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Cost Contingency Analysis using Polytopes

Risk II Track



Contingency is defined as a possibility that must be prepared for—an event that may occur but is not likely or intended. In the realm of cost estimation, contingency refers to a reserve above and beyond a baseline estimate that would be tapped if one or more unexpected events led to higher program costs. Ideally, the selection of a contingency amount should be based on achieving a particular level of confidence derived from cost uncertainty and risk analyses, but often a mere rule-of-thumb percentage (e.g. 15% of the baseline estimate) is applied. In either case, it is important for decision makers to understand exactly what possibilities might be covered by the contingency reserve selected, and what would not. Given a finite set of cost elements, this presentation discusses a method of exploring the entire “contingency space” providing the ability to generate and analyze the infinite set of scenarios (combinations of possibilities of the known unknowns) that would be covered, and hence those that would not.

In particular, we show how to model the set of possibilities with high-dimensional geometry, or equivalently, algebraically as a finite system of equalities/inequalities and variables. The variables correspond to individual cost elements and the equalities/inequalities model potential interdependencies, correlations, upper and lower bounds. The number of cost elements (or variables), d, defines the dimension of the contingency space—formally known as a d-dimensional convex polytope (a polytope is a generic term that can refer to a two-dimensional polygon, a three-dimensional polyhedron, or a generalization to higher dimensions). Convex polytopes have a dual, V-representation to the defined system of equalities/inequalities. This V-representation represents the convex polytope as a finite set of points V. The notion of this representation is that every point of a polytope (of which there are infinitely many) can be generated by taking a special linear combination of the points of the finite set V.

Applying convex polytope theory to the contingency space allows us to characterize and generate scenarios (combinations of possibilities) to inform and provide greater awareness to decision makers. Every point of the polytope associated with the contingency space represents a possible scenario. To illustrate the theory, examples based on select Canadian Department of National Defence acquisition programs (e.g. F-35A) are presented.


Dr. Bohdan L. Kaluzny
Defence Research & Development Canada
Bohdan L. Kaluzny is a defence scientist with Defence Research & Development Canada, Centre for Operational Research and Analysis. Dr. Kaluzny obtained his doctorate degree in computer science from McGill University and his research interests include polyhedral computation, combinatorial optimization, multi-criteria decision analysis, data mining and operations research.