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Modeling Short-Range Ballistic Missile Defense and Israel's Iron Dome System
Abstract
This paper develops a model of short-range ballistic missile defense and uses it to study the performance of Israel's Iron Dome system. The deterministic base model allows for inaccurate missiles, unsuccessful interceptions, and civil defense. Model enhancements consider the trade-offs in attacking the interception system, the difficulties faced by militants in assembling large salvos, and the effects of imperfect missile classification by the defender. A stochastic model is also developed. Analysis shows that system performance can be highly sensitive to the missile salvo size, and that systems with higher interception rates are more "fragile" when overloaded. The model is calibrated using publically available data about Iron Dome's use during Operation Pillar of Defense in November 2012. If the systems performed as claimed, they saved Israel an estimated 1,778 casualties and $80 million in property damage, and thereby made preemptive strikes on Gaza about eight times less valuable to Israel. Gaza militants could have inflicted far more damage by grouping their rockets into large salvos, but this may have been difficult given Israel's suppression efforts. Counter-battery fire by the militants is unlikely to be worthwhile unless they can obtain much more accurate missiles.
Supplementary resource (1)
... Decision-making models have been widely used for battle analysis and evaluation, from fire optimization to strategic assessment or battle economic cost. The classic applications from operational research (Morse and Kimball, 1946;Bracken et al., 1995;Armstrong, 2014) are listed, giving an overview of its impact on military decision making: the evaluation of the courses of action (COAs), theater of operations scenario analysis, relationship between intelligence and attrition for the increment of precision in decision-making models, as well as the optimization of war efforts, fire optimization, computer-assisted exercises (CAX), economic cost calculation and force replacement, and others of similar consideration. Representative examples of application that allows us to realize how decision-making models are applied, not included so far, can be identified in the literature review: Sheeba and Ghose (2008a) used the Square Law for optimization of the defensive tactics associated with the disposition and distribution of resources. ...
Casualties in conflicts and wars are the subject of study and research, which allows us to quantify the odds of victory and tries to minimize one's losses. There have been numerous attempts to develop battle dynamics models that determine the evolution of war as a competitive advantage. Understanding which decision types are applied, how those decisions are made, and what empirical evidence confirms the proposed approaches' value becomes fundamental in their application in system aid decisions. The current state of knowledge and the variety of research show a degree of convergence and conclusion, very useful to provide through this study a reference for the application of battle prediction models in decision-making.
... Stochastic versions of the Lanchester's model have been analyzed by Hausken and Moxnes (2002). These kinds of models have been applied to simulate a larger number of historical battles, from the Battle of Gibeah, which took place sometimes between 1200 and 1000 BC (David 1995) to the study of Israel's Iron Dome (Armstrong 2014). A series of battles that have attracted considerable interest in the literature is the Ardennes Campaign, studied by Bracken (1995), Fricker (1998), Chen and Chu (2001), and Hausken and Moxnes (2000, 2005. ...
This paper proposes a Lanchester-type combat model to simulate battles in which one or two of the opposing sides cannot use all the forces simultaneously due to some physical restriction (i.e., topographic constraints, transforming the battlefield into a bottleneck). We show that this model, when the bottleneck restriction applies to both sides, leads to the Lanchester’s linear law for both aimed- and unaimed-fire, but the rate of change over time is a constant. The main characteristics of the bottleneck combat model are the following: (1) the topographic constraint makes the quality (fighting effectiveness) and size of the restriction the more relevant factors for the outcome of the battle, reducing the relative importance of quantity; (2) the bottleneck transforms the Lanchester’s square law into the linear law under direct-fire; and (3) if quality is similar among foes, the topographic bottleneck restriction is irrelevant for victory. The model is used to simulate the Battle of Thermopylae and shows that if the bottleneck restriction had persisted and was not removed, the Persian army would have been defeated.
... Lots of efforts have been done to evaluate the effectiveness of BMDS [4]. Some scholars studied on the multilayer missile defense system which considers the combat effectiveness and operational costs by using traditional deterministic modeling methods [5]. Others explored the use of system engineering thinking [6,7] or the computational experiments approach [8] to solve the complex BMDS modeling and simulation problem. ...
Ballistic missile defense system (BMDS) is important for its special role in ensuring national security and maintaining strategic balance. Research on modeling and simulation of the BMDS beforehand is essential as developing a real one requires lots of manpower and resources. BMDS is a typical complex system for its nonlinear, adaptive and uncertainty characteristics. The agent-based modeling method is well suited for the complex system whose overall behaviors are determined by interactions among individual elements. A multi-agent decision support system (DSS), which includes missile agent, radar agent and command center agent, is established based on the studies of structure and function of BMDS. Considering the constraints brought by radar, intercept missile, offensive missile and commander, the objective function of DSS is established. In order to dynamically generate the optimal interception plan, the variable neighborhood negative selection particle swarm optimization (VNNSPSO) algorithm is proposed to support the decision making of DSS. The proposed algorithm is compared with the standard PSO, constriction factor PSO (CFPSO), inertia weight linear decrease PSO (LDPSO), variable neighborhood PSO (VNPSO) algorithm from the aspects of convergence rate, iteration number, average fitness value and standard deviation. The simulation results verify the efficiency of the proposed algorithm. The multi-agent DSS is developed through the Repast simulation platform and the constructed DSS can generate intercept plans automatically and support three-dimensional dynamic display of missile defense process.
... Washburn (Washburn and Kress 2009) suggested that CEP references a circular normal firing error, but the definition of CEP applies to any kind of twodimensional error distribution. CEP is frequently used in target location estimates such as parachute payload, precision guided munitions, and ballistic missiles, providing a measure of the search area in which the true target location can be expected to be found (Didonato 2007;Armstrong 2014). In Nelson's analyses (Nelson 1988), he stated that CEP is a measurement of real target detection for an area search. ...
The internet of things (IoT) has become a trend in interactive environments for providing information to decision-makers. Anti-submarine warfare (ASW) is a typical pursuit and evasion (PE) game that is a very complicated process. The ASW helicopter is assigned to execute the final phase of hunting the submarine with a torpedo attack. In most cases, a single helicopter is assigned to detect the submarine by dipping sonar, and then drops a torpedo. Once the dipping sonar goes off, uncertainty takes over, with the possible result of losing track of the submarine. To prevent this problem, using the IoT concept to create a wireless sensor network (WSN) in the area of interest for keeping ears on the evading submarine is a potential solution. The objective of this paper is to gain insights into this PE scenario so as to quantify the interaction result in order to demonstrate the effectiveness of the helicopter in terms of hunting the submarine. Monte Carlo simulation has been developed as the analytical tool, and ANOVA was used to verify the significance of the output measure of effectiveness (MOE) before analysis. The results show that a slow, unalerted submarine has a very low chance of survival. An alerted submarine has very high chance of survival, but when the proposed sonobuoy WSN is in place, this situation benefitting the submarine will be reversed. The WSN has been proved to be effective in a single helicopter carrying out its ASW task.
... We note that the accuracy of the Iron Dome missile interception unit, employed by Israel in November 2012 during the Pillar of Defense operation, is reported to be 80% (see Jerusalem Post 2013; for further discussion on the efficiency and accuracy of the Iron Dome see Armstrong 2014). This accuracy is similar to that of Terrorist rockets in equation (8). ...
In the last two decades, terrorism has become a major issue around the world. We analyze a continuous conflict between a terrorist (Terrorist) and a passive defender (Defender). Defender is passive because her actions can influence only the costs (damages) when Terrorist attacks. We focus on high-trajectory fire attacks and passive responses to them in the conflict between Israel and the various terrorist groups in the Gaza Strip in accordance with scientific principles. We first consider three sources of data on this part of the conflict and based on these data we make several observations on the sustainability of cease-fires, the use of technologies, different attacks and the frequency of attacks using high-trajectory fire. To explain these observations, we present several simple game theoretical models. Specifically, we consider single-and multi-period games. In each period, Terrorist may attack Defender, who may in turn try to prevent damage. We show that given this conflict's political situation our models are in agreement with the observations from the data. We also present an extension that considers an active defender. This extension also agrees with the observations from the data. This article is protected by copyright. All rights reserved.
... We note that the accuracy of the Iron Dome missile interception unit, employed by Israel in November 2012 during the Pillar of Defense operation, is reported to be 80% (see Jerusalem Post 2013; for further discussion on the efficiency and accuracy of the Iron Dome see Armstrong 2014). This accuracy is similar to that of Terrorist rockets in equation (8). ...
In the last two decades, terrorism has become a major issue around the world. We analyze a conflict between a terrorist (Terrorist) and a passive defender (Defender) using a simple game theoretical model. Defender is passive as her actions can only influence the costs (damages) when Terrorist attacks. We consider single and multi period games. In each period, Terrorist may attack Defender who may try to prevent damage. The games take into account the available technologies for Terrorist and his potential learning. Based on the equilibrium in these games, we make several conjectures related to political events that may change the level of violence, the technology used by Terrorist, and possible cease-fire agreements. We use three sources of data on the conflict between Israel, as Defender, and terrorist groups from the Gaza strip, as Terrorist. Based on these data, we estimate parameters for the models and present numerical examples. We show that for this conflict's political situation our data do not reject our conjectures. In addition, our conjectures provide insights into long term conflicts in general.
A common network security approach is to create a De‐Militarized Zone (DMZ) comprising two layers of network defense. The DMZ structure provides an extra layer of security between the sensitive information in a network (e.g., research and development files) and the component of the network that must interface with the general internet (e.g., the mail server). We consider a cyber‐attack on a DMZ network where both attacker and defender have limited resources and capabilities to attack and defend, respectively. We study two optimization problems and one game‐theoretic problem. Given that the attacker (defender) knows the potential capabilities of the defender (attacker) in the two layers, we obtain the optimal allocation of resources for the attacker (defender). The two‐optimization problems are not symmetrical. Absent any knowledge regarding the allocation of the adversary's resources, we solve a game‐theoretic problem and obtain some operational insights regarding the effect of combat (e.g., cyber) capabilities and their optimal allocation.
This paper uses Israel’s technologically advanced Iron Dome short‐range missile defence system as a deep empirical case study to examine how affective atmospheres mediate the relationship between state power and the agency of technological objects deployed to govern (in)security. Drawing theoretically from productive tensions between more‐than‐human theories of Object‐Oriented‐Ontology, Actor‐Network‐Theory and affect theory, it evaluates Iron Dome as a scintillating “bright object” with variable capacities and limitations that exceed the sum of its components. Iron Dome both overreaches and contradicts the intentions and governance logics of state elites within a spatio‐temporally distributed array of architectures, infrastructures and practices. Within this milieu, its affective power is ambivalent and capricious, and can both enhance and undermine the atmospheric production of security in ways that exceed both instrumental functionality and human intentions. These findings indicate how a geography of security that attends to the affective dimension of object politics can account for the complex and non‐causal ways in which technologies can either reinforce or inhibit security as an anthropocentric endeavour.
Over the last three decades, the Israeli missile defense concept has evolved significantly along with the emerging threats. As a result of long-term strategy, implemented in the 1990s, contemporary missile defense of Israel has been developed into a complex, multi-tier system aimed to match the country’s unique security needs. The purpose of this study is to determine the main directions of the development of the Israeli missile defense concept that have led to its current form. The study identifies key elements of the multi-layered missile defense architecture as well as their role in overall defense strategy of Israel. With the use of theoretical research methods, the period of 1991-2020 was examined in order to trace the development of new missile defense systems and the evolution of Israeli posture on the subject in relation to changing international situation. The author concludes that comprehensive anti-missile shield, consisting of Arrow-3, Arrow-2, David’s Sling and Iron Dome systems, provides Israel with solid defense against short, medium and long range missile threats and presents potential to be developed into one of the most advanced missile defense systems in the world.
Firing multiple artillery rounds from the same location has two main benefits: a high rate of fire at the enemy and improved accuracy as the shooter's aim adjusts to previous rounds. However, firing repeatedly from the same location carries significant risk that the enemy will detect the artillery's location. Therefore, the shooter may periodically move locations to avoid counter‐battery fire. This maneuver is known as the shoot‐and‐scoot tactic. This article analyzes the shoot‐and‐scoot tactic for a time‐critical mission using Markov models. We compute optimal move policies and develop heuristics for more complex and realistic settings. Spending a reasonable amount of time firing multiple shots from the same location is often preferable to moving immediately after firing an initial salvo. Moving frequently reduces risk to the artillery, but also limits the artillery's ability to inflict damage on the enemy.
This empirical article studies rocket attacks and defenses in Israel during operations Protective Edge, Pillar of Defense, and Cast Lead, and also during the Second Lebanon War. It analyzes publicly available counts of rockets fired, fatalities, casualties, and property damage. The estimates suggest that interceptor deployment and civil defense improvements both reduced Israel’s losses slightly during Pillar of Defense and substantially during Protective Edge. They also imply that interceptor performance
during Pillar of Defense may have been overstated. Ground offensives were the most expensive way to prevent rocket casualties. Interceptors were at least as cost-effective as military offensives, and their advantage improved over time.Without its countermeasures, Israel’s rocket casualties could have been more than fifty times higher during Operation Protective Edge. These results imply that Israel’s rocket concerns were more justified than critics admit, but its military operations were less worthwhile than intended.
Summarizes:
Armstrong MJ, 2014. “Modeling short-range ballistic missile defense and Israel’s Iron Dome system”. Operations Research 62 #5, 1028-1039.
This paper develops a probabilistic model that can be used to determine the technical performance required for a defense to meet specific political/military objectives. The defense objective is stated as a certain probability that no warheads leak through the defense. The technical performance is captured by the interceptor single‐shot probability of kill and the warhead detection, tracking, and classification probability. Attacks are characterized by the number of warheads and undiscriminated decoys. Barrage and shoot‐look‐shoot firing modes are examined, with the optimal interceptor allocation derived for the shoot‐look‐shoot mode. Applications of this model for sizing national and theater missile ballistic missile defenses are discussed.
We treat a situation in which a terminal defense wishes to minimize the expected fraction of a target destroyed. Attacking reentry vehicles (RVs) arrive simultaneously, and therefore the defense knows the attack size. Several sequential engagements of the RVs by defensive interceptors are possible, and the defense has the capability to shoot-look-shoot between these engagements. Using techniques of stochastic dominance and stochastic dynamic programming, we determine optimal defensive tactics as functions of the numbers of RVs, interceptors, and engagements remaining; the various single-shot kill probabilities are parameters. We examine extensions of the basic model that determine both the expected number of interceptors remaining after the attack and the minimum number of interceptors needed to provide a desired level of protection of the target.
During counterinsurgency operations, government forces with superior firepower confront weaker low-signature insurgents. Under what conditions should government (Blue) forces attack insurgent (Red) strongholds? How should the government allocate its force across different strongholds when the insurgents' threat to the Blue civilian population must be taken into account? How should the government respond to "smart" insurgents who anticipate the government's optimal plan of attack and prepare accordingly? How do the results change when the government takes Red civilian casualties resulting from attacks on insurgent strongholds into account? This article addresses these questions. Using Lanchester models modified to account for imperfect intelligence, we formulate an optimal force allocation problem for the government and develop a knapsack approximation that has tight error bounds. We also model a sequential force allocation game between the insurgents and the government and solve for its equilibrium. When the government has perfect intelligence, in equilibrium the insurgents concentrate their force in a single stronghold that the government either attacks or not depending upon the resulting casualty count. Otherwise, under reasonable assumptions regarding the government's behavior and intelligence capabilities, it is optimal for the insurgents to "spread out" in a way that maximizes the number of soldiers required to win all battles. If the government worries about Red civilian casualties, the insurgents have a strong incentive to blend in with the Red civilian population, because this can prevent government attacks while allowing the insurgents to inflict casualties on Blue civilians. Such strategic behavior makes it harder for the government to protect its citizens from insurgent attacks.
We describe JOINT DEFENDER, a new two-sided optimization model for planning the pre-positioning of defensive missile interceptors to counter an attack threat. In our basic model, a defender pre-positions ballistic missile defense platforms to minimize the worst-case damage an attacker can achieve; we assume that the attacker will be aware of defensive pre-positioning decisions, and that both sides have complete information as to target values, attacking-missile launch sites, weapon system capabilities, etc. Other model variants investigate the value of secrecy by restricting the attacker's and/or defender's access to information. For a realistic scenario, we can evaluate a completely transparent exchange in a few minutes on a laptop computer, and can plan near-optimal secret defenses in seconds. JOINT DEFENDER's mathematical foundation and its computational efficiency complement current missile-defense planning tools that use heuristics or supercomputing. The model can also provide unique insight into the value of secrecy and deception to either side. We demonstrate with two hypothetical North Korean scenarios.
A methodology is introduced with which to compare the military worth of warship capabilities. It is based on a simple salvo model for exploratory analysis of modern combat characteristics. The “fractional exchange ratio” is suggested as a robust way to compare equal-cost configurations of naval forces, because we cannot know in advance how and where the warships will fight. To aid in exposition, definitions of all terms are included in Appendix A. The methodology is illustrated with important conclusions from parametric analysis, among which are
In spite of the bombardment of southern Israel by thousands of Qassam rockets fired from the Gaza Strip, comparatively few people have been injured or killed. This has led some observers to dismiss Qassams as more of a symbolic than lethal threat. However, southern Israeli towns feature robust civil defense systems that include safe rooms, bomb shelters, early detection alarms, and missile defense. How many casualties would occur were such systems not in place? This article applies shrapnel-casualty and spatial allocation models to the population of the southern Israeli town of Sderot to estimate casualties per randomly aimed rocket fired into the unprepared town; that is, in the absence of civil defense (technical details appear in the Appendix). Assuming an injury radius of only 5 meters from impact, the modeled expected casualties per rocket are between three (best-case) and nine (worst-case) times higher than Sderot's observed casualties-to-rocket ratio, suggesting that Qassam-like terror attacks on unprotected urban locations could prove much more serious than what one would expect based solely on the observed number of casualties in Sderot.
In this article, we examine a model of terrorism that focuses on the tradeoffs facing a terrorist organization that has two qualitatively different attack modes at its disposal. The terrorist organization's objective is to successfully attack at least one target. Success for the target government is defined as defending all targets from any and all attacks. In this context, we examine how terrorist entities strategically utilize an efficient but discrete attack technology — e.g., suicide attacks — when a more conventional mode of attack is available, and the optimal anti-terrorism measures.© 2012 Wiley Periodicals, Inc. Naval Research Logistics, 2012
This article considers two related questions of tactics in the context of the salvo model for naval missile combat. For a given set of targets, how many missiles should be fired to produce an effective attack? For a given available salvo size, how many enemy targets should be fired at? In the deterministic version of the model I derive a simple optimality relationship between the number of missiles to fire and the number of targets to engage. In the stochastic model I employ the expected loss inflicted and the probability of enemy elimination as the main performance measures and use these to derive salvo sizes that are in some sense “optimal.” I find that the offensive firepower needed for an effective attack depends not only on a target's total strength but also on the relative balance between its active defensive power and passive staying power. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2007
The authors study the choice of a deployment and firing doctrine for defending separated point targets of (potentially) different values against an attack by an unknown number of sequentially arriving missiles. They minimize the total number of defenders, subject to an upper bound on the maximum expected value damage per attacking weapon. It is shown that the Greedy Algorithm produces an optimal integral solution to this problem.
This paper reviews the missile-allocation-problem literature. The problem considered is: given an existing weapon force and a set of targets, what is the optimal allocation of weapons to targets? References are organized by type, characterized by submodel, discussed, and annotated. It is proposed that this review methodology by applied to other appropriate areas.
This thesis develops a modified version of Hughes' Salvo Model and employs it to analyze the tactical disposition (concentration or dispersion) of a small, but modern, navy whose adversary is numerically superior but technologically inferior. It also identifies tactical factors and develops insights that are critical to the success of small navies when fighting outnumbered. Quantitative results indicate that the smaller navy must fight dispersed and win by outscouting the enemy and attacking him effectively first. This requires a superior scouting capability, effective command, control, and communications (C3), and the ability to deliver sufficient striking power. To ensure the delivery of sufficient striking power, a small navy must put greater emphasis on offensive firepower to compensate for small force size. To be successful in battle, small navies must show initiative, and be willing to implement bold tactics. These attributes have been demonstrated by small, but successful, naval forces in the history of naval warfare. In addition, innovative tactical thinking can allow small navies to take advantage of useful tactical phenomenon like the "missile- sump effect" and to design the most appropriate type of combat craft for their respective operating environments.
A methodology is introduced with which to compare the military worth of warship capabilities. It is based on a simple salvo model for exploratory analysis of modern combat characteristics. The “fractional exchange ratio” is suggested as a robust way to compare equal-cost configurations of naval forces, because we cannot know in advance how and where the warships will fight. To aid in exposition, definitions of all terms are included in Appendix A. The methodology is illustrated with important conclusions from parametric analysis, among which are 1Unstable circumstances arise as the combat power of the forces grows relative to their survivability. (Stable means the persistence of victory by the side with the greater combat potential.)2Weak staying power is likely to be the root cause when instability is observed.3Staying power is the ship design element least affected by the particulars of a battle, including poor tactics.4Numerical superiority is the force attribute that is consistently most advantageous. For example, if A's unit striking power, staying power, and defensive power are all twice that of B, nevertheless B will achieve parity of outcome if it has twice as many units as A. © 1995 John Wiley & Sons, Inc.†
Utilizing elementary geometric and probability considerations, we estimate the effect of crowd blocking in suicide bombing events. It is shown that the effect is quite significant. Beyond a certain threshold, the expected number of casualties decreases with the number of people in the arena. The numerical results of our model are consistent with casualty data from suicide bombing events in Israel. Some operational insights are discussed. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2005.
When the stochastic version of the salvo combat model was designed, several assumptions and approximations were made to keep its mathematical structure relatively simple. This paper examines the impact of those simplifications by comparing the outputs of the stochastic model to those from a Monte Carlo simulation across 486 scenarios. The model generally performed very well, even where the battle size was relatively small or the damage inflicted by each missile was not normally distributed. The model’s accuracy did decrease where missiles were positively correlated instead of independent.
We formulate and analyze a spatial queueing model concerning a terrorist who is attempting to drive a nuclear or radiological weapon toward a target in a city center. In our model, imperfect radiation sensors form a circular wall around the periphery of the city, and vehicles setting off sensor alarms (representing a terrorist or a nuisance alarm) arrive randomly at the perimeter of a circle (representing the wall of sensors) and drive toward the center of the circle. Interdiction vehicles, one in each wedge of the circle, chase the alarm-generating vehicles. We derive an accurate mathematical expression for the mean damage inflicted by a terrorist in this system in terms of the arrival rate of alarm-generating vehicles and the number of interdiction vehicles. Our results suggest that detection-interdiction systems using current technology are capable of mitigating the damage from a nuclear weapon made of plutonium, but not one made of uranium or a radiological weapon.
This article presents the first models developed specifically for understanding the infiltration and interdiction of ongoing terror plots by undercover intelligence agents, and does so via novel application of ideas from queueing theory and Markov population processes. The resulting "terror queue" models predict the number of undetected terror threats in an area from agent activity/utilization data, and also estimate the rate with which such threats can be detected and interdicted. The models treat terror plots as customers and intelligence agents as servers. Agents spend all of their time either detecting and infiltrating new terror plots (in which case they are "available"), or interdicting already detected terror plots (in which case they are "busy"). Initially we examine a Markov model assuming that intelligence agents, while unable to detect all plots, never err by falsely detecting fake plots. While this model can be solved numerically, a simpler Ornstein-Uhlenbeck diffusion approximation yields some results in closed form while providing nearly identical numerical performance. The transient behavior of the terror queue model is discussed briefly along with a sample sensitivity analysis to study how model predictions compare to simulated results when using estimated versus known terror plot arrival rates. The diffusion model is then extended to allow for the false detection of fake plots. Such false detection is a real feature of counterterror intelligence given that intelligence agents or informants can make mistakes, as well as the proclivity of terrorists to deliberately broadcast false information. The false detection model is illustrated using suicide bombing data from Israel.
In this paper, we propose a stochastic version of the salvo model for modern naval surface combat. We derive expressions for the mean and variance of surviving force strengths and for the probabilities of the possible salvo outcomes in forms simple enough to be implemented in spreadsheet software. Numerical comparisons of the deterministic and stochastic models indicate that while the two models tend to provide similar estimates of the average number of ships surviving a salvo, this average by itself can be highly misleading with respect to the likely outcomes of the battle. Our results also suggest that a navy's preferences for risk (variability) and armament (offensive versus defensive) will depend on not only its mission objectives but also on whether it expects to fight from a position of strength or of weakness.