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The Genetic Algorithm

Based System
for Economic De
sign of Multiple Control Charts
Chen

Fang Tsai
Department of Industrial Engineering Management Aletheia University, R. O. C.
No.32, Chien

Li St., Tam

Sui, Tai

Pei Shien, Taiwan
Tel: 02

26212121 Fax: 26212
121

5512
Email: tsai@email.au.edu.tw
Abstract
In mixed

production, quality

control operations are necessary, and multiple
control charts are required to monitor the production processes. Traditional
approaches to the Economic Design of Multiple Control Cha
rts provided only a
local optimum. After a period of experiments the author found that a genetic
algorithm approach to the Economic Design of Multiple Control Charts provided
the best solution to the global optimum problem. This research was verified using
data from the Tien

Huang Aluminum Corporation. Thus, in this research the
author is proposing a novel approach to the problems of the Economic Design of
Multiple Control Charts using the technologies of genetic algorithms with generic
adjustment models.
K
eywords:
Economic Design of Multiple Control Charts; Genetic algorithm
基因演算法系統下的多重管制圖經濟設計
蔡振芳
真理大學工管系
摘
要
混合式生產中，製程品管需多種和多階的管制圖同步監測及控管。多管
制圖經濟性的設計，在傳統方法的研究中僅能達成局部的最佳解。本研究嘗
試使用基因演算法，可提供近似全面性的最佳解。在研究中，提出數種動態
基因演算法模式，並利用理論與實務數據得到証明，本研究模式具有改善多
管制圖經濟性設計的效果。進而証明此
一新嘗試，可成為多管制圖經濟性設
計的一般化調整模式。
關鍵字
:
管制圖經濟性設計
;
基因演算法
蔡振芳
基因演算法系統下的多重管制圖經濟設計
104
1
.
Motivation
Chemical production processes usually deal with a number of products
simultaneously. When Economic Design of Control Charts is used in the chemical
manufacturing industries, this is called mixe
d

production. Hence, process engineers
need to find a new design that can handle multiple control charts for mixed

production.
This leads to the Economic Design of Multiple Control Charts (EDMCC). The
Economic Design of Multiple Control Charts is a search
and optimization problem and
there are various approaches that can be used to solve it. Many of these have been
applied to the simpler EDCC problem.
In the last decade, many new approaches have been developed within this field.
These models attempt to enha
nce the effectiveness of control charts, especially by
shortening the response time in decision making. Expert systems have made a
significant contribution to the Economic Design of Control Charts by speeding up this
reaction time. However, some situations
need multiple control charts to handle complex
problems within a single process. The design of these control charts is called the
Economic Design of Multiple Control Charts. As this has several objectives, which need
to be optimized simultaneously, it is
difficult for conventional rule

based expert systems
to handle (Awadth, 1994). Therefore, this research aims to develop a system to optimize
the solutions for the Economic Design of Multiple Control Charts using genetic
algorithms.
2. Background of the Eco
nomic Design of
Multiple
Control Charts (EDMCC)
The aim of the section is to provide background information about the EDCC and
techniques of searching and optimization so that the implications of a GA system in
EDMCC can be understood. Theories of statisti
cal quality control have been utilized as
a control mechanism for process control in various industrial applications. Control
charts are a robust statistical process control tool. They are applied to the detection of
the out of control status, analysis of
the process stability and rectification of the process
situation. They are designed to allow a certain level of noise within a process; but when
a test point taken from the process falls outside the control limits, it is taken as a sign of
a problem within
the process (Pan, 1988).
The design of any control chart is a problem of economy. Attempts to find the best
method of achieving such economy have led to the field of the Economic Design of
Control Charts. In this section, the effects of control charts on
the process, taking into
account different kinds of maintenance and revision policies, are considered, in order to
produce the most cost effective solution. Hence, process engineers need to find a new
design, which can handle multiple control charts for mi
xed

production.
2.1
Control Charts
Control charts are designed to allow a certain level of noise within a process; but
when a test point taken from the process falls outside the control limits, it is taken as a
sign of a problem within the process. Control
charts may be classified by the
characteristic being tested. There are four basic types (Pruett, 1993):
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1.
chart

This takes the average of the measurements in the sample. Averages are
used because they are more sensitive to change t
han individual values. The average
measures the aim or centering of a process.
2.
R chart

This takes the range of the measurements in the sample. The range
measures variation within the process. The spread, or variation, for the range of
charts can also be calculated using the population standard deviation (
charts).
3.
P chart

This calculates the percentage of defective products in the sample, which
is the number of defective items rather than the total number o
f defects.
4.
C chart

This uses the number of defects that are yielded in the sample, where a
defect is a specific type of error in a product.
The other category is for discrete data, which are called attributes. The P chart and
NP chart are both based on
the binomial statistical distribution. They measure discrete
data such as the proportion (P chart) and number (NP chart) of defective items or
sub

standard items (Hutchins, 1991). The U chart and C chart are based on the Poisson
statistical distribution. T
he U chart is usually used for non

constant sample sizes,
whereas the C chart is used when the range over which defects are possible is fixed. If
the product data is variable and relates only to an individual, then CUSUM charts
should be used. Figure 1 (Pr
uett, 1993) shows the various charts and the situations for
which they are most suitable.
Figure 1 The framework of control charts
After the process engineers have selected the appropriate control charts, they then
have to consider the
process design and the states of the process. This is because the
design of the process state is a crucial factor that will affect the eventual performance of
the control charts.
2.2 The Process State
The cycle time plays an important role in determining t
he cost factors of the
Economic Design of Control Charts. Usually, it consists of four time periods (See
Figure 2): firstly, there is the expected time of an in

control period (T1, t1); secondly,
there is the expected time taken up by a process failure bef
ore any action is taken to
detect the fault (T2); thirdly, we have the expected time for inspection to take place, in
order to detect the failure of the process, and the expected time taken to diagnose the
assignable causes which exist (T3); and finally,
there is the expected time it takes to
revise the process and correct the problems (T4).
The length of the down

time period is crucial to the decision maker, because this
will effect the total cost indicated by the EDCC. Usually, there are two main polici
es,
which are described as follows. Policy 1 takes the down time from the beginning of t3
to the end of t4. Policy 2 takes it from the beginning of t4. The difference between these
蔡振芳
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106
two policies is the cost of shutting down the process during the time perio
d (T3).
Figure 2 A comparison between Policy 1 and 2
註：
TC: total cost
T: total cycle time
T1, t1: expected time in

control
T2: expected time out of control
T3: expected time down time
t2: expected time before detecting the fault
t3: expected time to find the causes
t4: expected t
ime for revising action
The cost functions are defined as follows: the cost of setting up and maintaining
the control charts (C1 + C2 * N); the cost of searching false alarms (C3); the cost of
searching for assignable causes (C4); the cost of production lo
ss (C5); and the cost of
sales loss (C6). The total cost (TC) is formulated as follows:
TC = (C1 + C2 * N) + C3 + C4 + C5 + C6
(1)
C1: fixed cost of operating control chart
C2: variable cost of control chart
C3: cost of searching fa
lse alarm
C4: cost of searching assignable cause
C5: cost of production loss
C6: cost of sales loss
N: sample
size
The total cost function is derived from the cost of four cycle time states, which
include the in

control state (T1); cause happe
ned state (T2); cause searching state (T3)
and revising state (T4) (see Equation 1). The total cycle time is composed of these four
states and it is an important factor in the total cost of quality management. Therefore it
must be taken into account when d
esigning the control charts. Having now introduced
the main components of the process design for the EDCC, this process will now be
discussed in detail in the following sections.
2.3
Economic Design of Control Charts (EDCC)
The cost functions which make up
the EDCC are based upon the operation cost of
the control chart; the searching cost for false or real alarms; the testing cost required to
determine the assignable cause; the cost of stopping during the running of the process
(Loss of sales, loss of produ
ction); and the cost of revising actions. Some research has
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produced several generalized optimization functions from previous studies based upon
some assumptions, which relate to different kinds of changing situations as in the
following examples (See Tabl
e 1):
Table 1 Optimization functions for various control charts

R
chart
:
CUSUM chart
:
C chart
: model 1
C chart
: model 2
註：
TC: total cost of whole time cycle
C1: fixed cost of operating control chart
C2: variable cost of operating control chart
C3: cost of searching false alarm
C4: cost of searching assignable cause
C5: penalty cost of each defective product
C6: penal
ty cost of revising action
C7: penalty cost of stopped production
C8: cost of mean moving to lower bound
T: total cycle time
t1: searching time for false alarm
t2: searching time for assignable cause
t3: average total time of searching action
t4:
average total time of revised action
T1: average time of process in control
T2: average time of process out of control
T3: average time of down t ime
R1: t ime rat io of process moving to upper
bound
R2: time ratio of process moving to lower b
ound
P: defective rate of production
Pr: production rate
C5u: cost of mean moving to upper bound
C5L: cost of mean moving to lower bound
H: sampling interval
N: sample size
fn: number of samples between false alarms
m0:
number of defects when process in

control
m1: number of defects when out of control
: process parameter of Exponential distribution
: probability of error type
of control
charts
These are only a design for a single control chart (EDCC). However, in
practice,
production processes usually require more than one chart for their purposes. Therefore,
because most process designers deal with multiple objectives based on using multiple
control charts, a specific design for the EDMCC is required.
3. The Formu
lation of EDMCC
This problem domain is the EDMCC in mixed

production within the aluminum
industry. These functions are based on different maintenance policies, sampling size and
time interval. Multiple objective problems can be represented as a group of si
ngle
EDCC objective functions. The representation for the minimization function is: min
{TCn}. For example, the three EDCCs could be: an

R chart, a CUSUM chart and a
C chart. The example (EDMCC) has three control charts (See Table 2
):
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Table 2 The total minimum cost function for a multiple control chart
3.1 EDMCC in the Chemical Industry
The Tien

Huang Aluminum Corporation is one of the largest treated aluminum
manufacturers in Taiwan. The process involves a chem
ical technique, which treats the
surface finishing of aluminum alloys. Aluminum and its alloys have some inherent
resistance to atmospheric corrosion because of an oxide coating, which is spontaneously
formed when the metal comes into contact with the air.
The main processes of
aluminum surface treatment, including the degreasing (To clean the dirty surface),
etching (To remove the original oxide coating), neutralizing (To neutralize the etching
solution), anodizing (To form the new oxide coating), electrol
ytic coloring (To add color
to the surface) (See Table 3).
Table 3 Processing steps in the treatment of aluminum
Process 1
Process 2
Process 3
Process 4
Process 5
Process Steps
Degreasing
Etching
Neutralizing
Anodizing
Coloring
Composition
Detergent
Na
OH
HNO
3
H
2
SO
4
CuSO
4
Concentration
35
5g/l
35
5g/l
85
15g/l
180
5g/l
90
5g/l
Temperature
65
5C
55
5C
25
5C
18
2C
25
5C
Processing Time
3 min
5 min
3min
30 min
20 min
The main processes of aluminum surface trea
tment are classified into three types:
organic, inorganic and electroplate coating. Different finishes are required for different
markets (See Figure 3).
Figure 3 Production processes of the aluminum alloy treatment
In practice, howeve
r, anodizing and coloring are related processes; bad control in
anodizing will produce bad effects in the coloring stage. Again, traditional EDCC does
not take account of these special inter

relationships between products. Hence, a new
approach, which coul
d handle the multiple variables, was needed, and the result was
EDMCC. The EDMCC needs a genetic algorithm search technique to find single
near

optimal, global

optimal or multiple objective solutions.
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4. Genetic Algorithms
Genetic algorithms generate a seq
uence of populations using a selection
mechanism and use crossover (I.e. combining solutions) and mutation (I.e. modifying
existing solutions) as search mechanisms (See Figure4).
Identify process attribute from external demand (EDMCC) by the control chart
selector
Initialize operators and parameters by the GA manager
While (Termination condition = FALSE) do
Begin
Population initialization
Selection
Crossover
Mutation
Replacement
End
Set chromosome structure and assign sub

goal weighting
by the process para
meter controller
Refine GA’s operators and parameters by the GA manager
Figure
4 A flow chart for a simple genetic algorithm
Genetic algorithms have had success in both single function optimization and
multiple objective optimization (Tamaki, 1996). The
following sections will describe
the inner mechanisms by which genetic algorithms obtain their final solutions.
4.1 Components of the GA
The simple genetic algorithm, which is used as a function optimizer, is composed
of three main parts: a population of e
ncoded strings, a fitness function and a
reproduction process (Goldberg, 1989). The reproduction process is in turn defined by
three main operators: selection, crossover and mutation (Kolonko, 1995). The initial
population is randomly generated according t
o a uniform random distribution over the
set of possible solutions (Or chromosomes). A chromosome consists of a string of bits.
A number of bits are concatenated together to form a gene; several genes are
combined together to produce the chromosome. The p
opulation evolves, adapting to the
forces operating upon it. The population has to be sized such that there is a reasonable
probability that the important genes exist somewhere in the initial population
(Mandavilli, 1997).
The fitness function is the only
component, which has to be adapted to the domain
of the optimization problem. The evaluation function is a process which, when
presented with the chromosome, returns the corresponding solution and measures how
closely the chromosome meets the optimization
criteria. The fitness value is used by the
genetic algorithm to control the selection mechanism in the reproduction process. In the
case of aluminum industrial manufacturing where a cost is being optimized, the fitness
function is the cost function.
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5. Sys
tem Architecture
The prototype system we created for the optimization of EDMCC employs a novel
architecture. It consists of four main components. (1). A control chart selector (An
expert system), which identifies the process situation and selects the appro
priate control
charts for the problem. (2) A GA manager which selects the optimal operator and
parameter settings for the GA
evolutions, using orthogonal arrays with dynamic
parameter setting. (3) A process parameter controller (PP Controller), which refin
es the
chromosome structure of the genetic algorithm and assigns suitable weights for each
individual objective function using the fitness ratios. (4) A simple genetic algorithm,
which improves the optimization of EDMCC (See Figure 5).
Figure 5 The system architecture
This experimental framework has two main objectives: the first is to apply a simple
genetic algorithm to find better solutions for EDMCC; the second objective is to utilize
three proposed approaches to improv
e the behavior of the genetic algorithm. The
operations of this experimental framework are divided into three stages (Product and
process identification; policy selection; and control chart setting). In the initial stage, the
control chart selector identif
ies the product type in the production process and the
failure model of the machine operation and then selects the production and maintenance
policies accordingly. Once the situation has been defined, the control charts are then
chosen for the EDMCC.
Durin
g the next stage, the GA manager alters the operators and parameters in order
to achieve a balance between exploration and exploitation in the search space. A better
combination of operators and parameters will help the genetic algorithm to keep pace
with
changing problem situations and hence improve its performance. At the same time,
the process parameter controller utilizes evidence gathered from the GA
evolutions to
refine the chromosome structure and assign weights for the sub

goals.
Finally, the exper
imental simulations are performed in the genetic algorithm
module to verify the efficiency of these approaches. Thus it can be seen that this system
architecture is an approach, which combines existing methods with the methodologies.
For the experiment, th
e author selected four kinds of control charts, the

R, P, C and
CUSUM charts, with four failure process mechanisms, two production policies and two
maintenance models. These are X

E

N

Y; CS

W

N

Y; C

P

S

Y; and P

B

N

N. A more
detail
ed explanation is shown in Table 4.
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Table 4 The four EDCC which are combined in the EDMCC test

bed
Test

beds
Chart type
X

E

N

M

R chart;
E
xponential distribution;
N
on

stop

production;
M
aintenance
CS

W

N

M
C
u
S
um chart;
W
eibull mod
el;
N
on

stop

production;
M
aintenance
C

P

S

M
C
chart;
P
oisson model;
S
top

production;
M
aintenance
P

B

N

N
P
chart;
B
inomial model;
N
on

stop

production;
N
o maintenance
5.1 The methodologies of dynamic genetic algorithms
To prove the effectiveness of this
system, the author developed seven models with
different degrees of preservation and deterioration to match the different problem
situations. Models 1 to 6 are instances of dynamic parameter settings. Model 7 is an
instance of static parameter setting, bu
t can also be considered to be an extension of the
other six models, where the settings are the same in every region.
Model 1 is one slope equation (Rc and Rm) for the crossover and mutation rates
(CR and MR). It starts from region 1 to region 6, which the
crossover rate starts from a
value of 0.0 to 1.0 and the mutation rate starts from a value of 0.0 to 0.5. Model 2 is a
two stage operation and the crossover rate starts from a value of 0.0 to 0.85 and the
mutation rate starts from a value of 0.0 to 0.425.
Where the region 6 is assigned a
constant crossover rate (1.0) and mutation rate (0.5).
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Model 3 follow Model 2 but with the constant value
s being applied to an
increasing number of regions. In Model 7, the crossover and mutation rates are constant
in all regions. The designer can select the crossover rates from the range 0.75 to 0.95
and mutation rates from 0.001 to 0.005. These ranges are t
he experimental results from
previous researches.
This research tries to measure the responses of crossover and mutation, the generic
mathematics model for genetic algorithms search. The equation models (1,2,3), which
this research was
designed, are follows:
(
) (n=1, 2, 3)
(1)
(
) (n=1, 2, 3)
(2)
=
+ n
(3)
where
h :a schema.
F
l
:the fitness value.
:the average fitness value of the population.
R
c
= the range ratio for crossov
er operation.
R
m
= the range ratio 6 for the mutation operation.
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f
max
= the maximum value in the population.
f
min
= the minimum value in the population.
F
l
: the length of the solution.
pf
cn
= the policy factor from region 1 to region n.
pf
mn
= the polic
y factor from region 1 to region n for the mutation operation.
In the above equations, we conclude the generic schema theorem for the dynamic
genetic algorithms for all values of pfcn and pfmn.
This approach is inspired the concept of Quality Control by
Control Chart, the
crossover and mutation rates are subject to the quality levels of the parents and offspring.
The designer can select the policy factors (PFc and PFm) to tune the deterioration rate
to the particular situation. The work of Srinivas (1994)
is an instance of this approach.
Hence, this can be considered to be a generic approach which can therefore be applied
to other problems.
An important innovation in this research has been the development of this series of
models. These models are a generi
c approach, which encompasses a wide range of
possible situations. The idea behind these seven models was inspired by the concept of
quality control using control charts (E.g.
chart) (Besterfield, 1990). In effect, we
altered the cro
ssover and mutation rates, subject to the quality levels of the parents and
offspring. The designer can select the situation factors to tune the deterioration rate to
the particular situation.
5.2 The experimental results of EDMCC by dynamic
parameter sett
ings models
This research presents and analyses the experimental results of the empirical
test

beds. The results in this section enable a comparison between static and dynamic
parameter settings with different kinds of GA operators and parameters. There ar
e three
main ways in which the performance of each method was compared:
1.
The value of the static setting is compared to the individual value of each dynamic
model, D1 to D6, and counts the number of ‘wins’ (The winner is the one with the
best performance) f
or each model and experiment, E1 to E8. This comparison can
provide an indication of how much better the dynamic approach is.
2.
The value of the static setting is compared to the mean value of all of the dynamic
parameter setting models (Dmean), for a given
experiment. This comparison is
between general static and dynamic settings.
3.
The mean value of the static setting is compared to the mean value of each
dynamic model over all of the experiments. This comparison shows the average
performance of the static mo
del and the six dynamic models.
The following sections will now discuss the experimental results of the empirical
test

beds. In these experiments, there are several combinations for the EDMCC,
containing two or more objective functions. In the following co
mparisons of the
theoretical test

beds, we use five types of symbol, which are E1

8, D1

6, Dmean, Static
and Dynamic. They are explained as follows:
E1

8: experiment 1 to experiment 8 in the orthogonal array (L8)
D1

6: model 1 to model 6 of the dynamic par
ameter setting
Static: the static parameter setting
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Dynamic: the dynamic parameter setting
Once the experiments had been carried out, the results were compared. For every
combination of the operators and parameters listed in the tables (Table 5), 20 trial
s were
performed. All results presented are averages of the 20 trials for each of the eight
experiments given by the orthogonal arrays. For the experiment on EDMCC, provides a
comparison of the static setting and the dynamic parameter settings on the EDMCC
,
using the two most appropriate dynamic models (Models D3 and D6) to show the
significance performance in a GA. In the following comparisons of the EDMCC
test

beds, the following terms are used: E1

8, D3 or D6, Static, Dynamic. Table 5 shows
the experimen
tal configuration for the dynamic setting. Both setting are identical,
except that for the dynamic setting the crossover and mutation rates are selected
automatically by the transformation function of the dynamic parameter setting.
Table 5 The dynamic set
tings for the experiments
Experiment No.
Population
Selection
Replacement
Condition
E
1
20
roulette
uniform
if

improved
E
2
20
tournament
tournament
annealing
E
3
60
roulette
uniform
annealing
E
4
60
tournament
tournament
if

improved
E
5
120
roulette
tourn
ament
if

improved
E
6
120
tournament
uniform
annealing
E
7
180
roulette
tournament
annealing
E
8
180
tournament
uniform
if

improved
Table 6 provides the number of wins for the static and dynamic settings. It wins 14
out of the 16 contests, the dynamic set
tings performed better than the static setting.
(E1

8: experiment 1

8; D1

6: model 1 to 6 of the dynamic parameter setting; Static: the
static parameter setting; Dynamic: the dynamic parameter setting)
Table 6 The number of wins for the static and dynamic
settings for EDMCC
EDMCC
E
1
E
2
E
3
E
4
E
5
E
6
E
7
E
8
Total
Static
0
1
0
1
0
0
0
0
2
Dynamic
2
1
2
1
2
2
2
2
14
Table 7 shows the total cost for the static settings compared to the minimum total
cost for models D3 and D6 of the dynamic settings. In experime
nts 1 and 3, the
dynamic settings both performed better than the static setting. In the rest of the
experiments both methods were equally effective, although D3 performed relatively
badly in experiment 2. This implies that the moderate deterioration model
(D3) can only
perform well on simple functions, and that model D6 is more suitable for the EDMCC
function. When the static setting is given a large population size, its performance is
similar to model D6 on the EDMCC function.
Table 7 A comparison of the
static and dynamic settings for the EDMCC
EDMCC
E
1
E
2
E
3
E
4
E
5
E
6
E
7
E
8
Static
300
360
390
360
355
355
350
350
D
3
230
235
225
215
205
208
205
206
D
6
210
200
205
200
203
203
180
187
Having shown that dynamic settings are generally better than the static
setting for
the smaller populations. Table 8 shows the average result for the static settings
compared to the average results for Models D3
and D6 of the dynamic settings.
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Table 8 A comparison of the static and dynamic settings for the EDMCC
EDMCC
E
1
E
2
E
3
E
4
E
5
E
6
E
7
E
8
Static
320
370
420
370
365
375
340
330
D
3
220
385
235
355
370
380
335
325
D
6
200
375
225
370
360
370
330
320
In Experiments 1 and 3, the dynamic settings both performed better than the static
setting. In the rest of the experiments bo
th methods were equally effective, although D3
performed relatively badly in Experiment 2 (see Table 8).
6. Conclusion
In this research, we utilized a Genetic Algorithm approach, which can vary all of
the process parameters in order to achieve the global o
ptimum in the Economic Design
of Multiple Control Charts. However, an inherent problem of Genetic Algorithms is that
most researchers waste too much time tuning the selection of the operators and
parameters. Hence, in this research, we present new methodol
ogy to improve the
performance of the Genetic Algorithm. The methodology is a dynamic function, which
assists users in selecting optimal parameters for their problems.
In this research, the GA’s parameters (Crossover, mutation, etc.) and process
variables
(Sample size, width of control limits, etc.) are considered in an integrated way,
for different situations. The selection of operators and parameters for the GA search is
always a long and tedious trial. Therefore, a fast and simple method would be very
us
eful and many researchers have attempted to develop such a method. One of the
contributions of this paper was the production of a function, which can improve GA
performance in the Economic Design of Multiple Control Charts. This research also
presents a pr
ocedure, utilizing related domain knowledge, which can be used to identify
and preserve the beneficial building blocks in Genetic Algorithms. The results of this
approach have shown, through empirical analysis of the GA on the test

beds of the
Economic Des
ign of Multiple Control Charts. It is successful in identifying and
preserving the beneficial blocks, thus preventing disruptive recombination.
Through the simulation offered by this system, we are able to attain more
alternatives than previous research wa
s able provide, in relation to the dynamic
situations that are encountered within contingent management. We have also attempted
to classify and refine these alternatives so that they can become generic rules that can
provide guidance to process designers.
The experimental results also show that models
D3 (Which is a moderate deterioration model) and model D6 (Which is the strongest
deterioration model) of the dynamic parameter settings achieved the better results for
the EDMCC. The experimental results indi
cate that model D3 is good at the simpler
functions and model D6 is particularly good at coping with difficult and complex
functions.
The results from the Tien

Huang Aluminum Corporation show that both the defect
rate and reworking costs have been signific
antly reduced, and output has been increased
by use of our proposed system. It is hoped that similar financial benefits can be made in
the future by adapting this architecture to other mixed

production processes that also
require quality control of complex
multiple attributes.
蔡振芳
基因演算法系統下的多重管制圖經濟設計
116
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