Designing of parameters plays an important role in economic design of control charts for lowering the cost and time. Manipulating sample size (n) and sampling interval (h), the effect of double exponentially weighted moving average (DEWMA) model was studied for the Economic Design (ED) of X̅ control chart. Optimum sizes and level were obtained when the characteristics of an item possesses DEWMA model. When shifts are uncertain the optimal design for DEWMA chart should be more conservative and should be implemented for benefiting the consumers as well as producers

Khanday, Manzoor A.

Singh, J. R.

English

Digital Commons@Wayne State University

Journal of Modern Applied Statistical Methods Volume 18 Issue 2 Article 27 10-2-2020 Economic Design of X ̅ Control Chart under Double EWMA Manzoor A. Khanday Vikram University Ujjain, manzoorstat@gmail.com J. R. Singh Vikram University Ujjain Follow this and additional works at: https://digitalcommons.wayne.edu/jmasm  Part of the Applied Statistics Commons, Social and Behavioral Sciences Commons, and the Statistical Theory Commons Recommended Citation Khanday, M. A., & Singh, J. R. (2019). Economic design of X ̅ control chart under double EWMA. Journal of Modern Applied Statistical Methods, 18(2), eP2793. doi: 10.22237/jmasm/1604189520 This Regular Article is brought to you for free and open access by the Open Access Journals at DigitalCommons@WayneState. It has been accepted for inclusion in Journal of Modern Applied Statistical Methods by an authorized editor of DigitalCommons@WayneState. Economic Design of X ̅ Control Chart under Double EWMA Cover Page Footnote Acknowdgements I am highly thankful to reviewers and editorial board that they have suggested the concrete comments for the modification of the paper and same has been carried out by me in the above paper. This regular article is available in Journal of Modern Applied Statistical Methods: https://digitalcommons.wayne.edu/jmasm/vol18/iss2/27 Journal of Modern Applied Statistical Methods November 2019, Vol. 18, No. 2, eP2793. doi: 10.22237/jmasm/1604189520  Copyright © 2020 JMASM, Inc. ISSN 1538 − 9472    doi: 10.22237/jmasm/1604189520 | Accepted: February 26, 2018; Published: October 2, 2020. Correspondence: Manzoor A. Khanday, manzoorstat@gmail.com    2 Economic Design of X̅ Control Chart under Double EWMA Manzoor A. Khanday Vikram University Ujjain Ujjain, India J. R. Singh Vikram University Ujjain Ujjain, India   Designing of parameters plays an important role in economic design of control charts for lowering the cost and time. Manipulating sample size (n) and sampling interval (h), the effect of double exponentially weighted moving average (DEWMA) model was studied for the Economic Design (ED) of X̅ control chart. Optimum sizes and level were obtained when the characteristics of an item possesses DEWMA model. When shifts are uncertain the optimal design for DEWMA chart should be more conservative and should be implemented for benefiting the consumers as well as producers.  Keywords: Control chart, economic design of control chart, DEWMA  Introduction Control charts are the essential statistical tools for evaluating process presentation, dealing process capability, observing processes and refining processes. During the observation, the statistical control chart is employed to detect the occurrence of assignable cause. The Shewhart X̅ chart assumed to be the best statistical tool in Statistical process control. However, it has limitations int the detection of small and moderate process mean shifts. Therefore, a widespread control chart used for small alterations of process mean moved to the EWMA by Roberts (1959). The exertion to rise the sensitivity of EWMA control chart for identifying small alterations and logic in a process, a double EWMA (DEWMA) control chart was developed. Zhang (2002) considered the DEWMA control charts for the mean. The traditional EWMA and DEWMA control charts for watching process means were developed under the likelihood of normality. Other extensions of control charts to improve process monitoring were Saghaei et al. (2014), Amiri et al. (2015), and Lee et al. (2014). Akhavan Niaki et al (2013a, b) presented the ESD of the VSI KHANDAY & SINGH 3 X̅ control charts for the correlated and non-normal data. Akhavan Niaki et al. (2013c) studied the ESD of the VSI X̅ control charts in the existence of the multiple assignable causes with normal and correlated data. The design of a control chart depends on sample size (n), the control limits (k), and the time interval between samples (h). Various costs are linked with control chart usage, which includes sampling cost and testing, the cost related with examining out-of-control signals and modifying the assignable causes, and the costs of allowing non-conforming units reaching the customer. Khanday and Singh (2015) studied the effect of Markoff’s model on Economic design of X̅ control charts under independent observations. Duncan’s Model for the Cost Function Duncan (1956) obtained an approximate function for the average net income per hour of using the control chart for mean of normal variables as   ( )0BI1 BM T h W b cnVh  + + += − −+.  (1)  Duncan’s cost model indicates  (i) The rate of an out-of-control conditions; (ii) The rate of false alarms; (iii) The rate of finding an assignable cause; and (iv) The rate of sampling inspection, evolution, and plotting.  The average cost per hour involved for maintaining the control chart is (b + cn) / h. The average gross income per hour of the process under the investigation of the control chart for mean can be redrafted as I = V0 – L, where   ( )BL1 BM T h W b cnh  + + += −+.  (2)  L can be treated as the per hour cost due to the surveillance of the process under the control chart. The probability density function is denoted by P′ and α′, which are determined from the sampling distribution of mean and are written as  ECONOMIC DESIGN OF CONTROL CHART UNDER DOUBLE EWMA 4  ( )P 1 Φ  = − ,  (3)   ( )α α 2ΦN k = = − ,  (4)  where ( )k n = − . Derivation for Ideal Value of Sample Size n and Sampling Interval h The optimum value of sample size (n) and sampling interval (h) may be obtained either by maximizing the gain function I or by minimizing the cost function L. Differentiating L with respect to n and h and equating to zero,   ( )( )2B α α B1 B BL01 BT TM M Wcn h n h nn h          + + − + +         = + = +,  (5)   ( )( )22 2B α α B1 BL01 BT TB M M Wb cnh h h hh h         + − − + +    +      = − =   +,  (6)  where   2B P B 1 1 α, , 0P P 2 6h hcn n h n  −   = + = − + =    ,  (7)   ( )P2n n =.  (8)  The solutions of the equations (5) and (6) for n and h yield the required optimum values, and may be rewritten  KHANDAY & SINGH 5  ( )2α B α0 BB αB 1 BT Th M M Wh n h nTM cn h n       = − − − +     + + + +     (9)   ( )( )( )222 2α B0 B α 1 BBB 1 BTh M M W Th nh M b cnn      = − − − − +  + − + +  (10)  If η is small, and the optimum h is crudely of order of 1  , ignore terms with ηB containing ηWc, (α′T) / h, and the terms equating higher powers of η. The equations (9) and (10) are simplified:   22pα 0ph MT cn− − + = ,  (11)   ( )21 1α 0P 2Mh T b cn − − + + =  .  (12)  From equation (12),   12α1 1P 2T b cnhM   + + =    −    .  (13)  By eliminating h from equation (11),   2α pα 01 1PP 2T b cnT cn − + + − + =  −  .  (14)  ECONOMIC DESIGN OF CONTROL CHART UNDER DOUBLE EWMA 6 The values of n for which equation (14) holds true gives the optimum value of sample size n. Using this value of n in equation (13), obtain the optimum value of the sampling interval (h). Derivation of the Optimum Values of Sample Size (n) and Sampling Interval (h) under DEWMA Suppose a process is on target μ initially and successive measurements X̅t, (t = 1, 2, 3,…) are taken it may be average of several measurements taken at time ‘t’ to check whether there is a shift from the target. To use a control chart based on the statistic   ( ) ( )1 11 and 1t t t t t tY X Y Z Y Z   − −= + − = + − ,  (15)  such that 0 < λ < 1 and Y0 = Z0 = μ0,   ( )( ) ( ) ( )2 0 011 1 1 1tt j t tt jjZ t j X t Y Z    −== − + − + − + −   (16)  where E(Zt) = μ0 and   ( ) ( ) ( )( ) ( )( )( )2 2 2 2 2 42 22 40321 1 1 2 2 1 1 11 1tt t tZt t t     + + + − − + + − − − − =  − −  .  (17)  The upper and lower control limits for DEWMA control chart are   ( )( )( )( )020 320 32 2L22 2L2UCLUCLUCL      =− += +−− += −−  (18)  KHANDAY & SINGH 7 Assume Xt is drawn independently from a normal distribution with variance σ2 and t is sufficiently large. Unfortunately, λ is arbitrary and lies between 0 and 1. Suppose a machine whose performance can be effectively represented by a single unknown quality μ is inspected regularly to see whether the quality of performance is deteriorated. The successive performance level μ1, μ2, μ3,…, μt are tracked by the observations x1, x2, x3,…, xt. The operation continues until a decision is made to overhaul it in which case the level is set to zero instantaneously and the whole sequence begins again. This resetting after overhaul may be subject to error. Assuming μ0 is N(0, σ2 / n) and each subsequent state of repair is drawn independently from this distribution:   ( ) ( )( )( )22 220 32 2E and Var2t tZ Z gn n   − += = =−,   where   ( )( )2232 22g  − +=−.  (19)  For the DEWMA model, the probability density function is represented as   ( )P 1 Φ for 0e e  = −  ,  (20)   α αe Ne = ,  (21)  where   ( ),α 2e Nek n kg g−  −= =   ,   and equations (5) and (6) are reduced to  ECONOMIC DESIGN OF CONTROL CHART UNDER DOUBLE EWMA 8  ( )( )2α αB B1 B BL01 Be eTTM M Wcn h n h nn h         + + − + +         = + = +  (22)   ( )( )2 2α αB B1 B BL1 B0e eT TM M Wb cnn h h nh h         + − − + +    +      = −    +=  (23)  where   ( )2P 1 1, ,P P 2 6α P0,2ee ee eeB h B hcn n hn n ng  − = + = − +     = =   (24)  By solving equations (22) and (23),   12α1 1P 2eeT b cnhM   + + =    −      (25)  and   2α Pα 01 1PP 2e eeeeT b cnT cn + + − − + =  −  .  (26)  The values of n for which the equation (26) holds true yield us the required optimum value of sample size n. Substituting this value n in equation (25) yields the ideal value of the sampling interval (h) under non-normality for DEWMA model. KHANDAY & SINGH 9 Numerical Illustration In order to illustrate, let k = 1.0, 1.5, 2.0, 2.5, and 3.0; δ = 1.0, 1.5, and 2.0; η = 0.01, M = 100, W = 25, T = 50, C = 0.05, D = 2, b = 0.5, c = 0.1, and λ = 1, 0.8, 0.6, and 0.2 to determine the optimum values of sample size and sampling interval. The values of n and h are presented in Table 1, it is clearly seen that for given k and δ the sample size n and sampling interval h decrease with the decrease in the values of λ. However, the sample size n has a dramatic effect while decreasing of k, while the sampling interval h has dramatic effect decreasing k. When the rate of occurrences of assignable cause is fixed, the value of sample size and sampling interval are different for different values of λ (shifts). The effect is more serious for DEWMA model for different parameters. In general, the in-control ARL acts of DEWMA control charts were more vigorous. The gradation of strength of DEWMA control chart increases for smaller values of flattening parameter. Thus, the performance of DEWMA is conservative and is recommended as it performs better results than that of EWMA model for smaller shifts. The DEWMA chart employed together with economic design of the control chart scheme when minor shifts to restrained shifts in the mean of the controlled parameter are expected. From an economic point of view DEWMA chart performs better when there are small shifts and spoiled data thus, we recommend DEWMA model in such cases.   Table 1. Optimum sample size (n) and sampling interval (h) under DEWMA model for different values of λ    k = 3.0  k = 2.5  k = 2.0  k = 1.5  k = 1.0 λ δ n h   n h   n h   n h   n h 1.0 1.0 23 2.3371  20 2.4134  19 3.0258  35 4.6216  29 6.2076  1.5 11 1.8026  9 1.9963  9 2.7064  16 4.1978  14 5.9551  2.0 6 1.5648  6 1.8157  6 2.5740  9 4.0314  8 5.8593                 0.8 1.0 19 2.1217  15 1.9265  11 1.8554  10 2.4322  17 4.3902  1.5 9 1.6199  7 1.5147  5 1.5448  5 2.2319  8 4.1850  2.0 5 1.3972  4 1.3342  3 1.4116  3 2.1495  5 4.1063                 0.6 1.0 16 2.0240  12 1.8264  9 1.6409  6 1.5734  6 2.6078  1.5 7 1.5565  6 1.4445  4 1.3437  3 1.3669  3 2.4991  2.0 4 1.3514  3 1.2794  3 1.2170  2 1.2802  2 2.4559                 0.2 1.0 13 1.8646  9 1.6780  7 1.5029  4 1.3440  2 1.2088  1.5 6 1.4562  4 1.3517  3 1.2567  2 1.1736  1 1.1058   2.0 3 1.2810   2 1.2147   2 1.1557   1 1.1049   1 1.0646  ECONOMIC DESIGN OF CONTROL CHART UNDER DOUBLE EWMA 10 References Akhavan Niaki, S. T., Gazaneh, F., & Toosheghanian, M. (2013a). A parameter-tuned genetic algorithm for economic-statistical design of variable sampling interval X-bar control charts for non-normal correlated samples. Communications in Statistics – Simulation and Computation, 43(5), 1212-1240. doi: 10.1080/03610918.2012.732176 Akhavan Niaki, S. T., Gazaneh, F., & Toosheghanian, M. (2013b). Economic design of variable sampling interval X-bar control charts for monitoring correlated non normal samples. Communications in Statistics – Theory and Methods, 42(18), 3339-3358. doi: 10.1080/03610926.2011.628769 Akhavan Niaki, S. T., Gazaneh, F., & Toosheghanian, M. (2013c). Economic design of VSI X̅ control chart with correlated nonnormal data under multiple assignable causes. Journal of Statistical Computation and Simulation, 83(7), 1279-1300. doi: 10.1080/00949655.2012.658395 Amiri, A., Moslemi, A., & Doroudyan, M. H. (2015). Robust economic and economic statistical design of EWMA control chart. The International Journal of Advanced Manufacturing Technology, 78 511-523. doi: 10.1007/s00170-014-6667-9 Duncan, A. J. (1956). The economic design of X chart used to maintain current control of a process. Journal of the American Statistical Association, 51(274), 228-242. doi: 10.1080/01621459.1956.10501322 Khanday, M. A., & Singh, J. R. (2015). Effect of Markoff’s model on economic design of X- bar control chart for independent observations. International Journal of Scientific Research in Mathematical and Statistical Sciences, 2(2), 1-6. Retrieved from https://www.isroset.org/journal/IJSRMSS/archive_issue.php?pub_id=59 Lee, S. H., Park, J. H., & Jun, C. H. (2014). An exponentially weighted moving average chart controlling false discovery rate. Journal of Statistical Computation and Simulation, 84(8), 1830-1840. doi: 10.1080/00949655.2013.769540 Roberts, S. W. (1959). Control charts tests based on geometric moving average. Technometrics, 1(3), 239-250. doi: 10.1080/00401706.1959.10489860 Saghaei, A., Ghomi, S. M. T. F., & Jaberi, S. (2014). Economic design of exponentially weighted moving average control chart based on measurement error KHANDAY & SINGH 11 using genetic algorithm. Quality and Reliability Engineering International, 30(8), 1153-1163. doi: 10.1002/qre.1538 Zhang, L. Y. (2002). EWMA control charts and extended EWMA control charts [Unpublished doctoral dissertation]. University of Regina, Saskatchewan, Canada. ECONOMIC DESIGN OF CONTROL CHART UNDER DOUBLE EWMA 12 Appendix V0 = the rate per hour at which income accrues from operation of the process is in control and process average is μ, V1 = the rate per hour at which income accrues from operation of the process when process is not in control and process average is μ′ = μ + δσ, M = V0 – V1, η = the average number of times the assignable cause occurs within an interval of time, starting in a state of control at time t = 0, the probability that the process will still be in control at time t1 is 1te − , B = ah + Cn + D, 1 12 12haP= − + , h = the interval between samples measured in hours, e = the rate at which the time taken between the sample and plotting of a point on the X̅ chart increases with the sample size n, Cn = the time required to take and inspect a sample of size n, D = average time taken to find the assignable cause after a point plotted on the chart falls outside the control limits, P = probability of detecting an assignable cause when it exists, ( ) ( )( )g g1 Φ for 0knknx dx x dxk n  −−+ = + − −    where g(x̅ / μ′) is the density function of x̅ when the true mean μ and Φ(x) is the normal probability, α = probability of incorrectly indicating the occurrence of assignable cause ( ) ( )g =2Φknknx dx k+−= − , T = the cost per cases of looking for an assignable cause, W = the average cost per event of finding the assignable cause when it exists, b = per-sample cost of sampling and plotting, that is independent of sample size, and c = the cost per unit of calculating an item in a sample. 

Economic Design of X̅ Control Chart under Double EWMA

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Economic Design of X̅ Control Chart under Double EWMA

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