Designing a multiple dependent state sampling plan based on the coefficient of variation
 Aijun Yan^{1}Email author,
 Sanyang Liu^{1} and
 Xiaojuan Dong^{1}
Received: 22 October 2015
Accepted: 16 August 2016
Published: 30 August 2016
Abstract
A multiple dependent state (MDS) sampling plan is developed based on the coefficient of variation of the quality characteristic which follows a normal distribution with unknown mean and variance. The optimal plan parameters of the proposed plan are solved by a nonlinear optimization model, which satisfies the given producer’s risk and consumer’s risk at the same time and minimizes the sample size required for inspection. The advantages of the proposed MDS sampling plan over the existing single sampling plan are discussed. Finally an example is given to illustrate the proposed plan.
Keywords
Background
Nowadays, quality is one of the most important consumer decision factors. It has become one of the main strategies to increase the productivity of industries and service organizations. Therefore, the companies are trying to enhance the quality of their products by using various statistical techniques and tools. Acceptance sampling plans are important tools that have been widely used for lot sentencing in the industries. The inspection of the final product is always done on the basis of acceptance sampling scheme. There are two major types of acceptance sampling plans: attribute sampling plans and variable sampling plans. The major advantage of a variable sampling plan is that it has the same protection as an attribute acceptance sampling plan with a smaller sample size. When destructive testing is employed, variables sampling is particularly useful in reducing the costs of inspection. For more detail about the applications of the acceptance sampling plan can be found in Wu (2012), Liu et al. (2014), Kurniati et al. (2015), Yen and Chang (2009), and Sheu et al. (2014).
The coefficient of variation (CV), which is defined as the ratio of the standard deviation to the mean, is widely used to measure the relative variation of a variable to its mean. CV has been widely used in many practical applications. It is used as a measure of the reliability of an assay in chemistry and medicine (Reed et al. 2002), to quantify the riskiness of stocks in finance (Miller and Karson 1977), in clinical trials to account for baseline variability of measurements (Pereira et al. 2004), in physical therapy to determine sincerity of effort (Robinson et al. 1997), in quality control to seek production processes with minimal dispersion (Box 1988). Recently, Parsons et al. (2009) concluded that it was important to use CVs to assess the quality of metabonomics datasets. Kang et al. (2007) developed a Shewharttype control chart for monitoring the CV using rational subgroups and showed the CV to be a very attractive tool in quality control.
In the literature, either the mean or the standard deviation (SD) of the quality characteristics are usually considered to measure the quality of products. However, in certain scenarios, the practitioner is not interested in the changes in the mean or the standard deviation but is instead interested in the relative variability compared with the mean (see for Yeong et al. 2015). This relative variability is called the CV. Verrill and Johnson (2007) have pointed that building materials are often evaluated not only on the basis of mean strength but also on relative variability, but laboratory techniques are often compared on the basis of their CVs. In many laboratories, the variability of the chemical assay that produces continuoustype values is summarized not by the SD but by the CV, because the SDs of such assays generally increase or decrease proportionally as the mean increases or decreases (refer to Reed et al. 2002). Therefore, acceptance sampling plans considering the CV as the reliability parameter can complement each other with the other acceptance sampling plans, so as to control the product quality and improve the management level.
CV can be applied not only characteristic analysis of ultimate strength or fatigue limit, failure rates and structural/material reliability, but also for both the reliabilitybased design of mechanical systems or components and the evaluation of an existing product (see for He and Oyadiji 2001). In the fields of materials engineering and manufacturing, Castagliola et al. (2015) have stated that some quality characteristics related to the physical properties of products often have a standard deviation that is proportional to their population mean. Tool cutting life and several properties of sintered materials are some typical examples. In such scenarios, the CV remains constant even though the mean and standard deviation may change from one sample to another. Zhang (1989) pointed that the CV can be predetermined from the long term of engineering practice in the research of structural reliability design, evaluation, and inspection.
CV is a good measure of the reliability of experiments, that is, the smaller the CV value, the higher the reliability (Steel and Torrie 1980; Taye and Njuho 2008). Recently, Ma and Zhang (1997) deduced the CV method for structural reliability inspection using the CV as the quality control parameter, under the condition of the CV being known. The inspection efficiency of CV method is higher than S method and \(\sigma\) method. Tong and Chen (1991) proposed a variable single sampling plan using CV to evaluate the quality stability of normally distributed products. Yan et al. (2016) developed a variable two stage sampling plan based on CV, which is more efficient than the single sampling plan proposed by Tong and Chen (1991).
In advanced manufacturing processes, supplier production is frequently continuous, so the quality of preceding and/or successive lots is expected to be homogeneous and dependent (Kuraimani and Govindaraju 1992). But the single sampling plan and the two stage sampling plan only consider the present state of a lot, that is, they accept or reject a lot based on the present lot quality. In order to compensate for this weaknesses, Wortham and Baker (1976) introduced the multiple dependent state (MDS) sampling plan, which examines a lot based on not only the sample information from the current lot but also the quality of preceding lots. So the MDS sampling plan can be used in the case that lots are submitted for inspection serially. Recently, Balamurali and Jun (2007) proposed MDS sampling plan by variables for the assessment of normally distributed quality characteristics. Aslam et al. (2015) proposed a mixed MDS sampling plan using the process capability index, and Aslam et al. (2014) considered MDS sampling for the development of a new attribute control chart. To the best of our knowledge, there exist no studies about the MDS plan based on the CV. Therefore, assuming that the quality characteristic follows the normal distribution, we will develop the MDS sampling plan using the CV with expectation that it is more efficient than the single plan proposed by Tong and Chen (1991) in this article.
Multiple dependent state (MDS) sampling plan

Step 1: Choose the values of \((v_{1} ,v_{2} )\) based on the CV at producer’s risk \(\alpha\) and consumer’s risk \(\beta\).

Step 2: Select a random sample of size n, (\(X_{1} ,X_{2} , \ldots ,X_{n}\)), from the lot, then compute the sample CV \(\hat{\gamma }\) defined in (2).

Step 3: Accept the entire lot if \(\hat{\gamma } \le k_{a}\), reject the lot if \(\hat{\gamma } > k_{r}\); if \(k_{a} < \hat{\gamma } \le k_{r}\), then accept the current lot provided that the proceeding m lots have been accepted under the condition of \(\hat{\gamma } \le k_{a}\), otherwise reject the lot. Note that \(k_{a}\) and \(k_{r}\) are acceptance constant and rejection constant, respectively.
The proposed plan is characterized by four parameters \(k_{a}\), \(k_{r}\), m and n. If \(k_{a} = k_{r}\), then it reduces to an ordinary variable single sampling plan proposed by Tong and Chen (1991) .
Determination of the proposed sampling plan parameters
The proposed plan parameters under (α, β) = (0.05, 0.10), (0.10, 0.05), (0.10, 0.10) (m = 1)
v _{1}  v _{2}  (α, β)  

(0.05, 0.10)  (0.10, 0.10)  (0.10, 0.05)  
k _{ a }  k _{ r }  n  k _{ a }  k _{ r }  n  k _{ a }  k _{ r }  n  
0.05  0.06  0.05318  0.05849  85  0.05166  0.06078  95  0.05216  0.05888  70 
0.07  0.05561  0.06530  28  0.05342  0.06279  30  0.05336  0.06758  22  
0.08  0.05693  0.07151  15  0.05412  0.06777  17  0.0557  0.0667  13  
0.09  0.05771  0.08538  10  0.05466  0.07324  12  0.05759  0.06885  9  
0.10  0.05867  0.09885  8  0.05475  0.09235  9  0.05665  0.08375  7  
0.11  0.05970  0.10620  7  0.05611  0.09931  8  0.05527  0.09801  6  
0.12  0.06081  0.12030  6  0.05821  0.10770  7  0.05941  0.07568  5  
0.06  0.07  0.06314  0.06938  120  0.06181  0.06947  132  0.06247  0.06793  95 
0.08  0.06516  0.07975  36  0.06311  0.07612  40  0.06381  0.07678  29  
0.09  0.06732  0.08352  19  0.06470  0.07958  22  0.06485  0.08342  16  
0.10  0.06850  0.09399  13  0.06588  0.08128  14  0.06787  0.08078  11  
0.11  0.06975  0.10900  10  0.06528  0.10460  11  0.0661  0.1106  9  
0.12  0.07055  0.11010  8  0.06558  0.10500  9  0.06854  0.1005  7  
0.07  0.08  0.07302  0.08102  159  0.07232  0.07648  172  0.07233  0.07918  128 
0.09  0.07551  0.08876  46  0.07370  0.08416  52  0.07395  0.0871  38  
0.10  0.07792  0.09311  25  0.07471  0.08945  27  0.07599  0.09193  21  
0.11  0.07916  0.10670  16  0.07535  0.09831  17  0.07595  0.1029  13  
0.12  0.08030  0.11050  12  0.07572  0.11050  13  0.07692  0.1063  10  
0.08  0.09  0.08329  0.08939  203  0.08204  0.08777  218  0.08249  0.0885  161 
0.10  0.08558  0.09992  60  0.08342  0.09748  64  0.08397  0.1000  48  
0.11  0.08772  0.10650  30  0.08491  0.10060  33  0.08519  0.1085  25  
0.12  0.09045  0.10820  19  0.08521  0.11070  22  0.08751  0.1056  16  
0.09  0.10  0.09318  0.09987  255  0.09218  0.09861  273  0.09244  0.09902  197 
0.11  0.09646  0.10590  73  0.09346  0.10820  78  0.09414  0.1087  57  
0.12  0.09829  0.11530  37  0.09476  0.11360  40  0.09587  0.1149  30  
0.10  0.11  0.10320  0.10990  307  0.10210  0.10840  330  0.1023  0.1103  240 
0.12  0.10580  0.12160  89  0.10380  0.11560  94  0.1046  0.1162  69 
The proposed plan parameters under (α, β) = (0.05, 0.10), (0.10, 0.05), (0.10, 0.10) (m = 2)
v _{1}  v _{2}  (α, β)  

(0.05, 0.10)  (0.10, 0.10)  (0.10, 0.05)  
k _{ a }  k _{ r }  n  k _{ a }  k _{ r }  n  k _{ a }  k _{ r }  n  
0.05  0.06  0.05361  0.06971  87  0.05257  0.06966  97  0.05298  0.08101  72 
0.07  0.05656  0.10760  28  0.05437  0.06490  32  0.05524  0.07025  23  
0.08  0.05925  0.09898  16  0.05540  0.08372  18  0.05681  0.08024  13  
0.09  0.06141  0.10630  11  0.05707  0.08895  13  0.05754  0.08661  9  
0.10  0.06247  0.09962  8  0.05765  0.10510  9  0.06024  0.07335  7  
0.11  0.06400  0.11040  7  0.05776  0.11090  8  0.05898  0.1057  6  
0.12  0.06360  0.11140  6  0.06097  0.11960  7  0.05969  0.1041  5  
0.06  0.07  0.06372  0.07712  121  0.06269  0.07227  134  0.06313  0.07553  98 
0.08  0.06673  0.08421  39  0.06455  0.08179  41  0.06534  0.1322  31  
0.09  0.06893  0.09754  20  0.06634  0.09493  23  0.06716  0.1025  17  
0.10  0.07102  0.09844  13  0.06739  0.10130  16  0.06908  0.09683  11  
0.11  0.07373  0.10640  10  0.06842  0.11550  11  0.07035  0.1206  9  
0.12  0.07435  0.11480  8  0.06888  0.12110  9  0.07174  0.1052  7  
0.07  0.08  0.07374  0.08281  162  0.07260  0.08323  175  0.07309  0.1294  129 
0.09  0.07715  0.08884  48  0.07481  0.09545  53  0.07545  0.1254  40  
0.10  0.07950  0.10380  25  0.07640  0.09878  29  0.07711  0.1305  22  
0.11  0.08203  0.12330  17  0.07739  0.11040  19  0.07909  0.1011  14  
0.12  0.08407  0.12380  12  0.07914  0.12240  14  0.07968  0.1198  11  
0.08  0.09  0.08393  0.09348  206  0.08274  0.09897  227  0.08312  0.1015  161 
0.10  0.08725  0.10110  61  0.08477  0.10510  67  0.08584  0.1188  49  
0.11  0.08962  0.12350  31  0.08678  0.11840  35  0.08735  0.114  26  
0.12  0.09241  0.12200  20  0.08791  0.12830  23  0.08927  0.1231  17  
0.09  0.10  0.09382  0.14880  257  0.09275  0.10150  277  0.09314  0.1126  201 
0.11  0.09735  0.11020  74  0.09501  0.11530  81  0.09572  0.12  58  
0.12  0.10060  0.11770  38  0.09698  0.12370  42  0.09813  0.1413  32  
0.10  0.11  0.10400  0.11200  312  0.10270  0.11970  334  0.1032  0.1441  246 
0.12  0.10760  0.1286  90  0.10510  0.12140  97  0.1061  0.121  72 
The proposed plan parameters under (α, β) = (0.05, 0.10), (0.10, 0.05), (0.10, 0.10) (m = 3)
v _{1}  v _{2}  (α, β)  

(0.05, 0.10)  (0.10, 0.10)  (0.10, 0.05)  
k _{ a }  k _{ r }  n  k _{ a }  k _{ r }  n  k _{ a }  k _{ r }  n  
0.05  0.06  0.05395  0.06719  92  0.05288  0.06051  103  0.05339  0.05934  74 
0.07  0.05710  0.06966  29  0.05489  0.0814  33  0.05566  0.0761  24  
0.08  0.05949  0.08085  16  0.05670  0.08502  19  0.05782  0.0805  14  
0.09  0.06236  0.09867  11  0.05753  0.09749  13  0.05911  0.07918  9  
0.10  0.06327  0.09177  8  0.05951  0.10750  10  0.06029  0.1294  7  
0.11  0.06551  0.11210  7  0.05955  0.11660  8  0.06088  0.1195  6  
0.12  0.06732  0.12820  6  0.06107  0.12420  7  0.06602  0.08333  6  
0.06  0.07  0.06414  0.07668  127  0.06297  0.07011  142  0.06342  0.1292  101 
0.08  0.06735  0.08271  39  0.06518  0.08149  44  0.06608  0.09012  32  
0.09  0.06995  0.09504  21  0.06706  0.09177  24  0.06793  0.1465  18  
0.10  0.07306  0.09381  14  0.06839  0.09786  17  0.06969  0.09208  12  
0.11  0.07423  0.11220  10  0.06969  0.10520  12  0.0723  0.1028  9  
0.12  0.07568  0.12390  8  0.07129  0.12060  10  0.07275  0.09233  8  
0.07  0.08  0.07419  0.08536  167  0.07303  0.08775  189  0.07346  0.09107  135 
0.09  0.07792  0.08845  52  0.07556  0.09699  58  0.07637  0.1125  41  
0.10  0.08034  0.09316  27  0.07740  0.10450  31  0.07831  0.1091  22  
0.11  0.08359  0.11960  17  0.07901  0.11100  20  0.08043  0.09617  15  
0.12  0.08486  0.11550  12  0.08128  0.12160  15  0.08309  0.1181  11  
0.08  0.09  0.08421  0.10350  217  0.08304  0.09478  240  0.08354  0.09341  171 
0.10  0.08770  0.10430  63  0.08551  0.10920  70  0.08666  0.1078  52  
0.11  0.09088  0.11450  32  0.08806  0.11210  38  0.08893  0.1376  27  
0.12  0.09319  0.12680  20  0.08932  0.12470  24  0.09183  0.1431  18  
0.09  0.10  0.09426  0.10830  266  0.09307  0.11120  299  0.09357  0.1264  214 
0.11  0.09768  0.11610  78  0.09557  0.11660  86  0.09656  0.1206  62  
0.12  0.10130  0.12490  39  0.09777  0.12680  45  0.09903  0.1442  32  
0.10  0.11  0.10430  0.11740  327  0.10310  0.11850  366  0.1036  0.1188  262 
0.12  0.10800  0.12940  92  0.10570  0.12870  105  0.1066  0.1477  74 
Advantages of the MDS plan
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 MDS plan over the single plan proposed by Tong and Chen (1991).
OC curves
Sample sizes required for inspection
The comparison of sample size of two sampling plans with (α, β) = (0.05, 0.10), (0.10, 0.05), (0.10, 0.10)
v _{1}  v _{2}  \(\alpha = 0. 0 5\), \(\beta = 0.10\)  \(\alpha = 0. 1 0\), \(\beta = 0. 0 5\)  \(\alpha = 0. 1 0\), \(\beta = 0. 1 0\)  

m = 1  m = 2  m = 3  n  m = 1  m = 2  m = 3  n  m = 1  m = 2  m = 3  n  
0.05  0.06  85  87  92  131  95  97  103  134  70  72  74  101 
0.07  28  28  29  39  30  32  33  41  22  23  24  31  
0.08  15  16  16  20  17  18  19  23  13  13  14  17  
0.09  10  11  11  14  12  13  14  15  9  9  9  12  
0.10  8  8  8  11  9  9  10  12  7  7  7  9  
0.11  7  7  7  9  8  8  8  9  6  6  6  8  
0.12  6  6  6  7  7  7  7  8  5  5  6  7  
0.06  0.07  120  121  127  182  132  134  142  186  95  98  101  142 
0.08  36  39  39  53  40  41  44  56  29  31  32  42  
0.09  19  20  21  28  22  23  24  28  16  17  18  22  
0.10  13  13  14  17  14  16  17  20  11  11  12  15  
0.11  10  10  10  14  11  11  12  15  9  9  9  11  
0.12  8  8  8  11  9  9  10  12  7  7  8  9  
0.07  0.08  159  162  167  242  172  175  189  248  128  129  135  188 
0.09  46  48  52  69  52  53  58  72  38  40  41  55  
0.10  25  25  27  35  27  29  31  36  21  22  22  28  
0.11  16  17  17  23  17  19  20  25  13  14  15  19  
0.12  12  12  12  17  13  14  15  17  10  11  11  14  
0.08  0.09  203  206  217  311  218  227  240  318  161  161  171  242 
0.10  60  61  63  88  64  67  70  91  48  49  52  69  
0.11  30  31  32  44  33  35  38  46  25  26  27  35  
0.12  19  20  20  28  22  23  24  30  16  17  18  23  
0.09  0.10  255  257  266  327  273  277  299  336  197  201  214  303 
0.11  73  74  78  109  78  81  86  112  57  58  62  85  
0.12  37  38  39  55  40  42  45  56  30  32  32  44  
0.10  0.11  307  312  327  335  330  334  366  343  240  246  262  335 
0.12  89  90  92  132  94  97  105  136  69  72  74  103 
An illustrative example
To illustrate the proposed MDS plan for practical applications, we use the actual data as discussed by Aslam et al. (2013). The data is about concrete which is widely used to construct buildings, roads, and a variety of other structures. The compressive strength of concrete is the most common quality measure used by the engineer in designing buildings and other structures. In the contract formulated from the producer and the consumer, suppose that the producer requires the probability of accepting the concrete at least 95 % if the CV of the compressive strength is less than 0.08, and the consumer require that the probability of accepting the concrete would be no more than 10 % if the CV of the compressive strength is larger than 0.12. That is, the values of \(v_{1}\) and \(v_{2}\) are set to 0.08 and 0.12 with the producer’s risk \(\alpha\) = 0.05 and the consumer’s risk \(\beta\) = 0.10. Therefore, the problem is the determination of the acceptance constants and the inspected sample sizes that provide the desired levels of protection for both producers and consumers.
The compressive strength of 20 concrete mixture specimens
36.3  40.1  31.8  33.6  34.9  31.2  32.8  25.8  30.8  32.9 
30.9  31.9  35.6  30.9  27.8  24.9  31.6  27.9  33.7  38.4 
Conclusions
In this paper, a multiple dependent state (MDS) sampling plan for accepting a lot whose quality characteristic follows a normal distribution based on the coefficient of variation (CV) is presented. Several tables are given for practical use. By comparison with the single sampling plan propose by Tong and Chen (1991) in terms of the required sample size and the OC curve, which show that our proposed MDS plan has a better performance than the single plan. Hence, the industrialists can save the inspection cost if they use the proposed MDS plan. Finally, a real example shows the application of the proposed plan in various industries. The present study can be extended for nonnormal distribution as future research.
Declarations
Authors’ contributions
AY conceived this project and wrote this manuscript. XD helped to write a part of R code. All authors read and approved the final manuscript.
Acknowledgements
The authors would like to thank the National Natural Science Foundation of China (Nos. 61373174, 71271165 and 11502184) and the Fundamental Research Funds for the Central Universities under Grant (No. 7214475301).
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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