Designing of Variables Modified Chain Sampling Plan Based on the Process Capability Index with Unknown Mean and Variance

SYNOPTIC ABSTRACT

This article proposes a variables-modified chain sampling plan based on the process capability index C_pk for the inspection of normally distributed quality characteristics with unknown mean and variance. The modified chain sampling plan is one of the conditional sampling procedures and this plan under variables inspection will be very useful, particularly in compliance testing. Tables are also constructed to select the optimal parameters of the variables modified chain sampling plan for specified two points on the operating characteristic curve by minimizing the average sample number. Symmetric and asymmetric cases based on the fraction nonconforming by the lower and the upper specification limits are also considered. The optimization problem is formulated as a nonlinear programming problem 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.

KEY WORDS AND PHRASES:

• Conditional sampling by variables
• decision making
• process capability index
• quadratic programming
• quality control

1. Introduction

In recent years, process capability indices (PCIs) are generating much interest, and there is a growing body of statistical process control literature. The PCIs are considered a practical tool by many advocates of the statistical process control industry. They are used to determine whether a manufacturing process is capable of producing units within a specified tolerance range. A capability index is a dimensionless measure based on the process parameters and the process specification designed to quantify, in a simple and easily understood way, the performance of the process. Several PCIs have been developed, among which the basic indices are C_p, C_pk, and C_pm (see Kane, 1986; Chan, Cheng, & Spring, 1988). A detailed review of the PCIs can be seen in Kotz and Johnson (2002) and Yum and Kim (2011). The index C_p is defined as the ratio of the allowable process output range to the natural process spread of the concerned process. The index C_p and its estimator are defined, respectively, as, (1) and (2) where USL and LSL are the upper and lower specification limits, respectively. Whenever the process variance σ² is not known, the unbiased sample variance S² is used (see Kane, 1986) to estimate the capability index. Since the index C_p fails to reflect the impact of the location of the process mean μ, the index C_pk was developed. The index C_pk which is relevant to our study is defined as follows. (3)

The PCI C_pk can also be defined in terms of C_p as C_pk = C_p(1 − k), where, and M is the midpoint of the specification limits USL and LSL. The estimator of C_pk is defined as follows. (4) Where and S are the sample mean and standard deviation, respectively (see Kane, 1986). For a C_pk index at level 1, one would expect that not more than 2700 parts per million (ppm) fall outside the specification limits. At a C_pk level of 1.33, the rate of non-conforming comes down to 66 ppm. A C_pk level of 1.67 is required in order to achieve less than 0.544 ppm non-conforming rate. At a C_pk level of 2, the likelihood of the non-conforming rate drops to 2 parts per billion (ppb). It has also been observed that when C_pk increased from 1.0 to 1.33, there is a drastic reduction in the fraction non-conforming (refer to Pearn and Kotz, 2006).

2. Sampling Plans Based on Process Capability Indices

The concept of acceptance sampling constitutes one of the major areas of statistical quality control. The acceptance sampling is very useful in situations where testing is destructive, the cost of inspection is extremely high, the time required for inspection will be too long, the inspection error is too high, or the product liability risks are serious, and so on. Also, inspection of raw materials or finished products using acceptance sampling is one aspect of quality assurance. It is well known that the acceptance sampling plans are used to reduce the cost of inspection. Acceptance sampling plans are designed to develop decision rules in order to accept or reject products based on sample information. When the products are being supplied by a manufacturing process in a series of lots, the entire lot is customarily accepted or rejected on the basis of the empirical information contained in a random sample of items drawn from the concerned lot.

According to Dodge (1969), there are four major classifications of acceptance sampling; namely: (i) lot-by-lot sampling by the method of attributes; (ii) lot-by-lot sampling by the method of variables; (iii) continuous sampling of a flow of units by the method of attributes; and (iv) special purpose sampling plans, including chain sampling, skip-lot sampling, etc. Attribute sampling constitutes one of the vital areas of acceptance sampling. There are several sampling plans available in the literature for the inspection of attribute quality characteristics. Some of them are called conventional sampling plans and some are called special purpose plans. Single sampling plan (SSP), double sampling plan (DSP), and sequential sampling plan are called conventional sampling plans. The acceptance sampling plans have been investigated by many authors on various aspects; see for example, Aslam, Balamurali and Arif (2016); Aslam and Wang (2017); Balamurali et al. (2017); Balamurali and Usha (2017a); Gui and Aslam (2017); Rao and Aslam (2017); Subramani and Balamurali (2016); and Vijayaraghavan and Uma (2016). Various special purpose plans have been developed to serve certain special purposes. Some of the special purpose plans are chain sampling plan (ChSP), multiple dependent state (MDS) sampling plan, repetitive group sampling (RGS) plan, etc. The MDS plan was first proposed by Wortham and Baker (1976). The concept of ChSP was developed by Dodge (1955), and the RGS plan was proposed by Sherman (1965). In MDS plan, the inspected lot will be accepted if the number of nonconforming items in the sample (d) is less than or equal to c₁, the acceptance number (d ≤ c₁). If d > c₂, then the lot will be rejected. If c₁ < d ≤ c₂, then the current lot can be accepted provided preceding m lots have been accepted with the condition that d ≤ c₁. ChSP is a particular case of MDS plan with c₁ = 0, c₂ = 1 and m = i. The RGS plan can be operated in a similar way with a slight difference. Through the RGS plan, the lot will be accepted or rejected as in the MDS plan. But, if c₁ < d ≤ c₂, then the sampling can be repeated until reaching a decision on the disposition (either acceptance or rejection) of the lot. For further details about the MDS sampling plan, one may refer to Aslam, Azam, and Jun (2016); and Balamurali, Jeyadurga and Usha (2016, 2017, 2017b). Variables sampling plans also constitute another major area 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 attribute sampling plan. Thus, a variables acceptance sampling plan would require less sampling. The variables sampling provides more information than the attribute 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.

Most of the variables sampling plans available in the literature of acceptance sampling solely rely on process performance and do not take into account the engineering specifications while determining the required parameters of the sampling plans (see for example, Balamurali and Usha, 2015). In order to overcome this drawback, it has become mandatory to develop sampling plans based on the process capability index. Based on this idea, some of the researchers have developed sampling plans based on the PCI C_pk (refer to Pearn and Wu, 2007; Itay, Parmet, & Schechtman, 2009, 2010; Aslam, Wu, Azam, & Jun, 2013a, 2013b, 2013c; Balamurali and Usha, 2014, 2017b; Wu & Wang, 2017; Fallah Nezhad, & Seifi, 2016, 2017; Wu, Aslam, & Jun, 2017; and Hussain et al., 2017). Some of the authors have proposed mixed sampling plans based on the PCI C_pk (see for example, Aslam et al., 2015a, 2015b; Balamurali, Aslam, & Jun, 2016). With this motivation, we develop the variables modified chain sampling plan (MChSP) based on the PCI C_pk since no variables MChSP based on the PCI C_pk is available in the literature.

3. Variables Modified Chain Sampling Plan Based on the PCI C_pk

The concept of chain sampling was first introduced by Dodge (1955) for the application of attribute quality characteristics. The characteristics and the properties of the ChSP were investigated by many authors (see for example Soundararajan, 1978). The ChSP belongs to the group of conditional sampling plans and in these, acceptance or rejection of a lot is based not only on the sample from that particular lot, but also on sample results from past lots. The ChSP 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 ChSP can be seen in Dodge (1955). As pointed out earlier, the ChSP is a particular case of the MDS sampling plan proposed by Wortham and Baker (1976). Govindaraju and Balamurali (1998) extended the concept of chain sampling to variables inspection. Recently, Balamurali and Usha (2013b) proposed the designing methodology of variables ChSP for the application of normally distributed quality characteristics having a single specification limit. The conditions of application and the operating procedure of the variables ChSP can be seen in Govindaraju and Balamuali (1998) and Balamurali and Usha (2013b).

In the orginal version of ChSP of Dodge (1955), chaining of preceding lot results will not take place when the sample from the current lot consists of no non-conforming items. It means that preceding lot information is not fully used for the disposition of the current lot even though there is evidence of good quality history. It is to be noted that any sampling plan which makes use of past lot information for disposition of the current lot will have a minimum sample size. In light of this, Govindaraju and Lai (1998) proposed a modified version of ChSP which uses the information of preceding lots compulsorily for the disposition of current lot and the plan is designated as modified ChSP (MChSP). Govindaraju and Lai (1998) have shown that a MChSP plan has a smaller sample size compared to the ChSP of Dodge (1955). In this article, we extend the concept of MChSP to the variables inspection. When the quality characteristic under study has double specification limits such as USL and LSL, no MChSP is available in the literature. Hence, we propose an MChSP based on double specification limits and the PCI C_pk. The conditions of application of the proposed variables MChSP are the same as that of variables ChSP of Govindaraju and Balamurali (1998).

Suppose that a quality characteristic of interest has double specification limits; namely, USL and LSL, and that an item having the quality characteristic beyond these limits is considered to be nonconforming. Further, it is also assumed that the quality characteristic under study follows a normal distribution with unknown mean and unknown standard deviation. Then, the operating procedure of the variables MChSP based on C_pk is proposed as given below.

• Step 1. From each submitted lot, take a random sample of size n, say, (x₁, x₂, …, x_n) and compute the estimate of C_pk as shown in (4(4) ).

• Step 2. (i) Reject the lot if .

• (ii) Accept the lot if provided preceding i samples (from i lots) also satisfy the condition except in one sample may lie in between k_a and k_r, i.e., . Otherwise reject the lot. (Note: k_a > k_r).

Thus, the proposed unknown sigma variables MChSP based on C_pk is characterized by four parameters namely n, i, k_a and k_r.

4. Operating Characteristic Function of the Proposed Plan

Govindaraju and Lai (1998) derived the operating characteristic function of the attribute MChSP. Based on Govindaraju and Lai (1998) and Govindaraju and Balamurali (1998), the OC function of the variables MChSP is derived as follows. (5) where p is the fraction non-conforming, P_a is the probability of accepting a lot based on a single sample with parameters (n, k_a), P_r is the probability of rejecting a lot based on a single sample with parameters (n, k_r), and i stands for the number of preceding lots used for conditionally accepting the current lot. 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. It is also to be pointed out that when i = 0 and/or k_a = k_r, the OC function given in (5(5) ) reduces to the OC function of variables SSP based on C_pk with parameters (n, k_a). Based on the PCI C_pk, P_a and P_r, respectively, are written as follows (see Aslam et al., 2013b). and

It is known that when n is large, is approximately normally distributed with mean μ ± kE(S) and variance (see Duncan, 1986; Balamurali et al., 2005). Based on this assumption, the probabilities P_a and P_r, respectively, can be written as follows. (6) and (7) where

Similarly,

If we define p_U and p_L as the fraction non-conforming when the item falls outside USL and LSL, respectively, then,

If we let and , then the probabilities P_a and P_r can be obtained, respectively, as follows. (8) and (9) where,

5. Determination of Optimal Parameters

5.1. Case of Symmetric Fraction Non-conforming

Let p_U and p_L be the upper and lower limits of the fraction non-conforming and are defined as the fraction non-conforming when the item falls outside USL and LSL, respectively. In the case of symmetric fraction non-conforming, we first assume that,

The OC function of the proposed plan under symmetric fraction nonconforming is given in (5(5) ) and the probabilities P_a and P_r will be obtained, respectively, as follows. (10) and (11) where,

In order to find the optimal parameters of the proposed sampling plan, we use two points on the OC curve approach; 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. A well designed sampling plan must provide at least (1- α) probability of acceptance of a lot when the process fraction nonconforming is at AQL, and the sampling plan must also provide not more than β probability of acceptance if the process fraction nonconforming is at the LQL. Thus, the acceptance sampling plan must have its OC curve passing through two designated points (AQL, 1- α) and (LQL, β). For given AQL and LQL, the parametric values of the variables MChSP, namely, n, i, k_a and k_r, are determined by satisfying the required producer and consumer conditions. Alternatively, we can determine the above parameters of the proposed plan to minimize the ASN at AQL, which is analogous to minimizing the ASN in the variables RGS plans (see Balamurali et al., 2005). The ASN for the MChSP is the sample size n only. Therefore, the following optimization problem is considered to determine those parameters. (12)

where Φ^{− 1}( · ) is inverse normal cumulative distribution function, L (p₁) is the probability of acceptance of the proposed MChSP at AQL (p₁), and L (p₂) is the probability of acceptance at LQL (p₂).

5.2. Case of Asymmetric Fraction Non-conforming

In some situations, the fraction non-conforming below the lower specification limit and those above the upper specification limit are different. The asymmetric case can be used when the fraction non-conforming beyond LSL and USL are believed (by quality engineers in practice) to be not equal. In such situations, it will be assumed that for p_L + p_U = p,

In case of asymmetric fraction non-conforming, the OC function of the proposed plan is same as in (5(5) ), but the probabilities P_a and P_r will be different. The probabilities P_a and P_r respectively will be determined as, (13) and (14) where,

In the case of asymmetric fraction non-conforming, the similar optimization problem as given in (12(12) ) can be used to determine the optimal parameters.

We may determine the optimal parameters of the MChSP based on C_pk by solving the nonlinear equation given in (12(12) ) for both symmetric and asymmetric fraction non-conforming cases. There may exist multiple solutions since there are four unknowns with only two equations. Generally, a sampling plan would be desirable if the required sample size or ASN is small. So, in this article, 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. By solving the nonlinear problem mentioned above, the parameters (n, i, k_a and k_r) can be determined. For the specified AQL and LQL values, we can determine the optimal parameters of the proposed plan by a search program with n = 2 (1(1) )5000, k_r = 0.1 (0.0005)3.5, k_r = k_a + 0.0005 (0.0005)3.5, and i = 1 (1(1) )10 for both symmetric and asymmetric fraction non-conforming cases. Out of these combinations, the optimal plan parameters which satisfy both producer and consumer risks (that is, the probability of acceptance at AQL is exactly 0.95 or just greater than 0.95 and the probability of acceptance at LQL is exactly 0.10 or just less than 0.10) and have minimum ASN at AQL will be chosen and tabulated. For the symmetric case, we consider p_U = p_L = p/2 and for asymmetric fraction non-conforming we consider two cases, namely, (p_L = p/3, p_U = 2p/3) and (p_L = p/4, p_U = 3p/4). Table 1 presents the optimal parameters of the MChSP plan for symmetric fraction non-conforming, and Tables 2 and 3 give the optimal parameters of the proposed plan for (p_L = p/3, p_U = 2p/3) and (p_L = p/4, p_U = 3p/4), respectively.

Table 1. Optimal parameters of variables modified chain sampling plan based on C_pk (symmetric fraction non-conforming).

p1	p2	n	i	ka	kr	Pa (p1)	Pa (p2)
0.001	0.002	267	8	0.98299	0.96299	0.95012	0.09519
	0.003	90	8	0.91200	0.88200	0.95005	0.09882
	0.004	51	8	0.86100	0.82600	0.95023	0.09745
	0.006	27	7	0.79700	0.75200	0.95015	0.09535
	0.008	18	8	0.73000	0.70000	0.95036	0.09869
	0.010	13	8	0.68600	0.65100	0.95042	0.09838
	0.015	8	8	0.60101	0.57601	0.95019	0.09849
	0.020	6	7	0.55601	0.53101	0.95014	0.09762
0.0025	0.005	185	8	0.88400	0.85900	0.95048	0.09869
	0.010	37	8	0.75200	0.72200	0.95095	0.09783
	0.015	19	7	0.67900	0.64400	0.95043	0.09958
	0.020	12	7	0.62201	0.57701	0.95087	0.09551
	0.025	8	8	0.55701	0.51201	0.95035	0.09918
	0.030	7	7	0.53201	0.49701	0.95091	0.09806
	0.050	4	6	0.42901	0.40901	0.95074	0.09813
0.005	0.010	143	8	0.80400	0.77900	0.95052	0.09566
	0.015	48	8	0.72100	0.69100	0.95023	0.09911
	0.020	26	8	0.66000	0.62500	0.95039	0.09866
	0.030	13	8	0.56901	0.53901	0.95061	0.09999
	0.040	8	8	0.50201	0.46701	0.95018	0.09838
	0.050	6	8	0.44801	0.42301	0.95080	0.09685
	0.100	3	4	0.36301	0.31801	0.95069	0.09937
0.010	0.020	106	7	0.72100	0.69100	0.95079	0.09737
	0.030	37	8	0.62700	0.60700	0.95008	0.09568
	0.040	20	6	0.58000	0.54000	0.95035	0.09832
	0.050	13	7	0.51700	0.48700	0.95081	0.09680
	0.100	4	7	0.33701	0.30701	0.95074	0.09958
	0.150	3	3	0.34101	0.26101	0.95012	0.09942
	0.200	3	1	0.33901	0.32401	0.95018	0.09903
0.030	0.060	58	7	0.56300	0.53300	0.95029	0.09995
	0.090	19	8	0.45200	0.43200	0.95012	0.09529
	0.120	9	8	0.37101	0.33601	0.95036	0.09652
	0.150	6	7	0.31601	0.28101	0.95033	0.09938
	0.300	3	2	0.22301	0.21301	0.95073	0.09535
0.050	0.100	43	6	0.48600	0.45100	0.95027	0.09782
	0.150	12	8	0.35600	0.32100	0.95056	0.09598
	0.200	6	8	0.26500	0.22500	0.95030	0.09649
	0.250	4	8	0.18600	0.16100	0.95014	0.09954
	0.300	3	7	0.13801	0.10801	0.95031	0.09813

Table 2. Optimal parameters of variables modified chain sampling plan based on C_pk (asymmetric fraction non-conforming).

p1	p2	n	i	ka	kr	Pa (p1)	Pa (p2)
0.001	0.002	280	4	1.18199	1.10199	0.95066	0.09539
	0.003	105	7	0.92000	0.88500	0.95094	0.09600
	0.004	57	7	0.87100	0.83100	0.95061	0.09727
	0.006	28	8	0.78800	0.75300	0.95079	0.09889
	0.008	18	8	0.73400	0.69400	0.95074	0.09671
	0.010	14	7	0.70200	0.66200	0.95070	0.09942
	0.015	8	8	0.60701	0.57201	0.95088	0.09654
	0.020	6	7	0.55901	0.52901	0.95042	0.09959
0.0025	0.005	228	8	0.88000	0.85500	0.95008	0.09742
	0.010	40	8	0.75300	0.72300	0.95048	0.09948
	0.015	19	8	0.67100	0.63600	0.95056	0.09871
	0.020	12	8	0.60901	0.57401	0.95004	0.09942
	0.025	9	8	0.56201	0.53201	0.95067	0.09583
	0.030	5	7	0.47001	0.44001	0.95062	0.09752
	0.050	4	6	0.43601	0.40601	0.95036	0.09717
0.005	0.010	174	8	0.79900	0.77400	0.95002	0.09522
	0.015	54	8	0.72100	0.69100	0.95045	0.09915
	0.020	29	8	0.66100	0.63100	0.95074	0.09995
	0.030	15	7	0.58301	0.55801	0.95052	0.09742
	0.040	9	7	0.52001	0.48501	0.95038	0.09548
	0.050	6	7	0.46801	0.41801	0.95012	0.09546
	0.100	3	4	0.37101	0.31101	0.95075	0.09888
0.010	0.020	126	8	0.71000	0.68500	0.95054	0.09744
	0.030	38	8	0.62500	0.59500	0.95022	0.09956
	0.040	21	7	0.51700	0.53600	0.95083	0.09612
	0.050	13	7	0.51800	0.47800	0.95016	0.09772
	0.100	4	8	0.31801	0.30301	0.95013	0.09814
	0.150	2	8	0.20601	0.15601	0.95011	0.09735
	0.200	2	4	0.20701	0.19701	0.95031	0.09767
0.030	0.060	70	7	0.55600	0.52600	0.95035	0.09618
	0.090	20	8	0.44800	0.42300	0.95070	0.09785
	0.120	10	6	0.39500	0.34000	0.95035	0.09786
	0.150	6	8	0.30100	0.27100	0.95093	0.09903
	0.300	3	2	0.23500	0.20000	0.95007	0.09900
0.050	0.100	48	8	0.46300	0.43800	0.95097	0.09612
	0.150	13	8	0.34900	0.31900	0.95023	0.09948
	0.200	7	7	0.27100	0.24600	0.95024	0.09727
	0.250	4	8	0.18200	0.15200	0.95013	0.09884
	0.300	3	7	0.13500	0.10000	0.95029	0.09648

Table 3. Optimal parameters of variables modified chain sampling plan based on C_pk (asymmetric fraction non-conforming).

p1	p2	n	i	ka	kr	Pa (p1)	Pa (p2)
0.001	0.002	317	8	0.96899	0.94399	0.95042	0.09721
	0.003	110	8	0.90900	0.87900	0.95095	0.09962
	0.004	63	7	0.87200	0.83200	0.95080	0.09557
	0.006	29	8	0.79000	0.75000	0.95013	0.09927
	0.008	19	8	0.73700	0.69700	0.95039	0.09730
	0.010	15	7	0.70500	0.67000	0.95022	0.09971
	0.015	8	8	0.61201	0.56701	0.95056	0.09624
	0.020	6	7	0.56801	0.52301	0.95058	0.09720
0.0025	0.005	229	8	0.86900	0.84400	0.95051	0.09703
	0.010	42	8	0.75300	0.71800	0.95092	0.09594
	0.015	20	8	0.67200	0.63700	0.95035	0.09897
	0.020	13	8	0.61201	0.58201	0.95056	0.09773
	0.025	9	7	0.58001	0.52001	0.95021	0.09704
	0.030	4	6	0.44201	0.40201	0.95023	0.09765
	0.050	7	6	0.63400	0.57400	0.95287	0.09667
0.005	0.010	180	8	0.78600	0.76600	0.95049	0.09566
	0.015	22	7	0.68300	0.65300	0.95086	0.09887
	0.020	13	8	0.61201	0.58201	0.95056	0.09773
	0.030	7	8	0.52301	0.48801	0.95089	0.09759
	0.040	5	8	0.44701	0.43701	0.95055	0.09811
	0.050	4	6	0.44201	0.40201	0.95023	0.09765
	0.100	3	2	0.40901	0.39901	0.95025	0.09665
0.010	0.020	125	8	0.69700	0.67200	0.95060	0.09551
	0.030	39	8	0.62000	0.58500	0.95021	0.09564
	0.040	21	8	0.55600	0.52600	0.95084	0.09861
	0.050	13	8	0.50400	0.46900	0.95054	0.09907
	0.100	4	8	0.32201	0.29701	0.95066	0.09759
	0.150	3	4	0.29701	0.27201	0.95014	0.09932
	0.200	3	5	0.40301	0.34801	0.95498	0.09856
0.030	0.060	68	7	0.54000	0.51000	0.95064	0.09613
	0.090	20	7	0.44900	0.40900	0.95073	0.09918
	0.120	10	8	0.35900	0.33400	0.95063	0.09943
	0.150	6	8	0.29700	0.26200	0.95076	0.09725
	0.300	4	1	0.31000	0.21500	0.95076	0.09502
0.050	0.100	44	8	0.44500	0.41500	0.95064	0.09562
	0.150	14	8	0.32900	0.31900	0.95034	0.09959
	0.200	7	7	0.26000	0.23500	0.95092	0.09898
	0.250	4	8	0.17300	0.14300	0.95017	0.09759
	0.300	3	7	0.12300	0.09300	0.95036	0.09834

6. Examples

The first two examples show the use of tables for designing plans, and the third example shows an industrial application.

Example 1. Symmetric Fraction Non-conforming Case

Suppose a quality characteristic of interest follows a normal distribution and has double specification limits as LSL = 75 and USL = 125. The inspector wishes to adopt a MChSP, where AQL at α = 0.05 and LQL at β = 0.1 are specified by p₁ = 0.03 and p₂ = 0.06, respectively. Then, Table 1 gives the optimal parameters as n = 58, i = 7, k_a = 0.56300, and k_r = 0.53300. For this optimal plan, P_a (p₁) = 0.95029 and P_a (p₂) = 0.09995.

Example 2. Asymmetric Fraction Non-conforming Case

Suppose one wants to determine the optimal parameters of variables MChSP when the quality characteristic of interest follows a normal distribution with asymmetric fraction non-conforming for the specified AQL and LQL conditions. Table 2 and Table 3 can be used to obtain the optimal plans. For example, let AQL and LQL be specified as p₁ = 0.05 and p₂ = 0.10 at α = 0.05 and at β = 0.1 respectively. If we assume asymmetric fraction nonconforming as p_L = p/3 and p_U = 2p/3, then Table 2 gives the optimal parameters as n = 48, i = 8, k_a = 0.46300 and k_r = 0.43800. For this optimal plan the probability of acceptance at AQL and at LQL respectively are P_a (p₁) = 0.95097 and P_a (p₂) = 0.09612.

Example 3. Industrial Application

In order to illustrate the implementation of the proposed MChSP in industries, we consider a case study data on super twisted nematic (STN) – liquid crystal displays (LCD) (STN-LCD) manufacturing process as given in Wu and Pearn (2008). As pointed out by Wu and Pearn (2008), STN-LCD is used in various products, such as the Notebook personal computer. Suppose that an industry produces various models of LCD. The USL of a glass thickness is 0.77 mm and the LSL of the glass thickness is 0.63 mm. Suppose that the quality bench marking levels of the producer and the consumer based on the PCI C_pk are specified as AQL = 0.05 and LQL = 0.10, respectively, and the specified producer's and consumer's risks are α = 0.05 and β = 0.10, respectively, under asymmetric fraction non-conforming case as given above in Example 2. For the specified quality requirements, the plan parameters are obtained from Table 2 as n = 48, i = 8, k_a = 0.46300 and k_r = 0.43800. The plan is implemented as follows.

• Step 1. Take a random sample of size 48. As given by Wu and Pearn (2008), the measurements of glass thickness are as follows.

  0.717     0.698     0.726     0.684     0.727     0.688     0.708     0.703     0.694     0.713

  0.730     0.699     0.710     0.688     0.665     0.704     0.725     0.729     0.716     0.685

  0.712     0.716     0.712     0.733     0.709     0.703     0.730     0.716     0.688     0.688

  0.712     0.702     0.726     0.669     0.718     0.714     0.726     0.683     0.713     0.737

  0.740     0.706     0.726     0.688     0.715     0.704     0.724     0.713

• Step 2. For this data, calculate and .

• Step 3. Calculate

• Step 4. Since , the current lot is accepted provided at least 7 of 8 preceding samples (from 8 lots) satisfy , and in at most one sample lies between 0.43800 and 0.46300, i.e., .

7. Efficacy of the Proposed Plan

In this section, we compare the efficacy of the proposed MChSP with some other existing sampling plans developed based on the PCI C_pk in the case of symmetric fraction non-conforming. Any sampling plan with smaller sample size or ASN would always be more desirable in order to reduce the inspection costs particularly for compliance testing. Hence, we compare the ASN of MChSP with the ASN of SSP, the sampling plan for resubmitted lots (resampling scheme) and RGS plan for both symmetric fraction non-conforming and asymmetric fraction nonconforming (p_L = p/4, p_U = 3p/4) cases. For this purpose, we provide Tables 4 and 5, which give the ASN of the above-mentioned sampling plans under symmetric and asymmetric fraction non-conforming cases, respectively, for some selected combinations of p₁ and p₂. Tables 4 and 5 The ASN of the sampling plan for resubmitted lots is obtained from Aslam et al. (2013a) and the ASN of RGS plan is from Aslam et al. (2013b). From Tables 4 and 5, it is clearly observed that the proposed sampling plan achieves minimum ASN compared to all other sampling plans tabulated in the case of the symmetric fraction non-conforming case. For example, in the case of a symmetric fraction non-conforming, when p₁ = 0.001 and p₂ = 0.002, Table 4 shows the ASN of SSP is 1134, the ASN of resampling scheme is 1041.53, and the ASN of RGS plan is 452.77, whereas the ASN of the proposed MChSP is just 267. The same observation can be made from Table 5 for the case of an asymmetric fraction non-conforming case except for very few combinations of AQL and LQL. Hence, the proposed MChSP is more efficient compared to other sampling plans in terms of minimum ASN, so that inspection effort, time, and cost of inspection will also be greatly reduced.

Table 4. ASN of proposed modified chain sampling plan and other sampling plans based on C_pk (symmetric fraction non-conforming case).

		ASN
p1	p2	SSP	Resampling Scheme	RGS	MChSP
0.001	0.002	1134	1041.53	452.77	267
	0.003	351	374.06	168.17	90
	0.004	166	216.86	89.06	51
	0.006	74	113.87	43.47	27
	0.008	47	78.88	29.26	18
	0.010	34	59.68	23.55	13
0.0025	0.005	822	750.74	319.83	185
	0.010	118	152.14	61.23	37
	0.015	53	78.60	29.07	19
	0.020	32	51.60	19.28	12
	0.025	24	40.67	13.37	8
	0.030	18	32.53	11.05	7
0.005	0.010	623	567.65	246.49	143
	0.015	186	195.21	79.39	48
	0.020	87	111.19	49.51	26
	0.030	37	56.95	22.99	13
	0.040	23	38.09	13.55	8
	0.050	17	27.12	9.41	6
0.010	0.020	449	407.29	187.96	106
	0.030	132	138.38	59.71	37
	0.040	61	75.93	32.88	20
	0.050	37	51.52	21.45	13
	0.100	11	19.97	5.80	4
	0.150	6	10.85	3.95	3
0.030	0.060	240	214.18	88.13	58
	0.090	68	67.81	29.40	19
	0.120	31	35.24	17.43	9
	0.150	18	24.41	10.02	6
	0.300	5	8.13	3.35	3
0.050	0.100	167	149.13	67.91	43
	0.150	46	46.08	19.71	12
	0.200	20	24.39	10.07	6
	0.250	12	13.55	6.68	4

Table 5. ASN of proposed modified chain sampling plan and other sampling plans based on C_pk (asymmetric fraction non-conforming case).

		ASN
p1	p2	SSP	Resampling Scheme	RGS	MChSP
0.001	0.003	350	86.87	172.31	110
	0.004	169	62.34	66.42	63
	0.006	74	40.67	30.85	29
	0.008	47	32.52	22.85	19
	0.010	35	24.39	17.00	15
0.0025	0.010	119	43.36	47.66	42
	0.015	51	27.10	21.23	20
	0.020	32	21.70	15.67	13
	0.025	23	16.27	11.49	9
	0.030	18	13.55	8.98	4
0.005	0.015	189	46.15	53.40	22
	0.020	87	32.59	32.90	13
	0.030	38	18.99	16.67	7
	0.040	23	13.56	11.82	5
	0.050	17	10.84	8.79	4
0.010	0.030	135	32.54	52.14	39
	0.040	61	21.68	25.56	21
	0.050	37	16.26	15.79	13
	0.100	11	8.13	5.84	4
	0.150	6	5.42	3.94	3
0.030	0.090	68	16.27	23.49	20
	0.120	31	10.84	13.01	10
	0.150	18	8.13	8.74	6
	0300	5	3.80	3.26	4
0.050	0.150	46	10.84	18.47	14
	0.200	13	5.42	9.90	7
	0.250	12	5.42	6.45	4

8. Conclusions

In this article, we have developed variables MChSP based on the process capability index C_pk to deal with the product acceptance decision making problem. The proposed plan can be used when the quality characteristic of interest is normally distributed with unknown mean and unknown standard deviation and has two specification limits; namely, the lower and upper specification limits. Procedures and methodologies of determining the optimal parameters of the proposed plan have been developed. We have constructed tables for both symmetric and the asymmetric fraction nonconforming cases for the unknown sigma variables MChSP based on the process capability index C_pk, as the process capability indices are becoming standard tools for quality reporting. The implementation of the proposed plan has also been explained with an example. It has been shown that the proposed MChSP based on C_pk is more efficient than the existing sampling plans based on C_pk in terms of minimum ASN, so that the cost and time of inspection and errors due to sampling can be reduced.

Acknowledgments

The authors would like to thank the reviewers, the associate editor, and the editor for their constructive comments and valuable suggestions, which significantly improved the presentation of this work.

Additional information

Funding

This research was supported by the Department of Science and Technology-Science and Engineering Research Board (DST-SERB), India through the project (SR/S4/MS:790/12).