Air defense missile-target allocation models for a naval task group

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Abstract

In this study, we address the issue of allocating the air defense missiles to incoming air targets in order to maximize the air defense effectiveness of a naval task group. The shoot-look-shoot engagement policy for missile allocation is assumed to be used. Two integer linear programming models are developed and analysis of an example problem is presented. Computational results show that large instances of the proposed models can be solved within a few seconds optimally. The possible use and the extensions of the models are also discussed.

Introduction

This study addresses the issue of allocating surface-to-air missiles (SAMs) to incoming air targets in a coordinated way within a naval task group (TG), which is a collection of naval combatants and auxiliaries that are grouped together for the accomplishment of one or more missions. The competing technologies of anti-ship missiles (ASMs) and defending SAM systems force the navies to update the systems and to develop new tactics continuously. All modern navies devote considerable resources to ASM defense systems [1]. The sinking of Israeli destroyer Elath by four Styx ASMs by the Egyptian Navy in 1967 was a first in naval history and the demonstration of the potential threat posed by ASMs. Six years later in 1973, 54 ASMs launched by the Syrian and Egyptian Navies failed to hit their intended targets due to defensive tactics developed by the Israeli Navy [1]. Exocet ASMs sank the British destroyer HMS Sheffield during the Falklands War in 1982. The ASM attack on the US Navy frigate Stark in Persian Gulf in 1987 is another example of the fragility of ASM defense.

Nations spend billions of dollars for their navies. However, it is still prohibitively expensive to equip all the platforms (ships) with adequate air defense systems. For many navies, equipping all the platforms with air defense systems is clearly not the best and cost effective solution. A number of navies acquire area air defense (AAD) platforms that can provide air defense support to the other ships that have limited or no effective air defense capability. Allocation of the capability of AAD ship(s) to other units in the TG is an important problem to be solved for efficient use of these platforms.

The aim of this study is to develop a model for TG air defense that captures the reality of ASM defense, generates an efficient allocation plan and measures the effectiveness of the air defense under a given scenario. A scenario is defined by the information on attacking ASMs, defending SAM systems and the formation of the TG.

Proposed model can be categorized as a special type of weapon-target allocation (WTA) problem that can be used for evaluating the air defense effectiveness of a naval TG. WTA problem can be stated as the optimal allocation of existing weapons to a set of targets. Matlin [2], and Eckler and Burr [3] review the literature on the WTA problem. However, Matlin focuses on the problem from the attacker's perspective. Bracken and Brooks [4] argue that the WTA is not addressed much in the literature in an analytic sense after the 1972 Anti-Ballistic Missile Treaty.

The literature on WTA problem can be classified into three groups. Defense allocation models allocate defensive weapons to targets without taking into account the behavior of the opposing side. The actions of the opposing side and threat to defense are included in the scenario as a given input. Burr et al. [5], Shumate and Howard [6], Soland [7], Bertsekas et al. [8], Wacholder [9], and Jaiswal [10] consider WTA from the defenders point of view. Ahuja et al. [11] propose exact and heuristic approaches for WTA problem that minimizes the expected survival value of the targets.

The second class of approaches takes into account the opposing side's moves as well as the defensive moves. These models employ the two-person-zero-sum-game concept from game theory in the solution process. They reach the solution value of the game by assuming best defensive and offensive moves. The defense wants to minimize the maximum offense's return while the offense acts to maximize the minimum expected return. This approach is more suitable when the inventories of the opposing sides are known to some degree. Randolph and Swinson [12], Soland [13], O’Meara [14], O’Meara and Soland [15], [16], [17], Bracken et al. [18], [19], and Soland [20] model both defender and attacker sides.

Rest of the literature addresses different aspects and questions within the WTA context. Simulation models and layered defense models are included in this category. Nguyen et al. [21] introduce the idea of using generating functions for evaluating the effectiveness of an air defense system. Nguyen and Reding [22], [23] and Nguyen et al. [24] are the extensions of same approach. Nunn et al. [25], Orlin [26], and Mutairi et al. [27] analyze the effectiveness of layered defense systems. Interested readers are referred to Karasakal [28] for a recent survey of literature on WTA problem.

Proposed models introduce some extentions to generic WTA such as explicit resource coalescence under a shoot-look-shoot (SLS) engagement policy for maximizing the probability of no-leaker (i.e. shooting down all threat ASMs) for a naval TG. Proposed models can be seen as a customized extention of WTA problem for naval applications. Revised linear formulations can be solved using any ordinary solver for quick evaluation of air defense effectiveness of a naval TG. In this context, existing analysis methods, which can directly be used for evaluating the air defense effectiveness of a naval TG mainly consist of computer models that simulate the ASM defense. SEAROAD [29], [30], JASMINE [31], SADM [32], and SAADS [33] are the examples of such models. According to our knowledge, the only analytical model that addresses resource allocation within a naval TG is Nguyen's study [34]. In his work, Nguyen studies the quantification of benefit from resource allocation for a naval TG having perfect coordination between its assets. SAMs are assumed to cover all the other ships of the TG and are capable of defending the ships within range. Other geometric and the defense system limitations such as distances of the ships, bearing and range of attacking ASM, effective range and speed of SAMs are not considered.

Griffiths et al. [35] considers a very restricted air defense scenario for a naval TG. They assume identical aircraft in line-ahead formation (i.e. aircrafts are flying towards the same bearing one after the other with a specified distance between each other) attacking a naval TG that is composed of warships with identical air defense weapons and obtain a difference equation for ship and aircraft damage. They report that their model has been used to approximate more complex scenarios as a screening process for detailed simulations. Beare [36] presents the utility of integer programming to determine the most effective mix of air defense systems. Proposed mathematical programming model functions as a selection tool to identify the most promising air defense weapon mix for scrutinized evaluation using simulation model.

The organization of this article is as follows. In Section 2, we give the problem description and explain the assumptions used. In Section 3, a detailed formulation of the problem is presented. Section 4 contains the implementation issues of the proposed models. Results are presented using a simple example. Computational results are given in Section 5. The last section contains some concluding remarks and possible use of the developed models.

Section snippets

Problem description

Consider a naval TG, composed of several ships with variable air defense capabilities, defending itself against an air attack. These ships may be either equipped with one or more SAM systems or none at all. The air defense capability may be limited to self-defense or may extend to area defense that the other ships within range can be defended. In a naval TG, the individual ships function together as a team to provide mutual support and defense against opposition to assigned missions. These

Formulation of the problem

Indices:

i: Number of incoming ASMs, indexed iN={1,,n}.

j: Number of SAM systems onboard the warships composing the naval TG, indexed jM={1,,m}.

v(i,j): Valid combinations of the ASM and the SAM systems (i.e. SAM system j can engage ASM i).

Parameters:

uij: The maximum number of missiles that can be launched from SAM system j against ASM i, (i,j)v(i,j) using a SLS tactic. Time taken by each feasible engagement is determined by the speed of the attacking ASM, the speed of the defending SAM, the

Implementation of the models

Models, (P1) and (P2), have been implemented using GAMS (General Algebraic Modeling Language) mathematical programming package and solved using OSL Solver [39].

We show the results of the proposed models (P1) and (P2) on a simple example. The example is depicted in Fig. 3. Ship 1 has only self-defense SAM system, and Ship 2 has both self defense and area defense SAM systems (SAM2 is the area defense system). Assume that all necessary calculations for generating the input data have been performed.

Computational results

We randomly generated test problems for specified numbers of ASMs, self-defense SAM systems and area defense SAM systems. We assume that an area defense SAM system can engage ASMs within its effective range, while a self-defense SAM system can only engage to the ASM that is a direct threat to itself.

We created a sample single shot kill probability matrix for ASM and SAM systems from a uniform distribution in the interval [0.05, 0.80]. The number of available rounds on SAM systems was generated

Conclusion and future work

In this study, we develop realistic models for TG air defense problem. We make use of the formation information such as relative bearings and distances between ships as well as the specifics of attacking missiles. A generic engagement policy, shoot-look-shoot is assumed. Different types of attacking ASMs and different types of defending SAM systems are allowed. These assumptions are reasonable when TG operates in a formation and encounters an immediate air attack by ASMs. Considering the fact

Acknowledgment

A part of this research was carried out while the author was visiting the Operational Research Division in the Canadian Department of National Defense. The author would like to thank the anonymous referees for their valuable comments and suggestions for improving the paper.

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