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Journal of Statistical Modeling and Analytics
Vol. 1 No. 1, 69-79, 2010
Mean Chart and Median Chart with
K – Control Rules
Wong WK1
Universiti Tunku Abdul Rahman, Malaysia
Email: [email protected]
Sim CH2
Universiti of Malaya, Malaysia
ABSTRACT
In the common control charting procedure, an out-of-control signal
is detected when a single point falls beyond the upper control
limit ( UCL ) or below the lower control limit ( LCL ) of a control
chart. Control charts based on this criterion are easy to construct
and implement by the quality control engineers. They are able to
detect large shifts on the process mean and variation quickly. The
drawback of this simple criterion is that the resulting charts are
ineffective in detecting small shifts in either process mean or
variation. To overcome this drawback, we propose a control rule
that detects an out-of-control signal when K consecutive points
all fall above the upper probability limits or all fall below the
lower probability limits of the constructed control chart. We shall
restrict our discussion to the construction and performance of the
mean chart and median chart in detecting shift in process mean
based on the K − control rule.
Keywords: Statistical Quality Control, K − Control Rule, Probability
Limits, Average Run Length.
ISSN 2180-3102
© 2010 Malaysia Institute of Statistics, Faculty of Computer and Mathematical Sciences,
Universiti Teknologi MARA (UiTM), Malaysia
69
Journal of Statistical Modeling and Analytics
Introduction
In the common control charting procedure, an out-of-control signal is
detected when a single point falls beyond the upper control limit (UCL )
or below the lower control limit ( LCL ) of a control chart. However, the
drawback of this criterion is that the resulting charts are unable to
detect small shifts in process mean and variation, i.e. they tend to have
large out-of-control average run length ( ARL1 ) values when shifts in
the process mean or variation are small.
Procedures to improve the performance of the classical Shewhart
charts in detecting small shifts in either the process mean or variation
have been widely discussed in the literature. One of the well-known and
established methods is to perform selected supplementary sensitizing
rules (Western Electric (1956), Nelson (1984)) simultaneously to the
Shewhart control charts. Two of the commonly used sensitizing rules
are (i) two out of three successive points fall beyond two-sigma limits,
(ii) four out of five successive points fall beyond one-sigma limits. These
tests show positive results and are able to reduce the ARL1 values in
detecting small shifts in process mean and variation. However, it was
pointed out by Champ and Woodall (1987) that these additional
supplementary tests would significantly increase the undesirable false
alarm rate. Indeed, Montgomery (2005) suggested that the sensitizing
rules should be used cautiously, as an excessive number of false alarms
can be harmful to an effective statistical process control program.
Instead of using the supplementary sensitizing rules in improving the
performance of the 3 − sigma x − chart, Derman and Ross (1997)
proposed the following two alternating control rules.
Rule I: An out-of-control signal is detected if (i) two successive points
fall above a specially designed UCL , or (ii) two successive points
fall below a specially designed LCL , or (iii) either one of the two
successive points is above the UCL and the other is below the LCL .
Rule II: An out-of-control signal is detected if any two of three
successive points fall (i) above the newly specified UCL , or (ii)
below the newly specified LCL , or (iii) beyond either one of the
newly specified control limits.
Their study revealed that their proposed schemes increase the
performance of the x − chart in detecting moderate shifts in process
mean as compared to the Shewhart x − chart.
70
Mean Chart and Median Chart with K − Control Rules
Klein (2000) has simplified the above two control rules of Derman
and Ross (1997) by eliminating part (iii) of both rules. His study shows
that his simplify schemes achieve the objective of increasing the
sensitivity of detecting small to moderate shifts of the process mean
from µ to µ + δσ , 0 ≤ δ ≤ 2.6 .
In this study, we consider the control rules where an out-of-control
signal is declared when K consecutive points are above a specified
upper probability limit or K consecutive points are below a specified
lower probability limit where the value K can either taken to be 2, 3 or
4. We shall call this the “K − control rule” in the sequel. Note that for the
case when K = 2, the K − control rule” is the first out-of-control rule of
Derman and Ross modified by Klein (2000). We shall restrict our
discussion to the construction and performance of the mean chart ( x p −
~
chart) and median chart ( X p − chart) in detecting shift in process mean
based on the K − control rule and for samples taken from the
Normal(γ , β 2 ) and Laplace(γ , β ) processes. The construction of
probability limits of the K − control rule is discussed in Section 2.
~
Section 3 discusses the performance of the x p − chart and X p −
chart when the K − control rule is employed.
The Probability Limits of the K – Control Rule
Let W be a sample statistic that measures some quality characteristic
of interest. Control chart of the sample statistic W usually consists of
an upper control limit ( UCL ), a central line ( CL ) and a lower control
limit ( LCL ). These limits divide the set of possible values of W into
three quality zones.
They are:
•
•
•
Target zone ( T ) : consists of values of W between UCL and LCL .
Upper action zone ( A + ) : consists of values of W beyond UCL .
Lower action zone ( A −) : consists of values of W below LCL .
In evaluating the average run length (ARL ) , we follow the argument
of Page (1955). Let the probability that a point falls in each of the
three regions T , A + and A − be p , q1 and q 2 , respectively, where
p + q1 + q 2 = 1 . Action will be taken if K consecutive points fall in to
the zones A + or A − . Let L denote the ARL of the stopping rule, and L+i
( L−i ), i = 1, 2, ..., K − 1 , be the additional average number of points
71
Journal of Statistical Modeling and Analytics
needed before an action is taken when the last i sample points have
fallen in zone A + ( A − ). By taking expectations conditional upon the
result of the first sample, we have
L = p(1 + L ) + q1 (1 + L1+ ) + q 2 (1 + L1− ) .
= 1 + pL + q1 L1+ + q 2 L1−
(1)
By taking expectations conditional on the results of the next sample
when the last i sample points have fallen in zone A + , we have
L+i = 1 + pL + q1 L+i +1 + q 2 L1− , i = 1, 2, ..., K − 2
L+K −1 = 1 + pL + q 2 L1− .
(2)
Similar argument gives
L−i = 1 + pL + q1 L1+ + q 2 L−i +1 , i = 1, 2, ..., K − 2
L−K −1 = 1 + pL + q1 L1+ .
(3)
L+K −1 ,
Solving the set of ( K − 1 ) equations in (2) for
( K − 1 ) equations in (3) for L−K −1 , ..., L1− , we obtain
...,
L1+
, and set of
L1+ =
1 − q1K −1
(1 + pL + q 2 L1− )
1 − q1
(4)
L1− =
1 − q 2K −1
(1 + pL + q1 L1+ )
1− q2
(5)
and
respectively. Finally, by solving equations (4) and (5) for L1+ and L1− in
term of L , and by substituting them in (1) yields the solution
L=
(1 − q1K )(1 − q 2K )
(1 − q1 )(1 − q 2 ) − q1 q 2 (1 − q1K −1 )(1 − q 2K −1 ) − p(1 − q1K )(1 − q 2K )
.
(6)
Under the common assumption that the probability of a sample
statistic falls above the UCL is equal to the probability that it falls below
the LCL , i.e. by taking q1 = q2 = q which leads to p = 1− 2q , equation
(6) is then reduced to
L=
(1 − q K ) 2
(1 − q ) 2 − q 2 (1 − q K −1 ) 2 − (1 − 2q )(1 − q K ) 2
.
(7)
By fixing L at its desired value, q can then be found by solving
equation (7). For example, by taking q1 = q2 = q, the ARL of a control
chart with K − control rule when K = 2 is given by
72
Mean Chart and Median Chart with K − Control Rules
ARL =
1+ q
2q 2
.
By taking the desired ARL value to be 370.37 and K = 2 , we have
~
q = 0.0374235. The lower and upper probability limits of the X p − chart
and x p − chart given in equations (8) and (9) with K − control rule
( K = 2 ) should then be constructed using q = 0.0374235. Note that,
from equation (7), K − for control rule with K = 3 , we have q =
0.1150585; whereas for K = 4 , we have q = 0.20277.
The sampling distributions of the sample mean x (Sim, 2000) and
~
sample median X (Wong, 2007) are required To evaluate the probability
~
limits for both the x p − chart and X p − chart with K − control rule.
By taking a desired value of ARL and by estimating q from equation
~
(7), the corresponding lower and upper probability limits of the X p −
chart are given as
LCL X~ ( K ) = µ + Aq;K ,n σ
UCL X~ ( K ) = µ + A1− q; K ,n σ
(8)
and the probability limits of the x p − chart are
LCL x ( K ) = µ − B q; K , n
UCL x ( K ) = µ + B1− q; K , n
σ
n
σ
n
(9)
where Aα ;K ,n and Bα ;K ,n are factors determined based on the αth
percentile of the sampling distribution of the sample median and sample
mean, respectively.
~
In this study, the X p − chart and x p − chart with K − control
rule for the Normal(γ , β 2 ) and Laplace(γ , β ) populations are
considered. Note that under the normality assumption, both the factors
B q; K ,n and B1− q;K ,n depend only on K but not the sample size n and
that Bq;K , n = B1− q;K ,n . The values of factors, Aq;K ,n , A1− q;K ,n , Bq;K ,n
and B1− q;K ,n evaluated with ARL = 370.37 , corresponds to a false alarm
rate of 0.0027, are given in Table 1 for Normal (0, 1) and Laplace(0, 1)
distributions. Selected sample size of up to 20 are given in Table 1
as only samples of small size are usually used in industry applications.
73
Journal of Statistical Modeling and Analytics
~
Performance of the X p − Chart and x p − Chart with
the K − Control Rule
~
To assess the performance of the X p − chart and x p − chart with K −
control rule, we study the average run length ARL of the corresponding
charts when the process mean shifts from µ to µ + δσ . The values of
~
ARL for the X p − chart and x p − chart constructed with K − control
rule ( K = 2, 3, 4 ) are tabulated in Table 2 and Table 3 for samples taken
from Normal (γ , β 2 ) and Laplace(γ , β ) populations, respectively. For
~
comparison purpose, the values of ARL for the X p − chart and x p −
chart constructed with the conventional out-of-control rule (stated as
K = 1 ) are also given in the tables.
~
Table 1: Factors for Constructing the Probability Limits for the X p − Chart
and x p − Chart with K − Control Rule, K = 2, 3, 4 , when Samples of Size
n are Taken from Normal (0, 1) and Laplace(0, 1) Populations. The Values
of Factors are Evaluated with a False Alarm Rate of 0.0027. Note that q
Takes the Values 0.0374235, 0.1150585 and 0.20277 when K Takes the
Values 2, 3 and 4, Respectively.
Normal (0, 1)
n
2
3
4
5
7
10
15
20
Aq; K , n
A1− q;K ,n
Aq; K ,n
A1− q;K , n
Aq; K ,n
A1− q;K ,n
Aq; K , n
A1− q;K ,n
Aq; K ,n
A1− q;K , n
Aq; K ,n
A1− q;K ,n
Aq; K , n
A1− q;K ,n
Aq; K ,n
A1− q;K , n
K =2
K =3
K =4
–1.259642
1.259634
–1.193750
1.193747
–0.972948
0.972945
–0.954366
0.954363
–0.817408
0.817405
–0.662638
0.662635
–0.568152
0.568150
–0.482789
0.482786
–0.848570
0.848567
–0.801976
0.801974
–0.654501
0.654500
–0.641362
0.641361
–0.549541
0.549540
–0.445802
0.445800
–0.382300
0.382299
–0.324964
0.324962
–0.588148
0.588148
–0.555165
0.555164
–0.453356
0.453353
–0.444056
0.444055
–0.380552
0.380552
–0.308809
0.308808
–0.264842
0.264841
–0.225152
0.225151
74
B q; K , n
B1− q;K ,n
B q; K , n
B1− q;K , n
B q; K , n
B1− q;K ,n
B q; K , n
B1− q;K ,n
B q; K , n
B1− q;K , n
B q; K , n
B1− q;K ,n
B q; K , n
B1− q;K ,n
B q; K , n
B1− q;K , n
K =2
K =3
K =4
1.781401
1.781401
1.781401
1.781401
1.781401
1.781401
1.781401
1.781401
1.781401
1.781401
1.781401
1.781401
1.781401
1.781401
1.781401
1.781401
1.200058
1.200058
1.200058
1.200058
1.200058
1.200058
1.200058
1.200058
1.200058
1.200058
1.200058
1.200058
1.200058
1.200058
1.200058
1.200058
0.831768
0.831768
0.831768
0.831768
0.831768
0.831768
0.831768
0.831768
0.831768
0.831768
0.831768
0.831768
0.831768
0.831768
0.831768
0.831768
Mean Chart and Median Chart with K − Control Rules
Laplace( 0, 1)
n
2
3
4
5
7
10
15
20
Aq; K ,n
A1− q;K , n
Aq; K ,n
A1− q;K ,n
Aq; K , n
A1− q;K ,n
Aq; K ,n
A1− q;K , n
Aq; K ,n
A1− q;K ,n
Aq; K , n
A1− q;K ,n
Aq; K ,n
A1− q;K , n
Aq; K ,n
A1− q;K ,n
K =2
K =3
K =4
–1.282225
1.282225
–1.031343
1.031343
–0.829554
0.829554
–0.763044
0.763044
–0.624110
0.624110
–0.487152
0.487152
–0.397563
0.397563
–0.329454
0.329454
–0.782965
0.782966
–0.609107
0.609107
–0.511887
0.511887
–0.460647
0.460647
–0.381976
0.381976
–0.305590
0.305590
–0.249949
0.249949
–0.209799
0.209799
–0.511538
0.511538
–0.386667
0.386667
–0.336847
0.336847
–0.297033
0.297033
–0.248645
0.248645
–0.202956
0.202956
–0.165669
0.165669
–0.140484
0.140484
B q; K , n
B1− q;K , n
B q; K , n
B1− q;K ,n
B q; K , n
B1− q;K ,n
B q; K , n
B1− q;K , n
B q; K , n
B1− q;K ,n
B q; K , n
B1− q;K ,n
B q; K , n
B1− q;K , n
B q; K , n
B1− q;K ,n
K =2
K =3
K =4
1.813341
1.813341
1.803094
1.803094
1.797354
1.797354
1.793800
1.793800
1.789755
1.789755
1.786849
1.786849
1.784756
1.784756
1.783793
1.783793
1.107284
1.107284
1.133953
1.133953
1.148546
1.148546
1.157826
1.157826
1.169001
1.169001
1.177827
1.177827
1.184977
1.184977
1.188650
1.188650
0.723425
0.723425
0.756597
0.756597
0.774307
0.774307
0.785306
0.785306
0.798208
0.798208
0.808100
0.808100
0.815907
0.815907
0.819844
0.819844
~
Table 2: ARL of the X p − Chart and x p − Chart with K − Control Rule
( K = 1, 2, 3, 4 ) in Detecting Shifts of the Process Mean from µ
to µ + δσ where Samples of Size n are Taken from a Process with
Normal(γ , β 2 ) Distribution.
n
δ
~
Xp
K =1
xp
~
Xp
K =3
K =2
xp
~
Xp
xp
~
Xp
K =4
xp
3 0.0 370.3743
370.3907
370.3708 370.3712
370.3685
370.3759
370.3694 370.3715
0.2 258.2143
227.7224
209.4670 178.7858
186.3262
157.0062
173.3582 145.1200
0.4 125.1807
94.0443
80.4875
59.4618
65.7723
48.7933
59.1640
44.2010
0.6
59.2234
40.0324
33.9647
23.4342
27.5941
19.6058
25.3224
18.4412
0.8
29.6878
18.7862
16.5883
11.2352
14.0805
10.0809
13.5692
10.1144
1.0
15.9769
9.7647
9.3152
6.4164
8.5125
6.3315
8.7236
6.8229
1.5
4.6471
2.9081
3.6593
2.8516
4.1681
3.5610
4.9330
4.4188
2.0
2.0649
1.4734
2.4018
2.1477
3.2359
3.0721
4.1623
4.0428
4 0.0 370.3661
370.3907
370.3677 370.3712
370.3711
370.3759
370.3683 370.3715
0.2 220.0608
200.0741
168.4214 150.2423
134.3960 118.6848
146.2159
129.5332
0.4
87.0951
71.5523
53.1985
43.6284
43.2984
35.7572
39.2083
32.6263
0.6
35.9398
27.8213
20.4571
16.2757
17.1589
14.0015
16.2725
13.5440
0.8
16.4983
12.3826
9.7576
7.7946
8.8905
7.4059
9.0621
7.7673
1.0
8.4633
6.3030
5.6248
4.6119
5.6964
4.9215
6.2631
5.5855
Continued
75
Journal of Statistical Modeling and Analytics
Continued from Table 2
n
5
δ
~
Xp
K =1
xp
~
Xp
K =3
K =2
xp
~
Xp
xp
~
Xp
K =4
xp
1.5
2.5352
2.0000
2.6408
2.3922
3.4046
3.2293
4.2932
4.1554
2.0
1.3475
1.1886
2.0939
2.0405
3.0424
3.0154
4.0240
4.0077
0.0 370.3416
370.3907
370.3697 370.3712
370.3750
370.3759
370.3739 370.3715
0.2 218.2345
177.7323
165.0617 128.7586
142.5051
109.5196
130.6299
99.7750
0.4
85.3881
56.5932
51.2249
33.7466
41.4827
27.7872
37.5331
25.6078
0.6
34.8985
20.5636
19.5261
12.2093
16.3637
10.8377
15.5597
10.7772
0.8
15.8972
8.8558
9.2964
5.9410
8.5074
5.9603
8.7213
6.4966
1.0
8.1119
4.4953
5.3779
3.6734
5.4941
4.1899
6.0851
4.9501
1.5
2.4297
1.5665
2.5769
2.1881
3.3583
3.0962
4.2572
4.0590
2.0
1.3120
1.0758
2.0795
2.0107
3.0350
3.0032
4.0196
4.0014
10 0.0 370.3743
370.3907
370.3677 370.3712
370.3672
370.3759
0.2 143.4612
370.3650 370.3715
109.9669
96.5370
71.3701
80.0888
58.8089
72.3348
53.1785
0.4
37.8519
24.1706
21.5859
14.2124
18.0455
12.3950
17.0475
12.1397
0.6
12.4086
7.4023
7.6437
5.1834
7.2613
5.3682
7.6358
5.9767
0.8
5.1585
3.1337
3.9604
2.9678
4.4061
3.6484
5.1374
4.4909
1.0
2.6846
1.7716
2.7159
2.2822
3.4593
3.1556
4.3371
4.1009
1.5
1.1814
1.0424
2.0375
2.0046
3.0143
3.0012
4.0072
4.0005
2.0
1.0092
1.0004
2.0005
2.0000
3.0001
3.0000
4.0000
4.0000
15 0.0 370.3743
370.3907
370.3656 370.3712
370.3659
370.3759
370.3702 370.3715
0.2 113.6483
76.2941
72.8682
46.8696
59.7593
38.3994
53.9237
34.9623
0.4
25.3471
13.6204
14.5586
8.4516
12.6058
7.9172
12.3095
8.2166
0.6
7.7560
4.0088
5.2835
3.4210
5.4345
3.9946
6.0339
4.7828
0.8
3.2557
1.8546
3.0051
2.3217
3.6746
3.1816
4.5137
4.1199
1.0
1.8198
1.2366
2.2975
2.0561
3.1659
3.0227
4.1090
4.0118
1.5
1.0475
1.0025
2.0054
2.0001
3.0015
3.0000
4.0006
4.0000
2.0
1.0006
1.0000
2.0000
2.0000
3.0000
3.0000
4.0000
4.0000
Examination of all these tables reveals that the values of the incontrol ARL , are close to the desired value, 370. Thus, fair
comparison can now be made based on the out-of-control ARL
( ARL1 ) values.
The ARL1 values of Table 2 and Table 3 decrease when K increases
even when the shift in the process mean is small. For example, for a
~
X p − chart, with samples of size n = 5 taken from a Laplace(γ , β )
population with mean µ = γ and variance σ 2 = 2β 2 , we have ARL1 = 273
when K = 1 ; while ARL1 = 163 when K = 2 in detecting a shift in mean
from µ to µ + 0.2σ . In another word, when the process mean shifts
~
from µ to µ + 0.2σ , the X p − chart will detect the shift on the average
76
Mean Chart and Median Chart with K − Control Rules
~
Table 3: ARL of the X p − Chart and x p − Chart with K − Control Rule
( K = 1, 2, 3, 4 ) in Detecting Shifts of the Process Mean from µ to
µ + δσ where Samples of Size n are Taken from a Process with
Laplace(γ , β ) Distribution.
n
δ
~
Xp
K =1
xp
~
Xp
K =2
3 0.0 370.3253
0.2 318.7948
370.3498
296.9734
0.4 218.4704
0.6 133.7128
176.7958
95.0529
92.6619
35.8670
xp
370.3718 370.3728
231.4998 211.6875
~
Xp
K =3
xp
~
Xp
K =4
xp
370.3728
165.8537
370.3672
161.5457
370.3737 370.3745
122.5727 133.6116
77.8259
29.7862
46.2000
14.6735
47.4699
17.4871
27.4004
10.9623
36.5580
14.6987
0.8
1.0
78.6098
45.6763
50.3482
27.0296
14.7746
6.7142
13.0123
6.7162
6.9478
4.7739
8.6252
5.5188
6.9564
5.4599
8.3858
6.0472
1.5
2.0
11.9043
3.3608
6.4873
2.1542
2.6319
2.1413
2.7691
2.1538
3.3604
3.0859
3.4531
3.0882
4.3185
4.0767
4.3757
4.0711
4 0.0 369.9904
0.2 276.2455
370.3416
266.8217
370.3682 370.3728
172.8521 175.5819
370.3663
118.2006
370.3746
131.5735
0.4 144.2606
0.6 68.4807
132.2144
60.8031
50.2614
16.3894
53.6733
18.8700
26.9279
9.3695
34.1333
12.5739
20.1056
8.7532
27.8774
11.5091
370.3717 370.3671
90.0675 109.7374
0.8
1.0
32.3723
15.6553
28.4694
13.9297
6.6850
3.7165
8.2609
4.5855
5.2531
3.9379
6.6055
4.5378
5.8301
4.7880
6.9255
5.2652
1.5
2.0
3.1307
1.2736
3.1459
1.3667
2.2188
2.0313
2.3674
2.0532
3.1287
3.0178
3.2152
3.0282
4.1082
4.0146
4.1713
4.0212
5 0.0 370.4070
0.2 273.2579
370.3334
239.3736
370.3718 370.3728
163.1698 148.2892
370.3703
105.2263
370.3750
110.2650
0.4 139.4163
0.6 64.5466
101.1578
41.1957
44.1341
13.4506
39.6050
13.3647
21.1214
7.1862
26.3383
9.8683
14.7727
7.1767
22.4422
9.5529
370.3737 370.3671
74.4619 92.7299
0.8
1.0
29.7144
13.9642
17.6904
8.2324
5.1600
3.0925
6.0430
3.6095
4.4710
3.6055
5.4870
3.9959
5.2237
4.5188
6.0440
4.8127
1.5
2.0
2.5784
1.1603
2.0108
1.1414
2.1313
2.0174
2.1873
2.0187
3.0778
3.0100
3.1048
3.0090
4.0668
4.0085
4.0794
4.0063
10 0.0 370.2109
0.2 160.5076
370.3743
145.1555
370.3794 370.3723
68.4999 77.7484
370.3725
41.1276
370.3685
58.1741
370.3691 370.3766
31.0170 50.4465
0.4
0.6
41.4372
11.5422
36.1074
10.7122
10.7843
3.5982
14.9245
5.1902
7.3708
3.9203
11.8369
5.1614
7.4403
4.7642
11.3905
5.7861
0.8
1.0
3.8420
1.7252
4.0818
2.0619
2.3875
2.0990
2.9367
2.2756
3.2276
3.0553
3.5959
3.1554
4.1858
4.0437
4.4657
4.1119
1.5
2.0
1.0259
1.0007
1.0639
1.0016
2.0026
2.0000
2.0076
2.0001
3.0013
3.0000
3.0030
3.0000
4.0009
4.0000
4.0018
4.0000
15 0.0 370.3743
0.2 113.2463
370.4070
96.6079
370.3723 370.3728
42.7855 49.5619
370.3729
25.3196
370.3711
37.8402
370.3700 370.3685
19.2680 33.5736
0.4
0.6
21.3256
5.0516
18.0096
4.9165
5.8931
2.6057
8.5813
3.3972
5.0840
3.3599
7.6675
3.9339
5.7064
4.2930
7.9360
4.7422
0.8
1.0
1.7663
1.1518
2.0661
1.2927
2.1134
2.0195
2.3154
2.0598
3.0627
3.0098
3.1796
3.0274
4.0485
4.0073
4.1269
4.0169
1.5
2.0
1.0020
1.0000
1.0047
1.0000
2.0001
2.0000
2.0003
2.0000
3.0001
3.0000
3.0001
3.0000
4.0000
4.0000
4.0000
4.0000
77
Journal of Statistical Modeling and Analytics
after 273 samples have been taken under the conventional out-of-control
rule, whereas, when K − control rule, K = 2 is applied, this shift can be
detected on the average after 163 samples have been taken. This is a
~
significant improvement. The performance of the X p − chart can be
improved further by employing the K − control rule where K equals to 3
or 4, as its ARL1 value is 105 and 74, respectively.
~
When comparing the performance of the X p − chart and x p − chart
with K − control rule, we found that for samples taken from a
Normal(γ , β 2 ) population, the x p − chart has the lowest ARL1 values
for all magnitude of shift in process mean. Whereas, for samples taken
~
from a Laplace(γ , β ) population, in general, X p − chart outperforms the
x p − chart in detecting shifts in process mean when a K − control rule
with K = 3 or K = 4 is applied.
Summary
In our studies, we propose to apply the K − control rule ( K = 2, 3, 4 ) on
~
X p − chart and x p − chart, in which an out–of–control signal is obtained
when K consecutive points fall beyond same side of one of the
probability limits. The results given in Section 3 show that the ARL1
values decrease as K increases for the both charts considered. The
proposed K − control rule is effective in detecting small to moderate
shifts in the process mean. For industry applications, for ease of
implementation and interpretation, K − control rule with K equals to 2
or 3 is recommended.
References
[1] Champ, C. W. and Woodall, W. H. (1987). Exact Results for
Shewhart Control Charts with Supplementary Runs Rules,
Technometrics, 29, 393-399.
[2] Derman, C. and Ross, S. M. (1997). Statistical Aspects of Quality
Control. Academic Press, San Diego.
[3] Klein, M. (2000). Two Alternatives to the Shewhart X Control
Chart, Journal of Quality Technology, 32, 427-431.
78
Mean Chart and Median Chart with K − Control Rules
[4] Montgomery, D. C. (2005). Introduction to Statistical Quality
Control, 5th edition. John Wiley, New York.
[5] Nelson, L. S. (1984). The Shewhart Control Chart – Tests for Special
Causes, Journal of Quality Technology, 16, 237-239.
[6] Page, E. S. (1955). Control Charts with Warning Lines, Biometrika,
42, 243-257.
[7] Sim, C. H. (2000). X Charts With Warning Limits for Non–Gaussian
Processes, Technical Report 2/2000, IMS, UM.
[8] Western Electric (1956). Statistical Quality Control Handbook.
Western Electric Corporation, Indianapolis.
[9] Wong, W. K. (2007). Control Charts with Probability Limits,
Unpublished PhD Thesis.
79