# A Lagrange Relaxation Method for Solving Weapon-Target Assignment Problem

Authors: Mingfang Ni; Zhanke Yu; Feng Ma; Xinrong Wu
Journal: Mathematical Problems in Engineering
 (2011)

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## Abstract

We study the weapon-target assignment (WTA) problem which has wide
applications in the area of defense-related operations research. This problem calls for
finding a proper assignment of weapons to targets such that the total expected damaged
value of the targets to be maximized. The WTA problem can be formulated as a nonlinear
integer programming problem which is known to be NP-complete. There does not
exist any exact method for the WTA problem even small size problems, although several
heuristic methods have been proposed. In this paper, Lagrange relaxation method is
proposed for the WTA problem. The method is an iterative approach which is to decompose
the Lagrange relaxation into two subproblems, and each subproblem can be easy
to solve to optimality based on its specific features. Then, we use the optimal solutions
of the two subproblems to update Lagrange multipliers and solve the Lagrange relaxation
problem iteratively. Our computational efforts signify that the proposed method
is very effective and can find high quality solutions for the WTA problem in reasonable
amount of time.


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## Body

## 1. Introduction

Weapon-target assignment (WTA) problem is a fundamental problem arising in defense-related applications, which involve calling for finding a proper assignment of weapons to targets such that the total expected damaged value of the targets is to be maximized. It is a specific case of the more general resource allocation problem.WTA problem has been of interest to many researchers for several decades [1–4]. This problem can be formulated as a nonlinear integer programming problem which is known to be NP-complete. Ravindra et al. provided a brief and comprehensive survey of the relevant literature in this area and proposed a network flow-based construction heuristic algorithm [2]. There does not exist any exact methods for the WTA problem even relatively small size problems, and much research has focused on developing heuristic algorithms based on metaheuristic techniques, such as neural networks [5], genetic algorithms [6–9], tabu search algorithm [10], simulated annealing algorithm [11], and other expert systems [12]. These heuristic methods may not be able to produce an acceptable or even feasible solution, which is not allowed in a warfare scenario. Moreover, since no exact algorithm is available to solve WTA problem, it is unavailable to estimate the quality of solutions produced by such heuristics. A feasible solution corresponds to a WTA schedule. The purpose of this paper is to find feasible solutions in a reasonably fast time to help decision makers to make proper scheme on the battlefield.In this paper, we present a Lagrange relaxation method to solve WTA problem. First, a discrete variable is introduced in the formulation to transform the nonlinear integer programming problem into a linear optimization problem with the objective function is linear and the constraints respect to the original variables are linear and the introduced variable is nonlinear. Then we construct a Lagrange relaxation problem where the constraints are relaxed with Lagrange multipliers. The relaxation problem can be decomposed into two: one concerned with the introduced discrete variable is nonlinear optimization problem and the other concerned with the original variables is linear. We get the advantages of working with these two natural subproblems: the former subproblem can be decomposed into several one-dimensional discrete optimization problems which can be parallel processed easily. The latter subproblem’s constraint matrix is totally unimodular and thus can be solved by applying the simplex method. Therefore, the Lagrange relaxation problem is quite easy to solve, and any optimal integer solutions to the linear programming with respect to the original variables is feasible to the WTA problem. Then we use the optimal solutions of the two subproblems to update Lagrange multipliers and solve the Lagrange relaxation problem iteratively. The algorithm terminates due to a lack of improvement in the best solution over a number of generations. Computational results show that this method can be very successful for WTA problem.This paper is organized as follows: in Section2, we describe WTA problem and its formulation. In Section 3, we present a Lagrange relaxation method for solving WTA problem. Then, the results of employing the proposed algorithm to solve WTA problem are presented in Section 4. Finally, in Section 5 we give some concluding remarks and possible future work in this area.


## 2. The WTA Problem

To formulate the weapon-target assignment problem, we use the following notation and variables.W: The number of weapon types.T: The number of targets that must be engaged.uj: The value of target j. This is determined during the threat evaluation phase and used to priorities target engagement.wi: The number of weapons of type i available to be assigned to targets.tj: The minimum number of weapons required for target j.pij: The probability of destroying target j by a single weapon of type i, also referred to as the kill probability for weapon i on target j. It’s known for all i and j.xij: an integer decision variable indicating the number of weapons of type i assigned to target j.Then the WTA problem may now be modeled as the following nonlinear integer programming formulation in terms of the above variables [1, 4], (2.1)max⁡∑j=1Tuj(1-∏i=1W(1-Pij)xij),(2.2)s.t.∑j=1Txij≤wii=1,2,…,W,(2.3)∑i=1Wxij≥tjj=1,2,…,T,(2.4)xij≥0,integral,i=1,2,…,W,j=1,2,…,T.Sincepij is the destroying probability for weapon i on target j, the 1-pij term in (2.1) therefore denotes the probability of survival for target j if weapon i is assigned to it. Objective function (2.1) maximizes the probability of the total expected damaged value of the targets. Equation (2.2) provides a constraint to the problem that ensures that the total number of weapons used does not exceed what is available, (2.3) provides a constraint to the problem that ensures that the total number of weapons used should exceed the minimum number of weapons required for target j. Equation (2.4) provides a constraint that ensures that the number of weapons assigned to target j is nonnegative and discrete.


## 3. Lagrange Relaxation Method

WTA is computationally intractable because of the nonlinear of objective function and the integrality of variable. For problems involving small numbers of weapons and targets, these problems can be solved using any general purpose nonlinear integer programming package. For larger problems, faster algorithms are desired. So we are interested in solving these problems using Lagrange relaxation method as follow. Forj=1,2,…,T, let (3.1)yj=1-∏i=1W(1-Pij)xij,(3.2)Dj={yj∣yj=1-∏i=1W(1-Pij)xij,xij=0,1,2,…,wi}.
Then Dj is a discrete set. Let
(3.3)(x,y)=(x11,x12,…,x1T,x21,x22,…,x2T,…,xW1,xW2,…,xWT,y1,…,yT).
Then for (x,y), we have
(3.4)∑i=1W(ln⁡(1-Pij)xij)=ln⁡(1-yj),j=1,2,…,T.
WTA problem can be transformed as follows:(P)max⁡∑j=1Tujyj,s.t.(x,y)∈{(x,y)∣(x,y)satisfy(2.2),(2.3),(2.4),and(3.4),yj∈Dj},
We observed that problem (P) remove the nonlinear of objective function by increasing T discrete variable yj and will lead to some difficulties in the calculation. To avoid the problem we guide searching with yj by mapping between yj and xij in (3.1). We first derive the Lagrangian function as follows: (3.5)(L(λ))max⁡L(x,y,λ),s.t.(x,y)∈{(x,y)∣(x,y)satisfy(2.2),(2.3),(2.4),and(3.4),yj∈Dj},
where (3.6)L(x,y,λ)=∑j=1Tujyj+∑j=1Tλj(∑i=1W(ln⁡(1-Pij))xij-ln⁡(1-yj))=∑j=1T(ujyj-λjln⁡(1-yj))+∑j=1T∑i=1Wλj(ln⁡(1-Pij))xij,
where λ=(λ1,λ2,…,λT) is Lagrange multiplier, λi∈R,i=1,2,…,T, L(x,y,λ) is Lagrange function. It is obvious that (P) can be decomposed into two subproblems as follows: (3.7)(L1(x))max⁡∑j=1T∑i=1Wλj(ln⁡(1-Pij))xij,s.t.x∈{x∣forfixedy,xsatisfy(2.2),(2.3),and(2.4)},(3.8)(L2(y))max⁡∑j=1T(ujyj-λjln⁡(1-yj)),s.t.y=(y1,y2,…,yT)∈{y∣yj∈Dj,j=1,2,…,T}.Theorem 3.1.
Letx*,y* be optimal solutions of the two subproblems L1(x) and L2(y), respectively. If x*,y* satisfy (3.4), then x* is optimal solution of WTA.Proof.
Sincex*,y* are optimal solutions of the two subproblems L1(x) and L2(y), respectively so x*,y* are also feasible solutions of the two subproblems L1(x) and L2(y), respectively. If x*,y* satisfy (3.4), then x* is optimal solution of WTA. Let x is any feasible solution of WTA, for j=1,2,…,T, let
(3.9)yj=1-∏i=1W(1-Pij)xij.
Then (x,y) also satisfy (3.4) and we have
(3.10)∑j=1Tuj(1-∏i=1W(1-Pij)xij*)=∑j=1Tujyj*=∑j=1Tujyj*+∑j=1Tλj(∑i=1W(ln⁡(1-Pij))xij*-ln⁡(1-yj*))=∑j=1T(ujyj*-λjln⁡(1-yj*))+∑j=1Tλj(∑i=1W(ln⁡(1-Pij))xij*)≥∑j=1T(ujyj-λjln⁡(1-yj))+∑j=1Tλj(∑i=1W(ln⁡(1-Pij))xij)=∑j=1Tujyj+∑j=1Tλj(∑i=1W(ln⁡(1-Pij))xij-ln⁡(1-yj))=∑j=1Tujyj=∑j=1Tuj(1-∏i=1W(1-Pij)xij).
Hence, x* is optimal solution of WTA.Obviously, subproblemL2(y) can be decomposed into T 1-dimensional discrete problem as follows (j=1,2,…,T):(3.11)(L2(y))max⁡(ujyj-λjln⁡(1-yj)),s.t.yj∈{yj∣yj∈Dj}.Theorem 3.2.
Forj=1,2,…,T, let
(3.12)αj=1-(1-(min⁡1≤i≤WPij))Tj,βj=1-(1-(max⁡1≤i≤WPij))∑i=1Wwi,
one has αj≤yj≤βj.Proof.
Forj=1,2,…,T, we have
(3.13)yj=1-∏i=1W(1-Pij)xij≤1-∏i=1W(1-(max⁡1≤i≤WPij))xij=1-(1-(max⁡1≤i≤WPij))∑i=1Wxij≤1-(1-(max⁡1≤i≤WPij))∑i=1Wwi=βj.
Note that
(3.14)yj=1-∏i=1W(1-Pij)xij≥1-∏i=1W(1-(min⁡1≤i≤WPij))xij=1-(1-(min⁡1≤i≤WPij))∑i=1Wxij≥1-(1-(min⁡1≤i≤WPij))Tj=αj.
Thus, αj≤yj≤βj.
Consider the relaxation form of 1-dimensional discrete problemL2j(yj) as follows:
(3.15)(SL2j(yj))max⁡,(ujyj-λjln⁡(1-yj)),s.t.yj∈{yj∣αj≤yj≤βj}.
For given uj and λjSL2j(yj) is a convexity programming on a closed interval that can be solved easily, when λj≠0, the optimal solution is
(3.16)yjs={αj,1+λjuj≤αj,1+λjuj,1+λjuj∈(αj,βj),βj,1+λjuj≥βj,
when λj=0, the optimal solution is
(3.17)yjs=βj.Theorem 3.3.
Letyj1,  yj2∈Dj, yj1=max{yj∣yj≤yjs,yj∈Dj}, yj2=min{yj∣yj≥yjs,yj∈Dj}, then one has
(3.18)yj*=argmax{ujyj1-λjln⁡(1-yj1),ujyj2-λjln⁡(1-yj2)},
which is the optimal solution of 1-dimensional discrete problem L2j(yj).
SinceSL2j(yj) is a convexity programming on a closed interval, Theorem 3.3 is obvious.Next, we prove that the subproblemL1(x,y,λ) can be solved as a linear programming.Definition 1 (see [13]).
Anm×n integral matrix A is totally unimodular (TU) if the determinant of each square submatrix of A is equal to 0, 1, or −1.Corollary 3.4 (see [13]).
If A is TU, thenP(b)={x∈R+n:Ax≤b} is integral for all b∈Zm for which it is not empty.Corollary 3.5 (see [13]).
If the (0,1,−1) matrix A has no more than two nonzero entries in each column and if∑iaij=0 if column j contains two nonzero coefficients, then A is TU.Theorem 3.6.
The optimal solution of integer linear programmingL1(x) can be obtained by solving L1(x) as a linear programming using simplex method.Proof.
Integer linear programmingL1(x) can be described as follows:
(3.19)max⁡∑j=1T∑i=1Wλj(ln⁡(1-Pij))xij,s.t.∑j=1Txij≤wii=1,2,…,W,∑i=1W(-xij)≤-tj,j=1,2,…,T,xij≥0,integral,i=1,2,…,W,j=1,2,…,T.Note that the constraint matrix is a (0,1,−1) matrix A. Without loss of generality, assume that the column of A corresponds with the variablexij and contains two nonzero elements, for example, the i element is 1 and the w+j element is −1. Thus, the sum of two nonzero elements is zero. By Corollary 3.5, constraint matrix A is TU. Hence, the optimal solution of integer linear programming L1(x) can be obtained by solving L1(x) as a linear programming using simplex method.


## 4. Proposed Algorithm and Numerical Result

By Theorem3.1, for given Lagrange multiplier λk=(λ1k,λ2k,…,λTk), assume that xkyk are the optimal solutions of two subproblems L1(x) and L2(y), if xk and yk satisfy (3.4), then xk is the optimal solution of WTA. Otherwise, update Lagrange multiplier. Let (4.1)gj(xk,yk)=∑i=1W(ln⁡(1-Pij))xijk-ln⁡(1-yjk).
Update λ with (4.2)λjk+1={-1,gj(xk)>0,λjk,gj(xk)=0,1,gj(xk)<0.The proposed Lagrange relaxation algorithm solving WTA problem can be summarized as follows.Step 1.
Set the maximum iterate numberK, Lagrange multiplier λ0=(λ10,λ20,…,λT0),λi0=0(i=1,2,…,T) and k=0, z=0, x*=0.Step 2.
For fixedλk, solve L1(x) as linear programming using simplex method, obtain integral optimal solution xk and optimal value zk if zk≥z, then z=zk,  x*=xk.Step 3.
Ifk≥K, stop.Step 4.
SolveL2(y) and get the optimal solution yk. Compute gj(xk,yk) by (4.1). If
(4.3)gj(xk,yk)=0,j=1,2,…,T
Stop. Otherwise, update Lagrange multiplier λk with (4.2) and get λk+1, set k:=k+1, return Step 2.The proposed algorithm can make the computation more efficient. Because solving the relaxation linear programming is much easier than solving binary integer linear programming in Step2. In Step 4, we first calculate the optimal solutions of T 1-dimensional discrete problem SL2j(yj), then solve L2(y) with dichotomy. In order to narrow the search space, the algorithm guide searching by updating Lagrange multiplier using mapping between yj and xij in {-1,0,1}T.Our algorithm has been coded in Matlab 2009 and implemented on an Intel Core 2, CPU 2.53 Ghz, RAM 2 GB, Windows XP-System. We generate twenty random datasets in Matlab 2009. The settings are as follows:uj∈[0.3,0.9],  Pij∈[0.2,0.9],  wi∈[10,15],  tj∈[1,4], K=100. T, and W are given in Table 1. The number of integral variables in datasets from 28 to 40000. The numerical results are given in Table 1.Table 1
Numerical results of Lagrange relaxation method.
DatasetMetrics (W,T)Number of integral variablesInitial objective valueBest objective valueImprove rateComputing time (sec)Initial solutionBest solution1W=4,T=7283.374.199524.61%0.0020.0052W=20,T=3060011.795913.775316.78%0.0030.0123W=30,T=40120014.627117.320618.41%0.0030.0154W=40,T=50200018.709322.118618.22%0.0030.0155W=40,T=70280030.085234.095313.33%0.0040.0166W=40,T=70280025.603130.62519.61%0.0040.0187W=30,T=100300040.076745.100712.54%0.0070.0158W=30,T=100300040.32145.25512.24%0.0060.0169W=50,T=100500042.518847.036310.62%0.0080.03810W=50,T=100500041.51145.27539.07%0.0090.02411W=100,T=1001000040.626444.778510.22%0.0310.06112W=100,T=1001000045.009249.815710.68%0.0160.04113W=100,T=1501500062.903167.22826.88%0.0470.70314W=100,T=1501500067.016572.33437.94%0.0630.83515W=150,T=1502250069.543374.12156.58%0.0781.92716W=150,T=1502250068.960774.98368.73%0.0621.00617W=100,T=30030000122.0562129.90456.43%0.0942.31318W=100,T=30030000133.6946145.44278.79%0.0933.14219W=200,T=2004000092.330798.69236.89%0.1259.31520W=200,T=2004000096.3919106.796110.79%0.1129.637From the numerical results, we observe that our algorithm can find a good initial feasible solution quickly and make some improvements based on the obtained initial feasible solutions after some iteration. With the increasing of problem scales, the computing time required is acceptable. Hence, for large-scale problems, our algorithm can find a good feasible solution in reasonable computing time and is an efficient method to deal with WTA problem on the battlefield.


## 5. Conclusion

In this paper, Lagrange relaxation method is proposed for WTA. The method first decompose the Lagrange relaxation problem into two subproblems, and each subproblem can be easy to solve to optimality based on its specific features. Then use the optimal solutions of the two subproblems to update Lagrange multipliers and solve the Lagrange relaxation problem iteratively. The proposed method can find a good initial feasible solution quickly and make some improvements. The computational results obtained show that the proposed method is efficient. One direction of further research is to solveL2(y) in Theorem 3.3 accurately. The solution obtained by dichotomy usually is near optimal solution. This can lead to some difficulties in solving WTA and increases the number of iteration in calculating Lagrange relaxation problem.



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