ISyE 512 Chapter 7. Control Charts for Attributes. Instructor: Prof. Kaibo Liu. Department of Industrial and Systems Engineering UW-Madison

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1 ISyE 512 Chapter 7 Cotrol Charts for Attrbutes Istructor: Prof. Kabo Lu Departmet of Idustral ad Systems Egeerg UW-Madso Emal: klu8@wsc.edu Offce: Room 3017 (Mechacal Egeerg Buldg) 1

2 Lst of Topcs Chapter 7 P Cotrol Chart for ocoformg ut NP Cotrol Chart for ocoformg ut C Cotrol Chart for ocoformtes U Cotrol Chart for ocoformtes 2

3 How to descrbe a product ot meet Qualty requremets? Nocoformty A departure of a qualty characterstc from ts teded level or state that occurs wth a severty suffcet to cause a assocated product or servce ot to meet a specfcato requremet. Nocoformg ut A ut of product or servce cotag at least oe ocoformty. Defect A departure of a qualty characterstc from ts teded level or state that occurs wth a severty suffcet to cause a assocated product or servce ot to satsfy teded ormal, or reasoably foreseeable usage requremets. Defectve (Defectve Ut) A ut of product or servce cotag at least oe defect, or havg several mperfectos that combato cause the ut ot to satsfy teded ormal, or reasoably foreseeable, usage requremets. Nocoformty=Defect Nocoformg ut=defectve ut 3

4 Revew of Bomal Dstrbuto Let x = # defectve ut a sample of sze where defectve ut follow a Beroull Process (two outcomes, p-costat, x - depedet) f (x) x p x (1 p) x E(x) p V(x) p(1 p) 4

5 Sample Estmato of p Sample statstc Let pˆ x Proporto of ocoformg parts, Probablty of ocoformg for each dvdual part/observato, r.v. E( pˆ) E( x) p p V ( pˆ) 1 2 V ( x) p(1 p) 5

6 Revew of the Basc Model of a Cotrol Chart Let w be a sample statstc that measures some qualty characterstc of terest, ad suppose that the mea of w s w ad the stadard devato of w s w. The the ceter le, the upper cotrol lmt, ad the lower cotrol lmt become UCL = w + L w Ceter le = w LCL = w L w where L s the "dstace" of the cotrol lmts from the ceter le, expressed stadard devato uts 6

7 Fracto Nocoformg Cotrol Chart (p-chart) For p large pˆ ~ N p, p(1 p) Assume p s kow Bomal (p>10, 0.1<=p<=0.9) ormal UCL pˆ p Ceterle 3 p p(1 p) LCL pˆ p 3 p(1 p) If LCL p <0, set LCL p =0 Remarks: Whe data pots are plotted below LCL, they geerally do ot represet a real mprovemet. Actually, they are ofte caused by errors the specto rather tha a process mprovemet 7

8 How to Establsh a p-chart? m=20-25 samples for costructg tral cotrol lmts pˆ p If p ukow, coduct a test ad tral cotrol lmts wth m D 1 p m E(p) p m 1 m pˆ UCL p 3 Ceterle LCL p 3 p(1 p) p p(1 p) Tral Cotrol Lmts Is there a assgable cause for out-of-cotrol pots or a oradom patter? If so, fd the root causes ad delete these pots, ad the update cotrol lmts. Whe a pot s ON a cotrol lmt, t s cosdered as ether outof-cotrol or -cotrol depedg o how the problem asks

9 Example: The followg data gve the umber of ocoformg ROM chps samples of sze 200. Costruct a p chart for these data. Assume that ay values beyod the cotrol lmts have a assgable cause ad revse the cotrol lmts as approprate. Sample Nocoformg Sample Nocoformg

10 p = 200 sample Nocoformg p UCL LCL p-bar p-char (Example) Samples Good or NOT? 10

11 Useful Approxmato Cotrol Chartg Example 13-1: A fracto ocoformg cotrol chart wth ceter le 0.10, UCL p = 0.19, ad LCL p = 0.01 s used to cotrol a process. (Samples o the cotrol lmts are cosdered as out-of-cotrol) a. If 3-sgma lmts are used, fd the sample sze for the cotrol chart. b. Use the Posso approxmato to the bomal to fd the probablty of type I error. c. Use the Posso approxmato to the bomal to fd the probablty of type II error f the process fracto defectve s actually p =

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13 Varable Sample Sze Varable wdth of cotrol lmts Dfferet samples, p the same ceter le UCL correspodg to each sample sze ot approprate for oradom patter check 1 m p m 1 D p 3 total total # p(1 p) ; Costat wdth of cotrol lmts usg average sample sze future sample sze should ot dffer greatly Stadardzed Cotrol Chart Z Stadardzato # of defects of observatos LCL ca be used to check a oradom patter o referece to the actual process fracto defectve pˆ p p(1 p) ; p p p; p 3 p(1 p) UCL 3; Defects m 1 LCL 3; m P Chart U sg Crcketgraph III 10 CL 0 Row Numbers UCL UWL MEAN LW L LCL Dfferet samples, dfferet CL

14 z p p Table 7.5, P314 I () D() p=d()/()sgma=sqrt(pbar*(1-pbar)/()) LCL () UCL() LCL ( bar) UCL( bar) Z LCL(stad)UCL(stad) sum average 98 pbar= p bar=total defectve/total samples Fg 6-6 cotrol chart wth varable sample sze sample dex stadardzed cotrol chart dex 14

15 p Cotrol Chart (The umber of ocoformg tems) Rather tha plottg the fracto ocoformg, we plot the umber of ocoformg tems wth a p Chart : UCL X = p + 3 p(1 p) Ceter le = p LCL X = p 3 p(1 p) If LCL X <0, set LCL X =0 Np ad p cotrol charts ca be trasferred to each other. From p to p, multple to the UCL, CL, ad LCL; From p to p, dvde to the UCL, CL, ad LCL. 15

16 Example: The umber of trasmsso cases that requred deburrg a 16-day sample of 100 each was as follows: Day Number Day Number Prepare a p chart wth tral cotrol lmts. Assume that ay pots plottg out of cotrol have assgable causes, ad cotue to refe the cotrol lmts utl o pots plot out of cotrol. 16

17 I D UCL(tral) LCL(tral) UCL LCL sum 88 UCL(tral) LCL(tral) p bar=88/(100*16)= set to zero p elmate pot 11 pbar=(88-15)/(100*15) UCL LCL p set to zero p chart wth tral cotrol lmts p chart after elmate outlers D UCL(tral) LCL(tral) UCL LCL 17

18 Advatage p Chart Propertes p chart s a scalg of the vertcal axs by the costat, provde the same formato as p chart p chart eeds less calculato ( o eed to calculate D / ) ofte used whe s costat ad p s small Lmtato ot easy for terpretato whe s vared (UCL LCL ad Ctr le all vary) 18

19 OC Curve ad ARL Type II error for the p chart β = P p < UCL p p 1 P p LCL p p 1 = P D < UCL p p 1 P D LCL p p 1 ARL 0 =ARL -cotrol =1/ ARL 1 =ARL out-of-cotrol =1/(1-) 19

20 Example: A cotrol chart s used to cotrol the fracto ocoformg for a plastc part maufactured a jecto moldg process. Te subgroups yeld the followg data: Sample Number Sample Sze No. Nocoformg a. Set up a cotrol chart for the umber ocoformg samples of = 100. (Samples o the cotrol lmts are assumed to be -cotrol.) b. For the chart establshed part (a), what s the probablty of detectg a shft the process fracto ocoformg to 0.30 o the frst sample after the shft has occurred? 20

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22 Example: Cosder the 3 sgma cotrol chart (CL = 0.10, UCL = 0.19, LCL = 0.01). Fd the average ru legth f the process fracto ocoformg shfts to

23 Cotrol Charts for Nocoformtes (Defects) - C ad U Charts Why eed t: Cotrol the total umber of ocoformtes a sample or the average umber of ocoformtes per ut ocoformty/defect: Each specfc pot at whch a specfcato s ot satsfed, e.g., weld spots o a car pat det o a car body A ut may ot be ocoformg, eve though t has several ocoformtes. So, ocoformg defects or ocoformtes Assumpto: The occurrece of ocoformtes a specto ut (or a sample) of costat sze s well modeled by the Posso dstrbuto. The umber of potetal locato for ocoformtes ca be very large, but the probablty of occurrece of a ocoformty at ay locato s small ad costat 23

24 What s a Ispecto Ut A specto ut could be a sgle ut of product Or t ca be a group of several uts, e.g., 144 mcroprocessors, 5 cars It s NOT ecessarly teger MUST BE THE SAME # UNITS IN EACH INSPECTION UNIT (OR SAMPLE) 24

25 Statstcal Bass of C Chart Let X = # of ocoformtes a specto ut Assume X ~ Posso ( E(X) = C ) For large C X ~ N ( E(X) = C, Var(X) = C ) p( x) e c x! c x x=0,1,2,... c chart: total umber of defects a specto ut or sample Nocoformtes 25

26 Cotrol Charts for Nocoformtes - c Chart Cotrol lmts for the c chart wth a kow c (kow mea ad varace) UCL c 3 c CL c LCL c 3 c If LCL<0, set LCL=0 If ukow c, c s estmated from prelmary samples of specto uts for costructg tral cotrol lmts ĉ c m 1 m c total # of defects umber of Nocoformtes all samples samples UCL c 3 c LCL c 3 c The prelmary samples are examed by the cotrol chart usg the tral cotrol lmts for checkg out-of-cotrol pots CL c 26

27 Example (Textbook Ex 7.3): 26 successve samples of 100 PCB s What s the specto ut? 100 boards 27

28 c 3 c c c 3 c UCL c 3 c C. L c LCL c 3 c

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30 Statstcal Bass of u Chart X: # ocoformtes a sample of specto uts ( s ot ecessarly teger), X ~ Posso (c), E(X)=V(X)=c UCL/LCL=c±3 c CL= c Y = average # of ocoformtes per specto ut a sample = X / E ( Y ) = c / = u Var ( Y ) = c / 2 = u / u chart: average umber of defects per specto ut, a sample sze of specto uts 30

31 Cotrol Charts for Nocoformtes Per Ut - u Chart c: total ocoformtes a sample of specto uts ( s ot ecessary be teger) u: average # of ocoformtes per specto ut a sample u c ; u u m 1 m u LCL u 3 CL u UCL u 3 If ukow u, s estmated from prelmary samples of specto uts for costructg tral cotrol lmts u u 31

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34 Example Fd the 3-sgma cotrol lmts for: a. A c chart wth process average equal to four ocoformtes. b. A u chart wth c = 4 ad = 4. 34

35 Varable Sample Sze of Cotrol Charts for Nocoformtes If sample sze vares, oe should use a u chart rather tha a c chart Approaches Cotrol lmts vares wth each sample sze, but the ceter le s costat LCL u 3 UCL u 3 Use a cotrol lmts based o a average sample sze m 1 m Use a stadardzed cotrol chart (ths s preferred opto), wth UCL=3, LCL=-3, Ceter le=0. Ths chart ca be used for patter recogto Z u ; u u u u ; CL u 35

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38 Number of No-coformtes Example: Followg are the umber of ocoformtes 20 samples of 50 letter-qualty prter cases. Develop the tral cotrol lmts for a c chart. If ay values are out of cotrol, assume that the cause s assgable. Modfy the cotrol lmts accordgly. Sample Nocof. Sample Nocof c chart for Example Sample Number 38

39 Example A paper mll uses a cotrol chart to motor the mperfecto fshed rolls of paper. Producto output s spected for 20 days, ad the resultg data are show below. Use these data to set up a cotrol chart for ocoformtes per roll of paper. What kd of cotrol chart you should use ad why? What are the CL ad UCL/LCL? Day Rolls Produced Number of Imperfectos Day Rolls Produced Number of Imperfectos

40 z-score Example (Cot d) Cotrol Chart for Paper Imperfectos [LCL, UCL ] [0.1088, ] [0.1392, ] U [0.1527, ] [0.1653, ] 24 [0.1881, ] Day Stadardzed u chart Sample Number 40

41 OC Curve ad ARL for c ad u Charts Type II error for the c chart (OC curve see Fg 7-19, P332) (f assume pots of the cotrol lmts are cosdered as out-of-cotrol) P{ x UCLc c1} P{ x LCLc c1} Type II error for the u chart (f assume pots of the cotrol lmts are cosdered as out-of-cotrol) P{ y P{ x UCL UCL u u u u 1 1 } } P{ y P{ x LCL LCL u u u1} u } 1 ARL 0 =ARL -cotrol =1/ ARL 1 =ARL out-of-cotrol =1/(1-) 41

42 Example (Textbook Problem 7-56) A cotrol chart s to be establshed o a process producg refrgerators. The specto ut s oe refrgerator, ad a cotrol chart for ocoformtes s to be used. As prelmary data, 16 ocoformtes were couted spectg 30 refrgerators. (Assume pots of the cotrol lmts are cosdered as out-of-cotrol) a. What are the 3-sgma cotrol lmts'? b. What s the -error for ths cotrol chart? c. What s the -error f the average umber of defects s actually 2 (.e., f c = 2.O)? d. Fd the average ru legth f the average umber of defects s actually 2. 42

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