On The Character and Complexity of Certain Defensive Resource Allocation Problems

In this article we consider two classes of static defensive resource allocation problems, these are, the static "target-value based" weapon target allocation and the static "asset-value based" weapon allocation problem. It is shown that the target-value based problem can be recast, (using indicator functions), into an instantiation of the so-called transportation problem. The transportation problem can be solved by numerous polynomial-time algorithms and has received considerable attention in the literature. We also consider the so-called "asset-based" weapon target allocation problem. This problem is shown to be somewhat more difficult than the target value based problem. A simulation study is presented for the target-value allocation problem, with emphasis upon sensitivity to uncertain target-elimination probabilities. RELEASE LIMITATION Approved for public release Published by DSTO Systems Sciences Laboratory PO Box 1500 Edinburgh South Australia 5111 Australia Telephone: (08) 8259 5555 Fax: (08) 8259 6567 © Commonwealth of Australia 2004 AR-013-067 March 2004 APPROVED FOR PUBLIC RELEASE On The Character and Complexity of Certain Defensive Resource Allocation Problems Executive Summary The primary aims of this report are to introduce and explain, certain technical issues concerning defensive resource allocation problems. However, the core aim is to revisit the so-called weapon target allocation problem and consider its implications in the modern context of a networked defence. It is clear that the foremost issue arising from a networked defence, in our context, is that the number and diversity of available defensive resources will significantly increase. It is therefore timely to consider both the character and complexity of defensive resource algorithms. Quite apart from any particular context, optimal defensive resource allocation has two main classes of applications, these are, 1) the online scenario, that is, committing defensive resources in real time, during real engagements and 2) the offline scenario, that is, using allocation algorithms to simulate and model the effectiveness of defensive resources against a given threat scenario. The importance of the online scenario is immediate, however, the offline scenario also has significant value and can perceivably be used to aide acquisition, or to estimate a measure of preparedness. Further, a capability to consider offline scenarios will most likely enhance the development of online algorithms. This claim follows naturally from the inherent complexity in defensive resource allocation problems, which often necessitate unavoidable approximation for online applications. In this report we begin with a literature survey, starting from approximately 1950. Various models with various objectives are discussed. We also consider the diversity of approaches taken to solve defensive resource allocation problems. For a particular example, we consider the so-called static target-value based problem. A special case of this problem is shown to be amenable to a linear programming formulation and can be readily solved with the simplex algorithm. This algorithm is a standard algorithm in linear programming and is used to solve certain constrained optimisation problems. The special case we consider is indeed useful, as it provides a convenient means of studying weapon target allocation and potentially gaining insight in to more complex scenarios. Further, this special case is unique, in that despite being cast as a relaxed linear program, with decision variables ranging in the interval [0,1], its natural solutions are guaranteed to take integer values in the set {0,1}. Consequently, sensitivity analysis is also possible using this formulation. A computer simulation is provided, showing the cost of uncertainty in targetelimination probabilities. To make this report as self contained as possible, a proof of the fundamental Theorem of linear programming is provided in the Appendix.