Metal Foams: A Design Guide

208 Metal Foams: A Design Guide
3. A composite objective function or value function, V, is formulated; the solution
with the minimum value of V is the overall optimum, as in Figure 16.7.
This method allows true multi-objective optimization, but requires more
information than the other two. It is explored next.
Performance metric P 2
V 1
V 2
V 3
Decreasing,
V
V 4
Optimum
solution
Performance metric P 1
Contours of
value, V
Figure 16.7 A value function, V, plotted on the trade-off diagram. The
solution with the lowest V is indicated. It lies at the point at which the value
function is tangent to the trade-off surface
Value functions
Value functions are used to compare and rank competing solutions to multiobjective
optimization problems. Define the locally linear value function
V D ˛1P1 C ˛2P2 C ......˛iPi .... ⊲16.5⊳
in which value V is proportional to each performance metric P. The coefficients
˛ are exchange constants: they relate the performance metrics P1,P2 ...to V,
which is measured in units of currency ($, £, DM, FF, etc.). The exchange
constants are defined by
� �
∂V
˛1 D
⊲16.6a⊳
∂P1 P2,...Pi
� �
∂V
˛2 D
⊲16.6b⊳
∂P2
P1,...Pi
that is, they measure the change in V for a unit change in a given performance
metric, all others held constant. If the performance metric P1 is mass m (to be
minimized), ˛1 is the change in value V associated with unit increase in m. If