coefficients, trade- offs and utility functions.

On the other hand, many researchers have realized the need of an alternative approach. Savlukadze [20,211 and others consid- ered the use of utopia points as unattainable objective values representing some aspiration levels. Dyer

[6]

, ^{Kornbluth [I 31 and}

others introduced goal programming --the use of variable bounds on objective values in an interactive process of multicriteria optimization. Yet these and related works have not had the impact they deserved because of several reasons.

First, it was not clear whether it is possible to develop

a consistent, basic theory of multiobjective optimization and

decision making based on the use of reference objectives--

t h a t i s , any d e s i r a b l e a s p i r a t i o n l e v e l s f o r o b j e c t i v e s

--

r a t h e r t h a n w e i g h t i n g c o e f f i c i e n t s . I n o t h e r words, t h e neces- s a r y and s u f f i c i e n t c o n d i t i o n s , e x i s t e n c e c o n d i t i o n s , r e l a t i o n s t o p r e f e r e n c e o r d e r i n g s , e t c . , had t o be f o r m u l a t e d i n terms o f r e f e r e n c e o b j e c t i v e s . T h i s q u e s t i o n h a s been a t t a c h e d t o some o f my e a r l i e r works [ 2 2 , 2 3 , 2 5 ] ; a s y n t h e s i s and f u r t h e r develop- ment of r e l e v a n t r e s u l t s i s p r e s e n t e d i n t h e n e x t c h a p t e r of t h i s p a p e r .

Second, t h e u s e of r e f e r e n c e o b j e c t i v e s i m p l i e s a c h o i c e of d i s t a n c e o r norm i n o b j e c t i v e s p a c e and t h i s c h o i c e h a s been con- s i d e r e d , e r r o n e o u s l y , a s b e i n g e q u i v a l e n t t o t h e c h o i c e of

.

w e i g h t i n g c o e f f i c i e n t s . I n o r d e r t o work w i t h r e f e r e n c e o b j e c - t i v e s one h a s , a d m i t t e d l y , t o choose r e a s o n a b l e s c a l e s o r r a n g e s f o r a l l c r i t e r i a . But t h e c h o i c e o f a r e a s o n a b l e r a n g e i s i n - h e r e n t t o any c o m p u t a t i o n o r measurement and d o e s n o t n e c e s s a r i l y imply t h e c h o i c e of t r a d e - o f f s . A f t e r h a v i n g made a d e c i s i o n b a s e d on r e f e r e n c e o b j e c t i v e s , t h e c o r r e s p o n d i n g w e i g h t i n g co- e f f i c i e n t s can be a p o s t e r i o r i d e t e r m i n e d ( s e e n e x t c h a p t e r ) and examined. T h i s i s one o f t h e l i n k s between t h e t h e o r y b a s e d on r e f e r e n c e o b j e c t i v e s and t h e more c l a s s i c a l t h e o r y , b u t i t d o e s n o t impede t h e p r a c t i c a l u s e f u l n e s s o f r e f e r e n c e o b j e c t i v e s .

T h i r d , t h e u s e o f r e f e r e n c e o b j e c t i v e s h a s n o t been w i d e l y t e s t e d i n a p p l i c a t i o n s , and v a r i o u s problems r e l a t e d t o c o n s i d e r - a t i o n of u n c e r t a i n t i e s , t o g r o u p d e c i s i o n making, t o i n t e r a c t i v e p r o c e d u r e s o f d e c i s i o n making, e t c . , have n o t been s o l v e d y e t . Another c h a p t e r of t h i s p a p e r i s d e v o t e d t o t h e s e problems.

2 . B A S I C THEORY Fund am en&

L e t E o C E be a s e t o f a d m i s s i b l e d e c i s i o n s o r c o n t r o l s o r a l t e r n a t i v e s t o b e e v a l u a t e d . W e d o n o t s p e c i f y y e t t h e n a t u r e of s p a c e E . L e t G b e t h e s p a c e o f o b j e c t i v e v a l u e s o r performance i n d i c e s o r g o a l s . We assume t h a t G i s a H i l b e r t s p a c e , o u t of s e v e r a l r e a s o n s . F i r s t , some a b s t r a c t p r o p e r t i e s o f t h e H i l b e r t s p a c e - - m o s t l y t h e p r o p e r t i e s of a p r o j e c t i o n on a cone - - s i m - p l i f y t h e r e a s o n i n g and p r o o f s . Second, a H i l b e r t s p a c e i s t h e

least abstract one that includes trajectories of dynamical sys- tems or probability distributions and we would like to consider also dynamical trajectories or probability distributions as pos- sible goals of multiobjective optimization. Third, the Euclidean space E is a (finite-dimensional) n Hilbert space, and we can

therefore use graphical illustrations and intuition to comment on results.

Let a mapping Q :Eo + G be given, defining numerically the consequences of each decision or alternative. Let Qo

=

Q(EO) C G be the set of attainable objectives. To choose between them, suppose a partial preordering in G is given by means of a positive cone (any closed, convex, proper cone) D $ ^G:

A

corresponding strong partial preordering in G can be definedby:

Suppose,

t6 simplify theexpositioqthatwe areinterested inmini-

mizing all the objectives (losses, risks, etc.). In the Hilbert

space G, we define correspondingly a minimal element of Qo with respect to the partial preordering (1) or a D-minimal element of Qo

:

Let us denote by a0 the set of all D-minimal points in Qo. 1f

2 2 1 2

G = R a n d D

= R + =

{(q , q ) € R : q l > O t q - -

^>

01. then a D-minimal

point of Qo is Pareto-minimal, see Figure 1 . In fact, in finite-

dimensional cases we are mostly concerned with Pareto-minimal

points; the possibility of using other positive cones illustrates

only possible generalizations of infinite-dimensional spaces.

FIGURE 1. D-MINIMAL POINTS AND SETS:

a) PARETO-MINIMAL POINTS

b) MORE GENERAL D-MINIMAL POINTS

One of t h e most i m p o r t a n t p r o p e r t i e s o f p a r t i a l p r e o r d e r i n g s i n d u c e d by a p o s i t i v e cone i n H i l b e r t s p a c e i s t h a t we c a n g i v e a p r e c i s e answer t o t h e t y p i c a l q u e s t i o n : g i v e n two outcomes q l , q 2 which a r e i n c o m p a r a b l e ( t h a t i s , n e i t h e r q l < q 2 n o r q 2 < q l )

,

what i s t h e p a r t o f q 2 t h a t h a s improved w i t h r e s p e c t t o q l ? The answer r e s u l t s from t h e f o l l o w i n g lemma.

Lemma 7 ( P r o j e c t i o n s on Cones i n a H i l b e r t Space --Moreau 1962 [ I 6]).. G i v e n a H i l b e r t s p a c e G and a c l o s e d , c o n v e x c o n e D C G , e a c h e l e m e n t q E G c a n b e u n i q u e l y and o r t h o g o n a Z l y d e - composed i n t o i t s p r o j e c t i o n s o n t h e c o n e s -D and D* =

{q* E G : ( q * , q ) LO VqED):

n h e ~ e t h e p r o j e c t i o n s q-D and qD* a r e d e f i n e d b y :

-D

-

( 5 ) q = a r g min Ilq

-

-qll ; qD* = arg-min II q

-

^FJII

.

q E -D

q

^{E D*}

and ( . , = ) d e n o t e s t h e s c a l a r p r o d u c t , I I * I 1 d e n o t e s t h e norm.

The cone

D*

is called the dual cone;

-D

and

D*

are called

2 2

D* -

mutually polar.

If D =

R+, then

D* = D =

R+, and q

^-

1 A L

(max(0,q ),max(O,q

) )

is just the vector composed of the posi- tive components of the vector q. This is interpreted in Figure2, where Lemma

I

is applied to the difference a2 - ^ql

=

q in order to discern the part of q2 that has improved when compared to ql and the other part that is worse than ql.

The projection on a cone has several additional useful properties of norm-minimality, Lipschitz-continuity, ~rgchgt- differentiability of its square norm, etc. - see Wierzbicki and Kurcyusz 1977 [24] .

FIGURE 2. DETERMINATION OF THE NEGATIVE AND THE POSITIVE PART OF q2

-

q1 VIA PROJECTION ON CONES: a) D = R

_{+ '}

2

-

b) MORE GENERAL CASE

Ohdm- Phu ehving F u n d o n a and P&y S c d a h i z a t i o n ; Suddicient CondLtiona doh M L L e R ; i & ~ Optbn&.Ay

Now we approach the basic question in the use of reference

objectives in multicriteria optimization: given any aspiration

level or reference objective

~ E G ,

can we construct a scalar-

izing function s

: G

x Q O + R which is strictly order-preserving

1

in its second argument (thus can be considered as a type of

arbitrarily chosen utility function)

?

Recall that s (q,q) is

strictly order-preserving in q, iff:

and t h a t e a c h m i n i m a l p o i n t o f a s t r i c t l y D - o r d e r - p r e s e r v i n g f u n c t i o n i s a D-minimal p o i n t (see, e . g . , [ 5 , 2 3 1 ) :

( 7 )

6

⁼ a r g min s ( q , q )

( e - a ) n ~ ,

⁼

+ .

q E Q O

The a n s w e r t o t h e a b o v e q u e s t i o n i s n o t o n l y p o s i t i v e b u t a l s o vague: t h e r e a r e many s c a l a r i z i n g f u n c t i o n s t h a t a r e s t r i c t l y o r d e r - p r e s e r v i n g . F o r e x a m p l e , c h o o s e any v e c t o r o f p o s i t i v e w e i g h t i n g c o e f f i c i e n t s - - o r , more g e n e r a l l y , L a g r a n g e m u l t i - p l i e r s

X

€

*

^{= { q * € G : ( q* , q ) > 0 Yq €}

^51,

w h e r e

b*

^{i s} c a l l e d t h e q u a s i - i n t e r i o r o f D*

-

a n d d e f i n e t h e known l i n e a r f u n c t i o n s

(q,q)

^{= (}

^x

^{, q}

- q

⁾ which i n t h e s i m p l e s t c a s e i s j u s t t h e sum o f

n i i - i

w e i g h t e d o b j e c t i v e d i f f e r e n c e s

igl

^{X ( q}

-

q )

.

T h i s f u n c t i o n i s s t r i c t l y o r d e r - p r e s e r v i n g , a n d e a c h o f i t s m i n i m a l p o i n t s i s D-minimal, o r P a r e t o - m i n i m a l . But t h e m i n i m a l p o i n t s o f t h i s f u n c t i o n d o n o t depend on t h e i n f o r m a t i o n c o n t a i n e d i n

6

a n d r e q u i r e t h e i n f o r m a t i o n c o n t a i n e d i n A . T h e r e f o r e , w e s h o u l d l o o k f o r n o n l i n e a r s t r i c t l y o r d e r - p r e s e r v i n g f u n c t i o n s t h a t do n o t r e q u i r e t h e s p e c i f i c a t i o n o f w e i g h t i n g c o e f f i c i e n t s

X

a n d h a v e minima d e p e n d e n t o n t h e r e f e r e n c e o b j e c t i v e

q.

One s u c h f u n c t i o n h a s t h e f o l l o w i n g f o r m :

s ( 9 , q ) =

-Ilq-qll +

^{pll ( q}

- q )

D* 11

o r t o p r o v i d e f o r d i f f e r e n t i a b i l i t y

w h e r e p > 1 i s a n a r b i t r a r y s c a l a r c o e f f i c i e n t . T h e s e f u n c t i o n s a r e c a l l e d p e n a l t y s c a l a r i z i n g f u n c t i o n s . One o f t h e b a s i c p r o p e r t i e s o f t h e s e f u n c t i o n s i s t h e f o l l o w i n g :

L e m m a 2 ( W i e r z b i c k i 1 9 7 5 , [ 2 2 ] ) . . If G is a H i l b e r t s p a c e , D C G is a c l o s e d c o n v e x c o n e s a t i s f y i n g t h e c o n d i t i o n D E D * , a n d p > 1 , t h a n , f o r a n y

I E

^G, t h e f u n c t i o n s

(qlq)

d e f i n e d by

( 8 1 o r ( 8 a ) is s t r i c t l y o r d e r - p r e s e r v i n g .

O b s e r v e , f i r s t , t h a t t h e c o n d i t i o n D C D * i s n o t v e r y re- s t r i c t i v e , s i n c e i f D = E + ,

n

t h e n D* = D ~ D * ; g e n e r a l l y , t h e c o n d i t i o n means t h a t t h e c o n e D s h o u l d n o t b e " t o o b r o a d " . S e c o n d l y , o b s e r v e t h a t t h e lemma i s v a l i d f o r a n y E G a n d , t h e r e f o r e , g e n e r a l i z e s and p u t s two known a p p r o a c h e s i n t o a common f r a m e : u t o p i a p o i n t a p p r o a c h , where < $ ! Q ~ a n d Q O

c q +

D

( a p o i n t

q

s a t i s f y i n g t h e l a s t r e q u i r e m e n t i s c a l l e d D-preceding

Q o )

,

and g o a l programming a p p r o a c h , where

<

^{E Q ~ .} I n f a c t , ob- s e r v e t h a t i f Q0

cG +

^{D , t h e n} ~ - ~ E D _ C D * f o r a l l q E Q o , a n d

( q

-qlD*=

q - q ; t h u s , f u n c t i o n ( 8 a ) t a k e s t h e form s ( q , q )

-

= ( p

-

^{1 )} U q - i l l 2 a n d w e m i n i m i z e t h e d i s t a n c e f r o m p o i n t

q

^{t o Q O ,}

see F i g u r e 3 a . I f < E Q ~ i s a t t a i n a b l e , t h e n t h e r e are a l w a y s p o i n t s q E Q O s u c h t h a t

~q -

D , (CJ

-41D*

= 0 , a n d s ( < , q ) =

-

²

I

q

-

q

,

see F i g u r e 3b. Now, m i n i m i z i n g t h e minus norm o r m a x i m i z i n g t h e norm o f t h e o b j e c t i v e improvement q - q , s u b j e c t t o t h e c o n s t r a i n t

~ - G E - D

^{i s} a v a r i a n t o f goal-programming:

w e would l i k e t o g e t t h e b e s t p o i n t (& i n F i g u r e 3 b ) w e c a n o n c e t h e a s p i r a t i o n l e v e l s a r e s a t i s f i e d . But t h e b a s i c prop- e r t y o f t h e s c a l a r i z i n g f u n c t i o n ( 8 ) o r ( 8 a ) i s t h a t t h e a d d i - t i o n a l c o n s t r a i n t q - q ~ - D

-

n e e d n o t b e t r e a t e d as a h a r d con- s t r a i n t ; i t s v i o l a t i o n i s e x p r e s s e d by t h e p e n a l t y t e r m

p I1 ( q

- g)

^D* 11 2 , a s a s o f t c o n s t r a i n t . Even i f t h e a s p i r a t i o n l e v e l s

-

q a r e s l i g h t l y v i o l a t e d ( d e p e n d i n g o n t h e p e n a l t y c o e f f i c i e n t p , see F i g u r e 3b) a t a m i n i m a l p o i n t

6

o f s ( q , q )

,

t h e p o i n t i s D-minimal. And, f i n a l l y , i f n e i t h e r ~ E n o r Q Q O c < ~

+

D , see 7 i g u r e 3 c , t h e n t h e known a p p r o a c h e s c o u l d n o t u s e t h e i n f o r m a - t i o n c o n t a i n e d i n

q ,

w h e r e a s t h e m i n i m i z a t i o n of f u n c t i o n ( 8 ) a n d ( 8 a ) s t i l l r e s u l t s i n a D-minimal p o i n t .

Thus, a n y d e s i r a b l e r e f e r e n c e o b j e c t i v e p o i n t

<

c a n b e u s e d t o d e t e r m i n e a c o r r e s p o n d i n g D - m i n i m a l p o i n t

6 .

^The l a t t e r d e p e n d s c l e a r l y n o t o n l y o n r e f e r e n c e o b j e c t i v e

q,

b u t a l s o on t h e pen- a l t y c o e f f i c i e n t P a n d t h e p a r t i c u l a r norm c h o s e n ( o r o n t h e s c a l i n g o f s e p a r a t e o b j e c t i v e s ) . But t h i s d e p e n d e n c e h a s o n l y t e c h n i c a l c h a r a c t e r : w e do n o t assume t h a t a s c a l a r i z i n g f u n c t i o n of t h e f o r m ( 8 ) o r ( 8 a ) r e p r e s e n t s t h e u t i l i t y f u n c t i o n o f a

g i v e n d e c i s i o n m a k e r , w e r a t h e r u s e t h i s s c a l a r i z i n g f u n c t i o n t o a p p r o x i m a t e l o c a l l y h i s p r e f e r e n c e s ( a n d h i s u t i l i t y f u n c t i o n ,

i f he a c t u a l l y h a s o n e ) v i a a n i n t e r a c t i v e p r o c e d u r e , t h r o u g h a s k i n g him q u e s t i o n s h e u n d e r s t a n d s w e l l . An i l l u s t r a t i o n o f s u c h a p r o c e d u r e i s r e p r e s e n t e d i n F i g u r e 3 d . The c o r r e s p o n d i n g q u e s t i o n i s : "You h a v e a s k e d u s t o a t t a i n o b j e c t i v e l e v e l s

- - -1 -2

q i - ( q i , q i ,

...

^{) .} The b e s t w e c a n d o u n d e r t h e l i m i t a t i o n s o f A1 A2

o u r model i s

ai

⁼ ( q i , q i , . . . ⁾

.

^DO you a c c e p t t h i s , o r would you l i k e t o m o d i f y y o u r d e s i r e d l e v e l s t o some

ci+l

^{- -}

1 -2

t9i+l ^I

.

^{) ?} I n t h e l a t t e r case, p l e a s e s p e c i f y new de- s i r e d l e v e l s . " O b v i o u s l y , t h i s p r o c e d u r e c a n h a v e many v a r i a n t s : t h e a n a l y s t c a n r e s p o n d w i t h more t h a n o n e

Gi

t o a g i v e n

qi

^by

v a r y i n g t h e c o e f f i c i e n t p , o r t h e norm, o r e v e n by a p p l y i n g s p e c i a l l y d e s i g n e d v a r i a t i o n s

AC

i n o r d e r t o p r e s e n t t h e de-

i

c i s i o n maker w i t h more t h a n o n e a l t e r n a t i v e . B u t t h e b a s i c i d e a r e m a i n s t h e same: t o a s k t h e d e c i s i o n maker a b o u t a s p i r a t i o n l e v e l s a n d n o t a b o u t p r e f e r e n c e s

a

1

F I G U R E 3 . M I N I M A L P O I N T S O F THE PENALTY S C A L A R I Z I N G F U N C T I O N ( 8 ) : a) WHEN IS D-PRECEDING Q

-

^{b) WHEN} IS ATTAINABLE;

C ) WHEN q Is N E I T H E R D - P m & D I N G NOR A T T A I N A B L E ; d ) AN I N T E R A C T I V E PROCEDURE

Necendatry Candttiom, RdaLian lto ~ e i g k t i n g Coe66iciev~tb, and Exhltence 06 M L L e t j C h i t h O p i h n d Soha%ond

The scalarizing function

(8)

or (8a) has other useful prop- erties. The most important one is that of order-approximation:

the level set SO

=

{q

E G :

s(q,q) 1 0 ) approximates the set of improvement

< - D

from above and arbitrarily- closely for suf- ficiently large

p

- see Figure 4a. More precisely, the following lemma holds:

Lemma 3

(Wierzbicki 1977, [23]

)

.

^Denote

D E =

I ~ E G

:

dist(qrD) =llq

- D * ~ < ~ U q l l } . For a r b i t r a r i l y s m a l l E , c h o o s e p >

E-2.

Then t h e l e v e l s e t

So

o f t h e f u n c t i o n ( 8 ) o r ( 8 a ) s a t i s f i e s t h e f o l l o w i n g r e l a t i o n :

From this lemma, the following necessary condition of D-minimal- ity can be easily deduced:

Lemma 4

(General Necessary Condition of Multicriteria

Optimality).

If

G

i s a H i l b e r t s p a c e w i t h a p o s i t i v e c o n e DCD*, and i f

6

i s a DE-minimal p o i n t o f

QO

=

Q(E)

( t h a t i s , i f ( $

- E E ) ^{n Q o}

= @ w i t h

6 ,

= D E \ ( D ,

n-D,)

and D E d e f i n e d a s i n Lemma 3 1 , t h e n

( 1

0) min s(G,q)

=

0 q EQO

w h e r e

s(3,q)

i s d e f i n e d a s i n ( 8 ) o r ( 8 a ) w i t h p

>max(l , E - ~ )

and t h e minimum i n ( 1 0 ) i s a t t a i n e d a t

q

= $. Moreover,

i f $ E Q ~ i s a t t a i n a b l e b u t n o t DE-minimal, t h e n

min s ($,q) <O.

I f

Q G Q ~

i s n o t a t t a i n a b l e , t h e n

min s(e,q) > O . qEQo qEQ0

In contrast to the known necessary conditions of multi- criteria optimality via weighting coefficients

_{A ,}

Lemma 4 is easily applicable and valid even for nonconvex sets Qo of attainable objectives. Lemma 4, in fact, corresponds to sup- porting the set Qo at

$

by the set So contained in the cone

6 -

^{D E ,}

while the known necessary conditions of multicriteria

o p t i m a l i t y c o r r e s p o n d t o s u p p o r t i n g t h e s e t Qo a t

6

^{by a}

h y p e r p l a n e , c f . F i g u r e 4 a , b .

FIGURE 4. NECESSARY CONDITIONS OF MULTICRITERIA OPTIMALITY:

a) GENERAL CASE, WITH THE USE OF PENALTY SCALARIZING FUNCTIONS;

A

b) CONVEX CASE, WITH THE USE OF WEIGHTING COEFFICIENTS A .

I t i s a l s o i n t e r e s t i n g t o n o t e t h a t , i f t h e r e f e r e n c e ob- j e c t i v e

6

i s n o t D-minimal, t h e c o r r e s p o n d i n g m i n i m a l p o i n t

6

o f t h e s c a l a r i z i n g f u n c t i o n ( 8 a ) d e f i n e s u n i q u e l y a r e l a t e d v e c t o r

X

o f w e i g h t i n g c o e f f i c i e n t s :

Lemma 5 ( A P o s t e r i o r i D e t e r m i n a t i o n o f W e i g h t i n g Co- e f f i c i e n t s , 1 2 5 1 ) . S u p p o s e t h e a s s u m p t i o n s o f Lemma 2

a r e s a t i s f i e d a n d Z e t $ b e a m i n i m a l p o i n t o f t h e

f u n c t i o n ( B a ) ,

4 # <.

S u p p o s e Qo i s t o c a t t y c o n v e x i n a n e i g h b o r h o o d o f

6 .

T h e n :

i s a ( n o r m a l i z e d ) v e c t o r o f w e i g h t i n g c o e f f i c i e n t s a t

6,

A

t h a t i s , t h e s e t

SO

= { q E G : ( ~ , q

- 4 ) 2 O )

s u p p o r t s t o c a t t y t h e s e t QO a t

G.

A n o t h e r r e s u l t o f r a t h e r t h e o r e t i c a l i m p o r t a n c e , r e l a t e d t o t h e n o t i o n o f r e f e r e n c e o b j e c t i v e s , i s t h e f o l l o w i n g s i m p l e lemma :

Lemma 6

(Sufficient Conditions for the Existence of Multicriteria Optimal Solutions).

Suppose t h e r e e x i s t s a r e f e r e n c e o b j e c t i v e

q

s u c h t h a t t h e s e t

(q - ^{D) n Q O}

i s nonempty and ( w e a k l y ) c o m p a c t . Then t h e r e e x i s t D-minimal p o i n t s

6

o f t h e s e t

QO.

This lemma has been given in [22] under the additional assumption that the cone D* has nonempty quasi-interior

6*

=

{q* EG

:

(q*,q)

>

0 Vq €51, and was proved via consider- ation of the linear form

(

X,q) , X E ~ * . But we can omit the additional assumption, since the function (8a) is weakly lower semicontinuous, see [24], and thus has a minimum in ( q - ^{D) nQo}

under the assumptions of the lemma. This minimum is a D-mini- ma1 point of the set ( q - D ) "a0, hence also of the set QO.

If G is finite-dimensional with the Euclidean norm,

G =

E n ,

and D

= E+,

n then the penalty scalarizing function (8a) takes the form:

which might be convenient for nonlinear dependence of qi on the decision variables

x

E E O , but is not convenient for multicriteria

linear programming problems. However, penalty scalarizing func- tions based on other norms in R", that is, the sum of absolute values norm:

or the maximum (Chebychev) norm

(1

4) s(c,q)

=

- ^maxlqi - ~ + ~ pmax (0.q i l

- q

-i

)

i i

possess almost all properties of the function (12)

:

if

p >

1,

then the function (13) is strictly order-preserving in q for

any and the function (14) is order-preserving (hence her

minimal p o i n t s a r e P a r e t o - m i n i m a l e x c e p t i n some d e g e n e r a t e c a s e s ) . T h e s e f u n c t i o n s a r e a l s o o r d e r - a p p r o x i m a t i n g , see [ 2 5 1 . I f t h e d e p e n d e n c e o f q on d e c i s i o n v a r i a b l e s i ^{x E E o} i s l i n e a r , t h e n t h e m i n i m i z a t i o n o f f u n c t i o n s ( 1 3 )

,

( I 4 ) c a n b e r e d u c e d a f t e r t y p i c a l t r a n s f o r m a t i o n s i n t o l i n e a r programming p r o b l e m s . F o r p r a c t i c a l a p p l i c a t i o n s o f r e f e r e n c e o b j e c t i v e s i n m u l t i - c r i t e r i a l i n e a r programming, a c o m b i n a t i o n o f f u n c t i o n s ( 1 4 ) and

( 1 5 ) m i g h t b e a l s o u s e f u l , s e e [ 1 4 ] .

A n o t h e r p r a c t i c a l f o r m of p e n a l t y s c a l a r i z i n g f u r t c t i o n s i s r e l a t e d t o a t y p i c a l p r o c e d u r e i n g o a l programming, where o n e o f t h e o b j e c t i v e s i s m i n i m i z e d , s u b j e c t t o v a r i a b l e a t t a i n a b l e

l e v e l s o f a s p i r a t i o n f o r o t h e r o b j e c t i v e s t r e a t e d a s c o n s t r a i n t s . The u s e o f p e n a l t y s c a l a r i z i n g f u n c t i o n s r e s u l t s i n a more u n i - v e r s a l p r o c e d u r e o f t h i s t y p e , s i n c e t h e assumed l e v e l s f o r o t h e r o b j e c t i v e s d o n o t n e c e s s a r i l y h a v e t o b e a t t a i n a b l e when u s i n g p e n a l t y t e r m s . To r e p r e s e n t t h i s method, i t i s n e c e s s a r y t o s p l i t t h e s p a c e o f o b j e c t i v e s i n a C a r t e s i a n p r o d u c t o f t h e s p a c e R 1 o f v a l u e s o f t h e f i r s t o b j e c t i v e , and a s p a c e G r f o r o t h e r o b j e c -

1 1 1

t i v e s , G = R x G r , w i t h D = R+ x D a n d q = ( q , q r ) . Then t h e r

c o r r e s p o n d i n g p e n a l t y s c a l a r i z i n g f u n c t i o n i s :

o r , i f d i f f e r e n t i a b i l i t y i s i m p o r t a n t :

I f t h e s p a c e G r i s H i l b e r t a n d D , ~ D : , t h e n t h e f u n c t i o n s ( 1 5 ) , ( 1 5 a ) are o r d e r - p r e s e r v i n g f o r any p > 0 and t h e f u n c t i o n

( 1 5 ) i s o r d e r - a p p r o x i m a t i n g ( t o o b t a i n t h e o r d e r a p p r o x i m a t i o n 1 -1 p r o p e r t y i n t h e f u n c t i o n ( 1 5 a ) , o n e had t o s q u a r e a l s o q

-

q ) .

The r e f e r e n c e l e v e l

4'

m a t t e r s a c t u a l l y o n l y i n t h e o r d e r - a p p r o x - i m a t i o n p r o p e r t y s i n c e it d o e s n o t i n f l u e n c e t h e minimum o f t h e f u n c t i o n s ( 1 5 )

,

(1 5 a )

.

The r e f e r e n c e o b j e c t i v e

qr

^{E Gr} i s n o t n e c e s s a r i l y a t t a i n a b l e and p c a n b e s m a l l , p r o v i d e d it i s p o s i - t i v e ; n e v e r t h e l e s s , e a c h m i n i m a l p o i n t o f t h e f u n c t i o n s ( 1 5 ) ,

(1 5 a ) i s a D-minimal p o i n t .

n- 1 n- 1

If Gr

= R

and Dr

= R+

, then any norm can be used in (1 5) , ^{(1 5a)} . The functions:

n

(1 7) 1 -1 i -i

s(C,q)

=

q - ^q +

p

1 max (0,q - q

)

i=2

are strictly order-preserving, whereas (1 6) and (1 7) are also order-approximating, and the function

is order-preserving and order-approximating, see [25]. All these functions actually express

a

simple approach to goal pro-

2 n

gramming: treat the objectives

q

,...,q as constraints, given aspiration levels G ~ , . . . ^,q -n ^, and introduce penalty components for them. But new, compared to typical goal programming, is the fact that q2,. . . ,q need not be attainable and that the

-rl

penalty coefficient

p

need not be increased to infinity, nor other iterations on penalty terms need to be performed: even

2 -2

if some or all of the constraints

q

-

<

q , . . ^{. ,qn 5} cn ^are

violated, all minimal points of the functions (16), (16a), (17), (18) are Pareto-minimal.

Consider now a practical interactive procedure for choosing

a Pareto-minimal point, where the actual decisions are made by

a decision maker and the mathematical model of a given problem

and the optimization techniques serve only as a tool to help

him to recognize quickly a relevant part of the Pareto-minimal

set.

At the beginning, the decision maker is presented with all thc information about the model of the problem he desires

--

^for

example, with the minimal levels of objective functions when minimized separately, and with the corresponding decisions.

After that, he is asked to specify the vector of the desired

- - -1 -n

levels for all objective functions, qo - (qO

, . . . ,

^{qO) E R"}

(only the finite-dimensional case is considered here, although generalizations to the infinite-dimensional case are possible and even have applicational value).

For each desired objective vector qi, the mathematicalmodel and the optimization technique respond with:

1) The Pareto-minimal attainable objective vector Gi, ob- tained through a minimization of the function (12), and the

corresponding decision variable levels (any other penalty scalar- izing function from the previous paragraph can also be used,

depending on the particular nature of the model);

2) n other Pareto-minimal attainable objective vectors

A

qiIjf j = 1,

...,

n, obtained through minimization of the function (12) with perturbed reference points:

where di is the distance between the desired objective vector

-

qi and the attainable one bi, e is the jth unit basis vector, j

and a is a scalar coefficient. Only the case a = 1 is consid- ered in the sequel, which corresponds to the widest-spread additional information for the decision maker and is also more

. ^{l i} difficult to obtain convergence of the procedure.

To obtain any additional information at the beginning of the procedure, the decision maker can change

q

several times

0

(without counting it as iterations, i is kept equal 0) and

analyze the responses. Once he is ready for "real bargaining", he specifies a desired objective vector T I , i = 1, and the

iteration count begins. Now his modifications of the desired vector to

q

_{i+ 1} from

qi

are limited by the responses

Gi

cor-

I j responding to

qi

through two requirements:

I j

where B i s a p r e s p e c i f i e d p a r a m e t e r . The r e q u i r e m e n t (21) s t a t e s t h a t t h e d e c i s i o n maker h a s t o move a t l e a s t some p a r t of t h e d i s t a n c e t o t h e P a r e t o s e t , t h e r e q u i r e m e n t (20) l i m i t s h i s d i - r e c t i o n s of movement t o t h e simplex spanned by

qi

^{and Cji}

t j - A c t u a l l y , t h e d e c i s i o n maker s h o u l d n o t be b o t h e r e d by t e c h n i -

c a l i t i e s ( 2 0 ) , ( 2 1 ) ; it i s s u f f i c i e n t t h a t he i s informed a b o u t them and, a f t e r he h a s s p e c i f i e d any

q i c l t

a complementary auto- m a t i c p r o c e d u r e p r o j e c t s

qi+l - -

^{qi on Si} t o s a t i s f y ( 2 0 ) and

a d j u s t s i t s l e n g t h t o s a t i s f y ( 2 1 ) , i f n e c e s s a r y .

The above p r o c e d u r e and l i m i t a t i o n s of t h e a d j u s t m e n t s of t h e d e s i r a b l e o b j e c t i v e v e c t o r

qi

a r e d e p i c t e d i n F i g u r e 5 .

FIGURE 5. ILLUSTRATION OF THE INTERACTIVE PROCEDURE OF MULTIOBJECTIVE OPTIMIZATION. SHADED REGION DENOTES THE SET OF ADMISSIBLE

<.

~t i s c l e a r f r o m F i g u r e 5 t h a t , o n c e t h e d e c i s i o n maker h a s s p e c i f i e d

;ill

h e c a n u s u a l l y o b t a i n from t h i s p o i n t o n l y a l i m i t e d p a r t o f t h e P a r e t o - m i n i m a l s e t Q

0 ^' T h i s i s b o t h weakness and s t r e n g t h o f t h e p r o c e d u r e . The weakness c a n b e compensated by t h e i n i t i a l , e x p l o r a t o r y p a r t o f t h e p r o c e d u r e , where t h e d e c i - s i o n maker c a n g a t h e r a s much i n f o r m a t i o n a s h e w i s h e s . The s t r e n g t h c o n s i s t s p r e c i s e l y o f t h e l i m i t a t i o n o f t h e p r o c e d u r e t o t h e r e g i o n o f i n t e r e s t f o r t h e d e c i s i o n maker. O b s e r v e t h a t t h i s r e g i o n would d e c r e a s e i f ^a w e r e s m a l l e r t h a n 1 .

The c o n d i t i o n s o f t h e c o n v e r g e n c e f o r t h i s p r o c e d u r e a r e g i v e n by t h e f o l l o w i n g lemma:

Lemma 7 ( C o n v e r g e n c e o f t h e I n t e r a c t i v e M u l t i c r i t e r i a O p t i m i z a t i o n P r o c e d u r e ) . S u p p o s e t h e s e t o f a t t a i n a b l e

n n

o b j e c t i v e s Q o i s c o n v e x , G = E

,

D = E+ ( t h e norm u s e d i n s c a l a r i z a t i o n i s E u c l i d e a n ) . T h e n , f o r a n y a, B E ( 0 ; l

I ,

t h e p r o c e d u r e d e s c r i b e d a b o v e w i t h r e q u i r e m e n t s ( 1 9 ) , ( 2 0 ) , 1 2 1 ) i s c o n v e r g e n t , t h a t i s , l i m di = 0.

i + c n

The p r o o f o f t h i s new t h o u g h n o t v e r y a s t o n i s h i n g r e s u l t i s g i v e n i n t h e Appendix. The lemma c a n p r o b a b l y b e p r o v e d f o r o t h e r t h a n E u c l i d e a n norms i n R". O b s e r v e t h a t i f t h e r e q u i r e - ment ( 2 0 ) w e r e s u b s t i t u t e d by a s i m p l e r o n e , f o r e x a m p l e ,

- -

_n

q i + 1

-

q i ~ R + , o n e c o u l d d e v i s e m o v e s f o r t h e d e c i s i o n maker which would r e s u l t i n d i v e r g e n c e . But t h e s e moves would a l s o b e un- r e a s o n a b l e from h i s p o i n t o f view a n d , c o u n t i n g on h i s r e a s o n - a b i l i t y , w e c a n s i m p l i f y t h e r e q u i r e m e n t s ( 2 0 ) , ( 2 1 ) , o r e v e n s i m p l y d r o p them a s k i n g t h e d e c i s i o n maker t o move g e n e r a l l y i n t h e d i r e c t i o n o f t h e P a r e t o - s e t .

I n t h e lemma w e d i d n o t assume any p r e f e r e n c e - o r d e r i n g o r u n d e r l y i n g u t i l i t y f u n c t i o n d e s c r i b i n g t h e b e h a v i o r o f t h e de- c i s i o n maker, a n d w e d i d n o t c o n c l u d e a n y t h i n g a b o u t t h e f i n a l p o i n t o f t h e p r o c e d u r e ,

3,

⁼ l i m

G i ,

a l t h o u g h t h e e x i s t e n c e o f

l+='

s u c h a l i m i t i s e a s y t o p r o v e . From a p u r e l y m a t h e m a t i c a l p o i n t o f v i e w , i t would b e i n t e r e s t i n g t o examine u n d e r which assump- t i o n s on t h e d e c i s i o n m a k e r ' s b e h a v i o r w e c a n p r o v e t h a t

G,

a c t u a l l y maximizes ( o r m i n i m i z e s ) h i s u t i l i t y f u n c t i o n . From a

p r a g m a t i c a l p o i n t o f view, s u c h a n i n v e s t i g a t i o n would o n l y c o n f u s e t h e i s s u e s i n c e t h e u n d e r l y i n g m o t i v a t i o n o f t h e i n t e r - a c t i v e p r o c e d u r e i s t o f i n d a compromise d i r e c t l y i n t e r m s of g o a l s , n o t i n t e r m s of u t i l i t y f u n c t i o n s . Also, we do n o t e x p e c t t h e d e c i s i o n maker p l a y i n g w i t h t h e i n t e r a c t i v e p r o c e d u r e u n t i l

i + - i e x p e r i m e n t s show t h a t he v e r y soon a c c e p t s some

Gi,

p u t t i n g

- -

q i + I

- 4,

and t h u s s t o p p i n g t h e p r o c e d u r e . A t t h i s p o i n t , h e c a n a l s o be informed on t h e t r a d e - o f f s i m p l i e d by h i s d e c i s i o n :

h

w e i g h t i n g c o e f f i c i e n t s A i r e l a t e d t o t h e p o i n t

ai

c a n b e computed from e q u a t i o n ( 1 1 )

.

Observe a l s o t h a t t h e i n t e r a c t i v e p r o c e d u r e d o e s n o t depend on t h e s c a l i n g o r r a n g e s f o r s e p a r a t e o b j e c t i v e f u n c t i o n s . Nat- u r a l l y , t h e s c a l i n g must b e r e a s o n a b l e i n o r d e r n o t t o impede c o m p u t a t i o n a l e f f i c i e n c y n o r e x p o s i t i o n o f t h e r e s u l t s t o t h e d e c i s i o n maker, and it i s a d v i s a b l e t o u s e s c a l e s t h a t c o r r e s p o n d t o a p p r o x i m a t e l y e q u a l r a n g e s o f a t t a i n a b l e v a l u e s of o b j e c t i v e f u n c t i o n s . But t h i s r e q u i r e m e n t o f a r e a s o n a b l e s c a l i n g d o e s n o t imply an a p r i o r i s p e c i f i c a t i o n o f a v e c t o r o f w e i g h t i n g c o e f f i - c i e n t s , and t h e r e s u l t s a r e r e l a t i v e l y i n v a r i a n t t o t h e s c a l i n g t r a n s f o n n a t i o n ( a f t e r changing s c a l e s , b o t h

Gi

and a p o s t e r i o r i d e t e r m i n e d

Ri --

i f n o t n o r m a l i z e d

--

change p r o p o r t i o n a l l y ) .

3 . APPLICATION AREAS OF REFERENCE OBJECTIVES AND PENALTY SCALARIZATION

I n t y p i c a l m u l t i o b j e c t i v e o p t i m i z a t i o n , p e n a l t y s c a l a r i z a - t i o n c a n be u s e d n o t o n l y i n i n t e r a c t i v e p r o c e d u r e s o f d e c i s i o n making b u t a l s o i n a n a l y z i n g p o s s i b l e outcomes. F o r example, a

t y p i c a l q u e s t i o n : i s a g i v e n d e c i s i o n P a r e t o - e f f i c i e n t , o r n o t ? - can be c o n v e n i e n t l y r e s o l v e d by a p p l y i n g Lemma 4 , w h i l e a n ap- p l i c a t i o n of w e i g h t i n g c o e f f i c i e n t s r e s u l t s i n r a t h e r c o m p l i c a t e d p r o c e d u r e s .

One must b e a r i n mind however, t h a t a d e c i s i o n t h a t i s n o t P a r e t o - e f f i c i e n t i n t h e o p t i m i z a t i o n model m i g h t b e P a r e t o - e f f i c i e n t f o r t h e d e c i s i o n maker, f o r v a r i o u s r e a s o n s . F i r s t , t h e d e c i s i o n maker might c o n s i d e r o t h e r c r i t e r i a

--

f o r example,

o f a e s t h e t i c a l o r p o l i t i c a l n a t u r e - - t h a n t h o s e e x p r e s s e d by t h e model. S e c o n d , t h e d e c i s i o n maker m i g h t h a v e i n t u i t i v e l y a more

p r e c i s e a s s e s s m e n t o f v a r i o u s c o n s t r a i n t s , e t c . , o n l y i n a d e q u a - t e l y e x p r e s s e d by t h e model. C o n s e q u e n t l y , by l o o k i n g a t t h e o p t i m i z a t i o n model o n l y as a t o o l t o a i d t h e d e c i s i o n m a k e r , i t i s p o s s i b l e t o a n a l y z e t h e s e i n t e r e s t i n g q u e s t i o n s f u r t h e r , a n d r e f e r e n c e o b j e c t i v e s a r e c e r t a i n l y b e t t e r s u i t e d t h a n w e i g h t i n g c o e f f i c i e n t s f o r s u c h a n a n a l y s i s . However, much h a s t o b e d o n e y e t i n t h i s d i r e c t i o n o f r e s e a r c h .

A n o t h e r c o n v e n i e n t a p p l i c a t i o n o f p e n a l t y s c a l a r i z i n g f u n c - t i o n s i n t h e a n a l y s i s o f m u l t i c r i t e r i a p r o b l e m s i s t h e s c a n n i n g o f t h e P a r e t o s e t , n a t u r a l l y u n d e r t h e a s s u m p t i o n t h a t t h e num- b e r o f c r i t e r i a i s n o t t o o l a r g e . S c a l a r i z i n g f u n c t i o n s o f t h e t y p e ( 1 6 ) , ( 1 6 a ) , ( 1 7 ) , ( 1 8 ) c a n b e u s e d f o r t h i s p u r p o s e . An example o f a p p l i c a t i o n t o c o n t r o l e n g i n e e r i n g , see W i e r z b i c k i 1978 [ 2 5 ] , s h o w s t h a t t h e u s e o f w e i g h t i n g c o e f f i c i e n t s f o r t h a t p u r p o s e c a n l e a d t o d i s a s t r o u s r e s u l t s , w h i l e r e f e r e n c e o b j e c - t i v e s g i v e r e l i a b l e a n s w e r s . T h i s i s d e p i c t e d i n F i g u r e 6.

F I G U R E 6. REFERENCE O B J E C T I V E S VERSUS WEIGHTING C O E F F I C I E N T S WHEN SCANNING AN IRREGULARLY SHAPED PARETO - S E T

I n many a p p l i c a t i o n s o f dynamic m o d e l l i n g , s c a l a r - v a l u e d o b j e c t i v e f u n c t i o n s do n o t p r e c i s e l y e x p r e s s t h e g o a l s of a d e c i s i o n maker o r a m o d e l l e r . Of p r i m a r y i n t e r e s t i s o f t e n a f u n c t i o n o f t i m e , a t r a j e c t o r y o f t h e model. F o r example, a n e c o n o m i s t m i g h t w a n t t o compare t h e t r a j e c t o r i e s o f i n f l a t i o n r a t e s and o f GNP w h i l e n o t b e i n g r e a d y t o a v e r a g e them and t o u s e s c a l a r i n d i c e s , Thus, a function o f t i m e is a n equally reasonable goal in decision making as a scalar i n d e x , and ana- l y s t s a v o i d e d t h e u s e o f f u n c t i o n s as g o a l s o n l y b e c a u s e o f t h e l a c k o f a p p r o p r i a t e t e c h n i q u e s , However, t h e p o s s i b i l i t y o f u s i n g r e f e r e n c e o b j e c t i v e s i n a H i l b e r t s p a c e p r o v i d e s f o r a n appro- p r i a t e t e c h n i q u e . T h i s i s e x p l a i n e d i n F i g u r e 7 where, as a g o a l , an e c o n o m i s t s p e c i f i e d a d e s i r a b l e GNP a n d a r e a s o n a b l e i n f l a t i o n r a t e a s f u n c t i o n s o f t i m e . A model a f t e r an o p t i m i z a - t i o n , s a y , i n r e s p e c t t o t a x e s , r e s p o n d s by a t t a i n a b l e ( a n d , i n a s e n s e , P a r e t o - o p t i m a l ) f u n c t i o n s o f GNP and i n f l a t i o n , and t h e e c o n o m i s t can modify t h e n h i s r e f e r e n c e f u n c t i o n s i n o r d e r t o i n f l u e n c e t h e outcomes.

A

GNP I R

I

I

Model Response Inflation Rate

I

+ I

^{Time L}

FIGURE 7 . FUNCTIONS OF TIME OR TRAJECTORIES AS R E F E R E N a OBJECTIVES

I f t h e dynamic model i s t i m e - c o n t i n u o u s , a s d e p i c t e d i n

CI

F i g u r e 7 , t h e n we m i g h t c h o o s e , f o r e x a m p l e , t h e 'L [O;T] s p a c e f o r a n a l y s i s , w h i c h r e s u l t s i n t h e f o l l o w i n g e x p r e s s i o n f o r t h e s c a l a r i z i n g f u n c t i o n ( 8 a ) :

s ( q . q ) =

-1

⁽ ( q ( t ) 1

- 6'

^{( t ) )}

* ⁺

^{( q 2 ( t )}

^{- q2}

^{( t ) ) 2 , d t}

where ^{( a ) +} = m a x ( O I 8 ) and t h e c h a n g e t o ( ( t ) q 1 ( t ) ) + i n s t e a d o f (q 1 ( t )

- q l

( t ) ) + r e s u l t s f r o m t h e f a c t t h a t we maximize GNP and p e n a l i z e t h e G N P - t r a j e c t o r y i f it s t a y s below t h e d e s i r a b l e l e v e l . W e c a n a l s o u s e more g e n e r a l s p a c e s and o t h e r n o r m s - - f o r e x a m p l e , t h e Chebychev n o r m - - i f w e t a k e i n t o a c c o u n t t h e r e s u l t s p r e s e n t e d i n W i e r z b i c k i 1977, [23]

.

But m o s t p r a c t i c a l dynamic m o d e l s a r e t i m e - d i s c r e t e a n d a n

1 2

e c o n o m i s t m i g h t b e i n t e r e s t e d o n l y i n q ( t ) a n d q ( t ) f o r t = 0 1 T Then t h e p r o b l e m i s i n f a c t f i n i t e - d i m e n s i o n a l a n d w e d o n o t n e e d t h e H i l b e r t s p a c e f o r m u l a t i o n ; a l l f o r m s o f p e n a l t y s c a l a r i z i n g f u n c t i o n s d e s c r i b e d i n p r e v i o u s p a r a g r a p h s a r e a p p l i c a b l e i n s u c h a c a s e . - On t h e o t h e r h a n d , t h e number

1 2 2

o f o b j e c t i v e s q 0 q 1

,

q

( 1

^{I q} ( 0 ) ~ q 1 ) ^I

( 1

m i g h t be q u i t e l a r g e and i t i s c o n v e n i e n t t o t h i n k t h e n i n t e r m s o f d i s c r e t e - t i m e t r a j e c t o r i e s , n o t i n t e r m s o f s e p a r a t e o b j e c t i v e s .

The i d e a o f r e f e r e n c e t r a j e c t o r i e s h a s b e e n a p p l i e d and found u s e f u l i n a s t u d y o n t h e F i n n i s h f o r e s t r y i n d u s t r i a l s e c t o r ( K a l l i o and ~ e w a n d o w s k i , 1979, [1 4 1 )

.

T h e r e a r e many a p p r o a c h e s t o f i n d i n g p a r e t o - e q u i l i b r i a o r compromises i n c o o p e r a t i v e games. M o t i v a t e d by t h e o b s e r v a t i o n t h a t , i n r e a l i t y , a g r o u p o f d e c i s i o n makers p r e f e r s , f i r s t , t o d i s c u s s , b a r g a i n and a g r e e a b o u t g o a l s , a n a g r e e m e n t - a i d i n g

p r o c e d u r e b a s e d o n r e f e r e n c e p o i n t s i n t h e s p a c e o f o b j e c t i v e s h a s b e e n d e v i s e d .

T h i s p r o c e d u r e i s e x p l a i n e d i n F i g u r e 8 i n t h e c a s e o f two d e c i s i o n makers and two objective f u i ~ c t ion^. Faeh W ~ i e irrn makJ+-

h a s i t s "ownN o b j e c t i v e f u n c t i o n , k h e l e v e l s o f which however do

1 1 2 2

n o t o n l y depend on h i s d e c i s i o n s : q = q ( x l , x 2 ) and q = q 2 ( x 1 ,x ) . I t i s a l s o assumed t h a t t h i s i s a c a s e o f f u l l i n f o r m a t i o n and

e a c h d e c i s i o n maker f u l l y u n d e r s t a n d s t h e o b j e c t i v e and p o s s i b l e a c t i o n s of t h e o t h e r o n e . Moreover, t h e c h o i c e o f model d e c i s i o n s x1 , x 2 i s d e l e g a t e d t o an o p t i m i z a t i o n p r o c e d u r e b a s e d on t h e math- e m a t i c a l model; t h e d e c i s i o n makers o n l y s t a t e and t r y t o a g r e e o n r e f e r e n c e o b j e c t i v e l e v e l s . Each o f them i s a s k e d t h u s t o

-1 -2

s p e c i f y h i s r e f e r e n c e p o i n t

qiO

^{= (qiolqiO ) , i =} 1 , 2 , which ex- . p r e s s e s h i s judgment a b o u t h i s own a s p i r a t i o n l e v e l and t h a t of

h i s p a r t n e r ; l a t e r , t h e s e r e f e r e n c e p o i n t s a r e m o d i f i e d i n an i t e r a t i v e p r o c e d u r e t o

cij

u n t i l e i t h e r l i m

q

^{= l i m}

q2

o r t h e

,+a 1 j ,+a

n e g o t i a t i o n s b r e a k down d i s p l a y i n g an a d v e r s a r y s i t u a t i o n .

FIGURE 8. AN AGREEMENT-AIDING PROCEDURE BASED ON REFERENCE OBJECTIVES

S u p p o s e , t h e f i r s t d e c i s i o n maker w i t h a d o m i n a n t o b j e c -

1 -1 -2

t i v e t o m i n i m i z e q

,

h a s s p e c i f i e d

q1

= ( q l , q l 0 ) n a t u r a l l y a l l o t t i n g a r e l a t i v e l y l o w e r l e v e l f o r

4;

h i s 'own" o b j e c t i v e t h a n f o r q I 0 , t h e o n e of h i s p a r t n e r . -2 C o r r e s p o n d i n g l y , t h e

-1 -2

p o i n t

c20

⁼ ( q 2 0 , q 2 0 ) s p e c i f i e d by t h e s e c o n d d e c i s i o n maker

-1 -1 -2 -2

h a s > q 1 0 a n d q 2 0 < q l O , b e c a u s e h e i s i n t e r e s t e d i n m i n i m i z i n g h i s "own" o b f e c t i v e q 2

.

S i n c e w e c a n a s s u m e , a t t h e b e g i n n i n g , n o t h i n g e l s e t h a n t h e e q u i t y of e a c h d e c i s i o n m a k e r ' s r e q u i r e m e n t s , t h e a g r e e m e n t - a i d i n g p r o c e d u r e s i m p l y d e t e r m i n e s

%

a s t h e m i d d l e - p o i n t of

-

t h e segment [ c l 0 ; q 2 0 ] ( o r o f a c o r r e s p o n d i n g s i m p l e x i n c a s e o f more d e c i s i o n m a k e r s ) a n d r e s p o n d s t h r o u g h a m i n i m i z a t i o n o f o n e of t h e p e n a l t y s c a l a r i z i n g f u n c t i o n s from p r e v i o u s p a r a - g r a p h s by a P a r e t o p o i n t

Go

c o r r e s p o n d i n g t o

%

^{a s w e l l a s by}

P a r e t o p o i n t s Q1 $20 c o r r e s p o n d i n g t o

6,

0 , q 2 0 .

-

T h i s way b o t h d e c i s i o n m a k e r s h a v e a p r o p o s i t i o n of compromise and i n f o r m a t i o n a b o u t a t t a i n a b l e l e v e l s o f o b j e c t i v e s . The d i s t a n c e s d 1 0 =

- -

lk0 -

O1 I and d 2 0 = llQO

-

q2011 a r e a l s o d e t e r m i n e d .

Now b o t h d e c i s i o n m a k e r s have t o make c o n c e s s i o n s i n terms of two s c a l a r s a l , a 2 E [ B ; 1 ]

,

where B E ( 0 ; 1 ] i s a p r e s p e c i f i e d m i n i m a l c o n c e s s i o n l e v e l . The m o d i f i e d r e f e r e n c e p o i n t s

; i l l ,

-

92 1 a r e d e t e r m i n e d by

Thus, b o t h d e c i s i o n m a k e r s have t o move i n t h e d i r e c t i o n o f

6

a t l e a s t B t i m e s t h e d i s t a n c e d i j . I n F i g u r e 8 , it was assumed 1' t h a t t h e f i r s t d e c i s i o n maker made o n l y t h e m i n i m a l c o n c e s s i o n a l = B , w h i l e t h e s e c o n d d e c i d e d t o make a b i g g e r o n e , a 2 > B . When

G1 _,

_{j + l and G 2 , j+l} a r e d e t e r m i n e d , t h e p r o c e d u r e i s r e p e a t e d .

The mechanism o f t h i s p r o c e d u r e v e r y s t r o n g l y u r g e s b o t h d e c i s i o n makers t o r e a c h an a g r e e m e n t . T h e r e f o r e , a t some s t a g e of t h e p r o c e d u r e , o n e o r b o t h o f t h e d e c i s i o n makers c a n d e c i d e i f h e s h o u l d b r e a k t h e n e g o t i a t i o n s , t h a t i s , n o t making any

f u r t h e r c o n c e s s i o n s . Two f u r t h e r p o s s i b i l i t i e s c a n b e e n v i s a g e d :