Abstract The R chart is often used to monitor for changes in the process variability. However, the standard

Transcription

1 An Alternatve to the Stanar Chart chael B.C. Khoo an H.C. Lo School of athematcal Scences, Unverst Sans alaysa, 800 nen, Penang, alaysa Emal: & Abstract The chart s often use to montor for changes n the process varablty. Hoever, the stanar approach of plottng the range statstcs,,,, on the chart s slo n etectng sustanng shfts of small magntue n the process varablty. The am of ths paper s to propose a more effcent alternatve to the stanar control chart approach. The ne alternatve s base on the constructon of a movng average control chart base on the statstcs. Brefly, the movng average of th at tme s efne as smply the average of the most recent subgroup ranges. Varyng subgroup szes ll be consere. Comparsons n terms of the average run length (AL) performances beteen the stanar an the ne approaches are mae by means of smulaton. All smulaton programs are rtten n the SAS language. Note that AL s efne as the average number of ponts that must be plotte on a control chart before an out-of-control sgnal s gven. Snce the ne approach s more effectve than the stanar chart approach, t may be an attractve alternatve to the stanar approach. Ths paper scusses ho statstcal softare such as SAS s use n the evaluaton of the performances of control charts.. Introucton The am of ths paper s to propose a more superor alternatve to the classcal chart for the etecton of shfts n the process varablty. The next secton eals th the ervaton of the chart lmts, folloe by a scusson on the propose alternatve metho n secton. A comparson on the performances of the to fferent methos base on ther AL profles ll be gven n secton. Note that all the AL profles are compute usng SAS verson 6.. In secton 5, an example s gven to llustrate ho the propose alternatve metho s put to ork. In the concluson secton (see secton 6), a bref scusson hch summarzes the results of the paper s gven.. Dervaton of Chart Lmts The chart control lmts are base on the stanar σ approach,.e., µ ± σ here µ an σ are the mean an stanar evaton of the range statstc (see Alan, 000; Duncan, 97; Grant an Leavenorth, 980; an ontgomery, 996). The range statstc, for each subgroup s the fference beteen the largest an the smallest observatons n the subgroup. From Alan (000), the upper an loer control lmts (enote by U an L respectvely) of the chart are U () L

2 here s an estmate for µ or, equvalently the expecte value of, E(). Here, an are the control chart constants hose values epen on the sample sze, n (see Appenx). By efnng D an D, e obtan U D () L D For the case here the stanar evaton, σ, of the unerlyng strbuton, assume to be normal s knon, an snce σ, then from () the lmts for the chart are U σ σ () L σ σ. The Desgn of a ovng Average Chart Assume that all of the observatons for each subgroup are nepenently an entcally (...) ranom observatons th a N( µ, σ ) strbuton. Also assume that all of the subgroups are nepenent of one another. If the ranges for a seres of subgroups,,,...,,..., are compute here enotes the range of subgroup, then the movng average statstc of th at tme can be compute as... () j j, For peros <, there are not yet ranges to calculate the movng average of th. For these peros, the average of all the ranges up to pero efnes the movng average at tme,.e., For example, f, then The mean of for peros s j, < j (6), E( ) E j j (5)

3 E j E( j j ( ) σ σ (7) an ts corresponng varance s Var( ) Var j j Var j j ( σ ) σ (8) It can be shon easly that for peros <, the mean an the varance of the movng average are E( ) σ (9) an σ Var( ) (0) respectvely. The upper an loer control lmts (enote by U an L respectvely) an the center lne ( ) for the movng average chart for peros are U σ L σ σ () an for peros <, the lmts are L U σ σ σ L L L σ L σ here L s the control chart constant hose value s etermne base on the esre n-control AL. For cases here the value σ s unknon, the lmts n eqs. () an () are obtane by j σ ) σ ()

4 substtutng σ th ts estmate, here s the average range estmate from a prelmnary set of, say, m subgroup ranges an s gven by m j () m. Comparng the AL Performances of an ovng Average Charts AL s an mportant feature n evaluatng the performances of control charts. AL represents the average number of ponts untl a chart sgnals. Therefore, hen there s no change n the process varablty, a large AL s esre so that less frequent false out-of-control sgnals are gven. On the contrary, hen the process varablty ncreases to an unesrable level, a small AL s esre to enable a qucker etecton of the nstablty n the process. The AL profles for the to types of charts are compute usng SAS verson 6.. Here, e assume that the n-control observatons n each subgroup follo a N ( µ,σ 0 0 ) strbuton hereas the out-of-control observatons are N ( µ,σ 0 ) strbute, here σ δσ 0 an δ {.00,.05,,.50,.75,.00,.50,.,.00, 5.00}. The latter represents an ncrease n the process varablty f δ >. Note that f δ, there s no change n the process varablty, thus, t s the ncontrol case. Subgroup szes of n 5 an n 0 are consere. Ther corresponng AL profles for the varous schemes are splaye n Tables an respectvely. Note that the values of L hch are use n the computaton of the movng average chart lmts n eqs. () an () are etermne usng smulaton to acheve smlar n-control AL ( AL 0 ) value to the chart. Table. AL Profles for an ovng Average Charts for n 5 ovng Average Chart δ Chart (L.865) (L.79) (L.7)

5 Table. AL Profles for an ovng Average Charts for n 0 ovng Average Chart δ Chart (L.885) (L.88) (L.770) The results n both Tables an sho that the AL performances of the three (, an ) schemes of the movng average charts for δ > are more superor to the classcal chart. For example, from Table, f δ., the three AL values for, an are all loer compare to that of the chart. Therefore, the ncrease n the process varablty (δ > ) can be etecte faster usng the propose movng average chart for small to moerate shfts. Hoever, as the magntue of shft ncreases, say, δ >, both control charts perform equally ell. Snce usually more concern s gven to the etecton of small shfts, the propose metho serves the purpose ell. 5. Example In ths example, e ll assume that the n-control observatons follo a N(00, strbuton hereas the out-of-control observatons follo a N(00,5 ) strbuton,.e., the shft s ue to an ncrease n the process varance. All of the observatons are generate usng a poerful statstcal softare, ntab r. The n-control observatons consst of subgroups to 0, each of sze fve hle the out-of-control observatons belong to subgroups to 0. The number of observatons n each of the ten out-of-control subgroups s also fve. Tables an sho the n-control an the out-of-control observatons for the 0 subgroups together th ther corresponng ranges,, an movng averages,, for, an. The tral lmts for the an the movng average charts calculate base on the nformaton n Table are gven n Table 5. Note that the values of the control chart constant, L, for the movng average chart for, an are chosen to acheve an n-control AL of approxmately 7. The values of L are also shon n Table 5. If a negatve loer control lmt s obtane from the computaton usng the formulas gven n eqs. () an () then let L 0. None of the an statstcs n Table fall outse ther respectve charts lmts shon n Table 5. Therefore, no revson s requre. Thus, these tral lmts can be use to montor for future shfts n the process varance. )

6 Table. In-Control Observatons for the Frst 0 Subgroups Where Each Observaton Follos a N(00, ) Dstrbuton Subgroup Observatons ange, ovng Average, No., Table. Out-of-Control Observatons for Subgroups to 0 Where Each Observaton Follos a N(00,5 ) Dstrbuton Subgroup Observatons ange, ovng Average, No.,

7 Table 5. Tral Control Lmts of the an the ovng Average Charts Posson ovng Average Chart Chart For, (9.05) D L D U For, L U For, L U For, L U For, L U For, L U For, L U For, L U For, L U For, L U

8 For the montorng of future shfts, conser the ata for the ten subgroups n Table. Ths s the out-of-control stuaton, thus a quck etecton for shfts n the process varance s very mportant. Fgures sho the an the movng average control charts for, an respectvely ncorporatng all of the 0 subgroups. Although a permanent shft n the process varance occur from subgroup onars, the chart fals to etect t. On the contrary, the movng average chart for, an gve the frst out-of-control sgnal at subgroups, an 5 respectvely. Ths example clearly shos the superorty of the movng average chart over the classcal chart. 6. Concluson From the scussons n sectons an 5, t s event that the movng average chart s superor to the stanar chart. Due to ths reason, t s recommene that the propose movng average chart be use as an attractve alternatve by qualty control practtoners. eferences. Alan, L.C., Statstcal Process Analyss, Irn cgra-hll, Ne York, Duncan, A.J., Qualty Control an Inustral Statstcs, th e. Homeoo : char D. Irn, Inc, 97.. Grant, E.L. & Leavenorth,.S., Statstcal Qualty Control, 5th e. Ne York : cgra- Hll Book Company, ontgomery, D.C., Introucton to Statstcal Qualty Control, r e. Ne York : John Wley & Sons, 996.

9 Appenx Table 6. Factors for Control Chart Sample Sze, n D D (aapte from Alan, 000)