Backgrounds

OPERATIONS RESEARCH informs Vol. 53, No. 5, September–October 2005, pp. 745–763 issn 0030-364X
eissn 1526-5463
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0745 ® doi 10.1287/opre.1050.0231 © 2005 INFORMS OR PRACTICE A Two-Sided Optimization for Theater Ballistic Missile Defense Gerald Brown, Matthew Carlyle, Douglas Diehl, Jeffrey Kline, Kevin Wood Operations Research Department, Naval Postgraduate School, Monterey, California 93943 {gbrown@nps.edu, mcarlyle@nps.edu, ddiehl@nps.edu, jekline@nps.edu, kwood@nps.edu} 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. Subject classifications: missile defense; optimization; bilevel integer linear program; mixed-integer linear program. Area of review: OR Practice. History: Received June 2004; revision received September 2004; accepted March 2005. production facilities and prepared launch sites. Figure 2 depicts some of those launch sites and the areas they threaten. North Korea is developing longer-range intercontinental ballistic missiles (e.g., the Taep’o-Dong II) that will be capable of striking the western coast of the United States and Alaska (CIA 2001). Given that North Korea also claims to have developed fission weapons, it is vital that we understand how to best deploy (i.e., pre-position) interceptor platforms to defend against TBM attacks from that country. In response to such threats, this paper develops JOINT DEFENDER, a bilevel integer linear program for prepositioning theater ballistic missile defense (TBMD) assets, and demonstrates how to analyze scenarios using two hypothetical Korean examples. Before developing this new model, we first describe the interceptor platforms that have been fielded or are under development, and review the analytical tools currently used to plan deployment of these platforms. They may vex us with shot, or with assault. To intercept this inconvenience, a piece of ordnance ‘gainst it I have placed. Shakespeare, Henry IV 1. Theater Ballistic Missile Defense: Background This paper introduces JOINT DEFENDER, a new, bilevel (i.e., two-sided) optimization model to help plan the prepositioning of the defensive interceptor platforms that the United States and its allies are deploying to counter exigent theater ballistic missile threats. Solutions require only a few seconds or minutes on a personal computer and can yield important new insights. 1.1. The Theater Ballistic Missile Threat Theater ballistic missiles (TBMs) can deliver highexplosive, chemical, biological, or nuclear warheads over long distances. Although no potential adversary other than Russia possesses TBMs capable of striking the United States, both China and North Korea are developing missiles that will likely have that capability by 2015 (CIA 2001). Existing short-range and medium-range TBMs pose immediate threats in many regional conflicts, however, as demonstrated in the first and second Gulf wars. Figure 1 illustrates some TBMs that currently concern military planners. North Korea is particularly worrisome. It is known to be developing and exporting ballistic missiles and missile technology, and has numerous indigenous missile- 1.2. TBM Interceptor Platforms Figure 3 shows three components of the United States joint missile defense, which we will use as representative defensive platforms. The Army’s PATRIOT anti-missile missile system is currently deployed and has been used in Operation Iraqi Freedom. PATRIOT provides terminal defense against ballistic missiles, cruise missiles, and aircraft. It consists of 745 Brown et al.: A Two-Sided Optimization for Theater Ballistic Missile Defense 746 Figure 1. Operations Research 53(5), pp. 745–763, © 2005 INFORMS Current ballistic missile threats. Shown left to right are a North Korean Scud-B transporter-erector-launcher (TEL), a TEL firing a missile, and an Iranian fixed ballistic missile launcher. a mobile launcher, a phased-array air search-and-tracking radar, plus various command and support vehicles. It can shoot three types of interceptor missiles, the PAC-2, PAC-2 GEM, and PAC-3 (Jane’s 2003c). The Army is developing theater high altitude air defense (THAAD), which will provide a midcourse, high-altitude defense against ballistic missiles using a kinetic-kill interceptor. THAAD’s physical composition resembles that of PATRIOT (Jane’s 2003c). “Navy AEGIS” refers to deployed Ticonderoga-class guided-missile cruisers and Arleigh Burke-class guidedmissile destroyers. Each of these ships carries the AEGIS SPY-1 phased-array radar and can function as a TBM interceptor platform. These ships currently carry Standard Missile-2 (SM2) variants that provide terminal defense against cruise missiles and aircraft. The Navy is now developing the Standard Missile-3 (SM3), a kinetic-kill exoatmoFigure 2. Approximate maximum ranges of North Korean Scud-B, Scud-C, and No-Dong theater ballistic missiles. Note that all of Japan and Okinawa are vulnerable to the No-Dong missile. SCUD-B SCUD- C NO-DONG spheric interceptor, to provide a midcourse defense against TBMs (Jane’s 2003b). TBMD has become an important component of the Department of Defense research and development budget (Department of Defense 2004), and we may expect the United States to field a number of new TBMD systems in the next few years. Indeed, an air-based laser is already under development (Jane’s 2003e). We do not include this future system in our demonstration scenarios, but incorporating such innovations in JOINT DEFENDER is straightforward. 1.3. Current TBMD Planning Tools Effective pre-positioning of TBMD assets is crucial given that (a) a defensive interceptor has limited range, (b) it can destroy a TBM only at certain points in the TBM’s trajectory, and (c) that trajectory will depend on the type of TBM and its launch and target coordinates. Currently, joint forces commanders can plan pre-positioning using several analytical tools; we describe these next and point out their strengths and weaknesses. Area Air Defense Commander System. The Navy’s area air defense commander system (AADCS), AN/UYQ89, is currently deployed on command ships USS BLUERIDGE, USS MOUNT WHITNEY, the AEGIS cruiser USS SHILOH, and at the Joint National Integration Center in Colorado (Jane’s 2003a). In addition to modules for real-time tracking of assets and threats, AADCS contains a planning module that enables air-defense commanders to plan and “war-game” potential TBMD scenarios. AADCS uses 32-processor Silicon Graphics supercomputers to evaluate, using an enumeration-based myopic heuristic, a sequence of increasingly complex “defense plans” before committing to a good one (Silicon Graphics Incorporated 2003). For each target in a scenario, in priority order, AADCS enumerates every possible combination of (a) enemy launch site, (b) missile type, (c) interceptorplatform position on a fine geographical grid, and (d) intercept salvo (set of interceptors that might be shot at the TBM). For each of those combinations, it evaluates the probability of intercepting the TBM successfully. Once AADCS identifies the required platform(s), position(s), Brown et al.: A Two-Sided Optimization for Theater Ballistic Missile Defense 747 Operations Research 53(5), pp. 745–763, © 2005 INFORMS Figure 3. TBMD platforms, deployable or under development. Shown left to right are a THAAD launch vehicle, an AEGIS guided missile cruiser firing a standard missile, and a PATRIOT launch vehicle. and corresponding salvo(s) that yield a sufficiently high probability of intercept, the next target in the priority list is analyzed, more platforms are positioned, and so on. (Once a platform’s position is fixed, it is also considered for defending subsequent, lower-priority targets.) AADCS provides an estimate of defense coverage and an expected number of enemy missiles that will leak through the defense plan. AADCS’s brute-force enumeration determines an optimal defense for its highest-priority target, and that enumeration goes on to optimize a sequence of restricted problems. Several weaknesses are apparent: the procedure is essentially a sequential greedy heuristic with no guarantee of global optimality, it ignores the enemy’s strategy, and it requires an expensive computational platform. Theater Battle Management Core Systems (TBMCS). The U.S. Air Force’s air operations centers use theater battle management core systems (TBMCS) for theater-level planning in support of the area air defense commander. TBMCS supports strategic planning, air battle planning, and mission preparation, together with mission execution, reporting, and analysis; the last items are supported in near real-time as situations unfold. A module in TBMCS automates an overlay of potential “launch fans” by defensive “interceptor envelopes.” That is, the module evaluates a manuallyprepared, pre-positioning plan for defenses by analyzing the intersection of (a) the two-dimensional projection on the earth’s surface of the three-dimensional region that might be traversed by a TBM, and (b) a similar projection for the “kill zone” of an interceptor shot from a given position. This procedure suggests a plausible solution that indicates whether or not a hypothesized attacking missile can be struck by a pre-positioned interceptor, subject to the error induced by two-dimensional projections. TBMCS cannot optimize defense plans because it requires human intervention (i.e., guessing), it does not measure expected damage incurred by an attack, and it ignores the enemy’s strategy. Commander’s Analysis and Planning Simulation. Since 1993, the Missile Defense Agency has sponsored commander’s analysis and planning simulation (CAPS), which is currently hosted by theater ballistic missile- planning cells of Central Command, European Command, Pacific Command, Strategic Command, the Naval Postgraduate School, and others—a total of more than 50 sites. CAPS helps assess defense-system capabilities and positioning. The performance of manually-prepared defense plans can be evaluated against manually-prepared threat scenarios (Sparta 2004). The CAPS operator selects the “best-looking” defense plan that appears to protect defended assets (targets) with high probability and appears to maximize the number of missiles the defender can engage. CAPS does not make the two-dimensional approximations that TBMCS does, but it still requires human intervention and ignores the enemy’s strategy. All three fielded systems, AADCS, TBMCS, and CAPS, address the complex problem of TBM defense in very different ways, with wide variation in computational requirements, degrees of fidelity, and objectives. These systems can be used to search for “good” defense plans, but only through manual or automated heuristics. None of the systems account for how the enemy might change his strategy in response to observing pre-positioned TBMD assets, i.e., in response to observing an implemented defense plan. 2. A New Two-Sided Optimization for TBMD Planning We introduce a new paradigm for planning the prepositioning of TBMD assets. We first express enemy courses of action as an “inner” mathematical optimization that maximizes expected damage subject to known defensive positions and capabilities. An “outer” optimization minimizes that maximum by pre-positioning defensive platforms and committing to intercept strategies appropriately. We can most conveniently express our model for TBMD as a bilevel integer linear program (BLILP) (e.g., Moore and Bard 1990). Then, because of its special structure, we can convert our BLILP into a standard mixed-integer linear program (MILP) to actually solve it. With the roles of attacker and defender reversed, this general idea has been successfully used to model a number of networkinterdiction problems (Phillips 1993, Wood 1993, Israeli and Wood 2002; see Whiteman 2000 for details on an 748 application; see Fulkerson and Harding 1977, Golden 1978 for earlier, bilevel linear-programming models involving continuous interdiction effort; and see Salmeron et al. 2004 for an application of a bilevel optimization to interdicting electric power grids). In these network-interdiction problems, an interdictor uses limited offensive resources to attack and damage an adversary’s network (e.g., road system, communications network) to minimize the maximum benefit his adversary can obtain from it. Skroch (2004) and Brown et al. (2004) model the optimal disruption of a weapons development program by interdicting a project network. Their BLILP cannot be converted to a MILP, and is solved with a decomposition algorithm. 2.1. TBMD Terminology and Assumptions The following terminology and assumptions characterize JOINT DEFENDER. Both sides have full knowledge of the parameters and data described below. Each launch site for attacking missiles is located by latitude and longitude. Any finite number of dispersed launch sites may exist. Each target vulnerable to enemy attack is located by latitude and longitude. Any finite number of dispersed targets may exist, and each has a target value. Each candidate defender position is located by latitude and longitude. Any finite number of candidate positions may exist. The set of positions can include, for example, a discretized field of grid points with desired geographic fidelity. Each enemy missile has a minimum and maximum range, and can hit any target within this range of its launch site with a known probability of kill. This probability can depend on the missile type, target, and range from launch site to target. An enemy attack consists of a launch of a missile from an enemy launch site against a vulnerable target. The enemy’s goal is to launch a set of attacks that maximizes total expected target damage. Each defender class consists of a given number of individual platforms, each endowed with a loadout of a given number of each type of interceptor weapon (antimissile missiles are the only extant interceptors, but other types, like the air-based laser, are under development). Each defending platform may be located at any candidate defender position that is secure and compatible for its class. That is, ships may only be positioned at sea, land units on compatible terrain, and air defenders in safe airspace. A single attack (one missile from one launch site to one target) may be engaged by any defending platform with an intercept salvo of any number of any types of interceptor missiles available on that platform. For planning purposes, and as a matter of effective tactical doctrine, we assume that the planned intercept of each enemy missile will be executed by a single defending platform. (In execution, this would not preclude defending platforms from Brown et al.: A Two-Sided Optimization for Theater Ballistic Missile Defense Operations Research 53(5), pp. 745–763, © 2005 INFORMS providing a layered defense to defended targets, but we do not address this complication.) The probability of negation defines the probability that an intercept salvo will destroy the attacking missile; this varies by attack launch site, missile type, target location, defender position, defending salvo, and any synergy among the intercepting missiles in that salvo. The geometry of such an engagement can be depicted by an oblate spherical triangle, with apexes at the launch site, the target location, and the defender’s position. The probability of negation for an intercept salvo is then a function of (a) the relative positions of missile and interceptor, (b) the vulnerability of the attacking missile to the interceptor at the point of intercept—some interceptors can only strike a missile traversing its early- or middle-phase flyout trajectory, and some provide only terminal-phase defense—and (c) the combined effectiveness of the entire intercept salvo. In practice, JOINT DEFENDER uses probabilities of negation computed through a mathematical approximation, or through lookup and interpolation of engineering estimates in “cross-range” and “down-range” tables for each type of intercept salvo and missile altitude. The probability of negation for an interceptor salvo does not rely on an independence assumption among missiles in that salvo. 2.2. Mathematical Development of JOINT DEFENDER The attacker controls a set of launch sites s ∈ S, and possesses fixed m s missiles of type m ∈ M pre-positioned at site s, as well as a pool of mobilem missiles that can be transported to any capable receiving launch site. Transport of the mobile missiles may be restricted by move m s and/or movem s . Launch site s can launch no more than fixed m s + movem s missiles of type m. (Of course, if the defender knows that launch site s is incapable of launching missile type m, fixed m s = movem s = 0.) The defender guards a set of targets t ∈ T , with each target t having value val t . An attack a ∈ A consists of a launch from site sa ∈ S of a missile of type ma ∈ M at a target ta ∈ T . This attack will hit and destroy the target with probability of kill Pka , assuming that the defender takes no action. An upper bound missilest may be placed on the number of missiles the attacker will launch at target t. The attacker must decide which missiles to launch at which targets to maximize total expected target damage, weighted by target value. The defender controls a set of defending platforms p ∈ P , each of which is a member of platform class cp ∈ C. Each platform of class c can be pre-positioned at any one location g ∈ Gc ⊆ G. Each platform p carries loadout p i defensive interceptors of type i ∈ I. An attack a can be engaged with alternative defensive actions d ∈ D, where defense d launches salvoa c d i interceptors of type(s) i and succeeds in thwarting the attack with probability of negation Pna c g d . Each defensive engagement is conditional, meaning that if attack a is not launched, then no interceptor devoted to engaging that attack is launched. Brown et al.: A Two-Sided Optimization for Theater Ballistic Missile Defense Operations Research 53(5), pp. 745–763, © 2005 INFORMS The defender wishes to optimize defensive prepositioning for attack interception while assuming the attacker will observe these preparations and optimize his multimissile attack to exploit any weaknesses in these defenses. The defender’s objective is to minimize the maximum total expected damage to targets. We note that this model is a conservative one for the defender because he must protect against the worst possible attack. It is conservative for the attacker, because he must plan against the best possible defense. However, variants of the model we describe later enable analysis of a range of situations, from conservative to optimistic, for either opponent. Model JD-MINMAX: Minimize Maximum Expected Total Damage Indices and Index Sets Attacker m ∈ M attacking missile types s ∈ S attacker launch sites t ∈ T targets (“defended asset”) a ∈ A attacks (a single missile launched at a target) a ∈ Am s ⊆ A attacks launching a missile of type m from site s a ∈ At ⊆ A attacks a with target t sa launch site of attack a, sa ∈ S ma missile type launched in attack a, ma ∈ M ta target of attack a, ta ∈ T Defender p∈P c∈C cp g∈G defending platforms defending platform classes class of platform p, cp ∈ C candidate stationing positions for a defending platform g ∈ Gc ⊆ G candidate stationing positions for a defending platform of class c i ∈ I defensive interceptor types d ∈ D defense options Data units Attacker mobilem fixed m s movem s , movem s missilest valt Pka attacker’s total supply of mobile missile type m (missiles) attacker’s total supply of stationary type m missiles at launch site s (missiles) minimum and maximum number of mobile missile type m that attacker can transport to launch site s (missiles) maximum number of missiles that can attack target t (missiles) value of target t (value) probability that attack a hits and destroys its target ta if not intercepted, i.e., probability of kill (fraction) 749 Defender loadout p i number of type i interceptors carried by platform p (interceptors) salvoa c d i number of type i interceptors used against attack a by a class c platform exercising defense option d (interceptors) shoot p maximum number of interceptors platform p can shoot in an exchange (interceptors) Pna c g d probability that attack a is negated if platform p, class c = cp , in position g ∈ Gcp exercises defense option d, i.e., probability of negation (fraction) Variables units Attacker Wm s number of type m mobile missiles transported to launch site s (missiles) Ya 1 if attack a is launched, 0 otherwise (binary) (Y, the vector of attacks by individual missiles, is an “attack plan”) Defender Xp g 1 if platform p is positioned at g, 0 otherwise (binary) Ra p g d 1 if attack a is engaged by platform p from position g ∈ Gcp exercising defense option d, 0 otherwise (binary) Formulation of JD-MINMAX. We specify a set of dual variables, in square brackets, for each constraint of the inner (maximization) problem in JD-MINMAX. These duals are only defined (and used) for solutions to the linear programming relaxation of the integer linear program that results when X and R are fixed.           valt Pka 1− Pna cp  g d Ra p g d Ya A0  max    Y     t a∈At p g d              s.t. mobile ∀ m   A1 W
m m m s       s        −W + fixed ∀ ms   A2 Y
m s m s a m s ∗ Z = min a∈Am s  X R∈XR      Ya
missilest ∀ t t  A3        a∈At        
W
move and integer move   m s m s m s      ∀ ms − m s  m s  A4          0
Ya
1 and integer ∀ a !a " A5 The notation X R ∈ XR denotes all feasible prepositioning and interception plans for the defender. This feasible set is described in detail below. The attacker’s objective (A0) expresses total expected target damage, assuming a cumulative effect across targets, and for multiple missiles striking a single target. Constraints (A1) limit the number of mobile missiles of each type that can be transported to launch sites. Constraints (A2) limit the maximum number of missiles of each type, both mobile and fixed, that can be launched from each launch site. Constraints (A3) limit the number of missiles that can attack each target. Constraints (A4) limit the number of mobile missiles of each type that can be transported to each launch site. Brown et al.: A Two-Sided Optimization for Theater Ballistic Missile Defense 750 Operations Research 53(5), pp. 745–763, © 2005 INFORMS The objective (A0) expresses expected incremental target value damage inflicted as a consequence of each attacking missile. For an area target, such as a city or airfield, such a cumulative damage model is standard (e.g., Eckler and Burr 1972). But a point target might be destroyed by any single attacking missile, and the lack of a joint probability expression for surviving more than one hit means that the attacker can be over-credited with damage value. (This problem disappears if the attacker can launch no more than one missile at any target, which can be enforced through constraints (A3).) We believe that when it comes to weapons of mass destruction carried by TBMs, the damage to an economy and a society will continue to increase as the number of successful missile strikes increases. Thus, the cumulative model of damage is appropriate, although there might be some diminishing returns to an attacker as the number of successful strikes on a target (or in a target area) increases. Appendix A suggests how to modify the objective function for diminishing returns or point targets, should these issues arise. The defender’s actions are limited by X R ∈ XR, where XR is defined by the following set of constraints:  Xp g
1 ∀ p (D1) 2.3. Solving JD-MINMAX with JD-MILP Direct solution of a min-max model like JD-MINMAX is impossible with standard software. We could create a specialized decomposition algorithm for solving it, along the lines of Israeli and Wood (2002), but prefer a simpler method if one exists. In this case it does: Although the attacker’s decision vector W is integer and Y is binary, the constraint matrix involving W and Y is totally unimodular and all corresponding right-hand side data are integer. Thus, all solutions to the linear-programming relaxation of the attacker’s maximizing problem are intrinsically integer. Therefore, we can simply take the linear-programming relaxation of the inner problem to create an inner maximization that is a linear program. We then use the dual variables defined above, and take the dual of that inner maximization to create a “min-min” problem. This results in a simple, minimizing MILP, which we solve using standard optimization software. The MILP is JD-MILP    min mobilem m + fixed m s m s + missilest t
     X R m m s g  Xp g
1 ∀g (optional) (D2) ∀ a (D3) −  movem s t m s +  movem s m s + m s m s  a p  Ra p g d
1 p g d  a d + salvoa cp  d i Ra p g d
shoot p ∀p (optional) (D5) Ra p g d
Xp g m s  0 ∀ m s (T0) (T1) Pka val ta Pna cp  g d Ra p g d p g d ∀ p i g ∈ Gcp  (D4) a g d  m s ma  sa + ta + !a + salvoa cp  d i Ra p g d
loadout p i Xp g  s.t. m − m s − !a  Pka valta ∀ a (T2)  Xp g
1 ∀ p (T3)  Xp g
1 ∀ g (T4) g ∀ a p g (D6) d all Xp g  Ra p g d ∈ #0 1$" p (D7) Each constraint (D1) limits a platform to occupy at most one grid position; each constraint (D2) optionally limits a grid position to accommodate at most one platform; each constraint (D3) allows at most one interception of each attack; each constraint (D4) limits the number of interceptor engagements from each positioned platform and gridpoint combination; each constraint (D5) optionally limits the total number of interceptors that a platform can shoot in the short period of time that elapses in an exchange; each constraint (D6) permits an engagement only from an occupied platform and grid-point combination; and constraints (D7) require binary decisions. Note that constraints (D3) do not require a response for every attacking missile. Indeed, if defenses are overwhelmed, it may be impossible to intercept every missile launched, and we must allow for this eventuality. The attacker plans to maximize expected damage, and the defender plans to minimize the attacker’s maximum expected damage.  Ra p g d
1 ∀ a (T5) p g d  salvoa cp  d i Ra p g d − loadout p i Xp g
0 a d ∀ p i g ∈ Gcp  (T6)  salvoa cp  d i Ra p g d
shoot p ∀ p (T7) a g d  Ra p g d
Xp g ∀ a p g (T8) d all m  m s  a  m s  m s  0 all Xp g  Ra p g d ∈ #0 1$" (T9) The solution of JD-MILP yields an optimal defense prepositioning plan X∗ and interceptor-commitment plan R∗ . We recover the associated, optimal mobile-missile transport plan W∗ and attack plan Y∗ by fixing X = X∗ and R = R∗ in JD-MINMAX, and solving the linear program that results. Brown et al.: A Two-Sided Optimization for Theater Ballistic Missile Defense Operations Research 53(5), pp. 745–763, © 2005 INFORMS JD-MILP can be embellished with additional features as long as the modifications can be expressed linearly in X R ∈ XR, and the embellishments that modify the attacker’s constraints (A1)–(A5) do not destroy their total unimodularity. (If maintaining total unimodularity in the attacker’s optimization is too restrictive, more general solution methods apply, as mentioned above.) 2.4. Variants of JOINT DEFENDER to Assess the Value of Flexibility By tightening or relaxing constraints (D1)–(D7) on the defender and solving the resulting versions of JD-MILP, we can assess the value of flexibility, or the lack thereof, to the defender. For instance, a commander might not currently be able to place an AEGIS platform in a set of positions G′ that is threatened by the adversary’s coastal defenses. The commander could solve JD-MILP with and without G′ included in G, and determine whether or not it is worthwhile to neutralize those coastal defenses. (This comparison we envisage still assumes transparency between the sides, and that the attacker will know that his defenses have been neutralized and that the previously inaccessible positions are now available to the defender.) 2.5. The Value of Secrecy JD-MILP’s assumption of complete transparency between attacker and defender can lead to unappealing (but logical) outcomes. Suppose, for example, that a defender has two assets to defend, has two interceptors for that defense, and each interceptor has a Pn of 1. Further, assume that he is opposed by an attacker who has two missiles that can strike either target (asset), each with a Pk of 1. Because the attacker can see the defender’s preparations, he will destroy at least one target—with probability 1. This may be unappealing because, in the familiar setting of a two-person zero-sum game with randomized strategies, the defender can have a positive probability of losing neither of his assets. Of course, the game-theoretic setting requires opacity, i.e., each opponent must hide his intentions from the other. But, completely hiding missile launch sites and interceptor platforms such as ships is impossible. On the other hand, both attacker and defender probably do not have complete knowledge of their opponent’s plans. To handle this issue, we can modify JD-MINMAX, and JD-MILP correspondingly, to represent situations in which some of the defender’s assets are hidden from the attacker, and/or some launch sites or missiles are hidden from the defender. We refer to the defender being able to conceal part of his decision, fooling the attacker into basing his strategy on partial, or bogus data, and then taking advantage of that deception. This section discusses this case and its converse, where the attacker can conceal some information from the defender. The Value of Defender Secrecy. The following procedure will evaluate the advantage the defender can gain by 751 hiding the existence of a subset of his platforms from the attacker: (1) Solve the standard version of JD-MILP to determine total expected damage Z ∗ under the assumption that the defender’s platforms are all visible to the attacker. (2) Remove platforms whose existence the defender can hide; the attacker knows nothing whatsoever about these platforms. (3) Solve this modified version of JD-MILP for the visible defense strategy X∗  R∗ , and recover the attacker’s optimal strategy Y∗ . (4) Fix the “visible-defense strategy” X∗  R∗  and the unsuspecting attacker’s strategy Y∗ in JD-MINMAX and solve the defender’s minimization to determine the optimal strategy for the hidden platforms, and the total expected damage Z ∗∗ , given the attacker’s obviously suboptimal strategy. (5) Because of the attacker’s suboptimal strategy, Z ∗∗
Z ∗ , so that Z ∗ − Z ∗∗ may be viewed as “the value of secrecy” to the defender. If we hide all defending platforms, and use this procedure, we are estimating the “value of a total surprise defense.” (This emulates current planning tools.) The Value of Attacker Secrecy. Suppose that the defender has gained enough information to be able to, in essence, fix all the variables Wm s in JD-MINMAX. That is, he knows the exact location of every missile the attacker possesses. Both sides solve the resulting restricted version of JD-MILP and determine the total expected damage Z ∗∗ ; let X∗∗ represent the defender’s optimal pre-positioning plan for this situation. Now, if the attacker can transport his missiles from site to site without being observed, and do this optimally, he may be able to increase expected damage, because the defender has been fooled and will use his original, now suboptimal, pre-positioning plan X∗∗ . So, the attacker solves JD-MILP with X fixed at X∗∗ (for simplicity, we allow the defender to reoptimize interceptor commitments R), determines an optimal “missile-transport plan” W∗ , and optimal attack plan Y∗ with objective value Z ∗ . Clearly, Z ∗∗
Z ∗ , and the difference Z ∗ − Z ∗∗ represents the value of secrecy to the attacker. Suppose that the attacker can fool the defender into thinking he will never launch his missiles, or that he has none at all. In that case, X∗∗  R∗∗  = 0 0, i.e., no defense, is a reasonable response from the defender. If we fix X R = 0 0 in JD-MINMAX and solve the resulting linear program to obtain Z ∗ , we can determine the “value of a total surprise attack” by comparing Z ∗ to the optimal objective from JD-MILP for a baseline, nonsurprise scenario. 3. Case Study: North Korea, Circa 2010 We have developed two North Korean scenarios, circa 2010, which specify a North Korean arsenal of ballistic missiles and launch sites, a U.S. contingent of ballistic missile defense platforms, and a list of targets with associated Brown et al.: A Two-Sided Optimization for Theater Ballistic Missile Defense 752 Operations Research 53(5), pp. 745–763, © 2005 INFORMS target values. We use these scenarios, and variants, to demonstrate JOINT DEFENDER. In the basic scenario we put each North Korean missile at a specific launch site, but we also report cases in which the missiles are transportable. When no confusion results, we use the term JOINT DEFENDER to mean JD-MINMAX, or JD-MILP, or the full decision-support system that incorporates these models, prepares data for problem generation, solves the problem, and returns solutions in accessible format. Each diamond indicates a North Korean launch site. Figure 4. 0 50 100 Yong-don kilometers Chunggang-up China No-dong Korea, Democratic People's Republic of Toksong Kanggamchan 3.1. The Attacker’s Launch Sites Kanggye The attacker’s hypothetical missile launch sites are based on known North Korean missile facilities and bases taken from unclassified sources (Federation of American Scientists 2003). Table 1 lists these sites, and Figure 4 shows their approximate locations. Paegun Mayang Tokchlon Sunchon Ok'pyong Namgung-ri Pyongyang Mari'gyongdae-ri Sangwon 3.2. Attacker Missiles Chiha-ri Table 2 displays the selection of missiles from the North Korean inventory we model in this scenario, along with their approximate minimum and maximum ranges. These ranges have been compiled from unclassified sources (e.g., Jane’s 2003d). We assume that each missile hits and destroys its assigned target with perfect reliability, i.e., Pka = 1"0 for any a, if the missile is not intercepted. This expresses the worst-case situation. 3.3. Targets on a Defended Asset List (DAL) Table 3 displays the defended asset list (DAL) and target values for all scenarios, and Figure 5 displays target locations on an area map. We generate target values for the Table 1. Korea, Republic of DAL based upon a subjective assessment of the four factors currently used in air-defense planning: criticality, vulnerability, reconstitutability, and threat (e.g., Department of the Army Field Manuals FM 3-01.11 2000a and FM 44-100 2000b). Target t’s criticality ct judges the degree to which an asset is essential to the defender. A high value indicates that the asset is extremely critical, and a low value indicates otherwise. North Korean launch sites (after Federation of American Scientists 2003). These comprise current North Korean missile-production facilities and missile bases, and are used in our scenarios as potential launch sites. For fixed-launch-site scenarios, the maximum number and type of each North Korean ballistic missile is shown for each launch site. When we permit transporting mobile missiles, this same inventory of Scud-B, Scud-C, and No-Dong missiles is mobile. Missile types Launch sites Chiha-ri Chunggang-up Kanggamchan Kanggye Mari’gyongdae-ri Mayang Namgung-ri No-dong Ok’pyong Paegun Pyongyang Sangwon Sunchon Tokch’on Toksong Yong-don Latitude (N) 38 41 40 40 38 40 39 40 39 39 39 38 39 39 40 41 ′ 37 46′ 24′ 07′ 59′ 00′ 08′ 50′ 17′ 58′ 00′ 50′ 25′ 45′ 25′ 59′ Longitude (E) 126 126 125 126 125 128 125 129 127 124 125 126 125 126 128 129 ′ 41 53′ 12′ 35′ 40′ 11′ 46′ 40′ 18′ 35′ 45′ 05′ 55′ 15′ 10′ 58′ Scud-B Scud-C No-Dong 15 20 10 15 15 20 15 15 5 15 15 15 20 15 15 15 10 10 10 10 10 20 2 15 10 10 10 10 10 15 15 20 10 5 15 15 15 5 5 5 Taep’o-Dong I Taep’o-Dong II 1 1 1 1 1 1 Brown et al.: A Two-Sided Optimization for Theater Ballistic Missile Defense 753 Operations Research 53(5), pp. 745–763, © 2005 INFORMS Table 2. North Korean ballistic missile types with their range limits. The Scud-B, Scud-C, and NoDong missiles are operational today; the intercontinental Taep’o-Dongs are in development. Table 4. Interceptor Minimum Maximum 40 40 1350 2200 3500 330 700 1500 2900 4300 Scud-B Scud-C No-Dong Taep’o-Dong I Taep’o-Dong II Vulnerability vt represents the degree to which a target is susceptible to an air or missile attack or is vulnerable to surveillance. A high value indicates that the target is extremely vulnerable, i.e., unprotected and in the open; a low value indicates otherwise. Reconstitutability rt assesses the degree to which the target can recover from inflicted damage, and incorporates time, the need or lack of need for special repair equipment, and the amount of manpower required to resume normal operation. A high value indicates that the target would need considerable time, equipment and/or manpower to return to normal operation following an attack; a low value indicates otherwise. Threat ht subjectively estimates the probability of a target being attacked. A high value indicates that it is nearly certain that the enemy will attack this target. We combine these factors through val t = lnct × vt × rt × ht  + 1 250 160 160 70 120 1200 of target value can be replaced, but any alternative should address these four important components. Initially, we allow a defended target to be attacked at most once. We want results that are easy to visualize, so we present point targets, easily located on a map. 3.4. Defensive Platforms To evaluate the defender’s 2010 defense plan, we assume that two AEGIS cruisers are deployed, each with 10 SM3 and 20 SM2 interceptors, along with one AEGIS destroyer with 20 SM2 interceptors. Each AEGIS ship has been configured for ballistic-missile defense and deploys as an independent entity. The defender also has two land-based defensive assets. He can use one PATRIOT battery, which consists of eight mobile launchers (and support vehicles), each loaded with four PAC-3 missiles, two PAC-2 GEM missiles, and one PAC-2 missile. And, he can use one THAAD battery whose salient features comprise a mobile launcher and 10 interceptors. 3.5. Interceptor Ranges where all ct , vt , rt , and ht range from about 1 to about 10. The natural log function (ln) is chosen somewhat arbitrarily so that valt also ranges from about 1 to 10. Our definition Table 3. Maximum range (km) THAAD PAC-2 PAC-2GEM PAC-3 SM2 block III variants SM3 Range (km) Missile Ranges for each defender interceptor. Table 4 specifies the maximum range of the various interceptors used by defense platforms in our scenario. Ranges are gleaned from the open literature (Jane’s 2003b, c, e). Targets on a defended asset list (DAL). Targets are on this list because of their political or military significance and are spread out over South Korea, the main islands of Japan, and Okinawa. Each target is assigned four scores, reflecting criticality, vulnerability, reconstitutability, and threat. For example, Seoul has c v r h values of 4 8 5 9, which result in a target value of ln4 × 8 × 5 × 9 + 1 = 8"3. The example values shown here are completely arbitrary. Target Atsugi, JP Misawa, JP Okinawa, JP Sasebo, JP Tokyo, JP Yokosuka, JP Chinhae, ROK Inchon, ROK Kunsan, ROK Osan AB, ROK Pusan, ROK Seoul, ROK Latitude (N) 35 40 26 33 35 35 35 37 35 37 35 37 ′ 27 42′ 20′ 09′ 41′ 17′ 08′ 29′ 54′ 06′ 06′ 27′ Longitude (E) 139 141 127 129 140 139 128 126 126 127 129 126 ′ 27 25′ 47′ 44′ 00′ 40′ 41′ 38′ 37′ 02′ 02′ 57′ c v r h val 4 8 7 7 4 8 7 3 10 10 8 4 7 5 7 8 9 8 7 6 7 8 7 8 6 7 8 7 4 7 7 5 9 9 8 5 5 5 3 7 7 7 8 4 10 10 10 9 7.7 8.2 8.1 8.9 7.9 9.1 8.9 6.9 9.7 9.9 9.4 8.3 Brown et al.: A Two-Sided Optimization for Theater Ballistic Missile Defense 754 Figure 5. Operations Research 53(5), pp. 745–763, © 2005 INFORMS Candidate platform positions for the defender. Each circle in Japan and South Korea represents a target; each diamond in North Korea represents an attacker launch site; sea-based platforms can be located at any grid point at sea; land-based platforms can be based at any grid point on land excluding those in North Korea and China (the upper left-hand corner); land-based platforms can also be collocated with targets. For simplicity, grid points are placed at each integer value of latitude and longitude. In reality, the defender’s candidate locations could be specified with much greater freedom. 3.6. Interceptor Effectiveness: Probability of Negation (Pn) For simplicity, these test cases assume “reasonable engagement conditions,” which means that a nonterminal interceptor is within range of an attacking missile’s trajectory, or a terminal defender is, effectively, collocated with the target of an attack. When these conditions are met, we set the probability of negation (Pn) for an interceptor to a reasonable but hypothetical value between 0.7 and 0.9, and set it to 0.0 otherwise. Alternatively, JOINT DEFENDER could employ a set of (potentially enormous) tables that provide interceptor effectiveness indexed by interceptor type and engagement geometry as specified by cross-range and down-range proximity, and by the attacking missile’s altitude. (With respect to the great circle arc connecting the launch site to the target of the attacking missile, the “crossrange” proximity is the distance from the defending platform to the closest point of the arc, and the “down-range” proximity is the distance from this closest point to the target.) We derive the joint probability that a salvo of interceptors negates an attacker’s missile from the negation probabilities of the missiles that comprise a salvo. In this paper, we assume independence between interceptors in a salvo, but JOINT DEFENDER does not require this. 3.7. Candidate Defender Positions We discretize candidate defender positions into a latitude and longitude grid with increments of one degree, about 60 nautical miles; see Figure 5. This discretization yields 304 candidate grid locations for pre-positioning interceptor platforms, although geography precludes certain classes from being assigned to certain positions. Terminal defensive platforms can also collocate with targets. In addition to obvious restrictions to locate land units on land, and to position ships at sea, we have defined an optional, restricted set of sea positions that are at least 100 nautical miles from the North Korean coast. This puts the ships outside of the 60 nautical-mile range of North Korea’s shore-based HY-1 Brown et al.: A Two-Sided Optimization for Theater Ballistic Missile Defense 755 Operations Research 53(5), pp. 745–763, © 2005 INFORMS Silkworm and HY-2 Seersucker anti-ship missiles (Federation of American Scientists 2004, Department of the Army 1999). Our approach does not depend on the structure of the set of candidate locations; we only require that the list be finite. In a real scenario the area commander might nominate a list of candidate positions for JOINT DEFENDER to evaluate based on his expert knowledge of the theater and the capabilities of the platforms under his command. 3.8. Scenario Variants We develop a sequence of scenario variants to illustrate how the defender or attacker can evaluate flexibility in their strategy. We depict postures for the defender in which: (D1) The defender does nothing. This establishes a worst-case baseline for any surprise attack. (D2) The defender hides nothing, including his commitments to intercept each potential attacking missile. This is the completely flexible and transparent case where attacker and defender each have complete knowledge of each other’s plans. (D3) The defender lets his platform locations be seen, keeps his ships out of range of shore-based anti-ship missiles, and hides his interceptor commitments. (D4) The defender lets his platform locations be seen, suppresses shore-based threats to his ships as necessary, positions his ships as close to shore as he pleases, and hides his interceptor commitments. (D5) The defender hides the positions of his ships, does not hide the positions of his ground-based interceptors, but does hide all interceptor commitments. (D6) The defender hides everything so that the defense is a complete surprise to the attacker. This establishes a best-case baseline for whatever the attacker decides to do. (This is the case assumed by current TBMD planning tools.) Posture D2 is our baseline. JD-MINMAX represents this “perfectly transparent” case by allowing the attacker to see both the defender’s pre-positioning decisions X, as well as his interceptor commitments R. This constitutes a restriction of the defender’s capabilities in a real engagement, in which (a) the attacker would observe only the defender’s pre-positioning; (b) he would plan and launch an attack given that information; (c) the defender would observe the attack; (d) and only then, after knowing the details of the attack, would the defender need to commit (allocate) interceptors from his pre-positioned platforms. The unrestricted model would be, however, significantly harder to solve than JD-MINMAX. In postures D3 through D5, we relax our baseline posture to model the case in which the attacker has information about the locations of some or all of the defensive platforms but does not know specific interceptor commitments. Exact solution of this model would also be difficult, so we approximate this situation by (a) solving JD-MINMAX as if the defender were using posture D2 (revealing both position and intercept commitments), (b) fixing the resulting attack plan, and then (c) letting the defender re-allocate his interceptors to engage the attacking missiles more effectively. In addition, we evaluate postures D7 through D10 under the same assumptions as D4, with one defending platform of each type omitted from the theater. (Specifically, D7 omits CG48, D8 omits DDG68, D9 omits Pbat1, and D10 omits Tbat1; see platform names in Table 6.) We depict postures for the attacker in which: (A1) The attacker must use a fixed launch site for each missile, or (A2) The attacker transports mobile missiles in secret to any launch site he chooses, while deceiving the defender into expecting fixed launch sites. 4. Results and Analysis We generate JD-MILP using the general algebraic modeling system (GAMS) (Brooke et al. 1998) and solve it with CPLEX 9.0 (ILOG 2003) on a 2 GHz laptop computer operating under Windows XP (Microsoft Corporation 2004). The largest models encountered in analyzing the North Korean cases have, after filtering and presolve reductions, about 120,000 binary variables, 250 continuous variables, and 6,000 constraints. In our experience, posture D6 (a surprise defense, the case assumed by current TBMD planning tools) can be solved optimally in a few seconds. A good solution to the more nuanced cases, such as posture D2 (where attacker and defender have complete knowledge of each other), is discovered within a minute or two, although proving near optimality with a 1% relative tolerance can take a half hour or more. 4.1. Multimissile Attack with No Defense Table 5 lists an optimal multimissile attack that launches a single missile from each fixed launch site at an undefended target. This produces a total expected damage of 103.0 (each attacking missile is assumed to hit its target). In this “posture,” D1-A1, the defender does nothing and the attacker uses fixed launch sites. Figure 6 illustrates the tracks that attacking missiles would follow in this scenario. 4.2. An Optimal Defense Plan Assuming that the attacker does not observe defensive preparations, the defender positions his assets to intercept an optimal, theaterwide attack (this is posture D6-A1, with positions shown in Table 6), and reduces expected damage from 103.0 to 1.0. The defender knows in advance about all optimal attack opportunities, so he positions his defensive platforms and engages the attacker’s missiles with interceptors having high probabilities of negation. The Brown et al.: A Two-Sided Optimization for Theater Ballistic Missile Defense 756 Operations Research 53(5), pp. 745–763, © 2005 INFORMS Table 5. An optimal, theaterwide undefended attack plan. There are no defensive interceptions at all. Each target on the defended asset list is attacked with a single missile producing a combined expected damage of 103.0. (See map in Figure 6.) Target Launch site Atsugi, JP Misawa, JP Okinawa, JP Sasebo, JP Tokyo, JP Yokosuka, JP Chinhae, ROK Inchon, ROK Kunsan, ROK Osan AB, ROK Pusan, ROK Seoul, ROK Figure 6. Kanggamchan Kanggamchan Chiha-ri Chiha-ri Kanggamchan Kanggamchan Chiha-ri Chiha-ri Chiha-ri Chiha-ri Chiha-ri Chiha-ri Expected damage Missile type No-Dong No-Dong No-Dong Scud-C No-Dong No-Dong Scud-C Scud-B Scud-B Scud-B Scud-C Scud-B 7.7 8.2 8.1 8.9 7.9 9.1 8.9 6.9 9.7 9.9 9.4 8.3 Map of an optimal, theaterwide undefended attack plan. Maximal attacks are shown with at most one missile aimed at each target and with no interceptions. Maximum expected damage is 103.0. (See data in Table 5.) Misawa, JP Inchon, ROK Seoul, ROK Osan AB, ROK Tokyo, JP Kunsan, ROK Pusan, ROK Atsugi, JP Yokosuka, JP Chinhae, ROK Sasebo, JP Okinawa, JP Brown et al.: A Two-Sided Optimization for Theater Ballistic Missile Defense 757 Operations Research 53(5), pp. 745–763, © 2005 INFORMS Table 6. Optimal defender positions maintaining defense secrecy against an optimal theaterwide attack. From these (hidden) positions, defending platforms intercept all attacking missiles, but do not necessarily destroy every missile intercepted. The maximum expected damage is reduced to 1.0, or about one-tenth of an attacking missile leaking through. Defender class Platform Latitude (N) AEGIS CG AEGIS CG AEGIS DDG PATRIOT THAAD CG47 CG48 DDG68 Pbat1 Tbat1 35 34 36 37 40 expected damage, evaluating to 1.0, derives from approximately one-tenth of an attacking missile leaking through over all engagements. Figure 7 illustrates the defender’s positions relative to an optimal attack plan, and the subsequent, optimal engagements of the plan’s attacking missiles. 4.3. Assume Transparency: A Two-Sided Optimization If each side can observe everything the other intends to do, the attacker knows that the defender may commit an interceptor salvo to each candidate missile attack, and shoot it if he launches that missile. The defender knows that the attacker may get some of his missiles through. The defender’s objective is to minimize maximum expected damage, given the attacker can see and take advantage of pre-positioned, defensive forces. This is posture D2-A1. The two-sided attack and defense produces an optimal set of interceptor commitments against threatened launches and shots at missiles launched, as well as, perhaps, some launches against which there is no available defense. Here, the expected damage of 22.7 represents loss of undefended targets Inchon and Chinhae, but still represents an overall reduction in expected damage from a surprise attack of 78 percent. Inchon and Chinhae are undefended because available interceptors cannot cover all possible attacking missiles. In posture D4-A1 we model the situation where the attacker observes platform positions for the defender, but the defender keeps his interception decisions secret. Tables 7 and 8, respectively, illustrate such a defense and the attacking missiles engaged, and Figure 8 depicts the missile attacks, defense, and engagements on a map of the theater. 4.4. A More Stressful Scenario Showing How to Evaluate Partial Transparency, Secrecy, Deception, and the Incremental Value of Each Defender Platform Now consider a more stressful case for the defender that is too cluttered to illustrate on a map of the theater. Suppose that the attacker can launch as many as 00′ 00′ 00′ 06′ 42′ Longitude (E) 130 129 126 127 141 00′ 00′ 00′ 02′ 25′ three missiles of any type from any launch site, and that each target can be attacked as many as 10 times. Posture D3-A1 exhibits 96 attacks, defended by 53 intercepting salvos using 90 interceptors. Total expected target damage is 394.4, with 43 attacking missiles expected to leak through defenses. Maintaining total defender secrecy, D6-A1, reduces total expected damage to 152.2. Suppose that the defender can keep naval defensive platforms hidden from the attacker, but the attacker can observe all land-based defenses; this is posture D5-A1. The resulting expected damage changes from the upper bound of total transparency toward the lower bound of total defender secrecy; see Figure 9. The difference between the expected damage in the transparent solution and the expected damage of this solution is the value of partial defender secrecy. In practical terms, this value quantifies how an increase in information hiding either through funding, strategy, or a combination of both, will reduce the attacker’s ability to inflict damage. The value of partial defender secrecy is bounded by the difference between the expected damage of the completely transparent solution and the expected damage given complete defender secrecy. In the latter case, the defender knows which individual attacks will occur and hides the existence of all interceptors from the attacker (with resulting value 394"4 − 152"2 = 242"2). The attacker may gain some advantage from secretly transporting missiles to alternative launch sites. Defending ships are most affected by this deception. In contrast, the PATRIOT battery provides a terminal defense that is relatively insensitive to an incoming missile’s track, which depends on its origin. We present defensive postures D7 through D10 to assess the value of each defending platform. Here, we assume that posture D4 applies (platforms are seen by the attacker, but engagements are concealed), as one platform of each type is successively removed. Table 9 shows the value of each platform, estimated by comparison with all platforms available. The Patriot battery is valuable in both fixed and mobile attack-missile postures (∼90 units of expected damage); the Aegis cruiser is more valuable in the mobile attack posture (∼110 units of expected damage), but has low value in Brown et al.: A Two-Sided Optimization for Theater Ballistic Missile Defense 758 Figure 7. Operations Research 53(5), pp. 745–763, © 2005 INFORMS Map of optimal defender positions and engagements when defense secrecy is maintained against an optimal, theaterwide attack. From these (hidden) positions, defending platforms intercept all attacking missiles, but do not necessarily destroy every missile intercepted. The maximum expected damage is reduced to 1.0, which corresponds to the expected damage from one tenth of an attacking missile leaking through the defense. Tbat1 Pbat-1 DDG68 C347 CG48 Table 7. Optimal defending platform positions given that these positions are observed by the attacker, but defending interceptor commitments are kept secret. Each defending platform is positioned to minimize the attacker’s worst possible attack. The defender, while determining his platform positions, commits interceptors to thwart potential missile attacks that may not actually be launched, but will be intercepted if they are. Once positioned, the defender can intercept any attacking missile he chooses. (See map in Figure 8.) Defender class Platform Latitude (N) AEGIS CG AEGIS CG AEGIS DDG PATRIOT THAAD CG47 CG48 DDG68 Pbat1 Tbat1 34 36 35 37 40 00′ 00′ 00′ 27′ 00′ Longitude (E) 130 126 130 126 140 00′ 00′ 00′ 57′ 00′ Brown et al.: A Two-Sided Optimization for Theater Ballistic Missile Defense 759 Operations Research 53(5), pp. 745–763, © 2005 INFORMS Table 8. An optimal attack plan given that defending platform positions are observed by the attacker, but defending interceptor commitments are kept secret. Each target is attacked with at most one missile. The defender, while determining his platform positions, commits interceptors to thwart potential attacks that may not actually be launched, but will be intercepted if they are. Once positioned, the defender can intercept any attacking missile he chooses. Total defended asset list target value at risk is 103.0 and expected target damage is 1.0. (See map in Figure 8.) Target Atsugi, JP Misawa, JP Okinawa, JP Sasebo, JP Tokyo, JP Yokosuka, JP Chinhae, ROK Inchon, ROK Kunsan, ROK Osan AB, ROK Pusan, ROK Seoul, ROK Figure 8. Launch site Kanggamchan Kanggamchan Chiha-ri Chiha-ri Kanggamchan Kanggamchan Chiha-ri Namgung-ri Chiha-ri Chiha-ri Chiha-ri Chiha-ri Missile type Salvo option No-Dong No-Dong No-Dong Scud-C No-Dong No-Dong Scud-C Scud-C Scud-B Scud-B Scud-C Scud-C 2 2 2 2 2 2 2 2 2 2 2 2 SM3 SM3 SM3 SM2-III SM3 SM3 SM2-III PAC3 SM2-III PAC3 SM2-III PAC3 Salvo Pn Defender Expected damage 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.99 CG48 CG48 CG47 CG47 CG47 CG47 DDG68 Pbat1 CG48 Pbat1 DDG68 Pbat1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 Map of an optimal theaterwide attack given that defending platform positions are observed by the attacker, but defending interceptor commitments are kept secret. Each target is attacked with at most one missile. The defender, while determining his platform positions, commits interceptors to thwart potential attacks that may not actually be launched, but will be intercepted if they are. Once positioned, the defender can intercept any attacking missile he chooses. Total defended asset list target value at risk is 103.0 and expected target damage is 1.0. (See data in Tables 7 and 8.) Tbat-1 Pbat-1 CG48 DDG68 CG47 Brown et al.: A Two-Sided Optimization for Theater Ballistic Missile Defense 760 Figure 9. Operations Research 53(5), pp. 745–763, © 2005 INFORMS Minimized maximum expected target damage for 20 scenarios mixing defense and attack postures. From D1 to D6, the defender works harder and harder to intercept attacks and conceals more and more information from the attacker, while the attacker either has known, observed launch sites (vertical scale A1) or mobile launch sites hidden from the defender (vertical scale A2). For example, with defending ships hidden from the attacker, posture D5-A1 has expected damage 355.5 with known, fixed attack launch sites, or 394.7 if the attacking missiles can be transported to surprise launch sites D5-A2. Moving defending ships out of range of shore-based anti-ship missiles, D2, does not degrade the defense. The value of secrecy is the positive difference between the expected damage under that level of secrecy and the expected damage in the fully transparent model (e.g., the value of complete defender secrecy is 456"1 − 152"2 = 303"9). Value of Various Levels of Secrecy D1: Surprise attack 1,000 1,000 829.9 829.9 D2: Fully transparent 456.1 394.4 355.5 458.1 394.7 394.4 D3: Seen defenders, ships standoff D4: Seen defenders D5: Ships hidden 152.2 152.2 D6: Surprise defense 0 0 A1 Table 9. A2 Defensive postures D7 through D10 show the effect of removing one of each type of defender platform from the scenario. Posture D4 provides a baseline for comparison. In each scenario, the remaining platforms reposition and resort to defensive salvos using fewer interceptors. For instance, when launch locations are fixed, removing one Aegis cruiser (CG48) results in a moderate increase in expected target damage 398"5 − 394"4 = 4"1. But, when launch sites are mobile, removing the same platform results in a dramatic increase in expected target damage 504"3 − 394"4 = 109"9. A1: Fixed launch sites Posture D4: All platforms seen (baseline) D7: Remove CG48 D8: Remove DDG68 D9: Remove Pbat1 D10: Remove Tbat1 A2: Mobile launch sites Expected damage Increase from D4 Expected damage Increase from D4 394.4 0"0 394.4 0"0 398.5 432.6 482.6 404.4 4"1 38"2 88"2 10"0 504.3 432.6 482.6 404.4 109"9 38"2 88"2 10"0 Brown et al.: A Two-Sided Optimization for Theater Ballistic Missile Defense 761 Operations Research 53(5), pp. 745–763, © 2005 INFORMS the fixed-launch-site attack posture (∼4 units of expected damage). 5. Conclusions We have introduced JOINT DEFENDER, a new optimization-based decision-support tool for pre-positioning theater ballistic missile defense (TBMD) assets, i.e., missile-interceptor platforms. JOINT DEFENDER can model a scenario in which both attacker and defender have knowledge of the other’s strategy, it can model no defense at all, and it can model an optimal defense against an attack assuming that the attacker expects no defense. Existing defensive planning tools can only evaluate the last type of scenario, and then, only approximately. JOINT DEFENDER solves such problems exactly on a laptop computer in just a few seconds. JOINT DEFENDER can also model a more complicated scenario in which a defender first pre-positions his TBMD platforms to protect a set of targets. An attacker observes those defensive positions and, given that information, launches his missiles so as to maximize the total expected value of target damage. The defender can optimize his pre-positioning (and commitments of interceptors to attacking missiles), because he knows the attacker will behave to optimize his own objective function. The attacker cannot increase expected damage by using any other strategy. We develop JOINT DEFENDER as a bilevel integer linear program, but convert it to a standard mixed-integer linear program for solution purposes. We have demonstrated its practicability by solving a number of realistic scenarios involving North Korea, using data gleaned from public sources. We have also explored the “value of secrecy” to both sides of the conflict. JOINT DEFENDER identifies an optimal plan for a typical transparent scenario in a minute or two on a laptop computer, although we find instances that require one half hour to prove optimality. Two-sided mathematical models of military conflict have been studied since Lanchester (1916). Danskin (1967, p. viii) recounts that, in 1951, the Rand Corporation studied two-sided situations where “one side allocates anti-missile defenses to various cities. The other side observes this allocation and then allocates missiles to those cities.” In discussing defense against nuclear strikes, and in addition to using a dual reformulation from max-min to max-max, Owen (1969, p. 491) states: “It is, of course, assumed that the defender must deploy his hardware first; the attacker, in full knowledge of this deployment, will act next.” In Appendix B, we establish the relationship between our twosided model (JD-MINMAX) in JOINT DEFENDER and a game invented by von Stackelberg (1952). These seminal contributions, achieved solely with classical mathematics (i.e., with no computers) but only by asserting many simplifying assumptions (such as continuous activities) still offer prescient insight. We have now discovered how to actually formulate and solve such problems with realistic fidelity. In contrast to our bilevel optimization, a standard gametheoretic model would assume that the attacker does not observe the positioning of defensive assets before launching his attack, and the defender is unaware of the allocation of attacking missiles to targets (e.g., Matheson 1975), or that either side is unaware of the total number of assets (offensive or defensive) possessed by the adversary. Eckler and Burr (1972) discuss solutions for many versions of such games. Bracken et al. (1987) discuss solutions that are robust with respect to uncertain numbers of attacking assets. Diehl (2004) provides the contemporary (unclassified) foundation for JOINT DEFENDER, discusses the philosophy of target damage functions, and suggests some alternative solution strategies that we have not pursued here. JOINT DEFENDER represents a substantial technological advance over existing TBMD planning tools that employ heuristics, or expect the planner to guess at good defense plans, or require supercomputers for implementation. None of these existing tools assumes that the attacker can detect defensive preparations and respond accordingly—this is a key weakness addressed by JOINT DEFENDER. Planners are comfortable with a decision-support tool they can control, so JOINT DEFENDER accepts advice such as “fix this platform in this position,” “try this position first,” or “we have no advice to offer.” Similarly, JOINT DEFENDER accepts, but does not require, other advice on the details of an interception plan, including “evaluate this exact plan.” The Joint Task Force’s area air defense commander and regional air defense commander can use JOINT DEFENDER for initial defense planning and assessment, and for assessing the value of hiding information from the attacker. JOINT DEFENDER can also provide insight to TBMD program officers in Washington, DC. For instance, it can evaluate trade-offs between investing in a few, highly effective but expensive interceptors or in larger numbers of relatively inexpensive, but less effective interceptors. JOINT DEFENDER has been presented to Naval Warfare Development Command (NWDC), to the United States Strategic Command program and requirements staff, and has undergone additional proof testing with a number of scenarios of interest to these organizations. The NWDC’s air defense department head, Captain Garry Holmstrom, USN (ret), has stated, “This project has produced a most promising solution to the Joint as well as the Navy’s problem of BMD asset allocation, at minimal development and fielding cost.” JOINT DEFENDER is now under further development for NWDC in preparation for further testing and future integration into the TBMD planning environment, and for potential use in the Global Command and Control System-Maritime and/or the Area Air Defense Command System-lite. 6. Epilogue On or about 1 October 2004, the USS Curtis Wilbur, a destroyer of the U.S. 7th Fleet, began patrolling the Sea of Brown et al.: A Two-Sided Optimization for Theater Ballistic Missile Defense 762 Operations Research 53(5), pp. 745–763, © 2005 INFORMS Japan as our first step in building a missile shield for the United States and its allies (Army Times 2004). Appendix A. Variations on JD-MINMAX’S Objective Function The objective function (A0) models area targets that can be damaged more than once, but not point targets that can be destroyed just once (e.g., Eckler and Burr 1972). If we partition the set of targets T into area and point targets, i.e., T = #Tarea  Tpoint $, the following objective function models the situation more accurately:  val t min max Y X R∈XR t∈Tarea     · Pna cp  g d Ra p g d Ya Pka 1 − a∈At + Appendix B. Stackelberg Games p g d   valt 1 − t∈Tpoint a∈At  1 − Pka · 1−  Pna cp  g d Ra p g d p g d Ya  " (A0′ ) But, our linear-programming subproblem for the attacker no longer suffices. On the other hand, one can argue that, even in the case of area targets, the damage resulting from multiple successful strikes is not additive. For instance, the economy and welfare of a city might suffer almost as much from a single successful missile strike as from two. Ignoring variations in weapon types for simplicity, a sensible modeling technique makes the expected value of a set of successful strikes on a target a concave function of the expected number of such strikes. (Indeed, the simple point-target model fits this description.) We can accommodate this by subtracting larger and larger fractions of expected target value as the number of attacking missiles increases:  val t min max Y X R∈XR t∈Tarea     Pna cp  g d Ra p g d Ya Pka 1 − · a∈At p g d missilest −  Pka ftk Ytk′  (A0′′ ) k=1 where (a) Ytk′ is 1 if the number of missiles targeted at t is greater than or equal to k and is 0 otherwise, and (b) ftk , 0
ftk < 1 for all t and k, is an increasing function of k; this takes into account some approximation of the probability that each attacking missile is thwarted. The inner subproblem remains totally unimodular because we simply replace constraints (A2) with missilest  a∈At Ya −  Ytk′
0 k=1 0
Ytk
1 ∀ t k" ∀ t Thus, the inner subproblem remains a linear program with a dual formulation, and a mixed-integer linear program formulation for the overall problem can be created. If it becomes imperative to model point targets with an objective akin to (A0′ ), the linear objective of (A0) can be maintained by letting Ya represent a multiweapon strike and adjusting ka accordingly. Unfortunately, the constraints necessary to enforce this would result in a subproblem that is not totally unimodular. Brown et al. (2004) (see also the discussion on trilevel defense models in Israeli and Wood 2002) deal successfully with this issue by solving their analog of JD-MINMAX with a specialized decomposition algorithm. That technology also applies to our problem, at least in theory. The models in JOINT DEFENDER’s basic model comprises an instance of a Stackelberg game (von Stackelberg 1952; see Luo et al. 1996, pp. 11–15 for an overview), which we represent as a bilevel integer linear program (e.g., Moore and Bard 1990). The bilevel program converts to a standard MILP for solution purposes. The key ingredients of any Stackelberg game are a leader (our defender) and a follower (our attacker). The basic (“one-play”) version of the game we use has two phases: (a) The leader carries out a set of actions to coerce behavior from the follower, and (b) the follower observes the leader’s actions and how they have affected his ability to respond and/or the value of responding, and reacts by optimizing his own objective. The leader has an objective, too, which is based on the costs of his own actions and his evaluation of the follower’s responses. Because the leader understands and models the follower’s optimizing behavior precisely, he can and does optimize his own objective by coercing the follower appropriately. If the follower suboptimizes for some reason, or if the leader has assumed that the follower has more flexibility or ability than he actually does, the leader can guarantee the actual outcome will be no worse than one predicted by the game. The leader’s and follower’s objectives in a Stackelberg game need not be diametrically opposed, but are in our TBMD problem. Our attacker (follower) wishes to maximize total expected damage while our defender (leader) wishes to minimize that maximum. Our defender’s actions are constrained by the physical limits on deploying TBMD platforms and on the physical limits of his interceptors. Our attacker’s actions of firing TBMs are not constrained by our defender’s actions, but the values associated with firing TBMs are. 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