Multiple Constraints and Conflicting Objectives

Chapter 7Multiple Constraints

and Conflicting

Objectives

Materials Selection in Mechanical Design, 4th Edition, © 2010 Michael Ashby

Multiple Constraints and

Conflicting Objectives

The selection of a material or process must

satisfy several often conflicting constraints;

a second class of problem involves more

than one objective, and here the conflict is

more severe


Strategies for tackling selection with

multiple constraints and conflicting

objectives


Figure 7.1

Selection With Multiple Constraints

Nearly all material selection problems are overconstrained, meaning there are more constraints than free variables

Selection involves identifying the constraints and the objective and applying the following steps:



Figure 7.2

The screening stage imposes

constraints on properties, on

requirements such as corrosion

resistance, or on the ability to

be processed in a certain way

The candidate materials that

survive the screening stage

are ranked using property

charts

The simple process of screening and ranking

becomes more complex for the special case of a

single objective that can be limited by more than

one constraint


Example: The requirements of a tie-rod of minimum mass might

specify both stiffness and strength, leading to two independent

equations for the mass

If stiffness is the dominant constraint, the mass of the rod is m1;

if it is strength, the mass is m2

If the tie is to meet the requirements on both, its mass has to be

the greater of m1 and m2

One objective (here, minimizing mass) with two

constraints leads to two performance equations,

each with its own value of MMaterials Selection in Mechanical Design, 4th Edition, © 2010 Michael Ashby

Figure 7.3

We seek the smallest value of a metric that is the

larger of two or more alternatives

Analytical Method


Graphical Method

Coupled selection can be done using performance metrics as in (a), or using

material indices M and a coupling constant Cc as in (b)


Figure 7.4

Conflicting Objectives


Real-life materials selection almost

always requires that a compromise be

reached between conflicting objectives

Trade-Off Strategies

Strategy 1

A shortlist of materials can be identified by plotting the performance

metrics against one another;

solutions on or near the trade-off surface offer the best compromise,

the rest can be rejected


Figure 7.5

Figure 7.6

Strategy 2

One objective can be reformulated

as a constraint; in this example, an

upper limit is set on cost; however,

this is not a true optimization

The trade-off surface identifies the subset of solutions that offer the

best compromises between objectives. To obtain a single solution, we

must aggregate the various objectives into a single objective function,

formulated such that its minimum defines the most preferable solution.

To do this we define a locally linear penalty function Z


Figure 7.7


Relative Penalty Functions

When we seek a better material for an existing application, it is more helpful to compare the

new material choice with the existing one; To do this we define a relative

penalty function


Figure 7.8

An exchange constant is a measure of the penalty of unit increase in a performance metric, or it is the value or “utility” of a unit

decrease in the metric


A cost-mass trade-off plot for bicycles. The tangent to the trade-off surface at any point gives an estimate of the exchange

constant. It depends on the application. To a consumer seeking a cheap bike for shopping, the value of weight savings is low ($20/kg). To an enthusiast who wants performance, it can be

high ($2000/kg).


Figure 7.9

It is often the case that a single material (or subset of materials) is optimal over a wide range of values of

the exchange constant. Then approximate values for exchange constants are sufficient to reach precise

conclusions about the choice of materials.


Figure 7.10

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Multiple Constraints and Conflicting Objectives