# The Cost Constraint as a Cost Driver

**from the Perspective of Competitive Advantage**

**by**

Edwin B. Dean

For many years, in the parametric cost analysis community, the word cost driver has been used to denote a measure which is statistically significant in an equation which relates cost to a set of noncost measures. System weight, which is a measure of size, for example, is usually statistically significant for a hardware system. It was not until I began to address the minimization of cost that a second type of cost driver arose within a new and very important context, the existence of cost forces which arise from constraints.
The constrained cost minimization problem may be stated as

minimize f(x)

over x

subject to g(x) >= a

where f(x) is life cycle cost in terms of conventional cost drivers x_{i} which are components of the vector x, g(x) is a vector of system constraints g_{i}(x), a is a constant vector, and f(x) and g_{i}(x) are to be viewed as generalized coordinates.

At the minimum the following equation is satisfied.

D_{x}f(x) = r * D_{x}g(x)

where D_{x} represents the partial derivative operator, D_{x}h(x) is the gradient of the function h(x), r is the Lagrange multiplier vector, and * is the vector dot product.

This is Newtons third law for cost where r_{i} D_{x}g_{i}(x) is the ith reactive cost force with magnitude r_{i} |D_{x}g_{i}(x)| and direction D_{x}g_{i}(x) / |D_{x}g_{i}(x)|. D_{x}f(x) is the driving cost force with magnitude |D_{x}f(x)| and direction D_{x}f(x) / |D_{x}f(x)|. It is the gradient of the potential f(x) which is the life cycle cost. Note that r_{i} is nonzero at equilibrium (the minimum) only if g_{i}(x) = a_{i}. If g_{i}(x) = a_{i}, then g_{i}(x) is the generalized coordinate of the reactive cost force (driver) r_{i} D_{x}gi(x). To reduce the life cycle cost f(x), g_{i}(x)-a_{i} must be relaxed.
The cost force concept has been derived from many places, including personal experience in analogic thought as an analog computer programmer, but, in particular, from the very excellent description of mechanics by Lanczos (1949) and the very excellent description of generalized coordinates by Beyerly (1916).

#### References

- Beyerly, W. E. (1916). An Introduction to the Use of Generalized Coordinates in Mechanics and Physics, Ginn and Company, New York NY, republished in 1965 by Dover Publications, New York NY.
- Lanczos, C. (1949). The Variational Principles of Mechanics, University of Toronto Press, Toronto, Canada.

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