A MIXED CONTROL CHART ADAPTED TO THE TRUNCATED LIFE TEST BASED ON THE WEIBULL DISTRIBUTION

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1 O P E R A T I O N S R E S E A R C H A N D D E C I S I O N S No. 27 DOI:.5277/ord73 Nasrullah KHAN Muhammad ASLAM 2 Kyung-Jun KIM 3 Chi-Hyuck JUN 4 A MIXED CONTROL CHART ADAPTED TO THE TRUNCATED LIFE TEST BASED ON THE WEIBULL DISTRIBUTION The design of a new mixed attribute control chart adated to a truncated life test has been resented. It was assumed that the lifetime of a roduct follows the Weibull distribution and the number of failures was observed using a truncated life test, where the test duration was secified as a fraction of the mean lifesan. The roosed control chart consists of two airs of control limits based on a binomial distribution and one lower bound. The average run length of the chart was determined for various levels of shift constants and secified arameters. The efficiency of the chart is comared with an existing control chart in terms of the average run length. The alication of the roosed chart is discussed with the aid of a simulation study. Keywords: life test, Weibull distribution, attribute chart, binomial distribution, average run length. Introduction A control chart is one of the most imortant tools for monitoring a manufacturing rocess in industry. Time series of statistics of interest are lotted on the control chart to see whether the rocess is under control or out of control. If a lotted statistic is above Deartment of Statistics, Jhang Camus, University of Veterinary and Animal Sciences, Lahore 54, Pakistan, address: nas_shan@hotmail.com 2 Deartment of Statistics, Faculty of Sciences, King Abdulaziz University, Jeddah 255, Saudi Arabia, address: aslam_ravian@hotmail.com 3 Deartment of Industrial and Management Engineering, POSTECH, Pohang 37673, Reublic of Korea, address: k_kim@ostech.ac.kr 4 Deartment of Industrial and Management Engineering, POSTECH, Pohang 37673, Reublic of Korea, address: chjun@ostech.ac.kr

2 44 N. KHAN et al. the uer control limit (UCL) or below the lower control limit (LCL), the rocess is said to be out-of-control. If the statistic lies within the control limits, the rocess is said to be under control. There are many factors which might cause the rocess to shift from the under-control state to the out-of-control state. In this situation, quick indication is needed to bring the rocess back under control. Delay in indicating a roblem may cause defective roducts. Usually, control charts are designed under the assumtion that the quantity of interest follows the normal distribution. In industry, however, it may be ossible that the quantity of interest follows some non-normal distribution. In such situations, a control chart designed under the assumtion of normality may mislead an engineer when making a decision about whether the rocess is under control or not. Detailed studies about some control charts develoed for non-normal distributions can be found in [ 8]. Further, the selection of control charts according to the tye of data collected is an imortant issue. If the data are obtained from a counting rocess, they are called attribute data. On the other hand, data obtained from a measurement rocess are called variable data. For attribute data, an attribute control chart, such as an n chart is used, while for variable data, a variable chart such as an X-bar chart is used to monitor the rocess. An attribute control chart has the advantage of simlicity in designing the control limits because it usually involves a binomial or Poisson distribution. In general, however, variable data are more informative, so a variable control chart may be more effective. However a variable control chart requires knowledge about the robability distribution of the lotting statistic which cannot be derived for some non-normal distributions. Sometimes, there is a need to use a mixture of attribute and variable control charts to monitor a manufacturing rocess. The use of such a mixed control chart may enjoy the combined advantages of both attribute and variable control charts. Recently, [9] designed a mixed control chart to monitor manufacturing rocesses. [] designed a mixed chart using an exonentially weighted moving average statistic. [] roosed a mixed control chart for attribute data. By exloring the literature on control charts, we note that there is no work on designing a mixed control chart adated to the truncated life test. In this aer, we focus on designing a new mixed control chart adated to the time truncated life test based on a Weibull distribution. The alication of the roosed chart is discussed with the aid of a simulation study. The efficiency of the roosed chart is comared with the control chart roosed by []. 2. Design of the roosed control chart It is assumed that the lifetime of a roduct (denoted by the random variable X) follows the Weibull distribution with the scale arameter and the shae arameter and thus has the following cumulative distribution function (cdf):

3 A mixed control chart adated to the truncated life test 45 t F t;, ex, t () The average lifetime based on this Weibull distribution is given as follows: (2) If the shae arameter is known, then the transformed variable Y X follows an exonential distribution with mean. Let be the average lifetime when the rocess is under control, which corresonds to the scale arameter. It is assumed that the scale arameter changes to c, while the shae arameter remains unchanged, when the rocess is shifted. In this section, we will roose a mixed tye of control chart utilizing the failure data from a truncated life test. Ste. Take a samle of size n from the roduction rocess. Test them until the secified time t. Ste 2. Count the number of failures (d, say) in Ste. Declare the rocess to be out-of-control if d UCL or d LCL. Declare the rocess to be under-control if LCL d UCL Otherwise, go to Ste Ste 3. For the samle described in Ste, obtain the time to failure of item i (denoted by X i ). Set Xi t if item i has not failed by time t. Calculate Yi X i and Y. Declare the rocess as out-of-control if Y L. 3 Declare the rocess as under-control if Y L. 3 We call the roosed control chart a mixed control chart because Ste 2 is based on attribute data and Ste 3 is based on variable data. There are two airs of control limits in Ste 2 which will be described later, while there is one cutoff value in Ste 3. Using the two airs of control limits, a quick decision can be made as to whether the number of failures is small or large and the second decision will be made based on the failure times when the number of failures is moderate. 2.. Control limits and in-control ARL We will first derive the necessary measures used for the roosed control chart when the rocess is under control. It would be convenient to select the secified test time t as a fraction of the in-control mean, i.e. t, a where a is a constant. For

4 46 N. KHAN et al. examle, when a =.5, the test time is the half of the mean lifetime of a roduct when the rocess is under control. Thus, the number of failures by time t (denoted by d) follows a binomial distribution with arameters n and, where t / ex ex a (3) Therefore, we roose two airs of chart limits to be used in Ste 2 as follows: UCL n k n (4a) LCL max, n k n (4b) 2 2 UCL n k n (5a) LCL max, n k n 2 2 (5b) where k and k 2 are control coefficients to be determined. The distribution of Y in Ste 3 can be aroximated by a normal distribution according to the central limit theorem. The mean and the variance are derived as follows: E Y E Y E Y Y t P Y t E Y Y t P Y t (6) It should be noted that for the Weibull distribution we take a ower transform because the transformed variable follows an exonential distribution and the sum of Y s (or Y ) follows a gamma distribution. Thus, Eq. (4) can be written as t / y y/ t / t y E Y e dy t e dy e e a / (7)

5 A mixed control chart adated to the truncated life test 47 The variance of Y can be written as follows: Var Y Var Y E Y E Y n n 2 2 (8) 2 2 t y Var Y e e e n t y/ 2 y/ 2 t / dy t dy (9) After simlification, Var Y can be rewritten as 2t/ t/ t 2 Var Y e 2 e n () Therefore, the robability of the rocess being declared under control using the roosed chart when the rocess is actually under control is given as follows: where P f f f () in, 2 3 n UCL2 d nd f P LCL2 d UCL2 d LCL2 d f P UCL d UCL P LCL d LCL n n UCL LCL2 d nd d nd d UCL2 d d LCL d L3 E Y f3 P Y L3 P Z Var Y a/ / L3 / e 2 2 a 2 a/ / 2 e n / e / a/ /

6 48 N. KHAN et al. The erformance of the roosed control will be evaluated using the average run length (ARL), which is used to indicate the mean length of time that asses before the rocess is declared out-of-control. When the rocess is under control, the ARL (called the under-control ARL) should be large. However, when the rocess has shifted, the ARL (called the out-of-control ARL) should be small as ossible. The ARL for a rocess under control is denoted by ARL and given as follows: ARL f f f (2) Out-of-control ARL Next, we derive some necessary measures under the assumtion that the rocess has shifted. Suose that there has been a change in the scale arameter of the Weibull distribution, while the shae arameter remains unchanged. The scale arameter of the Weibull distribution is assumed to be shifted to c, where c is a shift constant smaller than. For the shifted rocess, the robability of being declared under control using the roosed chart is given as follows: where where P f f f (3) in, 2 3 n UCL2 d nd f P LCL2 d UCL2 d LCL2 d f P UCL d UCL P LCL d LCL n n UCL LCL2 d nd d nd d UCL2 d d LCL d 3 3 f P Y L (4) t / ex ex a c

7 A mixed control chart adated to the truncated life test 49 Table. The values of ARLs when = and = 5 a k k L c ARL Table 2. The values of ARLs when = and = a k k L Shift ARL The distribution of Y for the shifted rocess can be aroximated by a normal distribution with mean and variance of / e t (5) E Y

8 5 N. KHAN et al. Table 3. The values of ARLs when =.5 and = 5 a k k L Shift ARL Table 4. The values of ARLs when =.5 and = a k k L Shift ARL t / / Var t Y e 2t e n (6)

9 A mixed control chart adated to the truncated life test 5 Thus, Table 5. The values of ARLs when = 2 and = 5, n = a k k L Shift ARL f L3 E Y Var Y 3 (7) or equivalently f 3 a c / c L3 e / 2 a 2 / 2 a / c c ac c e 2 e n / / (8) Hence, the out-of-control ARL for the shifted rocess is given from Eq. (9). ARL f f f (9) 2 3

10 52 N. KHAN et al. The values of the ARLs for various values of the target mean, and samle size n = 3 are reorted in Tables 5. From the tables, we note the following trend in the ARL values. When all other arameters are fixed, the ARL increases as the target ARL increases. When all other arameters are fixed, the ARL increases as increases. When all other arameters are fixed, the ARL increases as increases. 3. Comarative study In this section, a comarison of the roosed chart with the one roosed by Aslam et al. [] is given. The efficiency of the roosed chart has been comared in terms of the ARLs. A control chart is said to be more efficient than another if it rovides smaller values of the ARLs for the same values of all the arameters secified. To save sace, we will resent the ARLs by Aslam et al. [] for r = 37, =.5, a =.5,.7,.9, and = (Table 6). Table 6. Comarison of the resent method with the one roosed by Aslam et al. [] a..5 Present Aslam et al. [9] Present Aslam et al. [9] Present Aslam et al. [9] k = k = k = k = k2 =.38 k = k2 =.73 k = k2 =.433 Shift n = 2 L3 =.5 n = 3 L3 = n = 49 L3 = ARL From Table 6, we note that the roosed control chart rovides smaller values of the ARLs as comared to the one roosed by Aslam et al. []. For examle, when c =.9 and a =.5, the ARL for the roosed chart is 87, while it is 44 for the existing control chart. The erformance of the roosed chart is better for all values of c when a >.5.

11 NUMBER OF DEFECTIVE (DI) A mixed control chart adated to the truncated life test Simulation study In this section, we discuss the imlementation of the roosed mixed chart for a time truncated life test using simulated data. The first 2 observations of subgrous of size n =3 are generated from the Weibull distribution with =.5 and = 55. so that = 5. The next 2 observations are generated from the Weibull distribution with =.5, and = Let us choose a =.. Thus, the test time will be t = a = 5.. Hence, we declare a failure if the lifesan generated is smaller than 5. We counted the number of failures in each subgrou, which is listed here: d s:,,,, 2,,,,,,,,,,,,,,,,,,,,,,,,, 2, 2,,,, 5, 2,,,,. From Table 3, the two control chart constants and L 3 are given by: k , k2.38, L3.5. Also, we have / ex a UCL=4; 4 Di; UCL2=2; 2 LCL=LCL2=; Fig.. The roosed control chart for simulated data Therefore, the two airs of control limits using the roosed chart for these simulated data are given as follows: UCL n k n

12 54 N. KHAN et al. LCL max[, n k n ] max, UCL n k n ] 2 LCL LCL max, n k n 2 2 max, We lotted the number of defects in Fig. along with the two airs of control limits. The roosed chart requires Ste 3 (calculating Y ) only when the number of failures is 3 (between 2 and 4) but there are no such cases for these data. We note that the roosed chart detects the rocess shift based on the 4th set of observations after the actual shift. 5. Concluding remarks A mixed control chart for a life test is roosed by assuming that the lifetime of a roduct follows the Weibull distribution. Extensive tables are rovided for industrial use. The alication of the roosed chart is discussed with the hel of simulated data. The efficiency of the roosed chart is comared with an existing chart and it is concluded that the roosed chart is more efficient in detecting a shift in the manufacturing rocess comared to the existing control chart. Proosals for charts based on other nonnormal lifesan distributions should be considered in future research. Acknowledgements The authors are deely thankful to the Editor and Reviewers for their valuable suggestions to imrove the quality of this manuscrit. This work was funded by the Deanshi of Scientific Research (DSR), King Abdulaziz University, Jeddah. The author, Muhammad Aslam, therefore, acknowledges with thanks the technical and financial suort from the DSR. The work by Chi-Hyuck Jun and Kyung-Jun Kim was a art of the roject titled Develoment of TCS System on ECO-Shi Technology, funded by the Ministry of Oceans and Fisheries, Korea. References [] AL-ORAINI H.A., RAHIM M., Economic statistical design of X control charts for systems with Gamma (λ, 2) in-control times, Com. Ind. Eng., 22, 43 (3),

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