IOSR Journal of Engneerng (IOSRJEN) e-issn: 225-32, p-issn: 2278-879, www.osrjen.org Volume 2, Issue (October 22), PP 5-6 Usng Cumulatve Count of Conformng CCC-Chart to Study the Expanson of the Cement Dr.Kawa M. Jamal Rashd, Dr.Abdul Rahm Khalaf Rah 2 Unversty of Sylamanya College Of Admnstraton and Economcs Department Of Statstcs 2 Unversty Of Mustansrya, College Of Admnstraton and Economcs Department Of Statstcs Abstract: In ths paper, we dscuss the need for process mprovement and problem that may be faced n controllng hgh Qualty process, we frst revew some basc concepts n statstcal process control and control charts, the focus s on the modfcaton of the tradtonal control charts, for attrbutes the cumulatve count of conformng (CCC) that s a powerful technque based on countng of cumulatve conformng tem between nonconformng ones. The CCC chart s very easy to mplement, effcent n detectng process changes.. Keywords: Tradtonal Control Charts, Modfcaton Of Control Chart, The Cumulatve Count of Conformng (CCC) chart I. Introducton Control chart are wdely used n ndustry as a tool to montor process characterstcs. Devatons from process targets can be detected based on evdence of statstcal sgnfcance. Statstcal process control orgnated n early twentes when Shewhart (926 ) presented deas of statstcal control charts for process characterstcs. The basc prncple s that processes are always subject to random varaton, whch s generally not controllable ordered dentfable. Thus some varaton wthn lmts are allowed n order to do ths we can plot successvely observed process characterstcs and make decsons based on what has actually been observed n the long run. II. Tradtonal Control Charts Control chart n ndustry are dvded nto two man types, varable charts and attrbute charts.for varable charts, the process or qualty characterstcs take on contnuous values whle for attrbute charts the data n ths form of dscrete counts. The P-chart s usually used to montor the proporton nonconformng n a sample the control lmts for the p or np Chart can be derved n the followng manner, a sample sze n follows the bnomal dstrbuton wth the parameter p, wth mean np and varance npq The upper control lmt and lower control lmt for p-cart gven by: UCL, LCL p 3 np( p) / n 2- UCL, LCL np 3 np( p) 2-2 And the central lmt s: Cl p And for np-chart,whch s used for the montorng number of non-conformng tems n samples of sze n,modfed lmts are as follows:. UCL, LCL np 3 np( p) 2-3 Cl np www.osrjen.org 5 P a g e
Usng Cumulatve Count Of Conformng CCC-Chart To Study The Expanson Of The Cement When the sample sze s fxed for all sample the p chart and np Chart are very smlar and the dfferent s only n the scale of the y-axs. Dfferent p-charts can be easly compared as the center lne for the process fracton nonconformng level.for np Chart the center lne s affected by the sample sze. When p or np Chart are used t s mportant to use the approprate lmt, because of the skewness of bnomal dstrbuton, the lower lmt based on 3 sgma concept may not exst as t s usually a negatve value, probablty lmt should be used when possble.on the other hand, smple modfcaton can be used to obtan better control lmt for p or np Chart. III. Modfcaton Of Control Chart Some mprovement made on tradtonal p and np Chart usng some transformatons so as to acheve hgh power n detectng np words shft n a process, wth a smple adjustments to the control lmts of the p-chart one can acheve equal or even better results form of the avalable methods are as follows: 3- Bnomal Q chart: By transformng the bnomal varable to standard normal varable, a Q-chart (Queensberry 997),could be drawn, wth upper and lower control lmts equal to 3 and -3 respectvely t s defned by: And u n k nk p ( p) k p Q ( u ) 3- Where (.) s the nverse functon of the standard normal varable. Plottng s on the chart, standard normal control chart wth a unform lmt of ( and ) could be mantaned. 3-2 Arcsne p-chart Ths s another nonlnear transformaton lke n the bnomal Q chart wth W sn 3 x 8 sn 3 n 8 p 2 n 3-2 Snce wth control lmts are of 3 and -3. W s the approxmately standard normal, a chart could be drawn by plottng www.osrjen.org W on the chart 3-3 Cornsh Fsher expanson method. Ths method havng some addtonal constant values n the three sgma lmts usng the normal dstrbuton. Defne Y X / n Plottng Y values on the chart t s very easy to detect the changes n p values wth the followng control lmt P( P) 4( 2P) UCL, LCL P 3 3-3 n 3n When the p s unknown, they have to be estmated and nserted n the above equatons. It s noted that the above equatons could be used for varable sample szes replacng n=n 3-4 Modfed lmts for np-chart Ryan and Schwartzman (997) proposed the followng regresson equatons of np and found them accurate. The proposed control lmts are. np and 52 P a g e
Usng Cumulatve Count Of Conformng CCC-Chart To Study The Expanson Of The Cement UCL.695.523 np 2. 983 np 3-4 UCL 2.9529..956 np 3. 2729 np 3-5 It s noted that these lmts are vald for (p<.3). Whle assumng the acceptable average run length (ARL), whch s the average number of ponts plotted before an alarm to be (37.4.).In comparng wth the exact lmt these lmts are less based n terms of ARL than the Shewhart p-chart. IV. The Cumulatve Count Of Conformng (CCC) Chart The Cumulatve Count of Conformng (CCC) chart or the count of conformng control chart s frst developed n Calvn (983) to montor ZD (zero-defects) process, It was further studed and gven the name CCC chart by Goh (987) the use of ccc type control chart has been further studed by Lucas (989), Bourke (99), Xe and Goh (992),and Wu et al (999-2). Ths s a powerful technques for process control when a larger of consecutve conformng tems are observed between two nonconformng ones. The dea behnd the CCC-charts s the fact that the number of conformng tems between two consecutve nonconformng ones changes when the fracton nonconformng shfts themselves n near ZD processes because they are usually vary small n the absolute magntudes. V. Settng Up Of The CCC Chart The settng of the CCC chart s smlar to the generc procedure of the settng up a Shewarthart control chart except that the measurement are the number of conformng tems after the last nonconformng one. Ths count should be plotted only when a new conformng tem s observed. Let (n) be the number of tems observed before a nonconformng one s found.e (n-) tems are conformng followed by n th tem whch s nonconformng.t s clear that ths count follows a geometrc dstrbuton. The determnaton of the control lmts based on ths. If the probablty nonconformng tem s equal to p then the probablty of gettng (n-) conformng ones followng a nonconformng one s: n p ( p) n=,2, 5- The mean of geometrc dstrbuton wth parameter p whch can be used as the center lne s : CL / P 5-2 Suppose that the acceptable of false alarm probablty s α, the UCL and LCL can be determned as: ln( / 2) UCL 5-3 ln( p) And ln( ( / 2)) LCL 5-4 ln( p) Respectvely control charts can be set up by ncludng the control lmts and the central lne. If the proporton of non-conformng tems a assocated wth the process s p, then the probablty that no nonconformng tem s found n n nspected s P(no nonconformng n n tems )=(-P) n 5-5 Ths prob. reflects the certanty we have when we judge the process to be out of control when a nonconformng tem has been found. Xe and Goh (992) ntroduced the concept of certanty level S, Whch s the probablty that the process s actually out of control? The certanty level, S related to false alarm probablty when nterpretng CCC chart sgnal. Wth one sde control lmt S=- α 5-6 The relaton between the proporton of nonconformng tems P and number of tems nspected n whch s gven as: ( p) n S...5-7 www.osrjen.org 53 P a g e
Usng Cumulatve Count Of Conformng CCC-Chart To Study The Expanson Of The Cement We can now determne the number of conformng tems nspected before a nonconformng one s allowed for the process to stll be consdered n control, ths can be calculated as: ln S n.. 5-8 ln( p) For practcal applcatons a decson graph can be constructed to facltate decsons on the state of control of process whenever nonconformng tem s observed. VI. Applcaton The mportant factor that descrbe the qualty of the cement s the (expanson of the cement - EXP) where the standard value of ths feature s less than mm. A set of CCC data s shown n table (6-) Table (6-) shows that the cumulatve data. Wth acceptable value of EXP s < Table (6-) Uncensored CCC_EXP data No CCC No CCC No CCC No CCC 2 3 5 2 8 2 22 32 4 3 3 2 23 33 7 4 4 3 24 34 3 5 6 5 25 35 6 6 26 4 36 7 7 27 45 37 3 8 8 8 4 28 5 38 3 9 6 9 2 29 3 39 3 2 3 4 4 4 Calculate the upper and lower of CCC-EXP of table (6-) by usng the equaton (5-3 and 5-4),(44.9,.56) respectvely n Fg(6-) shows the correspondng CCC chart s dsplayed. Fg (6-) Acceptance chart of CCC-EXP data In ths case two ponts out of the upper control lmt and (6) ponts are out of the lower control lmt and the process n out of the control. www.osrjen.org 54 P a g e
Usng Cumulatve Count Of Conformng CCC-Chart To Study The Expanson Of The Cement Table (6-2) shows that the frequency of occurrence of the 7 unque the classes are sorted accordng to the counts, wth the most frequently occurrng class frst. The hghest class s wth a count of 6, whch represents 39.244% of the total, and Fg (6-2) shows the Pareto chart. Table (6-2) Pareto Cumulatve Frequences of CCC-EXP data Class Label Rank Count CuSum Score Percent % Cusum Percent % 6 6 39.2 39.2 3 2 4 2 9.76 48.78 3 4 24 9.76 58.54 4 4 2 26 4.88 63.4 6 5 2 28 4.88 68.29 4 6 2 3 4.88 73.7 5 7 3 2.44 75.6 45 8 32 2.44 78.5 3 9 33 2.44 8.49 2 34 2.44 82.93 8 35 2.44 85.37 7 2 36 2.44 87.8 3 3 37 2.44 9.24 2 4 38 2.44 92.68 5 39 2.44 95.2 8 6 4 2.44 97.56 5 7 4 2.44 Total 4 Fg (6-2) Pareto Chart OF CCC-EXP data Transformaton: 6- Charts based on Transformaton www.osrjen.org 55 P a g e
Usng Cumulatve Count Of Conformng CCC-Chart To Study The Expanson Of The Cement The transformaton s useful when the dstrbuton s non-normal whch s the case for geometrc dstrbuton under the transformaton, the data become normally dstrbuted. There are a number of transformatons avalable, few of there are shows below: -The double square root Transformaton A double square root for the (fourth root) s a smple transformaton by: (.25) Y X, X The below table Show that the llustrate some possble transformaton approaches. (.25) Y X Ln(x) Table (6-3) Transformatons of CCC-EXP data (.25) Y X Ln(x) (.25) Y X Ln(x) (.25) Y X Ln(x).778 2.33..495.69 2.6 2.89.44.386.86 2.485 2.3 2.833.36.99.36.99.565.792.. 2.53 3.74 2.59 3.87.36.99.682 2.79 2.53 3.74 2.672 3.932.36.99.565.792 2.5 2.996.899 2.565 2.36 3.434.44.386 In Fg (6-3) the correspondng CCC chart s dsplayed n ths case, all ponts are wthn the control lmts and the process s n control compare wth fg.(6-2) Fg(6-3) The of Transformaton data chart of CCC_EXP 6-2 Quesenberry s Q-Transformaton Usng Q - to denote the nverse functon of standard normal dstrbuton, defne Q Q where u ( u ) x F( x, p) ( p) For,2,3,... Q Wll approxmately follow the standard normal dstrbuton, and the accuracy mproves as p approaches zero. www.osrjen.org 56 P a g e
X Usng Cumulatve Count Of Conformng CCC-Chart To Study The Expanson Of The Cement Table (6-4) shows the data of Queensberry s Transformaton of CCC-EXP data Q Q Q Q.2327 2.326.757.762 -.26 2.326 2.326.9385 2.326.257 2.326 -.2224 2.326.494 2.326.494.628 2.326 2.326.757 2.326.757 -.65 2.326 2.326 2.326 -.2873.494.436 -.65.4532.494.628 -.3836.426 2.326 -.83 2.326.92 2.326.757 5.8 X Chart for Quesenberry 3.8 UCL = 4.3.8 -.2 CTR =. -2.2 LCL = -2. 2 3 4 5 Observaton Fg (6-4) the ndvdual chart of CCC_EXP data wth Q- Transformaton From above chart of Q- Transformaton, t shows that all ponts are wthn the control lmts and the process s n control. 6-3 fndng the Sample Sze at dfferent fractonal non conformng level at certanly level of S To fnd the Sample Sze at dfferent fractonal non conformng level at certanly level of S value by usng the equaton (5-7 and 5-8) and depend on the P, and S value t seen the result n the below table (6-5). www.osrjen.org 57 P a g e
Usng Cumulatve Count Of Conformng CCC-Chart To Study The Expanson Of The Cement We gve some numercal values of (n) for some dfferent combnatons of values of p and S. If the cumulated count of conformng tem s less than the tabulated value, then the proporton of nonconformng tems s hgher than wth certanty S. Ths table can be used to determne the value of certanty level (s) for a gven level of proporton nonconformng P, as well as determne the value of (p) for gven level of certanty and see f t s hgher than the acceptable level for the process.fur there more for gven (p and s ), determne the mnmum number of conformng tem that must have been observed before a nonconformng one can be tolerate e, the process can stll be deemed under control even one nonconformng tem has been observed. Table (6-5) Sample Sze at dfferent fractonal non conformng level P value S=-α If S =.9 then n If S 2 =.95 then n If S 3 =.98 then n. 5.38 5.268 2.93.2 52.628 25.62.9.3 35.67 7.72 6.724.4 26.287 2.798 5.4.5 2.9.233 4.3.6 7.57 8.523 3.357.7 4.999 7.32 2.876.8 3.7 6.386 2.55.9.654 5.674 2.235..483 5.4 2.. 9.525 4.637.826.2 8.727 4.249.673.3 8.52 3.92.544.4 7.473 3.638.433.6 6.532 3.8.253.8 5.8 2.824.2.2 5.25 2.539. www.osrjen.org 58 P a g e
Usng Cumulatve Count Of Conformng CCC-Chart To Study The Expanson Of The Cement Fg( 6-5) A decson graph for the stat of control of a process Fg (6-5) Show the chart of the sample sze of dfferent fractonal non conformng level at certanty level, s Number Of non- conformng tems percent nonconformng. Snce usually p < for hgh- qualty products, a good approxmaton of Eq.(5-7) s np S e, P Hence, for hgh-qualty products, we have an expontonal relaton between the number of nspected tems n and the confdence that the process s out of control when the sngle nonconformng tem s found. Then exponental relaton s very convenent to use from another pont of vew. By takng logarthms, we can show that the relaton s equvalent to. Lnn Ln p Ln( LnS) VII. Concluson - The CCC-chart s easy to use and more senstve to short-term drfts n process parameter as well as enable ts user to judge the state of control. 2- The decson graph can be used for combnaton of proporton non-conformng, P and certanly level, S to determne the mnmum number of conformng tem. 3- Under the transformatons ether Q-transformaton or a double square root, the data of Expanson become normally dstrbuton and the accuracy mproves as well as the process s mproved also (all ponts are wthn control lmts). Reference []. Anscombe F,FJ (948)The transformaton of Posson, Bnomal, and negatve bnomal data [2]. Bourke,P.D(99)Detectng shft n fracton nonconformng usng run length control charts wth % nspecton. [3]. Bourke,P.D(2) Sample sze and Bnpmal Cusum control chart the case of % nspecton [4]. Crossley, M.L. (2) The Desk Reference of Statstcal Qualty Methods, ASQ Press, Mlwaukee, WI, USA. [5]. Ishkawa, K. (986) Gude to Qualty Control, Asan Productvty Assocaton, Tokyo, Japan. [6]. Johlan Oakland Sxth edton 28 Statstcal Process Control [7]. James R.; Koronack T.Jacek 22 by Chapman & Hall/CRC STATISTICAL PROCESS CONTROL The Demng Paradgm and Beyond,second edton [8]. Kemp, K.W. (962) Appled Statstcs, Vol., pp.6 3, The use of cumulatve sums for samplng nspecton schemes. [9]. Xe, TN Goh, V Kuralman (22) Statstcal Model and Control Chart for Hgh- Qualty Process. []. Oakland, J.S.(2)Total Qualty Management-Textand Cases, 2nd Edton, Butterworth -Henemann, Oxford, UK. www.osrjen.org 59 P a g e
Usng Cumulatve Count Of Conformng CCC-Chart To Study The Expanson Of The Cement []. Ognyan Ivanov 2 APPLICATIONS AND EXPERIENCES OF QUALITY CONTROL [2]. PETER W. M. JOHN (99) Statstcal Methods n Engneerng and Qualty Assurance [3]. Xe, M: Goh T.N and tang X,Y(2) Data Transformaton for geometrcally dstrbuted qualty characterstcs [4]. Yang, Z.I : Xe M and Goh (2) Process montorng of exponentally dstrbuted characterstcs through an optmal normalzng transformaton. www.osrjen.org 6 P a g e