Economic statistical design for x-bar control charts under non-normal distributed data with Weibull in-control time

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1 Journal of the Operational Research Society (2011) 62, Operational Research Society Ltd. All rights reserved /11 Economic statistical design for x-bar control charts under non-normal distributed data with Weibull in-control time FL Chen and CH Yeh National Tsing Hua University, Hsinchu, Taiwan, ROC This paper proposes an approach which simultaneously considers the properties of cost and quality based on the Burr distribution to determine three parameters (including sample size, sampling interval between successive samples, and the control limits) when an x-bar chart monitors a manufacturing process with Weibull failure characteristic and non-normal data. Also, the cost model of Banerjee and Rahim (1988) is used as the objective function, and the probability density function of the Burr distribution is applied to derive the statistical constraints of economic statistical design of the x-bar control charts for non-normal data. The example of Banerjee and Rahim (1988) is adopted to indicate the solution procedure and sensitivity analyses. Meanwhile, the design parameters of the x-bar control charts can be obtained through the grid search method. The results show that an increase of skewness coefficient (α 3 ) results in a slight decrease for sample size (n), but is robust to the control limit width (L). Also, an increase of kurtosis coefficient (α 4 ) leads to a wider control limit width. Journal of the Operational Research Society (2011) 62, doi: /jors Published online 12 May 2010 Keywords: control charts; variable sampling; non-normal distribution; economic design; Burr distribution 1. Introduction Statistical approaches have been recently used to solve the industrial process control problems. The control chart technique, first proposed by Dr Shewhart in 1924, is particularly a well-known statistical method, and is widely applied to monitor the variance of a manufacture process. Furthermore, Shewhart proposed a so-called heuristic design approach for determining the values of three parameters of a control chart in 1939 (Chou et al, 2000). However, this approach may lack the performance that a manager desires to find process shifts promptly, correctly and economically (Zhang and Berardi, 1997). Another decision rule of the three-parameter values is called statistical design. Statistically designed control charts are those in which control limits (which determine the Type I error probability, α) and power are previously selected at the desired levels by practitioners. These rules determine the sample size and, if the average time to signal is specified, the sample interval (Woodall, 1985). However, this approach does not take into account any cost objective when these parameters of control charts are determined under designing a control chart. An alternative method of designing control charts is called economic design. This approach, first completely presented Correspondence: FL Chen, Department of Industrial Engineering and Engineering Management, National Tsing Hua University, Hsinchu, Taiwan, ROC. by Duncan in 1956, is applied to x-bar control charts to control the average value of a manufacturing process. Meanwhile, Duncan also proposed the cost model, including the costs of searching for an assignable cause when an out-ofcontrol condition when occurs, and the cost of sampling, inspection, evaluation and plotting (Chou et al, 2000). Since then, many studies have subsequently been conducted for the optimal economic design of the three control charts parameters based on Duncan s (1956) model (eg, Goel and Wu, 1968; Knappenberger and Grandage, 1969; Gibra, 1971). Furthermore, there are also some literature of related topics in various control charts (eg, Montgomery, 1980; Vance, 1983; Ho and Case, 1994). Nevertheless, the major weakness of this method is that the full economic design ignores the statistical performance of control charts such that it may produce too many nonconforming products and false alarms when utilizing this approach to design control charts to monitor the manufacturing process (Woodall, 1985). Since the outcome from an economic design of control charts may lead to poor statistical properties (eg, low power; high Type I error), Saniga (1989) first considered an economic design of control charts with statistical constraints and then proposed the economic statistical design of the joint x-bar and R control charts for normal data. The objective of economic statistical designs is to minimize the expected total cost per unit time previously specified Type I error and power which rely on designer s need. In general, economic statistical

2 FL Chen and CH Yeh Economic statistical design for x-bar control charts 751 designs are more costly than economic designs due to the added statistical constraints (ie, minimum value of power and maximum value of Type I error). However, the tight limits on the statistical properties of control charts can result in low process variability which enhances output quality (Zhang and Berardi, 1997). McWilliams (1994) presented the economic statistical design of x-bar control charts. EWMA control charts had been developed with an economic statistical design approach by Montgomery, Torng, Cochran, and Lawence (Montgomery et al, 1995). McWilliams and Saniga (2001) developed a comprehensive computer program for an economic statistical design of the joint x-bar and R control or x-bar and S charts. Chou also applied an economic statistical design method into the multivariate control charts to monitor the mean vector and the covariance matrix (Chou et al, 2003). Duncan s economic design of x-bar control charts also assumed that the time that an assignable cause occurred belonged to a random variable of exponential distribution with parameter λ and constant hazard rates. However, some investigations concerning the number of periods of the process remaining in control do not have Poisson distributions (eg, Baker, 1971). Hu (1984) proposed a method of economic design of an x-bar control chart under a non-poisson distribution by following Duncan s (1956) approach. Meanwhile, Hu assumed that the length of the sampling intervals was fixed in the model. Banerjee and Rahim (1988) also presented an economic design of an x-bar control chart based on a Weibull failure mechanism and determined that the sampling interval was variable rather than constant. Moreover, they assumed an increasing hazard rate in systems. The assumption of Poisson failure mechanism was not always appropriate in practice, especially for processes in which machine wear occurs over time. Processes with increasing hazard functions are common in the mechanical and the electrical industries (Zhang and Berardi, 1997). Also, the concept of applying a non-uniform sampling scheme has received considerable attention (eg, Parkhideh and Case, 1989; Rahim, 1993, 1994; Rahim and Banerjee, 1993; Otha and Rahim, 1997; Rahim and Ben-Daya, 1998; Rahim and Costa, 2000; Al-Oraini and Rahim, 2003). The sampling subgroup measurements are conventionally assumed to be normally distributed when the three control chart parameters are set. Moreover, the statistic of sample mean is also normally distributed. This condition is based on the central limit theorem that the distribution of the sample mean is approximately normally distributed in large sample sizes. Unfortunately, raising the sample size increases the sampling costs in practical industries, so the sample size is never sufficiently large enough to meet the central limit theorem. When the sampling subgroup measurements taken are not normally distributed, and the sample size is not large enough to adopt the central limit theorem, the conventional approach of determining the three control chart parameters may reduce the ability of control charts to detect a mean shift in a manufacture process. Hence, Rahim (1985) proposed an economic model of x-bar charts by assuming that the data were non-normally distributed, and the time between the occurrences of the assignable cause was exponentially distributed. He employed the first four terms of an Edgeworth series to approximate the non-normal probability density function of the process, and considered the chart to be one-sided. However, Rahim s method is complicated and includes some non-closed-form expressions in the computations of error probabilities. Also, he does not take statistical constraints and discussion of the effect of non-normal into this economic design model (Chou et al, 2001). Likewise, Zhang and Berardi (1997) presented the economic statistical design of x-bar control charts for the manufacturing process that owns the Weibull failure mechanism. However, they did not incorporate the influence of non-normal into this economic statistical design. Burr developed the Burr distribution to describe the nonnormal probability density function of the process in-control time in Since then, the concept of applying the Burr distribution has received only a little attention (eg, Zimmer and Burr, 1963; Yourstone and Zimmer, 1992; Chou et al, 2000, 2001; Chen, 2003). Nevertheless, Chen and Cheng (2007) utilized the Johnson distribution and McWilliams s (1989) cost model to model an x-bar control chart economic statistical within Weibull distribution failure mechanism for non-normal data. Although there are many methodologies for developing a control chart within non-normal data, the primary advantage of the Burr distribution is that it has a closed-form cumulative distribution function, which simplifies the Type I error and Type II error probability calculations (Chen and Cheng, 2007). This research intends to develop the economic statistical design model of an x-bar control chart, assuming that the collected data from a manufacture process are not normally distributed, and employing that the Weibull failure mechanism. The Burr distribution and the cost model of the Weibull failure mechanism given in Banerjee and Rahim (1988) will be applied to establish the economic statistical design model of an x-bar control chart. Also, the computation of the Type I error and power of this model will be derived based on the Burr distribution and the upper bound of α risk and lower bound power that are previously specified as the statistical constraints. The cost model of economic statistical design extends the cost model of Banerjee and Rahim (1988) which are developed and presented in Section 2. Section 3 introduces the Burr distribution. Type I error probability and the power of the chart with the Burr distribution for the economic statistical design of an x-bar control chart presented with non-normal data are presented in Section 4. The cost model of Banerjee and Rahim is employed as the objective function, which is expected to be minimized (ECT) and is restricted by specifying the statistical constraints (ie, the upper bound of α risk and lower bound power) to design the three parameters of an x control chart. An example considered by Banerjee and Rahim (1988) is applied to describe the

3 752 Journal of the Operational Research Society Vol. 62, No. 4 solution procedure of the economic statistical design of an x-bar control chart for non-normal data under the Weibull failure mechanism in Section 5. Section 6 describes the sensitivity analyses that are conducted to evaluate the effects of non-normality, including the time, cost and Weibull s parameters, and skewness and kurtosis coefficients for data values. 2. The cost model The cost model of Banerjee and Rahim (1988) is based on Duncan s (1956) cost model as modified by Lorenzen and Vance (1986), and has the following features: (1) The in-control time of the manufacture process is based on Weibull failure mechanism. Restated, a mean shift of a manufacture process follows a Weibull distribution. The probability density function is expressed mathematically as f (t)=λωt (ω 1) exp{ λt ω }, t > 0, λ > 0, ω 1 (1) (2) The sample approach for the cost model of Banerjee and Rahim is to draw random samples of size n at times (h 1 + h 2 ), (h 1 + h 2 + h 3 ),..., and so on. (3) Production stops when searching and repairing occur. (4) The time for sampling and charting one item in the control chart is assumed to be negligible. (5) The length of sampling interval h i is assumed to vary with time in the manufacturing process. However, the hazard rate rises in the Weibull model, so the probability of a shift in an interval has to keep constant for all sampling intervals. Thus, the mathematical expression of the length of sampling interval is defined as h i =[i 1/ω (i 1) 1/ω ]h 1, i = 1, 2, 3 (2) Notably, h i has to fill two basic conditions, (1) h 1 h 2 h 3,and(2)lim m m j=1 h i =. The length of the sampling interval is constant, and the failure mechanism belongs to the Poisson distribution when ω = 1. In the cost model of Banerjee and Rahim (1988), the production cycle assumes that a new component is sited, and therefore production ceases until a shift via the component failure is detected. Replacing a component returns the process to an in-control state. The four possible process states are: out of control with alarm, out of control with no alarm, in control with no alarm and in control with false alarm. A general formulation for the economic statistical design model with a Weibull failure shock in the manufacturing process is defined as Min F(n, h 1, L) = E(C) E(T ) s.t. α α U, 1 β p L (3) where E(T ) and E(C) represent the expected cycle length and the total expected cost per cycle, respectively. The objective function that the Type I error probability (α) and power (1 β), respectively, subjected to the predetermined statistical constraints (ie, maximum value of Type I error (α U ) and minimum value of power (p L )) is minimized by determining the sample size (n), sampling interval (h 1 ) and the control limits (L), which can be expressed mathematically as E(T ) = Z 1 + αz 0 (1 p)/p + h 1 pa(1 p) +βh 1 p[pa(1 p) (1 β)a(β)]/(1 p β) (4) E(C)=(a+bn)[β/(1 β)+1/p]+αy (1 p)/p+p[d 0 D 1 ] (1/1λ) 1/ω Γ(1 + 1/ω) + D 1 h 1 p(1 p)a(1 p) + βh 1 D 1 p[pa(1 p) (1 β)a(β)] (5) where A(x) = (1 + ν) 1/ω x ν, p = 1 exp( λh ω 1 ) ν=0 These parameters in the objective function of the cost model are identified as follows. Time parameters Z 0 is the expected search time associated with a false alarm. Z 1 is the expected search time and repair time if a failure is detected. Cost parameters D 0 is the expected cost per hour caused by the production of a non-conforming item when the process is in control. D 1 is the expected cost per hour caused by production of a non-conforming item when the process is out of control. W is the expected cost of locating an assignable cause and repairing the process (including the cost of down time). Y is the expected cost of false alarms (including the cost of searching and that of down time if production ceases during the search). a is the fixed cost per sample. b is the cost per unit sampled. The Weibull distribution parameters λ is the scale parameter. ω is the shape parameter. 3. Review of the Burr distribution In 1942, Burr developed the Burr distribution that describes various non-normal distributions. This investigation illustrates various non-normal distributions with the Burr distribution. The probability density function of the Burr distribution is

4 FL Chen and CH Yeh Economic statistical design for x-bar control charts 753 mathematically expressed as f (y) = ckyc 1 (1 + y c ) k+1 for y 0; c 1, k 1 = 0 for y < 0 (6) Therefore, the mathematical expression of cumulative distribution function of the Burr distribution defined in (6) is 1 F(y) = 1 (1 + y c ) k for y 0; c 1, k 1 = 0 for y < 0 (7) where c and k denote the skewness and kurtosis coefficients of the Burr distribution. The first four moments, or the 3rd and 4th moments of an empirical distribution, can be utilized to determine values of c and k. They can also be applied to approximate the empirical distribution in the Burr distribution. Burr (1942) tabulated the mean and standard deviation as well as the coefficients of skewness and kurtosis for the family of Burr distribution, respectively, as shown in the Tables II and III of his article. These tables enable a user to perform a standardized transformation between a Burr variable (given by Q) and any random variable (given by X ) (Chou et al, 2000). The data are set as follows: (1) The skewness and kurtosis coefficients of the original sample data are estimated. (2) The skewness and kurtosis coefficients for the Burr distribution family are obtained from Table II according to the outcome of Step 1. (3) The mean and standard deviation for the family of Burr distribution is obtained from Table III according to the outcome of Step 2. (4) A standardized transformation between a Burr variable (given by Q) and any random variable (given by X) is performed according to the outcome of Step 1 and 2. This transformation is mathematically expressed as X X S X = Q M S where X and S X represent the value of sample average and standard deviation for the original sample data, and M and S denote mean and standard deviation, respectively, for the family of Burr distribution relative to the original sample data. 4. Development of the Type I and Type II error probabilities The Type I and Type II error probabilities for an x-bar chart under non-normal data are derived in this section. The two error probabilities are estimated differently in normal and nonnormal distributions, so are calculated by the Burr distribution for non-normal data in this investigation. However, the control limits of the x-bar charts have to be obtained to estimate the Type I and Type II error probabilities. The UCL and LCL indicate the upper and lower control limits of the x-bar charts, respectively, and may be mathematically expressed as UCL = μ 0 + L σ n LCL = μ 0 L σ (8) n where μ 0 is the average value of a manufacturing process in an in-control state, L and n represent the control limit width of the charts and the sampling size, respectively, and σ is the standard deviation value of a manufacturing process. The Burr random variate Q thus may be transformed to the sample statistic X based on the standardized transformation formula, which is given by X μ 0 σ/ n = Q M (9) S This function can be written as X = μ 0 + (Q M) σ/ n (10) S In other words, X may be represented by the Burr distribution based on Equation (10). The Type I error probability of an x-bar chart for data with a non-normal Burr distribution is α = Pr(X > UCL) + Pr(X < LCL) (11) From Equations (7) (10), α can be derived as follows: α = Pr (X > μ 0 + L n σ ) + Pr (X < μ 0 L n σ ) = Pr (μ 0 + (Q M) n σ > μ 0 + L n σ ) + Pr (μ 0 + (Q M) n σ < μ 0 L n σ ) = Pr(Q > M + LS) + Pr(Q < M LS) = [1 + (M + LS) C ] K 1 [1 + (M LS) C ] K (12) The mathematical expression of the Type II error probability of an x-bar chart is then given as β = Pr LCL X UCL μ 1 = μ 0 + δσ (13) where μ 1 is the new average value of the manufacturing process, which has shifted from the original process mean by δσ. The Type II error probability indicates that the process probability is in an out of control state but does not detect shift. The parameter δ is the size of a manufacturing process shift. When the process mean changes from μ 0 to μ 1,theBurr random variate Q may be transformed to the sample statistic X

5 754 Journal of the Operational Research Society Vol. 62, No. 4 Table 1 Comparison of the economic design of control charts with Burr and Normal distributions Distribution Design parameters Outcome n h 1 L α 1 β ECT Burr distribution (c = 5, k = 6) Normal distribution according to the standardized transformation formula, which is defined as X (μ 0 + δσ) σ/ n = Q M S The above-mentioned function can be expressed as (14) X = μ 0 + δσ + (Q M) σ/ n (15) S The Type II error probability of an x-bar chart for non-normal data with Burr distribution is derived using Equations (7), (8), (14) and (15) as follows: β = Pr (μ 0 L n σ X μ 0 L n σ ) ( = Pr μ 0 L σ μ n 0 +δσ+(q M) σ/ n μ S 0 +L σ ) n = Pr(M LS Sδ n Q M + LS Sδ n) 1 = [1 + (M LS Sδ n) C ] K 1 [1 + (M + LS Sδ n) C (16) ] K Equations (12) and (16) are employed to estimate the Type I and Type II error probabilities for the economic design of an x-bar chart model under non-normal data in this investigation. 5. An illustration of the solution procedure This section presents the procedure for obtaining the most economic statistical design of an x-bar chart for non-normal data by an example proposed from Banerjee and Rahim (1988) The description of the example The values of the time, cost, Weibull and shift parameters of the example are as follows: Z 0 = 0.25 h; Z 1 = 1.00 h; D 0 = $50.00; D 1 = $950.0; W = $ ; Y = $500.00; a=$20.00; b=$4.22; δ=0.50σ; λ=0.002; ω=3.0; α U =0.05; p L = 0.9. The assumptions for the example are as follows: (1) The process-failure mechanism of the example belongs to the Weibull distribution. (2) The sampling scheme of the example is a non-uniform approach. (3) The example has a non-decreasing failure rate for the Weibull type shock model. (4) The objective function of the example which 1 β and α are restricted by the minimum value of power (p L ) and the maximum value of Type I error (α U ), respectively, is the minimum value of the expected cost per hour (ECT) based on the overall optimal design parameters (n, h 1, and L) The solution procedure The grid-search method was employed to find the optimal combination of design parameters (n, h 1 and L) by measuring a wide range of possible solutions for the objective function of the example which is the smallest expected cost per hour while 1 β and α are limited by p L and α U, respectively. The MATLAB computer program was used to calculate the expected cost per hour and the corresponding Type I error (α) and power (1 β) on different combination of design parameters (n, h 1 and L) in Equations (3), (4), (5), (12) and (16) to find the optimal combination which has the minimum (ECT). However, the original sample data must first be computed by the skewness and kurtosis coefficients. The mean and standard deviation of the Burr distribution must also be determined before the above-mentioned calculation. To demonstrate that the Burr distribution can approximate a normal distribution, the economic design of control charts was established according to a normal assumption employing the Burr and Normal distribution in the above-mentioned example. Table 1 indicates that the Burr distribution with a skewness coefficient of 5 and kurtosis coefficient of 6 are close to the Normal distribution in Rahim s (1993) example. For the non-normality aspect of Table 2, we assume that the skewness and kurtosis coefficients for Rahim s (1993) example were taken to be approximately and 4.122, respectively, as by the Burr distribution with c = 2 and k = 10. As for the normal part of Table 2, we also assume that the Burr distribution with a skewness coefficient of 5 and kurtosis coefficient of 6 are close to the Normal distribution. From the results, we find out that using the economic design approach can obtain higher Type I error rate and lower power for detecting a shift in the process than the economic statistical design method. However, slightly higher expected cost will be

6 FL Chen and CH Yeh Economic statistical design for x-bar control charts 755 Table 2 Comparison of the economic and economic statistical design of control charts under normal and non-normality Distribution Design parameters Outcome n h 1 L α 1 β ECT Economic design * Economic statistical design * Economic design Economic statistical design normal. non-normal. Table 3 The outcome of the presented example under non-normal distribution (c = 3, k = 6) Design parameters Outcome n h 1 L α 1 β ECT needed for the economic statistical design no matter whether the data distribution of process is normal or non-normal. In addition, we also can conclude that the non-normal factor may reduce the ability of control charts for detecting a mean shift in a manufacture process from the outcome of Table 2. Furthermore, if we do not consider this in control charts design, the power for detecting a shift in the process would become worse and the overall cost could be higher no matter whether the economic statistical design or the economic design method is employed. This comparison of results indicates that we shall consider the non-normal factor when the economic statistical design is employed to develop an x-bar control chart that can gain the desired quality level of monitoring a manufacture process. Meanwhile, we only need to pay a slightly higher price than the economic design. In Table 2, the Burr distribution was utilized to approximate the economic statistical design and economic design of the x-bar control charts for non-normal distributions. Table 3 shows that the skewness and kurtosis coefficients for Rahim s (1993) example were taken to be approximately and , respectively, as by the Burr distribution with c = 3andk = 6. Furthermore, the grid-search method was then employed to find the optimal combination of design parameters (n, h 1,andL) by measuring a wide range of possible solutions for the objective function of the example which is the smallest expected cost per hour while 1 β and α are limited by p L and α U, respectively. For this example, the optimal design parameters of the x-bar control chart were n=40, h 1 = h (indicating that a sample was taken every min) and the sigma coefficient located at L = , yielding a minimum value of the expected cost per hour (ECT) of $ , as in Table 3. Moreover, the Type I error probability and power (1 β) rate were computed as α = 0.05% and 1 β = 90.34% relative to the optimal design parameters of the x-bar control chart in Table Sensitivity analysis This section discusses the effects of time, cost, and Weibull s parameters of cost model under the assumption that the manufacturing process has a non-normal distribution and its skewness and kurtosis coefficients are transformed by the Burr distribution with c = 3andk = 6 for three design parameters and expected cost of the x-bar control chart. In addition, by changing various combinations of skewness and kurtosis coefficients, we can realize how a non-normality factor affects these optimal values for x-bar control charts including design parameters (ie, n, h 1 and L), Type I error, power, average time to signal (ATS) as well as the expected cost. Therefore, we employ an orthogonal array experimental design and linear regression, in which the cost model parameters (ie, Z 0, Z 1, D 0, D 1, W, Y, a, b, δ, α U, p L and λ) are regarded as the independent variables. The optimal design parameters (ie, n, h 1 and L), Type I error, power, average time to signal (ATS) as well as the expected cost are considered as the dependent variables in this sensitivity analysis. Table 4 shows the level plan of 12 independent variables in the proposed cost model. Each variable is assigned with three levels where level 2 is the original level. We use the L27 orthogonal array to assign the 12 independent variables to the columns of the L27 array as presented in Table 5. There are 27 trials, that is, 27 different level combinations of the 12 independent variables in the L27 orthogonal array experiment. For each trial, these optimal dependent variables are determined by the approach from the perspective of economic statistical design. Simultaneously, the corresponding outputs of the grid search for each trial including the optimal design parameters, Type I error, power, ATS as well as the expected cost are also shown in Table 5. Then, we take the outcome of

7 756 Journal of the Operational Research Society Vol. 62, No. 4 the corresponding outputs for various level combinations of independent variables in each trial from Table 5 to implement a linear regression analysis. The results of linear regression analysis for dependent variables including design parameters (ie, n, h 1 and L) and the expected cost are shown in Equations (17) (20), respectively. We utilize the stepwise selection approach and the statistical software SPSS 11.5 to perform the regression analysis. Table 4 Levels plan for model parameters in the economic statistical design of the x-bar chart Model parameters Level 1 Level 2 Level 3 Z Z D D W Y a b δ α U p L λ For each dependent variable in the regression analysis, we present the outcome of the statistical hypothesis testing with regression coefficients from the output of SPSS. Equation (17) shows the effect of model parameters for the sample size (y 1 ) of the x-bar chart under a non-normal situation. By examining Equation (17), we discover that both the shift size of a manufacturing process mean (x 1 ) and the maximum value of Type I error (x 2 ) significantly affect the sample size. Also, both the sign of x 1 and x 2 are negative, indicating that an x-bar chart is used to monitor a manufacturing process in which a tiny shift magnitude of its mean could be detected and a slight maximum value of Type I error is permitted that tends to need a larger sample size. y 1 = x x 2 (17) Equation (18) presents the effect of model parameters for the sampling interval (y 2 ) of the x-bar chart under a non-normal condition. It can be seen from the regression coefficients of Equation (18) that the shift size of a manufacturing process mean (x 1 ), the expected cost per hour caused by production of a non-conforming item (x 3 ) when the process is out of control, and the scale parameter of Weibull distribution (x 4 ) significantly influence the sample interval. Also, all the signs of estimation for x 1, x 3 and x 4 are negative, implying that a tiny shift magnitude of process mean could be detected and Table 5 Experiment design of L27 orthogonal array for model parameters within optimal design of the x-bar chart under non-normality Trail Model parameters Design parameters Outcome Z 0 Z 1 D 0 D 1 W Y a b δ α U p L λ n h 1 L α 1 β ECT

8 FL Chen and CH Yeh Economic statistical design for x-bar control charts 757 Table 6 Effect of scale parameter of Weibull for the optimal design of the x-bar chart under non-normality Case Weibull parameters Design parameters Outcome λ ω n h 1 L α 1 β ATS ECT (1) (2) (3) a slight amount of cost for non-conforming item yield in out of control state when an x-bar chart is employed to handle a manufacturing process that requires longer sampling interval. Simultaneously, a decrease in the scale parameter value for failure mechanism of a manufacturing process with a Weibull distribution also lengthens sampling interval. y 2 = x x x 4 (18) Equation (19) demonstrates the effect of model parameters for the control limit width (y 3 ) of the x-bar chart under a nonnormal state. According to Equation (19), it is noticed that only the maximum value of Type I error (x 2 ) significantly affects the control limit width. Meanwhile, the sign of the estimate for the coefficient is negative. In other words, an increase in the x 2 value results in a narrower control limit width when an x-bar control chart is employed to monitor the process mean. y 3 = x 2 (19) Equation (20) displays the effect of model parameters for the expected cost per hour (y 4 ) of the x-bar chart that incorporates a non-normal condition. It may be seen from Equation (20) that five model parameters, including the expected search time and repair time if a failure is detected (x 5 ), the shift magnitude of process mean (x 1 ), the cost per unit sampled (x 6 ), the expected cost per hour caused by production of a non-conforming item when the process is out of control (x 3 ) as well as the expected cost of locating an assignable cause and repairing the process (x 7 ), influence the expected cost per hour. Simultaneously, we also find that the signs of estimation for x 5 and x 1 are negative. However, the signs of x 6, x 3 and x 7 are positive. Both a larger shift size and an increase in the expected search time and repair time for a failure being detected lead to a decrease in the expected cost per hour. Nevertheless, a higher unit sampling cost, a higher amount of cost for non-conforming items, and a higher expected cost of locating an assignable cause and repairing the process will all result in an increase in the expected cost per hour. y 4 = x x x x x 6 (20) Table 6 shows the effect of changing the shape parameter (ω) of the Weibull parameters for the optimal design of the x-bar chart under non-normal distribution. The average value of the Weibull parameters were assumed to be maintained at 7.09 before any change of the scale parameter (λ) and the shape parameter (ω) of the Weibull parameters. An increase in the shape parameter (ω) leads to a decrease in the expected cost per hour (ECT) by 9.57%, but results in increasing ATS and the sampling interval by 37.23%, respectively. Thus, increasing ω significantly increases ATS, but decreases the sampling frequency. Table 7 shows the various combinations of skewness and kurtosis coefficients for different non-normally distributed data on the optimal design of the x-bar chart based on the example of Banerjee and Rahim (1988). To investigate the effect of economic statistical design of the x-bar chart under the various combinations of skewness and kurtosis coefficients, Table 7 is divided into six groups. Furthermore, the following conclusions can be drawn from Table 7. (1) Increasing the skewness coefficient (α 3 ) from to decreases the sample size (n). Meanwhile, when the kurtosis coefficients exceed 4.0, the sampling interval increases and the ECT gradually decreases. (2) The skewness coefficient (α 3 ) has no significant effect on the control limit width (L). (3) Increasing the kurtosis coefficient (α 4 ) from to results in a wider control limit width (L). Meanwhile, when the kurtosis coefficients are less than 3.646, the sampling size gradually decreases. (4) When α 3 > 1.06 and α 4 > 7.22, simultaneously increasing α 3 and α 4 result in a decrease in sample size, sampling intervals and ECT. (5) When α 4 > 4.106, increasing α 3 leads to a decrease in sample size and sampling intervals. (6) When α 4 > 7.22, increasing α 4 does not affect the control limit width (L). 7. Conclusions This investigation considered the effect of the economic statistical design of an x-bar control chart for non-normally distributed data under the Weibull failure mechanism. Also, we can conclude from this study as follows: (1) The shift magnitude of a manufacturing process mean (δ) has a significant effect on the sample size, the sampling intervals and expected cost. When a tiny shift of manufacturing process mean has to be detected, a larger sample

9 758 Journal of the Operational Research Society Vol. 62, No. 4 Table 7 Effect of non-normal distribution on the optimal design of x-bar chart c k α 3 α 4 Design parameter Outcome n h 1 L α 1 β ATS ETC Group I (α 3 : from to +, α 4 : close to normal) Group II (α 3 : close to normal, α 4 : increasing) Group III (α 3 : from to +, α 4 : near a constant) Group IV (α 3 : increasing, α 4 : near a constant and > 4.0) Group V (α 3 : close to one, α 4 : increasing) Group VI (α 3 : increasing and 0, α 4 : increasing and 0) size and longer sampling interval will be required, which also result in higher expected cost per hour. (2) The maximum value of Type I error (α U ) significantly affects the sample size and the control limit width (L). It results in an increase in the sample size and the control limit width (L) becomes wider when α U is lower. (3) The scale parameter of Weibull distribution (λ) significantly influences the sample interval. Also, an increase in the scale parameter value (λ) will require frequent sampling. (4) The shape parameter (ω) of the Weibull parameters significantly affects the sample interval and the expected cost per hour. Meanwhile, an increase in ω results in decreasing the sampling frequency and the expected cost per hour. (5) The expected cost per hour caused by the non-conforming items when the process is out of control (D 1 ) has an effect on the sample interval and the expected cost per hour. Furthermore, a higher D 1 results in the increasing sampling frequency and the expected cost per hour. (6) The expected search time and repair time if a failure is detected (Z 1 ), the cost per unit sampled (b), and the expected cost of locating an assignable cause and repairing the process (W) significantly affect the expected cost per hour. An increase in Z 1 leads to a decrease in the expected cost per hour. Nevertheless, a higher b and

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