# Optimal Designing of Variables Chain Sampling Plan by Minimizing the Average Sample Number

Authors: S. Balamurali; M. Usha
Journal: International Journal of Manufacturing Engineering
 (2013)

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## Abstract

We investigate the optimal designing of chain sampling plan for the application of normally distributed quality characteristics. The chain sampling plan is one of the conditional sampling procedures and this plan under variables inspection will be useful when testing is costly and destructive. The advantages of this proposed variables plan over variables single sampling plan and variables double sampling plan are discussed. Tables are also constructed for the selection of optimal parameters of known and unknown standard deviation variables chain sampling plan for specified two points on the operating characteristic curve, namely, the acceptable quality level and the limiting quality level, along with the producer’s and consumer’s risks. The optimization problem is formulated as a nonlinear programming where the objective function to be minimized is the average sample number and the constraints are related to lot acceptance probabilities at acceptable quality level and limiting quality level under the operating characteristic curve.


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## Body

## 1. Introduction

Although in recent years process control techniques and off-line quality control methods have taken important roles, acceptance sampling procedures remain as a major tool for many practical quality control systems. In general, acceptance sampling is a statistical tool used to help make decisions concerning whether or not a batch or lot of products should be released for consumer use. That is, sampling plans are widely used in industries for estimating lot or process characteristics and for determining the disposition of lots. Variable sampling plans constitute one of the major areas of the theory and practice of acceptance sampling. The principle prerequisite for variables sampling is that the quality characteristic of interest is measured on a continuous scale. The primary advantage of the variables sampling plan is that the same operating characteristic (OC) curve can be obtained with a smaller sample size than would be required by an attributes sampling plan. Thus, a variables acceptance sampling plan would require less sampling. The variables sampling provides more information than the attributes sampling, and therefore the same protection is attained with partly considerable smaller sample size. When destructive testing is employed, variables sampling is particularly useful in reducing the costs of inspection.However, it is to be pointed out that the variables sampling plans have some demerits compared to attributes sampling plans. The variables sampling plans are, of course, little bit more tedious to apply than the attributes sampling plans and the assumptions on which they are based may not be met or may not be known to be met. It is also obvious that the cost of inspecting each item by attributes is less than by variables. The lower cost may also be a result of the simpler procedure of recording only the number of units failing a test rather than recording variables measurements. However, in modern quality control systems, these reasons are no longer valid because measurements are made by automated test equipment. Even the simplest automated test equipment usually has the capability of the simple calculations and data accumulation needed for variables sampling. In one of his papers, Collani [1] criticized the variables sampling plans and argued that the acceptance sampling by variables is inappropriate if one is interested in the fraction nonconforming in incoming batches. But, at the same time, Seidel [2] has proved in one of his subsequent papers that the sampling by variables is always optimal.The conventional variables sampling plans have been investigated by many authors and are available in the literature of acceptance sampling. Lieberman and Resnikoff [3] developed tables for the selection of plan parameters for various acceptable quality levels (AQL) for MIL-STD 414 scheme. Owen [4] developed variables sampling plans based on normal distribution when the standard deviation of the process is unknown. Bender Jr [5] considered variables sampling plans for assuring the product quality when the quality characteristic of the product follows normal distribution with unknown standard deviation and provided a procedure for calculating the noncentral t-distribution. Hamaker [6] has given a procedure for finding the parameters of the unknown sigma variables sampling plans from the known sigma variables sampling plans. Govindaraju and Soundararajan [7] developed tables for selecting the parameters of variables single sampling plans that match with the OC curves of MIL-STD 105D schemes. Bravo and Wetherill [8] developed a method for designing double sampling plans (DSP) by variables with OC curves matching with the OC curves of the equivalent single sampling plans. Schilling [9] has written an exclusive book on acceptance sampling which also deals with the conventional variables sampling plans. Baillie [10] developed tables for variables double sampling plans when the process standard deviation is unknown. Govindaraju and Balamurali [11] extended the chain sampling for variables inspection. However, they have not provided any tables for easy implementation of the plan. Also, they have restricted their discussion only with known standard deviation. Hence, this paper attempts to develop tables for easy application of chain sampling to variables inspection for a normally distributed quality characteristic for both known and unknown standard deviation cases.


## 2. Chain Sampling Plan for Variables Inspection

The concept of chain sampling was first introduced by Dodge [12] for the application of attribute quality characteristics. The characteristics and the properties of the chain sampling plan were investigated by many authors (see, for example, Soundararajan [13]). The chain sampling plan belongs to the group of conditional sampling procedures like multiple dependent (deferred) sampling plans (see Vaerst [14] and Wortham and Baker [15]). In these procedures, acceptance or rejection of a lot is based not only on the sample from that lot, but also on sample results from past lots. The chain sampling plan is applicable in the case of Type B situations (i.e., sampling from a continuous process) where lots expected to be of the same quality are submitted for inspection serially in the order of production. The operating procedure and characteristics of the attributes chain sampling plan can be seen in Dodge [12].The performance of an acceptance sampling plan can be measured by its OC curve which quantifies the risks of producer and consumers. It would be of interest to examine whether or not the shape of the variables plan is concave for good quality levels. If the variables plan possesses an unsatisfactory OC curve similar to that of the zero acceptance number attributes plan, then it is desirable to improve the shape of the curve. A detailed discussion on the shape of the OC curve of a single sampling variables plan with known standard deviation can be seen in Govindaraju and Kuralmani [16]. It is always necessary to increase the discriminating power of the OC curve. This can be achieved by increasing the sample size, but this may not be always possible particularly in costly and destructive testing. An alternative solution is to develop a conditional sampling procedure. Based on this idea, Govindaraju and Balamurali [11] extended the concept of chain sampling to variables inspection. As pointed out earlier, they have not provided any tables for the selection of parameters and also they have dealt only with known standard deviation case. This paper attempts to provide tables for the easy industrial application of this plan for both known and unknown standard deviation cases. It is also to be pointed out that Govindaraju and Balamurali [11] have given an approximate solution for finding the plans of unknown standard deviation case. But in this paper, we provide a different procedure for finding the unknown sigma chain sampling plan. The major advantage of this plan is to achieve better protection to the producer with minimum average sample number (ASN) and these aspects will be discussed in a separate section. The following assumptions should be valid for the application of the variables chain sampling plan.(i)
Lots are submitted for inspection serially in the order of production from a process that turns out a constant proportion nonconforming items.(ii)
The consumer has confidence in the supplier and there should be no reason to believe that a particular lot is poorer than the preceding lots.In addition, the usual conditions for the application of single sampling variables plans with known or unknown standard deviation should also be valid. The operating procedures of the variables chain sampling plan are given in the next section.


## 3. Operating Procedures of Variables Chain Sampling Plan

### 3.1. Known Sigma Case

Suppose that the quality characteristic of interest has the upper specification limitU and follows a normal distribution with unknown mean μ and known standard deviation σ. Then, the following two procedures of the variables chain sampling plan are proposed.Step  1. From each submitted lot, take a random sample of size nσ, say (X1,X2⋯Xnσ), and compute V=(U-X-)/σ,  where X-=(1/nσ)∑i=1nσXi.Step  2. Accept the lot if V≥kσ and reject the lot if V<kσ′. If  kσ′≤V<kσ, then accept the current lot provided that the preceding iσ lots were accepted on the condition that V≥kσ but reject, otherwise (note: kσ>kσ′).In the case of lower specification limitL, the operating procedure is as follows.Step  1. From each lot, take a random sample of size nσ, say (X1,X2⋯Xnσ), and compute V=(X--L)/σ, where X-=(1/nσ)∑i=1nσXi.Step  2. Accept the lot if V≥kσ and reject the lot if V<kσ′. If  kσ′≤V<kσ, then accept the current lot provided that the preceding iσ lots were accepted on the condition that V≥kσ but reject, otherwise.Thus, the proposed variables chain sampling plan is characterized by four parameters, namely,nσ,iσ,kσ, and kσ′. If kσ=kσ′, then the proposed plan will reduce to the variables SSP. Also, when iσ tends to infinity, the proposed plan becomes variables SSP with parameters nσ and kσ. It is to be pointed out that the chain sampling plan can be applied for inspection of lots which are submitted serially in the order of production or in the order of being submitted. The decision of current lot depends on the results of preceding lots. So, when iσ ≥ 2, we need to keep the records of results of previous lots. Of course, maintaining records of preceding lots may be a drawback of the chain sampling plan over the single sampling plan; however, this can be compensated by minimizing the inspection efforts in terms of minimum ASN with desired protection.


### 3.2. Unknown Sigma Case

Whenever the standard deviation is unknown, we may use the sample standard deviationS instead of σ. In this case, the plan operates as follows.Step  1. From each submitted lot, take a random sample of size nS, say (X1,X2⋯XnS), and compute V=(U-X-)/S,  where X-=(1/nS)∑i=1nSXi and S=∑(Xi-X¯)2/(nS-1).Step  2. Accept the lot if V≥kS and reject the lot if V<kS′. If  kS′≤V<kS, thenaccept the current lot provided that the precedingiS lots were accepted on thecondition thatV≥kS.Thus, the proposed unknown sigma variables chain sampling plan is characterized by four parameters, namely,nS,iS,kS, and kS′. If kS=kS′, then the proposed plan will reduce to the variables single sampling plan with unknown standard deviation.


## 3.1. Known Sigma Case

Suppose that the quality characteristic of interest has the upper specification limitU and follows a normal distribution with unknown mean μ and known standard deviation σ. Then, the following two procedures of the variables chain sampling plan are proposed.Step  1. From each submitted lot, take a random sample of size nσ, say (X1,X2⋯Xnσ), and compute V=(U-X-)/σ,  where X-=(1/nσ)∑i=1nσXi.Step  2. Accept the lot if V≥kσ and reject the lot if V<kσ′. If  kσ′≤V<kσ, then accept the current lot provided that the preceding iσ lots were accepted on the condition that V≥kσ but reject, otherwise (note: kσ>kσ′).In the case of lower specification limitL, the operating procedure is as follows.Step  1. From each lot, take a random sample of size nσ, say (X1,X2⋯Xnσ), and compute V=(X--L)/σ, where X-=(1/nσ)∑i=1nσXi.Step  2. Accept the lot if V≥kσ and reject the lot if V<kσ′. If  kσ′≤V<kσ, then accept the current lot provided that the preceding iσ lots were accepted on the condition that V≥kσ but reject, otherwise.Thus, the proposed variables chain sampling plan is characterized by four parameters, namely,nσ,iσ,kσ, and kσ′. If kσ=kσ′, then the proposed plan will reduce to the variables SSP. Also, when iσ tends to infinity, the proposed plan becomes variables SSP with parameters nσ and kσ. It is to be pointed out that the chain sampling plan can be applied for inspection of lots which are submitted serially in the order of production or in the order of being submitted. The decision of current lot depends on the results of preceding lots. So, when iσ ≥ 2, we need to keep the records of results of previous lots. Of course, maintaining records of preceding lots may be a drawback of the chain sampling plan over the single sampling plan; however, this can be compensated by minimizing the inspection efforts in terms of minimum ASN with desired protection.


## 3.2. Unknown Sigma Case

Whenever the standard deviation is unknown, we may use the sample standard deviationS instead of σ. In this case, the plan operates as follows.Step  1. From each submitted lot, take a random sample of size nS, say (X1,X2⋯XnS), and compute V=(U-X-)/S,  where X-=(1/nS)∑i=1nSXi and S=∑(Xi-X¯)2/(nS-1).Step  2. Accept the lot if V≥kS and reject the lot if V<kS′. If  kS′≤V<kS, thenaccept the current lot provided that the precedingiS lots were accepted on thecondition thatV≥kS.Thus, the proposed unknown sigma variables chain sampling plan is characterized by four parameters, namely,nS,iS,kS, and kS′. If kS=kS′, then the proposed plan will reduce to the variables single sampling plan with unknown standard deviation.


## 4. Designing Methodology of Variables Chain Sampling Plan

Generally, variables sampling plans are designed based on two points on the OC curve, namely,(p1,1-α) and (p2,β), where p1 is called the acceptable quality level (AQL), p2 is the limiting quality level (LQL), α is the producer’s risk, and β is the consumer’s risk. Any well-designed sampling plan must provide at least (1-α)% probability of acceptance of a lot when the process fraction nonconforming is at AQL level and the sampling plan must also provide not more than β% probability of acceptance if the process fraction nonconforming is at the LQL level. Thus, the acceptance sampling plan must have its OC curve passing through two designated points (AQL, 1-α) and (LQL, β). Some other strategies are also followed to design the sampling plans besides the statistical based paradigm, which include Bayesian approach and economic based approach. For further details, readers may refer Chen and Lam [17], Ferrell and Chhoker [18], Chen [19], Chen et al. [20], Balamurali and Subramani [21], Vijayaraghavan and Sakthivel [22], Balamurali et al. [23], and Fallahnezhad and Aslam [24]. In this paper, we have followed the designing methodology based on two points on the OC curve approach. The variables chain sampling plan is designed based on the two points on the OC curve in the following manner.The fraction nonconforming in a lot is given as(1)p=1-Φ(U-μσ),
where Φ(Y) is the cumulative distribution function of standard normal distribution and is given by
(2)Φ(Y)=∫-∞Y12πexp(-Z22)dZ.
Here, the quality characteristic of interest is normally distributed with mean µ and standard deviation σ, and the unit is classified as nonconforming if it exceeds the upper specification limit U. So, the unknown mean μ can be determined if p is specified. Let us define the standardized quality characteristic corresponding to the fraction conforming as
(3)Zp=Φ-1(1-p).Then, the OC function of the variables chain sampling plan, which gives the proportion of lots that are expected to be accepted for given product quality,p, is given by (see Govindaraju and Balamurali [11])
(4)Pa(p)=Pr(V≥k)+[Pr(V≥k′)-Pr(V≥k)][Pr(V≥k)]i,
where Pr(V≥k) is the probability of accepting a lot based on a single sample with parameters (n,k) and Pr(V<k′) is the probability of rejecting a lot based on a single sample with parameters (n,k′). Under Type B situation (i.e., a series of lots of the same quality), forming lots of N items from a process and then drawing random sample of size n from these lots is equivalent to drawing random samples of size n directly from the process. Hence, the derivation of the OC function is straightforward.The probability of acceptance of the chain sampling plan can also be written as(5)Pa(p)=Φ(w2)+[Φ(w1)-Φ(w2)][Φ(w2)]i,
where w1=(Zp-k′)n  and  w2=(Zp-k)n.It is to be pointed out that the parameters of the known sigma variables chain sampling plan are denoted by (nσ, kσ′, and kσ) and for unknown sigma plan they are denoted by (nS, kS′, and kS). The determination of parameters (nS, kS′, and kS) of unknown sigma variables chain sampling plan is explained as follows. It is known that, for large samples, X¯±kSS is approximately normally distributed with mean μ  ±  kSE(S) and variance (σ2/n)+kSVar(S) (see Duncan [25], and Balamurali and Jun [26]). That is,
(6)X¯+kSS~N(μ+kSσ,σ2n+kS2σ22n).
Therefore, the probability of accepting a lot at each repetition is given by
(7)P(V≥kS)=P{X¯≤U-kSS∣p}=P{X¯+kSS≤U∣p}=Φ(U-kSσ-μ(σ/nS)1+(kS2/2))=Φ((Zp-kS)nS1+(kS2/2)).
If we let
(8)w2S=((Zp-kS)nS1+(kS2/2)),thenPr(V≥kS)=Φ(w2S).
Similarly, if we let w1S=((Zp-kS′)nS/(1+(kS′2/2))), then we have
(9)Pr(V<kS′)=1-Φ(w1S).
Hence, the lot acceptance probability for sigma unknown case is given by
(10)Pa(p)=Φ(w2S)+[Φ(w1S)-Φ(w2S)][Φ(w2S)]iS,
where w1S=(Zp-kS′)nS  and  w2S=(Zp-kS)nS.If (AQL,1-α) and (LQL, β) are prescribed, then the OC function can be written as
(11)Φ(w21S)+[Φ(w11S)-Φ(w21S)][Φ(w21S)]iS=1-α,Φ(w22S)+[Φ(w12S)-Φ(w22S)][Φ(w22S)]iS=β.
Here, w11S is the value of w1S at p=p1,w21S is the value of w2S at p=p1, w12S is the value of w1S at p=p2, and w22S is the value of w2S at p=p2.We obtainw11S,w21S,w12S, and w22S, respectively, by
(12)w11S=((Zp1-kS′)nS1+(kS′2/2)),w21S=((Zp1-kS)nS1+(kS2/2)),w12S=((Zp2-kS′)nS1+(kS′2/2)),w22S=((Zp2-kS)nS1+(kS2/2)),
where Zp1 is the value of Zp at AQL and Zp2 is the value of Zp at LQL. For given AQL and LQL, the parametric values of the unknown sigma variables chain sampling plan, namely, nS,iS,kS, and kS′, are determined by satisfying the required producer and consumer conditions. Alternatively, we can determine the above parameters of the variables chain sampling plan to minimize the ASN at AQL, which is analogous to minimizing the average sample number in the variables repetitive group sampling plans and multiple dependent state sampling plan (see Balamurali et al. [27] and Balamurali and Jun [26]). Some of the authors have investigated the designing of sampling plans by using some other optimization techniques which are available in the literature (see, for example, Feldmann and Krumbholz [28], Krumbholz and Rohr [29, 30], Krumbholz et al. [31], and Duarte and Sariava [32, 33]. The ASN for the chain sampling plan is the sample size only. Therefore, the following optimization problem is considered to determine those parameters. 						(13)MinimizeASN(p1)=nSpppppppppnS,iS,kS,kS′SubjecttoPa(p1)≥1-αppppppppppPa(p2)≤βppppppppppnS≥2,iS≥1,kS+δ≥kS′+δ≥0,ppppppppppδ≥0,nS∈N,iS∈N.
In the case of known sigma variable chain sampling plan, the OC function under the specified AQL and LQL conditions can be written as
(14)Φ(w21σ)+[Φ(w11σ)-Φ(w21σ)][Φ(w21σ)]iσ=1-α,Φ(w22σ)+[Φ(w12σ)-Φ(w22σ)][Φ(w22σ)]iσ=β.
Here, w11σ is the value of w1σ at p=p1, w21σ is the value of w2σ at p=p1,w12σ is the value of w1σ at p=p2, and w22σ is the value of w2σ at p=p2. That is,
(15)w11σ=(Zp1-kσ′)nσ,w21σ=(Zp1-kσ)nσ,w12σ=(Zp2-kσ′)nσ,w22σ=(Zp2-kσ)nσ,
where Zp1 is the value of Zp at AQL and Zp2 is the value of Zp at LQL. The following optimization problem is considered to determine the optimal parameters of known sigma variables sampling plan such as nσ,iσ,kσ, and kσ′. 						(16)MinimizeASN(p1)=nσpppppppppnσ,iσ,kσ,kσ′SubjecttoPa(p1)≥1-αpppppppppPa(p2)≤βpppppppppnσ≥2,iσ≥1,kσ+δ≥kσ′+δ≥0,pppppppppδ≥0,nσ∈N,iσ∈N.
We may determine the parameters of the unknown sigma chain sampling plan and the known sigma chain sampling plan by solving the nonlinear equation given in (13) and (16), respectively. There may exist multiple solutions since there are four unknowns with only two equations. Generally, a sampling would be desirable if the required number of samples is small. So, in this paper, we consider the ASN as the objective function to be minimized with the probability of acceptance along with the corresponding producer’s and consumer’s risks as constraints. To solve the above nonlinear optimization problems given in (13) and (16), the sequential quadratic programming (SQP) proposed by Nocedal and Wright [34] can be used. Here, it is to be pointed out that the optimization problems given in (13) and (16) include continuous values of kσ′ (or kS′) and kσ (or kS) and integer values of nσ (or nS) and iσ (or iS). This kind of formalization leads to a problem of mixed integer programming (MINLP) type. The SQP method does not allow directly solving MINLP problems but only the nonlinear programming (NLP) structures. The strategy we have followed is to relax the discrete variables first and solve the NLP problem and then fix them to the upper integers and resolve the problem again now just to find kσ′ (or kS′) and kσ (or kS). In such a way, we have solved the NLP problem given above. The SQP is implemented in Matlab software using the routine “fmincon.” By solving the NLP based on the procedure described above, the optimal parameters (nσ, iσ, kσ, and kσ′) for known sigma plan and the parameters (nS, iS, kS, and kS′) for unknown sigma plan are determined and these values are tabulated in Table 1.Table 1
Variables chain sampling plans indexed by AQL and LQL forα = 5% and β = 10%.
p
1
p
2
Known sigma
Unknown sigma
n
σ
i
σ
k
σ
′
k
σ
n
S
i
S
k
S
′
k
S
0.001
0.0020
119
1
2.883
3.013
650
1
2.888
3.013
0.0025
68
2
2.810
2.965
361
1
2.834
2.984
0.003
46
1
2.776
2.961
242
1
2.774
2.964
0.004
31
2
2.740
2.885
147
1
2.716
2.921
0.005
21
2
2.610
2.860
103
1
2.637
2.897
0.006
19
4
2.597
2.807
84
2
2.599
2.829
0.007
15
3
2.534
2.789
72
3
2.532
2.792
0.008
13
3
2.495
2.765
62
3
2.518
2.768
0.009
12
3
2.521
2.736
51
2
2.470
2.765
0.010
12
6
2.517
2.697
45
2
2.425
2.750
0.012
9
4
2.331
2.686
40
3
2.414
2.694
0.015
7
3
2.237
2.657
34
4
2.387
2.637
0.020
7
6
2.414
2.539
27
5
2.335
2.565
0.025
6
6
2.370
2.485
20
3
2.197
2.552
0.030
5
6
2.264
2.454
19
5
2.231
2.476
0.0025
0.004
230
2
2.658
2.738
1131
2
2.676
2.736
0.005
105
2
2.598
2.703
474
2
2.585
2.705
0.006
64
2
2.530
2.675
282
1
2.533
2.698
0.0075
39
2
2.441
2.641
171
1
2.464
2.669
0.010
26
3
2.379
2.579
106
1
2.430
2.610
0.012
19
2
2.335
2.555
76
1
2.321
2.601
0.015
16
4
2.296
2.491
55
1
2.237
2.572
0.020
12
4
2.260
2.425
38
1
2.139
2.529
0.025
10
6
2.206
2.366
32
2
2.163
2.413
0.030
8
5
2.114
2.334
29
5
2.107
2.342
0.035
7
6
2.307
2.297
25
5
2.092
2.302
0.040
6
6
1.884
2.274
21
5
1.952
2.282
0.005
0.0075
259
1
2.434
2.524
1113
2
2.442
2.512
0.010
86
1
2.326
2.486
347
1
2.348
2.483
0.012
53
1
2.277
2.457
211
1
2.299
2.454
0.015
33
1
2.211
2.421
128
1
2.233
2.418
0.020
24
4
2.161
2.316
75
2
2.100
2.345
0.025
16
3
2.046
2.281
55
2
2.083
2.293
0.030
14
6
1.989
2.224
43
3
1.974
2.249
0.035
12
6
2.002
2.182
36
3
1.979
2.209
0.040
10
4
1.976
2.156
31
3
1.964
2.174
0.050
8
5
1.893
2.098
25
4
1.925
2.105
0.0075
0.010
505
2
2.339
2.384
1817
1
2.329
2.394
0.012
173
1
2.259
2.369
690
2
2.278
2.353
0.015
78
1
2.185
2.335
286
1
2.182
2.337
0.020
43
2
2.146
2.251
133
1
2.083
2.293
0.025
29
5
2.008
2.198
83
1
2.003
2.258
0.030
19
2
1.968
2.178
61
1
1.984
2.214
0.035
17
5
1.888
2.123
47
1
1.925
2.190
0.040
13
3
1.837
2.107
41
3
1.851
2.111
0.050
10
3
1.792
2.052
28
2
1.756
2.081
0.060
9
5
1.792
1.982
26
4
1.815
1.990
0.010
0.015
248
1
2.212
2.257
786
1
2.187
2.267
0.020
74
1
2.090
2.220
247
1
2.071
2.226
0.025
42
1
2.027
2.177
135
1
2.011
2.186
0.030
27
1
1.906
2.161
93
2
1.954
2.124
0.035
21
2
1.846
2.096
66
1
1.900
2.125
0.040
18
3
1.804
2.054
53
2
1.846
2.066
0.045
15
3
1.762
2.027
46
3
1.811
2.026
0.050
13
3
1.731
2.001
39
2
1.824
2.004
0.060
11
5
1.662
2.942
30
3
1.730
1.955
0.070
9
5
1.573
1.903
26
4
1.728
1.898
0.080
7
3
1.486
1.891
21
5
1.579
1.869
0.02
0.030
187
2
1.896
1.976
528
1
1.886
1.991
0.035
96
2
1.835
1.945
272
2
1.831
1.946
0.040
59
1
1.779
1.939
169
2
1.770
1.920
0.045
45
2
1.749
1.889
117
1
1.734
1.919
0.050
34
2
1.693
1.868
88
1
1.683
1.903
0.060
22
2
1.563
1.833
63
2
1.670
1.820
0.070
18
3
1.544
1.779
44
2
1.577
1.792
0.080
14
3
1.434
1.749
36
3
1.525
1.745
0.090
13
4
1.502
1.697
27
2
1.423
1.737
0.10
10
2
1.436
1.691
22
1
1.386
1.766
0.11
10
4
1.457
1.632
23
4
1.476
1.636
0.12
7
1
1.235
1.725
17
1
1.379
1.699
0.13
8
5
1.360
1.580
17
4
1.356
1.596
0.15
6
4
1.110
1.560
14
5
1.279
1.544
0.030
0.040
341
2
1.771
1.821
1027
5
1.771
1.816
0.045
157
1
1.711
1.811
416
2
1.704
1.799
0.050
96
1
1.650
1.795
253
1
1.679
1.789
0.060
52
2
1.566
1.736
126
1
1.585
1.760
0.070
33
1
1.508
1.728
81
1
1.539
1.724
0.080
26
3
1.427
1.657
62
2
1.514
1.659
0.090
23
5
1.449
1.609
44
2
1.401
1.641
0.100
17
3
1.378
1.593
36
2
1.384
1.609
0.110
13
2
1.268
1.588
29
2
1.305
1.590
0.120
11
1
1.270
1.605
26
3
1.274
1.549
0.130
11
4
1.204
1.514
23
3
1.289
1.519
0.150
10
6
1.273
1.443
18
3
1.230
1.475
0.200
6
5
0.950
1.365
12
5
1.134
1.354
0.04
0.060
145
2
1.568
1.663
339
2
1.564
1.664
0.070
73
1
1.513
1.643
166
1
1.507
1.647
0.080
50
2
1.478
1.588
102
1
1.425
1.625
0.090
34
2
1.394
1.564
71
1
1.374
1.600
0.100
27
3
1.324
1.529
53
1
1.315
1.580
0.110
24
5
1.309
1.489
42
1
1.294
1.554
0.120
18
3
1.233
1.478
34
1
1.234
1.539
0.130
14
2
1.130
1.475
31
2
1.271
1.461
0.140
14
4
1.158
1.423
27
3
1.193
1.433
0.150
11
2
1.108
1.428
23
3
1.130
1.415
0.170
10
4
1.070
1.360
17
2
1.061
1.396
0.200
8
5
0.990
1.295
15
4
1.120
1.290
0.050
0.070
188
1
1.487
1.582
427
2
1.485
1.570
0.080
93
1
1.408
1.558
203
1
1.422
1.557
0.090
60
2
1.359
1.509
127
2
1.365
1.510
0.100
46
3
1.331
1.471
86
1
1.331
1.506
0.110
33
2
1.288
1.453
63
1
1.261
1.491
0.120
26
1
1.270
1.450
49
1
1.213
1.473
0.130
20
1
1.157
1.452
45
3
1.235
1.395
0.140
19
3
1.160
1.375
34
2
1.148
1.393
0.150
17
4
1.108
1.348
30
2
1.165
1.365
0.160
13
2
1.011
1.356
25
1
1.134
1.384
0.170
12
2
1.045
1.330
24
3
1.112
1.312
0.200
11
6
1.059
1.229
17
3
1.023
1.258


## 5. Designing Examples

### 5.1. Selection of Known Sigma Variables Chain Sampling Plan for Specified AQL and LQL

Table1 is used to determine the parameters of the known σ variables chain sampling plan for specified values of AQL and LQL when α = 5% and β = 10%. For example, if p1 = 2%, p2 = 7%, α = 5%, and β = 10%, Table 1 gives the parameters as nσ = 18, iσ = 3, kσ′ = 1.544, and kσ = 1.779.For the above example, the plan is operated as follows.From each submitted lot, take a random sample of size 18 and computeV=(U-X-)/σ,  where X-=(1/18)∑i=118Xi.Accept the lot ifV≥1.779 and reject the lot if  V<1.544. If 1.544≤V<1.779, then accept the current lot provided that the preceding 3 lots were accepted on the condition that V≥1.779 with the sample size of 18.


### 5.2. Selection of Unknown Sigma Variables Chain Sampling Plan for Specified AQL and LQL

As mentioned earlier, the unknown sigma variables chain sampling plan is operated as a known sigma variables chain sampling plan, but the parametersnS, iS, kS, and kS′ are used in the place of nσ, iσ, kσ, and kσ′, respectively. Table 1 can also be used for the selection of the parameters of the unknown σ variables chain sampling plan for given values of AQL and LQL. Suppose that AQL = 0.0075, LQL = 0.035%, α = 5%, and β = 10%. From Table 1, the parameters of the variables chain sampling plan can be determined as nS = 47, iS = 1, kS′ = 1.925, and kS = 2.190.


### 5.3. Illustrative Example

To illustrate the implementation of the proposed sampling plan for the example given in Section5.2, we consider case study data on STN-LCD manufacturing process given in Wu and Pearn [35]. The upper specification limit is given as 0.77 mm. Here, we consider only 47 values randomly taken from the original data given in Wu and Pearn [35]. The data are shown below:(17)0.7170.6980.7260.6840.7270.6880.7080.7030.6940.7130.7300.6990.7100.6880.6650.7040.7250.7290.7160.6850.7120.7160.7120.7330.7090.7030.7300.7160.6880.6880.7120.7020.7260.6690.7180.7140.7260.6830.7130.7370.7400.7060.7260.6880.7150.7040.724The implementation of the plan is shown below.Step  1. Take a random sample of size 47.  The data are given above.Step  2. For this data, we calculate X-=(1/n)∑i=1nSXi=0.708915 and S=∑(Xi-X¯)2/(n-1)=0.017583.Step  3. Calculate V=(U-X-)/S=(0.77-0.708915)/0.017583=3.4741.Step  4. Since  V=3.4741>kS=2.190, we accept the current lot without considering the result of past lots.Just for the sake of discussion, let us assume that theV value (for different data sets) is calculated as 2.15. In this case, we can accept the current lot provided previous lot must have been accepted with the condition that V>kS′=1.925. Otherwise, the current lot is rejected.Further, it is to be pointed out that the proposed variables chain sampling plan is more efficient in terms of minimum ASN than the variables SSP for low values of producer’s risk (α) and consumer’s risk (β).  In order to prove this, we provide two tables. Table 2 gives the optimal parameters of variables chain sampling plan for some selected combinations of AQL and LQL and for α = 1% and β = 1% and  Table 3 gives the parameters of variables SSP under the same set of conditions. By comparing these two tables, one can easily observe that variable chain sampling plan involves minimum ASN compared to the variables SSP.Table 2
Variables chain sampling plans indexed by AQL and LQL forα = 1% and β = 1%.
p
1
p
2
Variables chain sampling plan (known sigma)
n
σ
i
σ
k
σ
′
k
σ
P
a
(
p
1
)
P
a
(
p
2
)
ASN
0.001
0.005
79
1
2.82801
2.83801
0.99007
0.00991
79
0.0025
0.01
91
1
2.56203
2.57203
0.99024
0.00956
91
0.005
0.02
77
1
2.31004
2.32004
0.99012
0.00974
77
0.0075
0.03
69
1
2.15205
2.16205
0.99003
0.00976
69
0.01
0.04
64
1
2.03504
2.04504
0.99007
0.00929
64
0.02
0.05
124
1
1.84403
1.85403
0.99018
0.00997
124
0.03
0.06
193
1
1.71300
1.73200
0.99007
0.00977
193
0.04
0.08
172
1
1.57299
1.58299
0.9906
0.00986
172
0.05
0.10
156
1
1.45798
1.46798
0.99017
0.00999
156Table 3
Variables single sampling plans indexed by AQL and LQL forα = 1% and β = 1%.
p
1
p
2
Variables single sampling plan (known sigma)
n
σ
k
σ
P
a
(
p
1
)
P
a
(
p
2
)
ASN
0.001
0.005
82
2.83305
0.99006
0.00992
82
0.0025
0.01
94
2.56703
0.99001
0.00980
94
0.005
0.02
80
2.31505
0.99016
0.00970
80
0.0075
0.03
72
2.15806
0.99004
0.00932
72
0.01
0.04
66
2.03907
0.99015
0.00957
66
0.02
0.05
130
1.84907
0.99018
0.00996
130
0.03
0.06
205
1.71806
0.99009
0.00971
205
0.04
0.08
182
1.57805
0.99007
0.00982
182
0.05
0.10
165
1.46305
0.99025
0.00984
165


## 5.1. Selection of Known Sigma Variables Chain Sampling Plan for Specified AQL and LQL

Table1 is used to determine the parameters of the known σ variables chain sampling plan for specified values of AQL and LQL when α = 5% and β = 10%. For example, if p1 = 2%, p2 = 7%, α = 5%, and β = 10%, Table 1 gives the parameters as nσ = 18, iσ = 3, kσ′ = 1.544, and kσ = 1.779.For the above example, the plan is operated as follows.From each submitted lot, take a random sample of size 18 and computeV=(U-X-)/σ,  where X-=(1/18)∑i=118Xi.Accept the lot ifV≥1.779 and reject the lot if  V<1.544. If 1.544≤V<1.779, then accept the current lot provided that the preceding 3 lots were accepted on the condition that V≥1.779 with the sample size of 18.


## 5.2. Selection of Unknown Sigma Variables Chain Sampling Plan for Specified AQL and LQL

As mentioned earlier, the unknown sigma variables chain sampling plan is operated as a known sigma variables chain sampling plan, but the parametersnS, iS, kS, and kS′ are used in the place of nσ, iσ, kσ, and kσ′, respectively. Table 1 can also be used for the selection of the parameters of the unknown σ variables chain sampling plan for given values of AQL and LQL. Suppose that AQL = 0.0075, LQL = 0.035%, α = 5%, and β = 10%. From Table 1, the parameters of the variables chain sampling plan can be determined as nS = 47, iS = 1, kS′ = 1.925, and kS = 2.190.


## 5.3. Illustrative Example

To illustrate the implementation of the proposed sampling plan for the example given in Section5.2, we consider case study data on STN-LCD manufacturing process given in Wu and Pearn [35]. The upper specification limit is given as 0.77 mm. Here, we consider only 47 values randomly taken from the original data given in Wu and Pearn [35]. The data are shown below:(17)0.7170.6980.7260.6840.7270.6880.7080.7030.6940.7130.7300.6990.7100.6880.6650.7040.7250.7290.7160.6850.7120.7160.7120.7330.7090.7030.7300.7160.6880.6880.7120.7020.7260.6690.7180.7140.7260.6830.7130.7370.7400.7060.7260.6880.7150.7040.724The implementation of the plan is shown below.Step  1. Take a random sample of size 47.  The data are given above.Step  2. For this data, we calculate X-=(1/n)∑i=1nSXi=0.708915 and S=∑(Xi-X¯)2/(n-1)=0.017583.Step  3. Calculate V=(U-X-)/S=(0.77-0.708915)/0.017583=3.4741.Step  4. Since  V=3.4741>kS=2.190, we accept the current lot without considering the result of past lots.Just for the sake of discussion, let us assume that theV value (for different data sets) is calculated as 2.15. In this case, we can accept the current lot provided previous lot must have been accepted with the condition that V>kS′=1.925. Otherwise, the current lot is rejected.Further, it is to be pointed out that the proposed variables chain sampling plan is more efficient in terms of minimum ASN than the variables SSP for low values of producer’s risk (α) and consumer’s risk (β).  In order to prove this, we provide two tables. Table 2 gives the optimal parameters of variables chain sampling plan for some selected combinations of AQL and LQL and for α = 1% and β = 1% and  Table 3 gives the parameters of variables SSP under the same set of conditions. By comparing these two tables, one can easily observe that variable chain sampling plan involves minimum ASN compared to the variables SSP.Table 2
Variables chain sampling plans indexed by AQL and LQL forα = 1% and β = 1%.
p
1
p
2
Variables chain sampling plan (known sigma)
n
σ
i
σ
k
σ
′
k
σ
P
a
(
p
1
)
P
a
(
p
2
)
ASN
0.001
0.005
79
1
2.82801
2.83801
0.99007
0.00991
79
0.0025
0.01
91
1
2.56203
2.57203
0.99024
0.00956
91
0.005
0.02
77
1
2.31004
2.32004
0.99012
0.00974
77
0.0075
0.03
69
1
2.15205
2.16205
0.99003
0.00976
69
0.01
0.04
64
1
2.03504
2.04504
0.99007
0.00929
64
0.02
0.05
124
1
1.84403
1.85403
0.99018
0.00997
124
0.03
0.06
193
1
1.71300
1.73200
0.99007
0.00977
193
0.04
0.08
172
1
1.57299
1.58299
0.9906
0.00986
172
0.05
0.10
156
1
1.45798
1.46798
0.99017
0.00999
156Table 3
Variables single sampling plans indexed by AQL and LQL forα = 1% and β = 1%.
p
1
p
2
Variables single sampling plan (known sigma)
n
σ
k
σ
P
a
(
p
1
)
P
a
(
p
2
)
ASN
0.001
0.005
82
2.83305
0.99006
0.00992
82
0.0025
0.01
94
2.56703
0.99001
0.00980
94
0.005
0.02
80
2.31505
0.99016
0.00970
80
0.0075
0.03
72
2.15806
0.99004
0.00932
72
0.01
0.04
66
2.03907
0.99015
0.00957
66
0.02
0.05
130
1.84907
0.99018
0.00996
130
0.03
0.06
205
1.71806
0.99009
0.00971
205
0.04
0.08
182
1.57805
0.99007
0.00982
182
0.05
0.10
165
1.46305
0.99025
0.00984
165


## 6. Advantages of the Variables Chain Sampling Plan

This section describes the advantages of the variables chain sampling plan over the conventional variables single sampling plan. Two acceptance sampling plans will be called equivalent when they possess nearly identical OC curves. A customary procedure for achieving such equivalency consists of constructing the sampling plans so that their OC curves coincide in two suitably chosen points, namely,(p1,1-α) and (p2,β). Suppose that, for given values of p1 = 0.5%, α = 5%, p2 = 1.5%, and β = 10%, one can find the parameters of the known sigma variables chain sampling plan from Table 1 as(i)
nσ = 33, iσ = 1, kσ′ = 2.211, and kσ = 2.421; ASN = 33.For the same values of the AQL and LQL, we can determine the parameters of the single and double sampling variables plan (from Sommers [36]) as(ii)
nσ = 53 and kσ = 2.35; ASN = 53,(iii)
nσ = 39, kaσ = 2.41, and krσ = 2.31; ASN = 43.It can be observed that variables chain sampling plan achieves a reduction of over 38% in sample size than the variables single sampling plan and a reduction of 23% over double sampling plan with the same AQL and LQL conditions. This indicates that the variables chain sampling plan achieves the same OC curve with minimum sample size compared to the variables single and double sampling plans. Further, Figure1 shows the OC curves of the variables chain sampling plans with parameters nσ = 18, kσ′ = 1.544, and kσ = 1.779 for different values of iσ. This figure apparently shows that the variables chain sampling plan increases the probability of acceptance in the region of principal interest, that is, for good quality levels, and maintains the consumer’s risk at poor quality levels. This is also an important feature of the variables chain sampling plan.Figure 1
OC curves of a variables chain sampling plan for differentiσ values.Further, it is also to be noted that the variables chain sampling plan is economically superior to the double sampling plans in terms of average sample number (ASN). Obviously, a sampling plan having smaller ASN would be more desirable. The variables double or multiple sampling plans are not practically very useful. Variables sampling standards avoid presenting such plans due to increased complexity involved in operating them, but the variables chain sampling plan has no such complexity. Table4 shows the ASNs for variables single sampling plan, variables double sampling plan along with variables chain sampling plan for some arbitrarily selected combinations of AQL and LQL. These ASN values are calculated at the producer’s quality level for the known sigma plans. The ASN of the variables single and double sampling plans can be found in Sommers [36].Table 4
ASN values of the variables SSP, DSP and variables chain sampling plans.
p
1
p
2
Average sample number
Known sigma
Unknown sigma
SSP
DSP
ChSP
SSP
DSP
ChSP
0.001
0.003
74
59.4
46
381
302.4
242
0.0025
0.0075
62
50.1
39
267
214.2
171
0.005
0.015
53
43.0
33
196
157.6
128
0.0075
0.025
39
31.3
29
129
102.1
83
0.01
0.05
19
14.9
13
54
41.5
39
0.02
0.08
21
16.9
14
50
39.6
36
0.03
0.09
30
24.0
23
66
52.5
44
0.04
0.10
39
31.3
27
82
65.1
53
0.05
0.12
39
32.0
26
76
60.9
49From this table, one can easily understand that the variables chain sampling plan will have minimum ASN when compared to the single and double sampling plans. Similar reduction in the ASN can be achieved for any combination of AQL and LQL values. This implies that variables chain sampling plan will give desired protection with minimum inspection so that the cost of inspection will greatly be reduced. Thus, the variables chain sampling plan provides better protection than the variables single and double sampling plans.


## 7. Average Run Length of Variables Chain Sampling Plan

Schilling [37] has pointed out that average run length (ARL) is a missing and meaningful measure for characterizing and evaluating the sampling plans under Type B situations as in the process control procedures. The ARL gives an indication of the expected number of samples until a decision is made. The ARL can be easily calculated once the probability of acceptance (Pa(p)) of the plan is known for any process fraction nonconforming, p. It is clear that the distribution of the run length, L, follows the geometric distribution with probability mass function
(18)fG(L)=[Pa(p)]L-1[1-Pa(p)].
Its mean and variance are, respectively, given by
(19)ARL=E(L)=1(1-Pa(p)),Var(L)=Pa(p)(1-Pa(p))2,
where (Pa(p)) is given in (5).Table5 gives the values of ARL of the chain sampling plan with nσ = 16, kσ′ = 1.501, and kσ = 1.841 for different values of iσ. This table apparently shows that when the process fraction nonconforming is small, the ARL is high and for the increased values of fraction nonconforming, the ARL is low. By comparing the ARL values for different iσ values, when iσ increases, the ARL values reduce even for the lower fraction nonconforming. Also, it is clear from the table that 95% of the lots will be accepted at fraction nonconforming 2% by the variables chain sampling plan (iσ = 1) at an average rate of 20 inspections, whereas with the single sampling plan (iσ =  ∞), at 2% nonconforming, only 80% of lots will be accepted at an average rate of 5 inspections. Also, 90% of the lots will be rejected at the fraction nonconforming 7% by the variables chain sampling plan at an average rate of 1.11 inspections, and at the fraction nonconforming, 93% of the lots will be rejected by the single sampling plan at the rate of 1.08 inspections.Table 5
ARLs of variables chain sampling plan for differentiσ values.
p
Average run length
i
σ
=
1
i
σ
=
2
i
σ
=
3
i
σ
=
4
i
σ
=
∞
0
∞
∞
∞
∞
∞
0.005
88768.34
71697.5
60133.4
51781.5
607.80
0.01
868.88
554.66
410.18
327.19
38.29
0.015
81.89
50.25
37.21
30.13
10.64
0.02
20.07
12.66
9.77
8.25
5.06
0.025
8.12
5.42
4.41
3.92
3.15
0.03
4.41
3.14
2.70
2.50
2.29
0.035
2.88
2.19
1.98
1.89
1.83
0.04
2.14
1.73
1.62
1.58
1.56
0.045
1.73
1.47
1.41
1.39
1.39
0.05
1.48
1.32
1.28
1.28
1.28
0.055
1.33
1.22
1.20
1.20
1.20
0.06
1.23
1.15
1.15
1.14
1.14
0.065
1.16
1.11
1.11
1.11
1.11
0.07
1.11
1.08
1.08
1.08
1.08
0.075
1.08
1.06
1.06
1.06
1.06
0.08
1.06
1.04
1.04
1.04
1.04
0.085
1.04
1.03
1.03
1.03
1.03
0.09
1.03
1.02
1.02
1.02
1.02
0.10
1.02
1.01
1.01
1.01
1.01


## 8. Conclusions

The purpose of this paper is to develop conditional sampling procedures for the inspection of normally distributed quality characteristics. Variables sampling plans generally require a smaller sample size than do attributes plans. If the OC curve of the variables plan is unsatisfactory, then its shape can be improved by chaining the past lot results. The proposed variables chain sampling plan is one of the variables conditional sampling plans which also ensure the protection against the consumer point of view. This plan is also simple to apply rather than double and multiple sampling variables plans. Also, this plan provides better protection than the conventional single and double sampling variables plans with minimum sample size. Such a variables chain sampling plan will be effective and useful for compliance testing. However, it is also to be pointed out that the variables chain sampling plan developed in this paper is based on the assumption that the quality characteristic of interest follows a normal distribution. Whenever the normality assumption is not true or invalid, using this variables chain sampling plan can be quite misleading. Hence, this study can be further extended for nonnormal distributions as future research.



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*Source: 751807-2013-12-17.xml*