Order-representing achievement functions

We consider here achievement functions first for the case when D = RP+; for the more complicated positive cone (4.2), order-representing functions are anyway not interesting, since their maxima would occur a t any admissible decision and attainable outcome, all attainable outcomes being weakly efficient for positive cones without inte- rior. If D = RP+, a general form of an order-representing function can be written as fol- lows:

where v: RP

-

^R' is a strongly monotone value (or utility) function with the property that v(q-Q) = 0 for all q - Q E D\int D , and any norm in RP can be taken t o define the distance. If we take a multiplicative form of v

-

for example, the Nash (1950) compromise function

-

and use the norm lk with k 2 2, then the function s(q,Q) is differentiable except for q - Q E D\int D:

where (fi ^- qi)+ = max(0, Qi - 9,).

Another form of order-representing function is piece-wise linear and can be inter- preted as a n exact penalty function for the characterization (4.31) of efficient solutions:

with some ai

>

0. This function is determined by the sum only for such q -

q

E D that

which is possible only when p

>

1

-

see Figure 4.8.

The above function is useful when applied t o linear vector optimization problems, where

Q,

is a convex polyhedral set. In such cases, we rewrite the problem of maximizing (4.42) by using additional variables zi = ai(qi - Q,), i = l,..p, zp+, = s(q,Q), t o the fol- lowing form:

Observe t h a t the additional variables z,, zp+l are not restricted in sign; hence, when using standard linear programming codes, additional modifications might be needed. This func- tion has been used in the DIDAS system of decision support - see Lewandowski e t al.

(1982)) Grauer et.a1.(1984) and further chapters. Similar transformations are possible for all concave or concave-like

-

see Jahn (1984) - piece-wise linear functions s(q,Q), such as (4.35)) (4.37) or their further modifications given below.

Figure 4.8. Level sets of the order-representing function (4.42).

The prototype order-representing function (4.35) has also several modifications in case when additional information about Q, is available. The function (3.15a,b), or the Chebyshev norm (4.24) under transformation (4.25), after slight modifications can be written in the form:

~ ( q y f ) = I ~ , i ~ p ( q i - f i ) / ( C , m a z -

b)

^(4.44)

where q'i,maz > tji,moz, i = l,.. .p, is an upper bound higher than utopia point component for the i-th outcome or objective function. In applications t o interactive decision support, when the user can change the controlling parameter f arbitrarily, an important considera- tion is t h a t fi should be always smaller than, and

-

for computational reasons

-

not too close t o q',,,,,. This can be practically secured by selecting an additional scaling point q' such that Bi,maz < Q;. < q'i,maz

-

a reasonable choice might be, for example,

and by using this scaling point as an upper bound for aspiration levels. The user should be informed t h a t his aspirations qi must not exceed

6 ,

but the decision support system should also automatically take fi = q', if the user specifies fi >

>,.

This restriction might be considered as a drawback of function (4.44); however, the function has several other advantages. Firstly, the function has a cardinal form (it is independent of positively monotone affine transformations of the outcome space) and can be thus used as an

approximation t o a cardinal utility function of the user. Secondly, the weighting coefficients ai = l/(cj,,,, - tj,), implied by the aspirations tji specified by the user, represent the relative importance of various outcomes or criteria t o him: the more close

qi

t o g,,,, or

&,

the more important is the i-th outcome. When using achievement function (4.44), he can much more easily influence the selection of an efficient outcome t j = 4(tj) by changing the controlling parameter tj, than when using functions (4.35) or (4.37), see Fig- ure 4.9.

Figure 4.9. Controllability properties of order-consistent achievement functions: (a) functions (4.35), (4.37); (b) functions (4.44), (4.49); observe the difference in

4"'

= $J(Q"') in these cases.

Another modification of the prototype order-representing function (4.35) is function (3.14a,b), based on two general information points

-

an upper bound point ij,, > ,,) and a lower bound point <

imi, -

as well as on two reference points spec~fied by the user

-

a reservation point Q' and a n aspiration point $"' whereas

ijilmin < fit <

q,"

<

4irnaz,

^{i = 1,.}

.

. p

For the convenience of the reader, we repeat here the definition of this function:

~(q,tj',Q") = min pi(qi,ijit ,Qif' ) ^(4.45) l s i s p

T h e main controlling parameter, corresponding t o the role of Q in the definition of order-representing achievement functions, is here the reservation level point Q'. However, the use of both aspiration and reservation levels as controlling parameters by the user of a decision support system expresses an important aspect of user's uncertainty in aspira- tions. Because of the form of this function, the user obtains important information about attainability of his aspirations and reservations, contained in the maximal values of (4.45) over q E Q,. Since

the user knows t h a t his reservation levels are attainable and the aspiration levels are not, if the maximal values of this function are contained between 0 and 1.

The above function can be used in decision support systems with subjective evalua- tion of merits of discrete decision alternatives by a committee of experts

-

for example, the SCDAS system, see Lewandowski et al. (1985); in such applications, the parameters P , r > 0 can be chosen arbitrarily to obtain a scale of achievement that is easily interpret- able by the user (for example,

P

⁼ 7 = 1 , which results in achievement values -1 for the lower bound, 0 for reservation levels, 1 for aspiration levels and 2 for the upper bound). In applications t o decision support systems with a substantive model of the outcome m a p ping q=f(z), however, an important consideration is t h a t all functions p i should be con- cave in q, which simplifies computational aspects of maximizing s(q,~',q") over q E Q,.

This can be achieved by selecting appropriate values of the parameters 0 , ~ and addition- ally restricting the selection of aspiration and reservation levels. Such additional restric- tions (which should obviously be communicated t o the user but also imposed automati- cally by the system) might take, for example, the form:

q," 5

^,ji"

where

and

where

Q;.'

= di,rnin > Q;.,min - - ii,min - O.Ol(ii,ma ^- qi,min)

and, finally,

For the concavity of pi, it is sufficient then t o take

p

⁼ 0.01 or less and 7 = 100 or more.

If the functions p i are concave and Q, is a convex polyhedral set, defined by a number of linear inequalities or equations, then the problem of maximizing (4.45) over q E Q, can be rewritten equivalently as a linear programming problem of maximizing a n additional variable z (not restricted in sign), where:

Im Dokument Aspiration Based Decision Analysis and Support Part I: Theoretical and Methodological Backgrounds (Seite 79-83)