# Designing a Repetitive Group Sampling Plan for Weibull Distributed Processes

Authors: Aijun Yan; Sanyang Liu
Journal: Mathematical Problems in Engineering
 (2016)

---

## Abstract

Acceptance sampling plans are useful tools to determine whether the submitted lots should be accepted or rejected. An efficient and economic sampling plan is very desirable for the high quality levels required by the production processes. The process capability indexC
L is an important quality parameter to measure the product quality. Utilizing the relationship between the C
L index and the nonconforming rate, a repetitive group sampling (RGS) plan based on C
L index is developed in this paper when the quality characteristic follows the Weibull distribution. The optimal plan parameters of the proposed RGS plan are determined by satisfying the commonly used producer’s risk and consumer’s risk at the same time by minimizing the average sample number (ASN) and then tabulated for different combinations of acceptance quality level (AQL) and limiting quality level (LQL). The results show that the proposed plan has better performance than the single sampling plan in terms of ASN. Finally, the proposed RGS plan is illustrated with an industrial example.


---

## Body

## 1. Introduction

In the manufacturing industries, acceptance sampling plan has been widely used for inspection purposes. It has played an important role in the inspection of raw materials, semifinished products, and finished products from product manufacture to marketing. Acceptance sampling plans provide the producer and the consumer with acceptance or nonacceptance criteria meeting both of their requirements for product quality, in which the decision is made on the sample information taken from the submitted lot. Because of human error and fatigue during the sampling inspection, there is a chance of making errors. The chance of rejecting a good lot is called the producer’s risk, and the chance of accepting a bad lot is called the consumer’s risk. One purpose of an acceptance sampling plan is to minimize the sample size so as to reduce the cost and time of the experiment while satisfying the producer’s risk as well as the consumer’s risk at the specified quality levels. So the use of an acceptance sampling plan earns good reputation of the organization and increases the profit. For more applications of the acceptance sampling plan can be found in Fernández [1], Wang [2], Yan et al. [3], Balamurali and Usha [4], and Wu et al. [5].During the inspection of the products, the producers care about the inspection cost which is directly related to the sample size. So the researchers want to propose a more efficient sampling plan to lower the inspection cost, time, and efforts. A single sampling plan is very popular in the industrial engineering because of the simplicity, but the decision of lot sentencing based only on the single sample may undermine good relations between the producers and the consumers in some cases (Liu and Wu [6]). Recently, Sherman [7] proposed the attributes of repetitive group sampling (RGS) plan whose operational procedure is similar to that of the sequential sampling scheme. Balamurali and Jun [8] extended the concept of RGS to variables’ inspection and then showed that it is more efficient than single sampling and double sampling in terms of the average sample number (ASN) while providing the desired protection to producers and consumers. The RGS plan has been used widely in the industries when the inspection is costly and destructive. Aslam et al. [9] designed the repetitive sampling plan using the process loss function. Liu and Wu [6] designed the repetitive sampling plan for unilateral specification limit. Yen et al. [10] proposed a variable repetitive group sampling plan based on one-sided process capability indices. Wu et al. [11] developed a variable repetitive group sampling plan based on the capability index C
p
m
k. Aslam et al. [12] presented three repetitive types of sampling plans using the generalized process capability index of multiple characteristics. A modified variables RGS plan based on the process capability index C
p
k for normally distributed processes was investigated by Lee et al. [13].Process capability analysis is an effective method to measure the performance and potential capability of process. In the manufacturing and services industry, process capability indices (PCIs) are utilized to examine whether product quality meets the consumers’ required level. Recently, acceptance sampling plans based on process capability index have attracted many researchers. Examples include Pearn and Wu [14], Aslam et al. [15], and Wu [16]. All of the above PCIs have been developed or investigated under the assumption of normality. Nevertheless, the normality is very questionable in many processes including manufacture process, service process, and business operation process [17]. The lifetime model of many products may generally follow a nonnormal distribution which includes Weibull, exponential, gamma, Rayleigh, and Burr XII or the other distributions.The process capability indexC
L (or C
P
L) proposed by Montgomery [18] is used to assess the lifetime performance of electronic components which have a larger-the-better type quality characteristic, where L is the lower specification limit. Recently, for some well-known nonnormal lifetime distributions, statistical inferences for C
L have been considered in the literature (see [19–22]).In this paper, we will firstly develop a repetitive group sampling plan based on theC
L index for Weibull distributed processes with the lower specification limit for product acceptance determination using the close relationship between the index C
L and the product nonconforming rate p. The plan aims to minimize the sample size required for inspection while controlling the nonconforming fraction or the number of nonconformities so as to meet the requirements of the producer and the consumer. The rest of this paper is organized as follows. In Section 2, the concept of the lifetime performance index C
L is introduced briefly and the maximum likelihood estimation (MLE) of C
L is also presented. The design and operating procedure of the proposed repetitive group sampling plan based on C
L is presented; moreover the plan parameters are determined by solving the optimization problem and a detailed analysis is also discussed in Section 3. In Section 4, we will compare the efficiency of the proposed RGS plan with the single sampling plan in terms of OC curve and the average sample number (ASN). Section 5 gives an example for illustration. Finally, some concluding remarks are made in the last section.


## 2. The Lifetime Performance IndexC
L

Montgomery [18] proposed a process capability index C
L for evaluating the larger-the-better quality characteristic. The C
L index can be defined as follows: (1)
C
L
=
μ
-
L
σ
,where μ and σ are the process mean and the standard deviation, respectively, and L is the lower specification limit.The Weibull distribution is commonly used for the lifetime or durability of diverse types of manufactured items, such as ball bearings, automobile components, and electrical insulation. Suppose that the quality of interestX follows a two-parameter Weibull distribution with the cumulative distribution function (cdf): (2)
F
x
;
λ
,
θ
=
1
-
exp
⁡
-
x
λ
θ
,
x
>
0
,where θ
>
0 is the known shape parameter and λ
>
0 is an unknown scale parameter. As stated in Jun et al. [23] and Aslam and Jun [24], “the shape parameter can be assumed as known because engineering experience with a particular type of application makes such an assumption reasonable. We may use the estimated value from the past failure data even though it is not known.” Since the mean and the standard deviation of the Weibull distribution are given by μ
=
λ
Γ
1
+
1
/
θ and σ
=
λ
A, respectively, the index C
L can be expressed as(3)
C
L
=
μ
-
L
σ
=
T
1
+
1
/
θ
-
L
/
λ
A
,
–
∞
<
C
L
<
Γ
1
+
1
/
θ
A
,where A
=
Γ
(
1
+
2
/
θ
)
-
Γ
2
(
1
+
1
/
θ
) and T
(
·
) is the complete gamma function.The probability that an item will not meet the specification is called the fraction defective or nonconforming rate. LetL denote the lower specification limit of an item from the Weibull distribution; then the nonconforming rate can be defined as(4)
p
=
P
X
<
L
=
1
-
exp
⁡
-
L
λ
θ
=
1
-
exp
⁡
-
T
1
+
1
θ
-
A
C
L
θ
,
–
∞
<
C
L
<
Γ
1
+
1
/
θ
A
.It is obvious that there is a strictly increasing relationship between nonconforming ratep and the index C
L for given θ. Therefore, utilizing the one-to-one correspondence relationship between p and C
L, the C
L index can be an effective tool for evaluating product quality.Suppose thatX
1
,
X
2
,
…
,
X
n is a random sample taken from the Weibull distribution with pdf as (2); the MLE of C
L is given as (see Lee [20]) (5)
C
L
^
=
Γ
1
+
1
/
θ
-
L
/
λ
^
A
=
Γ
1
+
1
/
θ
-
L
·
n
/
W
1
/
θ
A
,where W
=
∑
i
=
1
n
X
i
θ and A
=
Γ
(
1
+
2
/
θ
)
-
Γ
2
(
1
+
1
/
θ
).Wu and Kus [25] noticed that 2
W
/
λ
θ has a chi-squared distribution with 2
n degrees of freedom, that is, 2
W
/
λ
θ
~
χ
2
n
2. Denote the cumulative distribution function (cdf) of C
L
^ as (6)
P
C
L
^
≤
c
=
P
Γ
1
+
1
/
θ
-
L
·
n
/
W
1
/
θ
A
≤
c
=
P
2
W
λ
θ
≤
2
n
L
λ
θ
T
1
+
1
θ
-
A
c
-
θ
=
F
χ
2
n
2
2
n
L
λ
θ
T
1
+
1
θ
-
A
c
-
θ
,where F
χ
2
n
2
· is the cdf of the χ
2
n
2 distribution.Since(
L
/
λ
)
θ
=
-
ln
⁡
(
1
-
p
), which can be obtained from (4), the cdf of C
L
^ can be rewritten as (7)
P
C
L
^
≤
c
=
F
χ
2
n
2
-
2
n
ln
⁡
1
-
p
·
Γ
1
+
1
θ
-
A
c
-
θ
.


## 3. Design a Repetitive Group Sampling Plan Based onC
L

Suppose that the quality characteristic of interest follows a Weibull distribution and has a lower specification limitL. There is a one-to-one relationship between the C
L index and the nonconforming rate p, so we can use C
L as a quality benchmark for accepting a lot. Then the operating procedure of the repetitive group sampling plan based on C
L is stated as follows.Step 1.
Choose the values of (p
AQL, p
LQL) at producer’s risk α and consumer’s risk β.Step 2.
Select a random sample (X
1
,
X
2
,
…
,
X
n) from the lot; then compute C
L
^ in (5).Step 3.
Accept the lot ifC
L
^
≥
k
a; reject the lot if C
L
^
<
k
r; if k
r
≤
C
L
^
<
k
a repeat Step 2 by taking a new sample for further judgment, where k
a and k
r are acceptance constant and rejection constant, respectively.There are three parametersk
a, k
r, and n in the above proposed sampling plan. Note that the proposed RGS plan will reduce to the single sampling plan if k
a
=
k
r.The probability of accepting the lot based on the single sampling can be expressed as(8)
P
a
p
=
P
C
L
^
≥
k
a
=
1
-
F
χ
2
n
2
-
2
n
ln
⁡
1
-
p
·
Γ
1
+
1
θ
-
A
k
a
-
θ
.Similarly, the probability of rejecting the lot for the single sampling is given as(9)
P
r
p
=
P
C
L
^
<
k
r
=
F
χ
2
n
2
-
2
n
ln
⁡
1
-
p
·
Γ
1
+
1
θ
-
A
k
r
-
θ
.The probability required resampling is given by(10)
R
p
=
P
k
r
≤
C
L
^
<
k
a
∣
p
=
1
-
P
a
p
-
P
r
p
.Referring to Balamurali and Jun [8], the OC (operating characteristic) function of the RGS plan based on the C
L index can be derived as follows:(11)
π
p
=
P
Accepting  the  lot
∣
p
=
P
a
p
+
R
p
·
P
a
p
+
R
p
2
·
P
a
p
+
⋯
=
P
a
p
1
-
R
p
=
1
-
F
χ
2
n
2
-
2
n
ln
⁡
1
-
p
·
Γ
1
+
1
/
θ
-
A
k
a
-
θ
1
-
F
χ
2
n
2
-
2
n
ln
⁡
1
-
p
·
Γ
1
+
1
/
θ
-
A
k
a
-
θ
+
F
χ
2
n
2
-
2
n
ln
⁡
1
-
p
·
Γ
1
+
1
/
θ
-
A
k
r
-
θ
.The average sample number (ASN) means the expected number of sampled units per lots for making decisions. Thus the ASN of the proposed RGS plan based onC
L can be calculated as follows:(12)
ASN
p
=
n
1
-
R
p
+
2
n
R
1
-
R
p
+
3
n
R
2
1
-
R
p
+
⋯
+
m
n
R
m
-
1
1
-
R
p
+
⋯
=
n
1
-
R
p
=
n
1
-
F
χ
2
n
2
-
2
n
ln
⁡
1
-
p
·
Γ
1
+
1
/
θ
-
A
k
a
-
θ
+
F
χ
2
n
2
-
2
n
ln
⁡
1
-
p
·
Γ
1
+
1
/
θ
-
A
k
r
-
θ
.Denotep
AQL as the quality of the submitted lot at AQL (acceptable quality level) and p
LQL as the quality of the submitted lot at LQL (limiting quality level). Yen and Chang [26] stated “a well-designed sampling plan must provide a probability of at least 1
-
α of accepting a lot if the nonconforming rate of the lot is at p
=
p
AQL (in high quality), and a probability of no more than β of accepting a lot if the nonconforming rate of the lot is at p
=
p
LQL (in low quality).” For the specified values of α, β, p
AQL, and p
LQL, the proposed RGS plan parameters must satisfy the following two inequalities: (13)
π
p
AQL
=
P
Accepting  the  lot
∣
p
=
p
AQL
≥
1
-
α
,
π
p
LQL
=
P
Accepting  the  lot
∣
p
=
p
LQL
≤
β
.There are several combinations of the proposed plan parameters which satisfy (13), so we select those plan parameters which minimize 1
/
2
(
ASN
(
p
AQL
)
+
ASN
(
p
LQL
)
). Then, the proposed plan parameters (n, k
a, and k
r) can be obtained by solving the following optimization model:(14)
Minimize
1
2
ASN
p
AQL
+
ASN
p
LQL
Subject  to
1
-
F
χ
2
n
2
-
2
n
ln
⁡
⁡
1
-
p
AQL
·
Γ
1
+
1
/
θ
-
A
k
a
-
θ
1
-
F
χ
2
n
2
-
2
n
ln
⁡
1
-
p
AQL
·
Γ
1
+
1
/
θ
-
A
k
a
-
θ
+
F
χ
2
n
2
-
2
n
ln
⁡
1
-
p
AQL
·
Γ
1
+
1
/
θ
-
A
k
r
-
θ
≥
1
-
α
1
-
F
χ
2
n
2
-
2
n
ln
⁡
1
-
p
LQL
·
Γ
1
+
1
/
θ
-
A
k
a
-
θ
1
-
F
χ
2
n
2
-
2
n
ln
⁡
1
-
p
LQL
·
Γ
1
+
1
/
θ
-
A
k
a
-
θ
+
F
χ
2
n
2
-
2
n
ln
⁡
1
-
p
LQL
·
Γ
1
+
1
/
θ
-
A
k
r
-
θ
≤
β
.In order to determine the proposed RGS plan parameters, the Monte Carlo simulation using statistics softwareR is made to solve the above optimization problem. Tables 1–3 display the proposed plan parameters (
n
,
k
a
,
k
r
,
A
S
N
) for the three values of the shape parameters (
θ
=
1
,
1.97,3
) of the Weibull distribution, respectively, with commonly used risks α
,
β
=
(
0.05,0.10
), (
0.05,0.05
), and (
0.10,0.10
) and different quality levels (
p
AQL
,
p
LQL
). Thus, the practitioner can know the sample size required for inspection and the decisions whether accepting the submitted lot. For example, when the shape parameter θ
=
1, (
p
AQL
,
p
LQL
)
=
(
0.05,0.08
) and (
α
,
β
)
=
(
0.05,0.10
), the plan parameters (
n
,
k
a
,
k
r
,
A
S
N
)
=
(
21,0.9418,0.9178
,
31.143
) from Table 1. It implies that, taking 21 inspected measurements randomly from the lot, the entire lot will be accepted if C
L
^ is larger than 0.9418, and the entire lot will be rejected if C
L
^ is smaller than 0.9178. Otherwise, take a new sample from the lot for further judgment if 0.9178
<
C
L
^
<
0.9418. And the ASN required for making decisions on lot sentencing is 31.143.Table 1
The proposed plan parameters whenθ
=
1.
p
A
Q
L
p
L
Q
L
α
=
0.05, β
=
0.05
α
=
0.05, β
=
0.10
α
=
0.10, β
=
0.05
n
k
a
k
r
ASN
n
k
a
k
r
ASN
n
k
a
k
r
ASN
0.01
0.02
18
0.9863
0.9821
21.9
13
0.9859
0.9823
15.494
9
0.9885
0.9800
14.615
0.03
6
0.9857
0.9711
8.6793
5
0.9826
0.9717
6.5068
3
0.9874
0.9662
5.2036
0.04
3
0.9849
0.9548
4.9917
3
0.9807
0.9562
4.4119
2
0.9864
0.9318
4.4893
0.05
3
0.9819
0.9516
4.2998
3
0.9754
0.9543
3.787
2
0.9831
0.948
3.0405
0.06
2
0.9818
0.9268
3.3224
2
0.9743
0.9375
2.7382
2
0.9766
0.9450
2.6424
0.07
2
0.9753
0.9221
2.9413
2
0.9685
0.9364
2.4898
2
0.9726
0.9440
2.4449
0.08
2
0.9735
0.9349
2.514
2
0.9609
0.9333
2.3331
2
0.9666
0.9439
2.2715
0.09
2
0.9692
0.9383
2.3188
2
0.9550
0.9309
2.2389
2
0.9611
0.9484
2.1183
0.10
2
0.9605
0.9379
2.1855
2
0.9521
0.9308
2.1772
2
0.9625
0.9593
2.0244
0.02
0.03
30
0.9781
0.9704
47.052
25
0.9785
0.9688
44.72
27
0.9791
0.9718
41.903
0.04
13
0.9757
0.9618
18.931
7
0.9769
0.9515
14.153
9
0.9756
0.9642
12.226
0.05
9
0.9717
0.9591
10.914
6
0.9702
0.9504
8.3713
5
0.9744
0.9511
7.5425
0.06
6
0.9671
0.9436
7.9504
5
0.9670
0.9447
6.5408
4
0.9718
0.9453
5.6142
0.07
5
0.9647
0.9349
6.5976
4
0.9631
0.9355
5.1674
3
0.9705
0.9343
4.3163
0.08
4
0.9643
0.9334
5.0328
3
0.9635
0.9159
4.3488
2
0.9703
0.9027
3.3567
0.09
3
0.9588
0.9135
4.01
3
0.9547
0.9084
4.0215
2
0.9694
0.9001
3.1957
0.10
3
0.9544
0.9055
3.9127
2
0.9580
0.8676
3.2213
2
0.9640
0.8942
2.9726
0.15
2
0.9397
0.8761
2.4195
2
0.9260
0.8621
2.4095
2
0.9330
0.9090
2.1418
0.05
0.06
124
0.9480
0.9382
224.08
112
0.9464
0.9384
174.79
114
0.9480
0.9386
197.76
0.07
47
0.9434
0.9316
63.899
40
0.9428
0.9282
59.678
26
0.9498
0.9256
53.586
0.08
22
0.9417
0.9209
31.77
21
0.9418
0.9198
31.143
22
0.9421
0.9238
30.272
0.09
15
0.9407
0.9089
22.958
13
0.9386
0.9084
19.292
14
0.9416
0.9166
19.475
0.10
10
0.9404
0.8897
17.505
11
0.9304
0.9050
14.016
10
0.9369
0.9101
13.086
0.15
5
0.9222
0.8561
6.7725
5
0.9058
0.8557
6.2126
4
0.9293
0.8666
5.4255
0.20
3
0.9088
0.7796
4.3559
3
0.8843
0.8008
3.7471
2
0.9231
0.7554
3.2373
0.25
2
0.9028
0.6611
3.2337
2
0.8750
0.6677
2.9365
2
0.8869
0.7841
2.4505
0.10
0.15
26
0.8856
0.8397
42.076
21
0.8813
0.8359
33.26
14
0.8998
0.8208
33.081
0.20
8
0.8761
0.7651
13.84
8
0.8635
0.7728
12.211
7
0.8821
0.7871
11.208
0.25
5
0.8605
0.6867
8.4798
5
0.8456
0.6960
7.7256
4
0.8673
0.7016
6.5574
0.30
4
0.8356
0.6531
5.8202
4
0.8081
0.6731
5.1927
3
0.8451
0.6701
4.3006
0.35
3
0.8239
0.5647
4.4808
3
0.7907
0.5823
4.0697
2
0.8449
0.4802
3.4214
0.40
2
0.7724
0.5688
3.7834
2
0.7888
0.3009
3.3957
2
0.8068
0.5173
2.8152
0.50
2
0.7408
0.3418
2.7159
2
0.6790
0.3716
2.4992
2
0.7180
0.5817
2.2161Table 2
The proposed plan parameters whenθ
=
1.97.
p
A
Q
L
p
L
Q
L
α
=
0.05, β
=
0.05
α
=
0.05, β
=
0.10
α
=
0.10, β
=
0.05
n
k
a
k
r
ASN
n
k
a
k
r
ASN
n
k
a
k
r
ASN
0.01
0.02
10
1.663
1.5908
16.458
11
1.6607
1.5964
16.959
8
1.6725
1.59
14.2
0.03
4
1.6494
1.4998
7.1244
5
1.6129
1.5464
6.2595
4
1.6381
1.5448
5.5389
0.04
4
1.6044
1.5043
5.0553
3
1.6069
1.4591
4.3975
2
1.6452
1.4199
3.5794
0.05
3
1.5762
1.4614
3.7129
2
1.5867
1.3431
3.3056
2
1.6116
1.4403
2.8039
0.06
2
1.5767
1.3487
2.9532
2
1.5856
1.341
3.0521
2
1.5779
1.448
2.4443
0.07
2
1.5534
1.3667
2.5903
2
1.5091
1.3401
2.5397
2
1.5407
1.4613
2.2039
0.08
2
1.5476
1.3644
2.4721
2
1.4901
1.3781
2.2746
2
1.51
1.4684
2.0847
0.09
2
1.4957
1.3796
2.2356
2
1.4584
1.3706
2.1822
2
1.4934
1.463
2.0516
0.10
2
1.4608
1.3792
2.1392
2
1.4326
1.3878
2.0756
2
1.4765
1.4678
2.0123
0.02
0.03
33
1.5777
1.5317
49.045
24
1.5792
1.5205
40.844
23
1.5877
1.5308
38.264
0.04
12
1.564
1.4797
17.597
10
1.5523
1.4743
14.232
8
1.5751
1.4689
13.192
0.05
8
1.5284
1.4545
9.8811
5
1.55
1.367
9.4282
4
1.5711
1.3857
7.3731
0.06
4
1.5398
1.3475
6.5968
4
1.5176
1.3461
6.1958
4
1.5358
1.4158
5.3262
0.07
4
1.509
1.3555
5.4223
3
1.499
1.2826
4.7171
3
1.5153
1.3668
4.0192
0.08
3
1.5084
1.2608
4.7219
3
1.4701
1.2789
4.2003
2
1.4866
1.376
3.5767
0.09
3
1.4732
1.2728
4.0582
2
1.48
1.1072
3.6068
2
1.5061
1.2412
3.0002
0.10
3
1.4404
1.2912
3.6081
2
1.4531
1.122
3.1886
2
1.4746
1.2637
2.6436
0.15
2
1.3734
1.1425
2.4335
2
1.3024
1.1682
2.2361
2
1.3533
1.306
2.0693
0.05
0.06
116
1.4145
1.3685
217.43
116
1.4045
1.3713
175.95
71
1.4264
1.3582
182.68
0.07
36
1.4072
1.3239
64.232
35
1.3906
1.33
51.696
38
1.4004
1.3495
52.267
0.08
21
1.3874
1.2913
32.385
15
1.3939
1.2559
28.96
16
1.4028
1.291
26.693
0.09
15
1.3719
1.2679
21.152
10
1.3823
1.2144
18.281
9
1.4132
1.2329
17.328
0.10
9
1.3779
1.1977
15.256
10
1.3467
1.2297
13.692
7
1.3922
1.204
12.001
0.15
4
1.3169
1.0028
6.5325
3
1.2961
1.0361
5.9048
3
1.3514
1.0107
5.0282
0.20
3
1.2411
0.9237
4.1483
3
1.1796
0.9272
3.864
2
1.2868
0.8507
3.1257
0.25
2
1.2168
0.6689
3.1438
2
1.1587
0.6865
2.9261
2
1.1833
0.9131
2.444
0.10
0.15
22
1.1881
1.0343
39.075
25
1.1579
1.0557
35.526
20
1.1872
1.0593
31.445
0.20
9
1.1304
0.9064
13.612
6
1.1558
0.7844
12.333
7
1.1519
0.9199
10.695
0.25
5
1.0937
0.7432
7.8417
5
1.0436
0.7544
7.161
4
1.119
0.7708
6.1821
0.30
4
1.0259
0.6616
5.6353
3
1.0344
0.4845
5.1473
3
1.0609
0.6688
4.3262
0.35
3
0.9973
0.4818
4.4906
3
0.9463
0.5069
4.1959
2
1.0675
0.3828
3.3826
0.40
3
0.8819
0.5364
3.6763
2
0.9244
0.1276
3.3677
2
0.9666
0.4474
2.7629
0.50
2
0.8361
0.1600
2.754
2
0.6933
0.1948
2.5159
2
0.7766
0.5197
2.2246Table 3
The proposed plan parameters whenθ
=
3.
p
A
Q
L
p
L
Q
L
α
=
0.05, β
=
0.05
α
=
0.05, β
=
0.10
α
=
0.10, β
=
0.05
n
k
a
k
r
ASN
n
k
a
k
r
ASN
n
k
a
k
r
ASN
0.01
0.02
11
2.0435
1.9214
16.649
15
1.9978
1.968
16.258
9
2.0564
1.9384
13.318
0.03
5
1.6477
1.5483
7.0403
5
1.624
1.562
6.1122
4
2.013
1.756
6.7778
0.04
3
1.6365
1.4642
4.6702
3
1.9106
1.6983
4.0553
2
2.0303
1.607
3.7403
0.05
3
1.5901
1.4804
3.6433
2
1.606
1.3588
3.2927
2
1.9694
1.6307
2.949
0.06
2
1.8807
1.5128
2.9087
2
1.5745
1.371
2.7897
2
1.8647
1.6634
2.391
0.07
2
1.5631
1.3744
2.5823
2
1.7751
1.5448
2.4236
2
1.8153
1.6761
2.2104
0.08
2
1.7742
1.5441
2.3492
2
1.7089
1.5331
2.2721
2
1.7895
1.7007
2.1059
0.09
2
1.5125
1.3682
2.2991
2
1.4643
1.384
2.1608
2
1.7369
1.6911
2.0458
0.10
2
1.6839
1.548
2.1448
2
1.4302
1.3935
2.0611
2
1.7056
1.7038
2.0014
0.02
0.03
32
1.8796
1.8085
44.34
27
1.875
1.7949
39.665
24
1.8926
1.8106
35.328
0.04
11
1.5793
1.4856
16.816
8
1.5759
1.4548
14.142
9
1.8589
1.7281
12.537
0.05
6
1.8323
1.6026
9.4048
7
1.7699
1.6566
8.5377
5
1.8355
1.6485
7.036
0.06
5
1.5365
1.3999
6.9079
5
1.5113
1.4083
6.3292
4
1.7991
1.6116
5.2113
0.07
4
1.7486
1.519
5.2857
3
1.7516
1.4026
4.7796
3
1.7854
1.518
4.1665
0.08
3
1.5061
1.2967
4.302
3
1.4942
1.2916
4.2529
2
1.7926
1.3325
3.3448
0.09
3
1.7039
1.4229
3.8994
3
1.6333
1.4219
3.6597
2
1.7378
1.3618
2.879
0.10
2
1.4953
1.1148
3.4317
2
1.4752
1.128
3.2445
2
1.69
1.3881
2.5814
0.15
2
1.3706
1.1644
2.3686
2
1.4368
1.2133
2.2853
2
1.497
1.4206
2.0758
0.05
0.06
142
1.5926
1.5425
211.7
121
1.5855
1.5388
175.07
107
1.6011
1.5424
172.78
0.07
39
1.4071
1.34
60.006
33
1.4008
1.3313
51.63
32
1.5885
1.4872
48.931
0.08
19
1.5727
1.3989
32.591
17
1.5561
1.3938
27.827
17
1.5724
1.4366
25.124
0.09
12
1.3972
1.2424
20.761
12
1.3738
1.2526
18.009
9
1.5911
1.3365
16.464
0.10
8
1.5626
1.2429
15.512
10
1.4826
1.3295
13.182
6
1.5927
1.2353
12.207
0.15
4
1.3199
1.0319
6.1565
4
1.2793
1.0457
5.6084
4
1.4325
1.1997
5.0198
0.20
3
1.3334
0.9286
4.0843
3
1.2588
0.9563
3.7577
2
1.4319
0.8368
3.1732
0.25
2
1.2206
0.6531
3.1862
2
1.1586
0.6849
2.9158
2
1.2653
0.9024
2.4575
0.10
0.15
29
1.1682
1.0731
39.66
22
1.2336
1.0696
33.958
18
1.2843
1.0727
31.896
0.20
9
1.1893
0.8989
13.622
8
1.1165
0.8889
12.097
6
1.2583
0.8540
10.817
0.25
5
1.1026
0.7305
8.0712
5
1.0742
0.7248
7.0789
4
1.1732
0.7275
6.2323
0.30
4
1.0585
0.6079
5.6677
4
0.9631
0.6791
5.1694
3
1.1098
0.6213
4.3558
0.35
3
0.9855
0.4894
4.3949
3
0.8968
0.4429
4.0186
2
1.0921
0.2992
3.3459
0.40
3
0.8674
0.4524
3.7058
2
0.9196
0.1074
3.3934
2
0.9719
0.3589
2.7746
0.50
2
0.8106
0.1697
2.699
2
0.6956
0.1948
2.5155
2
0.7312
0.4498
2.2141Figure1 shows the OC curves of the proposed RGS plan against the nonconforming rate p for different sample sizes n
=
6, 10, 15, and 25 under θ
=
1.97, k
a
=
1.45, and k
r
=
1.30. It can be seen that the OC curve becomes more like the idealized OC curve shape (the slope is getting larger) as the sample size increases. It implies that the discriminatory power would become larger by increasing the sample size.Figure 1
OC curves of different sample sizes underk
a
=
1.45 and k
r
=
1.30.


## 4. Comparative Analysis

In this section, we will use these two criteria, the OC curves and the sample size required for inspection, to demonstrate the advantages of the proposed RGS plan over the single plan based on the indexC
L.


### 4.1. OC Curves

In order to examine the behaviour of the proposed RGS plan with different values ofθ, Figure 2 displays the OC curves of the RGS plan and the single sampling plan for two cases: (a) θ
=
1 and (b) θ
=
3, under (
p
AQL
,
p
LQL
)
=
(
0.05,0.08
) and (
α
,
β
)
=
(
0.05,0.05
).Figure 2
OC curves of the RGS plan and the single plan under two cases: (a)θ
=
1 and (b) θ
=
3.
(a)
(b)In Figure2, we can see that the two curves of the sampling plans are very similar in case (a) or in case (b), but the sample size required by the RGS plan is much fewer. For example, the single plan requires n
=
47 while the RGS plan requires ASN = 31.77 in case (a). In addition, all of the OC curves show that the probability of acceptance will become smaller as the nonconforming rate p increases, which is as expected from the theory. Since the RGS plan requires fewer sample size to give the desired protection, the cost of inspection will greatly be reduced. Therefore, it is reasonable to conclude that the proposed RGS plan has a better performance.


### 4.2. Sample Sizes Required for Inspection

In order to compare the sample sizes required for inspection in the RGS plan and the single plan with different values ofp
AQL and p
LQL, the p
AQL value is fixed at 0.05 and the p
LQL value increases from 0.06 to 0.25 when the risks are α
=
0.05 and β
=
0.10. The results are shown in Figure 3 (θ
=
1) and Figure 4 (θ
=
3). From Figures 3 and 4, we can note that the sample sizes required for both sampling plans decrease as the value of p
LQL rises from 0.06 to 0.25. Clearly, the sample size required is larger as the value of p
LQL is closer to the value of p
AQL. Moreover, it is obvious that the proposed RGS plan requires smaller sample size for inspection than the single sampling plan when p
LQL takes any value between 0.06 and 0.25. Therefore, the RGS sampling plan is a more cost-effective plan while the single plan is relatively uneconomical.Figure 3
ASN curves of two sampling plans withp
AQL
=
0.05 and p
LQL
=
0.06
~
0.25 when θ
=
1.Figure 4
ASN curves of two sampling plans withp
AQL
=
0.05 and p
LQL
=
0.06
~
0.25 when θ
=
3.On the other side, we also list the sample sizes required for the single sampling plan and RGS plan in Table4 with commonly used values of p
AQL and p
LQL when (
α
,
β
)
=
(
0.05,0.10
), (0.10, 0.05), and (0.10, 0.10) assuming that θ
=
1.97. From Table 4, it is obvious that the sample size required by the RGS plan is fewer than required by the single sampling plan for all cases. For example, when p
AQL
=
0.02, p
LQL
=
0.03, and (
α
,
β
)
=
(
0.10,0.05
), the sample size of the RGS plan is 38.264, while the single plan is 51. Therefore, the proposed sampling plan will give the desired protection with the less required sample size so that the RGS plan is economically superior to the single sampling plan.Table 4
The comparison of sample sizes for two sampling plans based onC
L (θ
=
1.97).
p
A
Q
L
p
L
Q
L
(α, β)
(0.05, 0.05)
(0.05, 0.10)
(0.10, 0.05)
n
A
S
N
R
n
A
S
N
R
n
A
S
N
R
0.01
0.02
23
16.458
19
16.959
18
14.2
0.03
10
7.1244
8
6.2595
8
5.5389
0.04
6
5.0553
5
4.3975
5
3.5794
0.05
5
3.7129
4
3.3056
4
2.8039
0.06
4
2.9532
4
3.0521
3
2.4443
0.07
4
2.5903
3
2.5397
3
2.2039
0.08
3
2.4721
3
2.2746
3
2.0847
0.09
3
2.2356
3
2.1822
2
2.0516
0.10
3
2.1392
3
2.0756
2
2.0123
0.02
0.03
65
49.045
53
40.844
51
38.264
0.04
23
17.597
19
14.232
18
13.192
0.05
13
9.8811
11
9.4282
10
7.3731
0.06
10
6.5968
8
6.1958
7
5.3262
0.07
8
5.4223
6
4.7171
6
4.0192
0.08
6
4.7219
5
4.2003
5
3.5767
0.09
5
4.0582
5
3.6068
4
3.0002
0.10
5
3.6081
4
3.1886
4
2.6436
0.15
3
2.4335
3
2.2361
3
2.0693
0.05
0.06
309
217.43
246
175.95
242
182.68
0.07
91
64.232
73
54.051
71
52.267
0.08
47
32.385
38
28.96
36
26.693
0.09
30
21.152
25
18.281
23
17.328
0.10
22
15.256
18
13.692
17
12.001
0.15
9
6.5325
8
5.9048
7
5.0282
0.20
6
4.1483
5
3.864
5
3.1257
0.25
5
3.1438
4
2.9261
4
2.444
0.10
0.15
59
39.075
47
35.526
46
31.445
0.20
20
13.612
17
12.333
16
10.695
0.25
12
7.8417
10
7.161
9
6.1821
0.30
8
5.6353
7
5.1473
6
4.3262
0.35
6
4.4906
5
4.1959
5
3.3826
0.40
5
3.6763
5
3.3677
4
2.7629
0.50
4
2.754
3
2.5159
3
2.2246
Note:n and A
S
N
R denote the sample size of the single sampling plan and VRGS plan, respectively.


## 4.1. OC Curves

In order to examine the behaviour of the proposed RGS plan with different values ofθ, Figure 2 displays the OC curves of the RGS plan and the single sampling plan for two cases: (a) θ
=
1 and (b) θ
=
3, under (
p
AQL
,
p
LQL
)
=
(
0.05,0.08
) and (
α
,
β
)
=
(
0.05,0.05
).Figure 2
OC curves of the RGS plan and the single plan under two cases: (a)θ
=
1 and (b) θ
=
3.
(a)
(b)In Figure2, we can see that the two curves of the sampling plans are very similar in case (a) or in case (b), but the sample size required by the RGS plan is much fewer. For example, the single plan requires n
=
47 while the RGS plan requires ASN = 31.77 in case (a). In addition, all of the OC curves show that the probability of acceptance will become smaller as the nonconforming rate p increases, which is as expected from the theory. Since the RGS plan requires fewer sample size to give the desired protection, the cost of inspection will greatly be reduced. Therefore, it is reasonable to conclude that the proposed RGS plan has a better performance.


## 4.2. Sample Sizes Required for Inspection

In order to compare the sample sizes required for inspection in the RGS plan and the single plan with different values ofp
AQL and p
LQL, the p
AQL value is fixed at 0.05 and the p
LQL value increases from 0.06 to 0.25 when the risks are α
=
0.05 and β
=
0.10. The results are shown in Figure 3 (θ
=
1) and Figure 4 (θ
=
3). From Figures 3 and 4, we can note that the sample sizes required for both sampling plans decrease as the value of p
LQL rises from 0.06 to 0.25. Clearly, the sample size required is larger as the value of p
LQL is closer to the value of p
AQL. Moreover, it is obvious that the proposed RGS plan requires smaller sample size for inspection than the single sampling plan when p
LQL takes any value between 0.06 and 0.25. Therefore, the RGS sampling plan is a more cost-effective plan while the single plan is relatively uneconomical.Figure 3
ASN curves of two sampling plans withp
AQL
=
0.05 and p
LQL
=
0.06
~
0.25 when θ
=
1.Figure 4
ASN curves of two sampling plans withp
AQL
=
0.05 and p
LQL
=
0.06
~
0.25 when θ
=
3.On the other side, we also list the sample sizes required for the single sampling plan and RGS plan in Table4 with commonly used values of p
AQL and p
LQL when (
α
,
β
)
=
(
0.05,0.10
), (0.10, 0.05), and (0.10, 0.10) assuming that θ
=
1.97. From Table 4, it is obvious that the sample size required by the RGS plan is fewer than required by the single sampling plan for all cases. For example, when p
AQL
=
0.02, p
LQL
=
0.03, and (
α
,
β
)
=
(
0.10,0.05
), the sample size of the RGS plan is 38.264, while the single plan is 51. Therefore, the proposed sampling plan will give the desired protection with the less required sample size so that the RGS plan is economically superior to the single sampling plan.Table 4
The comparison of sample sizes for two sampling plans based onC
L (θ
=
1.97).
p
A
Q
L
p
L
Q
L
(α, β)
(0.05, 0.05)
(0.05, 0.10)
(0.10, 0.05)
n
A
S
N
R
n
A
S
N
R
n
A
S
N
R
0.01
0.02
23
16.458
19
16.959
18
14.2
0.03
10
7.1244
8
6.2595
8
5.5389
0.04
6
5.0553
5
4.3975
5
3.5794
0.05
5
3.7129
4
3.3056
4
2.8039
0.06
4
2.9532
4
3.0521
3
2.4443
0.07
4
2.5903
3
2.5397
3
2.2039
0.08
3
2.4721
3
2.2746
3
2.0847
0.09
3
2.2356
3
2.1822
2
2.0516
0.10
3
2.1392
3
2.0756
2
2.0123
0.02
0.03
65
49.045
53
40.844
51
38.264
0.04
23
17.597
19
14.232
18
13.192
0.05
13
9.8811
11
9.4282
10
7.3731
0.06
10
6.5968
8
6.1958
7
5.3262
0.07
8
5.4223
6
4.7171
6
4.0192
0.08
6
4.7219
5
4.2003
5
3.5767
0.09
5
4.0582
5
3.6068
4
3.0002
0.10
5
3.6081
4
3.1886
4
2.6436
0.15
3
2.4335
3
2.2361
3
2.0693
0.05
0.06
309
217.43
246
175.95
242
182.68
0.07
91
64.232
73
54.051
71
52.267
0.08
47
32.385
38
28.96
36
26.693
0.09
30
21.152
25
18.281
23
17.328
0.10
22
15.256
18
13.692
17
12.001
0.15
9
6.5325
8
5.9048
7
5.0282
0.20
6
4.1483
5
3.864
5
3.1257
0.25
5
3.1438
4
2.9261
4
2.444
0.10
0.15
59
39.075
47
35.526
46
31.445
0.20
20
13.612
17
12.333
16
10.695
0.25
12
7.8417
10
7.161
9
6.1821
0.30
8
5.6353
7
5.1473
6
4.3262
0.35
6
4.4906
5
4.1959
5
3.3826
0.40
5
3.6763
5
3.3677
4
2.7629
0.50
4
2.754
3
2.5159
3
2.2246
Note:n and A
S
N
R denote the sample size of the single sampling plan and VRGS plan, respectively.


## 5. An Industrial Example

In this example, we use the data about the failure times of 25 ball bearings in endurance test, which have been discussed by Lee [20]. The following observations are the number of million revolutions before failure for each of 25 ball bearings (Lawless [27]):17.88, 28.92, 33.00, 41.52, 42.12, 45.60, 48.48, 51.84, 51.96, 54.12, 55.56, 67.80, 67.80, 67.80, 68.64, 68.64, 68.88, 84.12, 93.12, 98.64, 105.12, 105.84, 127.92, 128.04, and 173.40.Lawless [27] observed that the Weibull distribution is suitable for these failure times. The shape parameter θ of the Weibull distribution is supposed to be known. Following Lee [20] who used the max p value method to select the shape parameter θ, we take θ
=
1.97. The lower specification limit L is assumed to be 18.7681. In the contract formulated from the producer and the consumer, suppose that p
AQL
=
0.10 and p
LQL
=
0.15 with producer’s risk α
=
0.05 and buyer’s risk β
=
0.1. That is, the proposed RGS plan must provide a probability of at least 0.95 of accepting the lot if the quality of products is at p
AQL
=
0.06 and also provide a probability of no more than 0.10 of accepting the lot if the quality of products is at p
LQL
=
0.08. Therefore, the problem is the determination of the plan parameters that provide the desired levels of protection for both producers and consumers.From Table3, we can obtain the plan parameters as n
=
25, k
a
=
1.1579, k
r
=
1.0557, and A
S
N
=
35.526 according to the above values. Hence, we take 25 samples randomly from the entire lot and the observed measurements are displayed as above.Based on these observations, the estimation of theC
L is calculated as follows:(15)
C
L
^
=
Γ
1
+
1
/
θ
-
L
·
n
/
W
1
/
θ
A
=
1.3864
,where W
=
∑
i
=
1
25
X
i
θ
=
139525.2 and A
=
Γ
(
1
+
2
/
θ
)
-
Γ
2
(
1
+
1
/
θ
)
=
0.46974.SinceC
L
^
=
1.3864
>
k
a, the consumer should accept the entire lot.


## 6. Conclusions

Acceptance sampling plans can make a lot sentencing decision for the producer and the consumer a general rule in order to meet the desired quality requirement and protection. This paper presents a RGS plan based on the process capability indexC
L for accepting a lot whose quality characteristic follows a Weibull distribution. The optimal plan parameters of the proposed RGS plan are determined by minimizing the ASN function with two constraints required by the producer and the consumer. Then we use two criteria (the required sample size and OC curve) to compare the efficiency of the proposed RGS plan with the single plan proposed. The results imply that our proposed RGS plan requires the smaller ASN but provides the desired protection at the same time. So the industrialists can save the inspection cost if they use the proposed RGS plan. Finally, an example is also given to show the application of the proposed plan in various industries. For future research, it might be interesting to consider other sampling schemes, such as multiple dependent states (MDS) sampling based on the index C
L.



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*Source: 5862071-2016-09-18.xml*