Working Draft towards the Real Time Solution of Strike Force Asset Allocation Problems

The strike force asset allocation problem consists of grouping strike force assets into packages and assigning these packages to targets and defensive assets in a way that maximizes the strike force potential Integer programming formulations are developed semi real and random problems are built up and computational results via the CPLEX MIP Solver are presented Work Sponsored in part by the O ce of Naval Research grant numbers and Introduction Strike force planning is one of the fundamental problems faced by the armed services In a strike force planning problem there are a set of potential targets and a collection of assets that may be allocated to targets There may also be a set of threats which serve as the defenders of the targets The goal is to determine the allocation and routing of strike force assets so as to maximize strike e ectiveness while limiting potential damage to the strike force from the defenders The strike force planning problem is solved before the engagement although the problem of strike force management during the engagement is also a question of practical importance Current practice requires that the strike force planning problem be solved on a time scale of a few hours to one half a day With recent advances in information processing technology especially in parallel computation the potential for real time strike force planning on the order of a few minutes is now realizable The strike force planning problem includes routing and allocating the available assets The routing problem has received attention in several papers Boroujerdi et al Boroujerdi Zuniga et al Lee Zuniga and Gorman Zuniga and Gorman discuss strike route optimization and threat site over ight modeling They present some general graph theoretic techniques for route optimization Zuniga et al focus on the inter dependent joint routing problem They analyze routing a collection of assets to a collection of targets Lee constructs a network and uses a shortest path algorithm to obtain a very good route for the strike force aircraft As to the asset allocation problem Matlin Eckler and Burr and Cheong give reviews on the missile allocation problem Cho discusses how to allocate a given number of weapons to targets with target dependent kill probabilities While most of the literature models the weapon target allocation problem in a stationary way Hosein treats it as a dynamic nonlinear resource allocation problem Hosein and Athans give analytic results for the dy namic nonlinear weapon target allocation problem in simple cases They give asymptotic results when the number of targets becomes very large Work by Jakobovits highlights the core of the asset allocation problem Jakobovits separates a strike plan into an attack plan which focuses on primary targets and a suppression plan which seeks to suppress defensive threats He addresses the asset allocation problem in this general setting and solves some small problem instances but he points out that larger instances quickly become very di cult to solve In this paper we omit routing information and consider two models of the striking asset al location problem depending on whether or not suppression of defensive threats is involved In the rst model we formulate the problem as a large scale integer program consisting of a collection of knapsack constraints superimposed on a collection of disjoint Type special ordered sets SOS A knapsack constraint has the form ax b where x is a column vector of variables a is a nonnegative row vector and b and special ordered sets are sets of non negative variables that are required to sum to A Type SOS is when each variable is binary so the constraint is one of multiple choice Greenberg The second model further incorporates precedence constraints in addition to a collection of knapsack constraints and SOS constraints As the computational results demonstrate the problem structures a orded by these constraints yield integer programs that can be solved close to from the best bound of the optimal solutions very quickly via the Parallel MIP Solver MIP is an abbreviation for mixed integer programming of CPLEX even for relatively large problem binary variables and constraints instances In addition to the models presented and studied computationally a set of semi real problems has been developed based on data procured from the United States Geological Survey In addition a set of randomly generated problems were created It is hoped that other researchers will nd this set of test problems interesting and useful for future research in this problem area The problems and contour maps are available upon request The paper is organized as follows In Section the asset allocation problem formulations are described Section discusses generating semi real and random problems Section presents computational results for both models Conclusions and future research directions are presented in Section Formulations of Problems A practical aspect of strike force asset allocation is that often more than one asset is required to attack a target or suppress a threat We will use threats and defenders interchangeably in this paper There are several types of strike force assets represented in the form of di erent aircraft and missiles To illustrate a radar site might require two missiles of type I and a naval base might require one aircraft of type I and one missile of type II Moreover there might be other combinations which can be used to attack a target or suppress a threat In this paper the collection of assets capable of destroying a target and a threat will be referred to as an attack package and a suppression package respectively Model considers only targets This model is generalized to incorporate threat suppression as Model Strike Force Asset Allocation Problem without Suppression Let C be the set of all asset classes available and T be the set of all targets available Identical assets are referred to as the same asset type independent of location An asset class is an asset type with a speci c location For example two identical aircraft in di erent locations are of the same asset type but di erent asset classes The cost of assigning the same asset at di erent locations to a given target may be di erent due to the distance between the target and the asset For simplicity assume that an asset close to its corresponding target will have less opportunity for asset loss therefore the distance between each asset in an attack package and the corresponding target contributes location dependent values associated with the attack package value Before stating the formulation de ne some notation as follows A the set of all possible target attack packages At a subset of A which consists of all attack packages associated with target t where the cardinality of At is jAtj gta the pro t associated with assigning attack package a to target t where a At ecta number of class c assets contributing to attack package a for target t where a At b a column vector of size jCj where bc is the number of assets available in asset class c xta equals if attack package a is assigned to target t and equals otherwise where a At The goal is to maximize the damage yield which is both target and package dependent by the assignment of attack packages subject to asset class con gurations and availability constraints Each attack package has an associated binary variable in the model and each attack package a ects the objective value in an additive manner according to its coe cient if the attack package is chosen each attack package contributes nothing to the objective value if it is not chosen It is assumed each target is assigned at most one attack package This model can be described as follows