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_xf(x) = r * D_xg(x)

where D_x represents the partial derivative operator, D_xh(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_xg_i(x) is the ith reactive cost force with magnitude r_i |D_xg_i(x)| and direction D_xg_i(x) / |D_xg_i(x)|. D_xf(x) is the driving cost force with magnitude |D_xf(x)| and direction D_xf(x) / |D_xf(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_xgi(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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