Chapter9

# Chapter9 - Multiple constraints and compound objectives 9.1...

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Multiple constraints and compound objectives 9.1 Introduction and synopsis Most decisions you make in life involve trade-offs. Sometimes the trade-off 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 trade-off 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 wing-spar, weight must be minimized, with constraints on stiffness, fatigue strength, toughness and geometry. In the design of a disposable hot-drink 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 wing-spar 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 weight-factors, 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

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Multiple constraints and compound objectives 21 1 Fig. 9.1 The procedures for dealing with multiple constraints and compound objectives.
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## This note was uploaded on 04/26/2008 for the course M&AE 212 taught by Professor Miller during the Spring '07 term at Cornell.

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Chapter9 - Multiple constraints and compound objectives 9.1...

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