Can consolidation create sustainable long-term value, or will it prove seductive but, ultimately, disappointing to shareholders, employees, customers, and management alike?
Appendix: Opening the Black Box
Mathematical Structure of the Methodology
In this appendix to “Opening the Black Box,” ( Fortnightly, January 2014), we briefly describe the basic components of the models for managing aging assets: how to represent the condition of such assets and the outcome of replacement, maintenance, and testing decisions.
The model structure is one of optimal control with dynamic state variables and uncertainty. Let
= asset population that is in state s at time k.
The subscript indexes chronological time, measured in years, during the analysis period, which is can be represented as the interval , for some arbitrary number of years, T f , called the planning horizon .
Generally, there are three important aspects of the formulation of asset state dynamics. First, the distribution of the asset population among the states in the next year is a function of the current distribution. For instance, in the simple case that the state is the asset’s service age, and assets do not fail, the population at service age t in year k equals the population at service age t+1 in year k+1
Second, the migration of assets among the states occurs probabilistically. Again, for simplicity, let the state s be the age t. If assets fail at age t at the rate h(t), then the surviving population of age t+1 in year k+1 is
The function h(t) is the hazard rate for assets. The hazard rate is determined empirically, using some generally applied functional forms, such as a piecewise linear or a Weibull nonlinear hazard function. In general, the members of the surviving population in any year migrate from state to state according to a collection of state transition probabilities . For example, it is simplest to think of annual changes in state, such that a transformer known to be in good condition at the beginning of a given year can be in either good, fair, or poor condition at the end of the year. Three transition probabilities determine the relative likelihood of these transitions: pgoodgood, p goodfair, and pgoodpoor. These probabilities sum to 1.
Third, the migration of assets among the states can depend on some action taken with respect to a particular asset in a particular state. For instance assets that are to be replaced as a consequence of the state-dependent optimal policy or that have failed may be replaced with new assets, so that the population of new (that is, age 0) assets at the beginning of the next year is the total of the assets of all ages that are replaced in the current period.
The condition of an asset can be tested, but all tests are imperfect. Hence, the consequence of the test is to revise the