Criterion Functions and the Role of Values in the Engineering Design Process

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:

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.

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 eachcould 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 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.

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 cost[1] (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.

Criterion Comparison Table
CRITERION / A / B / C / D / E
A
B
C
D
E
Totals

Fig. 4

A pair-wise comparison of the criteriais 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

Let us further assume the following determinations are made through the same process of pair-wise 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
Totals / 53/15 / 133/21 / 19 / 143/15 / 91/9

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
CRITERION / 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

Fig. 7

One can then determine the normalized weights by dividing the individualrow 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 5.077to obtain the corresponding “Normalized Weights” column.

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.

Automobile Decision Matrix / Weight / Design 1 / Design 2 / Design 3 / Design 5
Aggregated Score / ? / ? / ? / ? / ?
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:

CriterionScale

Safety0-10

Cost0-1000

Comfort0-1.0

Style1-4

Mileage1-10

the following revised table is constructed:

Automobile Decision Matrix / Weight / Design 1 / Design 2 / Design 3 / Design 5
Aggregated Score / ? / ? / ? / ? / ?
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.

[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.

[2] Note: Each of these design criteria can be disaggregated into sub-criteria. It is highly desirable that any design criteria be thus decomposable.