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Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/copyright Author's personal copy Computational Statistics and Data Analysis 53 (2009) 3412–3425 Contents lists available at ScienceDirect Computational Statistics and Data Analysis journal homepage: www.elsevier.com/locate/csda Statistical computation and analyses for attribute events Yafen Liu a , Zhen He a , Lianjie Shu b , Zhang Wu c,∗ a School of Management, Tianjin University, 300072, China b Faculty of Business Administration, University of Macau, 999078, Macau c School of Mechanical and Aerospace Engineering, Nanyang Technological University, 639798, Singapore article info Article history: Received 8 June 2008 Received in revised form 11 February 2009 Accepted 16 February 2009 Available online 23 February 2009 a b s t r a c t This article studies the monitoring of the attribute events based on statistical computation and analyses. The size of an attribute event is an integer rather than a continuous variable. For example, the detection of a product lot containing defectives is an attribute event, the size of which is the number of defectives found in this lot. While many control charts have been developed for monitoring the time interval (T ) between the occurrences of an event, many other attribute charts may be employed to examine the size (C ) of the event. However, these two types of control charts have been investigated and applied separately with limited syntheses in Statistical Process Control (SPC). This article presents a single SPC chart (called the rate chart for attribute, or rate chart in short) for simultaneously monitoring the time interval T and size C of an attribute event based on the ratio between C and T . Our studies show that the new chart is more effective for detecting the out-of-control status of the attribute event compared with an individual t chart or an individual c chart, as well as a combined t & c chart. More profound is that the rate chart performs more uniformly than other charts for detecting both T shift and C shift, as well as the joint shift in T and C . The rate chart has demonstrated its potential for both manufacturing systems and nonmanufacturing sectors (e.g., supply chain management, office administration and health care industry), especially for the latter. © 2009 Elsevier B.V. All rights reserved. 1. Introduction In general, there are two types of events: a negative one and a positive one. A negative event may be the rejection of a lot of a product, a failure of a system, or a traffic accident; whilst a positive event may refer to a purchase order of product lots, the success of a program, the arrival of a tourist group. For example, in a manufacturing production line, a critical sensing device may go wrong (a negative event) at a random time. Each malfunction of this device will result in several (C ) defective products. The time between two malfunctions is a random variable T . It is different from the constant sampling interval h used by many control charts. The size C of an event is the number of items that are incurred by an event occurrence and also decide the characteristic of the event. It may refer to the number of defects in a rejected lot, the death toll in a traffic accident, the number of lots sold in a product order, and so on. An SPC system can continuously analyze and monitor the collected data of an event which involves the time interval T and the size C , and accordingly decides whether the situation is under control or out of control. Both T and C are random variables, but T is a continuous variable, while C is an integer one. In this article, T and C are assumed to be independent of each other. An upward shift (i.e., a decrease of T and/or an increase of C ) or an increase of the ratio C /T (called event ratio and denoted by R) indicates a move in a loss direction for a negative event. It means a higher rate in damage, cost, or loss incurred by the ∗ Corresponding author. Tel.: +65 67904445; fax: +65 67911859. E-mail address: mzwu@ntu.edu.sg (Z. Wu). 0167-9473/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.csda.2009.02.010 Author's personal copy Y. Liu et al. / Computational Statistics and Data Analysis 53 (2009) 3412–3425 3413 occurrences of the event. However, an upward shift represents a move in a gain direction for a positive event, suggesting that the gain or success has been achieved more often and in a larger scale. On the contrary, a downward shift (i.e., an increase of T and/or a decrease of C ) or a decrease of the event ratio R indicates a gain movement for a negative event, showing that the severity of the problem is alleviated or improved. But a downward shift represents a loss movement for a positive event, meaning that the gain and profit is made less frequently and in a smaller amount. In SPC practices, one is more interested in detecting the upward R shifts for a negative event and the downward R shifts for a positive event, because these shifts represent a transit to a worse status, and the users should be alarmed as soon as possible. Many control charts have been developed to monitor the time interval T of an event, including the TBE (Time Between Events) or exponential chart (Chan et al., 2000; Xie et al., 2002; Zhang et al., 2008), the Gamma chart (Wu et al., 2001; Zhang et al., 2007), the exponential CUSUM chart (Lucas, 1985; Vardeman and Ray, 1985; Gan, 1992; Borror et al., 2003; Liu et al., 2006), and the exponential EWMA chart (Gan, 1998). As a matter of fact, the effect (e.g., the amount of damage or loss) of an event in a certain time period is, however, not only dependent on T (or the frequency of occurrence), but also on the size C . For example, the loss of market competitiveness of an aged product is indicated not only by the decreased number of purchase orders but also by the decreased number of lots to be delivered in each order. Similarly, a task team for bird flu prevention and elimination must be enhanced not only when the outbreak of this infection becomes very frequent but also when the number C of poultry infected in each outbreak gets very high. Many control charts for attributes with different advanced features have been proposed recently by the researchers (Shepherd et al., 2007; Wu and Jiao, 2008; Chiu and Kuo, 2008; Chakraborti and Human, 2008; Höhle and Paul, 2008; Chiu and Sheu, 2008; Sim and Lim, 2008). The individual c chart used widely in SPC may be adopted to monitor the size C of an attribute event. Whenever the event occurs, the sample value of C will be checked against the control limits of the c chart, and the process status is decided. In SPC literature, little effort has been made to explore the algorithms for monitoring T and C concurrently. A combined t & x scheme consisting of a t chart and an x chart was recently investigated for examining both the time interval T and magnitude X of an event, in which X is a continuous variable (Wu et al., in press). It is similar to an x̄ & R (or x̄ & S) chart combination for monitoring the mean and variability of a variable (Khoo and Yap, 2005). In reality, the size of an event is an integer or count data in many SPC applications (Zhang and Lin, 2008; Rigby et al., 2008), especially in non-manufacturing sectors (e.g., health-care industry (Woodall, 2006), supply chain systems, language data process (Yu et al., 2003)). In these applications, a synthesized scheme is needed for monitoring the attribute events. This article investigates a single rate chart that monitors both T and C by checking a single statistic R, or the ratio between C and T . Two types of rate charts will be developed, one for detecting increasing R shifts of negative events, another for detecting decreasing R shifts of positive events. The single rate chart makes use of information about both T and C , and thus has higher detection effectiveness than an individual t chart or an individual c chart, or even a joint t & c chart. In this research, the effectiveness of a control chart is measured by the steady-state Average Time to Signal (ATS) which measures the average time required to signal a process shift once it occurs. 2. Data analysis for an attribute event A rate chart is operated by checking the sample value of the event ratio R (= C /T ). It has only one charting parameter: the upper control limit UCLr for detecting the upward shift (or an increase of the event ratio R) or the lower control limit LCLr for detecting the downward shift (or a decrease of R). The procedure to operate a rate chart and to analyze the sample data for detecting upward R shifts is outlined below: (1) When an occurrence of the event is detected, find the time interval T between the current and the last occurrences and meanwhile count the size C for this instance. (2) Calculate the event ratio R. C . (1) T (3) If R goes above UCLr , the process is thought to be out of control due to an increase in event ratio R. (4) If a R shift is detected, go to step (5). Otherwise, the process is considered to be in control, and go back to step (1) to wait for the next occurrence of the event. (5) Stop the process immediately, and subsequently investigate and remove the assignable cause. Afterwards, resume the process and start with step (1) again. R= The implementation of a rate chart detecting the downward R shifts (increasing T and/or decreasing C ) is quite similar except step (3) being modified as follows: (3) If R falls below LCLr , the process is considered as out of control due to a decrease in R. 3. Statistical computation of the parameters of a rate chart To design a rate chart, the following three specifications need to be determined: (1) The allowed minimum value, τ , of the in-control Average Time to Signal (ATS 0 ). The value of τ is decided with respect to the tolerable false alarm rate. The resultant (or actual) ATS 0 value must be no smaller than τ . Moreover, when comparing Author's personal copy 3414 Y. Liu et al. / Computational Statistics and Data Analysis 53 (2009) 3412–3425 the effectiveness of control charts, all charts are required to produce same or close ATS 0 values, so that all of them result in the similar false alarm rate and the comparison is fair (Montgomery, 1994; Reynolds and Stoumbos, 2004). Usually, this requirement can be met by adjusting the control limits of the charts. (2) The in-control values of the parameters associated with the probability distribution of T . Most of the reported works (Xie et al., 2002) assume that T follows an exponential distribution. Therefore, the only distribution parameter, λ, is the reciprocal of the mean value of T . The in-control value λ0 can be estimated from the in-control sample mean of T that is usually obtained by analyzing historical data. The probability density function and the cumulative probability function of an exponential distribution are given by, fT (T ) = λe−λT , (2) FT (t ) = Pr(T ≤ t ) = 1 − e −λt (3) . (3) The in-control values of the parameters of the probability distribution of C . The random number C is counted on a per event occurrence basis and usually follows a Poisson distribution. In fact, any random phenomenon that occurs on a per unit (or per unit area, per unit volume, per unit time, etc.) basis is often well approximated by the Poisson distribution (Montgomery, 2005). This distribution has only one parameter, g, which is the mean value of C . The in-control value g0 can be estimated by analyzing the average of the observed C values from m samples when the process is in control (Montgomery, 2005). g0 ≈ m P i=1 ci . m The probability mass function and the cumulative probability function of a Poisson distribution are given by, e−g g C p(C ) = C! (4) (5) , FC (c ) = Pr(C ≤ c ) = c X e−g g C C =0 C! (6) . When a process is in control, ATS 0 is determined by the following formula (Wu and Spedding, 1999): 1 ATS0 = λ0 · ARL0 = 1 λ0 · 1 α = 1 λ0 α (7) , where ARL0 is the in-control Average Run Length, and α is the type I error of the rate chart. If ATS 0 is made equal to τ , we obtain (from Eq. (7)), α= 1 λ0 · τ (8) . The cumulative probability function of the statistic R can be derived as follows: FR (r ) = Pr(R ≤ r ) = Pr = ∞ X = ∞ X C =0 C =0  Pr T ≥  exp − y r λC r  C T ≤r   = Pr T ≥  y = C p(C ) = −g  gC ≈ C! ∞ X Uc X C =0 C =0 C r   exp −  exp − λC r λC r  −g e−g g C  C! gC C! , (9) where p(C ) is the probability mass function of C given by Eq. (5). An upper limit Uc is used in Eq. (9) for the numerical computation. It ensures that the error  a specified level. In this article, Uc is determined as the smallest integer  is smaller than that makes the expression of exp − λ(Uc +1) r −g g Uc +1 (Uc +1)! equal to or smaller than 0.0000001. The design of a rate chart is simply to determine the control limit LCLr or UCLr . We can first calculate α corresponding to the specifications λ0 and τ (Eq. (8)). Then, LCLr for detecting downward R shifts is calculated by LCLr = FR−1 (α), (10) UCLr = FR−1 (1 − α). (11) where FR−1 () is the inverse function of the cumulative probability function FR () (Eq. (9)). For detecting upward R shifts, UCLr is determined by Author's personal copy Y. Liu et al. / Computational Statistics and Data Analysis 53 (2009) 3412–3425 3415 The above design algorithm for the rate chart has been implemented in a computer program (attr_rchart.c) in C language. The computer program is available from the authors upon request. The users only need to key in the specifications τ , λ0 and g0 , and the computer program will subsequently complete the design of a rate chart almost in no time in a personal computer. 4. Analyses of statistical characteristics The effectiveness of four control charts (the individual t chart and individual c chart, the t & c chart, and the rate chart) is compared in this section. The control limits of the individual t chart, c chart and t & c chart can be easily determined based on type I error α (see Eq. (8)). Individual t chart ln(1 − α) , (detecting upward R shifts) λ0 ln(α) . (detecting downward R shifts) UCL = − λ0 LCL = − (12) Individual c chart Its control limits can be obtained by rounding off the results from the following equation so that ATS 0 ≥ τ (or the resultant false alarm rate being no higher than the specified level). UCL = FC−1 (1 − α), −1 (detecting upward R shifts) (detecting downward R shifts) LCL = FC (α). (13) −1 where FC () is the inverse function of the cumulative probability function of the Poisson distribution (Eq. (6)). It is noted that the sampling interval for a conventional c chart is usually fixed. But when the c chart is adopted to monitor the size of an attribute event, the sampling interval must be equal to the random time interval T between two occurrences (i.e., taking a sample only when the event happens). Otherwise, the conventional control chart is almost incapable of detecting the changes of the size C of an event (let alone the detection effectiveness), because a sample taken at a fixed interval is unlikely to catch the occurrence of an event which happens randomly. t & c chart A t & c chart runs a t chart and a c chart simultaneously. If a t & c chart is to detect upward R shifts, it uses a t chart with a lower control limit LCLt and a c chart with an upper control limit UCLc . The t & c chart will produce an out of control signal when the sample value of T of an occurrence falls below LCLt and/or the sample value of C goes above UCLc . In contrast, a t & c chart for detecting downward R shifts uses a t chart with an upper control limit UCLt and a c chart with a lower control limit LCLc . It will produce an out of control signal when a sample value of T goes above UCLt and/or a sample value of C falls below LCLc . When a t & c chart for detecting upward R shifts is to be designed, the control limit UCLc of the c chart can be determined by UCLc = FC−1 (1 − αc ), (14) and LCLt of the t chart is determined by LCLt = − ln(1 − αt ) λ0 , (15) where FC−1 () is the inverse function of the cumulative probability function FC () for a Poisson distribution (Eq. (6)), and αt and αc are the type I errors allocated to the t chart and c chart, respectively. In this study, we make αt = αc = 1 − √ 1−α (16) so that the total type I error of the t & c combination is equal to the overall type I error α (see Eq. (8)). On the other hand, if a t & c chart is designed to detect the downward R shifts, the control limit LCLc of the c chart will be obtained by LCLc = FC−1 (αc ), (17) and the control limit UCLt of the t chart is determined as follows: UCLt = − ln(αt ) λ0 . (18) We first study the chart performance for detecting an increasing event ratio R, and then for detecting a decreasing R. Author's personal copy 3416 Y. Liu et al. / Computational Statistics and Data Analysis 53 (2009) 3412–3425 Table 1 ATS values of four control charts for increasing R shift. δλ Chart δg 1.00 1.10 1.20 1.30 1.41 1.51 10.00 t c t&c Rate 114.5 1229.7 148.1 115.6 114.5 665.7 142.8 106.4 114.5 391.7 134.6 98.7 114.5 247.1 123.6 92.2 114.5 165.2 110.6 86.6 114.5 116.2 96.6 81.8 8.20 t c t&c Rate 166.3 1499.6 214.6 167.6 166.3 811.9 205.2 153.9 166.3 477.7 191.2 142.4 166.3 301.3 173.0 132.8 166.3 201.5 151.9 124.5 166.3 141.7 130.2 117.3 6.40 t c t&c Rate 266.4 1921.4 341.6 268.0 266.4 1040.2 322.8 245.4 266.4 612.1 295.6 226.7 266.4 386.1 261.6 210.9 266.4 258.2 224.1 197.3 266.4 181.5 187.3 185.6 4.60 t c t&c Rate 502.8 2673.2 634.8 504.9 502.8 1447.2 588.0 461.4 502.8 851.6 523.5 425.2 502.8 537.1 447.5 394.6 502.8 359.2 369.7 368.4 502.8 252.5 298.5 345.8 2.80 t c t&c Rate 1322.6 4391.7 1600.4 1325.4 1322.6 2377.6 1422.3 1208.1 1322.6 1399.1 1199.3 1110.7 1322.6 882.4 965.1 1028.4 1322.6 590.1 752.2 957.9 1322.6 414.8 577.5 897.0 1.00 t c t&c Rate 10000.0 12296.7 10000.0 10000.0 10000.0 6657.3 7760.8 9082.7 10000.0 3917.5 5647.6 8320.2 10000.0 2470.8 3973.8 7676.4 10000.0 1652.3 2779.7 7125.7 10000.0 1161.5 1968.9 6649.0 Study 1: Detection of increasing event ratio R The comparison is first conducted with the following general condition: τ = 10 000, λ0 = 0.01, g0 = 4. (19) The specification of (λ0 = 0.01) means that, on average, the event takes place once for every 100 time units when the process is in control, and (τ = 10 000) implies that a false alarm is produced for every 100 occurrences of the event. Based on the above specifications, the four charts are designed and the charting parameters are listed below: Individual t chart: Individual c chart: t & c chart : Rate chart : LCL = 1.0050 UCL = 9 LCLt = 0.7207, UCLr = 3.9750. UCLc = 10 Since the mean value of C is equal to the parameter g, the Poisson distribution of C will move to the right when g becomes larger than g0 , and the C value as well as the event ratio R will get greater in general. Likewise, if the parameter λ increases, R becomes larger because of the decrease of T . We can set λ = δλ λ0 , g = δg g0 , (20) where δg represents the C shift in terms of g0 , and δλ indicates the change of T in terms of λ0 . A larger δg and/or δλ lead to a greater value of the event ratio R, and vice versa. The process is in control when (δg = δλ = 1). Then a domain of increasing R shift can be defined as follows: 1 ≤ δλ ≤ 10, 1 ≤ δg ≤ 1.51. (21) The maximum value δλ,max (= 10) of δλ and the maximum value δg ,max (= 1.51) of δg are determined so that the power of the individual t chart for detecting the pure decreasing T shift at (δλ = δλ,max ) is equal to the power of the individual c chart for detecting the pure increasing C shift at (δg = δg ,max ). The purpose is to leverage the sizes of the T shift and C shift in the performance study. Table 1 tabulates the ATS values of the four charts in the above shift domain (21). The formulae for calculating the ATS of the rate chart are given in the Appendix. There are a total of 35 out-of-control cells (combinations of the discrete values of δg and δλ ) and one in-control cell (δg = δλ = 1) at the bottom-left corner. Each cell contains the ATS values of the four charts for a particular combination of the values of δg and δλ . It is interesting to observe the followings: (1) Firstly, the t chart, t & c chart and rate chart generate an identical ATS 0 value equal to τ when the process is in control. However, the c chart generates an ATS 0 value larger than τ due to the discrete nature of the size of C . Any attempt to reduce the ATS 0 of the c chart (e.g., reduce the UCL of the c chart by one) will make ATS 0 smaller than τ and therefore is not allowed. Author's personal copy Y. Liu et al. / Computational Statistics and Data Analysis 53 (2009) 3412–3425 3417 (2) For detecting the pure C shift (see the out-of-control cells at the bottom row, corresponding to δg > 1 and δλ = 1), the individual c chart suggests itself to be most effective, followed by the t & c chart, and then the rate chart. The individual t chart is completely insensitive. (3) On the other hand, for detecting the pure T shift (see the out-of-control cells at the left-hand side column, where δg = 1 and δλ > 1), the individual t chart and the rate chart are most, and almost equally, effective. The other two charts are also capable of detecting the pure T shift, but the power, especially the power of the individual c chart, is very low. The ATS value of the c chart becomes shorter along with the increase of λ, because the sampling interval of the c chart is equal to T . (4) For detecting the joint (or simultaneous) shifts in T and C (see the out-of-control cells in which δg > 1 and δλ > 1), the rate chart outperforms other three charts in most of the cases. It is only slightly inferior to the individual c chart in a few cells where the C shift prevails. In order to have a more general and quantitative analysis of the relative effectiveness of the charts, the average of the ratios (AR) of ATS values across a shift region of interest is calculated for each chart. AR = m P i=1 ATS (δg ,i ,δλ,i ) ATSr (δg ,i ,δλ,i ) (22) , m where δg ,i and δλ,i denote the discrete values of δg and δλ in the ith cell and m is the number of out-of-control cells in the shift region. ATS (δg ,i , δλ,i ) is the value of ATS produced by a particular chart at (δg ,i , δλ,i ) and ATS r (δg ,i , δλ,i ) is the value generated by the rate chart in the same cell. Obviously, if the AR value of a chart is larger than one, this chart is generally less effective than the rate chart in the shift region, and vice versa. The AR values are studied in the following four regions: Region 1 (δg > 1 and δλ = 1): ARc is calculated in this region to compare the ATS performance against the pure C shifts, with m = 5 (see Table 1). Region 2 (δg = 1 and δλ > 1): ARt is calculated to compare the ATS performance against the pure T shifts, with m = 5. Region 3 (δg > 1 and δλ > 1): ARc +t is calculated to compare the ATS performance against the concurrent T and C shifts, with m = 25. Region 4 ARoverall is derived by merging regions 1, 2 and 3, and is used to compare the overall ATS performance over all the out-of-control cells in Table 1, with m = 35. The AR values for the individual t chart, individual c chart and t & c chart are calculated using the data in Table 1 and the results are listed in RUN 0 in Table 2. These AR values reveal that the rate chart outperforms the two individual charts and the t & c chart in region 3 (where δg > 1 and δλ > 1) and region 4 (the overall region). Specifically, the ARoverall values indicate that the rate chart is, on average, more effective than the individual t chart, individual c chart and t & c chart by 23.0%, 161.9% and 8.5%, respectively. Although the individual t chart is effective in the pure T shift region and the individual c chart and t & c chart are very sensitive to the pure C shift, each of them is either completely insensitive or very ineffective in some other circumstances. An apparent advantage of the rate chart is its uniform performance over the whole shift domain (sensitive to both T and C shifts). Next, a 23 factorial experiment (Montgomery, 2005) is carried out to further study the relative effectiveness of the charts under different conditions. The three specifications, τ , λ0 and g0 , are used as the input factors and each of them varies at two levels, resulting in eight runs. The low and high levels for each factor are decided below: τ g0 λ0 5000 2 0.003 20 000 8 0.03. (23) The eight resultant combinations of the values of τ , λ0 and g0 are listed at the left hand side of Table 2. The results of all eight runs are shown in Table 2. Similar to RUN 0, the rate chart is, on average, the most effective chart. The two individual charts excel only in a small region. The t & c chart has an ARoverall value slightly smaller than one in RUN 1 and RUN 2, but it is considerably less effective than the rate chart in other RUNs. Finally, a grand average AR that indicates the average of AR values encompassing all the eight runs is calculated. The results are listed at the bottom of Table 2. The values of ARoverall indicate that, from a most comprehensive viewpoint (covering all different values of τ , λ0 , g0 , δλ and δg ), the rate chart is more effective than the individual t chart, individual c chart and t & c chart by 23.0%, 174.6% and 14.7%, respectively. More importantly, the rate chart demonstrates a unique strength in achieving a balanced effectiveness for detecting both T shift and C shift, as well as the joint shift. Its performance is satisfactory across the entire shift domain. Conversely, the individual t chart is only effective for detecting a T shift yet fails to sense the C shift. Similarly, the individual c chart is only powerful for detecting a C shift but intolerably ineffective for a T shift. Study 2: Detection of decreasing event ratio R In this case, the four charts are first compared under the following condition: τ = 10 000, λ0 = 0.01, g0 = 13. (24) Author's personal copy 3418 Y. Liu et al. / Computational Statistics and Data Analysis 53 (2009) 3412–3425 Table 2 AR values of the control charts for increasing R shift. RUN τ λ0 g0 Chart ARc ARc +t ARoverall 0 10000 0.010 4 t c t&c 1.303 0.386 0.547 0.994 7.071 1.260 1.262 2.176 1.158 1.230 2.619 1.085 1 5000 0.003 2 t c t&c 1.711 0.469 0.639 0.953 4.586 1.080 1.311 1.413 1.042 1.317 1.731 0.990 2 5000 0.003 8 t c t&c 1.330 0.361 0.443 0.988 3.939 1.363 1.174 1.239 0.974 1.170 1.499 0.954 3 5000 0.030 2 t c t&c 1.396 0.460 0.401 0.992 8.707 2.151 1.353 2.672 1.429 1.308 3.218 1.385 4 5000 0.030 8 t c t&c 1.198 0.515 0.409 0.998 10.696 1.772 1.182 3.044 1.229 1.158 3.776 1.190 5 20000 0.003 2 t c t&c 1.472 0.357 0.595 0.981 5.367 1.213 1.360 1.807 1.205 1.322 2.109 1.119 6 20000 0.003 8 t c t&c 1.236 0.563 0.430 0.995 10.976 1.554 1.193 2.979 1.142 1.171 3.776 1.099 7 20000 0.030 2 t c t&c 1.311 0.470 0.402 0.998 9.532 2.145 1.302 2.937 1.439 1.260 3.527 1.392 8 20000 0.030 8 t c t&c 1.159 0.328 0.481 0.999 6.570 1.456 1.156 2.056 1.170 1.134 2.454 1.112 ARc +t ARoverall 1.255 2.258 1.199 1.230 2.746 1.147 ARc t c t&c 1.346 0.434 0.483 ARt ARt 0.989 7.494 1.555 Accordingly, the parameters of control charts can be designed as listed below: Individual t chart: Individual c chart: t & c chart: Rate chart: UCL = 460.5170 LCL = 5 UCLt = 506.9858, LCLr = 0.0229. LCLc = 5 The following domain for the decreasing R shifts is used (see Table 3): 0.10 ≤ δλ ≤ 1, 0.37 ≤ δg ≤ 1. (25) The in-control cell is now located at the top-right corner in Table 3. The minimum value (δλ,min = 0.10) of δλ and the minimum value (δg ,min = 0.37) of δg are determined so that the sizes of the T shift and C shift are nearly leveraged. Table 3 tabulates the ATS values of the four charts within the shift domain. It can be found that: (1) The t chart, the t & c chart and the rate chart generate an identical ATS 0 value equal to τ when δλ = δg = 1 at the top-right corner. However, the c chart again generates an ATS 0 value larger than τ due to the discrete nature of the size C. (2) For detecting the pure decreasing C shift (see the out-of-control cells at the top row where δg < 1 and δλ = 1), the t & c chart is most effective, followed by the individual c chart and the rate chart. The individual t chart is again insensitive at all. (3) For detecting the pure increasing T shift (see the out-of-control cells at the right-hand-side column where δg = 1 and δλ < 1), the individual t chart is most sensitive, followed by the t & c chart and the rate chart. However, the out-ofcontrol ATS values generated by the individual c chart are even larger than the in-control ATS 0 . It is because that the sampling intervals become larger when T is increasing. (4) For detecting the joint shift in T and C (see the out-of-control cells in which δg < 1 and δλ < 1), the rate chart outperforms the other three charts in general. Author's personal copy 3419 Y. Liu et al. / Computational Statistics and Data Analysis 53 (2009) 3412–3425 Table 3 ATS values of four control charts for decreasing R shift. Chart δλ δg 0.37 0.50 0.62 0.75 0.87 1.00 1.00 t c t&c Rate 10000.0 211.9 210.4 552.3 10000.0 438.1 429.0 985.7 10000.0 1059.9 999.6 1759.3 10000.0 2873.5 2447.0 3139.8 10000.0 8482.1 5555.9 5603.5 10000.0 26736.6 10000.0 10000.7 0.82 t c t&c Rate 5010.3 258.4 248.3 396.8 5010.3 534.3 490.7 693.5 5010.3 1292.6 1080.3 1204.8 5010.3 3504.2 2342.9 2065.3 5010.3 10344.0 4285.8 3495.3 5010.3 32605.7 6040.2 5855.9 0.64 t c t&c Rate 2752.4 331.1 307.6 392.9 2752.4 684.6 577.7 614.9 2752.4 1656.1 1141.7 963.5 2752.4 4489.8 2036.4 1497.9 2752.4 13253.3 2927.4 2306.7 2752.4 41776.0 3457.5 3522.7 0.46 t c t&c Rate 1638.4 460.7 398.6 419.2 1638.4 952.4 674.2 581.2 1638.4 2304.2 1109.1 811.1 1638.4 6246.7 1565.3 1129.7 1638.4 18439.3 1863.6 1565.6 1638.4 58123.1 1997.1 2157.8 0.28 t c t&c Rate 1162.9 756.9 566.8 527.7 1162.9 1564.7 800.0 642.7 1162.9 3785.4 1045.2 789.0 1162.9 10262.4 1213.9 970.8 1162.9 30293.2 1295.3 1193.7 1162.9 95488.0 1326.6 1465.2 0.10 t c t&c Rate 1475.5 2119.3 1221.6 1142.9 1475.5 4381.2 1369.1 1222.1 1475.5 10599.2 1469.2 1311.8 1475.5 28734.6 1519.2 1410.7 1475.5 84820.9 1539.6 1518.4 1475.5 267366.4 1546.9 1635.0 A similar 23 experiment is also carried out in which the low and high levels of the specifications are determined, as below: τ g0 λ0 5000 10 0.003 20 000 17 0.03. (26) For this 23 experiment, the AR values are calculated in the following four regions and the results are shown in Table 4: Region 1: Region 2: Region 3: Region 4: ARc for pure decreasing shift in C (δg < 1 and δλ = 1) ARt for pure increasing shift in T (δg = 1 and δλ < 1) ARc +t for joint shift in T and C (δg < 1 and δλ < 1) ARoverall for the rally of all above. As shown in Table 4, the AR values indicate that the rate chart outperforms the t & c chart slightly from an overall viewpoint. Usually, the rate chart is more effective for detecting joint shifts in T and C and/or when g0 is smaller; and the t & c chart is more sensitive to pure C or pure T shifts and/or when g0 is larger. Both the rate chart and t & c chart outperform the individual t chart and individual c chart to a degree much more significant than the cases for detecting upward R shifts. The individual t chart is too ineffective for detecting any decrease in C , whereas the individual c chart is extremely dull for detecting any increase in T . The values of the grand ARoverall at the bottom of Table 4 manifest that the rate chart is, on average, more effective than the individual t chart and individual c chart by 336.5% and 1410.2%, respectively. 5. Examples Example 1: Detection of increasing event ratio R A municipal department is developing a SPC tool to monitor the occurrence of the disastrous accidents. The SPC system can help decide when an immediate and reinforced action should be taken. In this case, T is the time interval between two consecutive accidents and C is the death toll in an occurrence. The department has the records of ti and ci for 25 accidents (Table 5) for a time period when the situation was thought normal (in an in-control status). It is found that the data of T can be fitted very well to an exponential distribution and the data of C to a Poisson distribution. The in-control parameters of the distributions can be estimated from the records: c1 + c2 + · · · + c25 = 4.0400, g0 ≈ C̄ = 25 1 1 = 0.0720. λ0 ≈ = (t1 + t2 + · · · + t25 )/25 T̄ Author's personal copy 3420 Y. Liu et al. / Computational Statistics and Data Analysis 53 (2009) 3412–3425 Table 4 AR values of the control charts for decreasing R shift. RUN τ λ0 g0 Chart 0 10000 0.010 13 t c t&c 7.781 0.772 0.631 0.818 54.611 0.958 2.791 7.073 1.102 3.222 12.964 1.014 1 5000 0.003 10 t c t&c 2.860 0.892 0.707 0.903 13.088 1.056 1.635 3.292 1.059 1.706 4.349 1.009 2 5000 0.003 17 t c t&c 2.317 0.514 0.596 0.945 8.107 1.082 1.509 2.107 0.992 1.544 2.737 0.949 3 5000 0.030 10 t c t&c 14.590 0.882 0.756 0.715 60.185 0.859 3.893 7.839 1.181 4.967 14.323 1.074 4 5000 0.030 17 t c t&c 7.908 0.356 0.561 0.842 35.059 0.958 2.822 4.332 1.070 3.265 8.154 0.982 5 20000 0.003 10 t c t&c 7.106 0.656 0.599 0.803 22.064 1.088 2.638 3.971 1.100 3.014 6.082 1.027 6 20000 0.003 17 7 20000 0.030 10 t c t&c t c t&c 4.646 0.409 0.567 49.536 1.250 1.047 0.890 19.734 1.012 0.552 207.176 0.629 2.157 3.167 1.029 7.885 19.368 1.351 2.332 5.140 0.961 12.787 43.609 1.204 8 20000 0.030 17 t c t&c 20.724 0.545 0.477 0.753 196.434 0.914 4.732 14.587 1.103 6.448 38.559 0.987 ARc ARt ARc +t ARoverall t c t&c ARc 13.052 0.697 0.660 ARt 0.802 68.495 0.951 ARc +t 3.340 7.304 1.110 ARoverall 4.365 15.102 1.023 The value of λ0 indicates that, on average, there is an accident for every 13.9 days. Other specifications are decided as τ = 365 days. The main concern of the department is to alarm any rise of event ratio R, that is, the increase in the frequency of the accidents and the increase of death toll in each occurrence. Using the computer program attr_rchart.c, the four charts aforementioned can be easily designed and the charting parameters are listed below: Individual t chart: Individual c chart: t & cchart: Rate chart: LCL = 0.5389 UCL = 8 LCLt = 0.2217, UCLr = 7.4600. UCLc = 8 The ATS values of the four charts are given in Table 6, where the in-control cell is located at the bottom-left corner. The performance comparison of the four charts is similar to that summarized in Study 1 in the section of analyses of statistical characteristics. The rate chart is generally more powerful than other charts except for prevailing C shifts. The values of the average ratio are ARc = 1.410, ARt = 0.983, ARc +t = 1.257, ARoverall = 1.240 for the t chart; ARc = 0.509, ARt = 7.603, ARc +t = 2.119, ARoverall = 2.672 for the c chart; and ARc = 0.428, ARt = 1.608, ARc +t = 1.119, ARoverall = 1.090 for the t & c chart. The AR overall values indicate that the rate chart is more effective than the individual t chart, individual c chart and t & c chart by 24.0%, 167.2% and 9.0%, respectively. Example 2: Detection of decreasing event ratio R An electronic company is planning to develop a SPC chart to monitor the selling of one of its main products, an integrated circuit chip. In this case, a positive event arises, referring to the purchase order made by customers. Here, T is the time interval (in working hours) between two consecutive orders and C is the number of lots purchased in an order. The company has the records of ti and ci for 50 orders (Table 7) received during a time period when the selling of the product was thought normal. The in-control parameters of the exponential distribution for T and the Poisson distribution for C can be estimated from the records: c1 + c2 + · · · + c50 g0 ≈ C̄ = = 11.4600, 50 Author's personal copy 3421 Y. Liu et al. / Computational Statistics and Data Analysis 53 (2009) 3412–3425 Table 5 Sample data of T and C in example one. No. ti (day) ci (death toll) 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 8.0 6.7 13.3 9.5 0.1 3.4 16.5 5.5 20.4 38.8 2.2 47.7 19.3 7.0 10.7 0.9 12.1 3.9 32.4 38.2 3.3 14.1 15.1 9.6 8.5 2 1 5 6 5 7 5 5 2 4 6 5 3 2 1 2 5 6 6 4 1 2 5 6 5 λ0 ≈ 1 T̄ = 1 (t1 + t2 + · · · + t50 )/50 = 0.0561. The value of λ0 indicates that, on average, there is one purchase order for every 17.8 h or 2.23 days (eight hours per day). Other specifications are decided as τ = 1440 h (or six months). The main concern of the company is to signal any decline of sales, that is, the decrease in the frequency of the purchase orders and the reduction in the number of lots of each order. The four charts are designed and the charting parameters are listed below: Individual t chart: Individual c chart: t & c chart: Rate chart: UCL = 78.2572 LCL = 5 UCLt = 84.0564, LCLr = 0.1161. LCLc = 4 The ATS values of the four charts are given in Table 8, where the in-control cell is located at the top-right corner. For this case, the rate chart is more powerful for detecting most of the shifts, except for prevailing T shifts or prevailing C shifts. The ARoverall values indicate that the rate chart is more effective than the individual t chart, individual c chart and t & c chart by 216.0%, 420.8% and 7.2%, respectively. 6. Conclusions This article discusses the computation of the parameters of a single rate chart for monitoring the time interval T and size C of an attribute event, and also the analyses of the statistical characteristics of this chart. Unlike the t & c combination, the rate chart manipulates only a single statistic R (event rate). But, it is also able to make use of information related to both T and C . The statistical performance of the rate chart significantly excels the individual t chart and individual c chart, especially for detecting decreasing R shifts. It also outperforms the combined t & c chart from a holistic viewpoint. In addition to high detection effectiveness, the merit of the rate chart lies in its uniform statistical performance (sensitive to both T and C shifts). This feature is important as T and C shifts happen simultaneously in most attribute events. Reynolds and Stoumbos (2004) pointed out that, in most of the cases, it is very difficult to predict the types and magnitudes of process shifts. Therefore a control chart should have an excellent overall performance for different shifts. Another potential merit of the rate chart is that control charts with even better statistical properties may be further developed based on this single rate chart. For example, similar to the CUSUM scheme for monitoring the process mean and variance used in variable SPC (Reynolds and Stoumbos, 2004; Zhao et al., 2005; Shu and Jiang, 2006), a CUSUM type rate chart for simultaneously monitoring T and C may be an avenue for further research. In this article, it is assumed that T and C follow the exponential distribution and Poisson distribution, respectively, and they are mutually independent. It would be interesting to carry out further statistical analyses on the rate chart for the cases in which T and/or C follow other probability distributions, or T and C are interdependent. For interdependent T and C , a joint probability distribution has to be used. Author's personal copy 3422 Y. Liu et al. / Computational Statistics and Data Analysis 53 (2009) 3412–3425 Table 6 ATS values of four control charts in example one. δλ Chart δg 1.00 1.14 1.28 1.56 1.70 10.00 t c t&c rate 5.7 61.5 9.5 5.8 5.7 30.5 8.5 5.4 5.7 17.3 7.3 5.0 1.42 5.7 10.9 6.1 4.7 5.7 7.4 5.1 4.5 5.7 5.4 4.2 4.3 8.20 t c t&c rate 7.9 75.0 13.3 8.1 7.9 37.2 11.6 7.4 7.9 21.1 9.8 6.8 7.9 13.3 8.0 6.4 7.9 9.1 6.5 6.1 7.9 6.6 5.3 5.8 6.40 t c t&c rate 12.0 96.1 20.2 12.2 12.0 47.7 17.2 11.1 12.0 27.1 13.9 10.2 12.0 17.0 11.1 9.6 12.0 11.6 8.8 9.0 12.0 8.5 7.1 8.5 4.60 t c t&c rate 21.4 133.7 35.2 21.7 21.4 66.4 28.5 19.6 21.4 37.6 22.1 17.9 21.4 23.7 16.9 16.6 21.4 16.2 13.1 15.5 21.4 11.8 10.3 14.6 2.80 t c t&c rate 53.1 219.7 79.2 53.4 53.1 109.1 59.2 47.7 53.1 61.8 42.8 43.3 53.1 38.9 31.1 39.7 53.1 26.6 23.1 36.8 53.1 19.4 17.8 34.4 1.00 t c t&c rate 365.0 615.2 365.0 365.0 365.0 305.4 229.2 320.8 365.0 173.2 146.6 286.4 365.0 108.9 98.3 258.8 365.0 74.4 69.6 236.1 365.0 54.4 52.0 217.2 Table 7 Sample data of T and C in example two. No. ti (h) ci (No. of lots) No. ti (h) ci (No. of lots) 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 9.6 8.0 16.0 11.4 0.1 4.1 19.7 6.6 24.4 46.5 2.7 57.3 23.2 8.4 12.8 1.1 14.6 4.7 38.8 45.9 3.9 17.0 18.1 11.6 10.2 8 6 13 15 13 16 13 12 8 11 15 14 9 8 6 8 13 14 15 10 5 7 13 14 12 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 7.4 20.3 32.1 72.9 29.5 8.8 12.2 4.1 12.5 2.7 2.7 11.0 2.2 11.0 2.0 1.2 6.5 29.4 9.3 46.3 1.3 88.8 11.1 22.2 26.4 12 8 13 16 13 14 19 17 7 6 11 10 10 9 20 12 7 14 11 17 9 12 12 6 10 It is noted that the rate chart is insensitive to the simultaneous changes of T and C in the same direction (e.g., both T and C increase at the same time). However, most of the users may not be interested in signalling such changes, as the event rate (in terms of the loss per unit time for a negative event or the gain per unit time for a positive event) will be untouched. However, in some applications, a user may want to signal any change of T regardless of C , then he should use a t chart. Similarly, if a user is interested in any change of C regardless of T , he may select a c chart. Author's personal copy 3423 Y. Liu et al. / Computational Statistics and Data Analysis 53 (2009) 3412–3425 Table 8 ATS values of four control charts in example two. Chart δλ δg 0.35 0.48 0.61 0.74 0.87 1.00 1.00 t c t&c Rate 1440.0 28.2 40.3 82.0 1440.0 49.4 84.4 145.5 1440.0 101.6 195.2 258.1 1440.0 235.3 452.1 457.7 1440.0 595.4 916.2 811.9 1440.0 1613.0 1440.0 1440.0 0.82 t c t&c Rate 742.4 34.4 47.0 60.2 742.4 60.2 94.2 104.3 742.4 123.9 201.9 180.9 742.4 287.0 403.6 309.8 742.4 726.1 657.4 524.0 742.4 1967.1 846.2 876.8 0.64 t c t&c Rate 423.6 44.1 57.6 61.9 423.6 77.2 108.1 96.2 423.6 158.8 203.5 150.4 423.6 367.7 332.0 233.9 423.6 930.3 439.3 360.4 423.6 2520.4 496.3 550.7 0.46 t c t&c Rate 262.2 61.3 73.4 68.6 262.2 107.4 122.0 94.5 262.2 220.9 189.3 131.7 262.2 511.5 250.1 183.5 262.2 1294.3 285.7 254.6 262.2 3506.6 300.9 351.4 0.28 t c t&c Rate 194.0 100.7 102.3 89.5 194.0 176.4 140.6 108.6 194.0 362.9 176.3 133.2 194.0 840.4 198.5 164.0 194.0 2126.4 208.7 201.8 194.0 5760.8 212.6 248.0 0.10 t c t&c Rate 257.0 281.9 217.4 200.4 257.0 493.9 240.5 214.1 257.0 1016.2 254.9 229.7 257.0 2353.1 261.8 247.1 257.0 5953.9 264.6 266.0 257.0 16130.3 265.6 286.6 Acknowledgments This research work is supported by Natural Science Foundation of China (No. 70572044). Appendix. Computation of ATS of the rate chart Firstly, let ATS z be the zero-state ATS of a rate chart. Under the zero-state mode, the process shift is assumed to take place from the beginning or immediately after an occurrence of the attribute event. ATS z is determined as follows (Wu and Spedding, 2000): ATSz = 1 λ · Pz , (27) where Pz is the power of the rate chart under zero-state mode. Computation of ATS of the rate chart for detecting upward R shifts For detecting upward R shifts, Pz = 1 − FR (UCLr ). (28) The cumulative probability function FR () is calculated by Eq. (9). Next, suppose that an R shift occurs at a time moment Ts between the (k − 1)th and the kth occurrences of the event. The time interval between the (k − 1)th occurrence and Ts is denoted as T0 and the time interval between Ts and the kth occurrence is indicated by T1 . The time interval T between the (k − 1)th and the kth occurrences is given as, T = T0 + T1 . The cumulative probability function of T in the above equation is calculated by FT (t ) = FT0 +T1 (t ) = Pr(T0 + T1 < t ) t = Z t = Z Pr(T0 + y < t |y = T1 ) · fT1 (T1 )dT1 0 0 Pr(T0 < t − y|y = T1 ) · fT1 (T1 )dT1 (29) Author's personal copy 3424 Y. Liu et al. / Computational Statistics and Data Analysis 53 (2009) 3412–3425 = Z t 1 − e−λ0 (t −T1 ) · λ · e−λT1 dT1   0    e−λ0 t (1 − e−(λ−λ0 )t )  1 − e−λt λ , − = λ λ − λ 0  1 − e−λ0 t − λ0 te−λ0 t , if λ = λ0 . if λ 6= λ0 , (30) It is noted that both T0 and T1 are exponentially distributed random variables with the in-control parameter λ0 and out-of-control parameter λ, respectively. Next, let Pk be the probability that the R shift is signalled at the kth occurrence of the event (or the first occurrence after the R shift). Then, (1 − Pk ) is the complementary probability that the R shift is not detected by the kth occurrence, but by a following one. For the first case, ATS is equal to 1/λ; and for the complementary case, ATS equals (1/λ + ATS z ). It is noted that, after the R shift occurs, λ always takes the out-of-control value and therefore the zero-state mode always come to play after the kth occurrence of the event. Now, the steady-state ATS can be calculated by (Wu and Spedding, 1999), ATS = Pk · 1 + (1 − Pk ) λ  1 λ  (31) + ATSz . Finally, the computation of Pk needs be elaborated. It is the probability that the sample value of C /T (or C /(T0 + T1 )) in the kth occurrence falls beyond the control limit. For detecting upward R shifts, Pk = Pr  = Pr  = ∞ X C =0 C > UCLr T C T0 + T1 FT0 +T1   > UCLr C UCLr    = Pr T0 + T1 < · p(C ) ≈ Uc X C =0 FT0 +T1 C UCLr  C UCLr   · p(C ), (32) where FT0 +T1 () is calculated by Eq. (30) and p(C ) by Eq. (5). The procedure for calculating the steady-state ATS of the rate chart detecting upward R shifts is summarized below: (1) Calculate the power Pz using Eq. (28). (2) Calculate the zero-state ATS z using Eq. (27). (3) Calculate the probability Pk using Eq. (32). (4) Finally, compute the ATS using Eq. (31). Computation of ATS of the rate chart for detecting downward R shifts The above four steps can also be used to calculate the steady-state ATS of the rate chart for detecting downward R shifts subject to some modifications for steps (1) and (3). In step (1), the power Pz will be calculated as follows: (33) Pz = FR (LCLr ). In step (3), the probability Pk is determined by Pk = Pr  = Pr  C < LCLr T C T0 + T1  < LCLr = ∞  X  ≈ Uc  X  C =0 C =0 1 − FT0 +T1 1 − FT0 +T1    C = Pr T0 + T1 > C LCLr C LCLr LCLr  · p(C )  · p(C ) All the formulae derived above have been verified by simulation results. (34) Author's personal copy Y. Liu et al. / Computational Statistics and Data Analysis 53 (2009) 3412–3425 3425 References Borror, C.M., Keats, J.B., Montgomery, D.C., 2003. Robustness of the time between events CUSUM. 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