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
Materials Selection in Mechanical Design, 4th Edition, © 2010 Michael Ashby
Strategies for tackling selection with
multiple constraints and conflicting
objectives
Materials Selection in Mechanical Design, 4th Edition, © 2010 Michael Ashby
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:
Materials Selection in Mechanical Design, 4th Edition, © 2010 Michael Ashby
Materials Selection in Mechanical Design, 4th Edition, © 2010 Michael Ashby
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
Materials Selection in Mechanical Design, 4th Edition, © 2010 Michael Ashby
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
Materials Selection in Mechanical Design, 4th Edition, © 2010 Michael Ashby
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)
Materials Selection in Mechanical Design, 4th Edition, © 2010 Michael Ashby
Figure 7.4
Conflicting Objectives
Materials Selection in Mechanical Design, 4th Edition, © 2010 Michael Ashby
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
Materials Selection in Mechanical Design, 4th Edition, © 2010 Michael Ashby
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
Materials Selection in Mechanical Design, 4th Edition, © 2010 Michael Ashby
Figure 7.7
Materials Selection in Mechanical Design, 4th Edition, © 2010 Michael Ashby
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
Materials Selection in Mechanical Design, 4th Edition, © 2010 Michael Ashby
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
Materials Selection in Mechanical Design, 4th Edition, © 2010 Michael Ashby
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).
Materials Selection in Mechanical Design, 4th Edition, © 2010 Michael Ashby
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.
Materials Selection in Mechanical Design, 4th Edition, © 2010 Michael Ashby
Figure 7.10