Journal of Information & Computational Science 11:16 (2014) 5793–5800
Available at http://www.joics.com
November 1, 2014
Modeling and Optimization of Multiple Launchers and
Weapons Fire Distribution Based on
Lower Damage Bound
Xiaoliang He ∗, Yiming Bi, Zhenxin He
The Xi’an Research Institute of High Technology, Xi’an 710025, China
Abstract
According to the characteristics of multiple launchers and weapons ﬁre distribution, the model of multiple
launchers and weapons ﬁre distribution is established, restricted by lower damage bound. Due to Genetic
Algorithm (GA) is good at search capability macroscopically, and Simulated Annealing (SA) algorithm
is good at seeking the best result microcosmically. Through combining their advantages, a new algorithm
is designed and used to solve multiple launchers and weapons ﬁre distribution. Simulation results show
that this method could gain the best solution rather quickly, and the ﬁre distribution results ﬁt with the
operational requirement well, verify that the model is useful and the algorithm is feasible.
Keywords: Fire Distribution; Simulated Annealing; Genetic Algorithm; Multiple Launchers and
Weapons
1
Introduction
In modern warfare, with the integrated development of weapon system, a single launcher can
support multiple types of weapon, which not only save resources, but also improve operational
eﬃciency. In carrying out tasks, the type of weapon is selected to match the target in accordance
with the needs of battleﬁeld, in order to achieve the objective of operation. Moreover the ﬁre
distribution is an important part of decision-making [1], when multiple launchers and weapons
are going to attack the targets; it is mainly that the limited number of weapons and ammunition
are assigned to the relevant targets to maximize operational eﬀectiveness, by using a reasonable
method of combinatorial optimization.
In the ﬁeld of ﬁre distribution, there is an extensive research literature by domestic and foreign
scholars, who mainly focus on model and algorithm. In the study of model, the research results
are mostly concentrated in the static model, and the essence of these models is the goal that a
single type of weapon maximizes damage to the targets [2]-[5]. However, in the case of multiple
launchers and weapons, through combinatorial optimization of them to achieve the best damage,
∗
Corresponding author.
Email address: [email protected] (Xiaoliang He).
1548–7741 / Copyright © 2014 Binary Information Press
DOI: 10.12733/jics20104857
5794
X. He et al. / Journal of Information & Computational Science 11:16 (2014) 5793–5800
the literature which focuses on this kind of problem is relatively less. In the study of algorithm,
with the development of artiﬁcial intelligence technology, previous classical algorithm has shown
insuﬃcient [6]-[9]. Intelligent algorithm brings new ideas for solving. Modeling multiple launchers
and weapons ﬁre distribution and developing accurate and eﬃcient intelligent algorithm, are the
key point and diﬃculty in research [10]-[12]. Literature [14] about maximization of the beneﬁtcost ratio established ﬁre distribution model, but the mix of ﬁre units was not considered in the
solution process, the practicability of model still need further validation. Literature [15] used
Lingo to solve ﬁre distribution problem; for the data was large in scale and variety, it existed the
problem which was operated inconveniently and diﬃcult to be mastered.
In order to obtain optimal operational eﬀectiveness as a starting point, this paper comprehensively considers the matching relationship between launcher, weapon and target. The ﬁre
distribution model is established by the restriction of lower damage bound. In addition, against
the large scale of data in the process of solving, GA is good at search capability macroscopically,
while SA algorithm is good at seeking the best result microcosmically; a hybrid genetic simulated
annealing algorithm combining both advantages is designed for the simulation calculation model;
the solving speed and accuracy is also improved.
2
Modeling Multiple Launchers and Weapons Fire Distribution Based on Lower Damage Bound
In order to discuss the problem conveniently, the concept of lower damage bound is introduced.
Within a speciﬁc operational time, one target is attacked by a certain number of weapons. If the
damage value is less than β ∈[0, 1], the target can quickly restore ﬁghting capacity; if more than,
the ﬁghting capacity cannot be restored in the special operational time. The interval of [β, 1] is
called for the eﬀective damage interval, β is the eﬀective lower damage bound.
Let the number of launchers is a, the number of weapon’s types supported by the ith launcher
is mi , i=1, 2, · · ·, a. There are n targets to attack, the threat coeﬃcient of corresponding target
is wk , k=1, 2, · · ·, n. From the j th weapon attached to the ith launcher, Single-shot damage
probability is pijk (j=1, 2, · · ·, mi ) to the k th target.
Draw into the decision matrix of ﬁre distribution

x11
x12
· · · x1n

 x21 x22 · · · x2n
X=

···

xm1 xm2 · · · xmn






Among them, xhk represents the∑
ﬁre units to attack the k th target by using the j th weapon
attached to the i th launcher; m =
mi , h=1, 2, · · ·, m. Then, multiple launchers and weapons
ﬁre distribution model can be described as:
Look for a set of solutions X, which satisfy the following objective function and constraint
conditions.
X. He et al. / Journal of Information & Computational Science 11:16 (2014) 5793–5800
Objective function
F (X) =
n
∑
[
wk 1 −
k=1
m
∏
5795
]
(1 − pijk )xhk
(1)
h=1
Constraint conditions
1−
xhk ∈ Z
(2)
0 ≤ pijk ≤ 1
(3)
m
∏
(1 − pijk )xhk ≥ βk
(4)
h=1
Among them, the formula (2) indicates that weapons’ damage on target meets the requirements
of the lower damage bound. The formula (3) shows the range of the damage probability on the
target by weapon. The formula (4) shows the value of the decision variables from the set of
integer Z.
As can be seen, multiple launchers and weapons ﬁre distribution is a combinatorial optimization
problem. If the amount of launchers and weapons is less, the problem can be solved by some simple
mathematical programming methods; but with the scale of solving larger, the classical algorithm
cannot give satisfactory results faster, then you can develop eﬃcient intelligent algorithm to solve
the problem.
3
Implementation Algorithm
GA and SA algorithm are excellent computational methods of modern intelligence. GA is an
intelligent heuristic algorithm with inherent parallelism and better global optimization capability,
has been widely applied in the ﬁeld of combinatorial optimization, but there is also easy to
converge to a local optimum, more time consuming, poor stability. The calculation process of
SA algorithm is simple, better robust, excellent ability to choose better; it ﬁts complex nonlinear
optimization problems, meanwhile there are also shortcomings, such as strongly dependent on
the original values, sensitive parameters and poor capability of global search for optimal solution.
Accordingly, as can be seen that both are highly complementary. Firstly, SA algorithm enables
GA more targeted to choose search space, to avoid premature local optimum. Secondly, in the
crossover and mutation steps, GA provides new solutions for SA algorithm, enhancing the overall
climbing characteristic. Finally, the solutions of SA algorithm provide new population for GA
selectively; it improves the convergence rate of GA, so that the hybrid algorithm obtains the
global optimal solution within a short time.
This paper reasonably embeds SA algorithm in GA, and put forward a new global optimization
algorithm, called hybrid genetic simulated annealing (SAGA) algorithm. The SAGA algorithm
plays an important role in the process of crossover and mutation for generating new populations,
resolves the diﬃculties of GA parameter selection, improves convergence, and makes the algorithm
more advantages. The steps of SAGA algorithm are shown in Fig. 1.
In Fig. 1, Tmin is the terminal condition of temperature falling in the process of simulated
annealing.
(1) Coding
5796
X. He et al. / Journal of Information & Computational Science 11:16 (2014) 5793–5800
Initialize
Code: generate initial population
Calculate individual fitness
Evolution generation
increased by 1
Adjust annealing
temperature
Select, Cross, Mutate
Simulated annealing control search
N
T ≥Tmin
Y
Best individual fitness
Export
Fig. 1: SAGA algorithm ﬂow chart
Obviously, multiple launchers and weapons ﬁre distribution is an integer programming problem. Here is the decimal coding, more intuitive performance, shorter coding length, simple and
practical. Chromosome coding can be expressed as [x1 , x2 , · · ·, xt , · · ·, xm∗n ]. xt is a mathematical description of ﬁre units used the hth weapon against the k th target. The logical relationship
between h, k and t can be determined by formula (5); h and k are respectively the number of
weapon’s types and targets.
{
h = ceil(t/n), h = 1, 2, · · · , m
If rem(t, n) = 0, k = n
(5)
k = rem(t, n), k = 1, 2, · · · , n
Among them, ceil and rem are respectively the functions in Matlab for integer and remainder.
(2) Population initialization
Due to the large scale of weapons and targets, the initial solution of population is caused
some diﬃculties. In order to quicken the convergence speed and increase the ability of searching
optimization, the initial solution of population adopts the following methods:
Step 1 Generate m*n null matrix. Through the function X =zeros(m, n), generate m*n null
matrix; m and n are respectively the number of weapon’s types and targets.
Step 2 Put one number from randperm(µi ) into the random column of the i th row of matrix
X . µi is the maximum number of ﬁre units of the ith weapon, the random column of the ith row
is found by the function randperm(n), n is the number of targets.
Step 3 Verify the updated matrix X from Step 2 whether meets the lower damage bound
to target. If it meets, the matrix X can be used as the initial solution; otherwise, go to Step
1. Calculate the damage probability of each column’s elements to the corresponding target, to
obtain the sum of damage probability of each column, and to determine whether the sum satisﬁes
the lower damage bound.
Step 4 Have on Steps 1, 2, 3 iteratively until meeting the requirement of initial population.
X. He et al. / Journal of Information & Computational Science 11:16 (2014) 5793–5800
5797
By the formula (5), the element of matrix X can correspond to the element’s position on the
chromosome.
(3) Fitness
Fitness is used to calculate the value of each chromosome, judge its quality, to be tradeoﬀs. In the early GA running, due to the large diﬀerence in the ﬁtness of chromosomes, when
using roulette selection, the possibility of oﬀspring produced is proportional to the size of parent
chromosome ﬁtness; So in the early, the entire population is prone to produce more individual
oﬀspring, early to fall into local optimal value. In the late, the ﬁtness of chromosomes tends to
consistency; when the good chromosomes produce oﬀspring, there are not obvious advantages,
so as to the stagnation phenomenon of the entire population exists. To solve these problems,
combining with multiple launchers and weapons ﬁre distribution, the ﬁtness stretching method
is used as follows:
exp(fi /T )
Fi = M
(6)
∑
exp(fi /T )
i=1
Among them, fi is the ﬁtness of the ith chromosome, the objective function can be directly used
as the chromosome ﬁtness function. M is the size of population. T is the current temperature.
The formula (6) is served as the new ﬁtness function in the algorithm. So early in the GA (high
temperature), the probabilities of producing oﬀspring are similar, while the ﬁtness of chromosomes is similar. With the temperature falling, stretching eﬀect highlights, the diﬀerences among
the similar ﬁtness of chromosomes can be ampliﬁed, so that make the advantages of excellent
chromosomes more obvious.
(4) Selection, crossover and mutation
At the selection, adopt the method of roulette, keep chromosomes which have large occurrence
probability of ﬁtness.
At the crossover, randomly select parent chromosomes needed to be crossed, use multi-point
crossover way to select individuals, for oﬀspring chromosomes, to ensure the diversity of population. If appear the same weapon to two targets at the same time, this kind of crossover result is
illegal. At this point, adopt Zeros method to eliminate illegal distribution, that is 0 in place of
xt , make the distribution legalized.
At the mutation, randomly ﬁnd multiple non-zero points on the chromosome, make them
randomly variation within the largest amount of ﬁre units of the corresponding weapon; it can
jump out of local search, accurately determine the optimal solution.
(5) Simulated annealing control search
SA algorithm has strong ability of local optimum, which can eﬀectively avoid the GA into local
extremum, eventually reach a global optimization. In order to ensure the organic combination of
GA and SA algorithm, to complement each other, during the crossover and mutation operations,
this paper uses simulated annealing to control the search. The speciﬁc steps are as follows:
Step 1 Crossover and simulated annealing. According to certain conditions, Genes a1 and a2
generate oﬀspring c1 and c2 ; calculate the ﬁtness f (ai ) and f (ci ), i=1, 2. If f (ci )≤f (ai ), ci takes
the place of ai ; if not, accept ci with the probability of exp(- (f (ci ) - f (ai ))/T ).
Step 2 Mutation and simulated annealing. The same way is with Step 1.
Step 3 Temperature falling. T = T0 × α, α is a constant in [0, 1].
5798
X. He et al. / Journal of Information & Computational Science 11:16 (2014) 5793–5800
Step 4 Judge termination condition of algorithm. T ≥ Tmin , if meet the termination condition,
output the optimal solution, the algorithm is over; otherwise, turn to Step 1.
4
Simulation Application
To test the performance of SAGA algorithm to deal with multiple launchers and weapons ﬁre
distribution problem, the following scenario data is introduced. Let there are three launchers
L1 -L3 , the number of weapon’s types each launcher supports and ﬁre units are shown in Table
1. There are ten main targets T1 -T10 , the damage probability of weapon to target and threat
coeﬃcient of target are shown in Table 2.
Table 1: The scale of launchers and weapons
Launcher
Types of weapon
Fire units
L1
2
[3 3]
L2
3
[2 4 2]
L3
3
[4 5 2]
Table 2: The damage probability and threat degree of target
Targets
T1
T2
T3
T4
T5
T6
T7
T8
T9
T10
T hreat
0.25
0.06
0.10
0.03
0.13
0.08
0.01
0.18
0.07
0.09
ML1−1
0.20
0.33
0.25
0.31
0.23
0.27
0.38
0.23
0.35
0.31
ML1−2
0.32
0.40
0.31
0.40
0.29
0.35
0.49
0.27
0.48
0.43
ML2−1
0.26
0.35
0.27
0.32
0.33
0.40
0.52
0.26
0.42
0.40
ML2−2
0.35
0.47
0.34
0.45
0.40
0.52
0.64
0.38
0.53
0.54
ML2−3
0.44
0.69
0.52
0.56
0.50
0.66
0.70
0.42
0.64
0.65
ML3−1
0.33
0.65
0.43
0.54
0.45
0.50
0.66
0.40
0.58
0.51
ML3−2
0.40
0.71
0.57
0.69
0.51
0.60
0.72
0.51
0.66
0.62
ML3−3
0.56
0.80
0.68
0.77
0.62
0.73
0.85
0.63
0.82
0.71
In Table 2, ML1−1 -ML3−3 are respectively corresponding to the types of weapon supported by
L1 -L3 . Before the simulation, it is still to set some parameters; give the lower damage bound
all as β=0.8; set GA parameters, give the size of initial population N =100, crossover probability
pc =0.8, mutation probability pm =0.1; set SA algorithm parameters, give the initial temperature
T0 =100, temperature falling coeﬃcient α=0.99, terminal condition Tmin =5. This paper adopts
Matlab to simulate and calculate, the number of iterations is 300, and the results are shown
in Fig. 2 and Table 3. As can be seen from Fig. 2, SAGA algorithm has better convergence
performance, can quickly ﬁnd the optimal solution; the objective function value converges to
0.81, that is the ﬁre distribution may destroy the enemy’s target threat by 0.81. As can be seen
from Table 3, the launchers concentrate ﬁre units on the target with greater threat, and combine
diﬀerent launchers’ weapons to strike on the same target, to meet the lower damage bound for
maximum damage. The calculation results show the ﬁre distribution plan not only meets the
operational requirement, but also largely destroys the threat of enemy targets, can provide a
viable reference for relevant operational units.
X. He et al. / Journal of Information & Computational Science 11:16 (2014) 5793–5800
5799
Fitness curve termination=300
0.9
0.8
Fitness
0.7
0.6
0.5
0.4
SAGA
GA
SA
0.3
0.2
0
50
100
150
200
Iteration
250
300
350
Fig. 2: Algorithm performance comparison
Table 3: The ﬁre distribution result
Damages targets
Launchers
W eapons
T2
T3
T5
T8
T10
L2 +L3
L1 +L2
L3
L2
L3
L1
ML1−3 : 2+ML3−2 : 3 ML1−1 : 3+ML2−1 : 2 ML3−1 : 3 ML2−2 : 4 ML3−3 : 2 ML1−2 : 3
Damageprobability
5
T1
0.93
0.92
0.89
0.87
0.86
0.81
Conclusion
The problem of multiple launchers and weapons ﬁre distribution is a complex system engineering,
needs to consider many factors, involves broad areas. According to this characteristic, this paper
establishes the model of multiple launchers and weapons based on SAGA algorithm; on the basis of
the previous maximum damage to target, the model adds in the factor of lower damage bound, and
put forward SAGA algorithm to solve the problem. Algorithm eﬀectively combines the intelligent
algorithm and simulated annealing algorithm, has rapid convergence and excellent optimization
ability. The simulation results show that the built model is ingenious about combining operational
mission with eﬀective damage to target, reﬂects the need of real battleﬁeld, high credibility.
Development of the algorithm is simple and reasonable, easy to realize the global optimization,
easy to promote.
References
[1]
Z. X. Wang, L. Chen, Implementation of three shooting modes for distributed ﬁre control system
of antiaircraft gun [J], Acta Armamentarii, 32(7), 2011, 795-800
[2]
H. Cai, J. Liu, Y. Chen et al., Survey of the research on dynamic weapon-target assignment
problem [J], Journal of Systems Engineering and Electronics, 17(3), 2006, 559-565
[3]
M. Jay, H. Su, H. Wang et al., The generalized weapon target assignment problem [C], Proc. of
5800
X. He et al. / Journal of Information & Computational Science 11:16 (2014) 5793–5800
the 10th International Command and Control Research and Technology Symposium on the Future
of C2, 2005, 2-11
[4]
Y. Wang, L. Qian, Z. Guo et al., Weapon target assignment problem satisfying expected damage
probabilities based on ant colony algorithm [J], Journal of Systems Engineering and Electronics,
19(5), 2008, 939-944
[5]
H. D. Chen, S. Z. Wang, H. Y. Wang, Research of ﬁrepower assignment with multi-launcher
and multi-weapon based on a hybrid particle swarm optimization [J], Systems Engineering and
Electronics, 30(5), 2008, 880-883
[6]
H. Naeem, A. Masood, An optional dynamic threat evaluation and weapon scheduling technique
[J], Knowledge-Based Systems, 23(4), 2010, 337-342
[7]
B. Dalila, B. Belaid, D. Habiba, Scatter search and genetic algorithms for MAX-SAT problems
[J], Journal of Mathematical Modeling and Algorithms, 7(2), 2008, 101-124
[8]
S. Andrew J., P. M. Dinesh, T. Ramakrishna, Identifying algorithmic vulnerabilities through
simulated annealing [J], Optimization Letters, 5(3), 2011, 479-490
[9]
E. L. Zhou, X. Chen, Sequential monte carlo simulated annealing [J], Journal of Global Optimization, 55(1), 2013, 101-124
[10] H. Bai, L. Luo, Distributing and optimizing forces based on hybrid genetic algorithm [J], Development and Application of Computer, 24(8), 2011, 4-10
[11] Y. J. Lu, Y. W. Chen, D. Z. Yang, Improved GA with particle swarm’s evolutionary strategy for
solving constrained optimization problems [J], Control and Decision, 27(10), 2012, 1441-1446
[12] J. Y. Chen, Y. Li, D. Y. Yang, A novel quantum particle swarm optimization algorithm based on
Euclidean-distance division policy [J], Journal of Information Computational Science, 11(3), 2014,
691-698
[13] W. P. Li, H. F. Zhang, H. Sun et al., Study for the ballistic missile ﬁre distributing model based
on strike sequence optimization [J], Command Control Simulation, 34(2), 2012, 66-69
[14] Necio Dirik, First Lieutenant, Maximizing Strike Planning Eﬃciency for a Given Class of Targets
[R], Ohio: Air Force Institute of Technology, 2010