IEOR 130 Midterm Examination 2023_sol

Solutions to IEOR Midterm Examination Fall 2023, Prof. Leachman Work all problems. 20 points for each problem, 100 points total. Your answers to the exam must be turned in via email to leachman@berkeley.edu and to jooseung_lee@berkeley.edu by 2pm Sunday Oct. 15. You can email me any questions you have. You also may talk to others in the class about the exam questions, but each student must write up his or her own answers. Grading will be based not just on the correctness of analysis and numerical results but also on the clarity of exposition. You don’t need to turn in spreadsheets of your calculations, but you should show all formulas that you used and the values of all input parameters to those formulas. 1. (20 points) The thickness of a film deposited on wafers at a particular process step is subject to statistical process control. The upper specification limit for the film thickness is 60 angstroms and the lower specification limit is 15 angstroms, i.e., wafers with film thickness more than 60 angstroms or less than 15 angstroms deposited on them are scrapped. At present, the process has considerable variability, with mean film thickness equal to 30 angstroms and standard deviation equal to 10 angstroms. (a) (3 points) What kind of control chart(s) should be used to track this parameter? We should use X bar and R chart. The X bar is used to monitor the mean film thickness, and the R chart is used to monitor the standard deviation of the film thickness. (b) (5 points) What is the process performance index for this step? ? ?? = ?𝑖? ( ??? − 𝜇 3𝜎 , 𝜇 − ??? 3𝜎 ) = min ( 60 − 30 3 × 10 , 30 − 15 3 × 10 ) = ?𝑖?(1, 0.5) = 0 .5 (c) (7 points) Assume the only yield loss mechanism at this process step is out-of-spec film thickness. What is the yield of this process step? ? = ? − 𝜇 𝜎 ? 𝐿𝑆𝐿 = 15 − 30 10 = −1.5 ? 𝑈𝑆𝐿 = 60 − 30 10 = 3 Using the standard normal distribution table, we have Z -1.5 = 0.0668 and Z 3 = 0.9986. ?𝑖??? = 0.9986 − 0.0668 = 0.9319 = 93 .18%

(d) (5 points) To raise the yield of this step to 95%, what value for the process performance index must be achieved? A yield of 0.9500 corresponds to about Z => ? ?? = 1.645. 2. (20 points) A stacked wafer map for a particular product with a die area of 0.5 sq cm has been prepared from 450 wafers believed not to have been involved in any defect excursions. The map shows the average yield by die site. Out of 300 total die sites, there are 250 die sites that seem to be free of edge loss or other spatial signatures. The best observed yield in the stack is 72% and this yield occurs at only two sites. (a) (8 points) What is the baseline random yield? What is the baseline defect density? ?? − ? ? = ?𝜎 = 𝜙 −1 (1 − ? ? ) √ ? ? (1 − ? ? ) ? = 𝜙 −1 (1 − 2 300 ) √ ? ? (1 − ? ? ) 450 0.72 = ? ? + 2.475 √ ? ? (1 − ? ? ) 450 ? ? = 0 .665 = 66 .5% ? 0 = − ln(?) 𝐴 = − ln(0.665) 0.5 ≈ 0.816 ??????? ? ? 2 (b) (2 points) The die yield for the product is 48%. What is the systematic mechanisms-limited yield? ? ? = ? ? ? = 0.48 0.665 ≈ 0.722 = 72 .2% (c) (7 points) The following systematic mechanisms have been identified by the yield engineers: Mechanism Fraction of wafers Fraction of dice Kill rate Parametric test fails Metal deposition 0.02 1.00 of dice on wafer 1.00 defect excursions 0.05 0.7 of dice on wafer 0.5* Edge loss Contact layer 1.00 0.12 of dice on wafer 1.00 pattern fails Implant layer 1.00 0.025 of dice on wafer 1.00 pattern fails 1.00 0.05 of dice on wafer 1.00

The yield engineer reports that, on average, 25% of particles deposited by CMP become fatal defects. The fab has one product in production, as follows: Product Die size Gross die per wafer A 1 sq cm 600 (a) (5 points) Suppose the baseline fatal defect density for the fabrication process excluding CMP particles is D 0 . Assuming a simple Poisson yield model, express the total good die output in the first n lots processed right after a slurry change. Assume lots contain 25 wafers and line yield is 100%. You don’t need to simplify the resulting expression. ?𝑖??? 𝑖 = ? −(𝐷 0 +0.02×0.25×(𝑖−1) = ? −(𝐷 0 +0.005(𝑖−1)) ?????𝑖? 𝑖 = ?𝑖??? 𝑖 × 25 × 600 = 15000?𝑖??? 𝑖 ?????𝑖? ???𝑎? = ∑ ?????𝑖? 𝑖 ?−1 𝑖=0 = 15000 ? −𝐷 0 ∑ ? −0 .005 (𝑖 −1 ) ? 𝑖 =1 (b) (10 points) The CMP department is wondering how often they should schedule slurry changes. Suppose the frequencies under consideration are: once every 80 lots, once every 100 lots, or once every 120 lots. Assume CMP is the fab bottleneck and the minimum planned idle time for CMP is one hour per day. Which of these three frequencies maximizes good die output? ???? 80 = 24 − 1 − 2 0.5 ∗ 80 + 8 = 21 48 ???? 100 = 24 − 1 − 2 0.5 ∗ 100 + 8 = 21 58 ???? 120 = 24 − 1 − 2 0.5 ∗ 120 + 8 = 21 68 ?????𝑖 ? 80 = 21 48 × 15000? −𝐷 0 ∑ ? −0.005(𝑖−1) 80 𝑖=1 = 433788? −𝐷 0 ?????𝑖? 100 = 21 58 × 15000? −𝐷 0 ∑ ? −0.005(𝑖−1) 100 𝑖=1 = 428458? −𝐷 0 ?????𝑖? 120 = 21 68 × 15000? −𝐷 0 ∑ ? −0.005(𝑖−1) = 419059? −𝐷 0 120 𝑖=1

Comparing all three frequency outputs, we found that 80 lots maximizes the good die output. (c) (5 points) Now suppose CMP is not the fab bottleneck, and the current fab starts rate is 20 lots per CMP machine per day. In this case, what frequency of slurry changes is best? Requirement min = 20 ∗ 0.5 = 24 − 1 − 2 − 20 × 8 ? ?? ???? ??𝑖?𝑖𝑧???𝑖? ???? ?? ?? ?? ????? ? = 14.546.Higher the better. ??𝑖?𝑖𝑧 ? 80 = 20 × 0.5 24 − 1 − 2 − 20 × 8 80 = 52.63% ??𝑖?𝑖𝑧? 80 = 20 × 0.5 24 − 1 − 2 − 20 × 8 100 = 51.55% ??𝑖?𝑖𝑧? 80 = 20 × 0.5 24 − 1 − 2 − 20 × 8 120 = 50.85% 4. (20 points) A company has the following information concerning demand and supply of a particular product: Period 1 2 3 4 5 6 7 8 9 10 Demand forecast 120 40 50 40 55 45 50 50 50 50 Orders on hand 120 40 50 10 55 5 0 0 0 0 Planned supply 145 45 50 50 50 50 50 50 50 50 Note: The demand forecast includes orders on hand plus an estimate of future customer orders yet to be received. The above figures are NOT cumulative over time. (a) (4 points) Calculate cumulative available-to-promise quantities for each period. ? ? = ∑ ? 𝑖 ? 𝑖=1 ? ? = ∑ ? 𝑖 ? 𝑖=1 ?𝑖?? ? = ? ? − ? ? 𝐴 ? = ?𝑖?{?𝑖?? 𝑖 | 𝑖 = ?, ? + 1, … , ?}

? ? 120 160 210 220 275 280 280 280 280 280 ?𝑖?? 25 30 30 70 65 110 160 210 260 310 𝐴 ? 25 30 30 65 65 110 160 210 260 310 ? ? 0 0 30 70 110 150 200 200 200 200 ? ? 0 0 30 65 65 110 160 200 200 200 ??? 𝐴 ? 0 0 0 0 0 0 0 10 60 110 (d) (5 points) Suppose the lead time for producing the product is 3 periods and the yield is 85%. The product supply in period 1 includes 100 units already in finished goods inventory plus 45 units projected output from work-in-process. The remaining work-in-process is projected to supply 45 units in period 2 and another 50 units in period 3. Perform requirements planning calculations to compute required production start quantities in each period satisfying the demand forecasts. Assume demand forecasts do not change as a result of the new customer inquiry. You can report fractional amounts to the nearest tenth of a unit. ???????? ? = ∑ ???????? 𝑖 ? 𝑖=1 ?𝐼? ? = ∑ ?𝐼? 𝑖 ? 𝑖=1 ???????? ?? = ??? (0 , ???????? ? − ?𝐼? ? 0.85 ) ????? ?? = ???????? ?? 𝑖+1 − ???????? ??𝑖 ???𝑖?? 1 2 3 4 5 6 7 8 9 10 ? ? 145 190 240 290 340 390 440 490 540 590 ? ? 120 160 210 220 275 280 280 280 280 280 ?𝑖?? 𝑆−? 25 30 30 70 65 110 160 210 260 310 𝐴 ? 25 30 30 65 65 110 160 210 260 310 ? ? 0 0 30 70 110 150 200 200 200 200

? ? 0 0 30 65 65 110 160 200 200 200 ??? 𝐴 ? 0 0 0 0 0 0 0 10 60 110 ???????? ? 120 160 210 250 305 350 400 450 500 550 WIP 145 190 240 240 240 240 240 240 240 240 ?𝑖?? ??−𝑊𝐼? -25 -30 -30 10 65 110 160 210 260 310 ???????? ?? 0 0 0 11.8 76.5 129.4 188.2 247.1 305.9 365.7 ???? ? ?? 0 0 0 11.8 64.7 52.9 58.8 58.9 58.8 59.8 (e) (5 points) Perform requirements planning calculations to compute required production start quantities in each period satisfying only the orders on hand, excluding the new customer inquiry. You can report fractional amounts to the nearest tenth of a unit. ????? ? = ∑ ????? 𝑖 ? 𝑖=1 ???????? ??? = ??? (0 , ????? ? − ?𝐼? ? 0.85 ) ????? ??? = ???????? ??? 𝑖+1 − ???????? ??? 𝑖 ???𝑖?? 1 2 3 4 5 6 7 8 9 10 ? ? 145 190 240 290 340 390 440 490 540 590 ? ? 120 160 210 220 275 280 280 280 280 280 ?𝑖?? 𝑆−? 25 30 30 70 65 110 160 210 260 310 𝐴 ? 25 30 30 65 65 110 160 210 260 310 ? ? 0 0 30 70 110 150 200 200 200 200 ? ? 0 0 30 65 65 110 160 200 200 200 ??? 𝐴 ? 0 0 0 0 0 0 0 10 60 110 ???????? ? 120 160 210 250 305 350 400 450 500 550 WIP 145 190 240 240 240 240 240 240 240 240

h. The best average yield observed among the die sites on a stacked wafer map occurs at a die site where there were only systematic yield loss mechanisms affecting the site and no defect loss mechanisms were evident. False i. If there were no systematic yield loss mechanisms, then a histogram of die yield per wafer should look like a normal distribution. True j. The Bose-Einstein yield model assumes fatal defects follow an exponential distribution in each critical layer. True k. The kill rate for a particular type of defect is estimated by checking the die sort test results for die for which such a defect was deposited on it according to in-line defect monitoring. The yield loss for that type of defect is estimated as the product of the kill rate and the probability such a defect is deposited on the die. False l. Kill rates estimated as in part i immediately above may be understated in the case where there are multiple yield loss mechanisms present. True m. One may estimate the total yield loss from loss mechanisms that are mutually exclusive by adding the estimated losses from each mechanism. If the mechanisms may simultaneously affect the same die, then one must first compute the product of the yields for each mechanism, and then subtract the result from unity. True n. Particle contamination is included in random defect density and so it should not be modeled as a systematic yield loss mechanism. True o. Suppose that a change is proposed to maintenance procedures on a given equipment type, and this change would increase the average time to perform maintenance (considering both PMs and repairs after failure) by 10%. Further, suppose that if the change is made, the average time between maintenance events (again, considering both failures and PMs) will increase by 15%. Then the expected availability of the equipment will be improved by adding the proposed maintenance procedures. True p. Total Productive Manufacturing (TPM) is an approach in which the manufacturing operators are trained to continually inspect the manufacturing equipment, note any deficiencies, correct the deficiencies which they are qualified to correct, and report those deficiencies for which they are not qualified to correct. True q. If utilization increases, then OEE increases. False r. Capacity of a machine type is not a fixed number but instead is a limit that is set based on a judgment of what is the maximum acceptable cycle time for steps performed by that machine

type. If a longer cycle time becomes acceptable, then capacity can be increased accordingly. True s. One should not use theoretical process times to develop the coefficients of capacity constraints used in production planning because quality and rate efficiency losses typically vary with product mix. False t. MRP logic is not applicable to the case where there are alternative downstream uses for an intermediate product. False