Air defense has been an important area for nations andtheir armed forces. Substantial resources have been devotedto develop both defensive and offensive weapon systems. Theeffective use of and defense against these weapon systems isof utmost importance. The proliferation of anti-ship missiles(ASMs) and the increasing frequency of littoral operationshave increased the threat posed by the ASMs to the navies.The competing technologies of ASMs and air defense sys-tems force the navies to update the systems and to developnew tactics continuously. All modern navies devote consid-erable resources to ASM defense systems [9]. Planning foreffective use of those systems in operations has to be studiedcarefully in order to be able to get the highest level of ben-efit from technological developments. One particular aspectof planning is coordinated allocation of air defense systemswithin a group of ships to attacking missiles. In naval ter-minology, a collection of naval combatants and auxiliariesthat are grouped together for the accomplishment of one ormore missions is called a task group (TG). Modern area airdefense missile systems can provide support to the collocatedships in a TG, and new technologies such as improved tacti-cal data links and cooperative engagement capability enable a
fully coordinated air defense within a TG. For many navies,equipping all the platforms with adequate air defense sys-tems is clearly not the best and most cost-effective solution.A number of navies acquire area air defense ships that canprovide air defense support to the other ships that have lim-ited or no effective air defense capability. Allocation of thearea defense ship capabilities to other units in the TG is animportant problem to be solved for effective use of theseplatforms.
For air defense engagements, there are several prescriptiveassignment models such as sector assignment, closest pointof approach assignment, maximum number of shots assign-ment, least engaged assignment, defense in depth assignment,and deliberate assignment of multiple overlapping systems(see [28]). However, they do not make use of the addi-tional capability provided by the coordination of defensiveresources and cohesion.
The problem we consider in this study is a specific typeof weapon-target allocation (WTA). A most generic formof the WTA problem is the following [30]. Given an exist-ing weapon force and a set of targets, what is the optimalallocation of weapons to targets? The WTA problem can beviewed both from an attacker’s and a defender’s perspec-tive. We restrict ourselves to the defense of the friendly navalforces with surface-to-air missiles (SAMs) and focus on theallocation of SAMs to incoming ASMs.
Matlin [29] and Eckler and Burr [11] review the literatureon the missile allocation problem. A number of articles havebeen published in open literature after those reviews. Burr etal. [8] develop the optimal integer solutions for Prim-Readdefense that uses the allowable damage per attacker insteadof the number of attacking missiles and builds the modelwithout explicitly knowing the attack size. Soland [36] con-siders the defense of a single target against a simultaneousattack. He assumes that the defense has interceptor missileswith a fixed number of engagements and uses shoot-look-shoot engagement policy. Soland’s [36] solution procedureis intractable as the size of the problem gets larger. Bert-sekas et al. [6] propose a solution method for large problemswith known attack size using neuro-dynamic programming.Krokhmal et al. [23, 24] consider uncertainty in WTA prob-lem using conditional value-at-risk (CVaR) risk measure. In[23], the authors develop models for one-stage and two-stagestochastic WTA problems with CVaR constraints. In [24],they assume known number of targets in the first stage andunknown number of targets with known probability distribu-tion of the number of second stage targets within a two stagerecourse problem setting. They take into account the proba-bility of running out of munitions on the second stage of theengagement using the CVaR risk measure. Al-Mutairi andSoland [3] analyze the effectiveness of partially coordinatedarea defense systems against a simultaneous attack. Due tothe computational complexity of the missile allocation prob-lem, a number of heuristic approaches have been suggested.Wacholder [37] proposes a solution for a one sided many-on-many missile allocation problem using artificial neuralnetworks combined with the Lagrange differential multipli-ers method. Jaiswal [17] investigates a similar problem usingsimulated annealing, genetic algorithms and artificial neuralnetworks in a layered defense context.
Nguyen et al. [33] introduce the idea of using generatingfunctions as a simple, consistent, and easily applied tool forevaluating the effectiveness of an air defense system. Thisapproach does not provide any interceptor allocation plan.The scenario considered is similar to that of Soland [36].The model described in [33] is based on four parameters:the total number of available interceptors, the total numberof attackers, the maximum number of engagement oppor-tunities against each threat, and a constant probability ofsuccessful interception. Al-Mutairi et al. [2] propose for-mulations for analyzing a layered defense. Simulation isone of the tools frequently used to evaluate the effective-ness of the air defense systems. Hoyt [16] reports a simpleMonte-Carlo simulation model of ballistic missile defensesystem. SEAROAD [7, 28], JASMINE [35] and SADM [10]are examples of naval air defense simulations. Beare [5] andGriffiths et al. [15] describe the use of analytical models toreduce the number of scenarios that would be investigated indetail by simulation models.
Karasakal [19] and Nguyen [32] investigate the cohesion ofair defense capability within a TG. In his work, Nguyen [32]quantifies the benefit from resource allocation for a navalTG having perfect coordination among its assets. The inter-ceptors are assumed to cover all the other ships of the TGand are capable of defending the ships within range. Othergeometric and defense system limitations are not considered.Karasakal [19] uses the geometric information such as rela-tive bearings and distances between ships and attacking mis-siles, and capabilities of different types of SAM systems andASMs such as effective ranges and speeds. Proposed modelsenforce an approximate shoot-look-shoot engagement policy.The author claims that the models can be used to investi-gate the composition of a naval TG and the effectiveness ofdifferent SAM systems to protect the TG.
Almeida et al. [1] present the impact of information onthe effectiveness of air defense in a time-constrained context.Sherali et al. [34] develop algorithms to schedule a set ofillumination radars to engage incoming targets using surface-to-air missiles in a naval TG. Armstrong [4] analyze the effectof lethality (i.e., the relative balance between the offensiveand defensive power of rival naval forces) to the outcome ofthe naval combat represented by an aggregate level missilesalvo model. Lucas and McGunnigle [26] investigate the util-ity of simple models to provide insights for military decisionproblems by comparing a simple salvo model with a complexsimulation of naval combat.
In this study, we address the issue of allocating air defensemissiles to incoming air targets in a coordinated way withina naval TG such that the available defense capability is usedin the most effective manner. We call our version of the mis-sile allocation problem MAP. Our aim is to develop a MAPmodel for TG air defense that captures the reality of ASMdefense, generates an efficient allocation plan for SAM sys-tems, and measures the effectiveness of air defense under agiven scenario. A scenario is composed of the information onthe attacking ASMs and the defensive SAM systems as wellas the relative positions of the ships in TG, which is calledthe formation of the TG.
MAP is a new treatment of an emerging problem fosteredby the rapid increase in the capabilities of ASMs, and thedifferent levels of air defense capabilities of warships againstthe ASM threat. In addition to allocating SAMs to ASMs,MAP also schedules launching of SAM rounds according toshoot-look-shoot engagement policy or its variations, con-sidering multiple SAM and ASM types. Our objective is tomaximize the probability that none of the ASMs can reachtheir target. MAP can produce the best course of action fordefending the TG against an immediate and simultaneousASM threat. It can also be used for analysis of the air defenseeffectiveness of warships under different positioning at sea.For a given attack scenario and a positioning at sea (i.e., anescort ship and a defended ship positioned at a specified range
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Karasakal, Özdemirel, and Kandiller: Anti-Ship Missile Defense 307
and bearing from the escort ship), the air defense coverageprovided by the escort ship can be calculated using MAP.The aggregated coverage values for a set of representativeattack scenarios and positions can be used to decide on thedisposition of the TG. Interested readers are referred to [20]and [21] for a detailed explanation of the integrated solu-tion approach to form a screen disposition of a TG usingMAP.
Although we provide a mathematical programming modelfor MAP, we do not explicitly use the model to solve MAP.Instead, we propose efficient and fast heuristic algorithms tosolve the problem. As for many other combinatorial optimiza-tion problems, first we try to construct a good initial feasiblesolution and then improve this solution further. We make useof some conventional ideas such as 2-opt exchange, which isused for typical ordering-based problems, e.g., the travelingsalesman and scheduling problems.
After introducing our original MAP formulation and solu-tion procedure, we discuss extensions for alternative objec-tives and for relaxing some of the assumptions including theshoot-look-shoot engagement policy.
The literature review on the MAP, according to our knowl-edge, shows that there is no model that can be used to solve theproblem defined in this article. The existing analysis methodsmainly consist of computer models that simulate the defenseagainst ASM attack. The Naval Studies Board’s [30] panelon Modeling and Simulation also identified the requirementfor analytical models in anti-air warfare area.
The rest of the article is organized as follows. In the nextsection, we describe MAP and present a non-linear 0–1 inte-ger programming model for the problem. In Section 3, wepropose efficient heuristic solution algorithms for MAP. InSection 4, we present and discuss the results of the compu-tational study undertaken to evaluate the performance of theproposed heuristics. We discuss some extensions of MAP inSection 5, followed by concluding remarks.
2. PROBLEM DESCRIPTION AND MODEL
Consider a naval TG, composed of several ships with vari-able air defense capabilities, defending itself against an airattack. These ships may either be equipped with one or moresurface-to-air missile (SAM) systems or none at all. Theirair defense capability may be limited to self-defense or mayextend to area defense, i.e., a ship may defend the other shipswithin its effective weapon range. Ships in a TG are typicallyarrayed into a formation, called a screen, in which the mostvaluable and important units (termed high value unit or HVU)are surrounded and protected by the escorting vessels. Withinthe screen, the escort ships are stationed in sectors away fromthe HVU. Figure 1 depicts a generic naval TG compositionand an air attack scenario. In this scenario, a TG composed
of four ships in formation, one HVU and three escort ships, isattacked by four ASMs. Ship 1 (HVU with no SAM systemonboard) is targeted by ASM2 and ASM3, Ship 2 is targetedby ASM1, and Ship 4 is targeted by ASM4. There is no ASMthreat to Ship 3. Ships 2, 3, and 4 have short-range self defenseSAM systems (such as the NATO Sea Sparrow SAM), andthe effective ranges are depicted by the circular areas aroundeach ship. These self defense SAM systems can only engagethe ASMs targeting their own ships. Ship 2 also has a long-range area defense system SAM2 (such as the SM-2 SAM),and part of its effective range is depicted by the dotted areaand the arc drawn in a dashed-line. Because SAM2 is an areadefense system, it can engage ASMs targeting the other shipsas well as Ship 2. ASM1 can be engaged by both SAM1 andSAM2. ASM2 and ASM3 can be engaged by only SAM2.Note that SAM4 cannot engage ASM3, even if some part ofthe ASM3’s flight path falls into the effective range of SAM4,since SAM4 is a self-defense system and can only engage theASMs that are a direct threat to it. ASM4 can be engaged byboth SAM2 and SAM4.
The TG air defense commander will maintain the air pic-ture and coordinate the response until the time when the shipsare forced to defend themselves using the weapons for lastline of defense, such as close-in-weapon systems and softkill systems. The air defense command and control ship will,in most cases, have to coordinate the TG response to an airthreat to ensure maximum efficiency and probability of suc-cess. In this role, a set of command decision tools is requiredto plan the air defense of the TG, and to schedule the forcedefense as an attack develops.
Maximizing the probability of shooting down all theincoming ASMs is an important objective for a TG air defensecommander at sea. (We consider some alternative objectivesin Section 5.) However, saving the maximum number ofSAMs (for possible future attacks) from the limited num-ber available onboard the ships and the high price tag of eachmissile have to be considered as well. The objective mightbe to maximize the probability of neutralizing the incom-ing ASMs with minimal SAM expenditure. Several missileengagement tactics have been developed to achieve a bal-ance between these conflicting objectives. Although we donot explicitly minimize SAM expenditures, we enforce theshoot-look-shoot (SLS) tactic to take this secondary objectiveinto account. The SLS tactic requires shooting at the targetfirst, then looking to see if it is killed, and shooting again onlyif it is necessary to achieve the kill. In this research, we ini-tially consider the case when the TG employs the SLS tactic.Variations of SLS are also discussed in Section 5.
The engagement process of a SAM system to an ASMcan be divided into four phases. These are the tracking ofthe target illumination radar, the solution of the fire controlproblem, the launch delay, and the flight time to the engage-ment. Each engagement of SAM systems takes a constant
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Figure 1. Composition of a naval TG and an air attack scenario. [Color figure can be viewed in the online issue, which is available atwileyonlinelibrary.com.]
setup time for the first three phases and a variable time forthe last phase, which is the flight time to the engagement. Weassume that the time required for checking whether or not theASM is hit at the end of one engagement overlaps with thesetup time of the next engagement, hence there is no need toconsider this time separately. Alternatively, the “look” timecan be added to the setup time. Each engagement takes lesstime compared with the one before as the attacking ASMapproaches its target.
The maximum distance at which an ASM intercept can takeplace is determined by the smallest of the following three: themaximum effective range of the SAM system, the horizon ofthe fire control radar against the incoming ASM, and the firstdetection range of the ASM.
When the SLS firing policy is used, there are few engage-ment opportunities (mostly less than 10) against each ASM.In reality, each engagement does not take the same setup time,since the target illumination radar may already be on track,or the fire control problem may have already been solved.However, we use a conservative approach and consider thateach engagement takes a constant setup time.
In summary, MAP is concerned with the allocation ofdifferent types of SAMs available for attacking ASMs and
scheduling the SAM launches under SLS firing policy, so as tomaximize the TG’s air defense capability, which is measuredby the probability of no leaker.
The basic assumptions that are needed to develop themissile allocation model are stated below.
1. The TG sees all of the air threats to intercept simul-taneously. Thus, we investigate the case where theattack size is known.
2. The ships in the TG are capable of coordinating theallocation of the air defense, i.e., C2 (command andcontrol) capability is assumed.
3. The TG has both point (or self) and area air defensemissile systems.
4. Both attacking ASMs and SAM systems onboardships may be of different types.
5. Different SAM systems may have different effectiveranges, i.e., layered defense is assumed. Layers mayoverlap.
6. Defense systems can predict the eventual target of theattacking ASMs, i.e., impact point prediction capa-bility is assumed. The track of each ASM is a straightline.
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Karasakal, Özdemirel, and Kandiller: Anti-Ship Missile Defense 309
7. Defense systems can distinguish the ASMs fromeach other, i.e., a SAM intended for a particular ASMactually locks on that ASM.
8. Missile engagement policy is SLS.9. The incoming ASMs are assumed to be classified in
terms of their speed (e.g., supersonic or subsonic) andattack profile (e.g., sea-skimmer, high diver). Thus,the single shot kill probability of each SAM againsteach ASM is known.
10. The relative positions of the ships within the TG donot change as the air raid continues. The ships arethought to be stationary. This is a reasonable assump-tion since the speed of the ships is very low comparedwith the speed of the ASMs.
11. There are no limitations on the number of SAMs inflight that are launched from the same SAM systemto engage different ASMs.
We discuss how to relax assumptions 6, 8, and 11 in Section 5.Suppose that there aren incoming ASMs, indexed i ∈ N =
{1, . . . , n} and there are m SAM systems on board of warshipscomposing the naval TG, indexed j ∈ M = {1, . . . , m}. Letti be the time taken by ASM i to reach its known target.Then H = maxi∈N {�ti�} is the problem horizon given by thehighest time-on-target. The interval [0, H ] may be dividedinto t non-overlapping slots, each of unit duration δ, indexedk ∈ K = {1, . . . , t}, and τk denotes the beginning time ofslot k, k ∈ K . Let V denote the valid combinations of ASMand SAM systems, i.e., (i, j) ∈ V if SAM system j canengage ASM i. Each ASM i has a specified engageabilityinterval [qij , rij ], which depends on the location and capabil-ity of the SAM system j , and a successful engagement canbe achieved only during this interval. Here, qij refers to theearliest beginning time of the first engagement, and rij refersto the latest ending time of the last engagement, i.e., the lat-est time the last round of SAM j should intercept ASM i.We assume that the problem data related with time have beenperturbed in such a way that each value is an integer multipleof the unit time δ. Time taken by each feasible engagement isdetermined as the sum of a constant setup time and a variableflight time to the engagement. Thus, each engagement takesa specified time according to the ASM and SAM combina-tion (i, j) ∈ V and the starting time of the engagement. Thisengagement duration is denoted by �ijk .
The above parameters are calculated based on the scenariospecifications. ti is found by dividing the distance betweenASM i and the ship it targets by the velocity of the ASM. qij
and rij are calculated similarly using the minimum and max-imum engagement ranges of SAMs and the initial detectiondistances of ASMs. The constant setup time included in �ijk
is assumed to be given. The variable flight time to engage-ment is found based on the distance between ASM i and thetarget ship that has SAM j at the beginning of time slot k,
as well as the velocities of the ASM and SAM pair and theSAM fly out trajectory.
To formulate the problem, recalling that τk denotes thebeginning time of slot k ∈ K , let us define for each validcombination of ASM i ∈ N and SAM j ∈ M , a set
Sij = {k ∈ K : (i, j) ∈ V and [τk , τk + �ijk] ⊆ [qij , rij ]}.
Note that Sij denotes the slots for which SAM j can bescheduled to engage ASM i. For example, suppose �ijk hasthe respective values 5,4,3 for k = 1, 2, 3. (The engagementduration decreases as the ASM approaches its target.) ThenSij = {1, 2, 3} means that [1,5], [2,5], and [3,5] are possibleengagement intervals.
Accordingly we define the binary decision variable xijk =1 if SAM j is scheduled to start the engagement processagainst ASM i at the beginning of Slot k, and xijk = 0 oth-erwise. Furthermore, to ensure that schedules of the SAMsagainst an ASM do not overlap in accordance with the SLStactic, let us define for each slot k ∈ K and for each ASMi ∈ N , the set
Note that for each i ∈ N and k ∈ K , Jik is the set of combi-nations (j , ρ) such that slot k for ASM i will be blocked (todisallow other SAMs being fired during slot k) if xijρ = 1.For example, Ji3 = {(j , 1), (j , 2), (j , 3)} indicates that Slot3 will be blocked if SAM j starts engaging ASM i in timeSlots 1,2,3.
We need the following additional notation and variables toformulate the TG air defense problem.
pijk: the single shot kill probability of SAM j against ASMi when the engagement begins at the beginning of slotk, (i, j) ∈ V and k ∈ Sij .
dj : the number of available rounds on SAM system j .uij : the upper bound on the number of engagements of
SAM system j against ASM i, (i, j) ∈ V within theengageability interval. uij is calculated by dividingthe engageability interval by the minimum durationof a single engagement according to the SLS tactic.
Then, the TG air defense problem MAP can be formulatedas the following nonlinear integer programming model.
Max∏i∈N
1 −
∏k∈K
j∈M|(i,j)∈V
(1 − pijk)xijk
(1)
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subject to
∑k∈K
i∈N |(i,j)∈V
xijk ≤ dj for all j ∈ M (2)
∑(j ,ρ)∈Jik
xijρ ≤ 1 for all i ∈ N and k ∈ K (3)
∑k∈Sij
xijk ≤ uij for all (i, j) ∈ V (4)
xijk ∈ {0, 1} for all (i, j) ∈ V and k ∈ Sij (5)
The objective function (1) maximizes the probability ofno-leaker for the whole TG. Constraint set (2) reflects therestriction on the number of rounds available for each SAMsystem. Constraints of type (3) ensure that there is no over-lap of the engagements against each ASM. Constraints oftype (4) limit the total number of rounds that can be fired foreach valid ASM and SAM combination. This constraint settightens the feasible space of the problem. Constraint set (5)imposes binary restriction on the decision variables.
3. SOLUTION PROCEDURE
The non-linearity in the model can be removed by usinglogarithms and exploiting the relation between the objectivefunction and the right-hand side of a newly added constraintset coming from the objective function [18]. However, evena small problem can become quite large in terms of the num-ber of binary decision variables as the unit duration, δ, getssmaller.
The WTA problem is hard to solve in terms of the com-putational complexity. Indeed, Lloyd and Witsenhausen [25]prove that the weapon allocation problem is NP-completeeven in its simplest form. WTA models in literature usesimplifying assumptions to reduce the problem to a levelof suitable mathematical tractability. Our MAP formulationuses parameters and assumptions required by the specificair defense scenario we consider. Thus, further simplifyingassumptions cannot be used in our model without sacrificingthe representation of the real situation.
MAP can be used to analyze TG air defense effectivenessin different scenarios and settings. Solving MAP for a largenumber of representative cases is a prerequisite for a thor-ough scenario analysis. Since this process requires runningMAP many times, we need fast solution procedures that yieldhigh-quality solutions.
The mathematical programming model presented in thepreceding section does not meet the solution time require-ment due to its computational complexity. Thus, we focuson heuristic solution procedures for MAP. We present heretwo greedy construction and two improvement algorithms forMAP.
The first of those construction algorithms, the best engage-ment construction heuristic, allocates SAM systems toincoming ASMs according to a measure called the engage-ment potential. Its aim is to find the best engagement pairs interms of the objective. In the quasi-uniform construction algo-rithm, we try to engage each ASM at least once to reduce therisk of a zero objective function value. Thus, we give prece-dence to the ASM that has the lowest number of SAM systemsthat can engage it. After finding a starting solution by usingthese construction algorithms, we try to improve the solutionby using improvement algorithms. One of the improvementalgorithms, opt-change (OC) algorithm, improves the initialfeasible engagement schedule by changing the target ASMor SAM system of an engagement in the engagement list. Inthe 2-opt exchange (2OX) algorithm, we aim to exchange thetarget ASMs of two engagements to improve the solution.We choose the best move in each iteration of improvementalgorithms.
By dividing the planning horizon into small time slotsand allowing engagements to start at the beginning of thesetime slots, our formulation approximates the continuous timeproblem. Naturally, solution of this discrete time formula-tion may not be optimal for the continuous time problem.However, we do not solve the formulation explicitly but findthe optimum with enumeration. Both the enumeration proce-dure and the heuristic algorithms come up with a sequence ofSAM engagements against incoming ASMs observing theSLS tactic. In both methods, an engagement can start assoon as the previous one ends according to the generatedengagement sequence. Therefore, they are not affected bythe discretization, and their results are directly comparable.
3.1. Best Engagement Construction (BEC) Algorithm
First, we present the additional notation and the variablesfor the construction algorithm below.
vai : speed of ASM i
vsj : speed of SAM j
rj : maximum effective range of SAM j
rj : minimum effective range of SAM j
�c : constant setup time for an engagementfi : initial detection distance of ASM i from its
known targetpfi : present distance of ASM i from its known
targetpij : single shot kill probability of SAM j against
ASM i, (i, j) ∈ V
epij : engagement potential of SAM j againstASM i
w1, w2, w3, w4 : weights of the components of the engage-ment potential
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Karasakal, Özdemirel, and Kandiller: Anti-Ship Missile Defense 311
Gi : set of engagement potentials of the SAMsthat can be used against ASM i
ti : time for ASM i to reach its target, i.e.,time-on-target (TOT), ti = fi/vai
T : set of TOTs of ASMs
In this algorithm, we allocate SAM rounds to ASMsaccording to the engagement potential, which is a measureof the defensive capability of a SAM system against a givenASM. We compare each SAM system with a hypotheticalSAM, which has the best features such as the largest singleshot kill probability, highest speed, longest effective range,and shortest effective range against a given ASM. We assignthe SAM with the highest engagement potential to the closestASM in terms of TOT at each step of the algorithm.
STEP 0: Determine the ideal SAM for each ASM.
Find the best features, vs∗i , r∗
i , r∗i , p∗
i for ASM i using allSAM systems that can be used against that ASM. Define anew SAM called ideal SAM with these best features. (Maxi-mum range may be limited to the initial detection distance ofASM if the detection distance is smaller than the maximumeffective range of SAM.)
vs∗i = max
j{vsj : (i, j) ∈ V },
r∗i = min
{fi , max
j{rj : (i, j) ∈ V }
},
r∗i = min
j{rj : (i, j) ∈ V },
p∗i = max
j{pij : (i, j) ∈ V }.
Initialize present ASM distances to initial detection dis-tances, pfi = fi ∀i ∈ N .
STEP 1: Determine the engagement potential of eachSAM system against each ASM if the engage-ment is feasible. Find the set of engagementpotentials of the SAMs that can be used againstASM i.
epij = w1vsj
vs∗i
+ w2 min
{rj
r∗i
, 1
}
+ w3r∗
i
rj
+ w4pij
p∗i
for (i, j) ∈ V ,
Gi = {epij : (i, j) ∈ V }.STEP 2: Determine the TOT for each ASM, and find
the set of TOTs of ASMs, T .STEP 3: If all ASMs have been engaged, then start
a new engagement wave. If T = { }, thenre-populate T = {ti : i ∈ N}.
This step ensures that the final engagementschedule is as uniform as possible and that theASMs are engaged with a more or less equalnumber of SAMs.
STEP 4: Find the ASM with minimum TOT, andremove its TOT from the engagement list, T .
a = arg mini
T , T = T \{ta}.
STEP 5: If there are no SAM missiles left that can beused against any of the ASMs, stop. If Gi = { }∀i ∈ N , then STOP.
STEP 6: If there is no SAM system that can be usedagainst ASM a, then return to Step 3; other-wise, find the SAM system with the maximumengagement potential against the ASM in theengagement order.
If Ga = { } then go to Step 3; otherwise,b = arg maxj Ga .
STEP 7: If there is at least one SAM round of type b andthe intercept distance is larger than the min-imum engagement range of SAM b, allocateSAM b to ASM a. Reduce the number of avail-able rounds of SAM b by one and go to Step3. Otherwise, update the set of engagementpotentials and go to Step 5.
If db ≥ 1 and
(pfa − vaa�c
−[pfa − vaa�c
vaa + vsb
]vaa
)≥ rb then,
If the intercept distance is larger thanthe maximum engagement range ofSAM b, then reduce the intercept dis-tance to the maximum engagementrange of SAM b, i.e.,
If
(pfa − vaa�c
−[pfa − vaa�c
vaa + vsb
]vaa
)
≥ rb, then pfa = rb,
else pfa = pfa − vaa�c
−[pfa − vaa�c
vaa + vsb
]vaa .
Allocate SAM b to ASM a and setdb = db − 1. Go to Step 3.Otherwise, Ga = Ga\{epab} and go toStep 5.
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In Step 7 of the algorithm, the net closing velocity of aSAM and ASM pair is calculated as if they have a 180 degreeangle of deflection. These computations are in fact much morecomplicated, involving not only the deflection angles but alsovarious sources of error due to external factors such as windand temperature, and internal factors such as device and mea-surement errors. These computations are beyond the scope ofour study, and we assume head-on intercepts for all engage-ments for simplicity. However, longer elapsed time is theonly implication when the deflection angle is ignored, andthis does not affect our methodology and workings of thealgorithms.
l[(n − 1)m + (m − 1)] different cases are consideredfor change and enhancement in each iteration of the algo-rithm, where l is the total number of engagements. Thecomputational complexity for OC algorithm is O(lmn) periteration.
3.2. Quasi-Uniform Construction (QUC) Algorithm
The BEC algorithm assigns the SAM with the highestengagement potential to the closest ASM in terms of TOT.However, if the number of missiles in magazine or launcheris limited, the assignment rule may produce unsatisfactoryresults. Note that the probability of no-leaker will be zeroby allocating anything less then one shot per ASM. This dis-continuity, the jump from zero to a positive probability ofno-leaker value as the last ASM in the first engagement waveis shot at, causes difficulties for our construction algorithm.If there is an engagement schedule that has at least one shotper ASM, then it is desirable to find that one. This variationmakes sure that we find the desirable engagement scheduleif there is one. To achieve this goal, we change Step 3 of theprevious algorithm as follows.
STEP 3: If T = { } and there exists at least one ASMwith no interceptor assigned, then disregard allassignments made so far and let TOTs be thecardinality of the corresponding set of engage-ment potentials, i.e., T = {ti = |Gi | : i ∈ N}.Else if T
= { }, then re-populate T = {ti : i ∈ N}.Here, ti = |Gi | gives the number of SAM systems that
can engage ASM i. In Step 4, by choosing the ASM havingminimum ti value, the algorithm tries to allocate at least oneSAM shot to each ASM.
3.3. Opt-Change (OC) Algorithm
Our purpose in this algorithm is to find the engagementsthat would increase the objective function value by (1) chang-ing the target ASM of an engagement under consideration
and (2) simultaneously considering the enhancement of thedefense effectiveness by increasing the total number of SAMmissiles launched against target ASMs. Changing the tar-get ASM means that while one ASM will get one less shot,another ASM will get one more shot. The ASM that gets oneless shot after the change is considered for an additional shotobserving the SLS tactic. We summarize the OC algorithmbelow and give its full description in Appendix A.
STEP 0: Select an initial feasible engagement list.STEP 1: For each engagement in the list, check the pos-
sibility of the change of target ASM. A changeof target ASM will degrade defense againstthe target ASM before the change, and willenhance the defense against the new targetASM. To improve the overall defense capa-bility, we simultaneously consider enhancingthe defense against the former ASM by usingthe remaining SAM rounds, if any.
STEP 2: Consider changing the defending SAM foreach engagement in the list.
STEP 3: Find the best change in Steps 1 and 2. If thereis an improvement, update the engagement listand go back to Step 1. Otherwise, stop.
l[(n − 1)m + (m − 1)] different cases are consideredfor change and enhancement in each iteration of the algo-rithm, where l is the total number of engagements. Thecomputational complexity for OC algorithm is O(lmn) periteration.
3.4. 2-Opt-Exchange (2OX) Algorithm
Our purpose in this algorithm is to find the engagementpairs that would increase the objective function value byexchanging the target ASMs in the pair. We also try toincrease simultaneously the number of engagements againstthe ASMs under consideration with each exchange. We sum-marize the 2OX algorithm below. The full description is givenin Appendix B.
STEP 0: Select an initial feasible engagement list.STEP 1: For each pair of engagements in the list,
check the possibility of the exchange of tar-get ASM. Simultaneously consider enhancingthe defense against both target ASMs using theremaining SAM rounds.
STEP 2: Consider exchanging all the scheduledengagements of the two ASMs.
STEP 3: Find the best exchange in Steps 1 and 2.Update the engagement list, if necessary. Ifthere is an improvement, go back to Step 1.Otherwise, stop.
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l(l−1)
2 (2m + 1) different neighboring engagement lists arechecked for exchange for each iteration of the algorithm. If anexchange is made, then the algorithm starts over again. Thealgorithm stops when there is no exchange possible. Note thatan undesirable exchange may become desirable after a changein the engagement list. Thus, we continue until no desirableexchange is left for the engagement list. The computationalcomplexity for 2OX algorithm is O(l2m) per iteration.
4. COMPUTATIONAL RESULTS
In the preceding section, we developed heuristic algo-rithms for MAP. Since heuristic algorithms do not guaranteean optimal solution, it is of interest to study the worst-caseperformance of the algorithms. Submodularity concept isused to prove the worst-case performance of greedy type algo-rithms (see [22]). Submodularity states that adding an ele-ment to a smaller set helps more than adding it to a larger set.Formally, F(A∪{j})−F(A) ≥ F(A∪{j}∪{k})−F(A∪{j})must hold for all A ⊆ I and j ∈ I\(A ∪ {k}) where F is aset function defined over a finite set I [31]. We can show thatour objective is not submodular using a simple counter exam-ple. Let us assume a scenario with three attacking ASMs,i ∈ N = {1, 2, 3}, one SAM system, j ∈ M = {1} havingone available missile, d1 = 1, and pi1 ≥ 0 and ui1 ≥ 1 for alli. In this case, our objective function value is 0. Adding onemore available missile does not change the objective functionvalue. However, adding the third missile suddenly producesa positive objective function value, which contradicts withsubmodularity.
Below, we empirically show the performance of our algo-rithms by comparing their results with the optimal solutionscalculated using implicit enumeration and with the upperbounds calculated when the optimal solution is not tractableusing implicit enumeration.
We randomly generated test problems using the informa-tion on real weapon systems in open literature. Interestedreaders are referred to [12, 13, 27] for more informationon ASM and SAM systems. We defined seven differentSAM systems, including four self-defense and three area airdefense SAM systems. Self defense SAM systems are SeaSparrow, Evolved Sea Sparrow (ESSM), Aster-15 and Barak.Area defense SAM systems are SM-1, SM-2, and Aster-30.We also assumed seven ASM systems including Harpoon,Exocet, Polyphem, Gabriel, Penguin, SS-N-26, and Maver-ick. We created a sample single shot kill probability matrix forSAM and ASM systems. We choose the single shot kill prob-abilities by considering the speed of both SAM and ASM,ASM’s flight profile (i.e., sea-skimming or high diving) andthe technological age of the weapons. We assume single shotkill probabilities between 0.15 and 0.9. For a given numberof ASM and SAM systems, we randomly generate the type
Table 1. Minimum, average and maximum elapsed time forenumeration.
aMinimum.bAverage.cMaximum.dCPU time (sec) on a personal computer with AMD Athlon 2000+CPU and 256 MB of RAM.
of ASMs, initial detection range of ASMs (between 5000 mand 40,000 m or maximum range of the specific ASM if itsrange is less than 40,000 m), type of ships (a ship has onlySD capability or both SD and AAD capabilities with equalprobability), and target ships of ASMs. We decide on thenumber of missiles on each SAM system randomly between1 and 10. We set the weights used for the components ofthe engagement potential to one in our experiments, as rel-ative values of all four components are between zero andone.
We developed an enumeration algorithm to find the opti-mal solution for MAP. Since complete enumeration is stillvery time consuming, we restrict the problem size to a maxi-mum of five SAM systems with a total of nine missiles in thelaunchers and five ASMs. There are 25 combinations for thenumber of SAM systems and ASMs. For each combinationwe generated five problems. Problem parameters includingthe types of SAM systems and ASMs, the initial number ofmissile of each SAM system, the target of the ASMs and theinitial detection ranges of ASMs were generated using dif-ferent random number streams. Table 1 shows the summaryof the computational time for the enumeration algorithm forall 125 sample MAPs.
Table 2 presents the summary of sample MAPs in terms ofthe minimum, average, and maximum percent gaps betweenthe optimal solution and the best of the two construc-tion heuristic solutions. We calculate the gap as 100 ×(optimal-heuristic)/optimal objective function value. The
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Table 2. Minimum, average and maximum % gap between thebest construction heuristic and optimal solutions.
QUC algorithm produced better engagement schedules for17 out of 125 cases compared to BEC algorithm. Thosecases where the QUC algorithm produced better results donot represent any identifiable pattern. Construction heuris-tics failed to produce the optimal solution in 38 out of 125cases. Although the construction heuristics attained the opti-mal solution in 70% of the test cases, we conclude that theymay frequently produce unsatisfactory results.
We ran our improvement algorithms for those 38 caseswhere the construction algorithms failed to produce the opti-mal solution. Two different combinations of the improvementalgorithms were also investigated. One of those combinations(OC + 2OX) is running OC first and then 2OX. The other(2OX + OC) is running 2OX first and OC next. The sum-mary results of improvement heuristics are given in Table 3.The last column of Table 3 depicts the best results of theimprovement heuristics. The best result may be viewed asanother heuristic that runs OC + 2OX and 2OX + OC, andtakes the best of the two solutions.
We provide some measures of solution quality for heuris-tics OC+2OX, 2OX+OC, and “Best” in Table 4. OC+2OXdominates 2OX + OC with respect to five measures givenin Table 4. “Best” provides a slight improvement on theOC + 2OX results. OC + 2OX attains the optimal solutionin 33 out of 38 problems. In one out of the remaining fivecases where OC + 2OX failed to achieve the optimal results,“Best” was found by 2OX + OC. We statistically compared“Best” and OC + 2OX against 2OX + OC heuristic usingWilcoxon signed rank test as described in Golden and Stew-art [14]. Wilcoxon tests showed that “Best” and OC + 2OX
heuristics are statistically better than 2OX + OC heurisitic atα = 0.05 significance level.
For all small test problems presented above, our MAPsolution procedure produced high-quality solutions while sat-isfying the run time requirement for MAP. We used smalltest problems to be able to compare the heuristic results withthe optimal results. We restricted the number of total SAMsto nine. Thus, the average number of missiles available on
Table 3. % Gap between optimal solution and the improvementheuristics for the problems where constructions heuristics failed tofind the optimal solution.
Best of Improvement algorithmsProblem constructionNumbera heuristics OC 2OX OC+2OX 2OX+OC Best
found 19 12 33 27 34aProblem Number: the Roman numeral shows the problem set num-ber, the 2nd and 3rd numerals show the number of ASMs and SAMsystems, respectively.
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Table 4. Comparison of OC + 2OX and 2OX + OC heuristics with best results.
OC + 2OX 2OX + OC Best
Number of times heuristic is best or tied for best 35 31 38Average percentage below optimal value 0.77 2.64 0.65Average rank among three results 1.05 1.37 1.00Worst ratio of solution to optimal value 0.93 0.75 0.93Number of times heuristic found the optimal solution 33 27 34
the magazines for the problems with five SAM systems fallsbelow two missiles per system. Since the average number ofavailable missiles for each system is low, using this valuableasset against one ASM may prevent the heuristics from usingit against another one more effectively in a later engagement.This argument is generally valid for construction algorithms.We expect that if we had a larger number of missiles perSAM system, construction algorithms could produce betterresults. This is because, as the number of SAMs carried getstoo large, solution of the problem becomes trivial; one sim-ply allocates as many of the most effective SAMs as possibleto all ASMs. The real challenge is solving the problem withlimited resources, and that is why we create test cases with asmall number of missiles per SAM system.
Improvement heuristics increased the solution quality sig-nificantly. However, up to this point, we investigated smalltest problems to be able to compare the results of heuristicswith the optimal solution. Since we are unable to generateoptimal results for problems larger than five SAM systemsagainst five attacking ASMs due to computational burdenof implicit enumeration, we develop an upper boundingscheme for large problems to show the quality of the heuris-tic solutions. Large test problems also enable us to test theperformance of heuristics in terms of elapsed time.
We develop the upper bound (UB) by relaxing constraintset (2), which reflects the restriction on the number of roundsavailable for each SAM system. Thus, we assume unlimitednumber of rounds available for each SAM system. Objec-tive function of the relaxed problem is an upper bound forthe original problem. By relaxing constraint set (2), the prob-lem becomes separable on each ASM i. Sub-problems can besolved separately and then the upper bound for the objectivefunction can be calculated using the probability of shootingdown each ASM i. Each sub-problem can be solved usingimplicit enumeration.
The upper bounding scheme will produce optimal resultsas long as we have sufficient rounds on SAM systems toschedule the best engagement against each ASM. If there isshortage on missile availability to schedule the best engage-ment for each ASM, then the gap between the objectivefunction value and the upper bound would be large. For exam-ple, assume that we have a scenario with one SAM systemagainst two ASMs. Let, d1 = 2, u11 = 1, u21 = 2, p11 = 0.4,
and p21 = 0.6. The optimal engagement for this scenario isone engagement against ASM1 and one engagement againstASM2 with an optimal objective function value of 0.24. Ourupper bounding scheme produces one engagement againstASM1 and two engagements against ASM2 with an objec-tive function value of 0.336. If we assume d1 = 3 roundsavailable on SAM system, then optimal objective functionvalue and upper bound are both 0.336. This simple exampleshows how the optimal objective function value gets closerto the upper bound as we increase the available rounds onSAM systems.
For large problems, we again use the best solution obtainedfrom the two construction algorithms as the starting point ofimprovement heuristics. We assume that 10 rounds are avail-able for each SAM system. Table 5 depicts the results forthose large test problems in terms of computation time. Weuse five sample problems for each ASM and SAM systemcombination. Each row in Table 5 shows the average resultsof those five problems. The largest average run time recordedfor heuristics is 1.01 s for the problem with 25 ASMs and20 SAM systems using OC + 2OX algorithm. Computationtimes of the heuristics are less than half a second in 434runs out of 450 (two construction heuristics, two individualimprovement heuristics and two permutations of improve-ment heuristics tried for each of the 15 scenarios for fivedifferent problems). Elapsed times are larger than 1 secondand less than 1.6 s for only four out of 450 runs. The last col-umn in Table 5 shows the average elapsed time for upperbound computation. The largest average run time for theupper bound is 5222.69 s.
Table 6 shows the minimum, average and maximum gapbetween the best heuristic solution and the upper bound.We declare optimality for 22 out of 75 problems as the gapbetween the heuristic solution and the upper bound is zero.The largest average gap is 0.048, while the largest maximumgap is 0.164. We can say that our heuristics work well forsample problems with 5 and 10 ASMs. We need further inves-tigation to determine the source of larger deviations with 15,20, and 25 ASMs. The source of deviation may be a largeoptimality gap for the upper bound or a large optimality gapfor the heuristic solution. We theoretically know that, if wehave unlimited rounds available on SAM systems, the upperbounding scheme produces optimal results. Hence, we expect
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Table 5. Performance of heuristics for large problems in terms of elapsed time.
Number Number of Elapsed timea (sec)
of ASMs SAM systems BEC QUC OC 2OX OC + 2OX 2OX + OC UB
aAverage CPU time for the algorithms on a personal computer with AMD Athlon 2000 + CPU and 256 MB of RAM.
that the optimality gap for the upper bound will get smaller aswe increase the number of rounds available on SAM systems.If we are able to reduce the gap between the heuristic solu-tion and the upper bound by increasing the number of roundsavailable on SAM systems, larger gaps in Table 6 can beattributed to the large optimality gap for the upper boundingscheme. Therefore, we increase the number of rounds avail-able on SAM systems to 20 and 30 for sample problems with15, 20, and 25 ASMs. The results are depicted in Table 7. Thelargest average gap is reduced to 0.014 for 20 rounds, and to0.007 for 30 rounds. Although we cannot claim optimality
Table 6. Average gap between the best heuristic solution and theupper bound.
Number Number of Minimum Average Maximumof ASMs SAM systems gapa gapa gapa
for those problems, results warrant that the heuristics workwell for all problem sizes in general.
We focused on the quality of the solutions up to here. Sensi-tivity of the solutions to the input parameters is another issuefor better understanding of the model and quantification ofthe relationships between input parameters and output. Weassumed static single shot kill probabilities for the randomlygenerated SAM systems in the preceding computations. Wechecked the sensitivity of the solutions to randomly gener-ated single shot kill probabilities. We used problem I.5.5 fromTable 3 with five ASMs and five SAM systems. We solvedthe problem 20 times, each time using a randomly generatedsingle shot kill probability matrix. All heuristic solutions pro-duced optimal results with different objective function valuesand engagement schedules as expected.
Table 7. Average gap for 20 and 30 rounds available on SAMsystems.
Number of Number of Average gapa
ASMs SAM systems 20 rounds 30 rounds
15 10 0.000 0.00015 0.014 0.00720 0.001 0.000
20 10 0.002 0.00115 0.013 0.00720 0.012 0.007
25 10 0.009 0.00115 0.011 0.00720 0.006 0.003
aGap = UB-best heuristic solution.
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Table 8. Solutions of the sample case problem and the revised problems.
p42 = 0.30 0.53 SAM2 SAM2 SAM2(3) SAM5(3) SAM2(2)aShows the difference from the original parameters of the sample case.bFigure within parenthesis shows the number of engagements if it is larger than one.
We present a sample case to show the effect of input para-meters. We assume a TG with three ships. Ship1 has self andarea defense SAM systems SAM1 and SAM2. Ship2 simi-larly has self and area defense systems SAM3 and SAM4.Ship 3 has only a self defense system SAM5. ASM1 andASM3 target Ship1, and ASM2, ASM4, and ASM5 targetShip3. Note that Ship2 is not a target of any ASM. Thus,SAM3 cannot engage any of the ASMs. Let the number ofavailable rounds on SAM systems be (dj ) = (8, 8, 8, 16, 8).The upper bounds on the number of engagements of SAMsystems against ASMs within the engageability interval andthe single shot kill probabilities are:
We solved this sample problem and additional eight prob-lems generated by changing the available SAM rounds andthe single shot kill probabilities one at a time. We report the
revised input parameter values, the objective function valuesand the engagements produced in Table 8.
The objective function value of the original sample prob-lem is 0.59 and the resulting engagements are depicted in thefirst row of Table 8. All available rounds of SAM2, which isthe most effective area air defense system, are used. Whenwe reduce the available rounds of SAM2 to d2 = 6, SAM1and SAM4 replace SAM2 against ASM3. Note that ASM3attacks Ship1 that is equipped with SAM2, and the other areaair defense system of the TG, which is SAM4 on board ofShip2, replaces SAM2 for engaging ASM3. When we furtherreduce the number of available rounds of SAM2 to 4, 2, and 0,self-defense SAM systems SAM1 and SAM5 replace SAM2at the cost of reduced probability of no leaker. When wereduce p12 to 0.75, we get the same engagement plan as in theoriginal sample case. Although p14 = 0.80 is larger thanp12,the model cannot use it because of the engageability inter-val restrictions (i.e., u14 = 0). When we reduce p12 to 0.35,SAM1 replaces SAM2. When we reduce p42 to 0.60, SAM5replaces one of the SAM2 engagements. All of the engage-ments of SAM2 are replaced by SAM5 when p42 = 0.30.Although p44 is larger than p42, the upper bound on the num-ber of engagements of SAM4 against ASM4 prevents SAM4from replacing SAM2. We conclude that changes in the inputparameters produce reasonable results. Experiments carriedout on the sample case show the importance of cohesionwithin a TG and convey the additional capability providedby a fully coordinated air defense of a TG.
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5. EXTENSIONS
5.1. Alternative Objective Functions
Our formulation maximizes probability of no leaker forall the ships in a TG, treating them as equally important. Ifthe defense commander places higher priority on defense ofthe HVUs than of the escort ships, he may prefer his areadefense systems to be used for protecting the HVUs ratherthan the escorts. An extreme case may be to maximize theprobability of shooting down only those ASMs targeting theHVUs, ignoring the rest. If one does not want area defenseSAM system j to engage ASM i targeting an escort ship,this can easily be implemented in our formulation by exclud-ing respective (i, j) pairs from the valid engagements set V ,thereby setting the respective xijk variables to zero. Hencethis is a special case of our formulation.
In the above treatment, only the HVUs are protected andthe escort ships are completely ignored, which means thatthey can only rely on their own self-defense capabilities. Ifone still wants to provide some area defense for the escorts,but with lower priority compared to the HVUs, this can beachieved by the following alternative objective function.
∑s∈T G
vs
∏i∈Ns
1 −
∏k∈K
j∈M|(i,j)∈V
(1 − pijk)xijk
(1′)
Here, TG is the set of ships in the TG, vs is the value orimportance weight assigned to ship s, and Ns is the set ofASMs targeting ship s such that
⋃s∈T G Ns = N . We expect
the vs parameter to take a higher value for a HVU than itdoes for an escort ship. This objective can be interpreted asmaximizing the total expected value of TG.
The probability of no leaker may be a very small figure,when there is a large number of attacking ASMs. In this case,we may consider minimizing the expected number of leak-ers or maximizing the expected number of ASMs shot down.This leads to the simple linear objective function below.
Max∑i∈N
∑k∈K
j∈M|(i,j)∈V
pijkxijk (1′′)
However, we lose control of which ASM to engage withthis objective. Some of the attacking ASMs may reach theirtargets with no scheduled SAM engagement. This may makethe probability of no leaker zero and HVU(s) may be the tar-get of those leaking ASMs. If defending against all the ASMsis desirable, we can maximize the minimum probability ofshooting down the attacking ASMs to have an engagementplan that leaves none of the ships in TG undefended againstany ASM if possible.
Other objectives could be maximizing the survival prob-ability by taking into account the single shot kill probabil-ities of the ASMs against the ships in TG, maximizing thetotal expected value of surviving ships, and minimizing thedamage sustained, which would require damage assessment.
5.2. Impact Point Predictability
Assumption 6 states that the target of each ASM can be pre-dicted. If a radar can detect an ASM it can also determine itsspeed and direction. Hence, target prediction is possible at thetime of detection as long as the ASM’s flight path is a straightline and the ships are not too close to each other. However,when an ASM follows pre-programmed way points, there isno way of predicting its eventual target before it is in the finalleg of the pre-programmed flight path. In this case, there isnothing to do or no problem to solve at the time of initialdetection.
If impact point prediction is impossible because of the longengagement ranges, our approach can still be used. When theengagement range is very long, we can reasonably assumethat the distances between ships are much smaller (they areclose together) compared with the distances of the ASMsfrom the TG as a whole. Then, all distances (and respectivetimes) can be computed from the ASM to the center of the TG.In this case, one can consider using only the long range areadefense systems, i.e., the valid engagements set V includesonly those (i, j) pairs such that ASM i is within the rangeof SAM j . Our formulation and heuristic algorithms remainunchanged. Naturally, this may result in some unnecessarymissile allocations and waste of defense resources, but thisis the price of uncertainty since the exact target of an ASMis unknown at the time of detection and allocation.
5.3. SLS Engagement Policy
As an alternative to the SLS policy (assumption 8) onemight want to use shoot-shoot-look (SSL), shoot-shoot-shoot-look or other variations. Here, we describe how ourapproach can be adapted for the SSL policy without loss ofgenerality. If the defense commander wants to fire two roundsof SAM j at ASM i before he stops to “look,” the engage-ment duration �ijk should include two constant setup timesand one variable flight time to interception. The overall missprobability of this double engagement is (1 − pijk)
2. In thiscase, our binary decision variable xijk indicates whether ornot a double engagement starts in time slot k. The objectivefunction and constraint set (2) need to be changed as follows.
Max∏i∈N
1 −
∏k∈K
j∈M|(i,j)∈V
(1 − pijk)2xijk
(1′′′)
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subject to∏k∈K
i∈N |(i,j)∈V
xijk ≤ �dj/2� for all j ∈ M (2′)
When two rounds are to be fired at the same ASM fromdifferent SAM systems, determination of the engagementduration before “looking” is not as straightforward as above.Instead, one can simply “duplicate” the incoming ASMs touse our formulation and heuristic algorithms as they are.Assuming there are two ASMs instead of one, our approachcan allocate two rounds of different SAMs to these ASMs,which in fact are the same ASM. This treatment also allowsusing different engagement policies against different ASMs.The commander may use the SLS tactic against some of theASMs and SSL against others.
5.4. Multiple SAMs in Flight
Assumption 11 states that there can be multiple SAMs inflight launched from the same SAM system to engage dif-ferent ASMs. Although some modern systems (e.g., Astermissile system where each missile has its own radar andEvolved Sea Sparrow Missile system–ESSM) are capable ofilluminating multiple ASMs simultaneously, this assumptionmay be restrictive for systems that must illuminate the targetASM by a traditional single-beam radar. We can relax thisassumption by defining a new set Ljk symmetrical to set Jik .
Ljk is the set of combinations (i, ρ) such that time slot k forSAM system j is blocked (to prevent the SAM system fromhaving multiple missiles in flight during slot k). To imposethe restriction, we need the new constraint set (6).
∑(i,ρ)∈Ljk
xijρ ≤ 1 for all j ∈ M and k ∈ K (6)
6. CONCLUSION
In this study, we introduced a new formulation for the airdefense problem and developed an efficient solution methodfor it. The mathematical programming model that was devel-oped was not explicitly used to solve MAP. Although mathe-matical programming models do guarantee an optimal solu-tion (without loss of generality), they usually take much morethan a few seconds in which we have to solve MAP for itsrepetitive application in scenario analysis.
Our solution approach for MAP uses construction andimprovement heuristics. We have developed two greedy con-struction algorithms for MAP. The first of those algorithms,
BEC heuristic, allocates SAM systems to incoming ASMsaccording to a measure called the engagement potential. Inthe QUC algorithm, we aim to engage each threat ASM atleast once. Thus, we give precedence to the ASM that has thelowest number of SAMs allocated.
We developed two improvement heuristics, OC and 2OX.Our purpose in the OC algorithm is to find the engagementsthat would increase the objective function value by changingthe target ASM of an engagement and simultaneously con-sidering the enhancement of defense by increasing the totalnumber of SAM missiles launched against target ASMs. Ourpurpose in the 2OX algorithm is to find the engagement pairsthat would increase the objective function value by exchang-ing the target ASMs of the SAMs. With each exchange, weagain try to increase the number of engagements against theASMs under consideration.
We tested our solution approach against the optimum using125 small sample problems. The procedure yielded highlysuccessful results. We attained 121 optimal solutions out ofthe 125 test problems. We generated 12 large test problemsto be able to test the performance of our heuristics in termsof computation time. The largest run time recorded was 1.17seconds. Computation times of the improvement heuristicswere less than half a second in 44 out of 48 runs made for 12large problems.
As the computation times of our construction and improve-ment algorithms are short, we suggest running both construc-tion algorithms since they do not dominate each other. Then,one can try to improve the better of the two constructionsolutions by running different combinations of improvementalgorithms. If one combination is to be selected, we suggestusing OC+2OX as it gives better results according to Table 4and the Wilcoxon test.
MAP can be used in the decision-making process for theprocurement of new air defense ships, and in evaluating thecapabilities of ships in inventory and the effectiveness ofpresent tactics. The coverage a ship provides for an HVUor other ships can also be estimated by using MAP throughrepetitive scenario analysis. When there is no or limited infor-mation on the attack direction, various representative sce-narios can be developed and solved using MAP. The resultscan be aggregated to estimate the coverage provided fromone sector to another on the sea surface. The ships can thenbe located in these sectors so as to maximize the cover-age provided for them [20]. According to the computationalstudies carried out, MAP also has the potential for on-lineand real-time use in realistic tactical situations as part of thethreat evaluation and weapon allocation module of a TG airdefense system.
The proposed solution approach can be enhanced alongseveral directions. A comparison of the proposed method-ology with the existing air defense policies and proceduresmay reveal more insight into the utility of the approach.
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In addition to computational time requirement, the solutionquality of MAP for large problems needs to be investigatedprovided that an exact solution procedure for large prob-lems is developed. Solution procedures may be developedto include other engagement policies. We solve MAP ina static environment, assuming simultaneous attack. How-ever, both simultaneous and sequential attack waves mayoccur in the dynamic environment of a real combat situa-tion. High resolution simulation models can be developed toinvestigate the best use of MAP solutions in such a dynamicenvironment.
APPENDIX A: OC ALGORITHM
STEP 0: Select an initial feasible engagement list, E.
E = {(i1, j1), (i2, j2), . . . , (il , jl)|k ∈ L = {1, . . . , l}, (ik , jk) ∈ V ,
t(ik ,jk) < t(ik′ ,jk′ )∀k ∈ L and ∀k′ ∈L ⇔ ik = ik′ and |k|< |k′|}
where i ∈ N , j ∈ M , t(ik ,jk) is the time of the engagement (ik , jk)
and V is the set of valid combinations of ASM and SAM sys-tems, i.e., (i, j) ∈ V if SAM system j can engage ASM i. Letthe corresponding objective function value be Z(E).
STEP 1: Set k = 0, E∗ = E, where E∗ is the best engagement schedulethat has been found so far. Set the logical variables “add1” and“add2” to “false.”
STEP 2: Set k = k + 1. i.e., take the next engagement in the engagementlist of E.
STEP 3: Check the possibility of the change of target ASM for theengagement (ik , jk) in the engagement list for all possible tar-gets except ik , i.e., set F = N\{ik} = {f1, f2, . . . , fn−1}. Leth ∈ H = {1, 2, . . . , n − 1} and set h = 1.
STEP 4: If h > 1, then set h = h+1. If (fh, jk) ∈ V , then go to Step 5 tofind a SAM missile for ASM ik to enhance the defense againstit, otherwise go to Step 11.
STEP 5: If there is at least one SAM system that can engage ASM ik andhas missiles left, then check the possibility of enhancement usingall possible SAMs, i.e., set G = M\{jk} = {g1, g2, . . . , gm−1}.Let t ∈ T = {1, 2, . . . , m − 1} and set t = 1. Set the logicalvariable “change” to “false”.
STEP 6: If t > 1, then set t = t + 1.STEP 7: If dgt > 0 and (ik , gt ) ∈ V , then go to Step 8; otherwise, go to
Step 9. Note that dgt is the number of available rounds on SAMsystem gt .
STEP 8: Define a new engagement list, E = {E\{(ik , jk)}} ∪{(fh, jk), (ik , gt )}. Note that, (ik , gt ) will be the last engagementof the engagement list. Check the feasibility of new engagementlist E, and calculate the objective function value, Z(E). If E isfeasible and Z(E) > Z(E∗), then change the engagement list,E∗ = E, update the objective function value Z(E∗) = Z(E).Set g− = gt , the variables “add1” and “change” to “true”, and“add2” to “false”.
STEP 9: If t = m − 1, then go to Step 10. Else, go to Step 6.STEP 10: If the variable “change” has value “false”, then define a new
engagement list, E = {E\{(ik , jk)}} ∪ {(fh, jk)}. Check the fea-sibility of new engagement list E, and calculate the objectivefunction value, Z(E). If E is feasible and Z(E) > Z(E∗), thenchange the engagement list E∗ = E, update the objective func-tion value Z(E∗) = Z(E). Set the variable “add1” and “add2”to “false”.
STEP 11: If h = n − 1, then go to Step 12. Else, go to Step 4.STEP 12: Consider changing the defending SAM for the engagement
(ik , jk). Set t = 1.STEP 13: If t > 1, then set t = t + 1.STEP 14: If dgt > 0 and (ik , gt ) ∈ V , then go to Step 15; otherwise, go to
Step 16.STEP 15: Define a new engagement list, E = {E\{(ik , jk)}} ∪ {(ik , gt )}.
Note that we change (ik , jk) to (ik , gt ) in the engagement listE. Check the feasibility of new engagement list E, and cal-culate the objective function value, Z(E). If E is feasible andZ(E) > Z(E∗), then change the engagement list, E∗ = E,update the objective function valueZ(E∗) = Z(E). Setg− = gt ,g+ = jk , the variables “add1” to “false” and “add2” to “true”.
STEP 16: If t = m − 1, then go to Step 17. Else, go to Step 13.STEP 17: If k = l, then go to Step 18. Else, go to Step 2.STEP 18: If Z(E) = Z(E∗), then stop.
Otherwise, set E = E∗, Z(E) = Z(E∗), if variable “add1”has value “true”, then set l = l + 1, dg− = dg− − 1, if variable“add2” has value “true”, then set dg− = dg− −1, dg+ = dg+ +1and go to Step 1.
APPENDIX B: 2OX ALGORITHM
STEP 0: Select an initial feasible engagement list, E.
E = {(i1, j1), (i2, j2), . . . , (il , jl)|k ∈ L = {1, . . . , l}, (ik , jk) ∈ V ,
t(ik ,jk) < t(ik′ ,jk′ )∀k ∈ L and ∀k′ ∈L ⇔ ik = ik′ and |k|< |k′|}
where i ∈ N , j ∈ M , t(ik ,jk) is the time of the engagement (ik , jk)
and V is the set of valid combinations of ASM and SAM sys-tems, i.e., (i, j) ∈ V if SAM system j can engage ASM i. Letthe corresponding objective function value of the engagementlist E be Z(E). Set E∗ = E, where E∗ is the best engagementschedule that has been found so far.
STEP 1: Set k = 1 and h = 1. Set the logical variables “add1” and “add2”to “false”. Those logical variables are used to control whetherthe best engagement schedule that may be found has additionallaunhes against ASMs exchanged or not.
STEP 2: Check the possibility of exchange of SAM allocation of theengagements k and k + h in the engagement list. If {(ik , jk+h) /∈V or (ik+h, jk) /∈ V } go to Step 18.
STEP 3: Define a new engagement list, E = {. . . , (ik , jk+h), . . . ,(ik+h, jk), . . .}. Check the feasibility of new engagement listE, and calculate the objective function value, Z(E). If E isinfeasible, then go to Step 18.
STEP 4: If Z(E) > Z(E∗), then reset the best engagement list, E∗ = E,update the objective function value, and set variables “add1” and“add2” to “false.”
STEP 5: Check for additional assignment against ASM ik , i.e., set G =M = {g1, g2, . . . , gm}. Let t ∈ T = {1, 2, . . . , m} and sett = 1. Note that we do not exclude SAM jk from consider-ation, since change in a previous engagement may enable usto launch the same engagement (ik , jk) as the last engagementagainst ASM ik . Set the logical variable “change” to “false.”The variable “change” is used to control whether ASM ik hasadditional launches against itself.
STEP 6: If t > 1, then set t = t + 1.STEP 7: If dgt > 0 and (ik , gt ) ∈ V , then go to Step 8; otherwise, go to
Step 10. Note that dgt is the number of available rounds on SAMsystem gt .
STEP 8: Define a new engagement list, E = E ∪ {(ik , gt )}. Note that,(ik , gt ) will be the last engagement of the engagement list. Check
Naval Research Logistics DOI 10.1002/nav
Karasakal, Özdemirel, and Kandiller: Anti-Ship Missile Defense 321
the feasibility of new engagement list E, and calculate the objec-
tive function value, Z(E). If E is feasible, then set the variable“change” to “true”, gchange = gt and go to Step 9; otherwise, goto Step 10.
STEP 9: If Z(E) > Z(E∗), then reset the best engagement list, E∗ = E,
update the objective function value Z(E∗) = Z(E). Set g1∗ =gt , the variable “add1” to “true” and “add2” to “false.”
STEP 10: If t = m, then go to Step 11. Else, go to Step 6.STEP 11: Check for additional assignment against ASM ik+h. Set t = 1.STEP 12: If t > 1, then set t = t + 1.STEP 13: If the variable “change” has value “true”, go to Step 14;
otherwise, go to Step 16.STEP 14: If gt = gchange and dgt > 1, then go to Step 15, else if
gt �= gchange and dgt > 0, then go to Step 15; otherwise, goto Step 17.
STEP 15: If (ik+h, gt ) ∈ V then define a new engagement list, E =E ∪ {(ik+h, gt )}. Check the feasibility of new engagement list
E, and calculate the objective function value, Z(E). If E is
feasible and Z(E) > Z(E∗), then change the engagement list,
E∗ = E, update the objective function value Z(E∗) = Z(E),set g2∗ = gt , the variables “add2” “true”; otherwise, go toStep 17.
STEP 16: If dgt > 0, then define a new engagement list, E = E ∪{(ik+h, gt )}. Check the feasibility of new engagement list E,
and calculate the objective function value, Z(E). If E is fea-
sible and Z(E) > Z(E∗), then change the engagement list,
E∗ = E, update the objective function value Z(E∗) = Z(E).Set g2∗ = gt , the variable “add2” to “true” and “add1” to“false”.
STEP 17: If t = m, then go to Step 18.Else, go to Step 12.
STEP 18: If k + 1 = l, then go to Step 19.Else,
if k + h = l then set t = 1, k = k + 1 and go to Step 2.if k + h < l then set h = h + 1 and go to Step 2.
STEP 19: If Z(E) = Z(E∗), then go to step 20.Otherwise, set E = E∗, Z(E) = Z(E∗), if variable “add1”
has value “true”, then set l = l + 1, dg1∗ = dg1∗ − 1, if variable“add2” has value “true”, then set l = l + 1, dg2∗ = dg2∗ − 1, andgo back to Step 1.
STEP 20: For each possible ASM pair, try changing all the engagements ofthose ASMs. If there is an improvement, update the engagementlist E, and go back to Step1; otherwise, stop.
ACKNOWLEDGMENT
The authors thank the associate editor and the anonymous reviewers fortheir insightful comments and valuable suggestions on improving this article.
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