"The best method for assessing a single attribute value function depends on its qualitative characteristics"
As explained on previous pages, von Neumann-Morgenstern utility theory provides a basis for creating a model for estimating the value of projects . The key is defining project objectives and performance measures that are mutually, preferentially independent. If this requirement is achieved, then project value may be computed using an additive equation consisting of single-attribute value functions (also called scaling functions) and weights:
V(x1,x2...xN) = w1V1(x1) + w2V2(x2) + ... + wNVN(xN)
On this page I describe how to determine the single-attribute value functions, the Vi in the above equation. Figure 32 identifies this step and shows how it relates to my 12-step process for creating a project selection model.
Figure 32: Steps for creating a project selection decision model.
Characteristics of Single-Attribute Value Functions
A single attribute value function, Vi, translates the level of performance achieved for an attribute, xi, expressed in whatever unit of measurement was chosen for it, into a number indicating how useful or desirable that level of performance is perceived to be by the decision maker. With the additive value function, every performance measure has its own separate value function. If the decision maker accepts the utility theory axioms, and performance measures have been defined to be mutually preferentially independent, the theory guarantees that a value function, Vi, will exist for each of the attributes defined for measuring project performance .
To Normalize or Not To Normalize
As noted previously, many authors recommend that single attribute value functions be normalized such that zero is assigned to the worst performance level obtained by any alternative and one (or one hundred) is assigned to the best performance level obtained by any alternative . By following this recommendation, each single-attribute function is standardized so that its value falls in the same 0-to-1 (or zero-to-100) interval. A benefit of such normalization is that there is a common interpretation for every weight; namely, each weight indicates the relative value obtained if performance for the corresponding attribute swings from the worst performance achieved for any alternative to the best performance achieved for any alternative.
A problem for using this form of normalization for a project priority system is that the worst and best performance levels are likely to change each time the system is applied. Suppose, for example, that a firm wishes to prioritize its sales representatives, and one performance measure is the sales person's "opportunity win rate," defined as the percentage of leads that are converted into sales. The range defined by the current year's worst and best win rates may not encompass the worst and best rates obtained next year. You could renormalize each year, but that won't let you compare prioritization metrics across years. You could select a larger range, for example, one based on the worst and best performance levels from a longer time span, but that still won't guarantee that performance levels in all future year applications will fall within the defined range. For this reason, if single attribute value functions are normalized for use in priority systems, the functions are typically defined and normalized over a range that encompasses all theoretically possible levels of performance.
Figure 33: Alternative performance ranges.
For project prioritization, chances are good that for at least one objective and its corresponding performance measure, maximum and/or minimum (negative) performance levels will be unbounded. For example, the net revenue produced by the project is not necessarily bounded by any value. In some such cases, it may be possible to define limits to performance based on practical considerations, in which case the associated single attribute value function can be normalized between zero and one. Alternatively, you can decide not to normalize and rely on the weights to make the different units of value comparable.
Linear versus Nonlinearly Single Attribute Functions
Regardless of whether or not single attribute value functions are normalized, an important role for the functions is to account for non-linearities in the relationship between units of performance and the value of that performance. Figure 34 illustrates some possible shapes for (normalized) single attribute value functions intended to serve as scaling functions .
Figure 34: Alternative shapes for scaling functions.
Figure 35 below shows a frequently seen shape for single-attribute value functions that is, perhaps, made intuitive given the "performance" that the function is measuring . Although the performance level is theoretically unbounded, obtaining additional units of performance produces decreasing marginal value, suggesting that the utility function asymptotically approaches its maximum value.
Figure 35: Intuitively shaped scaling functions.
Assessing Single Attribute Value Functions
The best method for assessing a single attribute value function depends on its qualitative characteristics. For this reason, as Keeney and Raiffa advise, begin by determining the qualitative characteristics of the function .
Establish Qualitative Characteristics First
Qualitative characteristics include such basic considerations as whether the performance measure is continuous or discrete, whether it is bounded at the low end or high end, and whether preference increases or decreases as the number expressing the level of the performance measure increases. You should also be able to determine whether the function is likely linear in units of the performance measure and, if not, whether changes in performance matter more for low or high performance levels.
To provide an example, suppose you are prioritizing projects that impact environmental objectives and that one of the performance measures is visibility. One measure for visibility used by meteorologists is termed prevailing visibility, defined as the maximum horizontal distance that can be seen throughout at least half the horizon circle. In ideal weather conditions, it is possible to be able to see moderately sized objects up to about 12 miles away, although at sea level the curvature of the earth gets in the way at about 3 miles. So a performance measure for visibility relevant to a ship's captain is bounded at the low end at zero miles and at the high end at about 3 miles. Whether the single attribute value function is concave or convex (as indicated in Figure 34 by the red or green curves, respectively) depends on whether an amount of visibility improvement, say equal to one-quarter mile, is more desirable near the low or high end of the scale. Most people, I assume, would find it more desirable to obtain an additional one quarter mile of visibility when visibility is very poor than when visibility is pretty good. Thus, a single-attribute value function for visibility is likely to be convex, similar to the red curve shown in Figure 34.
The answers to the basic qualitative characteristics for a single-attribute value function can usually be determined by the analyst without need to question a decision maker. However, if there is doubt, you can, of course, pose some preference questions for the decision maker. In my experience, so long as you have some understanding of the organization's basic interests and means of measurement, you can answer these sorts of questions on your own.
Methods for Determining Single-Attribute Value Functions
I use variations of four methods depending on the characteristics of the functions
The next page provides advice for creating a consequence model for simulating the consequences of conducting a project.