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Economic design of autoregressive moving average control chart using genetic algorithms
Expert Systems with Applications 39 (2012) 1793–1798
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Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
Economic design of autoregressive moving average control chart using genetic algorithms Sung-Nung Lin a, Chao-Yu Chou b,⇑, Shu-Ling Wang c, Hui-Rong Liu d a
Department of Industrial Engineering and Management, National Yunlin University of Science and Technology, Douliu 640, Taiwan Department of Finance, National Taichung Institute of Technology, Taichung 404, Taiwan c Department of Information Management, National Taichung Institute of Technology, Taichung 404, Taiwan d Department of Leisure and Recreation Management, National Taichung Institute of Technology, Taichung 404, Taiwan b
a r t i c l e
i n f o
Keywords: Autocorrelation Control chart Economic design Genetic algorithm Moving average
a b s t r a c t When designing control charts, it is usually assumed that the observations from the process at different time points are independent. However, this assumption may not be true for some production processes, e.g., the continuous chemical processes. The presence of autocorrelation in the process data can result in significant effect on the statistical performance of control charts. Jiang, Tsui, and Woodall (2000) developed a control chart, called the autoregressive moving average (ARMA) control chart, which has been shown suitable for monitoring a series of autocorrelated data. In the present paper, we develop the economic design of ARMA control chart to determine the optimal values of the test and chart parameters of the chart such that the expected total cost per hour is minimized. An illustrative example is provided and the genetic algorithm is applied to obtain the optimal solution of the economic design. A sensitivity analysis shows that the expected total cost associated with the control chart operation is positively affected by the occurrence frequency of the assignable cause, the time required to discover the assignable cause or to correct the process, and the quality cost per hour while producing in control or out of control, and is negatively influenced by the shift magnitude in process mean. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Statistical process control is an effective approach for improving product quality and saving production costs for a firm. Since 1924 when Dr. Shewhart presented the first control chart, various control chart techniques have been developed and widely applied as a primary tool in statistical process control. The major function of control charting is to detect the occurrence of assignable causes so that the necessary corrective action can be taken before a large quantity of nonconforming product is manufactured. The control chart technique may be considered as the graphical expression and operation of statistical hypothesis test. When a control chart is used to monitor a process, some test parameters should be determined, i.e., the sample size, the sampling interval between successive samples, and the control limits or critical region of the chart. Duncan (1956) first proposed a cost function for economically determining the test parameters for the average control chart that minimizes the average cost when a single out-of-control state (assignable cause) exists, which is called the ‘‘economic design’’ of control charts. Duncan’s cost function includes the cost of sampling
⇑ Corresponding author. E-mail address: [email protected] (C.-Y. Chou). 0957-4174/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2011.08.073
and inspection, the cost of defective products, the cost of false alarm, the cost of searching assignable cause, and the cost of process correction. Since then, considerable attention has been devoted to the economically optimal determination of the test parameters of control charts, e.g., see Montgomery (1980), Vance (1983), Ho and Case (1994a) and Chou, Chen, and Liu (2001). Lorenzen and Vance (1986) also introduced a unified approach for economic design of control charts. Application of Lorenzen and Vance’s approach may be found in Torng, Montgomery, and Cochran (1994), Ho and Case (1994b) and Chou, Chen, and Liu (2006). Traditionally, when designing control charts, it is usually assumed that the observations from the process at different time points are independent. However, this assumption may not be tenable in some production processes, e.g., the continuous chemical processes. The presence of autocorrelation in the process data can result in significant effect on the statistical performance of control charts. The major problem is that variations due to the autocorrelation may produce false out-of-control signals. Excessive false alarms may lead to unnecessary process adjustment and loss of confidence in the control chart as a monitoring tool. Jiang et al. (2000) developed a control chart, called the autoregressive moving average (ARMA) control chart, which has been shown suitable for monitoring a series of autocorrelated data.
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Various solution procedures for economically determining the optimal values of the test parameters of control charts have been developed and applied in the literature. The genetic algorithm (GA), based on the concept of natural genetics, uses the stochastic way (not deterministic rule) to guide the search direction of finding the optimal solution and is able to search for many possible solutions at the same time. Therefore, GA is considered as an appropriate way for solving the problems of combinatorial optimization and has been successfully applied in solution procedure of economic designs of control charts, e.g., see Chou, Wu, and Chen (2006), Chou, Cheng, and Lai (2008) and Lin, Chou, and Lai (2009). The present paper presents the economic design of ARMA control chart, in which the test and chart parameters are determined such that the average total cost associated with control chart operation is minimized. In the next section, a brief description of the use of ARMA control chart to maintain current control of an autocorrelated process is given. The cost function is then established by applying the cost function in Lorenzen and Vance (1986). The GA is employed to obtain the optimal values of the test and chart parameters for ARMA control chart, and an example is provided to illustrate the solution procedure. A sensitivity analysis is then carried out to investigate the effects of model parameters on the solution of the economic design. 2. The ARMA control chart The ARMA control chart was developed by Jiang et al. (2000) and has been shown to be effective for monitoring a process with autocorrelated measurements. Suppose that the variable xt is the measurement at time t from a normal distribution with mean l and variance r2 . According to Jiang et al. (2000), for an ARMA process, the measurement xt at time t can be expressed as a linear combination of the measurement at time t 1, the vibration factors at time t (denoted by at) and the vibration factors at time t 1 (denoted by at1 ), i.e., mathematically:
X t ¼ at v at1 þ uxt1 ;
for juj < 1 and jv j < 1;
ð1Þ
where ai’s at time i are normally and independently distributed with mean 0 and variance r2a , the constant u is the autoregressive parameter of the process, and the constant v is the moving average parameter of the process. It can be shown that:
r2 ¼
1 2uv þ v 2 2 ra : 1 u2
ð3Þ
where Z0 is generally the target of the characteristic, h and / are respectively the moving average parameter and the autoregressive parameter of the ARMA control chart, h0 ¼ 1 þ h / and b ¼ h=h0 . To guarantee that the monitoring process is reversible and stationary, we have the constraints that jbj < 1 and j/j < 1. It may be shown that the sample statistic in Eq. (3) has the mean l and a steady-state variance r2Z , where:
2ðh /Þð1 þ hÞ 1þh
r2 :
ð4Þ
Thus, the upper and lower control limits, abbreviated by UCL and LCL respectively, and the center line (CL) of the ARMA control chart can be calculated by
UCL ¼ l þ krZ ;
ð5Þ
CL ¼ l; LCL ¼ l krZ :
To simplify the mathematical manipulation of the cost function and its corresponding economic design, the following assumptions are made: (1) The measurements monitored by the ARMA control chart follow the first-order autoregressive and moving average process. That is, the design of ARMA control chart in the present paper is focused on the process of ARMA(1, 1). (2) In the start of the process, the process is assumed to be in the safe state; that is, l ¼ l0 . (3) The process mean may be shifted to the out-of-control region due to an assignable cause; that is, l ¼ l0 þ dr. (4) The process standard deviation r remains unchanged. (5) The time between occurrences of the assignable cause is exponentially distributed with a mean 1=k. (6) When the process goes out of control, it stays out of control until detected and corrected. (7) During each sampling interval, there exists at most one assignable cause which makes the process out of control. The assignable cause will not occur at sampling time. (8) The measurement error is assumed to be zero. (9) The cost function developed by Lorenzen and Vance (1986) is applied in the present paper to be the objective function for the economic design of the ARMA control chart. The expected cost (EC) per hour derived by Lorenzen and Vance (1986) includes the quality cost during production, the cost of false alarm, the cost of searching assignable cause, the cost of process correction, and the cost of sampling and inspection, and is mathematically expressed by
EC ¼
cQÞ s þ r1 t1 þ r2 t2 Þ þ sðYþ ARL0 h ARL1 s þ t1 þ t 2
c0 þ c1 ðnE þ h ARL1 k st 0 1 þ ð1 r 1 Þ ARL þ nE þ k 0
ð6Þ
s þ nE þ h ARL1 þ r 1 t 1 þ r 2 t2 þ W þ cQ h ; st0 1 þ ð1 r1 Þ ARL0 þ nE þ h ARL1 s þ t 1 þ t 2 k 1
ða þ bnÞ k þ
Z t ¼ h0 xt hxt1 þ /Z t1 ¼ h0 ðxt bxt1 Þ þ /Z t1 ;
3. The cost function
ð2Þ
The sample statistic used in the operation of an ARMA control chart at time t is defined as:
r2Z ¼
where k is the control limit coefficient and is one of the test parameters in the ARMA control chart to be determined in the economic design. The chart parameters h and / play important roles in the detection performance for an ARMA chart. In the present paper, the values of h and / will be also determined based on economic consideration.
ð7Þ where n = sample size, h = the sampling interval, a = fixed cost per sample, b = cost per unit sampled, c0 = quality cost per hour while producing in control, c1 = quality cost per hour while producing out of control (c1 > c0), E = time to sample and chart one item, Y = cost per false alarm, Q = productivity loss per process cease, W = cost to locate and correct the assignable cause, s = expected number of samples taken while in control, and it ekh may be shown that s ¼ 1e kh , s = expected time of occurrence of the assignable cause between two samples while in control, and it is shown that kh s ¼ 1ð1þkhÞe , kð1ekh Þ t0 = expected search time when the signal is a false alarm, t1 = expected time to discover the assignable cause,
S.-N. Lin et al. / Expert Systems with Applications 39 (2012) 1793–1798
t2 = expected time to correct the process, r1 = 1, if process continues during searches, r1 = 0, if process ceases during searches, r2 = 1, if process continues during correction, r2 = 0, if process ceases during correction, c = 0, if r1 + r2 = 2, c = 1, if r1 + r2 < 2, ARL0 = average run length while in control, ARL1 = average run length while out of control. If the measurements are independent, Markov chains approach was conventionally applied to obtain the values of ARL0 and ARL1. Since, in the present paper, the measurements are assumed to be autocorrelated, therefore simulation approach is, instead, used to get the approximate values of ARL0 and ARL1. A subroutine is coded for this purpose using Matlab. By considering the case that l0 ¼ 100 and r2 ¼ 10, the simulation procedure is summarized as follows: Step 1. Search for the possible combinations of r2a , u and v from Eq. (2), and then pick up one of them as an illustrative example. Step 2. Compute the value of r2Z from Eq. (4) for the specific values of h and /. Step 3. Calculate the upper and lower control limits based on Eqs. (5) and (6) for certain value of k. Step 4. Generate a series of normal random variates ai’s with mean 0 and variance r2a . Step 5. Let x0 ¼ l0 and a0 = 0. Obtain the time series xi from Eq. (1) for determining the value of ARL0. Obtain the time series xj ¼ xi þ dr for determining the value of ARL1 (where j = i). Step 6. Obtain the series of Zi’s from Eq. (3) to record the run length of the series (up to the point that is greater than the upper control limit or less than the lower control limit). Step 7. Repeat Step 4 through Step 6 for 500 times to have the desired average run length (ARL). Once the values of ARL0 and ARL1 are determined, these two values will be returned to the main program for continuation of searching for the optimal solution of economic design. The economic design of an ARMA control chart is to determine the five optimal values of n, h, k, h and / such that the expected cost (EC) per hour in Eq. (7) is minimized.
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t1 = 2 h, t2 = 2 h. The solution procedure is carried out using genetic algorithms with Matlab software to obtain the optimal values of n, h, k, h and / that minimize EC. The genetic algorithm (GA), based on the concept of natural genetics, is directed toward a random optimization search technique. The GA solves problems using the approach inspired by the process of Darwinian evolution. The current GA in science and engineering refers to the models introduced and investigated by Holland (1975). In the GA, the solution of a problem is called a ‘‘chromosome’’. A chromosome is composed of genes (i.e., features or characters). The solution procedure for our example using the GA is briefly described as follows: Step 1. Initialization. Thirty initial solutions that satisfy the constraint condition of each test parameter are randomly produced. The constraint condition for each test/chart parameter is set as follows:
1 6 n 6 20;
n is an integer;
0:001 < h < 5; 0:001 < k < 5; 0:001 < / < 1; 0:001 < h < 1: Step 2. Evaluation. The fitness of each solution is evaluated by calculating the value of fitness function. The fitness function for our example is the cost function shown in Eq. (7). Step 3. Selection. The survivors (i.e., 30 solutions) are selected for the next generation according to the better fitness of chromosomes. (In the first generation, the chromosome with the highest cost is replaced by the chromosome with the lowest cost.) Step 4. Crossover. A pairs of survivors (from the 30 solutions) are selected randomly as the parents used for crossover operations to produce new chromosomes (or children) for the next generation. In the present example, we apply the arithmetical crossover method with crossover rate 0.8 as follows:
D1 ¼ 0:8R þ 0:2M; D2 ¼ 0:2R þ 0:8M;
4. A numerical example and its solution procedure From examination of the cost components in Eq. (7), it can be seen that determining the economically optimal values of the five test and chart parameters for the ARMA control chart is not straightforward. To illustrate the nature of the solutions obtained from the economic design of ARMA control chart, a particular numerical example is provided. Suppose that a production process is monitored by the ARMA control chart. The process characteristic has a target value of 100 and its variance is assumed to be 10. The cost and model parameters are as follows: a = $0.5 per sample, b = $0.1 per unit sampled, c0 = $10 per hour, c1 = $100 per hour, E = 0.05 h, Y = $150 per false alarm, Q = $100 per process cease, W = $25, k = 0.01 t0 = 1 h,
where D 1 is the first new chromosome, D2 is the second new chromosome, and R and M are the parents chromosomes. If 30 parents are randomly selected, then there are 60 children that will be produced. Thus, the population size increases to 90 (i.e., 30 parents + 60 children) in this step. Step 5. Mutation. Suppose that the mutation rate is 0.1. In the present example, the non-uniform method is used to carry out the mutation operation. Since we have 90 solutions, we can randomly select 9 chromosomes (i.e., 90 0.1 = 9) to mutate some parameters (or genes). Step 6. Repeat Step 2 through Step 5 until the stopping criteria is found. In the present example, we use ‘‘50 generations’’ as our stopping criteria. After proceeding above steps ten times, computation time for each run is approximately between 25 and 35 min, depending on the starting parameter values. By choosing the best one among the 10 solutions, we obtain the optimal solution to the present example as:
n ¼ 2; h ¼ 0:435 h; k ¼ 2:283;
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h ¼ 0:455; / ¼ 0:898;
Table 3 SPSS output for the sample size (n).
and
(a) ANOVA table
EC ¼ 15:25:
5. The sensitivity analysis In this section, a sensitivity analysis is conducted to study the effect of model parameters on the solution of economic design of ARMA control chart. The sensitivity analysis is carried out using orthogonal-array experimental design and multiple regression, in which the model parameters are considered as the independent variables and the five test/chart parameters (i.e., n, h, k, h and /), as well as the average total cost (EC), are treated as the dependent variables. Table 1 Eleven model parameters and their level planning. Model parameter
Factor code
Level 1
Level 2
Level 3
Level 4
d k t1 t2 E c0 c1 Y W a b
A B C D E F G H I J K
0.5 0.01 2 2 0.05 5 80 100 20 0.5 0.1
1 0.03 10 10 0.25 10 100 150 25 2.5 0.5
1.5 0.05 20 20 0.5 15 120 200 30 5 1
3 – – – – – – – – – –
Source of variable
SS
df
MS
F
P-value
Regression Residual
5.42 29.58
1 34
5.42 0.87
6.23
0.018
Total
35.00
35
(b) Table of regression coefficients Independent variable
Coefficient
Std. Error
t-Value
P-value
Constant d
0.5809 0.3669
0.2799 0.1470
2.0750 2.4952
0.0456 0.0176
Eleven independent variables (i.e., the model parameters) considered in the sensitivity analysis and their corresponding level planning is shown in Table 1. The revised L36 orthogonal array is employed and the eleven independent variables are then assigned to the columns of the revised L36 array, as shown in Table 2. In the revised L32 orthogonal array experiment, there are 36 trials (i.e., 36 different level combinations of the independent variables). For each trial, the GA is applied to produce the optimal solution of the economic design, with two model parameters fixed: Q = 100 and t0 = 1. The output of the GA for each trial is also recorded in Table 2. To study the effect of model parameters on the solution of economic design of ARMA control chart, based on the data in Table 2, the statistical software SPSS is used to run the regression analysis for each of the six dependent variables, i.e., n, h, k, h, /, and EC. For
Table 2 Model parameter assignment in the revised L36 orthogonal array and the corresponding solutions. Trial
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
Coded model parameter
Solution
A
B
C
D
E
F
G
H
I
J
K
n
h
k
h
/
EC
1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 1 1 1 1 1 1 4 4 4 4 4 4
1 2 3 1 2 3 2 3 1 3 1 2 3 1 2 3 1 2 1 2 3 2 3 1 2 3 1 2 3 1 3 1 2 1 2 3
1 2 3 2 3 1 1 2 3 1 2 3 3 1 2 1 2 3 3 1 2 3 1 2 2 3 1 2 3 1 2 3 1 3 1 2
1 2 3 2 3 1 2 3 1 3 1 2 2 3 1 1 2 3 3 1 2 1 2 3 3 1 2 1 2 3 3 1 2 2 3 1
1 2 3 1 2 3 3 1 2 2 3 1 1 2 3 2 3 1 3 1 2 3 1 2 1 2 3 2 3 1 3 1 2 2 3 1
1 2 3 2 3 1 3 1 2 2 3 1 1 2 3 3 1 2 1 2 3 2 3 1 3 1 2 1 2 3 2 3 1 3 1 2
1 2 3 2 3 1 3 1 2 3 1 2 3 1 2 2 3 1 2 3 1 1 2 3 1 2 3 3 1 2 2 3 1 1 2 3
1 2 3 3 1 2 1 2 3 2 3 1 3 1 2 3 1 2 2 3 1 1 2 3 3 1 2 2 3 1 1 2 3 2 3 1
1 2 3 3 1 2 2 3 1 1 2 3 2 3 1 3 1 2 1 2 3 3 1 2 1 2 3 3 1 2 2 3 1 2 3 1
1 2 3 3 1 2 2 3 1 3 1 2 1 2 3 2 3 1 2 3 1 3 1 2 2 3 1 1 2 3 1 2 3 3 1 2
1 2 3 3 1 2 3 1 2 2 3 1 2 3 1 1 2 3 3 1 2 2 3 1 2 3 1 3 1 2 1 2 3 1 2 3
1 1 1 3 3 2 4 3 6 3 1 8 3 2 7 3 1 1 1 2 2 1 2 1 4 5 1 1 4 9 1 2 1 1 1 1
.215 .831 4.49 1.63 .494 .611 1.05 3.57 2.92 2.05 1.02 1.02 2.25 3.43 2.18 1.52 1.69 2.39 4.07 4.84 4.22 4.85 3.11 4.38 4.57 3.36 .357 1.29 3.47 2.65 3.92 4.43 3.91 4.20 3.62 4.56
.675 2.92 2.26 1.52 4.85 3.52 .420 .533 1.64 .508 .049 3.60 .205 2.18 1.53 1.75 .215 4.51 4.72 4.31 1.38 3.26 3.47 1.18 1.55 3.87 1.43 1.50 3.45 3.96 1.09 .259 .958 1.70 1.55 .157
.338 .884 .001 .150 .981 .996 .180 .114 .951 .309 .001 .287 .290 .076 .081 .656 .880 .799 .264 .583 .923 .095 .039 .738 .325 .953 .166 .882 .583 .803 .895 .194 .023 .260 .189 .574
.376 .120 .828 .865 .906 .137 .851 .626 .087 .448 .010 .912 .573 .505 .538 .712 .840 .518 .089 .877 .584 .729 .245 .394 .160 .941 .909 .530 .548 .145 .354 .314 .808 .660 .924 .238
11.001 48.62 89.49 31.44 74.75 25.85 52.17 54.11 27.70 73.35 24.51 54.55 79.38 24.12 40.65 33.27 27.76 49.10 33.33 22.76 48.70 39.63 47.90 32.44 49.57 59.23 24.34 37.15 56.14 35.99 64.72 34.99 26.74 31.50 43.77 51.79
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SS
df
MS
F
P-value
Regression Residual
22.38 12.62
4 31
5.60 0.41
13.75
1.48E06
Total
35.00
35
(b) Table of regression coefficients Independent variable
Coefficient
Std. Error
t-Value
P-value
Constant d t1 t2 a
2.0586 0.6211 0.0327 0.0262 0.1678
0.3284 0.1006 0.0144 0.0144 0.0578
6.2694 6.1766 2.2624 1.8135 2.9051
4.87 E07 6.65 E07 0.0308 0.0794 0.0067
each dependent variable, the output of SPSS includes an ANOVA table for regression and a table of regression coefficients, showing the corresponding information about statistical hypothesis testing. Table 3 is the SPSS output for the sample size (n). From the ANOVA in Table 3(a), if the significance level is set to be 0.05, then there are at least one model parameters that significantly affect the value of sample size. By examining Table 3(b), we find that the shift magnitude of process mean (d) significantly affects the value of sample size. It is noticed that the sign of the coefficients of d is negative, indicating that a larger shift magnitude in process mean generally requires a smaller sample size, which is consistent with the principle of statistical hypothesis testing. Table 4 is the SPSS outputs for the sampling interval (h). It can be seen from Table 4(b) that four model parameters (i.e., d, t1, t2 and a) significantly affect the value of h. A larger shift magnitude in process mean generally increases the sample interval, which is consistent with our intuitive reasoning. Meanwhile, if it requires longer time duration to discover the assignable cause or to correct the process, then the sampling interval tends to be larger. A higher fixed cost per sample always leads to a larger sampling interval. Table 5 is the SPSS output for the control limit coefficient (k). Table 5(b) indicates that three model parameters (i.e., a, c0 and c1) significantly affect the value of k. Based on Table 5, it may be seen that a higher fixed cost per sample results in a wider safe region. A higher quality cost per hour while producing in control also gives a wider safe region; however, a higher quality cost per hour while producing out of control leads to a tighter safe region. Table 6 is the SPSS output for the moving average parameter of the ARMA chart (h). Table 6 indicates that there is no model parameter that significantly affects the value of h. Table 7 is the SPSS output for the autoregressive parameter of the ARMA chart (/). From Table 7(b), it is obvious that a higher cost to locate and correct the assignable cause or a higher fixed cost per Table 5 SPSS output for the control limit coefficient (k). (a) ANOVA table Source of variable
SS
df
MS
F
P-value
Regression Residual
14.90 20.10
3 32
4.97 0.63
7.91
0.0004
Total
35.00
35
(b) Table of regression coefficients Independent variable
Coefficient
Std. Error
t-Value
P-value
Constant a c0 c1
2.2093 0.0821 0.0677 0.0565
0.9017 0.0324 0.0179 0.0324
2.4502 2.5367 3.7741 1.7453
0.0199 0.0163 0.0007 0.0905
Table 6 SPSS output for the moving average parameter of the ARMA chart (h). ANOVA table Source of variable
SS
df
MS
F
P-value
Regression Residual
2.11 32.89
1 34
2.11 0.97
2.19
0.1458
Total
35.00
35
Table 7 SPSS output for the autoregressive parameter of the ARMA chart (/). (a) ANOVA table Source of variable
SS
df
MS
F
P-value
Regression Residual Total
7.58 27.42 35.00
2 33 35
3.79 0.83
4.56
0.0178
(b) Table of regression coefficients Independent variable
Coefficient
Std. Error
t-Value
P-value
Constant W a
2.6638 0.0909 0.1465
0.9680 0.0372 0.0825
2.7519 2.4434 1.7753
0.0096 0.0201 0.0851
sample usually leads to a larger value of the autoregressive parameter of the ARMA chart. Table 8 is the SPSS output for the expected total cost per hour (EC). Based on Table 8(b), it is noted that a larger shift magnitude in process mean generally reduces the expected cost. This is probably because a large shift magnitude in process mean can be easily detected by the control chart and consequently can be corrected just in time. Also, frequent occurrence of the assignable cause (i.e., a larger k value) leads to the higher expected total cost. A longer time duration required to discover the assignable cause or to correct the process definitely produces a higher expected total cost. Meanwhile, a higher quality cost per hour while producing in control or out of control generally results in a higher expected total cost, too. 6. Conclusions The major function of control charting is to detect the occurrence of assignable causes so that the necessary corrective action can be taken before a large quantity of nonconforming product is manufactured. Traditionally, when designing control charts, it is usually assumed that the observations from the process at different time points are independent. However, this assumption may not be true in some production processes, e.g., the continuous chemical processes. The presence of autocorrelation in the
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Table 8 SPSS output for the expected cost per hour (EC). (a) ANOVA table Source of variable
SS
df
MS
F
P-value
Regression Residual
32.24 2.76
6 29
5.37 0.095
56.39
1.12 E14
Total
35.00
35
(b) Table of regression coefficients Independent variable
Coefficient
Std. error
t-Value
P-value
Constant d k t1 t2 c0 c1
4.3863 0.1387 40.2667 0.0542 0.0556 0.0365 0.0186
0.3791 0.0487 3.1504 0.0070 0.0070 0.0126 0.0032
11.5688 2.8504 12.7813 7.7532 7.9638 2.8983 5.9082
2.19 E12 0.0080 1.93 E13 1.5 E08 8.78 E09 0.0071 2.05 E06
process data can result in significant effect on the statistical performance of control charts. The ARMA control chart has been shown to be effective for monitoring a process with autocorrelated measurements. Two chart parameters, the autoregressive parameter (/) and the moving average parameter (h), play important roles in the detection ability of the ARMA chart. In the present paper, we develop the economic design of ARMA control chart to determine the optimal values of the five test and chart parameters of the chart (i.e., the sample size, the sampling interval, the control limit coefficient, / and h) such that the expected total cost per hour is minimized. The cost function is established based on the cost model in Lorenzen and Vance (1986) with the simulated average run length. An illustrative numerical example is provided and the GA is employed to search for the solution of the economic design. A sensitivity analysis is then carried out to study the effect of model parameters on the solution of the economic design. Based on the sensitivity analysis, the following important results are observed: (1) The shift magnitude of process mean (d) significantly affects the optimal values of sample size, sampling interval, and the expected cost. A large shift in mean generally increases the sample size and the sampling interval, but reduces the expected cost. This is perhaps due to the fact that a large shift magnitude in process mean may be fast detected by the control chart and consequently the process can be corrected just in time. (2) Frequent occurrence of the assignable cause always leads to the higher expected total cost. (3) If the time required to discover the assignable cause or to correct the process increases, then both the sampling interval and the expected cost also tend to increase. (4) A higher quality cost per hour while producing in control also gives a wider safe region and a higher expected total cost; however, a higher quality cost per hour while producing out of control leads to a tighter safe region and a higher expected total cost.
(5) The fixed cost per sample has a significant influence on the sampling interval, the control limit coefficient, and the autoregressive parameter of the ARMA chart. A higher fixed cost per sample usually results in a longer sampling interval, a wider safe region, and a larger autoregressive parameter of the ARMA chart. (6) A higher cost to locate and correct the assignable cause generally leads to a larger value of the autoregressive parameter of the ARMA chart.
References Chou, C. Y., Chen, C. H., & Liu, H. R. (2001). Economic design of X-bar charts for nonnormally correlated data. International Journal of Production Research, 39, 1931–1941. Chou, C. Y., Chen, C. H., & Liu, H. R. (2006). Economic design of EWMA charts with variable sampling intervals. Quality & Quantity, 40, 879–896. Chou, C. Y., Cheng, J. C., & Lai, W. T. (2008). Economic design of variable sampling intervals EWMA charts with sampling at fixed times using genetic algorithms. Expert Systems with Applications, 34, 419–426. Chou, C. Y., Wu, C. C., & Chen, C. H. (2006). Joint economic design of variable sampling intervals X-bar and R charts using genetic algorithms. Communications in Statistics Part B – Simulation and Computation, 35, 1027–1043. Duncan, A. J. (1956). The economic design of X-bar charts used to maintain current control of a process. Journal of the American Statistical Association, 52, 228–242. Ho, C., & Case, K. E. (1994a). Economic design of control charts: A literature review for 1981–1991. Journal of Quality Technology, 26, 1–18. Ho, C., & Case, K. E. (1994b). The economically-based EWMA control chart. International Journal of Production Research, 32, 2179–2186. Holland, J. H. (1975). Adaptation in Nature and Artificial System. Ann Arbor, Michigan, USA: The University of Michigan Press. Jiang, W., Tsui, K. L., & Woodall, W. H. (2000). A new monitoring method: The ARMA chart. Technometrics, 42, 399–410. Lin, H. H., Chou, C. Y., & Lai, W. T. (2009). Economic design of variable sampling intervals X-bar charts with A&L switching rule using genetic algorithms. Expert Systems with Applications, 36, 3048–3055. Lorenzen, T. J., & Vance, L. C. (1986). The economic design of control charts: A unified approach. Technometrics, 28, 3–10. Montgomery, D. C. (1980). The economic design of control charts: A review and literature survey. Journal of Quality Technology, 12, 75–87. Torng, J. C. C., Montgomery, D. C., & Cochran, J. K. (1994). Economic design of the EWMA control chart. Economic Quality Control, 9, 3–23. Vance, L. C. (1983). A bibliography of statistical quality control chart techniques, 1970–1980. Journal of Quality Technology, 15, 59–62.
Contents lists available at SciVerse ScienceDirect
Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
Economic design of autoregressive moving average control chart using genetic algorithms Sung-Nung Lin a, Chao-Yu Chou b,⇑, Shu-Ling Wang c, Hui-Rong Liu d a
Department of Industrial Engineering and Management, National Yunlin University of Science and Technology, Douliu 640, Taiwan Department of Finance, National Taichung Institute of Technology, Taichung 404, Taiwan c Department of Information Management, National Taichung Institute of Technology, Taichung 404, Taiwan d Department of Leisure and Recreation Management, National Taichung Institute of Technology, Taichung 404, Taiwan b
a r t i c l e
i n f o
Keywords: Autocorrelation Control chart Economic design Genetic algorithm Moving average
a b s t r a c t When designing control charts, it is usually assumed that the observations from the process at different time points are independent. However, this assumption may not be true for some production processes, e.g., the continuous chemical processes. The presence of autocorrelation in the process data can result in significant effect on the statistical performance of control charts. Jiang, Tsui, and Woodall (2000) developed a control chart, called the autoregressive moving average (ARMA) control chart, which has been shown suitable for monitoring a series of autocorrelated data. In the present paper, we develop the economic design of ARMA control chart to determine the optimal values of the test and chart parameters of the chart such that the expected total cost per hour is minimized. An illustrative example is provided and the genetic algorithm is applied to obtain the optimal solution of the economic design. A sensitivity analysis shows that the expected total cost associated with the control chart operation is positively affected by the occurrence frequency of the assignable cause, the time required to discover the assignable cause or to correct the process, and the quality cost per hour while producing in control or out of control, and is negatively influenced by the shift magnitude in process mean. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Statistical process control is an effective approach for improving product quality and saving production costs for a firm. Since 1924 when Dr. Shewhart presented the first control chart, various control chart techniques have been developed and widely applied as a primary tool in statistical process control. The major function of control charting is to detect the occurrence of assignable causes so that the necessary corrective action can be taken before a large quantity of nonconforming product is manufactured. The control chart technique may be considered as the graphical expression and operation of statistical hypothesis test. When a control chart is used to monitor a process, some test parameters should be determined, i.e., the sample size, the sampling interval between successive samples, and the control limits or critical region of the chart. Duncan (1956) first proposed a cost function for economically determining the test parameters for the average control chart that minimizes the average cost when a single out-of-control state (assignable cause) exists, which is called the ‘‘economic design’’ of control charts. Duncan’s cost function includes the cost of sampling
⇑ Corresponding author. E-mail address: [email protected] (C.-Y. Chou). 0957-4174/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2011.08.073
and inspection, the cost of defective products, the cost of false alarm, the cost of searching assignable cause, and the cost of process correction. Since then, considerable attention has been devoted to the economically optimal determination of the test parameters of control charts, e.g., see Montgomery (1980), Vance (1983), Ho and Case (1994a) and Chou, Chen, and Liu (2001). Lorenzen and Vance (1986) also introduced a unified approach for economic design of control charts. Application of Lorenzen and Vance’s approach may be found in Torng, Montgomery, and Cochran (1994), Ho and Case (1994b) and Chou, Chen, and Liu (2006). Traditionally, when designing control charts, it is usually assumed that the observations from the process at different time points are independent. However, this assumption may not be tenable in some production processes, e.g., the continuous chemical processes. The presence of autocorrelation in the process data can result in significant effect on the statistical performance of control charts. The major problem is that variations due to the autocorrelation may produce false out-of-control signals. Excessive false alarms may lead to unnecessary process adjustment and loss of confidence in the control chart as a monitoring tool. Jiang et al. (2000) developed a control chart, called the autoregressive moving average (ARMA) control chart, which has been shown suitable for monitoring a series of autocorrelated data.
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Various solution procedures for economically determining the optimal values of the test parameters of control charts have been developed and applied in the literature. The genetic algorithm (GA), based on the concept of natural genetics, uses the stochastic way (not deterministic rule) to guide the search direction of finding the optimal solution and is able to search for many possible solutions at the same time. Therefore, GA is considered as an appropriate way for solving the problems of combinatorial optimization and has been successfully applied in solution procedure of economic designs of control charts, e.g., see Chou, Wu, and Chen (2006), Chou, Cheng, and Lai (2008) and Lin, Chou, and Lai (2009). The present paper presents the economic design of ARMA control chart, in which the test and chart parameters are determined such that the average total cost associated with control chart operation is minimized. In the next section, a brief description of the use of ARMA control chart to maintain current control of an autocorrelated process is given. The cost function is then established by applying the cost function in Lorenzen and Vance (1986). The GA is employed to obtain the optimal values of the test and chart parameters for ARMA control chart, and an example is provided to illustrate the solution procedure. A sensitivity analysis is then carried out to investigate the effects of model parameters on the solution of the economic design. 2. The ARMA control chart The ARMA control chart was developed by Jiang et al. (2000) and has been shown to be effective for monitoring a process with autocorrelated measurements. Suppose that the variable xt is the measurement at time t from a normal distribution with mean l and variance r2 . According to Jiang et al. (2000), for an ARMA process, the measurement xt at time t can be expressed as a linear combination of the measurement at time t 1, the vibration factors at time t (denoted by at) and the vibration factors at time t 1 (denoted by at1 ), i.e., mathematically:
X t ¼ at v at1 þ uxt1 ;
for juj < 1 and jv j < 1;
ð1Þ
where ai’s at time i are normally and independently distributed with mean 0 and variance r2a , the constant u is the autoregressive parameter of the process, and the constant v is the moving average parameter of the process. It can be shown that:
r2 ¼
1 2uv þ v 2 2 ra : 1 u2
ð3Þ
where Z0 is generally the target of the characteristic, h and / are respectively the moving average parameter and the autoregressive parameter of the ARMA control chart, h0 ¼ 1 þ h / and b ¼ h=h0 . To guarantee that the monitoring process is reversible and stationary, we have the constraints that jbj < 1 and j/j < 1. It may be shown that the sample statistic in Eq. (3) has the mean l and a steady-state variance r2Z , where:
2ðh /Þð1 þ hÞ 1þh
r2 :
ð4Þ
Thus, the upper and lower control limits, abbreviated by UCL and LCL respectively, and the center line (CL) of the ARMA control chart can be calculated by
UCL ¼ l þ krZ ;
ð5Þ
CL ¼ l; LCL ¼ l krZ :
To simplify the mathematical manipulation of the cost function and its corresponding economic design, the following assumptions are made: (1) The measurements monitored by the ARMA control chart follow the first-order autoregressive and moving average process. That is, the design of ARMA control chart in the present paper is focused on the process of ARMA(1, 1). (2) In the start of the process, the process is assumed to be in the safe state; that is, l ¼ l0 . (3) The process mean may be shifted to the out-of-control region due to an assignable cause; that is, l ¼ l0 þ dr. (4) The process standard deviation r remains unchanged. (5) The time between occurrences of the assignable cause is exponentially distributed with a mean 1=k. (6) When the process goes out of control, it stays out of control until detected and corrected. (7) During each sampling interval, there exists at most one assignable cause which makes the process out of control. The assignable cause will not occur at sampling time. (8) The measurement error is assumed to be zero. (9) The cost function developed by Lorenzen and Vance (1986) is applied in the present paper to be the objective function for the economic design of the ARMA control chart. The expected cost (EC) per hour derived by Lorenzen and Vance (1986) includes the quality cost during production, the cost of false alarm, the cost of searching assignable cause, the cost of process correction, and the cost of sampling and inspection, and is mathematically expressed by
EC ¼
cQÞ s þ r1 t1 þ r2 t2 Þ þ sðYþ ARL0 h ARL1 s þ t1 þ t 2
c0 þ c1 ðnE þ h ARL1 k st 0 1 þ ð1 r 1 Þ ARL þ nE þ k 0
ð6Þ
s þ nE þ h ARL1 þ r 1 t 1 þ r 2 t2 þ W þ cQ h ; st0 1 þ ð1 r1 Þ ARL0 þ nE þ h ARL1 s þ t 1 þ t 2 k 1
ða þ bnÞ k þ
Z t ¼ h0 xt hxt1 þ /Z t1 ¼ h0 ðxt bxt1 Þ þ /Z t1 ;
3. The cost function
ð2Þ
The sample statistic used in the operation of an ARMA control chart at time t is defined as:
r2Z ¼
where k is the control limit coefficient and is one of the test parameters in the ARMA control chart to be determined in the economic design. The chart parameters h and / play important roles in the detection performance for an ARMA chart. In the present paper, the values of h and / will be also determined based on economic consideration.
ð7Þ where n = sample size, h = the sampling interval, a = fixed cost per sample, b = cost per unit sampled, c0 = quality cost per hour while producing in control, c1 = quality cost per hour while producing out of control (c1 > c0), E = time to sample and chart one item, Y = cost per false alarm, Q = productivity loss per process cease, W = cost to locate and correct the assignable cause, s = expected number of samples taken while in control, and it ekh may be shown that s ¼ 1e kh , s = expected time of occurrence of the assignable cause between two samples while in control, and it is shown that kh s ¼ 1ð1þkhÞe , kð1ekh Þ t0 = expected search time when the signal is a false alarm, t1 = expected time to discover the assignable cause,
S.-N. Lin et al. / Expert Systems with Applications 39 (2012) 1793–1798
t2 = expected time to correct the process, r1 = 1, if process continues during searches, r1 = 0, if process ceases during searches, r2 = 1, if process continues during correction, r2 = 0, if process ceases during correction, c = 0, if r1 + r2 = 2, c = 1, if r1 + r2 < 2, ARL0 = average run length while in control, ARL1 = average run length while out of control. If the measurements are independent, Markov chains approach was conventionally applied to obtain the values of ARL0 and ARL1. Since, in the present paper, the measurements are assumed to be autocorrelated, therefore simulation approach is, instead, used to get the approximate values of ARL0 and ARL1. A subroutine is coded for this purpose using Matlab. By considering the case that l0 ¼ 100 and r2 ¼ 10, the simulation procedure is summarized as follows: Step 1. Search for the possible combinations of r2a , u and v from Eq. (2), and then pick up one of them as an illustrative example. Step 2. Compute the value of r2Z from Eq. (4) for the specific values of h and /. Step 3. Calculate the upper and lower control limits based on Eqs. (5) and (6) for certain value of k. Step 4. Generate a series of normal random variates ai’s with mean 0 and variance r2a . Step 5. Let x0 ¼ l0 and a0 = 0. Obtain the time series xi from Eq. (1) for determining the value of ARL0. Obtain the time series xj ¼ xi þ dr for determining the value of ARL1 (where j = i). Step 6. Obtain the series of Zi’s from Eq. (3) to record the run length of the series (up to the point that is greater than the upper control limit or less than the lower control limit). Step 7. Repeat Step 4 through Step 6 for 500 times to have the desired average run length (ARL). Once the values of ARL0 and ARL1 are determined, these two values will be returned to the main program for continuation of searching for the optimal solution of economic design. The economic design of an ARMA control chart is to determine the five optimal values of n, h, k, h and / such that the expected cost (EC) per hour in Eq. (7) is minimized.
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t1 = 2 h, t2 = 2 h. The solution procedure is carried out using genetic algorithms with Matlab software to obtain the optimal values of n, h, k, h and / that minimize EC. The genetic algorithm (GA), based on the concept of natural genetics, is directed toward a random optimization search technique. The GA solves problems using the approach inspired by the process of Darwinian evolution. The current GA in science and engineering refers to the models introduced and investigated by Holland (1975). In the GA, the solution of a problem is called a ‘‘chromosome’’. A chromosome is composed of genes (i.e., features or characters). The solution procedure for our example using the GA is briefly described as follows: Step 1. Initialization. Thirty initial solutions that satisfy the constraint condition of each test parameter are randomly produced. The constraint condition for each test/chart parameter is set as follows:
1 6 n 6 20;
n is an integer;
0:001 < h < 5; 0:001 < k < 5; 0:001 < / < 1; 0:001 < h < 1: Step 2. Evaluation. The fitness of each solution is evaluated by calculating the value of fitness function. The fitness function for our example is the cost function shown in Eq. (7). Step 3. Selection. The survivors (i.e., 30 solutions) are selected for the next generation according to the better fitness of chromosomes. (In the first generation, the chromosome with the highest cost is replaced by the chromosome with the lowest cost.) Step 4. Crossover. A pairs of survivors (from the 30 solutions) are selected randomly as the parents used for crossover operations to produce new chromosomes (or children) for the next generation. In the present example, we apply the arithmetical crossover method with crossover rate 0.8 as follows:
D1 ¼ 0:8R þ 0:2M; D2 ¼ 0:2R þ 0:8M;
4. A numerical example and its solution procedure From examination of the cost components in Eq. (7), it can be seen that determining the economically optimal values of the five test and chart parameters for the ARMA control chart is not straightforward. To illustrate the nature of the solutions obtained from the economic design of ARMA control chart, a particular numerical example is provided. Suppose that a production process is monitored by the ARMA control chart. The process characteristic has a target value of 100 and its variance is assumed to be 10. The cost and model parameters are as follows: a = $0.5 per sample, b = $0.1 per unit sampled, c0 = $10 per hour, c1 = $100 per hour, E = 0.05 h, Y = $150 per false alarm, Q = $100 per process cease, W = $25, k = 0.01 t0 = 1 h,
where D 1 is the first new chromosome, D2 is the second new chromosome, and R and M are the parents chromosomes. If 30 parents are randomly selected, then there are 60 children that will be produced. Thus, the population size increases to 90 (i.e., 30 parents + 60 children) in this step. Step 5. Mutation. Suppose that the mutation rate is 0.1. In the present example, the non-uniform method is used to carry out the mutation operation. Since we have 90 solutions, we can randomly select 9 chromosomes (i.e., 90 0.1 = 9) to mutate some parameters (or genes). Step 6. Repeat Step 2 through Step 5 until the stopping criteria is found. In the present example, we use ‘‘50 generations’’ as our stopping criteria. After proceeding above steps ten times, computation time for each run is approximately between 25 and 35 min, depending on the starting parameter values. By choosing the best one among the 10 solutions, we obtain the optimal solution to the present example as:
n ¼ 2; h ¼ 0:435 h; k ¼ 2:283;
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h ¼ 0:455; / ¼ 0:898;
Table 3 SPSS output for the sample size (n).
and
(a) ANOVA table
EC ¼ 15:25:
5. The sensitivity analysis In this section, a sensitivity analysis is conducted to study the effect of model parameters on the solution of economic design of ARMA control chart. The sensitivity analysis is carried out using orthogonal-array experimental design and multiple regression, in which the model parameters are considered as the independent variables and the five test/chart parameters (i.e., n, h, k, h and /), as well as the average total cost (EC), are treated as the dependent variables. Table 1 Eleven model parameters and their level planning. Model parameter
Factor code
Level 1
Level 2
Level 3
Level 4
d k t1 t2 E c0 c1 Y W a b
A B C D E F G H I J K
0.5 0.01 2 2 0.05 5 80 100 20 0.5 0.1
1 0.03 10 10 0.25 10 100 150 25 2.5 0.5
1.5 0.05 20 20 0.5 15 120 200 30 5 1
3 – – – – – – – – – –
Source of variable
SS
df
MS
F
P-value
Regression Residual
5.42 29.58
1 34
5.42 0.87
6.23
0.018
Total
35.00
35
(b) Table of regression coefficients Independent variable
Coefficient
Std. Error
t-Value
P-value
Constant d
0.5809 0.3669
0.2799 0.1470
2.0750 2.4952
0.0456 0.0176
Eleven independent variables (i.e., the model parameters) considered in the sensitivity analysis and their corresponding level planning is shown in Table 1. The revised L36 orthogonal array is employed and the eleven independent variables are then assigned to the columns of the revised L36 array, as shown in Table 2. In the revised L32 orthogonal array experiment, there are 36 trials (i.e., 36 different level combinations of the independent variables). For each trial, the GA is applied to produce the optimal solution of the economic design, with two model parameters fixed: Q = 100 and t0 = 1. The output of the GA for each trial is also recorded in Table 2. To study the effect of model parameters on the solution of economic design of ARMA control chart, based on the data in Table 2, the statistical software SPSS is used to run the regression analysis for each of the six dependent variables, i.e., n, h, k, h, /, and EC. For
Table 2 Model parameter assignment in the revised L36 orthogonal array and the corresponding solutions. Trial
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
Coded model parameter
Solution
A
B
C
D
E
F
G
H
I
J
K
n
h
k
h
/
EC
1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 1 1 1 1 1 1 4 4 4 4 4 4
1 2 3 1 2 3 2 3 1 3 1 2 3 1 2 3 1 2 1 2 3 2 3 1 2 3 1 2 3 1 3 1 2 1 2 3
1 2 3 2 3 1 1 2 3 1 2 3 3 1 2 1 2 3 3 1 2 3 1 2 2 3 1 2 3 1 2 3 1 3 1 2
1 2 3 2 3 1 2 3 1 3 1 2 2 3 1 1 2 3 3 1 2 1 2 3 3 1 2 1 2 3 3 1 2 2 3 1
1 2 3 1 2 3 3 1 2 2 3 1 1 2 3 2 3 1 3 1 2 3 1 2 1 2 3 2 3 1 3 1 2 2 3 1
1 2 3 2 3 1 3 1 2 2 3 1 1 2 3 3 1 2 1 2 3 2 3 1 3 1 2 1 2 3 2 3 1 3 1 2
1 2 3 2 3 1 3 1 2 3 1 2 3 1 2 2 3 1 2 3 1 1 2 3 1 2 3 3 1 2 2 3 1 1 2 3
1 2 3 3 1 2 1 2 3 2 3 1 3 1 2 3 1 2 2 3 1 1 2 3 3 1 2 2 3 1 1 2 3 2 3 1
1 2 3 3 1 2 2 3 1 1 2 3 2 3 1 3 1 2 1 2 3 3 1 2 1 2 3 3 1 2 2 3 1 2 3 1
1 2 3 3 1 2 2 3 1 3 1 2 1 2 3 2 3 1 2 3 1 3 1 2 2 3 1 1 2 3 1 2 3 3 1 2
1 2 3 3 1 2 3 1 2 2 3 1 2 3 1 1 2 3 3 1 2 2 3 1 2 3 1 3 1 2 1 2 3 1 2 3
1 1 1 3 3 2 4 3 6 3 1 8 3 2 7 3 1 1 1 2 2 1 2 1 4 5 1 1 4 9 1 2 1 1 1 1
.215 .831 4.49 1.63 .494 .611 1.05 3.57 2.92 2.05 1.02 1.02 2.25 3.43 2.18 1.52 1.69 2.39 4.07 4.84 4.22 4.85 3.11 4.38 4.57 3.36 .357 1.29 3.47 2.65 3.92 4.43 3.91 4.20 3.62 4.56
.675 2.92 2.26 1.52 4.85 3.52 .420 .533 1.64 .508 .049 3.60 .205 2.18 1.53 1.75 .215 4.51 4.72 4.31 1.38 3.26 3.47 1.18 1.55 3.87 1.43 1.50 3.45 3.96 1.09 .259 .958 1.70 1.55 .157
.338 .884 .001 .150 .981 .996 .180 .114 .951 .309 .001 .287 .290 .076 .081 .656 .880 .799 .264 .583 .923 .095 .039 .738 .325 .953 .166 .882 .583 .803 .895 .194 .023 .260 .189 .574
.376 .120 .828 .865 .906 .137 .851 .626 .087 .448 .010 .912 .573 .505 .538 .712 .840 .518 .089 .877 .584 .729 .245 .394 .160 .941 .909 .530 .548 .145 .354 .314 .808 .660 .924 .238
11.001 48.62 89.49 31.44 74.75 25.85 52.17 54.11 27.70 73.35 24.51 54.55 79.38 24.12 40.65 33.27 27.76 49.10 33.33 22.76 48.70 39.63 47.90 32.44 49.57 59.23 24.34 37.15 56.14 35.99 64.72 34.99 26.74 31.50 43.77 51.79
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S.-N. Lin et al. / Expert Systems with Applications 39 (2012) 1793–1798 Table 4 SPSS output for the sampling interval (h). (a) ANOVA table Source of variable
SS
df
MS
F
P-value
Regression Residual
22.38 12.62
4 31
5.60 0.41
13.75
1.48E06
Total
35.00
35
(b) Table of regression coefficients Independent variable
Coefficient
Std. Error
t-Value
P-value
Constant d t1 t2 a
2.0586 0.6211 0.0327 0.0262 0.1678
0.3284 0.1006 0.0144 0.0144 0.0578
6.2694 6.1766 2.2624 1.8135 2.9051
4.87 E07 6.65 E07 0.0308 0.0794 0.0067
each dependent variable, the output of SPSS includes an ANOVA table for regression and a table of regression coefficients, showing the corresponding information about statistical hypothesis testing. Table 3 is the SPSS output for the sample size (n). From the ANOVA in Table 3(a), if the significance level is set to be 0.05, then there are at least one model parameters that significantly affect the value of sample size. By examining Table 3(b), we find that the shift magnitude of process mean (d) significantly affects the value of sample size. It is noticed that the sign of the coefficients of d is negative, indicating that a larger shift magnitude in process mean generally requires a smaller sample size, which is consistent with the principle of statistical hypothesis testing. Table 4 is the SPSS outputs for the sampling interval (h). It can be seen from Table 4(b) that four model parameters (i.e., d, t1, t2 and a) significantly affect the value of h. A larger shift magnitude in process mean generally increases the sample interval, which is consistent with our intuitive reasoning. Meanwhile, if it requires longer time duration to discover the assignable cause or to correct the process, then the sampling interval tends to be larger. A higher fixed cost per sample always leads to a larger sampling interval. Table 5 is the SPSS output for the control limit coefficient (k). Table 5(b) indicates that three model parameters (i.e., a, c0 and c1) significantly affect the value of k. Based on Table 5, it may be seen that a higher fixed cost per sample results in a wider safe region. A higher quality cost per hour while producing in control also gives a wider safe region; however, a higher quality cost per hour while producing out of control leads to a tighter safe region. Table 6 is the SPSS output for the moving average parameter of the ARMA chart (h). Table 6 indicates that there is no model parameter that significantly affects the value of h. Table 7 is the SPSS output for the autoregressive parameter of the ARMA chart (/). From Table 7(b), it is obvious that a higher cost to locate and correct the assignable cause or a higher fixed cost per Table 5 SPSS output for the control limit coefficient (k). (a) ANOVA table Source of variable
SS
df
MS
F
P-value
Regression Residual
14.90 20.10
3 32
4.97 0.63
7.91
0.0004
Total
35.00
35
(b) Table of regression coefficients Independent variable
Coefficient
Std. Error
t-Value
P-value
Constant a c0 c1
2.2093 0.0821 0.0677 0.0565
0.9017 0.0324 0.0179 0.0324
2.4502 2.5367 3.7741 1.7453
0.0199 0.0163 0.0007 0.0905
Table 6 SPSS output for the moving average parameter of the ARMA chart (h). ANOVA table Source of variable
SS
df
MS
F
P-value
Regression Residual
2.11 32.89
1 34
2.11 0.97
2.19
0.1458
Total
35.00
35
Table 7 SPSS output for the autoregressive parameter of the ARMA chart (/). (a) ANOVA table Source of variable
SS
df
MS
F
P-value
Regression Residual Total
7.58 27.42 35.00
2 33 35
3.79 0.83
4.56
0.0178
(b) Table of regression coefficients Independent variable
Coefficient
Std. Error
t-Value
P-value
Constant W a
2.6638 0.0909 0.1465
0.9680 0.0372 0.0825
2.7519 2.4434 1.7753
0.0096 0.0201 0.0851
sample usually leads to a larger value of the autoregressive parameter of the ARMA chart. Table 8 is the SPSS output for the expected total cost per hour (EC). Based on Table 8(b), it is noted that a larger shift magnitude in process mean generally reduces the expected cost. This is probably because a large shift magnitude in process mean can be easily detected by the control chart and consequently can be corrected just in time. Also, frequent occurrence of the assignable cause (i.e., a larger k value) leads to the higher expected total cost. A longer time duration required to discover the assignable cause or to correct the process definitely produces a higher expected total cost. Meanwhile, a higher quality cost per hour while producing in control or out of control generally results in a higher expected total cost, too. 6. Conclusions The major function of control charting is to detect the occurrence of assignable causes so that the necessary corrective action can be taken before a large quantity of nonconforming product is manufactured. Traditionally, when designing control charts, it is usually assumed that the observations from the process at different time points are independent. However, this assumption may not be true in some production processes, e.g., the continuous chemical processes. The presence of autocorrelation in the
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Table 8 SPSS output for the expected cost per hour (EC). (a) ANOVA table Source of variable
SS
df
MS
F
P-value
Regression Residual
32.24 2.76
6 29
5.37 0.095
56.39
1.12 E14
Total
35.00
35
(b) Table of regression coefficients Independent variable
Coefficient
Std. error
t-Value
P-value
Constant d k t1 t2 c0 c1
4.3863 0.1387 40.2667 0.0542 0.0556 0.0365 0.0186
0.3791 0.0487 3.1504 0.0070 0.0070 0.0126 0.0032
11.5688 2.8504 12.7813 7.7532 7.9638 2.8983 5.9082
2.19 E12 0.0080 1.93 E13 1.5 E08 8.78 E09 0.0071 2.05 E06
process data can result in significant effect on the statistical performance of control charts. The ARMA control chart has been shown to be effective for monitoring a process with autocorrelated measurements. Two chart parameters, the autoregressive parameter (/) and the moving average parameter (h), play important roles in the detection ability of the ARMA chart. In the present paper, we develop the economic design of ARMA control chart to determine the optimal values of the five test and chart parameters of the chart (i.e., the sample size, the sampling interval, the control limit coefficient, / and h) such that the expected total cost per hour is minimized. The cost function is established based on the cost model in Lorenzen and Vance (1986) with the simulated average run length. An illustrative numerical example is provided and the GA is employed to search for the solution of the economic design. A sensitivity analysis is then carried out to study the effect of model parameters on the solution of the economic design. Based on the sensitivity analysis, the following important results are observed: (1) The shift magnitude of process mean (d) significantly affects the optimal values of sample size, sampling interval, and the expected cost. A large shift in mean generally increases the sample size and the sampling interval, but reduces the expected cost. This is perhaps due to the fact that a large shift magnitude in process mean may be fast detected by the control chart and consequently the process can be corrected just in time. (2) Frequent occurrence of the assignable cause always leads to the higher expected total cost. (3) If the time required to discover the assignable cause or to correct the process increases, then both the sampling interval and the expected cost also tend to increase. (4) A higher quality cost per hour while producing in control also gives a wider safe region and a higher expected total cost; however, a higher quality cost per hour while producing out of control leads to a tighter safe region and a higher expected total cost.
(5) The fixed cost per sample has a significant influence on the sampling interval, the control limit coefficient, and the autoregressive parameter of the ARMA chart. A higher fixed cost per sample usually results in a longer sampling interval, a wider safe region, and a larger autoregressive parameter of the ARMA chart. (6) A higher cost to locate and correct the assignable cause generally leads to a larger value of the autoregressive parameter of the ARMA chart.
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