# A branch‐and‐price algorithm for a targeting problem

**ABSTRACT** In this paper, we consider a new weapon-target allocation problem with the objective of minimizing the overall firing cost. The problem is formulated as a nonlinear integer programming model, but it can be transformed into a linear integer programming model. We present a branch-and-price algorithm for the problem employing the disaggregated formulation, which has exponentially many columns denoting the feasible allocations of weapon systems to each target. A greedy-style heuristic is used to get some initial columns to start the column generation. A branching strategy compatible with the pricing problem is also proposed. Computational results using randomly generated data show this approach is promising for the targeting problem. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007

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**ABSTRACT:**Effect-based weapon-target pairing assigns weapons to targets for the given desired effects on such targets. The most obvious and natural effects on targets are represented by the percentages of damage of these targets. In this paper, we focus on the generation of input for effect-based weapon-target pairing optimization. One way to generate such input is based on the Joint Munition Effectiveness Manual (JMEM). JMEM allows the evaluation of the weapons. It is a database that contains many tables, and each table contains many different data fields. Because of the sheer size of JMEM, the optimization of weapon-target pairing based on JMEM is currently focused mainly on one target at a time. In other words, the optimization of weapon-target pairing for many targets and weapons is not directly supported by JMEM, although all the necessary data is there. In this paper, we derive an input based on the given JMEM and desired effect(s), which should be useful in the follow-on effect-based weapon-target pairing optimization that is not limited to a single weapon or target. In particular, effect-based weapon-target pairing will rely on the scanning of the attack guidance table that we derive from JMEM to determine a preferred set of weapon combinations for engaging a given set of targets.IEEE Transactions on Systems Man and Cybernetics - Part A Systems and Humans 01/2012; 42(1):276-280. · 2.18 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**In this paper, we present a novel optimization algorithm for assigning weapons to targets based on desired kill probabilities. For the given weapons, targets, and desired kill probabilities, our optimization algorithm assigns weapons to targets that satisfy the desired kill probabilities and minimize the overkill. The minimization of overkill assures that any proper subset of the weapons assigned to a target results in a kill probability that is less than the desired kill probability on such a target. Computational results for up to 120 weapons and 120 targets indicate that the performance of this algorithm yields an average improvement in quality of solutions of 26.8% over the greedy algorithms, whereas execution times remained on the order of milliseconds.IEEE transactions on cybernetics. 12/2013; 43(6):1835-1844. - SourceAvailable from: liu.diva-portal.org

Page 1

A Branch-and-Price Algorithm for a Targeting Problem

Ojeong Kwon,1Kyungsik Lee,2Donghan Kang,3Sungsoo Park4

1Combined Battle Simulation Center, ROK-US Combined Forces Command P.O. Box 181, Yongsan-dong 3-1, Yongsan-gu,

Seoul 140-701, Korea

2School of Industrial Information & Systems Engineering, Hankuk University of Foreign Studies, 89 Wangsan-ri,

Mohyeon-myeon, Yongin-si, Kyeonggi-do 449-791, Korea

3Information Management Team, Samsung Electro-Mechanics, 314 Maetan3-dong, Suwon-si, Kyeonggi-do 442-743, Korea

4Department of Industrial Engineering, Korea Advanced Institute of Science and Technology, 373-1 Guseong-dong,

Yuseong-gu, Daejeon 305-701, Korea

Received 29 June 2006; revised 4 May 2007; accepted 13 May 2007

DOI 10.1002/nav.20247

Published online 22 June 2007 in Wiley InterScience (www.interscience.wiley.com).

Abstract:

cost. The problem is formulated as a nonlinear integer programming model, but it can be transformed into a linear integer pro-

gramming model. We present a branch-and-price algorithm for the problem employing the disaggregated formulation, which has

exponentially many columns denoting the feasible allocations of weapon systems to each target. A greedy-style heuristic is used to

get some initial columns to start the column generation. A branching strategy compatible with the pricing problem is also proposed.

Computational results using randomly generated data show this approach is promising for the targeting problem.

Periodicals, Inc. Naval Research Logistics 54: 732–741, 2007

In this paper, we consider a new weapon-target allocation problem with the objective of minimizing the overall firing

© 2007 Wiley

Keywords:

targeting problem; weapon-target assignment; integer programming; branch-and-price

1.INTRODUCTION

Targeting problems are concerned with the strategy of

allocating weapon systems to targets so as to achieve the

desired objective while satisfying various tactical/technical

constraints. There are primarily two objectives. One is to

minimize the threat of the enemy after all scheduled firing is

terminated, and the other is to minimize overall firing cost.

The selection of the proper objective depends on the tactical

mission, weapon-target information, and combat situation.

The threat minimization is an important objective in both

offensiveanddefensiveoperations.Minimizingoverallfiring

cost is also an important objective considering the high price

ofeachweaponanditsoperation.Asweaponsystemsbecome

morelethalandexpensive,properdistributionoffirepoweris

recognizedasanimportantmilitaryinterest.Thus,wehaveto

consider the best strategy to allocate weapon systems against

targetstooperatepowerfulweaponsproperlyandeconomize

on the expensive modern weapon systems.

Correspondence to: S. Park (sspark@kaist.ac.kr)

We need information about the desired destruction levels

of the targets and the destroying probabilities for weapon-

target pairs for one unit firing to plan the targeting operation.

The desired destruction level of a target is generally decided

by analyzing the enemy’s combat order and the threat of the

target against friendly forces. The analytic hierarchy process

(AHP) and tactics discussions may be used to obtain the

desired destruction level of the targets. The destruction prob-

ability for a weapon-target pair for one unit firing can be

obtained by simulation or from the joint munitions effec-

tiveness manual (JMEM). The destruction probability for a

weapon-target pair depends on target size and range, charac-

teristics,locationerror,targetposture,aswellasthereliability

of the weapon system, destructive power, weapon system

deployment shape, the available amount of ammunition, and

ammunition type. Other factors such as weather condition

andterrainfeaturesarealsoconsidered.Theycanbeobtained

from the military database system. Targeting is not indepen-

dent of the other military support systems. In conclusion, all

available means of support should be combined to achieve

maximum effectiveness of the targeting operation.

© 2007 Wiley Periodicals, Inc.

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Kwon et al.: Branch-and-Price Algorithm for a Targeting Problem733

Targeting problems have been of interest to many

researchers for several decades. See, for example, Kwon

et al. [6] for a literature review of weapon-target assign-

ment (WTA) problems before 2000. Recently, traditional

WTA problems of maximizing damage done to targets were

dealt with using genetic algorithms (GA) embedded with a

local search approach like ant colony optimization (ACO)

[4, 7] or greedy eugenics [8]. Lu et al. [9] proposed the

GA with an improvement in terms of uniform creation of

initial population, selection based on fitness scaling and self-

adaptiveparametersforgeneticoperators,etc.Çetinetal.[3]

applied the WTA problem to media allocation in an adver-

tising campaign. Yost and Washburn [16] proposed a chal-

lengingweaponallocationmodelconsideringtheassessment

of current target state during air-to-ground operation: func-

tional, destroyed, or damaged. A linear programming model

was constructed and column generation approach was sug-

gested as a solution approach. Subproblems finding improv-

ing attack policies (columns) were modeled as a partially

observable markov Decision Process (POMDP) model.

The objective of minimizing firing cost has rarely been

considered in the literature despite the importance of con-

sideration of the high price of modern weapon systems and

their operation. In the previous work [6], we had dealt with a

targeting problem of minimizing the overall firing cost under

the constraints, which limit the enemy’s threat to a desired

level by setting minimum destruction probability of each tar-

get. We had also set the upper bounds on the rounds for each

weapon-target pair. In this paper, we consider a more general

targeting problem. We need to limit the number of targets on

whicheachweaponsystemcanfireduringtheoperation.The

upper bounds on the numbers had been set as |T| in the pre-

vious work [6]. Because of the restricted reaiming ability of

weapon systems, these constraints are necessary for achiev-

ing proper allocation of firepower. In this sense, the case in

which all targets are assigned to a single weapon system is

undesirable.

Weapon systems include filed artillery, rocket, MLRS

(multiple launcher rocket system), aircraft, UAV (unmanned

arielvehicle).Asinmostpapersonthetargetingproblem,we

assume that the following two statements hold. We assume

that all targets are fixed targets such as command posts,

communication sites, assembly areas, key terrain features,

or fire support units. We also limit our attention to plan-

ing a scheduled firing. Preparation fire and counter-fire are

typical examples of the scheduled firing. Preparation fire is

to carry out a preemptive strike against the enemy, while

counter-fire is to reply to the enemy’s fire. The scheduled fir-

ing requires target information on the type and the location

to be obtained and the needed weapons to be provisioned

beforehand. This implies that we have enough time to plan

the targeting operation. Thanks to recent advanced comput-

ing speed of PC or Workstation, the planning time has been

reducedconsiderably.Inthewargamesimulatedinthearmy,

destruction probabilities can be evaluated and applied in a

realtime.Themodelandsolutionapproachwouldbeapplied

in time-sensitive target or mobile target cases if the module

that destruction probability can be evaluated in a short time,

reflecting current target position.

The problem can be formulated as a nonlinear integer pro-

gram. Costs are incurred when assigning weapons to targets

as in [2]. We overcome the difficulties in treating nonlinear-

ity of constraints by transforming nonlinear constraints into

linear ones, which results in an integer program with 0-1

and general integer variables. Then we present a branch-and-

price algorithm as a solution approach. The results in this

paper are interesting in the sense that the targeting problem,

which can be also modeled as an integer program with gen-

eral integer variables, is solved successfully. The results may

be compared with those of the branch-and-price algorithms

for the one-dimensional cutting stock problem [13] in which

a general integer knapsack problem is solved for generating

a column.

The remainder of this paper is organized as follows. We

present two formulations for the same targeting problem

in Section 2. Section 3 deals with the branch-and-price

algorithm and subproblem optimization. Section 4 presents

the branching strategy applied in the algorithm. We report

computational results using randomly generated data in

Section 5.

2.FORMULATIONS

In the targeting problem, we search for a minimum cost

weapon-target allocation satisfying four types of constraints

on (1) the amount of available ammunition for weapon sys-

tems, (2) the number of targets on which a weapon system

can fire, (3) the number of rounds that can be fired for

each weapon-target pair, and (4) the desired destruction level

for each target, respectively. Constraints (2) are included

additionally in this paper.

We first propose a standard formulation of the problem,

which is a nonlinear integer programming model. We trans-

form it into an integer programming model and we propose

a disaggregated formulation with the variables denoting the

feasible allocations of weapon systems to each target.

2.1.The Standard Formulation

To formulate the targeting problem, we use the following

notation and variables:

W, set of weapon systems (n = |W|; we use these terms

interchangeably);

T, set of targets;

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A, set of arcs, where (i,j) ∈ A if and only if weapon

system i can fire on target j;

cij, the cost of firing one round or volley from weapon

system i to target j, (i,j) ∈ A;

fi, the number of rounds available for weapon system i

during the operation, i ∈ W;

qi, the upper bound on the number of targets on which

weapon system i can fire during the operation, i ∈

W;

pij, theprobabilityofdestroyingtargetj byoneroundof

fire from weapon system i, 0 < pij< 1,(i,j) ∈ A;

dj, theminimumdesiredprobabilityofdestroyingtarget

j, 0 < dj< 1, j ∈ T;

uij, the upper bound on the number of rounds from

weapon system i to target j, (i,j) ∈ A;

xij, an integer decision variable indicating the number

of rounds of weapon system i assigned to target

j, (i,j) ∈ A; and

yij, abinarydecisionvariableindicatingwhetherweapon

systemi firesontargetj ornot,i.e.,yij= 1ifweapon

i firesontargetj,andyij= 0,otherwise,(i,j) ∈ A.

Then, we can formulate the problem as the following nonlin-

ear integer programming model.

?

s.t.

xij≤ fi

?

xij− uijyij≤ 0

1 −

{i∈W|(i,j)∈A}

(TP1) min

(i,j)∈A

cijxij

?

{j∈T|(i,j)∈A}

for all i ∈ W, (1)

{j∈T|(i,j)∈A}

yij≤ qi

for all i ∈ W, (2)

for all (i,j) ∈ A,

(1 − pij)xij≥ dj

(3)

?

for all j ∈ T, (4)

(5)

(6)

xij≥ 0,

yij∈ {0,1}

integer for all (i,j) ∈ A,

for all (i,j) ∈ A.

Constraints (1) reflect the restriction on the amount of

ammunition available for each weapon system. Constraints

(2) ensure that each weapon system can fire on up to a lim-

ited number of targets during the operation. If qi≥ |T| for

all i ∈ W, the problem becomes the one considered in [6].

Constraints (3) represent the upper bound on the number of

roundsthatcanbefiredforeachweapon-targetpair.Theyare

alsolinkingconstraintsrepresentingtherelationshipbetween

the variable xijand the variable yij.

Constraints(4)meanthattheprobabilityofdestroyingtar-

get j should be at least a minimum desired level for that

target. These nonlinear constraints can be transformed into

linear inequalities by taking logarithms on both sides [2].

We need rational numbers on both sides of the constraints to

solve the problem. By multiplying a large number θ (100 in

our implementation) on both sides and rounding down, we

can approximate the feasible region and obtain integer coef-

ficients. This approximation scheme is acceptable from the

practical point of view because the values of coefficients in

the inequalities come from probabilistic estimates. Now, we

obtain an integer programming formulation, which is a close

approximation to TP1:

?

s.t.

(1),(2),(3),(5),(6),

?

where aij = ?−θ ln(1 − pij)? > 0 and bj = ?−θ ln(1 −

dj)? > 0.

WesolveTPinsteadofTP1inthisresearch.TPisNP-hard,

which can be easily verified by the fact that the knapsack

problem is a special case of TP, which is known as NP-hard

[5]. Setting T = {j} and uij= 1 for (i,j) ∈ A,fi= qi= 1

for i ∈ W, TP becomes the knapsack problem.

To solve TP, we use the branch-and-price approach

employing a new formulation which can be viewed as a dis-

aggregatedversionofTP,whichwillbepresentedinthenext

subsection.

(TP) min

(i,j)∈A

cijxij

{i∈W|(i,j)∈A}

aijxij≥ bj

for all j ∈ T,(7)

2.2. The New Formulation

By feasible allocation for a target, we denote the alloca-

tion of weapon systems to the target satisfying constraints

both on the upper bounds on the rounds and on the mini-

mum desired destruction level. Let Kjbe the set of feasible

allocations of weapon systems to target j. Then a feasible

allocation xj= (...,xij,...) ∈ Znj

for all i such that (i,j) ∈ A and?

integral n-dimensional vectors. Figure 1 shows an example

of a feasible allocation of three weapons to target j. There

areanexponentialnumberoffeasibleallocationsforatarget.

To reformulate TP, we use the following notation:

+in Kjsatisfies xij≤ uij

{i|(i,j)∈A}aijxij ≥ bj,

+is the set of nonnegative

where nj= |{i|(i,j) ∈ A}| and Zn

ξijk, the number of rounds from weapon i to target j in

the allocation k ∈ Kj;

Cjk, the cost of the allocation k ∈ Kj, i.e., Cjk =

?

otherwise;

λjk, abinarydecisionvariableindicatingwhetherthefea-

sible allocation k is selected for target j(λjk= 1) or

not (λjk= 0).

Note that ξijk= 0 for (i,j) / ∈ A.

{i|(i,j)∈A}cijξijk;

ρijk, an indicator defined to be 1 if ξijk > 0, and 0

Naval Research Logistics DOI 10.1002/nav

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Kwon et al.: Branch-and-Price Algorithm for a Targeting Problem735

Figure 1.

An illustration of a feasible allocation to a target.

Then, TP can be reformulated as follows.

?

s.t.

j∈T

?

?

λjk∈ {0,1}

Constraints (8) and (9) correspond to the constraints (1)

and (2), respectively. Constraints (10) and (11) ensure that

only one allocation is selected for a target.

Let RTPL (or TPL) denote the LP relaxation of RTP (or

TP). Let us discuss the optimal value of RTPL. RTPL is

obtained by relaxing binary conditions imposed on the vari-

ables. Wecandroptheconstraints,λjk≤1forallj ∈ T, k ∈

KjinRTPLastheyaresatisfiedimplicitlybyconstraints(10)

and the nonnegativity constraints.

Let us define

?

for (i,j) ∈ A,

{i|(i,j)∈A}

The convex hull of Q is denoted by conv(Q). Because

any feasible solution to RTPL is a convex combination

of extreme points of conv(Q) and satisfies constraints (1)

and (2), the optimal objective value of RTPL is equal to

min{?

decomposition ([15], Section 11.2) to TP, where constraints

(3), (7), (5), and (6) have been placed in the subproblem (see

also [12]). Also, the value provided by RTPL is equal to the

(RTP) min

j∈T

?

?

?

?

λjk= 1

k∈Kj

Cjkλjk

k∈Kj

ξijkλjk≤ fi

for all i ∈ W,(8)

j∈Tk∈Kj

ρijkλjk≤ qi

for all i ∈ W, (9)

k∈Kj

for all j ∈ T, (10)

for all j ∈ T,k ∈ Kj.(11)

Q =

x ∈ Z|W||T|

+

,y ∈ {0,1}|W||T|: xij− uijyij≤ 0

?

aijxij≥ bjfor j ∈ T

?

.

(i,j)∈Acijxij: (1),(2),(x,y) ∈ conv(Q)}.

Note that RTP is obtained by applying Dantzig–Wolfe

optimal value of the Lagrangian dual obtained by relaxing

constraints (1) and (2) (Nemhauser and Wolsey [10], Section

II.3.6).Consequently,RTPLprovidesaboundatleastastight

as the bound provided by the TPL.

Although RTPL has an exponential number of columns,

it can be solved efficiently by column generation technique.

Given a restricted formulation of RTPL with the column sets

K?

mal for RTPL if all reduced costs for columns k ∈ Kj\K?

j ∈ T are at least zero. The reduced cost for a column

k ∈ Kjis Cjk−?

ables associated with the ith constraint in (8), ith constraint

in (9) and jth constraint in (10), respectively. Therefore,

if any columns with negative reduced cost are found, they

can be added to the restricted formulation, which is then

re-optimized.Otherwise,wehavesolvedRTPLtooptimality.

The subproblem (pricing problem) is to generate a favor-

able feasible allocation to be added to the formulation as an

enteringcolumn.Theformulationofthesubproblemforgen-

erating the allocation with minimum reduced cost for target

j can be stated as follows:

?

−

{i∈W|(i,j)∈A}

s.t.

aijxij≥ bj

j⊂ Kj, j ∈ T, the current optimal solution is also opti-

j,

i∈Wξijkπ∗

jare optimal values of the dual vari-

i−?

i∈Wρijkµ∗

i− φ∗

j, where

π∗

i≤ 0,µ∗

i≤ 0, and φ∗

(SPj) min

{i∈W|(i,j)∈A}

cijxij−

?

{i∈W|(i,j)∈A}

?

π∗

ixij

µ∗

iyij− φ∗

j

(12)

?

{i∈W|(i,j∈A}

xij≤ uijyij

for all i ∈ W such that (i,j) ∈ A,

xij≥ 0,

such that (i,j) ∈ A,

yij∈ {0,1}

(13)

integer for all i ∈ W

(14)

for all i ∈ W such that(i,j) ∈ A.

(15)

Objective function (12) represents the reduced cost of

the generated column. Note that Cjk is expressed as

?

numberofrounds.Iftheoptimalvalueisnegative,thecolumn

(x∗

whereej=(0,...,1(j),0,...).Notethat(x∗

new (ξijk,ρijk) of the added column.

{i∈W|(i,j)∈A}cijxij. Since φ∗

sack problem with fixed cost (µ∗

jis a constant, SPj is a knap-

i) imposed upon a nonzero

ij,y∗

ij,ej)TareaddedtotherestrictedformulationofRTPL,

ij,y∗

ij)becomesa

3.ALGORITHM

In this section, we present the overall procedure of the

branch-and-price algorithm for RTP, subproblem optimiza-

tion, and a method for generating initial columns. The

branching strategy is presented in Section 4.

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Naval Research Logistics, Vol. 54 (2007)

3.1.Branch-and-Price Algorithm

Branch-and-price, a generalization of branch-and-bound

with LP relaxations, allows column generation to be applied

throughout the branch-and-bound tree. This approach has

been shown to be very effective to solve huge integer pro-

gramming problems that are difficult to solve using other

methods. Please refer to [1,14] for details of the algorithmic

issues and applications.

Theoverallprocedureisasfollows.Weinitiatearestricted

RTPL with an initial set of columns and apply column gener-

ationuntilanoptimalsolutiontoRTPLisfound.Wesolvethe

subproblem for each target to find an entering column. If the

finalsolutionisintegraloritcanbetransformedtoanoptimal

solution to TP (described in Section 4), we are done. Other-

wise, we use a suitable a branching strategy which allows us

to use column generation to solve the resulting problem. At

eachnodeofthebranch-and-boundtree,weapplythecolumn

generation to solve the LP relaxation at the node. Note that

the LP solution from a node is optimal across the entire sets

Kjs. Branch-and-price ends as soon as an optimal solution

is found while exploring the tree—branching and fathoming

as needed.

3.2.Subproblem Optimization

SPjdenotes the subproblem associated with target j ∈ T

(Section 2.2) and it is a knapsack problem with fixed cost

imposed upon a nonzero number of rounds. We solve the

subproblems via a dynamic programming algorithm. For

simplicity of exposition, we assume in this subsection that

(i,j) ∈ A for all i ∈ W and j ∈ T.

For m = 1,...,n(= |W|), define Nm≡ {1,...,m} and

−

i∈Nm

zm(d) = min

?

i∈Nm

(cij− π∗

i)xij

?

µ∗

iyij:

?

i∈Nm

aijxij≥ d,(13),(14),(15)

for d = 0,1,...,bj. (16)

Then zn(bj) − φ∗

that zm(d) cannot be obtained if?

0,1,...,bj. Then we proceed recursively to calculate zm(·)

from zm−1(·), where m = 2,...,n. The recursion is initial-

ized with

?

since π∗

jgives the optimal value of SPj. Note first

i∈Nmaijuij< d.

Westartthealgorithmwithm=1andcalculatez1(d),d =

z1(d) =

none (infeasible)

?d/a1j??c1j− π∗

1≤ 0.

if a1ju1j< d,

if a1ju1j≥ d,

1

?− µ∗

1

1,µ∗

If xmj=x∗

?

zm(d) =?cmj− π∗

where I(x) is defined to be 1 if x >0, and 0 otherwise.

Note that Lm≤x∗

L1=?d/a1j?andLm= max{0,?(d −?

Hence we obtain the recursion for m = 1,...,n and

d = 0,1,...,bjas follows:

where we have z0(d) = 0 for all d and zm(d) = 0 for all m

if d ≤ 0.

The number of calculations needed to get zm(d) for fixed

m is O(bjUj), which gives an overall running time of

O(nbjUj), where Uj = maxi∈Wuij. Given zm(d) and xmj-

level which yields zm(d) for all m and d, a recursion in

the opposite direction is used to determine an optimal solu-

tion (x∗

an optimal solution is dominated by the calculations in the

forward recursion (17).

mj

in an optimal solution to (16) then

i∈Nmaijuij≥ d and

?x∗

mmj− µ∗

mI?x∗

mj

?+ zm−1

?d − amjx∗

mj

?,

mj≤umj, where the lower bound is given as

i∈Nm−1aijuij)/amj?},

m≥2.

zm(d) =

none

min??cmj− π∗

integer?

if?

i∈Nmaijuij< d,

m

if?

?xmj− µ∗

i∈Nmaijuij≥ d,

mI(xmj)

+zm−1(d − amjxmj),Lm≤ xmj≤ umj,

(17)

j,y∗

j). The number of calculations required to obtain

3.3.Initial Feasible Columns

To start the column generation, we need a set of columns

giving an initial solution to the restricted LP of master prob-

lem, RTPL. First, we add a column with a sufficiently large

objective coefficient or an artificial variable to certify that

the initial RTPL has a feasible solution. In the column all

(2|W|+|T|)componentsareone.Severalfeasibleallocations

λjk’s are also generated and added to the restricted RTPL as

initial columns.

We start from the first target (j = 1). We solve the SPj

withtheobjectiveofminimizing?

system i to target j, we decrease qiby 1 since target j has

just been allocated by weapon system i. If qibecomes zero,

we also set uilto zero for all target l ≥ j + 1 to prevent the

weapon system i from firing on any remaining targets. Then,

we repeat the process for the remaining targets. The process

is terminated when all targets are visited or qi = 0 for all

i ∈ W. Consequently, the columns together are feasible with

respecttoqi-constraints(9).Thewholeprocedureforobtain-

ing initial columns is presented below. At most |T| columns

can be added initially by this procedure.

{i|(i,j)∈A}cijxij,whichwe

will call SP?

j. For a nonzero number of rounds from weapon

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Kwon et al.: Branch-and-Price Algorithm for a Targeting Problem737

3.3.1.Finding Initial Columns

Initialization: Set j := 1.

Step 1: Solve SP?

x∗

solution.

Step 2: Update data.

jfor target j and get a feasible column. Let

j= (...,x∗

ij,...) ∈ Znj

+bethex-partoftheoptimal

a. Set qi := qi− I(x∗

(i,j) ∈ A.

b. If qi = 0 for i ∈ W, set uil := 0 for all

l ≥ j + 1 such that (i,l) ∈ A.

Step 3: Termination test.

If j = |T| or qi= 0 for all i ∈ W, stop.

Otherwise, set j := j + 1 and go to Step 1.

ij) for i ∈ W such that

4.BRANCHING STRATEGY

Branching occurs when the resulting optimal solution to

RTPL is not integral. Suppose that we use a standard branch-

ing rule on the basis of variable dichotomy. We branch on a

fractional variable λjk. When λjkis fixed to zero in a node, it

is possible (and quite likely) that the column for the variable

will be generated again. In that case, it becomes necessary

to generate the column with the second lowest reduced cost.

At depth n in the branch-and-bound tree, we may need to

find the column with nth lowest reduced cost, which is gen-

erally intractable if the subproblem is NP-hard, as is the case

with SPj. Therefore, we need a branching strategy such that

subproblem can be modified so that the generated columns

will not be generated again and the subproblem will remain

tractable. In other words, we need the branching strategy

compatible with the pricing problem.

We adopt the hybrid branching policy as follows. First, we

use the weapon-target pair variables xij’s in TP for selecting

the branching variables. Second, we use the feasible allo-

cation variables λjk’s in RTP for applying the branching

decision. Similar approaches were used in [11,12]. Note that

the value of xijin TP can be obtained from the relationship:

xij=?

is selected for branching, where δ is a nonnegative integer

and 0 < ε < 1. We partition the feasible region of RTP into

two parts: one with xij ≤ δ and the other with xij ≥ δ + 1.

To apply xij≤ δ to RTP, we set the upper bounds to zero on

the variables λjk,k ∈ Kjsuch that ξijk> δ. Also, to apply

xij ≥ δ + 1 to RTP, we set the upper bounds to zero on the

variables such that ξijk≤ δ.

Here, we need to modify the algorithm for the subproblem

to reflect the branching decision at each node of the branch-

and-bound tree. Suppose that δ1 ≤ xij ≤ δ2is applied at

a node without loss of generality, where 0 ≤ δ1 ≤ δ2 ≤

uij. Since the dynamic programming algorithm presented

k∈Kjξijkλjk.

Suppose a fractional variable xijhaving the value of δ +ε

in Section 3.2 works when the lower bound on every vari-

able is zero, we should modify some values as follows:

uij := δ2− δ1and bj := bj− aijδ1. When δ1 > 0, yij

is fixed to one in the subproblem and we can set aside the

constantterm“−µ∗

omit the term “−µ∗

sion (17). When the algorithm is terminated, x∗

to be x∗

increased by (δ1(cij− π∗

solution.

Note that the branching strategy given above does not

necessarily guarantee an optimal integer solution to RTP as

shown in the simple example next.

i”intheobjectivefunction.Therefore,we

iI(xij)” in calculating zi(d) in the recur-

ijis updated

ij+ δ1and the value of the final solution should be

i) − µ∗

i), where (x∗

j,y∗

j) is the final

EXAMPLE 1: Consider an RTPL with two weapon sys-

tems and a target, where W = {1,2}, T = {1}, f1= 1,f2=

1,q1 = 1, and q2 = 1. Assume that the restricted master

problem has finally two variables after applying the column

generation.Thefirst(orsecond)columncorrespondstothree

rounds fired from weapon 1 (or weapon 2) at target 1 with a

cost of 3.

min

s.t.

3λ11+ 3λ12

3λ11+ 0λ12≤ 2,

0λ11+ 3λ12≤ 1,

1λ11+ 0λ12≤ 1,

0λ11+ 1λ12≤ 1,

1λ11+ 1λ12= 1,

λ11≥ 0, λ12≥ 0.

Note that the third and fourth constraints are redundant in

this case. Here, an optimal solution (λ∗

yields (x∗

On the other hand, if (x∗

in RTP for a feasible allocation and the associated cost is 3,

we are done with an integer optimal solution. Now, we show

that this replacement is always possible.

11,λ∗

12) = (2/3,1/3)

11,x∗

21) = (2,1), an integer solution.

11,x∗

21,1,1,1)Tcan be a column

PROPOSITION 1: If x∗

all integral for a fractional solution λ∗to RTPL, there exists

an optimal integer solution to RTP with the same objective

value.

ij=?

k∈Kjξijkλ∗

jk,(i,j) ∈ A are

PROOF: We show that the solution such that some vari-

ables λjk,k ∈ Kjfor target j are fractional can be replaced

by a new solution such that the (new) variable associated

with the column x∗T

j

= (x∗

may not appear in the current formulation – is one and the

rest are all zero.

By K??

λjk,k ∈ Kjfor target j. The jth constraint in (10) implies

Naval Research Logistics DOI 10.1002/nav

1j,x∗

2j,...,x∗

nj,...,1)T– which

j, we denote the set of indices of fractional variables

Page 7

738

Naval Research Logistics, Vol. 54 (2007)

that |K??

j| ≥ 2. We have that

?

{i|(i,j)∈A}

aijx∗

ij=

?

{i|(i,j)∈A}

?

?

aij

?

aijξijkλ∗

k∈K??

j

ξijkλ∗

jk

=

k∈K??

j

?

jk= bj.

{i|(i,j)∈A}

bjλ∗

jk

≥

k∈K??

j

Theinequalitycomesfromtheproperty,?

target j.

Moreover, the objective coefficient for this column (with

index, say k?) is

{i|(i,j)∈A}aijξijk≥

bj for k ∈ K??

j. So x∗T

j

represents a feasible allocation for

Cjk? =

?

{i|(i,j)∈A}

cijx∗

ij=

?

{i|(i,j)∈A}

cij

?

?

k∈K??

j

ξijkλ∗

jk

=

?

k∈K??

j

?

{i|(i,j)∈A}

cijξijk

λ∗

jk=

k∈K??

j

Cjkλ∗

jk.

In other words, the new solution such that λ∗

λ∗

given fractional solution.

Repeating this conversion for the rest of targets, we obtain

an integer solution to RTP.

jk? = 1 and

jk= 0, k ∈ Kj\{k?} has the same objective value as the

?

AccordingtoProposition1,whenthetransformedsolution

(x∗,y∗) to TP is integral at a node of the branch-and-bound

tree, we can prune it at the node even if λ∗is fractional.

When we branch, we select the fractional variable xijwith

fractional part closest to 0.5. In other words, we select the

variable that gives minimum |(x∗

tional values x∗

in selecting a node to solve next in the branch-and-bound

framework.

ij−?x∗

ij?)−0.5| for all frac-

ij, (i,j) ∈ A. We adopt the best bound rule

5.COMPUTATIONAL RESULTS

To test the algorithm, we coded a computer program in C

and generated random data for possible weapon-target pairs.

Tests were run on a Pentinum III PC (866 MHz, 256 Mb

RAM). We used Callable library of ILOG CPLEXTM7.0 as

LP and MIP (mixed integer programming) solvers.

Thedatageneratedcanbeclassifiedintosixgroupsaccord-

ing to weapon target sizes, namely 5 × 7, 5 × 15, 10 × 20,

10 × 40, 20 × 40, and 20 × 60. Data are generated so

that a weapon can fire at every target, i.e., the graph rep-

resenting the possible weapon–target pairs is a complete

bipartite graph. We considered the complete bipartite graph

to examine the performance of the algorithm in the worst

case though actual graph for the possible pairs is very sparse.

The weapon system sizes were taken considering the num-

ber of batteries that support various army echelons. We set

the number of targets to be about one to four times the num-

ber of weapon systems so as to describe the flexible combat

situation.

We generated the data according to the generation rule of

Kwon et al. [6] as follows. To see if the performance of our

algorithmissensitivetotheproblemdata,wepartitionedeach

groupfurtherintotwopartsaccordingtotherangeofthenum-

ber of rounds available for weapon systems (fis) : part 1 of

range [15, 30] and part 2 of range [5, 15]. We generated ten

test problems for each part in each weapon-target size. An

integer in the corresponding range was randomly selected

for fi, i ∈ W. Other data were also randomly generated in

their specific interval. An integer in the interval [1,|T|] was

generated for qi, i ∈ W. Integral costs were generated in

the interval [10, 30]. To reflect reality, destruction probabil-

ities of weapon-target pairs (pijs) and the minimum desired

destruction probabilities of targets (djs) were generated in

theinterval[0.1,0.4]and[0.2,0.7],respectively.Theseinter-

valssimulatetheactualdestructionlevelsofweaponsystems

and the desired destruction levels of targets in a real situa-

tion. Lastly, the upper bounds on the number of rounds for

weapon-target pairs (uijs) were randomly generated in the

interval [3, 8].

The test problems were solved by the branch-and-price

method with the RTP model first and were also solved,

for comparison, by an MIP routine of CPLEX with the

TP model. We limited the number of nodes to be gener-

ated to 100,000. We did not use any heuristic to get an

initial feasible solution nor cutting planes in CPLEX when

solving TP.

The results of tests for part 1 and 2 test problems using

two different approaches are summarized in Tables 1 and 2,

respectively. They show that maximum, minimum and aver-

age values of the gap, the number of nodes generated, the

number of generated columns including initial columns (in

solvingRTP’s)andtherunningtime.Componentsincolumns

“Gap” and “Time” are significant to the first digit below the

decimal point. Gap represents the percentage of the relative

ratio between the final objective value and lower bound pro-

vided by LP relaxation. In the tables, Gap1 is associated with

RTP, while Gap2 and Gap3 are associated with TP. Gap1 and

Gap2 are defined as follows:

Gap1 = (ZIP− ZR

Gap2 = (ZIP− ZT

LP)/ZIP× 100(%),

LP)/ZIP× 100(%),

where ZIPis the optimal value of TP; ZR

optimal value of RTPL (or TPL). On the other hand, Gap3

LP(or ZT

LP) is the

Naval Research Logistics DOI 10.1002/nav

Page 8

Kwon et al.: Branch-and-Price Algorithm for a Targeting Problem 739

Table 1.

Computational results for part 1 test problems.

Gap(%)#Nodes#Col.Time (sec.)

Problem sizeGap1Gap2Gap3RTPTP RTPRTPTP

5 × 7

Max.

Min.

Avg.

5 × 15

Max.

Min.

Avg.

10 × 20

Max.

Min.

Avg.

10 × 40

Max.

Min.

Avg.

20 × 40

Max.

Min.

Avg.

20 × 60

Max.

Min.

Avg.

0.0

0.0

0.0

28.5

9.5

21.2

0.0

0.0

0.0

0

0

0

10,898

154

1,661

12

8

9

0.0

0.0

0.0

2.7

0.1

0.4

0.1

0.0

0.0

31.9

11.7

19.8

3.8

0.0

1.4

1

0

0.1

100,000

4,208

90,421

43

18

26

0.1

0.0

0.1

57.6

1.7

42.7

0.0

0.0

0.0

25.9

13.7

18.8

15.3

0.6

7.3

0

0

0

100,000

100,000

100,000

27

21

23

0.2

0.1

0.1

118.8

102.7

110.3

0.0

0.0

0.0

22.9

15.3

19.4

26.9

15.4

20.4

0

0

0

100,000

100,000

100,000

65

41

47

0.3

0.1

0.2

253.2

224.9

240.7

0.0

0.0

0.0

21.3

14.5

18.2

46.6

16.1

32.2

1

0

0.1

100,000

100,000

100,000

49

41

43

0.8

0.2

0.4

639.5

582.0

598.1

0.0

0.0

0.0

19.8

15.2

18.4

35.6

17.4

31.0

0

0

0

100,000

100,000

100,000

78

61

66

0.9

0.4

0.6

1031.5

934.0

975.4

Table 2.

Computational results for part 2 test problems.

Gap(%) #Nodes#Col.Time (sec.)

Problem sizeGap1Gap2Gap3RTPTPRTP RTP TP

5 × 7

Max.

Min.

Avg.

5 × 15a

Max.

Min.

Avg.

10 × 20

Max.

Min.

Avg.

10 × 40

Max.

Min.

Avg.

20 × 40

Max.

Min.

Avg.

20 × 60

Max.

Min.

Avg.

0.0

0.0

0.0

31.5

8.8

20.1

0.0

0.0

0.0

0

0

0

22,194

139

2,722

220.1

0.0

0.0

5.9

0.1

0.7

8

11

0.9

0.0

0.1

29.0

15.0

21.2

6.2

0.0

2.4

6

0

0.9

100,000

11,175

85,105

59

16

30

0.2

0.0

0.1

59.2

4.8

42.2

0.0

0.0

0.0

26.1

15.0

19.2

9.2

1.4

5.7

2

0

0.2

100,000

100,000

100,000

39

21

29

0.3

0.1

0.1

126.8

93.7

107.0

0.1

0.0

0.0

22.4

14.3

18.6

27.7b

10.1b

17.5b

5

0

0.5

100,000b

100,000b

100,000b

121

52

71

0.8

0.2

0.4

257.2b

237.1b

247.5b

0.0

0.0

0.0

20.8

15.9

18.4

48.0

15.1

30.1

1

0

0.1

100,000

100,000

100,000

58

41

49

0.6

0.3

0.5

620.6

559.7

588.9

0.1

0.0

0.0

20.5

17.4

19.2

47.0

23.6

38.8

6

0

1.3

100,000

100,000

100,000

98

10

78

2.6

0.7

1.4

1001.7

930.2

973.9

aNine out of 10 problems have feasible solutions. Data compiled for the feasible problems only.

bOne out of 10 problems could not yield a feasible solution until 10,000th node is reached. Data compiled for the rest of problems.

Naval Research Logistics DOI 10.1002/nav

Page 9

740

Naval Research Logistics, Vol. 54 (2007)

Table 3.

An illustration of a weapon-target allocation for an instance: the third test problem of size 5 × 15 and the part 2 class.

Targets

Weapons123456789101112 13 1415

1

2

3

4

5

0

0

3

0

0

0

0

0

1

0

0

0

0

0

2

0

0

2

0

0

0

0

0

3

0

1

0

0

0

0

0

3

0

0

0

2

0

0

0

0

0

0

1

0

0

0

2

0

0

0

3

0

0

0

0

0

0

2

0

0

0

1

2

0

0

0

0

0

0

1

1

0

0

0

0

represents the ratio between the final objective value and the

optimal value, and it is defined as

Gap3 = (¯ZT

IP− ZIP)/ZIP× 100(%),

where,¯ZT

until the number of nodes reaches the node limit.

AsshowninTables1and2,ourbranch-and-priceapproach

applied to part 1 and part 2 test problems found optimal solu-

tions with a small number of nodes generated. We note that

RTPL gave much tighter bounds than TPL by comparing

Gap1 and Gap2. RTP was solved to optimality within 3 sec

for all test problems. However, we could not obtain optimal

solutions within 10,000 nodes using CPLEX’s MIP routine

for most of the test problems of size 5 × 15 or larger. Com-

ponents in the column “Gap3” for large-scale problems of

size 10 × 40 or more imply that the feasible solutions found

are far from optimal and many more nodes need to be gener-

atedinadditiontoreachoptimality.Thenumberofgenerated

IPis the objective value of the best solution found

columns,includinginitialcolumns,waswithin2|T|,butmore

columns were generated on average for part 2 problems than

part 1 problems.

We show a weapon-target allocation for an instance in

Table 3 for the third test problem of size 5×15 and the part 2

class.Theallocationmatrixissparseandaweaponsystemhas

been assigned to 3.2 targets on average. However, a weapon

system has not been assigned to one or two specific targets.

Unlike the Lagrangian relaxation approach of Kwon et al.

[6], we could not find clear difference in performance of

the branch-and-price algorithms applied to part 1 and part

2 problems. It is because the gaps (Gap1) are almost zero

and column generation procedure ends with a small num-

ber of columns generated. We compared the two approaches

using the part 2 test problems of size 10 × 30 generated

by Kwon et al. [6]. We set qi := |T| + 1, i ∈ W to

make constraints (2) redundant. In Table 4 we report the

computational results for Lagrangian relaxation approach

in which the third branching rule in [6] had been adopted.

Table 4.

relaxation (LR) [6].

Comparison of two approaches: branch-and-price vs. (RTP) branch-and-bound based on Lagrangian

Gap(%)#NodesTime (sec.)

No.Gap1Gap4 RTPLRRTPa

LRb

1

2

3

4

5

6

7

8

9

0.2

0.0

0.0

0.4

0.1

0.1

0.1

0.0

0.0

0.2

0.0

0.2

0.0

0.0

0.0

0.4

0.0

0.1

1.7

1.0

0.0

1.9

4.0

1.1

1.4

3.3

1.3

0.9

5.6

2.0

0.1

1.6

0.8

5.6

0.0

1.8

20916

250

1.5

0.3

0.1

1.0

0.4

0.3

0.8

0.4

0.3

0.8

0.4

3.8

0.1

0.4

0.2

3.8

0.1

0.7

8,558.6

2,406.30

000.6

121,606

1,144

144

306

14,288.5

10,851.9

1,378.3

3,083.5

799.7

4,485.7

2,261.2

5,102.9

4,657.2

20.5

2,326.4

1,121.3

14,288.5

0.6

4,089.5

2

1

4

2

0

4

1

86

394

216

558

556

10

11

12

13

14

15

81

02

1

0

224

110

1,606

0

434

Max.

Min.

Avg.

81

0

9

aTests were run on a Pentium III PC (866 MHz, 256 Mb RAM).

bTests were run on a HP9000/715 workstation (50 MHz, 32 Mb RAM).

Naval Research Logistics DOI 10.1002/nav

Page 10

Kwon et al.: Branch-and-Price Algorithm for a Targeting Problem 741

Gap4 is defined as

Gap4 = (ZIP− ZLD)/ZIP× 100(%),

where ZLDis the final value of Lagrangian dual obtained by

relaxing constraints (1). Gap1 should equal Gap4 for all test

problemssinceZLD= ZR

in Table 4, Gap1 is much smaller than Gap4 for most test

problems. It is because we terminated subgradient method

for Lagrangian dual after 100 iterations [6] (there is no guar-

antee that the subgradient method will be terminated with

the value ZLDin a finite number of iterations ([15], Section

10.3), and the method is usually terminated upon reaching

an iteration limit). The number of nodes generated indicates

that the branch-and-price algorithm converges much faster

than the branch-and-bound algorithm based on Lagrangian

relaxation.

LP(Section2.2).However,asshown

6. CONCLUDING REMARKS

We considered a new targeting problem with the objec-

tive of minimizing the total firing cost. The problem was

formulated as a nonlinear integer program. The constraints

reflect the amount of ammunition available for each weapon,

the upper bounds on the number of targets on which each

weapon system can fire, the upper bound on the number of

rounds that can be fired for each weapon-target pair, and the

desired destruction level of each target.

We overcame the difficulties in treating nonlinearity of

constraints by transforming nonlinear constraints into linear

ones. Then we reformulated the problem to get a better lower

bound on the optimal value and applied a branch-and-price

algorithm.Asaresult,wesolvedalltestproblemstooptimal-

itywithasmallnumberofnodesgeneratedinthebranch-and-

bound framework. The hybrid branching strategy was shown

to be very useful in getting an optimal solution to TP though

it is not compatible with RTP exactly. Since the performance

of the algorithm was hardly affected by the problem size, we

expect that our algorithm will still perform well for much

larger-scale problems than those considered in this paper.

ACKNOWLEDGMENT

ThisworkwassupportedbyHankukUniversityofForeign

Studies Research Fund.

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