# Resolving conflicting objectives: tradeoff methods

Just as with cars, real-life materials selection almost always requires that a compromise be reached between conflicting objectives. Table 8.2 lists nine of them, and there are more. The choice of materials that best meets one objective will not usually be that which best meets the others; the lightest material, for instance, will generally not be the cheapest or the one with the lowest embodied energy. To make any progress, the designer needs a way of trading weight against cost. This section describes ways of resolving this and other conflicts of objective.

Such conflicts are not new; engineers have sought methods to over­come them for at least a century. The traditional approach is that of using experience and judgment to assign weight factors to each constraint and objective, using them to guide choice in the following way.

Weight factors. Weight factors seek to quantify judgment. The method works like this: the key properties or indices are identified and their values Mi are tabulated for promising candidates. Since their absolute values can differ widely and depend on the units in which they are measured, each is first scaled by dividing it by the largest index of its group, (Mi)maxj so that the largest, after scaling, has the value 1. Each is then multiplied by a weight factor, wi, with a value between 0 and 1, expressing its relative importance for the performance of the component. This gives a weighted index Wi:

max

For properties that are not readily expressed as numerical values, such as weldability or wear resistance, rankings such as A to E are expressed instead by a numeric rating, A = 5 (very good) to E = 1 (very bad), then dividing by the highest rating value as before. For properties that are to be minimized, such as corrosion rate, the scaling uses the minimum value (Mi)min, expressed in the form

(M1) .

Wi = wi imin (8.6)

i i M1

The weight factors wi are chosen such that they add up to 1, that is: wi < 1 and Swi = 1. The most important property is given the largest w, the second most important, the second largest, and so on. The W’s are
calculated from Equations 8.5 and 8.6 and summed. The best choice is the material with the largest value of the sum

W = (8.7)

Sounds simple, but there are problems, some obvious (like that of sub­jectivity in assigning the weights), some more subtle. Experienced engineers can be good at assessing relative weights, but the method nonetheless relies on judgment, and judgments can differ. For this reason, the rest of this sec­tion focuses on systematic methods.

Systematic tradeoff strategies. Consider the choice of material to minimize both mass (performance metric P1) and cost (performance metric P2) while also meeting a set of constraints such as a required strength or durability in a certain environment. Following the standard terminology of optimiza­tions theory, we define a solution as a viable choice of material, meeting all the constraints but not necessarily optimal by either of the objectives. Figure 8.9 is a plot of P1 against P2 for alternative solutions, each bubble describing a solution. The solutions that minimize P1 do not minimize P2, and vice versa. Some solutions, such as that at A, are far from optimal; all the solutions in the box attached to it have lower values of both P1 and P2. Solutions like A are said to be dominated by others. Solutions like those at B have the characteristic that no other solutions exist with lower values of

FIGURE 8.9

both Pi and P2. These are said to be nondominated solutions. The line or surface on which they lie is called the nondominated or optimal tradeoff surface. The values of Pi and P2 corresponding to the nondominated set of solutions are called the Pareto set.

Just as with cars (Figure 8.2 ) , the solutions on or near the tradeoff surface offer the best compromise; the rest can be rejected. Often this is enough to identify a shortlist, using intuition to rank them. When it is not, the strategy is to define a penalty function.

Penalty functions. Consider first the case in which one of the objectives to be minimized is cost, C (units: \$), and the other is mass, m (units: kg). We define a locally linear penalty function1 Z (units: \$):

Z = C + a m (8.8)

Here a is the change in Z associated with unit increase in m and has the units of \$/kg. It is called the exchange constant. Rearranging gives:

m = – – C + – Z (8.9)

a a

This defines a linear relationship between m and C that plots as a fam­ily of parallel penalty lines, each for a given value of Z, as shown in Figure 8.10. The slope of the lines is the negative reciprocal of the exchange con­stant, – i/a. The value of Z decreases toward the bottom left: the best choices lie there. The optimum solution is the one nearest the point at which a penalty line is tangential to the tradeoff surface, since it is the one with the smallest value of Z.

Values for the exchange constants, a. An exchange constant is the value or "utility" of a unit change in a performance metric. In the example we have just seen, it is the utility (\$) of saving 1 kg of weight. Its magnitude and sign depend on the application. Thus the utility of weight saving in a fam­ily car is small, though significant; in aerospace it is much larger. The util­ity of heat transfer in house insulation is directly related to the cost of the energy used to heat the house; that in a heat exchanger for electronics can be much higher because high heat transfer allows faster data processing, [34]

Metric P1: cost, C Expensive

The penalty function Z superimposed on the tradeoff plot. The contours of Z have a slope of -1/a. The contour that is tangent to the tradeoff surface identifies the optimum solution.

something worth far more. The utility can be real, meaning that it meas­ures a true saving of cost. But it can also, sometimes, be perceived, mean­ing that the consumer, influenced by scarcity, advertising, or fashion, will pay more or less than the true value of the performance metric.

In many engineering applications, the exchange constants can be derived approximately from technical models for the life cost of a system. Thus the utility of weight saving in transport systems is derived from the value of the fuel saved or that of the increased payload, evaluated over the life of the system. Table 8.5 gives approximate values for a for various modes of transport. The most striking thing about them is the enormous range: the exchange constant depends in a dramatic way on the appli­cation in which the material will be used. It is this that lies behind the difficulty in adopting aluminum alloys for cars, despite their universal use in aircraft; it explains the much greater use of titanium alloys in military than in civil aircraft, and it underlies the restriction of beryllium (a very expensive metal) to use in space vehicles.

Exchange constants can be estimated approximately in various ways. The cost of launching a payload into space lies in the range of \$3,000 to \$10,000/kg; a reduction of 1 kg in the weight of the launch structure would allow a corresponding increase in payload, giving the ranges of a shown in

 Table 8.5 Exchange constants a for the mass-cost tradeoff for transport systems Sector: transport systems Basis of estimate Exchange constant, a US\$/kg Family car Fuel saving 1-2 Truck Payload 5-20 Civil aircraft Payload 100-500 Military aircraft Payload, performance 500-1000 Space vehicle Payload 3000-10,000

the table. Similar arguments based on increased payload or decreased fuel consumption give the values shown for civil aircraft, commercial trucks, and automobiles. The values change with time, reflecting changes in fuel costs, legislation to increase fuel economy, and the like.

These values for the exchange constant are based on engineering cri­teria. More difficult to assess are those based on perceived value. That of the performance/cost tradeoff for cars is an example. To the enthusiast, a car that is able to accelerate rapidly is alluring. He or she is prepared to pay more to go from 0 to 60 mph in 5 seconds than to wait around for 10, as we will see in Chapter 9.

There are other circumstances in which establishing the exchange constant can be more difficult still. An example is that of environmental impact—the damage to the environment caused by manufacture, use, or disposal of a given product. Minimizing environmental impact has now become an important objective, almost as important as minimizing cost. Ingenious design can reduce the first without driving the second up too much. But how much is unit decrease in impact worth?

Exchange constants for ecodesign. We explored this topic in Chapter 5. Interventions, as they are called, use taxes, subsidies, and trading schemes to assign a monetary value to resource consumption, energy, emissions, and waste, effectively establishing an exchange constant. As we’ll see in the next chapter, none is yet large enough to make big changes in the way we reduce any of these.