Two measures of organizational flexibility

Abstract

Since the second half of the past century, increasingly flexible organizational forms have been appearing among firms. However, while hierarchies are easily described, too few mathematical tools are available for flexible organizations. In this article, two measures are proposed in order to assess the state and trend of flexible organizations. The first of these measures is based on information waste, which occurs whenever information is classified into categories. The second measure is based on duplication of operations. The underlying idea is that firms have an endogenous drive towards organizational configurations where waste of information and duplication of operations are minimized. However, environmental uncertainty may require some flexibility, which is ensured by cognitive processes that discard some information as well as by parallel undertaking of similar actions.

Appendix: Lyapunov functions

A simple way to visualize stable equilibria is to think of a surface in a n + 1-dimensional space of n state variables, plus one variable for the height of this surface. Henceforth, this surface will be also called a landscape. The n-dimensional space of the state variables will be called state space.

The projection of a point of this surface on the n-dimensional state space represents a state that a system can attain. Thus, the state of a system may be represented by the position of a ball on the landscape.

According to one possible convention, the lower positions on the surface represent the most preferred states. Thus, a system has a tendency to move from higher positions to lower positions on the surface. Consequently, the bottom of valleys represent stable equilibria. Conversely, the peaks represent unstable equilibria.

According to the opposite convention, the upper positions on the surface represent the most preferred states. Thus, a system has a tendency to move from lower positions to higher positions on the surface. Consequently, the tops of mountains represent stable equilibria. Conversely, the bottoms of valleys represent unstable equilibria.

Figure 3 illustrates one such surface in a case where n = 2. The state space is the X − Y plane.

Fig. 3

A surface representing equilibria on a two-dimensional state space. According to one convention, the bottoms of valleys represent stable equilibria. According to the opposite convention, the tops of mountains represent stable equilibria

These conventions are mathematically equivalent to one another. In fact, minimizing a function F is equivalent to maximizing − F. Henceforth the convention will be used that a function F has to be minimized.

The Lyapunov theorem states that, given a system described by state variables x ₁, x ₂, ...x _N , the origin of axes is a stable equilibrium if:

• 1. $\exists F(\mathbf{x}) \in \mathcal{C}^{o}: \; F(0)=0, \; F(\mathbf{x})>0$ around the origin;

• 2. $\frac{\partial F}{\partial x_{1}} dx_{1} +\frac{\partial F}{\partial x_{2}} dx_{2} + \cdots + \frac{\partial F}{\partial x_{N}} dx_{N} < 0$.

The Lyapunov theorem can be expressed with respect to the origin of axes without any loss of generality. In fact, any point can be made the origin of the state space by means of a linear transformation.

The Lyapunov theorem says that, if we succeed in finding a basin-shaped function such that the state of the system tends to move towards the lowest position, then that position is a stable equilibrium. Note that several Lyapunov functions may be defined for a system, and that no standardized procedure exists to find one. Thus, the Lyapunov theorem requires considerable ingenuity in order to be applied. Furthermore, it is necessarily dependent on a point of view so far as it regards what variables are crucial for stability.

Note also that the Lyapunov theorem provides a sufficient, but not a necessary criterion for stability.⁵ Thus, if a Lyapunov function is found, we are certain that the equilibrium is stable. However, if no Lyapunov function is found, and even if it can be demonstrated that no Lyapunov function exists, we cannot conclude that the equilibrium is unstable, and not even that it is not an equilibrium point.