Criterion Functions and the Role of Values in the Engineering
Design Process
Engineering Ethics Assignment
Optimization is essential to engineering design. It seeks to adapt engineering artifacts to
particular goals and values, maximizing intended benefits and minimizing undesirable
consequences. Prior to WWII optimization was often confused with efficiency – the
maximization of output with respect to input – and treated as an inevitable consequence of
proper application of the design process. Methods were developed and deployed for maximizing
efficiency, but optimization was not treated explicitly. After WWII it became clear that optimal
designs were not necessarily the most efficient. Engineers searched for mathematical methods to
objectively establish optimal systems, and it was discovered that mathematical models of
engineering systems cannot ignore values.
Various methods now exist for representing design solutions mathematically. Such
representations are known as “criterion” or “objective functions.” Such a function represents a
solution as a sum of the various design criteria, ci, multiplied by a weighting coefficient, wi. Each
wi is assigned a numeric value on an arbitrary scale with each step on the scale representing the
relative importance of that criterion with respect to the other criteria. As Thomas Woodson notes,
such methods reveal the overriding importance of values and subjective decisions as part of the
design process.
As we look behind the scenes, we find major influences on decision-making coming
from the individual’s own value system, from that of his organization, and from the
culture, as well as from the technology. (Introduction to Engineering Design, 204)
The engineer must take responsibility for making such criteria explicit, assessing them, and
making appropriate decisions. If not, deleterious implicit assumptions may remain unanalyzed
and dominate a design solution or, what may be equally as bad, someone less technically
qualified will make the decisions for the engineer.
Formal procedures for enumerating design criteria and assigning weights provide a rational
means for deciding between competing, design solutions so positive outcomes can be maximized
and negative effects minimized. These procedures can be quite complex and even demand
extensive computational resources to adequately implement when many solutions with dozens or
even hundreds of design criteria are involved.
Often however, specific solutions or classes of solutions can be easily eliminated. This can be
demonstrated with the help of a hypothetical case. Consider the design problem with the criteria
ci, where i = 1, 2, 3, 4, and the set of possible solutions dj, where j = 1, 2, 3, 4, 5. This design
situation can be represented in tabular form as follow:
1
General Selection Problem
SOLUTIONS
CRITERIA
d1
d2
d3
d4
d5
c1
10
4
6
3
6
c2
600
700
580
500
660
c3
0.4
0.7
0.9
0.1
0.5
c4
2.1
3.4
2.4
2.0
1.9
c5
7
5.5
8
5
6
Fig. 1
In this problem, solution d4 is completely dominated by solution d2, that is, the value of every
criterion for d2 is larger than the corresponding criterion in d4. It would be irrational to select a
dominated solution. To do so would mean selecting a solution that is in every way inferior to at
least one of the alternatives. Of course, additional criteria might be discovered or included in the
selection problem which would make d4 an attractive alternative once again or possibly introduce
other possible solutions. These other solutions might dominate the remaining solutions. In any
case, if every relevant criterion is considered, then any dominated solution must be eliminated.
The remaining solutions constitute an efficient set in that no element of the set has all its
criterion values higher than those of any other element in the set, i.e., some values will be higher
while others are lower.
General Selection Problem
SOLUTIONS
CRITERIA
d1
d2
d3
d4
d5
c1
10
4
6
3
6
c2
600
700
580
500
660
c3
0.4
0.7
0.9
0.1
0.5
c4
2.1
3.4
2.4
2.0
1.9
c5
7
5.5
8
5
6
Fig. 2
With solution d4 eliminated, there still remain four alternatives to choose from. However, it is not
immediately obvious which solution is the best. If more criteria and possible solutions were
involved the choice would be even more inscrutable. The reason for this is that the criteria, i.e.,
the performance characteristics, are generally not independent. If they were, one could optimize
each criterion independently with the result that there would be one solution dominating all
others. In most situations though, optimizing involves a trade-off.
The relevant concepts are illustrated in Fig. 3. If criterion 1 and 2 are independent then each
could be maximized without affecting the other. The solutions obtained would be A and B
respectively. The ideal solution, I, would be obtained by combining the maximized performance
of both criteria. The area bounded by the dashed lines (BI and IA) and the lines BO and OA
2
represents the theoretically possible solutions. Various real-world and modeling constraints
reduce this to a smaller set of physically realizable solutions. These solutions are represented by
the shaded area under the dotted line or the Pareto front.
Ideal Solution
B
I
Pareto Front
Criterion 2
X
O
Criterion 1
A
Fig. 3
Not all physically realizable solutions are to be preferred though. Take the solution X. X is a
non-dominated solution with respect to every other solution in the purely shaded area under the
Pareto Front, i.e., at least one of X’s criteria, if not both, are better than every solution in the
purely shaded area. However, X is dominated by every solution in the hatched-area. As X
approaches the Pareto Front, the set of design solutions that dominates X grows smaller. When X
lies on the dotted line, no solution can be said to dominate it. This is true of every point on the
boundary line between the physically possible and that which is not feasible.
Points on the Pareto front then represent a set of non-dominating solutions. Improvement in one
criterion can only be achieved at the expense of the other criterion. As an example, consider the
design of a car. Let us assume that the only two criteria of importance are cost1 (criterion 1) and
safety (criterion 2). If cost was a primary consideration, then one would want to select a design
solution on the Pareto Front close to A. If you placed more value on safety, then you would
select a Pareto solution closer to B. The exact solution would depend on the relative weights you
assigned each criterion.
The only way then to select the “best” solution from this set is to impose a preference structure
upon the problem that embodies value judgments in such a way that design criteria are
differentially weighted. There are many different systems for doing this either directly,
indirectly, implicitly, or interactively. One such method will be considered below.
The Analytical Hierarchical Process (AHP) provides a systematic formal approach to developing
criterion weights. The method requires the construction of a comparison table in which criteria
are listed both vertically and horizontally as indicated in figure 4.
1
Note: All criteria can be thought of as maximizing. While cost is more naturally thought of as a loss or minimizing
quantity, it can easily be converted to a maximizing one by changing the sign.
3
Criterion Comparison Table
CRITERION
A
B
C
D
E
A
B
C
D
E
Totals
Fig. 4
A pair-wise comparison of the criteria is then made using preference ratings from the following
scale:
9. Absolutely more important/preferred
7. Very strongly more important/preferred
5. Strongly more important/preferred
3. Moderately more important/preferred
1. Equally as important/preferred
(Note: Even numbers are used for half-steps.)
If a row criterion is more important than the column criterion it is being compared to then the
appropriate whole number is entered. If a column criterion is more important then the reciprocal
is used. When a given criterion is compared to itself then ‘1’ is used.
As an example, consider the design problem with the following criteria: A, B, C, D, E and where
A is deemed “strongly more important” than B and “moderately more important” than E. A is
also held to be “equally as important” as both criterion C and D. Obviously, A is “equally as
important” as itself. Using the scale above, one would enter the following values.
Criterion Comparison Table
CRITERION
A
B
C
D
E
A
1
5
1
1
3
B
1/5
C
1
D
1
E
1/3
Totals
Fig. 5
4
Let us further assume the following determinations are made through the same process of pairwise comparison. (Note: the diagonal will always be ‘1’s.)
Criterion Comparison Table
CRITERION
A
B
C
D
E
A
1
5
1
1
3
B
1/5
1
3
7
3
C
1
1/3
1
1/5
1/9
D
1
1/7
5
1
3
E
1/3
1/3
9
1/3
1
53/15
133/21
19
143/15
91/9
Totals
Fig. 6
After all the cells are filled the column totals are computed.
The next step in computing the criteria weights is to normalize by columns. This is accomplished
by dividing each column element by its column total. Each row then should be summed to get
the individual row totals as shown below. These values are raw criterion weights.
COMPUTED CRITERION WEIGHTS
A
B
C
D
E
ROW
TOTAL
NRM
WGTS
RANK
A
0.283
0.789
0.053
0.105
0.297
1.527
0.301
1st
B
0.057
0.157
0.158
0.734
0.297
1.403
0.276
2nd
C
0.283
0.053
0.053
0.021
0.011
0.421
0.083
5th
D
0.283
0.023
0.263
0.105
0.297
0.971
0.191
3rd
E
0.094
0.053
0.474
0.035
0.099
0.755
0.149
4th
TOTALS
1.000
1.000
1.000
1.000
1.000
5.077
1.000
CRITERION
Fig. 7
One can then determine the normalized weights by dividing the individual row totals by the
“Row Total” column sum. Explicitly, the row totals for A = 1.527, B = 1.403, C = 0.421, D =
0.971, and E = 0.755 are divided by 5.077 to obtain the corresponding “Normalized Weights”
column.
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Aggregated Score
Design 5
Design 3
Design 2
Weight
Automobile
Decision Matrix
Design 1
Once the weights have been determined, they must be appropriately combined with the criterion
performance characteristics of the alternative solutions before a final determination can be made
concerning the “best” solution. To illustrate, assume the general selection problem represented in
Fig. 2 also characterizes an automobile design. The criteria A, B, C, D, and E of Figures 6 and 7
will be taken to map to c1, c2, c3, c4, and c5 which in turn represent safety, cost, comfort, style,
and mileage.2 These elements along with the non-dominating solutions and the criteria weights
can be displayed in a decision matrix.
?
?
?
?
?
c1: Safety
0.301
10
4
6
6
c2: Cost
0.276
600
700
580
660
c3: Comfort
0.083
0.4
0.7
0.9
0.5
c4: Style
0.191
2.1
3.4
2.4
1.9
C5: Mileage
0.149
7
5.5
8
6
Fig. 8
One final point must be considered before calculating the criterion functions for the alternate
design solutions, viz., the suitability of the scales being used to assign the specific numerical
values to each criterion, ci. Notice that in Fig. 8 the numerical values for the “Cost” criterion, c2,
is two to three orders of magnitude greater than for the “Safety” criterion, c1. This means either
c1 had relatively small impact on the design or that the scales being used to measure performance
need adjustment. Given that the weight for c1 is greater than that for c2 the former seems highly
unlikely. Thus, the design engineer is this case should adjust the maximum or minimum values
of his criterion variables so the assigned criterion weights play an appropriate role in determining
the “best” solution.
One might accomplish this by adjusting the individual scales so the maximum value is unity or
so that the mean value for each scale is the same. Using the former approach, and working with
the following knowledge derived from the design process concerning the criterion scales:
2
Note: Each of these design criteria can be disaggregated into sub-criteria. It is highly desirable that any design
criteria be thus decomposable.
6
Criterion
Scale
Safety
Cost
Comfort
Style
Mileage
0-10
0-1000
0-1.0
1-4
1-10
Aggregated Score
Design 5
Design 3
Design 2
Weight
Automobile
Decision Matrix
Design 1
the following revised table is constructed:
?
?
?
?
?
c1: Safety
0.301
1
0.4
0.6
0.6
c2: Cost
0.276
0.6
0.7
0.58
0.66
c3: Comfort
0.083
0.4
0.7
0.9
0.5
c4: Style
0.191
0.525
0.85
0.6
0.475
C5: Mileage
0.149
0.7
0.55
0.8
0.6
Fig. 9
From these values, the criterion functions for each design solution can be computed as follows:
d1 = (0.301)(1) + (0.276)(0.6) + (0.083)(0.4) + (0.191)(0.525) + (0.149)(0.7) = 0.704*
d2 = (0.301)(0.4) + (0.276)(0.7) + (0.083)(0.7) + (0.191)(0.85) + (0.149)(0.55) = 0.616
d3 = (0.301)(0.6) + (0.276)(0.58) + (0.083)(0.9) + (0.191)(0.6) + (0.149)(0.55) = 0.649
d5 = (0.301)(0.6) + (0.276)(0.66) + (0.083)(0.5) + (0.191)(0.475) + (0.149)(0.6) = 0.584
Thus, design d1 proves to be the optimal solution.
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