Open Access

Designing a multiple dependent state sampling plan based on the coefficient of variation

SpringerPlus20165:1447

https://doi.org/10.1186/s40064-016-3087-3

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

Multiple dependent state (MDS) sampling planNormal distributionCoefficient of variation (CV)Operating characteristics (OC) curve

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 Shewhart-type 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 continuous-type 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 reliability-based 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

The coefficient of variation (CV) is a statistic defined as the ratio of the standard deviation \(\sigma\) to the mean \(\mu\). Suppose that the quality of interest X follows a normal distribution with the mean of \(\mu\) and the variance of \(\sigma^{2}\), the CV of the random variable X is defined as
$$\gamma = {\sigma \mathord{\left/ {\vphantom {\sigma \mu }} \right. \kern-0pt} \mu }$$
(1)
Assume that \(X_{1} ,X_{2} , \ldots ,X_{n}\) is a sample of the normal distribution \(N(\mu ,\sigma^{2} )\), then the sample coefficient of variation is defined as
$$\hat{\gamma } = \frac{S}{{\bar{X}}}$$
(2)
where \(S = \sqrt {\frac{1}{n - 1}\sum\nolimits_{i = 1}^{n} {(X_{i} - \bar{X})^{2} } }\) is the sample standard deviation, \(\bar{X} = \sum\nolimits_{i = 1}^{n} {X_{i} /n}\) is the sample mean.
Iglewicz et al. (1968) noticed that the statistic \(\sqrt n /\hat{\gamma }\) follows the noncentral t distribution, i.e. \(\sqrt n /\hat{\gamma }\sim t(n - 1,\sqrt n /\gamma )\), where n − 1 is the degrees of freedom, and \(\sqrt n /\gamma\) is the noncentrality parameter. Denote the cumulative distribution function (cdf) of \(\hat{\gamma }\) as
$$F_{{\hat{\gamma }}} (u\left| {n,\gamma } \right.) = 1 - F_{t} \left( {\frac{\sqrt n }{u}\left| {n - 1,\frac{\sqrt n }{\gamma }} \right.} \right)$$
(3)
where \(F_{t} ( \cdot )\) is the cdf of the \(t(n - 1,\sqrt n /\gamma )\) distribution.
Steel and Torrie (1980), Taye and Njuho (2008) point that the CV is a good measure of the reliability of the experiment. Here we use the CV as the quality benchmark for acceptance of a product lot. Let \(v_{1}\) and \(v_{2}\) denote the quality level of AQL (acceptable quality level) and LQL (limiting quality level) based on the CV, respectively. Then the operating procedure of the proposed plan based on the CV is stated as follows:
  • 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) .

According to Balamurali and Jun (2007), the OC function of the proposed MDS sampling plan is
$$P_{a} (v) = P\left\{ {\hat{\gamma } \le k_{a} \left| {\gamma = v} \right.} \right\} + P\{ k_{a} < \hat{\gamma } \le k_{r} \left| {\gamma = v} \right.\} [P\{ \hat{\gamma } \le k_{a} \left| {\gamma = v} \right.\} ]^{m}$$
(4)
The lot acceptance probability using single sampling and the probability of rejecting the lot directly based on the CV are respectively given as follows
$$P\left\{ {\hat{\gamma } \le k_{a} \left| {\gamma = v} \right.} \right\} = 1 - F_{t} \left( {\frac{\sqrt n }{{k_{a} }}\left| {n - 1,\frac{\sqrt n }{v}} \right.} \right)$$
$$P\{ \hat{\gamma } > k_{r} \left| {\gamma = v} \right.\} = F_{t} \left( {\frac{\sqrt n }{{k_{r} }}\left| {n - 1,\frac{\sqrt n }{v}} \right.} \right)$$
So,
$$P\{ k_{a} < \hat{\gamma } \le k_{r} \left| {\gamma = v} \right.\} = P\{ \hat{\gamma } \le k_{r} \left| {\gamma = v} \right.\} - P\{ \hat{\gamma } \le k_{a} \left| {\gamma = v} \right.\} = F_{t} \left( {\frac{\sqrt n }{{k_{a} }}\left| {n - 1,\frac{\sqrt n }{v}} \right.} \right) - F_{t} \left( {\frac{\sqrt n }{{k_{r} }}\left| {n - 1,\frac{\sqrt n }{v}} \right.} \right)$$
Then the OC function of the MDS sampling plan can be rewritten as
$$P_{a} (v) = 1 - F_{t} \left( {\frac{\sqrt n }{{k_{a} }}\left| {n - 1,\frac{\sqrt n }{v}} \right.} \right) + \left[ {F_{t} \left( {\frac{\sqrt n }{{k_{a} }}\left| {n - 1,\frac{\sqrt n }{v}} \right.} \right) - F_{t} \left( {\frac{\sqrt n }{{k_{r} }}\left| {n - 1,\frac{\sqrt n }{v}} \right.} \right)} \right]\left[ {1 - F_{t} \left( {\frac{\sqrt n }{{k_{a} }}\left| {n - 1,\frac{\sqrt n }{v}} \right.} \right)} \right]^{m}$$
(5)

Determination of the proposed sampling plan parameters

Yen and Chang (2009) stated “A well-designed sampling plan must provide a probability of at least (\(1 - \alpha\)) of accepting a lot if the product quality level is \(v_{1}\) and a probability of no more than \(\beta\) of accepting a lot if the level of the product quality is \(v_{2}\).” Thus, the OC curve of the proposed variables MDS plan will be designed to pass through two designated points, (\(v_{1}\), \(1 - \alpha\)) and (\(v_{2}\), \(\beta\)). For the specified \(\alpha\), \(\beta\), \(v_{1}\) and \(v_{2}\), the proposed MDS sampling plan parameters must satisfy the following two inequalities
$$P_{a} (v_{1} ) = \, \Pr \{ {\text{Accepting}}\,{\text{the}}\,{\text{lot}}\left| {\gamma = v_{1} } \right.\} \ge 1 - \alpha$$
(6)
$$P_{a} (v_{2} ) = \Pr \{ {\text{Accepting}}\,{\text{the}}\,{\text{lot}}\left| {\gamma = v_{2} } \right.\} \le \beta$$
(7)
Since there are several combinations of the parameters for the proposed plans which satisfy the above two inequations, we choose the designed parameters which minimize the sample size. The parameters \(k_{a}\), \(k_{r}\) and n of the proposed plan can be obtained by solving the following optimization problem:
$$\begin{aligned}& \quad Minimize \quad n \\ & {\text{s.t}} \\ & \left\{ {\begin{array}{l} 1 - F_{t} \left( {\frac{\sqrt n }{{k_{a}}}\left| {n - 1,\frac{\sqrt n }{{v_{1} }}} \right.} \right) + \left[{F_{t} \left( {\frac{\sqrt n }{{k_{a} }}\left| {n - 1,\frac{\sqrt n}{{v_{1} }}} \right.} \right) - F_{t} \left({\frac{\sqrt n }{{k_{a}}}\left| {n - 1,\frac{\sqrt n }{{v_{1} }}}\right.} \right)}\right]\left[ {1 - F\left( {\frac{\sqrt n }{{k_{a}}}\left| {n -1,\frac{\sqrt n }{{v_{1} }}} \right.} \right)_{t} }\right]^{m} \ge1 - \alpha \\ 1 - F_{t} \left( {\frac{\sqrt n}{{k_{a} }}\left|{n - 1,\frac{\sqrt n }{{v_{2} }}} \right.} \right) + \left[ {F_{t}\left( {\frac{\sqrt n }{{k_{a} }}\left| {n -1,\frac{\sqrt n}{{v_{2} }}} \right.} \right) - F_{t} \left({\frac{\sqrt n }{{k_{r}}}\left| {n - 1,\frac{\sqrt n }{{v_{2} }}}\right.} \right)}\right]\left[ {1 - F_{t} \left( {\frac{\sqrt n}{{k_{a} }}\left| {n- 1,\frac{\sqrt n }{{v_{2} }}} \right.} \right)}\right]^{m} \le\beta \\ n \ge 2,v_{1} < v_{2} ,0 \le k_{a} \le k_{r} \\ \end{array} } \right.\end{aligned}$$
(8)
In order to investigate the effect of different m values on the required sample size of the proposed MDS sampling plan, we vary m from 1 to 8. Figure 1 shows the required sample size n varies with the m value under \((v_{1} ,v_{2} )\) = (0.05, 0.07), \((\alpha ,\beta )\) = (0.05, 0.10), (0.10, 0.05) and (0.10, 0.10). From Fig. 1, we see that the required sample size n decreases with the increase of \(\beta\) value (or \(\alpha\) value) for fixed the value of \(\alpha\)(or \(\beta\)). That is to say, the larger the risk tolerance, the smaller the sample size required to ensure the same quality level. In addition, the required sample sizes do not change much under the different m values for each set of risk values.
Fig. 1

Required sample size n of MDS sampling plan with m = 1–8

Referring to the values of CV selected by Kang et al. (2007) and Tong and Chen (1991), we consider \(v_{1}\) = 0.05, 0.06, 0.07, 0.08, 0.09, 0.10, \(v_{2}\) = 0.06 ~ 0.12 here. The proposed sampling plan parameters (n, \(k_{a}\), \(k_{r}\)) with schemes m = 1, 2, 3 are respectively displayed in Tables 1, 2 and 3 for \((\alpha,\,\beta )\) = (0.05, 0.10), (0.10, 0.05) and (0.10, 0.10). From the results of Tables 1, 2 and 3, we note that the corresponding sample size n decreases when \(v_{2}\) value increases for fixed values of \(\alpha\), \(\beta\) and \(v_{1}\). On the other hand, for fixed \(\alpha\), \(\beta\) and \(v_{2}\), the corresponding sample size n increases when \(v_{1}\) value increases. For example, when m = 3, \(v_{1}\) = 0.06, (\(\alpha\), \(\beta\)) = (0.05, 0.10), n = 127 as \(v_{2}\) = 0.07, and for all other same values, n = 8 when \(v_{2}\) = 0.12. On the other hand, when m = 3, \(v_{2}\) = 0.08, (\(\alpha\), \(\beta\)) = (0.05, 0.10), n = 16 as \(v_{1}\) = 0.05, and for all other same values, n = 167 when \(v_{1}\) = 0.07.
Table 1

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

Table 2

The proposed plan parameters under (αβ) = (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

Table 3

The proposed plan parameters under (αβ) = (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

In order to show the efficiency of the proposed sampling plan, Fig. 2 displays the OC curves of the MDS plan (m = 1, 2, 3) and the single sampling plan for two cases: (a) (\(v_{1}\), \(v_{2}\)) = (0.06, 0.09), (\(\alpha\), \(\beta\)) = (0.05, 0.10), (b) (\(v_{1}\), \(v_{2}\)) = (0.09, 0.12), (\(\alpha\), \(\beta\)) = (0.10, 0.05). In Fig. 2, we can see that the four curves of the sampling plans are very similar in case (a) or in case (b), but the sample size required by the MDS sampling plan is much fewer. For example, the single plan requires n = 28 while the MDS plan with m = 1 requires n = 19 in case (a). In addition, all of the OC curves show that the probability of acceptance will become smaller as the value of CV increases, which is as expected from the theory. Since the MDS sampling plan requires fewer sample size to give the desired protection, the cost of inspection will greatly be reduced. Therefore, it is reasonable to conclude the MDS plan has a better performance.
Fig. 2

OC curves of MDS plan (m = 1, 2, 3) and single plan for different quality and risk parameters: a (\(v_{1}\), \(v_{2}\)) = (0.06, 0.09), (\(\alpha\), \(\beta\)) = (0.05,0.10). b (\(v_{1}\), \(v_{2}\)) = (0.09, 0.12), (\(\alpha\), \(\beta\)) = (0.10,0.05)

Sample sizes required for inspection

In order to compare the sample sizes required for inspection in the MDS plan (m = 1, 3) and the single plan with different values of \(v_{1}\) and \(v_{2}\), the \(v_{1}\) value is fixed at 0.05 and \(v_{2}\) value increases from 0.06 to 0.12. The results are showed in Fig. 3 (\(\alpha\) = 0.05, \(\beta\) = 0.10) and Fig. 4 (\(\alpha\) = 0.10, \(\beta\) = 0.05). From Figs. 3 and 4, the required sample size n of three sampling plans all decreases as the value of \(v_{2}\) rises from 0.06 to 0.12. Clearly, the required sample size n is larger as the value of \(v_{2}\) is closer to the value of \(v_{1}\). Moreover, we also find that the single sampling plan requires more samples than the MDS plans when \(v_{2}\) takes any value between 0.06 and 0.12. Therefore, the MDS sampling plan is a more cost-effective plan while the single plan is relatively uneconomical.
Fig. 3

Required sample sizes of MDS plan (m = 1, 3) and single plan for \(\alpha\) = 0.05, \(\beta\) = 0.10, \(v_{1}\) = 0.05

Fig. 4

Required sample sizes of MDS plan (m = 1, 3) and Single plan for \(\alpha\) = 0.10, \(\beta\) = 0.05, \(v_{1}\) = 0.05

On the other side, we also list the sample sizes required for the single sampling plan and MDS plan (m = 1, 2, and 3) in Table 4 with commonly used values of \(v_{1}\) and \(v_{2}\) when \((\alpha ,\beta )\) = (0.05, 0.10), (0.10, 0.05) and (0.10, 0.10). From Table 4, it is obvious that the sample size required by the MDS plan is fewer than required by the single sampling plan for all cases. For example, when \(v_{1}\) = 0.08, \(v_{2}\) = 0.09, \((\alpha ,\beta )\) = (0.10, 0.05), the sample size of the MDS plan is 218 for m = 1, 227 for m = 2, and 240 for m = 3, while the single plan is 318. Therefore, the proposed sampling plan will give the desired protection with the less required sample size so that the MDS plan is economically superior to the single plan.
Table 4

The comparison of sample size of two sampling plans with (αβ) = (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.

Based on our proposed methodology, we can obtain the plan parameters as (n, \(k_{a}\), \(k_{r}\)) = (20, 0.09241, 0.122) from Table 2 considering the MDS plan with m = 2. Hence, the 20 inspected samples are taken from the lot randomly and the compressive strength of these 20 concrete mixture specimens is measured and displayed in Table 5. Aslam et al. (2013) have showed that these observed measurements are fairly close to the normal distribution. Based on the collected 20 measurements, we have
$$\overline{X} = 32.19,\quad S = 3.843,\quad {\text{and}}\quad \hat{\gamma } = S/\overline{X} = 0.1194.$$
Since \(k_{a} < \hat{\gamma } = 0.1194 < k_{r}\), the consumer will accept the lot provided that the proceeding m (= 2) lots have been accepted under the condition of \(\hat{\gamma } \le k_{a}\), otherwise, reject the lot. Moreover, we note that if the single sampling plan (Tong and Chen 1991) based on the CV are applied to this case, the sample size required for inspection is 28 under the same conditions.
Table 5

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 non-normal 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

(1)
School of Mathematics and Statistics, Xidian University

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Copyright

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