CAIV…NOT!!!

by Roy E. Rice, Ph. D.
MORS President Elect

Set the Stage

In today’s environment of Acquisition Reform, new terms and concepts are born daily. One term that has been around for a couple of years now and is getting abused more and more is Cost As an Independent Variable (CAIV). Let me be as blunt as possible. Cost is NOT an independent variable; period. I challenge all analysts (and pseudo analysts) to stop using this term. Acquisition decision makers, if anyone comes into your office and uses the term CAIV, I urge you to bounce them out.

Definition of Functions and Independent Variables

A function is defined as "a set of ordered pairs (x,y) of real numbers in which no two pairs have the same first element. In other words, to each value of x (the first member of the pair) there corresponds exactly one value of y (the second member). The set of all values of x which occur is called the domain of the function, and the set of all y which occur is called the range of the function."¹

Figure 1: Function with variables

Graphically, this means that if I drop a vertical line down through the curve labeled f(x) (shown in Figure 1) to an x value in the function’s domain, it will intersect the curve only once. Stated mathematically in set notation

f = {(x,y): x is in the domain of f and y = f(x)} ¹

The variable x is the independent variable and y is the dependent variable; the value that y takes on depends on the value of x chosen from the domain.²

Cost is a Dependent Variable

Let’s concentrate on the design of a weapon system (e.g., aircraft, ground combat vehicle, ship, radar, sensor, etc.). Once a design is fixed, (weight, speed, drag, gun system, bandwidth, revisit rate, sweep rate, etc.) the cost of the system can be calculated. These costs can be unit recurring flyaway costs, O&S costs, life cycle costs or any other costs. But the cost is a function of the design; cost depends on the design. Mathematically this is

cost = f(design).

The design parameters are the x values (independent variables) and cost is the y value (dependent variable). These various design variables correspond to one and only one value of cost. Conversely, if cost were truly an independent variable, a fixed cost could result in one and only one design. We all know this is not the case.

Tricks of the Analyst

Some analysts (and pseudo analysts) develop a graph like Figure 1 with cost as the dependent variable. Then they take the graph and rotate it about its diagonal so that they get something like Figure 2.

Figure 2: Rotate Graph of the Function About its Diagonal

Just because one flips a graph, does not, in any way, make the variable on the horizontal axis the independent variable. Decision makers, beware of this! The relation in Figure 2 shows that as y varies, so does x; that doesn’t change the fact that x is the independent variable and y is the dependent variable.

I purposely called this a relation. "Any set in the number plane, R₂, is called a relation."¹ Graphically, this means that if a vertical line is dropped through the curve (relation) as shown in Figure 2 down to the horizontal axis, it can intersect the curve in more than one place. This is shown in Figure 3 and shows the difference between a relation and a function.

Figure 3: Relation

So, what is cost?

Cost is actually a constraint. Think of this in terms of an optimization. I’ll show a notional example. Say, we’re trying to maximize a system’s capabilities subject to certain constraints, to include cost. Mathematically, as a linear program, we have

Maximize capability Z = f(X) (1)

subject to:

A [X] £ b, where bold letters represent vectors and matrices. (2)

Let the X vector represent the design parameters of the weapon system and the Capability (Z) be a function of X. Let b_i (one of the elements of the vector b) be the upper limit on the cost the program is willing to spend. So the ith constraint is the cost constraint. Then, if we vary only the value of b_i from a lowest value (for sensitivity purposes) over 4 specific values to a high value (again, for sensitivity purposes and as long as the problem remains feasible), we can get a (notional) relationship (not a function) like the following.

Figure 4: Capability vs. Cost

We can show with this type of analysis that, as the cost constraint is relaxed, we can obtain more (or, no less) capability from the resulting designs. Because of the properties of linear optimization, the capability will never decrease as the upper limit on cost increases. Again, this doesn’t mean that cost is an independent variable. Look at the constraint equations (2)…cost (b_i) is a function of the X independent variables.

We could also optimize by minimizing cost subject to a minimum level of capability to achieve. But, in this case too, the cost objective function would be a function of the design parameters — cost would still be a dependent variable.

We could also approach the problem without optimization. If we simply look at several designs that give us various levels of capability and each design has a cost associated with that design (again, cost is dependent), we would have the situation depicted in Figure 5.

Figure 5: Various System Designs

Now, we could plot Capability vs. Cost. As shown in Figure 6.

Figure 6: Non-optimized Designs

In the case of Figure 6, we have not optimized the design (maximized capability subject to a cost constraint) so it is possible that we have picked a design that is more capable and less expensive than other designs (Design 2 vs. Design 3).

Also, it is possible that we could examine another Design (say Design 7) that has the same cost at Design 2 but has the capability of Design n. This is shown in Figure 7.

Figure 7: Incorporation of Design 7

This shows that Cost 2 can have more than one capability. This is the case I made earlier in Figure 3 showing a relation. Again, it can’t be a function and cost can’t be an independent variable.

Bottom Line

We analysts can show (and decision makers should expect us to provide) the relationship of designs and capabilities to increasing and decreasing costs. But, cost is not an independent variable…so quit using the term CAIV. I encourage the CAIV community to write articles responding to my rantings.

REFERENCES

(1) Protter, M.H., Morrey, Jr., C.B., College Calculus with Analytic Geometry, 2nd Edition, Addison Wesley Publishing Co., 1970.

(2) Thomas, Jr., G.B., Calculus and Analytic Geometry, Addison Wesley Publishing Co., 1972.