Economic design of X̄ control charts for non-normal data using variable sampling policy

ARTICLE IN PRESS

Int. J. Production Economics 92 (2004) 61–74

Economic design of X% control charts for non-normal data using variable sampling policy Yan-Kwang Chen* Department of Business Administration, Ling Tung College, Taichung, Taiwan, ROC Received 27 April 2002; accepted 30 September 2003

Abstract Bai and Lee (Int. J. Prod. Econ. 54 (1998) 57) developed an economic model for a variable sampling interval (VSI) X% control chart. In developing the model they considered the process data with normality property in their design methodology. However, in practice the validity of the normal assumption is doubted in many situations. In this paper, % an alternative cost model that employs the Burr distribution is proposed to deal with the economic design of the VSI Xcharts when the normality assumption of process data is unacceptable. Besides non-normal process data, this model is also suitable for normally distributed data. Consequently, it is able to provide a more general application. The design procedure and sensitivity analysis on the process parameters, cost parameters, and degrees of skewness and kurtosis of population are illustrated through an industrial application. r 2003 Elsevier B.V. All rights reserved. Keywords: Control charts; Variable sampling interval; Non-normality; Economic design; Generalized control limits

1. Introduction The X% chart is an online control tool used to detect the mean shifts of a process. When a X% chart is used to monitor the process mean, three parameters are selected. These parameters include sample size, controllimit width, and sampling rate. Since the work by Duncan (1956), the economical design for control charts has received much attention (for example, Montgomery, 1980; Vance, 1983; Woodall, 1986; Pignatiello and Tsai, 1988). The usual approach to the economic design is to develop a cost model for a particular type of manufacturing process, and then derive the optimal parameters by minimizing the long-run expected cost per hour. Two models have been widely used in practice. One is the model proposed by Duncan (1956), in which the process remains in operation during the search for the assignable cause. Chiu (1975) proposes another popular model, in which the process is shut down during the search for the assignable cause. The sampling scheme in these models considers a fixed sample size and fixed sampling rate, which is referred to in the literature as the fixed-sampling-interval (FSI) scheme (Reynolds and Arnold, 1989).

*Tel.: +886-4-23895624. E-mail address: [email protected] (Y.-K. Chen). 0925-5273/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2003.09.011

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Y.-K. Chen / Int. J. Production Economics 92 (2004) 61–74

Recently, several modifications of using control charts with variable sampling interval (VSI) schemes have been suggested to improve traditional FSI schemes, and have been compared in the quality control literature (Cui and Reynolds, 1988; Reynolds et al., 1988; Reynolds, 1989, 1996; Runger and Pignatiello, 1991; Baxley, 1996). It has revealed that the control charts operating with VSI schemes give better performance than with FSI schemes in the sense of quick response to process change. Although there is extensive literature focused on VSI control chart, very little work has been done on the economic design of the VSI control chart. Bai and Lee (1998) proposed an economic model of VSI X% control chart, which basically followed Chiu’s model. Traditionally, when the issue on designing control chart is discussed, one usually assumes the measurements in each sampled subgroup (or say population) are normally distributed; therefore, the sample mean X% is also normally distributed. However, the assumption may not be acceptable in practice. For example, the distributions of measurements from chemical processes, semiconductor processes, or cutting tool wear process are often skewed (Chang and Bai, 2001). Surely, if the size of subgroup (sample size) is large enough, the statistic X% will be distributed normally according to the central limit theorem. However, this is often expensive. The non-normal behavior of measurements may imply that the traditional design approach is improper for the operation of control charts. Face to the impropriety, Rahim (1985) presented an economic model of symmetric FSI X% charts under non-normality assumption. Likewise, Chou et al. (2000) used the Burr distribution to construct an economic-statistical model for symmetric FSI X% control charts. Yourstone and Zimmer (1992) pointed out that the effect on the error probabilities could be large when the assumption of normality is not valid, and offered a statistical design of asymmetric FSI X% charts with the Burr distribution to represent various non-normal distributions. In this paper, an economic model of the VSI chart operating with generalized (asymmetric or symmetric) control limits for non-normal process data is developed using the Burr distribution. This model is an extension of the work by Bai and Lee (1998) and Chou et al. (2000). Via the genetic algorithms (GAs) searching technique, the optimal design parameters of this model can be found. In the next section the VSI control chart with generalized control limits will be sketched. Then, the Burr distribution representing various types of non-normal distributions is introduced in Section 3. The economic model based on the Burr distribution is constructed in Section 4, and the GAs based solution procedure is illustrated in Section 5. An industrial example will be presented to illustrate the solution procedure of economic design of an asymmetric VSI chart; the sensitivity analysis on process parameters, cost parameters, and the degrees of skewness and kurtosis in the individual data values are made in Section 6. Finally, concluding remarks make up the last section. 2. The VSI X% control chart Assume that the distribution of measurements from a process is non-normally distributed, and has the mean m and deviation s: When an X% controlpchart (with centerline m0 ; the upper control limit pffiffistandard ffi ffiffiffi 0 m0 þ k1 ðs= nÞ; and the lower control limit m0  k10 ðs= nÞ; where k1 and k1 are not necessarily equal) is used for monitoring the process, a sample of size n is taken and calculated its sample mean at each sampling point to judge whether or not the process remains in-control state. If the sample mean X% i plotted on the X% control chart goes beyond the control limits, then a signal will be given to inform the operator to search for the assignable cause. Otherwise, the process is considered being in-control, and the next sample is continually taken at next sampling point. In this situation, the control chart operates with fixed sampling interval (say h1 ) regardless of X% i which is said to be the FSI control chart. For VSI X% control charts, if the sample mean falls inside the control limits, the monitored process is also considered stable as FSI chart. However, with a difference to the FSI charts, the next sampling interval will be a function of this sample mean. That is, the next sampling rate depends on the current sample mean.

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Assume the VSI X% chart uses a finite number of sampling interval lengths, say h1 ; h2 ; y; hm ; where h1 oh2 o?ohm ; and mX2: The choice of a sampling interval can be made by a function hðxÞ of X% i when the value x of X% i is measured. If one partitions the region between the two control limits into I1 ; I2 ; y; Im ; the function hðxÞ can be defined as follows: hðxÞ ¼ hj ;

ð1Þ

xAIj :

Fig. 1 shows an example of a VSI chart using two interval lengths h1 and h2 ; where " ! !# s s 0 s 0 s I1 ¼ m0  k1 pffiffiffi; m0  k2 pffiffiffi , m0 þ k2 pffiffiffi; m0 þ k1 pffiffiffi ; n n n n ! s s I2 ¼ m0  k20 pffiffiffi; m0 þ k2 pffiffiffi ; n n

ð2Þ

where 0ok2 ok1 ; 0ok20 ok10 ; and h2 > h1 > 0: Traditional symmetric VSI charts can be easily obtained by setting k1 ¼ k10 : The sample means in the chart in Fig. 1 are plotted against the time on the horizontal axis. The first sample mean falls within I1 ; which is near control limits, so the next sampling interval h1 is shorter. So does the second one. However, the third one falls within I2 ; which is near the centerline, so a usual sampling interval is adopted to take the fourth sample. The way the VSI X% control charts work can improve the detection ability of FSI charts by shorten the length of time to give a signal. However, the complexity will increase when the number of partitioned regions becomes large.

3. The Burr distribution In this paper, we demonstrate the application of the generalized Burr distribution for the economic design of the VSI chart in monitoring non-normal process data. Before illustrating the economic model, we shall review the Burr distribution, which is discussed in detail by Burr (1942) to represent various types of non-normal distribution. The cumulative distribution of Burr distribution is 8 1 <1  for yX0; ð1 þ yc Þ r F ðyÞ ¼ : 0 for yo0;

Fig. 1. The asymmetrical VSI control chart when m ¼ 2:

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which can be rewritten as F ðyÞ ¼ 1 

1 ; ð1 þ maxf0; ygc Þr

ð3Þ

where c and r are greater than one. The first four moments of an empirical distribution may be used to determine values of c and r: These c and r then specify a member of the family of the Burr distributions, which approximates the observed empirical distribution. The family of Burr distributions covers a wide range of standardized third and fourth moments and can be used to fit a wide variety of practical data distributions like gamma, beta, normal distributions and so on. For example, for c ¼ 4:8621 and r ¼ 6:3412; it approximates a normal distribution. Two distributions with the same first moments will not necessarily be identical, but it is reasonable to assume that they will not differ in practicality (Yourstone and Zimmer, 1992). Burr (1942) tabulated the first two moments (Table 2) and the coefficients of skewness and kurtosis (Table 3) for the family of Burr distributions. These tables allow the user to make a standardized transformation between a Burr variable (say Y ) and any random variable (say X ) if X and Y have the same coefficients of skewness and kurtosis. For a set of data, once the skewness and kurtosis coefficients of this data set are estimated, then resting on the coefficients the mean M and standard deviation S of the corresponding Burr distribution can be found in Table 3 given by Burr (1942). Then, the standardized transformation between Burr variable Y and variable X can be simply expressed as X  X% Y  M ; ¼ sX S

ð4Þ

where X% and sX are the sample mean and standard deviation of this data set, respectively. For example, assume a data set is collected with sample statistics: the mean X% ¼ 10; standard deviation sX ¼ 2; skewness a# 3 ¼ 0:18; and kurtosis a# 4 ¼ 3:05: Then from Table 3, the Burr distribution with c ¼ 4 and r ¼ 6 approximates to this data set. Moreover, from Table 2 this Burr distribution has a mean of M ¼ 0:5951 and a standard deviation of S ¼ 0:1801: Finally, via the simple expression of (4), the relationship between Y and X can be expressed as X  10 Y  0:5951 ¼ ; 2 0:1801 which can be rewritten as X ¼ 11:105Y þ 3:391: Using the Burr distribution to approximate the distribution of sample mean, the economic model proposed in this paper is generally applicable to the process data whose sample means are not normally distributed.

4. Development of cost model In the cost model a process is assumed in an in-control state (m ¼ m0 ) initially. The process will be disturbed by a single assignable cause that causes a fixed shift in the process ðm ¼ m0 þ dsÞ: The inter-arrival time of the assignable cause disturbing the process is assumed following an exponential distribution with a mean of 1=l hours. A sample of size n is taken at each sampling point. If a sample mean falls inside the two control limits, its value will be used to decide the next sampling interval, so the sampling interval is changeable. If it goes beyond the two control limits, the process is stopped and a search starts to find the assignable cause and adjust the process.

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Let q0 and q1 be, respectively, the conditional probabilities that any sample mean X% falls outside the control limits, given that m ¼ m0 and m ¼ m0 þ ds: The conditional probabilities of q0 and q1 are also called the false alarm rate and the (failure-detection) power of a control chart, respectively. Moreover, p0j is the conditional probability that X% falls within Ij ; given that X% falls inside the two control limits when m ¼ m0 ; and p1j is the corresponding conditional probability when m ¼ m0 þ ds: If UCL and LCL are defined as the upper and lower control limits of an X% control chart: s UCL ¼ m0 þ k1 pffiffiffi ; n 0 s LCL ¼ m0  k1 pffiffiffi ; n

ð5Þ

then q0 can be represented as % q0 ¼ 1  PrðLCLoXoUCLjm ¼ m0 Þ % % ¼ PrðX > UCLjm ¼ m0 Þ þ PrðXoLCLjm ¼ m0 Þ that, replacing UCL and LCL by (5), yields ! ! X%  m0 X%  m0 pffiffiffi > k1 jm ¼ m0 þ Pr pffiffiffi o  k1 jm ¼ m0 : q0 ¼ Pr s= n s= n For the sample size of n; the values for the skewness and kurtosis of the sample means are a# 3 a# 3X% ¼ pffiffiffi n

ð6Þ

and a# 4X% ¼

a# 4  3 þ 3; n

ð7Þ

where a# 3 and a# 4 represent the estimated skewness and kurtosis coefficients of the population. Using the values of a# 3X% and a# 4X% as well as tables in Burr (1942), one can determine c; r; M; S for the distribution with the values closest to a# 3X% and a# 4X% by interpolation, which was interpreted in Burr (1942). Then, according to the standardized transformation of (4) and (3), we have



 Y M Y M > k1 þ Pr o  k1 q0 ¼ Pr S S ¼ PrðY > M þ Sk1 Þ þ PrðY oM  Sk10 Þ 1 1  : ¼1 þ ½1 þ maxf0; M þ Sk1 gc r ½1 þ maxf0; M  Sk10 gc r Likewise, q1 can be represented as % q1 ¼ 1  PrðLCLoXoUCLjm ¼ m0 þ dsÞ: Replacing UCL and LCL of (5) in that, we get q1 ¼ 1  Pr

k10

! pffiffiffi X%  ðm0 þ dsÞ pffiffiffi pffiffiffi  d no ok1  d n : s= n

ð8Þ

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Again, using the standardized transformation of (4) and (3),

 pffiffiffi Y  M pffiffiffi ok1  d n q1 ¼ 1  Pr k10  d no S pffiffiffi pffiffiffi 0 ¼ 1  PrðM  Sk1  Sd noY oM þ Sk1  Sd nÞ 1 1 pffiffiffi c r  pffiffiffi : ¼1 þ 0 ½1 þ maxf0; M þ Sk1  Sd ng  ½1 þ maxf0; M  Sk1  Sd ngc r

ð9Þ

Similarly, by formula of (4) and (3) the conditional probabilities of p0j and p1j are given by ! s s % % p0j ¼ Pr m0 þ kjþ1 pffiffiffioXom 0 þ kj pffiffiffi m ¼ m0 ; LCLoXoUCL n n ! s 0 s 0 % % þ Pr m0  kj pffiffiffioXom 0  kjþ1 pffiffiffi m ¼ m0 ; LCLoXoUCL n n

1 1 ¼  ½1 þ maxf0; M þ Skjþ1 gc r ½1 þ maxf0; M þ Skj gc r ! 1 1 þ  =ð1  q0 Þ 0 0 ½1 þ maxf0; M  Skj gc r ½1 þ maxf0; M  Skjþ1 gc r for j ¼ 1; 2; y; m  1;

ð10Þ !

s s 0 % % pffiffiffioXom p0m ¼ Pr m0  km 0 þ km pffiffiffijm ¼ m0 ; LCLoXoUCL n n

 1 1 ¼  =ð1  q0 Þ; 0 gc r ½1 þ maxf0; M  Skm ½1 þ maxf0; M þ Skm gc r ! s s % % p1j ¼ Pr m0 þ kjþ1 pffiffiffioXom 0 þ kj pffiffiffi m ¼ m0 þ ds; LCLoXoUCL n n

ð11Þ

! s s 0 % % þ Pr m0  kj0 pffiffiffioXom 0  kjþ1 pffiffiffi m ¼ m0 þ ds; LCLoXoUCL n n

¼

1 1 pffiffiffi c r  pffiffiffi ½1 þ maxf0; M þ Skjþ1  Sd ng  ½1 þ maxf0; M þ Skj  Sd ngc r

! 1 1 pffiffiffi pffiffiffi þ  =ð1  q1 Þ 0  Sd ngc r ½1 þ maxf0; M  Skj0  Sd ngc r ½1 þ maxf0; M  Skjþ1 for j ¼ 1; 2; y; m  1;

ð12Þ !

s s % % p1m ¼ Pr m0  km pffiffiffioXom 0 þ km pffiffiffi m ¼ m0 þ ds; LCLoXoUCL n n

! 1 1 pffiffiffi pffiffiffi ¼  =ð1  q1 Þ: 0  Sd ngc r ½1 þ maxf0; M  Skm ½1 þ maxf0; M þ Skm  Sd ngc r

ð13Þ

The economic design of VSI X% control charts is carried out by specifying a cost model, and searching the optimal design parameters for minimizing the cost model over a production cycle. The production cycle

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Cycle starts

67

Cycle ends

Process mean shift

Last sample before process mean shift

First sample after process mean shift

Out-of-control detected

Assignable cause detected

Assignable cause repaired

Y A (T1) In control period + (T3) Searching period due to false alarm

(T2) Out-of-control period, O

(T4) Time period for identifying and correcting assignable cause

Fig. 2. Production cycle considered in the cost model.

length is defined as the average time from the start of production until the first signal after the process shift. Once the expected cycle length is determined, the cost over the production cycle can be converted to an index—long run expected cost per hour (Ross, 1970). The optimal values of the design parameters based on the cost model can be determined by some optimization technique such as the grid search, nonlinear programming, or genetic algorithm. Fig. 2 depicts the production cycle, which is divided into four time intervals of in-control period, out-ofcontrol period, searching period due to false alarm, and the time period for identifying and correcting the assignable cause. Individuals are now illustrated before they are grouped together. (T1) The expected length of in-control period is 1=l: (T2) Let O be the out-of-control period, A be the length of the sampling interval in which an assignable cause occurs, and Y be the time interval between sample point just prior the occurrence of assignable cause and the occurrence itself. Reynolds et al. (1988) showed that m X EðOÞ ¼ EðAÞ  EðY Þ þ ðS1  1Þ hj p1j ; ð14Þ j¼1

where S1 represents the expected number of sample required to detect the assignable cause, which is a geometric random variable with parameter q1 ; so s1 ¼ ð1=q1 Þ: Reynolds et al. (1988) also assumed that the probability of the length of A being hj is m X PrðA ¼ hj Þ ¼ hj poj = hj poj ; ð15Þ j¼1

then from the result of Duncan (1956), the conditional expected value of Y ; given A ¼ hj is EðY jA ¼ hj Þ ¼

1  ð1 þ lhj Þelhj : lð1  elhj Þ

ð16Þ

Therefore, the expected length of the out-of-control period is obtained by using formulas (14)–(16): m m X X hj p0j ðlhj  1 þ elhj Þ 1  1Þ hj p1j : ð17Þ EðOÞ ¼ Pm þ ðS 1 ð j¼1 hj p0j Þ j¼1 lð1  elhj Þ j¼1

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(T3) Let S0 be the expected number of samples in the in-control state. Bai and Lee (1998) showed that Pm lhj m X j¼1 p0j e p0j ð1  elhj Þ: ð18Þ S0 ¼ Pm ð1  j¼1 p0j elhj Þ2 j¼1 Then, the expected searching period due to false alarm is ð19Þ

t1 q0 S0 ;

where q0 as aforementioned is the false alarm rate, and an average of t1 hours is spent if the signal is a false alarm. (T4) The time to identify and correct the assignable cause following an action signal is a constant t2 : Thus, the expected length of a production cycle, denoted by ET, can be aggregately represented as 1 ð20Þ ET ¼ þ EðOÞ þ t1 q0 S0 þ t2 : l If one defines c1 as the average search cost if the given signal is false, c2 the average cost to discover the assignable cause and adjust the process to in-control state, c3 the hourly cost associated with production in out-of-control state, c4 the fixed sampling cost for each sample and c5 the variable testing cost of sampling and testing for each sample; the expected cost of a production cycle, denoted by EC, that includes (a) the cost of false alarms; (b) the cost of detecting and eliminating the assignable cause; (c) the cost associated with production in the out-of-control condition; and (d) the cost of sampling and testing can be represented as the following: EC ¼ c1 ðq0 S0 Þ þ c2 þ c3 EðOÞ þ ðc4 þ c5 nÞðS0 þ S1 Þ:

ð21Þ

The economic design of a VSI X% control chart with generalized control limits is to determine the 0 appropriate values of design parameters ðn; k1 ; y; km ; k10 ; y; km ; h1 ; y; hm Þ such that the expected cost per hour, denoted by ECT, ECT ¼ EC=ET

ð22Þ

is minimized. Note that ECT is a function of the design parameters with its parameters: the process parameters ðM; S; c; r; l; d; t1 ; t2 Þ and the cost parameters ðc1 ; c2 ; y; c5 Þ: Since it is very cumbersome in solving the cost model to find the optimal design parameters— ðn ; k ; y; k ; k0 ; y; k0 ; h ; y; h Þ; the author will solve it based on the genetic algorithms, instead of 1

m

1

m

1

m

the conventional optimization method. The genetic solving algorithm will be briefly introduced in the next section.

5. Solution procedure In this paper, the procedure of finding the optimal design parameters for setting up asymmetric VSI chart has the same elements as that for setting up usual charts. That is, the optimal design parameters  ; k0 ; y; k0 ; h ; y; h Þ are derived by minimizing the ECT. However, the parameters M; S; ðn ; k1 ; y; km 1 m 1 m c; and r of ECT, out of the ordinary, are dependent on the sample size n: So, the author uses a two-stage optimization approach to find the optimal design parameters. Firstly, we find optimal values of 0 ðk1 ; y; km ; k10 ; y; km ; h1 ; y; hm Þ for a given n: In doing so, we apply the genetic algorithms. Genetic algorithms (GAs) (Davis, 1991; Goldberg, 1989) are global search and optimization techniques motivated by the process of natural selection in biological system. GAs are being applied to many engineering areas in industrial engineering, mechanical engineering, electronical engineering, aerospace engineering, architecture and civil engineering, etc. In recent years, several successful applications in industrial engineering have appeared in the literature, for example, job-shop scheduling problem (Kim et al.,

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2003; Pongcharoen et al., 2002), facility layout problem (Lee et al., 2003), vehicle routing problem (Baker and Ayechew, 2003), computer aided design (Renner, 2003), and optimization of production planning and control systems (Disney et al., 2000). GAs are different from other search procedures in the following ways (Karr and Gentry, 1993): (1) GAs consider many points in the search space simultaneously, rather than a single point; (2) GAs work directly with strings of characters representing the parameter set, not the parameters themselves; (3) GAs use probabilistic rules to guide their search, not deterministic rules. Because GAs consider many points in the search space simultaneously there is a reduced chance of converging to local optima. In a conventional search, based on a decision rule, a single point is considered and that is unreliable in multimodal space. The primary distinguishing features of GAs are an encoding, a fitness function, a selection mechanism, a crossover mechanism, a mutation mechanism, and a culling mechanism. The algorithm of GAs can be formulated as the following steps: (1) Randomly generate an initial solution set (population) of N individuals and evaluate each solution (individual) by fitness function. Usually, an individual is represented as a numerical string. (2) If the termination condition is not met, repeatedly do {Select parents from population for crossover. Generate offspring. Mutate some of the numbers. Merge mutants and offspring into population. Cull some members of the population.} (3) Stop and return the best fitted solution. When applying GAs to the first stage of solution procedure, a decimal encoding of individuals is adopted so that each individual in the form of decimal string represents a possible solution 0 ðk1 ; y; km ; k10 ; y; km ; h1 ; y; hm Þ: The fitness value of each individual is evaluated by the expected cost per hour ECT. Based on the ‘‘elitist’’ strategy of above algorithm, that is, the survival of the fittest, the evolution of a population of N individuals has been pursued. The termination condition is achieved when the number of generations is large enough or a satisfied fitness value is obtained. In the second stage of the solution procedure, we compare ECTs of potential values of n; and find n with  ; k0 ; y; k0 ; h ; y; h Þ: As the quotation cited in Yourstone and its corresponding parameters ðk1 ; y; km 1 m 1 m Zimmer (1992) from Shewhart, sample size in statistic process control methods tends to be small so that sample mean (or sample average) do not mask the process changes. Accordingly, the work of two-stage optimization approach is applicable in practice.

6. An industrial example To illustrate the solution procedure of the proposed model, the following industrial example whose process and cost parameters borrows directly from Chiu and Cheung (1977) is presented. 6.1. An industrial example A manufacturer produces a part. According to the previous runs the average time for occurrence of the process shift is about 100 hours (1=l ¼ 100). The manufacturer uses an X% control chart to monitor some vital quality characteristic of the part. Based on the analysis of operators and quality control engineers and the cost of testing equipment, it is determined that the fixed cost of taking a sample is $0.5 (i.e. c4 ¼ 0:5), while the variable cost is $0.1 per bottle (i.e. c5 ¼ 0:1). On the average, when the process goes to out-ofcontrol, the magnitude of the mean shift is about one standard deviation (i.e. d ¼ 1). The average time to

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Table 1 The optimal design parameters for FSI and VSI charts

FSI chart VSI chart (m ¼ 2) VSI chart (m ¼ 3)

n

k1

k2

13 8 8

2.29 2.99 2.72

0.98 1.07

k3

k10

k20

0.62

2.29 2.99 2.72

2.99 2.72

k30

h1

h2

2.72

1.85 0.01 0.00

1.66 1.27

h3

ECT

%

1.76

2.475 2.195 2.173

11.31 12.17

investigate an out-of-control signal that results in the elimination of an assignable cause is 0.3 hours (i.e. t2 ¼ 0:3), while the time spent to investigate a false alarm is 0.1 hours (i.e. t1 ¼ 0:1). An average search cost of $10 (i.e. c1 ¼ 10) is spent if the assignable cause does not exist. If the assignable cause exists, it takes an average of $30 (i.e. c2 ¼ 30) to discover it and correct the process to in-control state. The estimated cost associated with the production in the out-of-control period is $100 per hour (i.e. c3 ¼ 100). According to the collected data, the degrees of the skewness and kurtosis in the individual measurement are approximately 0.484 and 3.38, respectively (i.e. a# 3 ¼ 0:484 and a# 4 ¼ 3:380), which may be fitted by the Burr distribution with c ¼ 3 and r ¼ 6: The manufacturer wishes to design a VSI X% control chart to shorten the time responding to the process shift. However, he/she knows that, the degrees of the skewness and kurtosis in the distribution of sample means depend on the sample size (the relationship can be observed from Eqs. (6) and (7)). Therefore, the two-stage optimization approach in Section 4 is applied. In first stage, an optimization package (EVOLVER 4.0) is coded to minimize the cost function (22) for each sample size n (ranged from 1 to 30). The following setting of control parameters for the package manipulation has been employed: population size N ¼ 50; crossover probability=0.5; mutation rate=0.2; the number of generation=100,000. In stage 2, the ECTs for all sample size n is compared to determine the optimal sample size n and its corresponding design parameters. Table 1 shows the optimal design parameters of the VSI X% charts with two and three sampling interval lengths as well as the traditional X% charts. From Table 1, it is observed that the optimal VSI charts still have symmetric control limits. Besides, when a sample mean falls into their the lower side, a fixed sampling interval length of about 1.7 hours is suggested regardless of the value of the sample mean. On the contrary, variable sampling interval lengths are used for their upper side. Hereafter, this type of VSI charts is denoted as the half-VSI chart for convenience. In comparison with the traditional (FSI) chart, VSI control charts require to sample more often with larger control limits and smaller sample size. This result is identical to Bai and Lee (1998). Moreover, the ECTs of the VSI charts with m ¼ 2 and m ¼ 3 are nearly the same, and save the averaged cost about 11% in contrast with FSI chart. This percentages of cost reduction by VSI schemes in this study (for non-normal process data) is more significant than the percentage of 7.26% in Bai and Lee (1998), where the normal process data was considered. Even if the performances of VSI charts with m ¼ 2 and m ¼ 3 are near the same, in practice the former may be superior to the latter due to its lower complexity. Also, the result coincides with the recommendation in Reynolds and Arnold (1989). 6.2. Sensitivity analysis In the following, the effects of model parameters (process and cost parameters), non-normality, and statistical constraints on the solution of the aforementioned example will be discussed. Table 2 shows the effect of model parameters concerning the optimal design parameters for FSI and VSI X% control charts. Chiu and Cheung (1977) indicated that these model parameters in the 15 sets were

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Table 2 Effect of process and cost parameters on the optimal design of asymmetrical FSI and VSI X% charts for non-normal process data Process and cost parameters

VSI (m ¼ 2)

FSI

d

t1

t2

c1 c2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

2 2 2 2 2 2 2 2 2 2 1 1 1 0.5 0.5

0.1 0.1 0.1 0.1 0.1 0.5 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

0.3 0.3 0.3 0.3 0.3 1.5 2.1 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3

10 30 100 0.5 0.1 5 2.76 2.76 1.40 10 30 100 0.5 0.1 5 2.66 2.66 0.65 10 30 10 0.5 0.1 4 2.47 2.47 4.42 10 30 1000 0.5 0.1 5 2.82 2.82 0.44 25 75 100 0.5 0.1 5 2.29 2.84 1.44 10 30 100 0.5 0.1 5 2.76 2.76 1.39 10 210 100 0.5 0.1 5 2.76 2.76 1.42 1 3 100 0.5 0.1 3 2.02 2.02 1.23 10 30 100 5.0 0.1 6 2.62 2.62 3.42 10 30 100 0.5 1.0 3 2.17 2.17 2.51 10 30 100 0.5 0.1 13 2.29 2.29 1.85 10 30 20 0.5 0.1 13 2.37 2.37 4.11 10 30 100 0.5 0.1 12 2.22 2.22 0.86 10 30 100 0.5 0.1 30 1.86 1.86 2.40 10 30 5 0.5 0.1 27 1.68 1.68 11.83

0.01 0.05 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.05 0.01 0.01

c3

c4

c5

n

k1

k10

No. l

h

ECT n

k1

1.795 3 3.35 4.745 3 3.14 0.763 3 3.13 5.052 3 3.36 2.291 3 3.57 1.771 3 3.24 3.513 3 3.23 1.403 3 2.97 3.655 5 3.07 3.275 1 2.61 2.475 8 2.99 1.265 8 2.90 6.211 8 2.99 3.717 24 2.99 1.007 23 2.53

k2

k10

k20

h1

1.41 1.45 1.46 1.41 1.37 1.43 1.43 1.46 2.14 0.27 0.98 1.00 0.99 0.64 0.78

3.35 3.14 3.13 3.04 3.28 3.24 3.20 2.97 3.07 2.61 2.99 2.90 2.99 2.99 2.53

3.35 3.14 3.13 3.04 3.28 3.24 3.20 2.97 3.07 2.61 2.99 2.90 2.99 2.99 2.53

0.00 1.32 0.00 0.60 0.00 4.27 0.00 0.42 0.00 1.33 0.01 1.31 0.01 1.33 0.02 1.31 0.01 3.38 0.01 2.08 0.01 1.66 0.01 3.83 0.01 0.72 0.01 2.58 0.00 11.98

h2

ECT % 1.659 7.58 4.472 5.75 0.724 5.11 4.593 9.09 2.118 7.55 1.642 7.28 3.382 3.73 1.376 1.92 3.640 0.41 2.785 14.96 2.195 11.31 1.142 9.72 5.962 4.01 3.454 7.08 0.959 4.77

Fitting by the Burr distribution (c ¼ 3; r ¼ 6).

modified from Duncan (1956) and would provide fairly general variation. Several findings as illustrated in Table 2 are spelled out as follows. (1) The control charts with half-VSI schemes consistently have lower expected cost per hour than the charts with FSI scheme over the 15 sets. (2) For most cases (12 out of 15 sets), making use of the symmetric VSI charts seem more economical. Note that asymmetric chart limits only for large c2 or c3 : (3) Percent reduction in ECT is reduced when excessively large or small mean shifts (d ¼ 2:0 or 0.5) is encountered. (4) Percent reduction in ECT is significantly raised when c4 or l is small, or when one of the c3 or c5 is large. (5) The other parameters such as t1 and t2 perhaps are not sensitive factors to affect the percent reduction in ECT. Table 3 continues the above example and lists the economic design of X% control charts for various skewness and kurtosis coefficients of the population. To investigate the effect of non-normality on the optimal design parameters of the control charts, six groups of non-normally distributed populations following Chou et al. (2000) with little modification are illustrated in Table 3. In group I, III, and IV, the value of a# 4 is approximately fixed and a# 3 increases gradually. Oppositely, the values of a# 3 in group II and IV are close to zero and one individually when a# 4 increases. Finally, excessively large values of a# 3 and a# 4 are considered in group VI for possible non-normal populations. Several findings from the table are spelled out as follows. (1) Asymmetric VSI control charts are suggested when the population is negatively skewed or when the values of skewness and kurtosis coefficients are excessively large. Otherwise, symmetric and half-VSI charts are recommend. (2) When applying the VSI chart for negatively skewed population the control limits are wider relative to positively skewed population.

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Table 3 Effect of skewness and kurtosis of non-normal process data on the optimal design of the FSI and VSI X% charts VSI (m ¼ 2)

FSI c I

II

a# 3

r

6 11 0.254 6 6 0.147 5 5 0.04 4 7 0.136 3 11 0.329 4 11 5 5 10 2

0.050 0.04 0.044

a# 4 3.027 3.065 3.070 2.979 3.006

Note

n

k1

k10

h1

ECT n k1

k2

k10

k20

h1

h2

ECT %

5 10 15 10 13

2.04 1.89 2.09 2.16 2.38

3.77 4.72 2.09 2.16 2.38

1.35 1.70 1.67 1.63 1.80

2.127 2.355 2.589 2.495 2.486

1.46 1.00 1.00 1.00 0.96

3.80 4.33 2.95 2.95 3.01

3.79 4.26 2.95 2.95 3.01

0.00 0.00 0.01 0.00 0.00

1.45 1.60 1.66 1.67 1.69

1.928 9.36 a# 3 : from to + 2.165 8.07 a# 4 : close to normal 2.205 14.83 2.196 11.98 2.195 11.71

5 7 8 8 8

2.58 2.45 2.95 2.95 3.01

2.866 15 2.39 2.39 1.96 2.507 8 2.99 1.01 2.99 2.99 0.00 1.66 2.202 12.17 a# 3 : close to normal 3.070 15 2.09 2.09 1.67 2.589 8 2.95 1.00 2.95 2.95 0.01 1.66 2.205 14.83 a# 4 : increasing 3.646 12 2.05 2.05 1.85 2.496 9 2.95 0.93 2.95 2.95 0.00 1.70 2.175 12.86

III 10 10 0.519 10 3 0.208 3 6 0.484

3.462 9 1.86 4.98 1.59 2.332 7 2.39 1.06 4.99 4.90 0.00 1.56 2.158 7.46 a# 3 : from to + 3.418 10 1.89 4.44 1.71 2.359 7 2.44 1.01 4.86 4.63 0.00 1.61 2.163 8.31 a# 4 : near a constant 3.4 3.380 13 2.29 2.29 1.85 2.475 8 2.99 0.98 2.99 2.99 0.01 1.66 2.195 11.31

IV

6 5 2

2 2 7

0.434 0.635 1.014

4.106 13 2.24 2.24 1.88 2.473 8 2.95 0.98 2.95 2.95 0.00 1.71 2.188 11.52 a# 3 : increasing 4.630 13 2.33 2.33 1.84 2.477 8 2.95 0.96 2.95 2.95 0.00 1.71 2.182 11.91 a# 4 : near a constant and >4 4.707 13 2.27 2.27 1.86 2.457 7 2.95 0.83 2.95 2.95 0.00 1.69 2.164 11.93

V

2 2

8 6

0.958 1.094

4.443 13 2.18 2.18 1.91 2.456 7 2.89 0.85 2.89 2.89 0.00 1.61 2.166 11.81 a# 3 : close to 1 5.118 12 2.23 2.23 1.81 2.453 7 2.82 0.83 2.82 2.82 0.00 1.72 2.158 12.03 a# 4 : increasing

VI

2 2

4 3

1.432 7.356 12 2.10 2.10 1.89 2.440 7 2.77 0.83 2.72 2.62 0.00 1.72 2.145 12.09 a# 3 : b0 1.909 12.46 11 2.07 2.07 1.81 2.428 8 2.86 0.99 2.82 2.72 0.00 1.76 2.141 11.82 a# 4 : b3

Fixed process and cost parameters: l ¼ 0:01; d ¼ 1; t1 ¼ 0:1; t2 ¼ 0:3; c1 ¼ 10; c2 ¼ 30; c3 ¼ 100; c4 ¼ 0:5; c5 ¼ 0:1:

(3) No matter what the value of skewness coefficient is observed, the VSI or half-VSI chart has wider control limits and requires smaller sample size than traditional chart. But, sampling rate and percentage reduction in ECT are dependent on the value of skewness coefficient. The VSI chart for negatively skewed population seems to have smaller percentage reduction in ECT. (4) The optimal design parameters except sample size for both VSI and FSI schemes may be robust to the value of a# 4 : (5) When a process data is normally distributed (#a3 ¼ 0 and a# 4 ¼ 3), the Burr distribution with ðc; rÞ ¼ ð4:8621; 6:3412Þ will be used to approximate the normal distribution and conduct the economic design of symmetric VSI charts. The optimal solution of ðn; k1 ; k2 ; h1 ; h2 ; ECTÞ is (9, 2.69, 1.45, 0.00, 1.70, 2.311), which is very similar to the optimal solution (9, 2.70, 1.46, 0.00, 1.70, 2.310) found by Bai and Lee’s model that subject to the normality assumption.

7. Concluding remarks An economic design of VSI X% control charts has been presented when the distribution of sample means can not be assumed to be a normal distribution. For non-normal process data (population), the distribution of sample means is usually also non-normal especially for small sample size. Thus, the Burr distribution is used in this study to approximate the distribution of sample means. In designing control chart for nonnormal process data, control charts with asymmetric control limits are taken into account, and compared

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with traditional symmetric control charts from the economic point of view. An example is illustrated to present the solution procedure. According to the sensitivity analysis of this example, some important conclusions may be spelled out as follows. Firstly, from the 15 sets of Chiu and Cheung, the average percent reduction in ECT by VSI scheme for non-normal process data is about 6.69%, which is more significant than that for normal process data. Secondly, the improvements of ECT by VSI charts are sensitive to model parameters. Especially when the fixed sampling cost is low, the length of in-control period is large, or when the population is positively skewed, it become evident. Thirdly, the skewness of a non-normal population gives a more manifest effect on the optimal design of X% control charts, rather than the kurtosis. It is recommended that positively skewed process data require to be monitored by the half-VSI control chart with symmetric control limits. As for the process data whose distribution is negatively skewed or with immoderately large degree of positive skewness coefficient, the VSI chart with asymmetric control limits may be a cost-effective online tool. Additionally, the proposed model is also suitable when the process data are (near) normal, so it may provide a wider range of practical application than Bai and Lee’s cost model. Fourthly, when applying the genetic algorithm in the first stage of the solution procedure, it is recommended that at least 20,000 trials (the number of generation for termination condition) is used for FSI schemes while 100,000 trials for VSI schemes. Moreover, it would be interesting to conduct the research on the optimal design of VSI X% control charts for monitoring non-normal process data under the consideration of multiple assignable causes.

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