A Revised Double Sampling Control Chart

A REVISED DOUBLE SAMPLING CONTROL CHART Dradjad Irianto Department of Industrial Engineering Bandung Institute of Technology, Indonesia [email protected]

Abstract: Standard Shewhart control chart has been widely used, but it is slow in detecting small shift. A number of alternatives have been proposed to improve the sensitivity of control chart, for example by employing warning limits. The double sampling (DS) control chart is aimed at improving the ability to detect any out-of control condition by observing the second sample without any interruption. The DS control chart was firstly proposed by Croasdale (1974). Daudin (1992) proposed DS control chart that utilizes the information from both samples at the second stage. Irianto and Shinozaki (1998) analyzed both DS procedures, and found that Daudin’s procedure is better than Croasdale’s. Instead of minimizing the expected sample size, Irianto and Shinozaki (1998) maximized the power to detect a small shift of mean value. This optimization produced an out-of-control limit at the first sample as high as 3.6 standard deviation. In many applications in manufacturing companies, this high out-of-control limit is meaningless since it has only 0.3 per thousand opportunities of out-of-control condition. This result leads to a revised DS control chart, by neglecting the decision for out-of-control condition at the first sample. This implies to a reduction of limits parameter from three into two, which thus it simplifies the optimization procedure. Keywords: Double Sampling Control Chart, Optimization, Power of Control Chart, Sample Size.

1. Introduction Statistical process control is a well-known method to monitor process variability in order to control and improve quality of the process. Among the statistical process control tools, control chart is aimed at monitoring the process. A control chart is designed to identify variation in process, either as a result of unassignable causes, or as a result of assignable (or special) causes. In this respect, the standard Shewhart X control chart has been widely used, especially for detecting large shift. A number of alternatives have been proposed to improve the sensitivity of control chart for small shift, e.g. by employing warning limits. Reynolds et al. (1988) proposed a variable sampling interval (VSI) control chart in accordance to an out-of-control warning or signal. If an out-of-control warning or signal occurs, next sampling is taken in a shorter sampling interval; otherwise it is reasonable to take a longer sampling interval. Instead of sampling interval, Costa (1992) proposed a variable sample size (VSS) control chart in dealing with an out-of-control warning or signal. VSS has the same idea with VSI. The double sampling procedure (DS) uses the both ideas of VSI and

VSS. In case an out-of-control warning or signal occurs, in addition to the first sample, the second sample is observed with zero (the shortest) time intervals. The double sampling (DS) control chart was firstly proposed by Croasdale (1974). In this first DS control chart, information from the first and second samples is evaluated unconnectedly. Daudin, Duby and Trecourt (1990) and Daudin (1992) proposed DS control chart that utilizes the information from both samples at the second stage. Irianto and Shinozaki (1998) discussed both DS procedures and proposed the advantage of Daudin's DS control chart compared to Croasdale’s DS control chart. Daudin's DS control chart uses optimization of the expected sample size for obtaining the control chart parameters. Instead of minimizing the expected sample size, Irianto and Shinozaki (1998) maximized the power to detect a small shift of mean value. He at al. (2002), and Costa and Claro (2007) have made further development of double sampling control chart. 2. The DS Control Chart Procedures Croasdale’s DS control chart procedure is described as follows (its scheme is exhibited in Figure 1): 1. Take the first sample of size n1 , X 1i , i = 1,2,L.n1 and the second sample of size n 2 , X 2i , i = 1,2,L.n2 from a population with mean value µ 0 and a known standard deviation σ. n1

2. Calculate the sample mean X 1 = ∑ X 1i / n1 . i =1

3. If

X 1 − µ0

σ / n1

is in [− M 1 , M 1 ] , the process is considered to be under control, otherwise n2

observe the second sample and calculate the sample mean X 2 = ∑ X 2i / n 2 . i =1

X 1 − µ0

< − M 1 and

X 2 − µ0

< − M 2 , or if

X 1 − µ0

> M 1 and

X 2 − µ0

> M 2 , then σ / n1 σ / n2 σ / n1 σ / n2 the process is considered to be out-of-control, otherwise the process is considered to be under control.

4. If

For a shift from the mean value δ = ( µ 0 − µ ) / σ , the probability that the process is monitored as under control is given as follows: P = Φ[ M 1 + δ n1 ] − Φ[− M 1 + δ n1 ] + {1 − Φ[ M 1 + δ n1 ]} Φ[ M 2 + δ n 2 ] + Φ[− M 1 + δ n1 ] ⋅ {1 − Φ[− M 2 + δ n 2 ]} (1) where, Φ (⋅) is the cumulative distribution function of standard normal distribution respectively. We assume the characteristic of output of process follows a normal distribution N ( µ , σ 2 ) . The average run length is ARL = 1 /(1 − P) , and the expected total sample size is n1 + n 2 ⋅ (1 − P1 ) , where P1 = 1 − Φ[ M 1 + δ n1 ] + Φ[− M 1 + δ n1 ] .

2

X 1 − µ0 σ / n1

X

2

σ /

− µ0 n2

Out of control

Take second sample

M2

M1

Under control

Under control

0 - M1

-M2

Take second sample

Out of control

(Second sample)

(First sample)

Figure 1. Croasdale’s DS control chart procedure. With Daudin’s DS control chart, the second sample will be observed only if the first sample is signaling a warning of deviation of the mean value. The procedure is described as follows (and its scheme is exhibited in Figure 2): 1.Take the first sample of size n1 , X 1i , i = 1,2, L.n1 and the second sample of size n 2 , X 2i , i = 1,2,L.n2 from a population with mean value µ 0 and a known standard deviation σ . n1

2. Calculate the sample mean X 1 = ∑ X 1i / n1 . i =1

3. If ( X 1 − µ 0 ) /(σ / n1 ) is in I 1 , the process is considered to be under control. 4. If ( X 1 − µ 0 ) /(σ / n1 ) is in I 3 , the process is considered to be out-of-control. 5. If ( X 1 − µ 0 ) /(σ / n1 ) is in I 2 , observe the second sample. n2

6. Calculate the second sample mean X 2 = ∑ X 2i / n 2 . i =1

7. Calculate the total sample mean X = ( n1 X 1 + n 2 X 2 ) /( n1 + n 2 ) . 8. If

−L< X − µ0

X 1 − µ0

σ / n1

< − L1

or

L1 <

X 1 − µ0

σ / n1

< L,

and

if

X − µ0

σ / n1 + n 2

< − L2

or

> L2 , then the process is considered to be out-of-control, otherwise the σ / n1 + n 2 process is considered under control.

Let Z1 = ( X 1 − µ0 ) /(σ / n1 ) and Z = ( X − µ 0 ) /(σ / n1 + n2 ) , then the probabilities that the process is considered to be under control by the first sample and after observing the second sample can be formulated as Pa1 = Pr[ Z 1 ∈ I1 ] and Pa 2 = Pr[ Z1 ∈ I 2 and Z ∈ I 4 ] respectively, and the probability that process under control is Pa = Pa1 + Pa 2 . X − µ0

X1 − µ0 σ / n1

σ /

n1 + n 2

Out of control ( I 3 )

L

Out of control Take second sample ( I 2 )

L1

L2

Under control ( I 4 )

Under control ( I 1 )

0 - L1

Take second sample ( I 2 )

-L Out of control ( I 3 )

(First sample)

- L2 Out of control

(Second sample)

Figure 2. The Daudin’s DS control chart procedure. For a shift from the mean value δ = ( µ0 − µ ) / σ , the probability that the process is considered to be under control becomes: Pa = Φ[ L1 + δ n1 ] − Φ[− L1 + δ n1 ] +

∫ {Φ[cL

2

+ rcδ − z n1 n2 ] − Φ[−cL2 + rcδ − z n1 n2 ]}φ ( z )dz

(2)

z∈I 2*

where, φ (⋅) and Φ (⋅) are the density and cumulative distribution functions of standard normal distribution respectively, r = n1 + n 2 , c = r / n2 and I2* = [− L + δ n1 ,− L1 + δ n1 ) U ( L1 + δ n1 , L + δ n1 ] . The average run length is ARL = 1 / (1 − Pa ) , and the expected total sample size is n1 + n2 ⋅ Pr[Z1 ∈ I2 ] . Irianto and Shinozaki (1998) discussed both DS control chart procedures and proposed the advantage of Daudin's DS control chart compared to Croasdale’s DS control chart. 3. Estimating Control Chart Parameters

There are five parameters required to specify the Daudin’s DS control charts, i.e. L1 , L2 , L , n1 and n2 . Daudin et al. (1990) suggested an optimization procedure as follows :

Min

n 1 , n 2 , L, L1 , L 2

n1 + n2 ⋅ Pr[ X 1 ∈ I 2 | µ = µ 0 ]

(3)

Subject to: 4

(i) Pr[Out of Control |

µ = µ0 ] = α , that is

1 − {Φ[ L1 ] − Φ[− L1 ]} −

∫ {Φ[cL

2

− z n1 n2 ] − Φ[−cL2 − z n1 n2 ]}φ ( z )dz = α .

z∈I 2

(ii) Pr[Out of control | µ =

µ1 ] = β (for a given intended shift δ = µ1 − µ0 ), that is

1 − {Φ[ L1 + δ n1 ] − Φ[− L1 + δ n1 ]} −

∫ {Φ[cL

2

+ rcδ − z n1 n2 ] − Φ[−cL2 + rcδ − z n1 n2 ]}φ ( z )dz = β .

z∈I 2*

To find the solution, Daudin et al. (1990) proposed an algorithm as follows : (i) Determine n1 and n2 . (ii) For a given value of L , both constraints are used to determine the values of L1 and L2 . (iii) Find the optimal composition of L1 , L2 and L that minimize the objective function for all possible pairs of ( n1 , n2 ). This optimization procedure for minimizing sample size is mainly motivated by minimizing the inspection cost. Differently, Irianto and Shinozaki (1998) considered the power o capability of control chart in detecting deviation of the process' mean. Therefore the motivation is to minimize risk of not knowing that the process mean has deviated while setting sample sizes n1 and n2 so that the expected total sample size is fixed. The optimization is formulated as follows: Max 1 − {Φ[ L1 + δ n1 ] − Φ[− L1 + δ n1 ]} L,L1 ,L 2

−

∫ {Φ[cL

2

+ rcδ − z n1 n2 ] − Φ[−cL2 + rcδ − z n1 n2 ]}φ ( z )dz .

(4)

z∈I 2*

Subject to: (i) E[total sample size | µ = µ 0 ] = n, that is n1 + n2 ⋅ Pr[ Z 1 ∈ I 2 | µ = µ 0 ] = n ⇔ L = Φ −1

[

n −n1 2 n2

]

+ Φ[ L1 ] .

(ii) Pr[Out of Control | µ = µ0 ] = α , that is 1 − {Φ[ L1 ] − Φ[− L1 ]} −

∫ {Φ[cL

2

− n1 n2 z ] − Φ[−cL2 − n1 n2 z ]}φ ( z )dz = α

z∈I 2*

From the first constraint, L can be expressed in terms of L1 , which then it reduces the number of parameter. Since the left hand side of the second constraint is an increasing function of L2 , then L2 can be uniquely determined for fixed L1 and L . Since and then Φ( L1 ) = Φ ( L) − (n − n1 ) /(2n2 ) (1 − α / 2) ≤ Φ( L) ≤ 1 , n−n1 n − n −1 − 1 Φ 1 − 2 n2 − α2 ≤ L1 ≤ Φ 1 − 2 n21 . This range of L1 is quite small if α is small.

[

]

[

]

4. Numerical Results

Usually, standard Shewhart chart is used as the basis for comparison. The power Pa =Pr[Out of Control | µ = µ1 ] and the average run length (ARL) of the standard Shewhart X control chart (for n=5 and L=3) is shown in Table 1. The power Pa =Pr[Out of Control | µ = µ 1 ] of DS control chart (Irianto and Shinozaki, 1998) for some shift are presented in Table 2, where α is set at 0.0027 and sample sizes n1 = n2 =4 and n =5.

Table 1. ARL and Pa of Shewhart X Chart (for n =5 and L=3)

Pa

ARL

0.0027 0.0064 0.0228

370.4 155.2 43.9

Shift δ = µ1 − µ0 0 0.5 1

Table 2. Pa of Daudin’s DS Chart (for n1 = n2 =4 and n =5) Limits Power at δ = µ1 − µ0

L1

L

L2

0.0

0.2

0.4

0.5

0.6

0.8

1

1.15

3.8014

2.9924

0.0027

0.0076

0.0303

0.055

0.094

0.2253

0.4229

Clearly, the numerical result of the DS control chart gives better performance shown by higher power. Accordingly, the out of control signal occurs in a shorter interval than the standard Shewhart X control chart, thus further adjustment or improvement action can be performed sooner. However, it should be noted that the average sample size increases as the shift of process mean gets larger. 5. Revised DS Control Chart

Table 3 shows some control limits of DS control charts for some pairs of n1 and n2 but still give an expected sampling number n=5. Table 3. Pa DS control chart for some pairs of sample sizes. Power L1 L2 Sample size L δ = 0.5 δ = 1.0 0.673 3.3057 3.0720 0.0357 0.2766 n1 =4 0.674 3.6057 3.0149 0.0375 0.2882 n2 =2 0.6744 2.9999 0.0379 0.2910 ∞ n =5 n1 =4 0.966 3.3854 3.0557 0.0440 0.3459 0.967 3.7058 3.0087 0.0461 0.3577 n2 =3 0.9674 2.9961 0.0467 0.3606 ∞ n =5 1.280 3.4575 3.0135 0.0611 0.4662 n1 =4 1.281 3.7271 2.9754 0.0637 0.4762 n2 =5 1.2815 2.9593 0.0647 0.4801 ∞ n =5 1.381 3.4261 2.9966 0.0683 0.5069 n1 =4 1.382 3.6110 2.9590 0.0711 0.5158 n2 =6 1.3829 2.9292 0.0733 0.5225 ∞ n =5 This result shows that maximizing power leads to higher value of L1 with lower value of L2 . Based on the first constraint, the higher value of L1 is related to higher value of L, which is limited to L = ∞ . In this case, L is too big and thus it is no longer necessary. As a result, the DS control chart can be simplified eliminating L, and its scheme is shown in Figure 3. 6

X − µ0

X1 − µ0 σ / n1

L1 0 - L1

σ /

Take second sample ( I 2 )

n1 + n 2

Out of control

L2 Under control ( I 1 )

Under control ( I 4 )

- L2 Take second sample ( I 2 )

Out of control

(Second sample)

(First sample)

Figure 3. Revised DS control chart. Procedure for the revised DS control chart is as follows: 1. Take the first sample of size n1 , X 1i , i = 1,2, L.n1 and the second sample of size n 2 , X 2i , i = 1,2,L.n2 from a population with mean value µ 0 and a known standard deviation σ. n1

2. Calculate the sample mean X 1 = ∑ X 1i / n1 . i =1

3. If ( X 1 − µ 0 ) /(σ / n1 ) is in I 1 , the process is considered to be under control. 4. If ( X 1 − µ 0 ) /(σ / n1 ) is in I 2 , observe the second sample. n2

5. Calculate the second sample mean X 2 = ∑ X 2i / n 2 . i =1

6. Calculate the total sample mean X = (n1 X 1 + n 2 X 2 ) /(n1 + n 2 ) . −

−

−

−

X 1 − µ0 X 1 − µ0 X 1 − µ0 X 1 − µ0 7. If < − L1 or L1 < , and if < − L2 or > L2 , then σ / n1 σ / n1 σ / n1 + n2 σ / n1 + n2

the process is considered to be out-of-control, otherwise the process is considered under control. Accordingly the optimization of power can be formulated as follows: Max 1 − Φ[[ L1 + δ n1 ] + Φ[− L1 + δ n1 ] L1 , L 2

− {1 − Φ[ L1 + δ n1 ] + Φ[− L1 + δ n1 ]}.{Φ[ L2 + δ n1 + n2 ] − Φ[− L2 + δ n1 + n2 ]} (5) Subject to: (i) E[total sample size | µ = µ0 ] = n, and then n1 + n2 ⋅ Pr[ Z 1 ∈ I 2 | µ = µ 0 ] = n ⇔ n1 + n2 .{1 − Φ[ L1 ] + Φ[− L1 ]} = n . (ii) Pr[Out of Control | µ = µ 0 ] = α , and then

{1 − Φ[ L1 ] + Φ[− L1 ]}.{1 − Φ[ L2 ] + Φ[− L2 ]} = α . It is clear that by defining the sample size n1 , n2 and n, control limit L1 can be found from the first constraint and thus L2 can be found from the second constraint. Table 4 shows some control limits of DS control charts for some pairs of n1 and n2 but still give an expected sampling number n=5. Based on numerical example in Table 3 and Table 4, the revised DS control chart provides higher value of power for deviation δ = µ1 − µ0 of half and one process standard deviation. This result means that the revised DS control chart has higher capability in detecting shift in case there is a deviation δ = µ1 − µ0 from the designed process mean. Table 4. Pa of revised DS control chart for some pairs of sample sizes. Sample size n1 =4 n2 =2 n =5 n1 =4 n2 =3 n =5

n1 =4 n2 =5 n =5

n1 =4 n2 =6 n =5

L1

L2

0.672 0.673 0.674 0.6744 0.965 0.966 0.967 0.9674 1.279 1.280 1.281 1.2815 1.380 1.381 1.382 1.3829

2.7832 2.7827 2.7823 2.7821 2.6491 2.6486 2.6481 2.6478 2.4718 2.4712 2.4706 2.4703 2.4063 2.4057 2.4050 2.4043

Power δ = µ1 − µ0

δ = 0.5 0.040270 0.040280 0.040281 0.040286 0.049780 0.049783 0.049785 0.049775 0.066476 0.066471 0.066461 0.066460 0.073804 0.073788 0.073777 0.073766

δ = 1.0 0.336669 0.336770 0.336828 0.336879 0.424442 0.424494 0.424545 0.424579 0.536531 0.536484 0.536421 0.536399 0.567977 0.567860 0.567753 0.567647

6. Discussion

In the economic design of control charts, there are three categories of costs (Montgomery, 2002), i.e. costs for sampling and sample inspection, costs for investigating out-of-control signal and correcting the deviation, and costs of producing non-conforming products. Accordingly, there are three motivations in designing a control chart and in estimating of control chart parameter, i.e. (i) minimizing cost for inspection, (ii) maximizing capability or probability to detect out-of-control signal, and (iii) minimizing customer risk. Minimizing cost for inspection will lead to minimize the sample size (in single sampling) or to minimize the expected number of sample (in double sampling). This motivation was used by Daudin (1992) in order to estimate control limits as in equation (3). 8

Quick and accurate detection of mean shift is the main reason for the second motivation (He et al. 2002). Instead of using the first motivation, Irianto and Shinozaki (1998) used maximizing capability or probability to detect out-of-control signal of mean shift, as in equation (4), while keeping the estimate of sample size equal to 5 as commonly used. This paper estimates the parameter of DS control chart also by maximizing the power of control chart as in equation (5). Since power (often states in term of 1 − β ) measures the capability of correctly detecting that the process is out-of-control, maximizing power will also minimize risk at the customer because β also denotes the probability of delivering poor quality of product or allowing a process operating under deviation of process mean. 7. Conclusion

Despite of the advantage of giving higher capability in detecting out-of-control signal of mean shift, the DS procedure needs a complicated calculation. Efficiency of the calculation is improved by changing the optimization problem (3) and (4) into (5). Since the value of L1 is limited by the first constraint, numerical calculation in the optimization procedure is much less complicated. References Costa, A.F.B. (1994), X Charts with Variable Sample Size, Journal of Quality Technology, 26(3), p.155-163. Costa, A.F.B. and Claro, F.A.E. (2007), Double Sampling X Control Chart for a First-Order Autoregressive Moving Average Process Model, International Journal of Advanced Manufacturing Technology, on-line edition. Croasdale, R. (1974), Control Charts for a Double-Sampling Scheme Based On Average Production Run Lengths, International Journal of Production Research, 12(5), p.585592. Daudin, J.J., C. Duby, and P. Trecourt (1990), Plans de Controle Double Optimaux (Maitrise des Procedes et Controle de Reception), Rev. Statistique Appliquee, 38(4), p.45-59. Daudin, J.J. (1992), Double Sampling X Charts, Journal of Quality Technology, 24(2), p.7887. He, D., Grigoryan, A., and Sigh, M. (2002), Design of Double- and Triple-Sampling X-bar Control Chart Using Genetic Algorithms, International Journal of Production Research, 40(6), p.1387-1404. Irianto, D., and N. Shinozaki (1998), An Optimal Double Sampling X Control Chart, International Journal of Industrial Engineering – Theory, Applications and Practice, 5(3), p.226-234. Montgomery, D.C. (2002), Introduction to Statistical Quality Control, Fourth Edition, John Wiley & Sons, Singapore. Reynolds, M.R. Jr., R.W. Amin, J.C. Arnold, and J.A. Nachlas (1988), X Charts with Variable Sampling Interval, Technometrics, 30(2), p.181-192.