IEOR 130 Methods of Manufacturing Improvement Fall, 2017 Prof. Leachman Solutions to First Homework Assignment

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1 IEOR 130 Methods of Maufacturg Improvemet Fall, 2017 Prof. Leachma Solutos to Frst Homework Assgmet 1. The scheduled output of a fab a partcular week was as follows: Product 1 1,000 uts Product 2 2,000 uts Product 3 20,000 uts Product 4 10,000 uts Product 5 5,000 uts The actual output the week was as follows: Product uts Product 2 1,900 uts Product 3 21,000 uts Product 4 1,000 uts Product 5 5,500 uts The LIPAS (le-tem performace agast schedule) s a o-tme delvery metrc defed as the fracto of tems wth scheduled output a tme perod whose output the perod meets or exceeds the scheduled output. What s the LIPAS score for that week? O tme delvery score s the fracto of products delvered o tme. For ths case, products 3 ad 5 are o tme, whle products 1, 2 ad 4 are ot. So the LIPAS score s 2/5 = The LIPAS metrc for o-tme delvery preseted class (ad also wdely used dustry) s the fracto of all product types that were delvered o tme. (A product type s cosdered otme f actual output equals or exceeds scheduled output.) Ths metrc s sestve to the amout of shortage of each product. Develop a ew metrc for o-tme delvery that reflects the amout of shortage for each product. The metrc should have the propertes that (1) a score of 1.0 meas all product types were o tme, (2) o credt s gve for excess producto of ay product type, but there s o pealty for excess producto, ether, ad (3) f the shortage of ay product creases whle shortages of all other products are held costat, the the metrc score decreases. Frst, cosder the case of o-tme delvery a sgle tme perod. Let d deote the scheduled output quatty of product. Let p deote the actual output quatty of product. The fracto of product delvered o tme s f = M { d, p } / d. 1

2 We could do a weghted average of the o tme delvery scores for all products to arrve at a otme delvery score for the factory. That s, OTD = { d f } / d, or, equvaletly, OTD = M { d, p } / d Note the umerator sums over the porto of actual producto that was demaded, ad the deomator sums over all demad. Next, we cosder the case of a stream of tme perods whch surplus perod t-1 ca be used to make up for shortfall perod t. Let d(t) deote the demad for product perod t ad let p(t) deote the actual output of product perod t. Let I(0) deote the vetory (shortage f egatve) at tme 0,.e., the start of perod 1. The fracto of product delvered o tme perod t s f ( t) M d ( t), p ( t) Max 0, I (0) d ( t) t1 p ( ) t1 1 1 d ( ), ad the o-tme delvery metrc becomes OTD(t) = { d(t) f(t) } / d(t). 3. A cotrol chart s beg set up to track the resstvty after the o mplatato process step. Fve sample measuremets are made from oe wafer per lot. The average of the measuremets ad the rage of the measuremets are computed. Oe hudred observatos from 20 lots have bee collected. The average mea s 97.5, ad the average rage of the wafer measuremets s (a) Assumg the process was stable durg processg of the 20 lots, determe three-sgma cotrol lmts for R ad X-bar charts. = 97.5, R = 5.21, = 5 = 5.21 / d2 = 5.21 / = 2.24 X-bar chart: UCL = (2.24)/5 0.5 = LCL = (2.24)/5 0.5 = 94.5 R-chart: 2

3 LCL = d3* R = 0 UCL = d4* R = 2.1(5.21) = (b) Suppose the mea of the resstvty shfts to What s the probablty that the X-bar chart wll make a Type 2 error the ext lot? (A Type-2 error meas the process s out of cotrol but the cotrol chart does ot detect ths.) Prob {94.5 X = 100} = Prob { ( )/{2.24/5 0.5 } Z ( )/{2.24/5 0.5 } } = Prob { Z } = (0.4992) - (-5.490) = ) = (c) What s the probablty the X-bar chart wll make Type 2 errors each of the ext fve lots? (0.6914) 5 = A process s motored usg a X-bar chart wth UCL = 13.8 ad LCL = 8.2. The process stadard devato s estmated to be 6.6. If the X-bar chart s based o three-sgma lmts, (a) What s the estmate of the process mea? UCL= / 0.5, LCL= / 0.5 = (UCL + LCL) / 2 = 11 (b) What s the sze of each of the samplg subgroups? UCL - LCL = 6* / = (6*6.6) / ( ) = = A R-chart s used to motor the varato the weghts of packages of chocolate chp cookes produced by a large atoal producer of baked goods. A aalyst has collected a basele of 200 observatos to costruct the chart. Suppose the computed value of R (the average value of the rage) s (a) If subgroups of sze sx are used, compute the value of three-sgma lmts for ths chart. R = 3.825, = 6 3

4 = / d2 = / = 1.51 LCL = d3* R = 0 UCL = d4* R = 2*3.825 = 7.65 (b) If a X-bar chart based o three-sgma lmts s used, what s the dfferece betwee the UCL ad the LCL? UCL= / 0.5, LCL= / 0.5 UCL - LCL = 6* / = At a partcular maufacturg step, the mportat parameter for process cotrol s the deposto thckess. Ths s measured at fve pots o a sgle wafer from each maufacturg lot passg through the step. The mea of ths parameter s 380 ad the stadard devato s 54. (a) Assumg the estmates of the process mea ad stadard devato are vald (.e., the process was statstcal cotrol durg the tme data was collected to compute them), specfy upper ad lower cotrol lmts for X-bar ad R charts. I X-bar chart: UCL = I R-chart: 3 = (54)/2.236 = , LCL = R = d2() = 2.326(54) = LCL = d3 * R = 0, UCL = d4 * R = 2.11(125.6) = = 380 3(54)/2.236 = (b) Suppose the process mea suddely shfts by 27. What s the probablty that there wll be Type II errors occurrg for both of the ext two maufacturg lots? Prob of Type II error frst lot gve the mea shfts by s k P X E(X ) k k P X E(X )

5 P X k k E(X ) P{ k Z k } ( k ) ( k ). I ths case, k = 3, = 0.5 ad = 5. Usg the table the back of the otes, we fd (1.882) (-4.118) = 0.97 The probablty of a Type II error both of the ext two lots s 2 = The thckess of a flm deposted o wafers at a partcular process step s subject to statstcal process cotrol. The thckess s measured at fve pots o oe wafer per lot. The upper cotrol lmt s 132 agstroms ad the lower cotrol lmt s 96 agstroms. (a) What kd of cotrol chart(s) should be used to track ths parameter? Assume the followg questos that ths kd of chart s use. X-bar ad R charts should be used. (b) What are the mea ad stadard devato of the flm thckess? = ( )/2 = 114 agstroms = 36 = 6 / 5 or = (c) What s the average rage of the fve measuremets? R = d2() = 2.326(13.42) = (d) Suppose the process mea suddely shfts upward by 10 agstroms. What s the probablty the mea shft wll NOT be detected the ext fve lots? (Assume the oly cotrol rule s ordary UCL ad LCL for sgle-lot measuremets.) Prob of Type II error frst lot gve the mea shfts by s k P X E(X ) k k P X E(X ) 5

6 P X k k E(X ) P{ k Z k } ( k ) ( k ). I ths case, k = 3, = 10/13.42 = ad = 5. Usg the table the back of the otes, we fd (1.334) (-4.666) = The probablty of a Type II error all of the ext fve lots s 5 = (e) Suppose the process mea does ot shft. What s the probablty of a false alarm the ext lot? Prob of Type I error frst lot s k P X E(X ) k 2P X E( X ) X 2P k E( X ) 2( k ) 2( 3) 2(0.0014) Recet data o rework the photolthography process at a partcular fab are as follows. Shft # # of wafers processed # of wafers reworked Durg whch shfts was photo rework statstcal cotrol? 6

7 Ths s a p-chart problem wth varyg subgroup szes, so we ca apply the stadardzed varate Z o each shft ( p s for these data): Z p p. p(1 p) The calculatos of p ad Z for the four shfts are as follows: Shft # p Z /500 = /650 = /550 = /600 = All shfts have a Z value lower tha 3. Assumg 0.49 s a good estmate of the log-ru mea rework rate, rework was statstcal cotrol durg all four shfts. 9. A cotrol chart s beg set up to track the umber of partcles deposted o wafers after etchg. Partcles are couted o oe blak wafer usg a wafer surface scag mache before processg each lot. Oe hudred observatos have bee collected. The sample mea umber of partcles per wafer s 25. The sample varace s (a) Determe three-sgma cotrol lmts for a approprate cotrol chart. What kd of chart s ths? Use a c-chart, mea = 25. UCL = *( ) = 40 LCL = 25-3*( ) = 10 (but LCL may be rrelevat ths case.) (b) Suppose the mea shfts to What s the probablty the cotrol chart wll make Type 2 errors both of the ext two lots? Prob. of oe Type 2 error = Prob.{10 c 40 = 30} = Prob. { (10-30) / Z (40-30) / } = Prob. { Z }= Prob.{Z } = The prob. of two Type 2 errors a row s (0.966) 2 =

8 (c) Suppose oe of the observatos from the sample was 45 partcles. Was ths pot statstcal cotrol? How should the cotrol lmts be modfed? Mea was observatos. But the observato of 45 s above UCL ad so t was ot statstcal cotrol. We should re-calculate the mea after dscardg ths pot. The ew mea s ( ) / 99 = The ew cotrol lmts are UCL = *( ) = 39.74, or 40 LCL = *( ) = 9.86, or 10 (but LCL may be rrelevat ths case) So we do t eed to chage the cotrol lmts. 8