Multiple constraints and compound
objectives
9.1
Introduction and synopsis
Most decisions you make in life involve tradeoffs. Sometimes the tradeoff is to cope with
conflicting constraints: I must pay this bill but I must also pay that one

you pay the one which
is most pressing. At other times the tradeoff is to balance divergent objectives: I want to be rich
but I also want to be happy

and resolving this is harder since you must balance the two, and
wealth is not measured in the same units as happiness.
So it is with selecting materials. Commonly, the selection must satisfy several, often conflicting,
constraints. In the design of an aircraft wingspar, weight must be minimized, with constraints on
stiffness, fatigue strength, toughness and geometry. In the design of a disposable hotdrink cup,
cost is what matters; it must be minimized subject to constraints on stiffness, strength and thermal
conductivity, though painful experience suggests that designers sometimes neglect the last. In this
class of problem there is one design objective (minimization of weight or of cost) with many
constraints. Nature being what it is, the choice of material which best satisfies one constraint will
not usually be that which best meets the others.
A second class of problem involves divergent objectives, and here the conflict is more severe. The
designer charged with selecting a material for a wingspar that must be both as light
and
as cheap as
possible faces an obvious difficulty: the lightest material will certainly not be the cheapest, and vice
versa. To make any progress, the designer needs a way of trading off weight against cost. Strategies
for dealing with both classes of problem are summarized in Figure 9.1 on which we now expand.
There are a number of quick although subjective ways of dealing with conflicting constraints
and objectives: the
sequential index
method, the
method
of weightfactors,
and methods employing
fuzzy logic.
They are a good way of getting into the problem, so to speak, but their limitations must
be recognized. Subjectivity is eliminated by employing the
active constraint method
to resolve
conflicting constraints, and by combining objectives, using
exchange constants,
into a single
value
function.
We use the beam as an example, since it is now familiar. For simplicity we omit shape (or set
all shape factorrs equal
to
1); reintroducing it is straightforward.
9.2
Selection
by
successive application
of
property limits
and indices
Suppose you want a material for a light beam (the objective) which is both stiff (constraint 1)
and strong (constraint 2), as in Figure 9.2. You could choose materials with high modulus
E
for