An Artificial Bee Colony Algorithm for Solving the Weapon Target ...

An Artificial Bee Colony Algorithm for Solving the Weapon Target Assignment Problem Rafet Durgut

Hakan Kutucu

Sedat Akleylek

Karabuk University Karabuk University Ondokuz Mayıs University Department of Computer Engineering Department of Computer Engineering Department of Computer Engineering Karabuk, Turkey Karabuk, Turkey Samsun, Turkey +90 370 4333327 +90 370 4333327 +90 362 3121919

[email protected]

[email protected] [email protected]

ABSTRACT In this paper, we deal with the static weapon target assignment (WTA) problem which is a hard combinatorial optimization problem having some industrial applications. The aim of the WTA problem is to find an assignment of weapons to targets with the minimum total survival value of the targets. The WTA problem is known to be NP-complete problem. In this paper, we propose a novel artificial bee algorithm to give an efficient solution to the WTA problem. We test the proposed algorithm with benchmark problem instances and compare it with some other meta-heuristics in the literature. Computational tests show that our algorithm is competitive.

CCS Concepts • Mathematics of computing → Combinatorial optimization • Theory of computation → Optimization with randomized search heuristics

Keywords Weapon target assignment; artificial bee colony; design of algorithms

1. INTRODUCTION The weapon target assignment (WTA) problem, a special case of resource allocation problem, is an interesting combinatorial optimization problem which has some real-world applications. The problem deals with assigning weapons to targets properly to minimize (or optimize) total expected survival value of targets [2]. In other words, maximizing damage to the enemy through effective weapon target assignments. The WTA problems have two versions; static WTA and dynamic WTA. In the static WTA problem, all the inputs to the problem are fixed and the weapons are assigned to targets once. The dynamic version consists of many stages of assignment where the state of the system after each exchange of stage is considered in the next stage. The interested reader can refer to [4] and [5] for more information about the dynamic WTA problem. In this paper, we study the static WTA problem. For convenience, we will simply refer the static WTA problem to as the WTA problem.

Lloyd and Witsenhausen proved the NP-Completeness of the WTA problem by providing a natural reduction from 3-EXACTCOVER problem [3]. In the last decade, the WTA problem has attracted a lot of interest. In [6], Ma and Ni studied linear integer programming model of the WTA problem. In [7], Cetin and Esen gave an application of the WTA problem for the media industry. The proposed model is an integer nonlinear programming problem. In [8], Johansson and Falkman improved a variant of particle swarm optimization seeded with an enhanced greedy algorithm for the WTA problem. Their experiments shown that optimal or close to optimal solutions are obtained in real-time by the genetic algorithms and the particle swarm optimization algorithms on small-scale problems. In [9], Senay constructed different mathematical models for the needs of air-to-ground missiles using the WTA problem. In [1], Turan provided a comparison of several heuristics including search algorithms, maximum marginal return algorithms, evolutionary algorithms and bipartite graph matching algorithms for the WTA problem and posed a new hybrid algorithm consisting of particle swarm and random search to solve the problem efficiently. In [10], Ahner and Parson exploited the structure of the assignment problem using subgradient information to optimally solve a two-stage dynamic WTA problem. In [11], Li et al. designed a biobjective WTA optimization model which maximizes the expected damage of the enemy and minimizes the cost of missiles. They used a modified Pareto ant colony optimization algorithm to solve the problem. In [12], Madni and Andrecut presented two new heuristic algorithms based on Threshold Accepting and Simulated Annealing methods to solve the WTA problem. The paper is organized as follows: In Section 2, we first present a mathematical formulation of the WTA as a nonlinear integer programming problem. We give a review of the general artificial bee colony algorithm in Section 3. In Section 4, we present some computational results for the problem and finally, give some concluding remarks in Section 5.

2. MATHEMATICAL MODEL OF THE WTA PROBLEM Let n and m be the number of targets and weapon types, respectively. The other parameters and variables are as follows:

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Vj

:

the value of the target j,

Wi

:

the number of weapons of type i available,

pij

:

the destroying probability of target j by a single weapon of type i,

qij

:

the survivable probability of target j if a single weapon of type i is assigned to it, that is, qij=1-pij,

DOI: http://dx.doi.org/10.1145/12345.67890

xij

:

the number of weapons of type i to be assigned to target j.

The WTA problem consists of finding the number of weapons xij to minimize the total expected survival value of all targets. This problem can be formulated as the following nonlinear integer programming problem: 𝑛

𝑚

𝑀𝑖𝑛 𝑓(𝑥) = ∑ 𝑉𝑗 (∏ 𝑞𝑖𝑗 𝑥𝑖𝑗 ) 𝑗=1

(1)

𝑖=1

subject to n

x j 1

xij  0

ij

 Wi ,

and integer

 i  1,..., m

(2)

 i  1,..., m and  j  1,..., n (3)

3. ARTIFICIAL BEE COLONY ALGORITHM In this section, we recall the artificial bee colony (ABC) algorithm which is one of the most recent swarm intelligence algorithms proposed by Karaboga in 2005 that is inspired by the behavior of honey bees [13]. In the ABC algorithm, the colony consists of three types of honey bees namely employed, onlooker and scout bees. The foods positions representing individuals (solutions) are modified by the artificial bees with time and the aim of the bee is to discover the places of food sources with the highest nectar which corresponds to an optimal solution. The ABC algorithm puts together local search methods carried out by employed and onlooker bees with global search methods guided by onlooker and scout bees. This method has been applied to a huge number of not only combinatorial but numerical optimization problems. However, to the best of our knowledge, the ABC algorithm has not been used to solve the WTA problem. Figure 1 shows a general procedure of the ABC algorithm.

4. COMPUTATIONAL TESTS The ABC algorithm has some control parameters: swarm size, number of employed bees, number of onlookers, number of scouts, limit and maximum number of cycles. In the tests, the colony size is chosen as 50. The number of employed bees is taken equal to the number of onlooker bees and the number of scout bees is taken as one in the colony, i.e., 𝑘=25. The increase in the number of scouts encourages the exploration as the increase of onlookers on a food source increases the exploitation [14]. The value of limit is 1000 and the maximum number of cycles is 2.105. Our following algorithm have been implemented in Matlab

version R2014a and tested on i7-3930K machine with a 3.2 GHz processor and 32GB RAM. The test problems are taken from [15]. We perform 10 independent runs for each instance to get reliable statistical results. The worst, best result and average value are provided in Table 1. In all phases of ABC, the neighborhood operator is applied using the swap operator illustrated in Figure 2.

Figure 2. The neighborhood operators. 1. Randomly generate a set of solutions as initial food sources 𝑥𝑖 , 𝑖 = 1, … , 𝑘. Assign each employed bee to a food source; 2. Evaluate the fitness value 𝑓(𝑥𝑖 ) of each of the food sources 𝑥𝑖 using the formula in (1); 3. Set cycle=0 and 𝑙1 = 𝑙2 = ⋯ = 𝑙𝑘 = 0; 4. while (cycle < MaxCycles) a. for each food source 𝑥𝑖 i. Apply a neighborhood operator on 𝑥𝑖 → 𝑦; ii. if 𝑓(𝑦) > 𝑓(𝑥𝑖 ) then 𝑥𝑖 = 𝑦 and 𝑙𝑖 = 0; else 𝑙𝑖 = 𝑙𝑖 + 1; b. for each onlooker i. Select a food source 𝑥𝑖 using roulette wheel selection; ii. Apply a neighborhood operator on 𝑥𝑖 → 𝑦; iii. if 𝑓(𝑦) > 𝑓(𝑥𝑖 ) then 𝑥𝑖 = 𝑦 and 𝑙𝑖 = 0; else 𝑙𝑖 = 𝑙𝑖 + 1; c. for each food source 𝑥𝑖 if 𝑙𝑖 = 𝑙𝑖𝑚𝑖𝑡 then replace 𝑥𝑖 with a randomly generated solution and 𝑙𝑖 = 0; cycle=cycle+1; Figure 1. The flowchart of the ABC algorithm.

In Table 1, we provide a comparison between our proposed algorithm and a simulated algorithm (SA) improved by Sonuc et al on the results obtained with the 12 instances [15]. Sonuc et al. proposed a parallel SA algorithm for solving the WTA problem. They also present the results using a serial SA algorithm. In order to be a fair comparison, we use the results obtained by the serial SA algorithm. It is easy to see in Table 1 that our proposed ABC algorithm performs better than the SA algorithm. The SA algorithm and our proposed ABC found the same best values for WTA1, WTA2, WTA3 and WTA4. This is probably that these

values are the optimum values for the instances. Our ABC is faster than the SA for all instances. In Table 2, we also provide a comparison between all neighborhood operations showed in Figure 2. As it can be seen swap operator gives better results in terms of not only values of objective functions, but also CPU time. Swap operator is quite simple and effective. Insertion and Inversion operators require more processing and so computation time.

Table 1. Comparison of computational tests between the ABC algorithm and the SA algorithm. The Simulated Annealing of Sonuc et al.

The proposed ABC Algorithm

Problem

Weapon

Target

Best

Time (sec)

Best

Worst

Average

Time (sec)

WTA1

5

5

48.3640

2986

48.3640

48.3640

48.3640

390

WTA2

10

10

96.3123

2841

96.3123

96.3123

96.3123

417

WTA3

20

20

142.1070

2753

142.1070

142.8119

142.2480

473

WTA4

30

30

248.0285

2754

248.0285

249.2224

248.6854

532

WTA5

40

40

305.5016

2761

305.8729

307.4944

306.8570

585

WTA6

50

50

353.0767

2790

353.3794

356.8539

355.1488

654

WTA7

60

60

415.0528

2788

414.4555

420.1622

417.0145

712

WTA8

70

70

498.1049

2841

498.0948

504.3466

500.5102

786

WTA9

80

80

534.4408

2869

534.4742

541.8093

536.8911

831

WTA10

90

90

594.0639

2813

592.9167

598.3802

594.9403

889

WTA11

100

100

699.8357

2806

698.4465

707.7392

701.4467

954

WTA12

200

200

1306.9126

2902

1295.3142

1303.1223

1299.2044

1624

Table 2. Comparison of different neighborhood operations. Insertion

Swap

Inversion

Problem

Best

Worst

Time

Best

Worst

Time

Best

Worst

Time

WTA1

48.3640

48.3640

390

48.364

48.364

2847

48.364

48.364

3088

WTA2

96.3123

96.3123

417

96.3123

102.6295

3114

96.3123

106.3034

3347

WTA3

142.1070

142.8119

473

156.9845

183.519

3018

148.0404

160.1445

3262

WTA4

248.0285

249.2224

532

291.8235

326.2699

3120

267.1608

301.1441

3362

WTA5

305.8729

307.4944

585

376.3004

416.9571

3241

331.5172

367.0581

3501

WTA6

353.3794

356.8539

654

442.3354

501.582

3351

397.0861

428.7602

3604

WTA7

414.4555

420.1622

712

532.1615

584.1735

3427

468.9967

503.5623

3655

WTA8

498.0948

504.3466

786

653.3159

713.885

3534

572.2842

613.1252

3807

WTA9

534.4742

541.8093

831

712.7822

760.1409

3703

631.5822

665.986

4034

WTA10

592.9167

598.3802

889

767.929

841.2731

3947

694.3204

771.1402

4209

WTA11

698.4465

707.7392

954

937.9659

973.9638

4114

831.6605

895.5162

4402

WTA12

1295.3142

1303.1223

1624

1804.3493

1949.1664

5742

1668.3907

1759.2386

5745

5. CONCLUSION In this paper, we present an ABC algorithm for solving the WTA problem. The twelve benchmark instances generated in [15] are tested. We compare the results with the previous results. The computational experiments show that the proposed ABC algorithm is competitive, in general and the performance of the algorithm is efficient in terms of CPU time. Moreover, our algorithm finds better results for 5 instances. We also compared the effectiveness of the neighborhood operators named swap, inversion and insertion. We observed that swap operator is better in the ABC algorithm for the WTA problem. As a future work, it might be interesting to test the behavior of the ABC algorithm with the other neighborhood operators such as 3-Opt and 𝜆-interchange.

6. ACKNOWLEDGEMENT

[6] Ma, F., and Ni, M. 2015. An optimal assignment of multitype weapons to single-target, 2015 IEEE Advanced Information Technology Electronic and Automation Control Conference (IAEAC 2015). IEEE, 1-4. [7] Çetin, E. and Esen, S. T. 2006. A weapon-target assignment approach to media allocation", Applied Mathematics and Computation, 175(2). 1266-1275. [8] Johansson, F., and Falkman, G. 2011. Real-time allocation of firing units to hostile targets, J. Adv. Inf. Fusion, 6(2), 187199. [9] Senay, N. 2012. The strategic level optimization of air to ground missiles for turkish air force decision support system, Air Force Institute of Technology, 7. [10] Ahner, D. K., and Parson, C. R. 2015. Optimal multi-stage allocation of weapons to targets using adaptive dynamic programming Optimization Letters, 9(8) 1689–1701.

This work was supported by Research Fund of the Karabuk University. Project Number: KBÜBAP-17-YD-327

[11] Li, Y., Kou, Y., Li, Z., Xu A., and Chang, Y. 2017. A Modified Pareto Ant Colony Optimization Approach to Solve Biobjective Weapon-Target Assignment Problem, International Journal of Aerospace Engineering, 2017, 14.

7. REFERENCES

[12] Madni, A. M, and Andrecut. M. 2009. Efficient Heuristic Approaches to the Weapon–Target Assignment Problem. Journal of Aerospace Computing, Information, And Communication (6), 405–414.

[1] Turan A. 2012. Algorithms for the weapon-target allocation problem. Master Thesis. [2] Ahuja, R., Kumar, A., Jha K., and Orlin J. 2007. Exact and Heuristic Algorithms for the Weapon-Target Assignment Problem. Operations Research 55(6), 1136–1146.

[13] Karaboga, D. 2005. An idea based on honey bee swarm for numerical optimization, Tech. Rep. TR06, Erciyes University, Engineering Faculty, Computer Engineering Department.

[3] Lloyd, S. P., and Witsenhausen H. S. 1986. Weapon Allocation is NP-Complete [C]. Proc. 1986 Summer Compute. Simulation Conference.

[14] Karaboga D., and Basturk, B, 2008. On The Performance Of Artificial Bee Colony (ABC) Algorithm, Applied Soft Computing, 8(1), 687-697.

[4] Hosein, P., and Athans, M. 1990. Some analytical results for the dynamic weapon-target allocation problem, Tech. Rep., MIT, Laboratory for Information and Decision Systems,

[15] Sonuc E., Sen B., and Bayir S. 2017. A Parallel Simulated Annealing Algorithm For Weapon-Target Assignment Problem, International Journal of Advanced Computer Science and Applications (to appear in print).

[5] Cai, H., Liu, J., Chen, Y., and Wang, H. 2006. Survey of the research on dynamic weapon-target assignment problem, Journal of Systems Engineering and Electronics, 17(3), 559565.