Logical Decisions - Classweb

for all measures i except 1 and 2. Thus all of these terms cancel
out of the equation above leaving us with
kU(a) 1 1 1 + kU(a) 2 2 2 = kU(b) 1 1 1 + kU(b).
2 2 2
since we are looking for the weights k 1 and k 2,
we can rearrange
this to give k in terms of k :
so that
2 1
k 2(U 2(a 2) - U 2(b 2)) = k 1(U 1(b 1) - U 1(a 1))
k 2/k 1 = (U 1(b 1) - U 1(a 1))/(U 2(a 2) - U 2(b 2)).
If we know the individual SUFs, we can compute all of the terms
in the right hand side of the equation and thus we can find the
ratio of the weights for measures 1 and 2. If we establish a
tradeoff involving each of the measures and add the restriction
that the weights must sum to 1 we can solve for the exact values
of all the weights and thus completely define an additive MUF.
Example of Relative Weight Computation
Suppose we have two trucks, A and B that are identical except for
their price and horsepower. A costs $10,000 and has 120 hp and B
costs $12,000 and has 150 hp. Suppose also that we know that A
and B are equally preferred and that the SUFs for cost and
horsepower are known to give following SUF utilities:
U cost($10,000)
= 0.5
U ($12,000) = 0.4
cost
U hp(120)
= 0.2
U (160) = 1.0.
hp
This means that a change in the utility of cost of 0.5 - 0.4 = 0.1 is
just compensated for by a change in the utility of horsepower of
1.0 - 0.2 = 0.8. Thus the ratio of the weights for cost and
horsepower must be
k hp/k cost = (U cost(b cost) - U cost(a cost))/(U hp(a hp)
-
U hp(b hp))
= (0.4 - 0.5)/(0.2 - 1.0)
= (-0.1)/(-0.8)
A-2 Appendix