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Mathematical Problems in Engineering
Volume 2011, Article ID 873292, 10 pages
http://dx.doi.org/10.1155/2011/873292
Research Article

A Lagrange Relaxation Method for Solving Weapon-Target Assignment Problem

Institute of Communication Engineering, PLA University of Science and Technology, NanJing 210007, China

Received 18 August 2011; Accepted 9 October 2011

Academic Editor: Xing-Gang Yan

Copyright © 2011 Mingfang Ni et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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 [14]. 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 [69], 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 Section 2, 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], max𝑇𝑗=1𝑢𝑗1𝑊𝑖=11𝑃𝑖𝑗𝑥𝑖𝑗,(2.1)s.t.𝑇𝑗=1𝑥𝑖𝑗𝑤𝑖𝑖=1,2,,𝑊,(2.2)𝑊𝑖=1𝑥𝑖𝑗𝑡𝑗𝑥𝑗=1,2,,𝑇,(2.3)𝑖𝑗0,integral,𝑖=1,2,,𝑊,𝑗=1,2,,𝑇.(2.4)

Since 𝑝𝑖𝑗 is the destroying probability for weapon 𝑖 on target 𝑗, the 1𝑝𝑖𝑗 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. For 𝑗=1,2,,𝑇, let 𝑦𝑗=1𝑊𝑖=11𝑃𝑖𝑗𝑥𝑖𝑗,𝐷(3.1)𝑗=𝑦𝑗𝑦𝑗=1𝑊𝑖=11𝑃𝑖𝑗𝑥𝑖𝑗,𝑥𝑖𝑗=0,1,2,,𝑤𝑖.(3.2) Then 𝐷𝑗 is a discrete set. Let 𝑥(𝑥,𝑦)=11,𝑥12,,𝑥1𝑇,𝑥21,𝑥22,,𝑥2𝑇,,𝑥𝑊1,𝑥𝑊2,,𝑥𝑊𝑇,𝑦1,,𝑦𝑇.(3.3) Then for (𝑥,𝑦), we have 𝑊𝑖=1ln1𝑃𝑖𝑗𝑥𝑖𝑗=ln1𝑦𝑗,𝑗=1,2,,𝑇.(3.4) WTA problem can be transformed as follows:max𝑇𝑗=1𝑢𝑗𝑦𝑗,s.t.(𝑥,𝑦)(𝑥,𝑦)(𝑥,𝑦)satisfy(2.2),(2.3),(2.4),and(3.4),𝑦𝑗𝐷𝑗,(𝑃) 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: (𝐿(𝜆))max𝐿(𝑥,𝑦,𝜆),s.t.(𝑥,𝑦)(𝑥,𝑦)(𝑥,𝑦)satisfy(2.2),(2.3),(2.4),and(3.4),𝑦𝑗𝐷𝑗,(3.5) where 𝐿(𝑥,𝑦,𝜆)=𝑇𝑗=1𝑢𝑗𝑦𝑗+𝑇𝑗=1𝜆𝑗𝑊𝑖=1ln1𝑃𝑖𝑗𝑥𝑖𝑗ln1𝑦𝑗=𝑇𝑗=1𝑢𝑗𝑦𝑗𝜆𝑗ln1𝑦𝑗+𝑇𝑊𝑗=1𝑖=1𝜆𝑗ln1𝑃𝑖𝑗𝑥𝑖𝑗,(3.6) where 𝜆=(𝜆1,𝜆2,,𝜆𝑇) is Lagrange multiplier, 𝜆𝑖𝑅,𝑖=1,2,,𝑇, 𝐿(𝑥,𝑦,𝜆) is Lagrange function. It is obvious that (𝑃) can be decomposed into two subproblems as follows: 𝐿1(𝑥)max𝑇𝑊𝑗=1𝑖=1𝜆𝑗ln1𝑃𝑖𝑗𝑥𝑖𝑗,𝐿s.t.𝑥{𝑥forxed𝑦,𝑥satisfy(2.2),(2.3),and(2.4)},(3.7)2(𝑦)max𝑇𝑗=1𝑢𝑗𝑦𝑗𝜆𝑗ln1𝑦𝑗,𝑦s.t.𝑦=1,𝑦2,,𝑦𝑇𝑦𝑦𝑗𝐷𝑗.,𝑗=1,2,,𝑇(3.8)

Theorem 3.1. Let 𝑥,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. Since 𝑥,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 𝑦𝑗=1𝑊𝑖=11𝑃𝑖𝑗𝑥𝑖𝑗.(3.9) Then (𝑥,𝑦) also satisfy (3.4) and we have 𝑇𝑗=1𝑢𝑗1𝑊𝑖=11𝑃𝑖𝑗𝑥𝑖𝑗=𝑇𝑗=1𝑢𝑗𝑦𝑗=𝑇𝑗=1𝑢𝑗𝑦𝑗+𝑇𝑗=1𝜆𝑗𝑊𝑖=1ln1𝑃𝑖𝑗𝑥𝑖𝑗ln1𝑦𝑗=𝑇𝑗=1𝑢𝑗𝑦𝑗𝜆𝑗ln1𝑦𝑗+𝑇𝑗=1𝜆𝑗𝑊𝑖=1ln1𝑃𝑖𝑗𝑥𝑖𝑗𝑇𝑗=1𝑢𝑗𝑦𝑗𝜆𝑗ln1𝑦𝑗+𝑇𝑗=1𝜆𝑗𝑊𝑖=1ln1𝑃𝑖𝑗𝑥𝑖𝑗=𝑇𝑗=1𝑢𝑗𝑦𝑗+𝑇𝑗=1𝜆𝑗𝑊𝑖=1ln1𝑃𝑖𝑗𝑥𝑖𝑗ln1𝑦𝑗=𝑇𝑗=1𝑢𝑗𝑦𝑗=𝑇𝑗=1𝑢𝑗1𝑊𝑖=11𝑃𝑖𝑗𝑥𝑖𝑗.(3.10) Hence, 𝑥 is optimal solution of WTA.

Obviously, subproblem 𝐿2(𝑦) can be decomposed into 𝑇 1-dimensional discrete problem as follows (𝑗=1,2,,𝑇):𝐿2𝑢(𝑦)max𝑗𝑦𝑗𝜆𝑗ln1𝑦𝑗,s.t.𝑦𝑗𝑦𝑗𝑦𝑗𝐷𝑗.(3.11)

Theorem 3.2. For 𝑗=1,2,,𝑇, let 𝛼𝑗=11min1𝑖𝑊𝑃𝑖𝑗𝑇𝑗,𝛽𝑗=11max1𝑖𝑊𝑃𝑖𝑗𝑊𝑖=1𝑤𝑖,(3.12) one has 𝛼𝑗𝑦𝑗𝛽𝑗.

Proof. For 𝑗=1,2,,𝑇, we have 𝑦𝑗=1𝑊𝑖=11𝑃𝑖𝑗𝑥𝑖𝑗1𝑊𝑖=11max1𝑖𝑊𝑃𝑖𝑗𝑥𝑖𝑗=11max1𝑖𝑊𝑃𝑖𝑗𝑊𝑖=1𝑥𝑖𝑗11max1𝑖𝑊𝑃𝑖𝑗𝑊𝑖=1𝑤𝑖=𝛽𝑗.(3.13) Note that 𝑦𝑗=1𝑊𝑖=11𝑃𝑖𝑗𝑥𝑖𝑗1𝑊𝑖=11min1𝑖𝑊𝑃𝑖𝑗𝑥𝑖𝑗=11min1𝑖𝑊𝑃𝑖𝑗𝑊𝑖=1𝑥𝑖𝑗11min1𝑖𝑊𝑃𝑖𝑗𝑇𝑗=𝛼𝑗.(3.14) Thus, 𝛼𝑗𝑦𝑗𝛽𝑗.
Consider the relaxation form of 1-dimensional discrete problem 𝐿2𝑗(𝑦𝑗) as follows: 𝑆𝐿2𝑗𝑦𝑗𝑢max,𝑗𝑦𝑗𝜆𝑗ln1𝑦𝑗,s.t.𝑦𝑗𝑦𝑗𝛼𝑗𝑦𝑗𝛽𝑗.(3.15) For given 𝑢𝑗 and 𝜆𝑗𝑆𝐿2𝑗(𝑦𝑗) is a convexity programming on a closed interval that can be solved easily, when 𝜆𝑗0, the optimal solution is 𝑦𝑠𝑗=𝛼𝑗𝜆,1+𝑗𝑢𝑗𝛼𝑗,𝜆1+𝑗𝑢𝑗𝜆,1+𝑗𝑢𝑗𝛼𝑗,𝛽𝑗,𝛽𝑗𝜆,1+𝑗𝑢𝑗𝛽𝑗,(3.16) when 𝜆𝑗=0, the optimal solution is 𝑦𝑠𝑗=𝛽𝑗.(3.17)

Theorem 3.3. Let 𝑦1𝑗, 𝑦2𝑗𝐷𝑗, 𝑦1𝑗=max{𝑦𝑗𝑦𝑗𝑦𝑠𝑗,𝑦𝑗𝐷𝑗}, 𝑦2𝑗=min{𝑦𝑗𝑦𝑗𝑦𝑠𝑗,𝑦𝑗𝐷𝑗}, then one has 𝑦𝑗𝑢=argmax𝑗𝑦1𝑗𝜆𝑗ln1𝑦1𝑗,𝑢𝑗𝑦2𝑗𝜆𝑗ln1𝑦2𝑗,(3.18) which is the optimal solution of 1-dimensional discrete problem 𝐿2𝑗(𝑦𝑗).
Since 𝑆𝐿2𝑗(𝑦𝑗) is a convexity programming on a closed interval, Theorem 3.3 is obvious.

Next, we prove that the subproblem 𝐿1(𝑥,𝑦,𝜆) can be solved as a linear programming.

Definition 1 (see [13]). An 𝑚×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, then 𝑃(𝑏)={𝑥𝑅𝑛+𝐴𝑥𝑏} is integral for all 𝑏𝑍𝑚 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 programming 𝐿1(𝑥) can be obtained by solving 𝐿1(𝑥) as a linear programming using simplex method.

Proof. Integer linear programming 𝐿1(𝑥) can be described as follows: max𝑇𝑊𝑗=1𝑖=1𝜆jln1𝑃𝑖𝑗𝑥𝑖𝑗,s.t.𝑇𝑗=1𝑥𝑖𝑗𝑤𝑖𝑖=1,2,,𝑊,𝑊𝑖=1𝑥𝑖𝑗𝑡𝑗𝑥,𝑗=1,2,,𝑇,𝑖𝑗0,integral,𝑖=1,2,,𝑊,𝑗=1,2,,𝑇.(3.19)

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 variable 𝑥𝑖𝑗 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 Theorem 3.1, for given Lagrange multiplier 𝜆𝑘=(𝜆𝑘1,𝜆𝑘2,,𝜆𝑘𝑇), 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 𝑔𝑗𝑥𝑘,𝑦𝑘=𝑊𝑖=1ln1𝑃𝑖𝑗𝑥𝑘𝑖𝑗ln1𝑦𝑘𝑗.(4.1) Update 𝜆 with 𝜆𝑗𝑘+1=1,𝑔𝑗𝑥𝑘𝜆>0,𝑘𝑗,𝑔𝑗𝑥𝑘=0,1,𝑔𝑗𝑥𝑘<0.(4.2)

The proposed Lagrange relaxation algorithm solving WTA problem can be summarized as follows.

Step 1. Set the maximum iterate number 𝐾, Lagrange multiplier 𝜆0=(𝜆01,𝜆02,,𝜆0𝑇),𝜆0𝑖=0(𝑖=1,2,,𝑇) and 𝑘=0, 𝑧=0, 𝑥=0.

Step 2. For fixed 𝜆𝑘, solve 𝐿1(𝑥) as linear programming using simplex method, obtain integral optimal solution 𝑥𝑘 and optimal value 𝑧𝑘 if 𝑧𝑘𝑧, then 𝑧=𝑧𝑘, 𝑥=𝑥𝑘.

Step 3. If 𝑘𝐾, stop.

Step 4. Solve 𝐿2(𝑦) and get the optimal solution 𝑦𝑘. Compute 𝑔𝑗(𝑥𝑘,𝑦𝑘) by (4.1). If 𝑔𝑗𝑥𝑘,𝑦𝑘=0,𝑗=1,2,,𝑇(4.3) 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 Step 2. 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.

tab1
Table 1: Numerical results of Lagrange relaxation method.

From 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 solve 𝐿2(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.

Acknowledgment

This work supported by the National Nature Science Foundation of China (no. 70971136).

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