Logical Decisions - Classweb

Appendix
This appendix describes the mathematics of computing the
relative weights of two measures based on a tradeoff between
them. The process is illustrated with a simple example.
Computing Relative Weights from a Tradeoff
The overall utility of any alternative X is assumed to be the
weighted average of its SUF utilities as follows:
U(X) = k1U 1(X) + k2U 2(X) + ... + knU n(X),
where U(X) = the overall utility for alternative X
k i = the weight for measure i; also called the scaling
constant small k for measure i.
U i(X)
= the SUF utility on measure i for alternative
X.
Call the two alternatives in the tradeoff A and B and suppose they
differ only in measures 1 and 2. Then the overall utility for the
alternatives is
U(A) = k1U 1(a 1) + k2U 2(a 2) + ... + knU n(a n),
U(B) = k U (b ) + k U (b ) + ... + k U (b ).
1 1 1 2 2 2 n n n
Since the alternatives in the tradeoff are equally preferred, they
must have equal overall utilities. This means
so that
U(A) = U(B)
k1U 1(a 1) + k2U 2(a 2) + ... + knU n(a n))
=
k U (b ) + k U (b ) + ... + k U (b ).
1 1 1 2 2 2 n n n
but, since alternatives A and B only differ on measures 1 and 2,
we have
U i(a i) = U i(b i)
Appendix A-1