hps://doi.org/0.545/mjis.08.600 Exponenially Weighed Moving Average (EWMA) Char Based on Six Dela Iniiaives KALPESH S. TAILOR Deparmen of Saisics, M. K. Bhavnagar Universiy, Bhavnagar-36400 E-mail: kalpesh_lr@yahoo.co.in Received: December 5, 07 Revised: February 0, 08 Acceped: February 8, 08 Published online: March 0, 08 The Auhor(s) 08. This aricle is published wih open access a www.chikara.edu.in/publicaions Absrac A conrol char is a graphical device for represenaion of he daa for knowing he exen of variaions from he expeced sandard. The echnique of conrol char was suggesed by W.A. Shewhar of Bell Telephone Company based on hree sigma limis. M. Harry, he engineer of Moorola has inroduced he concep of six sigma in 980. In six sigma iniiaives, i is expeced o produce 3.4 or less number of defecs per million of opporuniies. Moderae disribuion proposed by Naik and Desai is a sound alernaive of normal disribuion, which has mean and mean deviaion as pivoal parameers and which has properies similar o normal disribuion. Naik and Tailor have developed various conrol chars based on his disribuion. In his paper an aemp is made o consruc a conrol char based on six dela iniiaives for exponenially weighed moving average char. Suiable Table for mean deviaion is also consruced and presened for he engineers for making quick decisions. Keywords Moderae disribuion, EWMA, Six Dela.. INTRODUCTION The echnique of qualiy conrol was developed by W. A. Shewhar (93). I was based on 3sigma conrol limis. Mikel Harry (980), he engineer of Moorola has inroduced he concep of six sigma. He developed mehods for problem solving ha combined formal echniques, paricularly relaing o measuremen, o achieve measurable savings in millions of dollars. The companies, which are pracicing Six Sigma, are expeced o produce 3.4 or less number of defecs per million opporuniies R.Radhakrishnan and P.Balamurugan (00, 00, and 06) have developed six sigma based conrol chars for he number of defecives, exponenially weighed moving average and sandard deviaion. Mahemaical Journal of Inerdisciplinary Sciences Vol-6, No-, March 08 pp. 7 35
Tailor, KS Naik and Desai have proposed an alernaive of normal disribuion called moderae disribuion, which has mean ( µ ) and mean deviaion (δ ) as pivoal parameers and which has properies similar o normal disribuion. Naik and Tailor (05, 06) have suggesed 3δ (3 mean deviaion) conrol limis based on moderae disribuion. On he basis of 3 conrol limis, hey have developed X -char, R-char, s-char and d-char. Tailor (06) has also developed moving average and moving range char and exponenially moving average char under moderaeness assumpion. Similar o six sigma concep, he conceps of six dela can be developed. So here an aemp is made o develop six dela conceps similar o six sigma concep. The six sigma conrol limis are based on normaliy assumpion and he conrol limis are deermined by using sandard deviaion (σ-sigma) of he saisic, whereas he six dela conrol limis are based on moderaeness assumpion and he conrol limis are deermined by using mean deviaion (δ-dela) of he saisic. In six dela iniiaives, i is expeced o produce.7 or less number of defecs per million of opporuniies. If he companies pracicing Six Dela iniiaives use he conrol limis, hen no poin fall ouside he conrol limis because of he improvemen in he qualiy of he process. Tailor has proposed sample sandard deviaion(s) char and sample mean deviaion (d) char based on six dela iniiaives. Here an aemp is made o consruc a conrol char based on six dela iniiaives for exponenially weighed moving average. Suiable Table for mean deviaion is also consruced and presened for he engineers for making quick decisions.. CONCEPTS AND TERMINOLOGIES A. Upper specificaion limi (USL) I is he greaes amoun specified by he producer for a process or produc o have he accepable performance. B. Lower specificaion limi (LSL) I is he smalles amoun specified by he producer for a process or produc o have he accepable performance. C. Tolerance level (TL) I is he difference beween USL and LSL, TL = USL-LSL D. Process capabiliy (Cp) This is he raio of olerance level o six imes mean deviaion of he process. Cp = (TL/ 6 π δ ) = (TL/0.6369δ) = (USL-LSL)/0.6369δ 8
E. Mean deviaion (δ 6δ ): For many purposes mean deviaion is he mos useful measure of dispersion of a se of numbers. I is he mean of absolue deviaion. F. Qualiy Conrol Consan (M md ) The consan M md is inroduced in his paper o deermine he conrol limis based on six dela iniiaives for sample mean deviaion. G. Consan Facor (α) I is a consan, 0<, which is used o calculae exponenially weighed moving average. Exponenially Weighed Moving Average (EWMA) Char Based on Six Dela Iniiaives 3. THREE DELTA CONTROL LIMITS FOR EXPONENTIAL WEIGHTED MOVING AVERAGE (EWMA) CHART Tailor has proposed exponenially weighed moving average (EWMA) char under he moderaeness wih 3δ conrol limis. Suppose ha he main variable of he process x follows moderae disribuion. The mean of x is E(x) = μ and mean deviaion of x is δ = x δ. The EWMA funcion is defined as, Z = xi + ( ) Z, where 0<. i If he individual X are independen random variables wih variance σ n, hen he variance of Z is defined as Therefore σ σ = α n ( ) σ σ σ = α = α n ( ) ( ) n Since we are assuming moderaeness, he mean error of Z is defined as () δ = α n ( ) () Thus, he 3δ- conrol limis of EWMA char can be deermined as follows Cenral line (C.L) = E( X ) Lower conrol limi (L.C.L) = E( X ) 3δ = X (3) 9
Tailor, KS = X 3 n ( α) δ = X 5. 384 ( α ) (4) n Upper conrol limi (U.C.L) = EX ( ) + 3δ = X + 3 n ( α) δ = X + 5. 384 n ( α) (5) Where X and δ are ypically esimaed from preliminary daa as sample mean and sample mean deviaion. As α is small and if increases, he effec of saring value soon dissipaes and he mean error converges o is asympoic value. i.e δ = n The conrol limis for EWMA char are usually based on he asympoic mean deviaion of he saisic. Hence asympoic 3-conrol limis for his char can be derived as following way, Cenral line (C.L) = EX ( ) Lower conrol limi (L.C.L) = EX ( ) 3δ = X (6) = X 3 n = X 5. 384 δ n (7) 30
Upper conrol limi (U.C.L) = EX ( ) + 3δ = X + 3 n Exponenially Weighed Moving Average (EWMA) Char Based on Six Dela Iniiaives = X + 5. 384 δ (8) n 4. SIX DELTA BASED CONTROL LIMITS FOR EXPONENTIAL WEIGHTED MOVING AVERAGE (EWMA) CHART Fix he olerance level (TL) and process capabiliy (C p ) o deermine he process mean deviaion δ(ermed as δ 6δ ), which is calculaed from Cp = (TL/0.6369δ). For a specified TL and C p of he process, he values of δ 6δ is calculaed, and presened in able. The value of M md is obained by using 6 P( Z M md )= α, where α = 7. 0 and Z is a sandard moderae variae. Thus, he conrol limis for six dela based conrol char for EWMA are deermined as, LCL UCL CLδ X 6 δ = (9) = X Mmd n ( α ) (0) 6 6 = X + M 6 md 6 n ( α) Similarly, asympoic 6δ-conrol limis for his char can also be derived. 5. AN EMPIRICAL STUDY FOR EWMA CONTROL CHART AND COMPARISION OF THREE DELTA LIMITS AGAINST SIX DELTA INITIATIVES () To illusrae he use of EWMA conrol char wih hree dela and six dela limis, a daa se is aken from Lucas J. M and Crosier R.B. The daa, ogeher wih he corresponding EWMA values, are shown in Table. The arge value is aken o be 0. Three dela and six dela conrol limis are compued from his daa se, and conrol chars are ploed under hese wo limis. 3
Tailor, KS Table : Daa se. Observed value EWMA = + (- ) 0-0.0.0 0.5-0.5 0.063 3 0.0 0.047 4-0.8-0.65 5-0.8-0.34 6 -. -0.543 7.5-0.03 8-0.6-0.74 9.0 0.9 0-0.9-0.35. 0.98 0.5 0.74 3.6 0.855 4 0.7 0.87 5. 0.887 6.0.66 7.4.4 8.9.393 9 0.8.45 (a) Three dela conrol limis (asympoic) for EWMA char: Here he parameers of he EWMA are chosen o be α = 0.5, δ =, n = and he arge mean is aken as zero. Hence, he hree dela conrol limis are found as, LCL = -.00, CL = 0 and UCL =.00 (b) Conrol limis (asympoic) based on six dela iniiaives for EWMA char: 3
For a given daa se USL =.393, LSL = -0.03, TL =.393+0.03 =.45.43 and C p =.5. The value of δ 6 δ = 0.0896, which is found from he Table, M md = 5.85 which is calculaed from P(Z M md ) = - α, where 6 α = 7. 0. The oher parameers are chosen o be α = 0.5, n = and he arge mean is aken as zero.. Hence, he conrol limis based on six dela iniiaives for EWMA char for a specified TL and M md are deermined as, CL = 0, LCL = 0. 969, UCL = 0. 969 6δ 6δ 6δ Exponenially Weighed Moving Average (EWMA) Char Based on Six Dela Iniiaives Table : Values of for a specified C p and TL TL Cp.4.4.43.44.45.0 0.36 0.335 0.344 0.354 0.363. 0.05 0.4 0. 0.3 0.39. 0.05 0. 0.0 0.8 0.36.3 0.00 0.07 0.034 0.04 0.049.4 0.0947 0.0954 0.0960 0.0967 0.0974.5 0.0884 0.0890 0.0896 0.0903 0.0909.6 0.088 0.0834 0.0840 0.0846 0.085.7 0.0780 0.0785 0.079 0.0796 0.080.8 0.0736 0.074 0.0747 0.075 0.0757.9 0.0698 0.0703 0.0708 0.073 0.077.0 0.0663 0.0667 0.067 0.0677 0.068. 0.063 0.0636 0.0640 0.0645 0.0649. 0.0603 0.0607 0.06 0.065 0.060.3 0.0576 0.0580 0.0585 0.0589 0.0593.4 0.055 0.0556 0.0560 0.0564 0.0568.5 0.0530 0.0534 0.0538 0.054 0.0545 (c) EWMA-chars for daa se given in Table based on hree dela and six dela limis: 33
Tailor, KS Figure SUMMARY AND CONCLUSION In his paper, EWMA char is discussed under hree dela and six dela conrol limis wih an illusraion. From figure, i can be seen ha he producion process is in saisical conrol in boh he cases. If we compare he UCL and LCL of boh he ypes of chars, six dela conrol limis are always smaller han he hree dela conrol limis. So i can be concluded ha he char under six dela conrol lifs are more effecive owards deecing he shif in he value of EWMA han he chars under hree dela conrol limis. This is a nex generaion conrol char echnique and i will replace exising six sigma echnique. So i is recommended ha he conrol chars under six dela conrol limis should be used for he bes resuls. REFERENCES [] Huner J. S. (986): The Exponenially Weighed Moving Average, Journal of Qualiy Technology, 8, 03-0. [] Lucas J.M. and Crosier R.B. (98): Fas Iniial Response for CUSUM Qualiy Conrol Schemes, Technomerics, 4, 99-05 [3] Lucas J.M. and Saccucci M.S. (990): Exponenially Weighed Moving average Schemes, Properies and Enhancemens, Technomerics, 3, -9 [4] K.S.Tailor(06): Moving Average And Moving Range Chars Under The Assumpion Of Moderaeness And Is 3 Conrol Limis, Sankhya Vignan, December-06,, 8-3 [5] K.S.Tailor(07): Exponenially Weighed Moving Average (EWMA)Chars Under The Assumpion Of Moderaeness And Is 3-dela Conrol Limis, Mahemaical Journal of Inerdisciplinary Sciences(MJIS), March-07, Vol. 5,, -9 34
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