X. H. WANG ET AL.

76

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).

2

2

2

1e

2π

i

ii

tμ

xσ

ii

i

Gx t

σ

d

k

i

k

i

(6)

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-

lowing.

1

112 2

kk k

ii iig

vwvcrpxcrpx

(7)

1kk

ii

xv

(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;

k

id

v

k

id

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 = pgi – △f, xj = pgi + △f, and keeping the other

units’ funds invariable.

4. A Case Study of Air Spares Support with

IPSO

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

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