COMPARISON OF THE EFFICIENCY OF S-CONTROL CHART AND EWMA-S 2 CONTROL CHART FOR THE CHANGES IN A PROCESS

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1 COMPARISON OF THE EFFICIENCY OF S-CONTROL CHART AND EWMA-S CONTROL CHART FOR THE CHANGES IN A PROCESS Supraee Lisawadi Departmet of Mathematics ad Statistics, Faculty of Sciece ad Techoology, Thammasat Uiversity, Bagkok, Thailad supraee@mathstat.sci.tu.ac.th Puamee Sachakamol Iteratioal Graduated Program for Idustrial Egieerig (IGP), Departmet of Idustrial Egieerig, Faculty of Egieerig, Kasetsart Uiversity, Bagkok, Thailad Abstract: Statistical Process Cotrol (SPC) is a method of quality cotrol which uses statistical methods to moitorig ad cotrollig the process to esure that the maufacturig lie operates at its full potetial. At its full potetial, the process ca be applied to make as much coformig product as possible with miimum waste. However, the quality of the process cotrol is determied by samples chose ad Cotrol Chart tools whereas this research aims to determie ad compariso the efficiecy of S-Cotrol Chart ad EWMA-S Chart. The determiatio factors are fluctuatio i variaces, differet sizes of samples, ad variatios of samplig method. From, iteratio ru i R Versio.. with Media-Set Samplig, variaces rage from. to 3., the EWMA-S Cotrol Chart yield higher efficiecy i every possible sceario tested. Keywords: cotrol chart, quality cotrol, productio, statistical process cotrol, radom samplig, stadard deviatio 63

2 . THE ORIGIN AND SIGNIFICANCE OF THE PROBLEM Geeral maufacturig process teds to always vary. This variatio will cause the quality of the product varies as well. The variatio sometimes is ot so much. Maufacturer allows this to happe because it does ot affect the overall quality of the product. However, if the variatio is severe ad is ot acceptable because it may make the product becomes a waste or caot be used eve though there have bee attempts to reduce or cotrol the variatio i each factor, but it still has the variatio evetually i the maufacturig process. Therefore, the variatio should be cotrolled, which is the quality cotrol to esure that the quality of products meets the required stadards or is i withi the acceptable limit. Statistical Process Cotrol or SPC is the implemetatio of statistical process to cotrol the maufacturig process so that the properties of products i the maufacturig process is cosistet as well as raisig maufacturig capacity or reducig the variatio of the maufacturig process util the properties of products meet or close to the stadards, which SPC ow is very popular for aalyzig the performace of the maufacturig process. (Shewhart, 939) has developed a cotrol chart, which is cosidered as the mai tool of SPC by relyig o statistical methods. The mai fuctio of cotrol chart is to detect variatios i the maufacturig process ad to be a tool to idicate how the maufacturig process operates, whether variatio occurs out of cotrol or is egligible, which is cosidered i cotrol. This cotrol chart, if the maufacturer uses to cotrol the maufacturig process regularly, will prevet quality problems well sice the cotrol chart will detect chages i the maufacturig process i a timely maer allowig maufacturers to rapidly kow maufacturig coditios. I the evet of a problem, it ca stop the maufacturig process ad improve it. I this study, 4 samplig plas will be used icludig Simple Radom Samplig (SRS), Systematic Samplig (SYS), Raked-Set Samplig (RSS) ad Media-Set Samplig (MSS) by usig R software package for simulatig radom samplig of each pla as a represetative for moitorig ad create a cotrol chart. From the study of the chart above, the researcher was iterested to study the process with little chage i the stadard deviatio with stadard deviatio cotrol chart ad expoetially weighted movig average cotrol chart for examiig the variatio of the process fiacial. The study foud that there was o compariso of the two cotrol charts uder the circumstaces of the maufacturig process i the stadard deviatio with egligible chage at the same time ad the researcher will study differet sample sizes i order to compare the performace of all cotrol charts with the Average Ru Legth() at each samplig pla.. RELATED THEORIES AND RESEARCHES I this study, the researcher was iterested i several types of radom samplig to create S cotrol charts ad expoetially weighted movig average cotrol chart for measurig the variace of the process (EWMA-S Cotrol Chart) uder data with a ormal distributio by comparig cotrol charts obtaied from various samplig methods. The related theories ad researches are as follows. (Maravelakis & Castagliola, 9) foud that EWMA cotrol chart is more effective tha other charts if is low for ay costat i chages ( ) τ whe are equal i all charts i which low λ value is suited to detect small to medium chages ad high λ value is suited for large-scale chage τ.8. ( ) More researches are also compared the stregth betwee the expoetially weighted Shewhart average cotrol chart ad sythetic by usig Average Ru Legth () that is out of cotrol as a basis o makig a decisio from the coclusios. The distributios made the expoetially weighted average cotrol chart the most effective. 64

3 (Tawatira & Mayureesawa, ) studied the cotrol charts by Raked-set Samplig ad compared the value from data simulatio of 5 charts which were x chart, x RSS Chart, MRSS Chart, x MSS Chart, ad MMSS Chart ad showed the results at some m values adσ at all k levels ad at specified δ value i the experimet pla. The results that the average of the maufacturig process has ot chaged (δ = ) was foud that at all levels of k, m, σ the MRSS-Chart had the best ability to moitor the maufacturig process sice it gave the value closest to the target value of 37 while x RSS, Chart, x MSS Chart, ad MMSS Chart และ x Chart has subsequetly lower ability to moitor the maufacturig process, havig the value more differet from 37, respectively. 3. RESEARCH OPERATION I this research the researcher wated to study the performace of S-cotrol charts ad expoetially weighted movig average cotrol chart for measurig the variace of a process (EWMA-S Cotrol Chart) for a process with a chage i the stadard deviatio of various samplig methods by simulatig data with differet samplig methods. The the average ru legth () was calculated usig R software to compare the performace of both charts. The followigs are the procedure to coduct the research. 3.. Procedure The procedure is divided as follows: Data simulatio: Data simulatio was performed usig R software package versio 3.. to simulate data with ormal distributio, average equals to, variace rages from. to 3. ad icreases at. icremet(.,.,.4,.6,.8,.,.,.4,.6,.8,.,.,.4,.6,.8, 3.) for samples (m = ). Sampligs were doe usig the followig methods:. Simple Radom Samplig (SRS). Systematic Samplig (SYS) 3. Raked-Set Samplig (RSS) 4. Media-Set Samplig (MSS) The sample size was desigated as 5,, 5, 3, respectively ( = 5,, 5, 3). Specificatio of Cotrol Rage. Cotrol Chart (S Cotrol Chart): The cotrol rage for S cotrol chart was LCL = B3 S UCL = B4 S i which B 3 ad B 4 ca be opeed from the appedix of the Factors for Costructig Variables Cotrol Charts.. Expoetially Weighted Movig Average Cotrol Chart for Moitorig the Process Variace (EWMA-S Cotrol Chart): The cotrol rage for EWMA-S Cotrol Chart was LCL = UCL = σ + c λ λ i which c is the critical value ad k is the degree of freedom ad k = -. Average Ru Legth () is calculated from M = RL i M i= Where M is the umber of experimet i each situatio i order to fid, RLi is the umber of samples moitored util the maufacturig process is out of cotrol at experimet. σ k 65

4 To fid, Mote Carlo simulatio techique was used with the followig procedure. Cotrol Chart(S Cotrol Chart). Simulate data ~ N( σ ) X, j, i with variace from. to3. at. icremet at = 5,, 5, 3 for samples.. Specify the lower cotrol limit LCL = B3 S ad upper cotrol limitucl = B4 S 3. Calculate statistical values from the chart from s i = s i = j= j= ( x x) i, j x i, j or ( xi, j ) j= ( ) 4. Cosider statistical value LCL < s i < UCL (the statistical value is uder cotrol) whe t = util LCL < s i ad s i > UCL (the statistical value is out of cotrol) ad cout the ru legth (RL). 5. Repeat steps, 3, ad 4 util M =,. 6. Calculate. Expoetially Weighted Movig Average Cotrol Chart for Moitorig the Process Variace (EWMA- S Cotrol Chart). Simulate data X i, j ~ N(, σ ) with variace from. to3.at. icremet at = 5,, 5, 3 for samples.. Specify the upper cotrol limit UCL σ + c λ λ σ k = where c = 3 ad k = - Specify the lower cotrol limit LCL = 3. Calculate Expoetially Weighted Movig Average Cotrol Chart for Moitorig the Process Variace (EWMA) zi = ( λ) zi + λvi, i where λ =. ad v i = l s i. 4. Cosider statistical value LCL < zi < UCL (the statistical value is uder cotrol) whe t = util LCL < ad z i > UCL (the statistical value is out of cotrol) ad cout the ru legth zi (RL). 5. Repeat steps, 3 ad4util M =,. 6. Calculate. 3.. Compariso of the chart efficiecy: I this study, was used to compare the performace of S Cotrol Chart ad expoetially weighted movig average cotrol chart (EWMA-S Cotrol Chart) for measurig the process variace for a process that has a chage i the stadard deviatio for various sample methods, where is the average umber of samples to be moitored util the discovery of the out of cotrol ad ca be determied as follows: If is high the the chart has low efficiecy. If is low the the chart has high efficiecy. 4. RESULTS This research aims to study ad compare the performace of S Cotrol Chart ad Expoetially Weighted Movig Average Cotrol Chart (EWMA-S Cotrol Chart) for measurig the process variace i differet situatio such as whe the data has a chage i the stadard deviatio at differet levels, the data has differet sample sizes, ad the data has differet samplig methods i 66

5 order to draw coclusios about the performace of S Cotrol Charts ad Expoetially Weighted Movig Average Cotrol Chart (EWMA-S Cotrol Chart) for measurig the process variace usig the average ru legth () as a value to compare the performace of both charts, which was preseted i the form of tables ad graphs showig the average ru legth (). From Tables -4, it was foud that EWMA-S Cotrol Chart had the average ru legth () less tha S Cotrol Chart at all levelsof stadard deviatio ad all samplig plas except RSS ad MSS samplig methodidicatig that EWMA-S Cotrol Chartis more efficiettha S Cotrol Chart whe the sample size is 3. Table : from cotrol chart of differet samplig sizes =5 Sig ma S Cotrol Chart SRS SYS RSS MSS EWM A-S S Cotrol Chart EWM A-S S Cotrol Chart EWM A-S S Cotrol Chart EWMA -S = = Figure : i the SRS ad SYS EWMA-S (SRS) SRS =5 SRS = SRS =5 SRS =3 EWMA-S (SYS) SYS =5 SYS = SYS =5 SYS =

6 From figure, it was foud that i the Simple Radom Samplig (SRS) ad Systematic Samplig (SYS) will icrease whe the stadard deviatio rises ad be costat whe the stadard deviatio is betwee. to.6 ad decrease whe the stadard deviatio is greater tha.6. From Figure, Raked-Set Samplig (RSS) ad MSS give higher whe the stadard deviatio icreases. Figure : i the RSS ad MSS EWMA-S (RSS) EWMA-S (MSS) RSS =5 RSS = RSS =5 RSS =3 RSS =5 RSS = RSS =5 RSS = Figure 3: S Cotrol Chart i the SRS, SYS, ad MSS 6 S Cotrol Chart (SRS) SRS =5 SRS = SRS =5 SRS =3 68

7 S Cotrol Chart (SYS) SYS =5 SYS = SYS =5 SYS = S Cotrol Chart (MSS) MSS =5 MSS = MSS =5 MSS =3 From figure 3, it was foud that the sample size does ot affect of S-Chart, meaig the sample size does ot affect the performace of S-Cotrol Chart. Figure 4: S Cotrol Chart i the RSS, S Cotrol Chart (RSS),5,, RSS =5 RSS = RSS =5 RSS =3 From figure 4, it was foud that Raked-Set Samplig (RSS) gives = at all sample sizes. This is because Raked-Set Samplig (RSS) is a samplig method that sorts the value from low to high ad uses the lowest umber as the first sample ad repeat util all samples are used. Therefore, sample umber is the sample that has the lowest observatio. The lowest observatio will be out of the cotrol rage whe plotted o a graph resultig i =. 69

8 The other figure are plots but ot listed i this paper. The coclusios ca be draw as below:. It was foud that SRS ad SYS are more efficiet tha other samplig methods whe the stadard deviatio is high. If the stadard deviatio is lower, MSS will be more efficiet.. It was foud that samplig method did ot affect the performace of both charts because they gave the same for all samplig methods. 5. CONCLUSIONS AND RECOMMENDATIONS The fidigs i this research are to study ad compare the performace of S Cotrol Chart ad Expoetially Weighted Movig Average Cotrol Chart (EWMA-S Cotrol Chart) for measurig the process variace i differet cases as follows. Data chaged the stadard deviatio at differet levels. Data had with differet sample sizes. Samplig method was differet for the data. The coclusios about the performace of S Cotrol Chart ad Expoetially Weighted Movig Average Cotrol Chart (EWMA-S Cotrol Chart) for measurig the process variace usig the average ru legth () as a value to compare the performace of two charts were that whe cosiderig the appropriate samplig method it was foud that i the case of a mior variace (σ.) Media-Set Samplig (MSS) was more efficiet tha other samplig methods, whe applied to a large variace (σ >.) Simple Rasom Samplig (SRS) ad Systematic Samplig (SYS) were more efficiet tha other samplig methods, ad for whe the sample size was larger, the cotrol charts were also efficiet cotrol chart as well. I all cases, EWMA-S Cotrol Chart was more efficiet tha S Cotrol Chart. Table : The coclusio of the efficiecy of the cotrol charts at various sample sizes. Sample Size() Most Efficiet Cotrol Chart 5 EWMA-S Cotrol Chart EWMA-S Cotrol Chart 5 EWMA-S Cotrol Chart 3 EWMA-S Cotrol Chart Table 3: The coclusio of the efficiecy of the cotrol charts whe the samplig method was differet. Samplig Method SRS SYS RSS MSS Most Efficiet Cotrol Chart EWMA-S Cotrol Chart EWMA-S Cotrol Chart EWMA-S Cotrol Chart EWMA-S Cotrol Chart I this study, ormal distributio was used to simulate the data for the compariso of S Cotrol Chart ad Expoetially Weighted Movig Average Cotrol Chart (EWMA-S Cotrol Chart) for measurig the process variace. If ayoe is iterested i coductig further study, other kids of distributios may be used such as U-Shaped distributio or J-Shaped distributio, etc. REFERENCE LIST. Crowder, S., & Hamilto, M. (99). A EWMA for moitorig stadard deviatio. Joural of Quality Techology, 4, -.. Maravelakis, P., & Castagliola, P. (9). A EWMA chart for moitorig the process stadard deviatio whe parameters are estimated. omputatioal Statistics ad Data Aalysis, 53, Shewhart, W. A. (939). Statistical method from the viewpoit of quality cotrol. (W. Edwards Demig). Graduate School, the Departmet of Agriculture, Tawatira, N., & Mayureesawa, T. (). The cotrol chart by arraged order samplig. Joural of KigMogkut's Uiveristy Techology North Bagkok, 3,