Economic design of control charts

Economic design of control charts with variable sample size and sampling interval under the assumption of non- normal data

Abstract:

Schewhartcontrol chart is a univariate control method to detect the process mean shift. In this control chart is assumed sample size and sampling interval are constant (e.g. n=4 or 5, h=1) and speed of detecting small changes in the process mean vector is slow. One of method to eliminate this problem is use of adaptive charts. recent studies have shown that the chart with variable sample size and sampling interval (VSSI) are faster than standard schewhart (SS) control chart in detecting process mean shifts. On the other hand, the usual assumption to design a control chart is that the Measurements are normally distributed. However, this assumption may not be correct for some processes. One way for solving this problem is use of the burr distribution that has been used to represent various non-normal distributions (Burr 1942). In this paper we considers an economic design of the VSSI chart under non- normality. Based on the study, VSSI and SS chart are compared with respected to the expected loss per hour. At the end of, an example is given to illustrate the model procedure and Sensitivity analysis is carried out on the input parameter (i.e. the cost and process parameter) of the model.

1. Introduction

Statistical quality control is an efficient method for improving the quality of product and saving the cost of production in an individual company. Therefore when schewhart (1924) represented the first mean control chart, different control chart techniques were developed and were used widely as a primary instrument in statistical process control. The main function of control charts is to detect the occurrence of assignable causes in order that the necessary corrective actions can be taken before the manufacturing of a large number of non-conforming products. When a control chart is applied to check a process, three parameters are selected, the sampling size, the sampling interval and the control limit coefficient. The common methods for designing the control charts are economic or statistical. In statistical design, design parameters are determined based on statistical observations like error type I and error type II. In an economic design, selecting of design parameters is based on loss function. charts with variable sample size, variable sampling interval, variable sample size and sampling interval and also variable parameters have shown that are basically faster than Schewhart charts in detecting process changes (Such as Reynolds et al. 1998, Runger and Pingnatiello 1991, Prabhu et al. 1993, Reynolds 1996a, b, Costa 1994, 1997, 1998, 1999a, b, Zimmer et al. 1998, 2000, Bai and Lee 2002).

These researches are based on the assumption that data have normal distribution. However, this assumption may not be correct in some manufacturing processes. If measurements are distributed normally, chart will also have normal distribution. If measurements have non normal distribution, charts will have normal distribution only then the sample size is large enough (based on central limit theory), but when control charts are applied for process observations, the sample size can not be large enough the most of time. Therefore if process data do not have normal distribution, old methods will not be appropriate for designing control charts. Burr (1942) introduced the Burr distribution has been used to present various non-normal distributions. Burr, Zimmer (1963) offered the statistical design of VSI chart under non-normality. Also, Yourstone and Zimmer (1992) used Burr distribution for representing different non normal distributions. Recently Lin and Chou (2005) have represented design of VSSIstatistical chart and Chen and Cheng (2007) have represented design of VP statistical chart both with assumption of non normal data. On the other side, Montgomery (1980) and Woodall (1986) attended highly to economic design of control charts, which two mentioned models were applied expendably in practice. One of these models was represented by Duncan (1956) for control chart, which minimizes the expected cost per each time unit when a single assignable cause exists. It is also assumed that in this method the process go on its operation in a search time for assignable cause. Chiu (1975) proposed another popular model in which the process is stopped during the search for assignable cause. Costa (2001) considered the economic design of VPchart by using Markov chain. Also, Chen e.g. (2007) presented the economic design of VSSI chart by using Markov chain. Bai and Lee (1998) used Chiu (1975) model and Chen (2004) also applied the developed model of Bai and Lee (1998) for economic design of VSI chart under the assumption of non normal data (Burr distribution). This article is a development of papers by Costa (2001), Chen (2007), and Chen (2004) in which economic design of VSSIchart is applied under the assumption of non- normal data. Markov chain method allowed us to access the properties of chart very faster than methods used by Park and Reynolds (1999). In section 2 and 3 VSSIchart and Burr distribution have been explained briefly. The assumptions and equations of the economic model have been represented in section 4, a solution method based on genetic algorithm have been explained in section 5, an example for sensitivity analysis of solution procedure for VSSIchart have been represented in section 6 and finally the results have been mentioned in the last section.

2. The review of VSSI chart

2.1 The VSSI charts with symmetric limits

The idea of variable sample size and sampling interval for improving the function of control chart represented by Montgomery, Prabhu and Runger (1994, 1997), Costa (1997) and Das and Jain (1997) for monitoring the process mean in Schewhart chart. For simplicity in applying and performing the proposed procedure, the VSSI chart chooses two variable sample sizes and sampling intervals, that

Where n0 is the average sample size and usually set as 4 or 5, h0 is the fixed sampling interval and usually set as 1.

In designing of VSSI chart we have to determine the coefficient of action limit, k, and the coefficient of warning limit, w. In statistical statement of these control limits we have:

Where UCLi and LCLi are defined as the upper and lower action limits and UWLi and LWLi are defined as the upper and lower warning limits. The VSSI chart applies the action limit and warning limit for dividing the chart to three regions, central region, warning region and action region.

If the sample point is located in the central region, we will use the (n1, h1) pair in the next step.

If the sample point is located in the warning region, we will use the (n2, h2) pair in the next step.

And if the sample point is located in the action region, the process will be out of control.

2.2 The VSSI charts with asymmetric limits

Yourstone and Zimmer (1992) offered a procedure for designing Schewhart Standard charts with asymmetric limits, by the time distribution of sample means is not normal. This article will develop this concept for VSSI chart. To state these control limits we have:

Where 0<wi1<ki1 and 0<wi2<ki2.

The traditional VSSI symmetric chart is acquired simply with the assumption of ki2=ki1 , i=1, 2.

3. Burr distribution

Burr (1942) represented a simple distribution function which was capable to estimate various types of distributions. The mathematical statement of probability function of density for Burr distribution is as follows:

And for cumulative distribution of this function we have:

orWhere c and r are greater than one. The first four moments of the empirical distribution can be used to determine the values of c and r. taking into consideration the different combinations of c and k, this distribution can cover an extensive range of the skewness and kurtosis coefficients of variform probability density functions(e.g., Normal, Gamma, Beta and ….). For instance for c=4.8621 and k=6.3412 the Burr distribution approximates the normal distribution Yourstone and Zimmer (1992). Also Zimmer and Burr (1963) used Burr distribution for development of variable sampling method in non normal population. After that Chou et al. (2000, 2001) employed Burr distribution for economic design of charts under the assumption of non normality. Burr (1942) tabulated the first two moments (Table 2) and the coefficients of skewness and kurtosis (Table 3) for the family of Burr distribution. These tables allow users to establish a standardized transformation between the variable of Burr distribution (y) and any random variable (x) when they have the similar coefficients of skewness and kurtosis.

The standardized transformation between variable of Burr distribution, y, and variable x is declared as follows:

Where S and M are the sample mean and standard deviation of Burr distribution respectively and and sx are sample mean and standard deviation of data set (random variable).

4. Development of cost model

4.1. Model assumptions

1. Sampling interval, h, and sample size, n, change between maximum value and minimum value. Also

2. In the beginning, the process is assumed in control with the average of, but by passing the time, the process's mean changes from (on target) to (off target) at some random times.

3. It is assumed that the time length before the first assignable cause has the exponential distribution with λ parameter.

4. After the shift, the process's mean will remain off target until the assignable cause is eliminated.

5. During the search of an assignable cause, the process remains shut down.

4.2. Markov chain method

The statistical efficiency criterion of a control chart is a speed which detects mean shifts. When the interval between samples is constant, the speed can be measured by the average of rear length (ARL), but when the interval is variable, it has to be measured by adjusted average time to signal (AATS). AATS is the time mean from the process mean shift to the time which chart signals. The average time cycle (ATC) is the time mean from the beginning of production to the first signal after the process shift. Therefore we have:

Because of memoryless property of exponential distribution we can compute ATC by the help of markov chain.

At each sampling, one of the four following transition states can happen:

1. The process is in control and the sample is small.

2. The process is in control and the sample is large.

3. The process is out of control and the sample is small.

4. The process is out of control and the sample is large.

Table (1) - the states of markov chain

ith sample

(i+1)st sample

sample point

situation

process mean status¬

state of the

markov chain

Warning

on target

2

Warning

off target

4

Central

on target

1

Central

off target

3

¬On target means (in control)

¬Off target means (out of control)

When the sample point is located in action region, the control chart generates signal.

If the new state is 1 or 2, the signal will be a false alarm.

If the new state is 3 or 4, the signal will be a correct alarm.

And the absorbing state, the fifth state, will be obtained when the correct alarm is occurred.

The transition probability matrix is defined as follows:

Where Pij denotes the transition probability that i is the prior state and j is the new state.

According to the initial properties of markov chain (Cinlar 1975):

Where is the vector of initial probability, I is the identity matrix of order 4, Q is the transition matrix which its elements related to absorbing state have been eliminated and is the vector of sampling intervals.

* Note that Calculations of transition probabilities with the assumption of asymmetrical control limits is brought in Appendix.

4.3. The loss function

The process cycle involve the following phases:

In control, out of control, detecting the assignable cause and repair. As the result of, average length of a production cycle is as follows:

Where is the average amount of consumed time for searching the assignable cause when the process is in-control and is the average time of detecting and correcting the assignable cause.

Also E (FA) is the average number of false warnings at each production cycle.

And for the average net profit of production cycle we have:

V0= the profit earned at each hour when the process is in control

V1= the profit earned at each hour when the process is out of control

C0= the average cost based on the false warning

C1= the average cost of detecting and removing the assignable cause

s= the cost for each inspected item

E (M) = the average number of inspected items at each cycle

Where. Hence the loss function E (L) is as follows:

5. An example and solution procedure

The loss function denoted in equation (15) is a function of process parameters, cost parameters and design parameters .

The economic design of control chart is applied for a special use to reach design parameters which minimum E (L).

In this study, we use the genetic algorithm, because the minimization problem has a nonlinear objective function with the following constraints:

If the methods of non-linear programming are employed for this kind of optimization problem, they may be time-consuming and inefficient.

To illustrate the solutions acquired from economic design of VSSI control charts, a numerical example is afforded in Section 5.1 and the genetic algorithm (GA) is used to search for the optimal solution of the economic design. As mentioned, When the GA is used, four control parameters have to be first specified (Chen e.g. 2006): the crossover rate (CR), the Mutation rate (MR), the Generation number (GN), and the population size (PS), that usually the quality of the solution acquired from GAs depends on these four control parameters. Determination of these control parameters is through orthogonal-array experiments, which is described in Section 5.2.

5.1. An example

For solving the problem, we assume that s=5, C0=500, C1=500, V0=500, V1=50, t0=5, t1=1,. The solution method is achieved using the GA to obtain the optimal values that minimize E (L).

Genetic algorithms are optimization techniques and global search which have been derived from the processes of natural selections in biological system (Goldberg 1989, Davis 1991). The genetic algorithms have the following differences in compare of other search algorithms:

1. Consider many points, rather than a single point, in the search space concurrently.

2. Work directly with strings of characters representing the parameter set, not the parameters themselves.

3. Apply the probabilistic rules instead of deterministic rules to guide their search.

Because the genetic algorithm considers many points in the search space concurrently, there is a diminished chance for reaching to local optimal. The initial remarkable characteristics of genetic algorithms are encoding, the fitness function, the selection mechanism, the crossover mechanism, the mutation mechanism, and the culling mechanism.

During the use of genetic algorithm to minimize the problem, decimal encoding of solutions is selected so that any solution in the form of decimal string shows a possible solution for. The fitness value of each solution is evaluated and measured by E (L). in consideration of genetic algorithm strategy, the survival of the fittest, the evolution of the population of N solutions is followed. The stopped condition is acquired when the number of repetitions is large enough or the requested fitness value is obtained.

After the several times repetition above steps, we obtain the following optimal solution:

=4, =12, =3.5968, =0.0238, =1.0125, =4.9802, =1.4535, =3.5160, =3.1938, =4.3277, =2.8729, =3.2212, =31.6553.

5.2. Determining the values of the control parameters in the GA

The quality of solution acquired from GA Algorithm depends on four parameters: population size, crossover probability, number of generation and mutation rate. In order to achieve these optimal parameters, we use orthogonal array experiment (as used by Chou et al 2006). In the orthogonal array experiment, three levels of each parameter were defined as shown in Table 2.

Table 2 -Parameters and levels in the GA

Parameters

Level 1

Level 2

Level 3

PS

50

75

100

CP

0.5

0.6

0.7

MR

0.1

0.15

0.25

GN

20,000

40,000

60,000

The L9 Orthogonal array was used to assign the four Control parameters. In the experiment of L9 Orthogonal array, there are entirely nine array (or 9 Different level combinations of the four parameters). For each array, three cost values (E (L)) are acquired from GA (denoted by y1, y2, y3) that the responses are brought in Table 3. Considering that expected total cost is a characteristic of smaller-the-better characteristic, the suitable signal-to-noise ratio (SN) for assessment and measurement the experiment results (Taguchi, 1987) is

Where m is the total number of E (L) values in each assay.

Table 3- Experimental layout of L9 orthogonal array and results

Assay

PS

CP

MR

GN

SN

1

1

1

1

1

31.7875

32.7846

32.0538

-30.160190

2

1

2

2

2

33.2731

32.7040

32.0091

-30.281961

3

1

3

3

3

32.6601

32.5078

32.7626

-30.27598

4

2

1

2

3

31.4807

30.9842

31.8487

-29.949613

5

2

2

3

1

35.9639

36.1853

34.5044

-31.018996

6

2

3

1

2

32.3651

32.7712

32.6166

-30.260283

7

3

1

3

2

32.1808

33.4928

32.8641

-30.330779

8

3

2

1

3

30.9674

31.8311

31.5630

-29.954044

9

3

3

2

1

32.9486

33.1535

33.5145

-30.424425

Table 4 -Response table of S/N's for the three control parameters in the GA

Levels

PS

CP

MR

GN

Level 1

-90.718131

-90.440582

-90.374517

-91.603611

Level 2

-91.228892

-91.255001

-90.656000

-90.873023

Level 3

-90.709248

-90.960688

-91.625755

-90.179637

After the calculation of S/N's through EQ. (16), then from the S/N's in Table 3, a response table of S/N at each level for each control parameter may be acquired and the results are represented in Table 4. Taking into Consideration present information in Table 4, the optimal level combination of the four control parameter in the GA is: PS= 100, CP=0.5, MR=0.1, GN= 60000.

6. Numerical comparisons

In this section, the control charts with the SS scheme and the VSSI scheme are compared with respect to the Loss function. In the SS scheme the Loss can be easily accrued by setting h1 = h2, n1=n2 and w = 0, which implies p11=p21=0 and p13=p23=p33=p43=0.

In selecting the process and cost parameters, we use the 13 parameter sets, which its cost parameters and process parameters have been chosen from Chen (2007) (Table 5).

In doing so, we use the genetic algorithms (GAs) because the minimization problem has a non-linear objective function.

Table 6 demonstrates the minimum costs and the optimal design parameters for the SS and VSSI charts. From the result of Table 6, we have the following findings:

Table 5. The cost parameters and process parameters

NO.

s

1

5

500

500

500

50

5

1

0.01

1

2

10

500

500

500

50

5

1

0.01

1

3

5

250

500

500

50

5

1

0.01

1

4

5

500

50

500

50

5

1

0.01

1

5

5

500

500

250

50

5

1

0.01

1

6

5

500

500

500

100

5

1

0.01

1

7

5

500

500

500

0

5

1

0.01

1

8

5

500

500

500

50

2.5

1

0.01

1

9

5

500

500

500

50

5

10

0.01

1

10

5

500

500

500

50

5

1

0.05

1

11

5

500

500

500

50

5

1

0.01

1.5

12

5

500

500

500

50

5

1

0.01

0.5

13

5

500

500

500

50

5

1

0.01

2

(1) In all of trials, the Loss values of the VSSI control schemes are consistently smaller than that of the SS control scheme.

(2) In Comparison with the SS schemes, the corresponding VSSI scheme need more often sampling with a wider upper control limit and a smaller sample size.

Table 6. The optimum design of an x- bar chart with fixed and variable sample size and sampling interval under the Burr distributions (c=2, k=4)

NO.

FSSI

VSSI

n

h

AATS

AATS

%

1

16

6.1888

2.967

2.967

4.6295

4.6295

42.1012

0.1111

3.9199

4

12

3.6

0.02

1.01

4.98

1.45

3.52

3.19

4.33

2.873

3.221

31.6553

0.0356

2.3724

24.8114

2

13

8.0393

2.645

2.645

4.5747

4.5747

52.1693

0.1738

5.2557

4

9

6.51

0.05

0.94

4.96

1.07

3.07

4.16

4.8

3.658

4.723

40.3476

0.0433

4.2082

22.6603

3

16

6.1929

2.941

2.941

4.6001

4.6001

41.8303

0.1175

3.8817

5

10

3.95

0.03

1.29

4.88

1.08

3.38

4.86

4.99

2.849

2.872

31.7726

0.0338

2.6657

24.0441

4

17

6.3667

3.001

3.001

3.8144

3.8144

37.884

0.0988

3.8999

5

14

3.87

0.04

1.19

4.92

1.63

3.48

4.68

4.77

3.389

4.487

27.7729

0.0275

2.4904

26.6896

5

14

8.9481

2.728

2.728

3.6353

3.6353

27.4636

0.1292

5.7284

4

11

6.01

0.02

1.06

4.86

1.31

3.16

4.48

4.98

3.718

3.771

21.6718

0.0356

4.024

21.0891

6

17

6.8188

2.999

2.999

3.5915

3.5915

40.1986

0.0925

4.1762

4

9

5.22

0.03

0.83

4.99

0.96

3.41

3.36

3.63

4.645

4.678

30.6543

0.0363

3.1952

23.7429

7

17

6.0556

2.999

2.999

4.2102

4.2102

43.9212

0.1045

3.7052

4

9

3.51

0.03

1.1

4.99

0.96

3.43

4.43

4.89

3.841

4.855

32.6702

0.0437

2.445

25.6163

8

15

6.0272

2.78

2.78

4.181

4.181

40.5975

0.1723

3.7262

5

11

4.54

0.02

1.03

4.99

1.24

3.27

4.59

4.89

3.528

4.965

31.4029

0.0359

2.7082

22.6482

9

17

6.7092

2.98

2.98

4.9082

4.9082

78.1082

0.098

4.0808

5

13

4.54

0.02

1.15

4.99

1.48

3.41

2.44

2.5

3.303

4.215

69.291

0.0256

2.8351

11.2884

10

16

3.0367

2.906

2.906

3.8071

3.8071

111.4724

0.0494

1.9006

5

12

1.79

0.02

1.27

4.99

1.45

3.18

3.31

4.27

3.593

4.018

94.3834

0.0163

1.2057

15.3303

11

10

4.9195

3.452

3.452

4.4809

4.4809

33.556

0.0586

2.829

4

8

3.27

0.02

1.7

5

1.93

3.83

4.78

4.95

4.541

4.557

26.2948

0.0248

1.847

21.6391

12

41

10.165

2.312

2.312

3.5615

3.5615

63.8281

0.2376

6.8734

10

27

5.32

0.01

0.93

4.99

0.93

2.79

3.48

4.54

3.446

4.562

49.9909

0.0624

4.527

21.6789

13

7

401273

3.847

3.847

3.1048

3.1048

28.9897

0.0381

2.2611

3

9

3.2

0.02

2.01

4.99

3.44

4.39

3.38

4.31

3.805

3.872

24.0503

0.0302

1.7173

17.0385

(3) All the cases from the above tables declare that the optimal value of h2 is close to zero, which means if put in the warning area the process should be sampling rapidly.

(4) Smaller AATS indicates the VSSI control schemes suggest a quicker speed for detecting a mean shift, whereas Lower E (FA) indicates that the VSSI control schemes suggest better protection against false alarm than the Corresponding SS scheme.

(5) By the time,, t1,C1 are small, V0 is large in comparison with V1, or s is small, Percent reduction is large.

Conclusions:

In this paper, we develop the economic design of the control chart with variable sample size and sampling interval under the assumption of non- normal data, which has been shown to give substantial improvement of the traditional control chart on the speed of detecting small changes in the process mean vector.

An economic design of VSSI control charts has been presented whereas the distribution of sample means can not be assumed to be a normal distribution. Taking into consideration this subject, we use the Burr distribution to approximate the distribution of sample means. In designing control chart for non-normal process data, control charts with asymmetric control limits are taken into account, and compared with traditional asymmetric control charts from the economic point of view. A cost model was derived by the Markov Chain method (Costa's model 2001), and genetic algorithms were used to determine the optimal design parameters.

An example is illustrated to present the solution procedure. According to the result of this example, some important conclusions may be spelled out as follows:

1- VSSI control charts consistently lower loss than the SS control charts over the 13 sets.

2- When the profit obtained per hour during in-control period is higher or the time the

Process remains in-control is shorter; the minimum cost will be occurred.

3- A longer time spent on detection and elimination of an assignable cause leads to a

Higher expected hourly loss.

4- All of cases, the AATS and E (FA) of VSSI scheme are lower than SS scheme.

Appendix A. Calculations of transition probabilities with the assumption of asymmetrical control limits

Also, according to Dodge and Rousson (1999) the coefficients of skewness and kurtosis for sample mean are respectively:

Where α3 and α4 are coefficients of skewness and kurtosis estimated by population. Using the value of and the tables available in Burr (1942), it's possible to calculate k, c, S and M for the distribution with the value near to by interpolation.

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