J. Software Engineering & Applications, 2009, 2: 350-353
doi:10.4236/jsea.2009.25046 Published Online December 2009 (http://www.SciRP.org/journal/jsea)
Multi-Objective Incomplete Probability
Information Optimization Reliability
Design Based on Ant Colony Algorithm
Qiang ZHANG1,2, Shouju LI2, Ying TIAN1,3
1Liaoning Technical University, Fuxin, China; 2State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian Uni-
versity of Technology, Dalian, China; 3Fuxin Miming Company, Fuxin, China.
Received April 19th, 2009; revised June 3rd, 2009; accepted June 15th, 2009.
ABSTRACT
In view of incomplete probability information multi-objective question, it used probabilistic perturbation method and
Edgeworth series technique to study reliability optimization design. The first four moments of basic random variables
are known under condition. It used the Ant Colony Algorithm to design cutting head roadheader, the optimized result
indicated that cutting head load fluctuation and compared energy consumption were reduced obviously at the same
time. This result enhanced roadheader operational reliability and energy effectively.
Keywords: Optimization Reliability, Multi-Objective, Probabilistic Perturbation Method, Incomplete Probability In-
formation
1. Introduction
Cutting head is the key component and important part of
roadheader which affects the roadheader efficiency and
load exceptional change. There are many published re-
ports on the cutting head’s reliability optimization re-
search overall the world, such as Zhang Xin from Shan-
dong Scientific and Technical University has studied on
cut truncation tooth arrangement and multi-objective
optimizations [1,2]; Li Xiao-huo from Liaoning Tech-
nology University has studied on cut head swinging cut-
ting and cut head design [3–5]; Ma Hong-wu from Da-
tong coal mining has studied on the cut head diameter
influence to roadheader [6]; Guo wu from IMM company
has studied on S150 cutting head [7]. The majority of
scholars use the conventional routes to research cut head
reliability optimization design, these methods are under
design variable and parameter to follow the normal dis-
tribution. Due to the complexity in real project and insuf-
ficient data message, it is prone to make errors by con-
ventional design [8–10]. This paper uses the stochastic
perturbation method and Edgeworth progression me-
thod [11–14] to discuss cutting multi-objective reliability
optimization which distributes the random parameter
obedience willfully.
2. The Reliability Design of Perturbation
Method
To calculate reliability or failure probability, it needs to
know the probability of density function or joint prob-
ability of density function. As lack of the tentative data,
it is mostly very difficult to obtain reliability or failure
probability by integral computation. Under the situation
that unable to determine the distribution generally, the
first to fourth moments (average value, variance and co-
variance, third-order moment, fourth-order moment) of
design variable are determined easily as it has enough
data, then it achieves reliable target, unknown function of
state probability distribution is transformed to the stand-
ard normal distribution expression by application fourth-
order moment technology, Edgeworth progression and
corresponding experience correlation formula [14], fi-
nally it may determine the structural element reliability.
2
235
34 3
11 1
()() ()()3 ()()
624 72
gg g
gg g
FyyyHyH yH y
 
 

 

 
 

 
 



Multi-Objective Incomplete Probability Information Optimization Reliability Design Based on Ant Colony Algorithm351
()
j
H
y is j Stage Hermite multinomial, its recurrence
relation is：
11
01
() ()()
() 1,()
jjj
H
yyHyjHy
HyHy y





The real distribution of random parameter could be
approached precisely by the Edgeworth progression. It
usually takes the progression of first to fourth items to
obtain good approximation. But it only take the progres-
sion of first to fourth items, sometimes it will cause reli-
ability to present R>1 situation because the approximate
distribution function and the real distribution function
exist deviation. The computation practice indicated that
following experience correlation formula obtained result
to approach in Monte Carlo numerical simulation under
R >1 situation; the Edgeworth progression may obtain
precise solution enough on condition of R﹤1.
 
() ()
() ()
1()()
R
RR
R








As known above, this method is fit for random pa-
rameter distribution generally during the equation devel-
opment. It is more useful for the actual project.
3. Styling Ant Colony Algorithm and
Optimize Calculate
When the design requirements is , it
is , satisfies the probability
values of the restraint, R obtains by the fore-mentioned
Edgeworth progression or experience correlation formula.
The probability optimization design model may be
solved by model transformation as follows.

0
() 0PgX R

0
() 0RPgXR
0
R
min

()() ()
f
X EfXfX
St.
0
() 0,(1,,)
() 0,(1,,
i
j
RR
qX il
hX jm


 

 


1) Initialize the population. Produce N to answer at
random in the independent variable defined area to build
the population, calculate adaptation degree of the indi-
vidual, arrange order according to the size, and to give
the same pheromone initial value to them. Produce M
ants, including G overall ants, L some ants.
2) Overall Search. Seek overall ants to operate explor-
ing. Through crossed and variable operating, produce
new G ants to replace the present population G ants.
3) Some Search. Seek some ant to operate excavating
one by one, calculate N individuals to choose probability
Pi (x), choose L individual as the goal, optimize and
search for the goal individual, and lead individuals to
better position.
4) Pheromone is evaporated. Carry out pheromone to
volatilize after changing and taking.
5) Check the condition of stopping. Finish one search,
population is checked to be satisfied with disappearing
terms. If it is satisfied, export and solve optimally at pre-
sent. Otherwise, it transfers to the step 2). The next is to
be searched, until disappearing or meeting the request of
schedule.
4. Multi-Objective Optimization Reliability
Design for Cutting Head of Roadheader
Model
4.1 Design Variable Choose
Transversal spacing top, circumferential distribution an-
gle δ, way speed v, rotational speed n are design vari-
ables:
X=[x1, x2, x3, x4] T= [top, δ, v, n] T
4.2 Objective Function Establish
In the design of cutting head according to the load fluc-
tuation and compared to the energy consumption com-
putational method, it makes the smallest cut angle of Ra,
sway resistance Rb, longitudinal force Rc, the load tor-
que coefficient of variation Mc, energy consumption Hw.
It takes objective function of cutting head multi-objective
optimization reliability design.
This article uses the linear weighted sum law to trans-
form the multi-objective optimizations for the simple tar-
get optimization. Considered various simple targets have
the equal status. It realizes the multi-objective standardi-
zations. The transformed objective function expression
is:
() ()()()()
() abcc w
abc c
RRRMH
RRRMH
w
f
xfxfxfxfx
Fx ffff f 
The formulas: ()
a
R
f
x, ()
b
R
f
x, ()
c
R
f
x, ()
c
M
f
x,
()
w
H
f
xare respectively simple goal optimization func-
tions. The denominator is the simple goal optimum
value.
4.3 Constraints Determine
Considering the actual situation, it establishes fuzzy re-
liability constraints.
1) Bolt between cutting head and the cutting head axis
is intensity fuzzy reliable restraint:
2
1
4P / (md)    (1)
Use the stochastic perturbation method and Edgeworth
progression method, then the max shear stress reliability
design restraint is:

Multi-Objective Incomplete Probability Information Optimization Reliability Design Based on Ant Colony Algorithm 352
Table 1. Optimal solution and goal function value
top δ v n
3.601 42.021 2.937 38.265
routine calculation Ra routine calculation Rbroutine calculation Rcroutine calculation Mcroutine calculation Hw
54.734 28.611 27.327 23.136 4.861
the optimize Ra the optimize Rb the optimize Rc the optimize Mc the optimize Hw
51.101 25.457 26.65 22.726 3.685
1
g(X)=R 0R


0
(2)
2) Truncation tooth fatigue fuzzy reliable restraint:
When the load changes function of number N＞106, by
reliability design request 1
R

, it obtains reliability re-
strain:
2
g(X)=R0
cR  1

 (3)
3) Transversal spacing nominal optimum value scope:
31
g(X)=x2

＞0 (4)
4
g(X)=6 x1
2
＞0 (5)
4) Circumferential angular interval scope:
5
g(X)=15 x

＞0 (6)
62
g(X)=x 60

＞0 (7)
5) Sway speed scope:
73
g(X)=x 1

＞0 (8)
8
g(X)=6x3
4
＞0 (9)
6) Rotational speed scope:
9
g(X)=15 x

＞0 (10)
10 4
g(X)=x 80

＞0 (11)
7) Truncation tooth distributed limit:
11 m
g(X)= 85


＞0 (12)
8) Cutting power limit:
12e j
g(X)=P P

＞0 (13)
9) The installs truncation tooth spacing limit:
13 k
g(X) =L80

＞0 (14)
4.4 Optimizations Resolvent
According to the algorithm described above, it optimized
and asked for resolving. The parameter is established
based on the most different long λ corresponding half
step by step, the overall situation searches for step count
k =40, Utilize Mab7.0 software to establish calculating
algorithm procedure. The experiment indicated that the
spiral drill weight reduced 16.77% and transported the
efficiency enhanced 7.05% through the optimization de-
sign.
It carried on multi-objective optimization design to cut-
ting head of roadheader by compilation optimization de-
sign procedure, obtained optimal solution and the opti-
mized goal function value in Table1.
5. Conclusions
This article proposed a practical efficacious device to
design cutting head which obeyed the willfully distribut-
ing random parameter reliability optimization. The cut-
ting head reliability optimization multi-objective reliabil-
ity optimization design method can reduce design cycles
and save experimental funds, raise design level, and en-
hance forecast accuracy. The practice indicated that cut-
ting head load fluctuation and compared energy con-
sumption were obviously reduced at the same time. This
result enhanced roadheader operational reliability and
energy effectively.
6. Acknowledgments
The study is partially financial supported by the program
Ministry of Education Doctor Fundation of China
(20060147001); Chinese coal machine equipment com-
pany scientific research foundation; Outstanding youth
scientific research fund in Liaoning technology univer-
sity; Scientific research plan of outstanding young tea-
cher in Liaoning technology university mechanical engi-
neering institute; Graduate student scientific research
fund in Liaoning technology university; Dalian Science
and Technology University of structure state key labora-
tory open fund (G0818); Liaoning Province safety pro-
duction development plan 2009.
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