Download Joint statistical design of double sampling X

Joint statistical design of double
sampling X-bar and s charts
指導教授： 童超塵 老師
作者：David He *, Arsen
Grigoryan
主講人：廖乃毅
Contents
Introduction
The joint DS X-bar and s charts
Formulation of joint statistical
design of the DS X-bar and s charts
Solving the optimization problem
using genetic algorithm
Performance of the joint DS X-bar
and s charts
Conclusions
國立雲林科技大學 工業工程與管理所
Introduction-Abstract
The statistical design of the joint DS X-bar
and s charts is defined and formulated as an
optimization problem and solved using a
genetic algorithm.
the joint DS X-bar and s charts offer a better
statistical efficiency in terms of ARL than
combined EWMA and CUSUM schemes,
omnibus EWMA scheme over certain shift
ranges.
In comparison with the STD, TSS and VSS Xbar and R charts, the joint DS charts offer a
better statistical efficiency for all ranges of
the shifts.
國立雲林科技大學 工業工程與管理所
Introduction-EWMA(EEu)
What’s EEu?
-EEU was obtained by running a two-sided
EWMA mean chart and a high-side EWMA
variance chart simultaneously.
-two-sided EWMA mean chart：
against sample number t for t=1, 2, . . .
(Crowder, 1987a,b, 1989; Lucas and Saccuci, 1990)
-high-side EWMA variance chart：
against sample number t for t=1, 2, . . .
•
(Crowder andHamilton, 1992)
the sample variance.
國立雲林科技大學 工業工程與管理所
Introduction-EWMA(EE)
What’s EE?
-EE consists of a two-sided EWMA mean chart
and a two-sided EWMA variance chart.
-two-sided EWMA variance chart：
against sample number t for t=1, 2, . . ..
and
(When process
variance is equal to the target variance )
國立雲林科技大學 工業工程與管理所
Introduction-combined two-sided
CUSUM(CC)
For the mean chart：
St=
and
Tt=
against sample number t
For the variance chart：
2
Vt= max 0,V t 1  log  S t   K VU
and
2
min
0,

log
 K t1  S t   K VL
Kt=
against sample number t
國立雲林科技大學 工業工程與管理所
Introduction-omnibus EWMA
The omnibus EWMA：
for i=1,2…,where 0<λ<1
選擇上式中的一些α，其主要是當σ≧ σ0時，展現局
部的敏感度，以及在分散中增加敏感度。而當σ≦σ0
時探索均數的小偏移是較有效的。
國立雲林科技大學 工業工程與管理所
Introduction-STD X-bar and R charts
X-bar control chart：
warning limits:
action limits :
where 0<w<k
R control chart：
warning limits:
action limits :
where wR(ni) and kR(ni)是標準差相關範圍的數目
(R/σ)。
國立雲林科技大學 工業工程與管理所
Introduction-twos-tage sampling (TSS)
samples of size n0 are taken from the process
at regular time intervals.
If one item’s X value of the sample is close to
the target ,then the sampling is interrupted.
Otherwise the sampling goes on to the
second stage.
the X-bar and R values are computed based
on the whole sample size n0 .
國立雲林科技大學 工業工程與管理所
Introduction-DS X-bar & DS S chart
DS X-bar chart was developed to improve the
statistical efficiency (in terms of ARL)
without increased sampling, or alternatively,
to reduce the sampling without reducing the
statistical efficiency. Daudin (1992) and He and Grigoryan (2002, 2003)
the DS s charts result in a significant
reduction in average sample size without
decreasing the out-of-control ARL in
comparison withthe traditional s charts. He and
Grigoryan (2002, 2003)
國立雲林科技大學 工業工程與管理所
The joint DS X-bar and s charts
取n2得Y-bar
取n2得S12
國立雲林科技大學 工業工程與管理所
Formulation of joint statistical design
of the DS X-bar and s charts(一)
the out-of-control ARL of the joint DS X-bar and s charts.
發出警報的機率
發出警報的機率
國立雲林科技大學 工業工程與管理所
Formulation of joint statistical design
of the DS X-bar and s charts(二)
解釋上述最佳模式：
-the probability of taking the 2’nd sample:
-取n2的集合:
-P(A∪B∪C)=
-α=在製程均數及變異數沒有偏移時，卻下在管制外的結
論。
-β=在製程均數及變異數有偏移時，卻下在管制內的結論。
國立雲林科技大學 工業工程與管理所
Formulation of joint statistical design
of the DS X-bar and s charts(三)
Note that since ARL1=   the
optimization model (1)–(4) becomes:
1
1
X
S
where Pa1 X,Pa2 X,Pa1s,Pa2s :第一、二階層在管制內的機
率。
國立雲林科技大學 工業工程與管理所
Solving the optimization problem
using genetic algorithm(一)
optimization problem formulated by model
(5)–(9) is characterized by mixed continuousdiscrete variables, and discontinuous and
non-convex solution space.
The operation of the genetic algorithm : (a)
create a random initial solution; (b) evaluate
fitness, i.e., the objective function that min
ARL;(c) reproduction and mutation; (d)
generate new solutions.
GA find a global optimum solution with a high
probability.
國立雲林科技大學 工業工程與管理所
Solving the optimization problem
using genetic algorithm(二)
Crossover is made up in hope that new
chromosomes will have good parts of old
chromosomes and maybe the new
chromosomes will be better. However it is
good to leave some part of population survive
to next generation.
Mutation is made to prevent the search
falling into local extremes, but it should not
occur very often, because then GA will in fact
change to random search.
In this paper the population size=1000,
crossover probability=0.5 and mutation
probability=0.06.
國立雲林科技大學 工業工程與管理所
Performance of the joint DS X-bar
and s charts
Comparison with combined EWMA chart,
combined CUSUM chart, and omnibus EWMA
chart :
- The ARL was calculated using computer
simulation.(10000 independence runs)
- Joint DS-1 was optimized for detecting the
shift with δ=0.169 and λ=1.188.
- All tabulated data for joint DS X-bar and s
charts were confirmed by using Monte Carlo
simulation with MATLAB
國立雲林科技大學 工業工程與管理所
Performance of the joint DS X-bar
and s charts
for detecting the shift with δ=0.169 and λ=1.188
國立雲林科技大學 工業工程與管理所
Performance of the joint DS X-bar
and s charts
Joint DS-1較佳
Joint DS1
較
佳
國立雲林科技大學 工業工程與管理所
Performance of the joint DS X-bar
and s charts
in process mean with δ≧0.75 and shifts in
process standard deviation with λ≧1.3 the
joint DS scheme is better.
the combined EWMA and CUSUM outperform
the DS for shifts with δ≦ 0.5 and λ ≦1.2.
EWMA,CUSUM在反應偏移上可能有延遲出現，這種延
遲稱為”inertia problem”(慣性問題)。因此
EWMA,CUSUM在探索小偏移勝過DS的優點必須忽略
掉此問題(inertia problem)才可。
國立雲林科技大學 工業工程與管理所
Performance of the joint DS X-bar
and s charts
Comparison with joint STD, VSS, and TSS
charts :
-The design parameters of all the schemes
were chosen such that the in-control ARL
= 433, and the average sample size=5
-Joint DS-2 was optimized for detecting the
shift with δ=0.5 and λ=1.1.
-Joint DS-3 was optimized for detecting the
shift with δ=0.75 and λ=1.5.
 joint DS X-bar and s charts result in a better
statistical performance than the rest of the
charts.
國立雲林科技大學 工業工程與管理所
Performance of the joint DS X-bar
and s charts
國立雲林科技大學 工業工程與管理所
Performance of the joint DS X-bar
and s charts
國立雲林科技大學 工業工程與管理所
Conclusions
The results of the comparison with the
combined EWMA and CUSUM, the omnibus
EWMA show that in process mean with
δ≧0.75 and shifts in process standard
deviation with λ≧1.3 the joint DS scheme is
better.
In comparison with the joint STD, TSS and
VSS X and R charts,the results show the
proposed joint DS X-bar and s chart scheme
outperforms these schemes for all shifts in
process mean with 0<δ≦1.0 and shifts in
process standard deviation with 1.0<λ≦2.0.
國立雲林科技大學 工業工程與管理所