Journal of Transportation Technologies, 2012, 2, 75-77 Published Online January 2012 (
Study to Cost of Air Spares Support Based on IPSO
Xiaohua Wang, Aiqin Mu, Fuhong Wang, Zhongbing Tang
The Fundamental Courses Department, Xuzhou Air Force College, Xuzhou, China
Email: {plaxz, muaqin}
Received October 11, 2011; revised November 9, 2011; accepted November 19, 2011
Air spares support is general term of using and repairing of aircrafts which is the material foundation of aero technical
support, its effectiveness influences operational effectiveness and equipments of aircrafts directly. Based on particle
swarm optimization algorithm, a new model is proposed to optimize the distribution of the cost of air spares, it take the
funds as resource and the improvement of performance efficiency as objective and deduces the expressions to get the
best distribution plan. The results of experiments indicate that this model can make full use of the limited funds and
obtain the highest efficiency of air spares support.
Keywords: Air Spares Support; Fund; Particle Swarm Optimization
1. Introduction
The aim of air spares support is supplying aircraft’s fly-
ing and repairing with enough equipment. Its primary
work is satisfying military’s need of air spares immedia-
tely and exactly. There are several processes in air spares
support, such as preparation, storage, supply and mana-
gement [1], and all of these need sufficient funds. As rep-
orted in [2], the air force of USA spends about 0.8 billion
to buy more than 500,000 consumptive materials annua-
lly, and an F-15 flight squadron spends 31,000,000 doll-
ars for its air spares support. It has being a major issue
for how to improve the whole efficiency of air spares
support thoroughly by making use of limit funds.
Many algorithms are using to improve the efficiency
of air spares support. Reference [3] deduced expressions
with the marginal analysis method, and obtained the
scheme to get best effect cost of air spares support with
the numerical calculate method. In [4], the sequential
solution in dynamic programming is applied in solving
the problem. This paper proposes a new model based on
improved particle swarm optimization. In Section 2, the
model of air spares support is constructed. Then an im-
proved algorithm is presented in Section 3 to overcome
local optimization of PSO. In Section 4, a simulated case
is using to test the performance of new algorithm by
comparing the results with other algorithm. Finally, con-
clusions are given in section 5.
2. A Model of the Cost of Air Spares Support
In order to analysis the cost of air spares support quanti-
tatively, several indicators are introduced. The support
efficiency represents details that air spares support sys-
tem has finished the goals of support task in a certain
condition, which reflects the effect of system’s support
ability in the process of running it. It is synthetic calcula-
tion of the ability and military profits of air spares sup-
port [5]. The good rate of support is defined as follows.
 (1)
in which, f—the total days and numbers of aircrafts in
good state, Y—the total days and numbers of all aircrafts.
If the total days and numbers of all aircrafts in one air-
port is fixed, the good rate is generally in proportion to
total days and numbers of aircrafts in good state, and it
represents the ability of support. For these sakes, this
paper take consider of total days and numbers of aircrafts
in good state as effectiveness indicator of optimal model
which presented as follows.
Assuming W is the sum of funds of all support units, xi
is the funds of support unit i, Yi is the total days and
numbers of one aircraft supported by unit i in airport,
Gi(xi) is good rate, and fi is the total days and numbers of
one aircraft in good state. Obviously, fi is function of
funds x
i, the optimal distribution model of air spares
support and be constructed as follows.
x (4)
YGx (5)
opyright © 2012 SciRes. JTTs
Because the good rate has consequentially regular law
with the investment of funds, which means the more
funds invested, the higher good rate will be reached. But
the increment of good rate would decrease if funds in-
vested over a number. It is obviously that the funds obey
the standard normal distribution, and good rate is its dis-
tribution function, increment of good rate is its probabil-
ity density, as in (6).
Gx t
in which, μi—mean value of the i-th unit’s funds, σi
mean square of the i-th unit’s funds.
Although traditional algorithms such as dynamic pro-
gramming can solve the model presented above, the pro-
cess is inconvenient and times cost for computing in-
creases with size n and they have a positive negative ex-
ponential relation, the workload increases rapidly.
3. Improved Particle Swarm Optimization
PSO (particle swarm optimization) [6], a new intelligent
optimization algorithm proposed by psychologist Ken-
nedy and Dr. Eberhart in 1995, which intimates the bird
swarm behaviors. It is a high efficiency algorithm and
features many new functions and characteristics, such as
concise conceptions, convenient for running, fast con-
vergence and less parameters. At present, PSO is widely
applied in many fields, such as functions optimization,
training of neural network, fuzzy system control.
In PSO algorithm, each individual is called “particle”,
which represents a potential solution. When algorithm is
running, all particles fly at random. Each of them keeps
the best position “pbest” found by itself, besides, it also
remembers all particles’ best positions “gbest”. Particles
adjust their directions and velocities according two best
positions found before.
In 1998, Shi proposed a new idea in [7], the weighting
of inertia w is introduced to improve PSO algorithm’s
performance. It is a perfect method and brings PSO algo-
rithm to completion, which is called standard PSO algo-
rithm now. Larger weighting of inertia will lead to high
searching efficiency of global area of PSO, on the con-
trary, local area’s efficiency will be more clearly. The
renewed formulas of standard PSO are described as fol-
112 2
kk k
ii iig
  (7)
 (8)
in which, xi = (xi1, xi2, ···, xiD) is the position of i-th par-
ticle in D-dimension; vi = (vi1, vi2, viD) is particle’s veloc-
ity which represents its direction of searching; i = 1, 2, ···,
n, n is the number of particles; k is iteration times; r1 and
r2 are random values ranging between 0 and 1, which are
used to keep the diversity of particle swarm; c1 and c2 are
learning coefficients, which are also called acceleration
coefficients; is the d-th component of velocity of
particle i in the k-th iteration;
is the d-th component
of position of particle i in the k-th iteration; pid is the d-th
component of the best position particle i has ever found;
pgd is the d-th component of the best position the particle
swarm have ever found.
For avoiding being entrapped in local optimization, the
idea of simulated annealing algorithm is used to im-
proved PSO’s performance. First, gets pg by running
PSO one time, then runs simulated annealing algorithm
in pg’s neighborhoods and calculates a new solution of x’.
Comparing the fitness of pg and x’, if x’ is better than pg,
runs PSO another time and renews pg. Otherwise, a new
value of x’ is produced in pg’s neighborhoods based on
simulated annealing algorithm. There is at most L new
solutions can be found, in which, L is length of Markov
chain in simulated annealing.
According to the problem in this paper, the way of
generating a new value of x’ is described as follows. First,
assuming pg = (pg1, pg2, ···, pgn), x’ = (x1, x2, ···, xn), two
support units i and j are selected randomly, meanwhile, a
random value of support funds is assigned to unit i, then
setting xi = pgif, xj = pgi + f, and keeping the other
units’ funds invariable.
4. A Case Study of Air Spares Support with
Assuming an armed force has six units of air spares sup-
port, each of them supplies a flying regiment. According
to statistical information in history, each unit’s funds and
total days and numbers of one year are shown in Table 1.
Using IPSO (improved particle swarm optimization)
algorithm proposed in this paper, the parameters are set-
ting as follows. The number of particles is 40, the max
iteration time is 500, learning factors c1 and c2 equal to
1.5, initial weighting wma x is 1.5, final weighting wmin is
0.6, the max flying velocity vmax is 3, the min flying ve-
locity vmin is –3, the length of Markov chain L is 10, and
particle’s position and velocity are initialized randomly.
In order to make it easy to apply the distribution
scheme of air spares support in practice, this paper as-
sumes that the funds allocate to each unit does not in-
clude decimal part. Program based on this algorithm is
carried out in Matlab 50 times independently and the
results are shown in Table 2.
From Table 2, it is shown that the maximum support
days and numbers is 67,748 by using IPSO algorithm,
and its good rate of support is 93.43%. In [3], marginal
analysis method is applied to solve the same problem and
the result is 67,672 days and numbers of support, and the
number of 67,697 is obtained in [4] by using dynamic
Copyright © 2012 SciRes. JTTs
Copyright © 2012 SciRes. JTTs
Table 1. The funds and total days and numbers of each air
spares support unit.
air spares support unit
1 2 3 4 5 6
mean value of funds
(per ten thousands) 67 67.3 59.4 71.8 71.2 62.8
mean square of funds 4.4271 4.6306 6.3749 6.7052 6.8234 4.7074
total days and numbers
of a year 10,950 10,220 13,140 12,410 11,680 14,110
Table 2. The running results based on IPSO.
the best distribution scheme 74,74,69,81,81,71
the maximum days and numbers of support 67,748
times of getting the best solution 50
the maximum iteration times 87
the minimum iteration times 22
average times of iteration 53
programming algorithm. It is clearly that using IPSO
algorithm can support more days than using other algo-
rithm proposed in [3,4], and IPSO algorithm can achieve
the best solution every time. Furthermore, the maximum
iteration times of this case is 86 and the minimum itera-
tion times is 22, it can get the best solution on average 53
times iteration, which indicates that IPSO algorithm has a
feature of fast convergence velocity.
By running programs 50 times independently, the re-
sults of simulated annealing (SA), PSO and IPSO are
compared in Table 3. It can be found that the best solu-
tion appears 16 and 41 times in experiments using SA
algorithm and PSO, however, IPSO algorithm can get
best solution every time. Furthermore, using PSO, it
needs at least 242 times of iterations to get the best solu-
tion, but the maximum iteration times of IPSO is 100
only, which indicates new algorithm improved speed and
rate of convergence.
5. Conclusion
It is easy to find an optimal distribution scheme on tradi-
tional optimize algorithm for air spares support, however,
if the sum of funds is invariable, the performance of sup-
port can not achieve the best solution. As a new intelli-
Table 3. The comparion of using SA, PSO and IPSO.
initial temperature 10000
ending temperature 1
annealing velocity 0.87
length of Markov chain 10 10
max iteration times 1500 100
the maximum days and numbers67,748 67,74867,748
the minimum days and numbers 67203 64612 67,748
average of days and numbers 67,701.42 67,06667,748
times of getting the best solution16 41 50
gent optimization algorithm, IPSO algorithm can solve
this problem successfully and features many other char-
acteristics, such as fast convergence speed and very con-
venient for running, all of these implies that IPSO has a
better performance in searching best solution of nonlin-
ear programming, and it has some practical value in
solve the optimization of large and medium-sized ex-
pense support problems.
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