1. IE 469 Manufacturing Systems 469 صنع نظم التصنيع II - Modeling Systems Tutorial

2. Question 1 A production line is used to produce a product. Number of products at the start of the line is 400 units per week. The line consists of four stations. It is arranged such that an inspection station (I) is placed after three process stations (S1, S2, S3). The processing time of stations (S1, S2, S4, I) are (8, 10, 13, 8) minute respectively. Inspection station (I) has defect rate of 10%. 4% of the defects are scraped and the reminders are returned to station (S2). Find : (a) Determine the effective arrival rate at each station. (b) Determine the number of machines at each station. (c) Find the average work in process. (d) Find the throughput time. Problem #1 400 4% S1S2IS3 6% Question Graph (NOT GIVEN!):

3. S1S2S3I Problem #1 Solution Arrival Rate, λ unit/hr10 Service Rate, m unit/hr7.564.6153857.5 Arrival Rate, λ’ unit/hr1010.6383 Utilization Ratio, r=λ’/ m1.3333331.773052.3049651.41844 Number of machines, n2232 At this stage a decision to be taken; 1st alternative to have single production line with multi station at all staion, OR 2nd two have two production lines [each produce half the weekly production at 200. in this case the solution will be: Arrival Rate, λ = R p, unit/hr5555 Arrival Rate, λ’ = λ/(1-p), unit/hr55.319149 Service Rate, m =1/T p, unit/hr7.564.6153857.5 Utilization Ratio, r=λ’/ m0.6666670.8865251.1524820.70922 Number of machines, n =int(ρ)+11121 Utilization Ratio, r=λ’/n m0.6666670.8865250.5762410.70922 Po =probability0.3333330.1134750.2688410.29078

4. Problem #1 Solution v= λ’/ l11.06383 Lq = length of queue1.3333336.9259750.572931.729805 L =System Queue, unit27.81251.72542.439024 Total Queue in system =ΣL, units13.98 Wq =queue throughput,0.271.300.110.33 W = System Throughput, hr0.41.468750.3243770.458537 v*w0.41.56250.3450820.487805 W=Snw, hr2.7954

5. ProductDemand / Week Routing Data [Centers, Process Time in hr] 1234 115C3, 1C2, 2C1, 1C4, 2 210C4, 3C1, 3 C3, 1 36C4, 4C3, 2C1, 2 410C3, 2 C2, 2C1, 1 58C3, 3C2, 1C1, 2 Problem #2 Question 2 Five parts are processed using four machining centers (C1, C2, C3, and C4) as indicated in Table (1). (a) Determine the effective arrival rate at each center. (b) Determine the average processing time at each center. (c) Determine number of machines in each center (d) Find the average number of jobs in process. (e) Find the throughput time for each parts.

6. a- Calculate Arrival Rate Product, (i) Effective arrival Rate λ’ C1 λ’ C2 λ’ C3 λ’ C4 115 2100 36066 4 0 58880 Sum49334931 Problem #2 Solution Process time data, Tp, Hr Product,(i) Machine centers C1C2C3C4 11212 23013 32024 41220 52130

7. c) Number of machines StationλmM(c)=l/μrPoLLqWWq C14923.6130.692 0.099 3.1521.0770.0640.022 C23322.7620.725 0.159 3.0571.6070.0930.049 C34924.230.675 0.107 2.9730.9480.0610.019 C43114.7630.7 0.096 3.2491.1490.1050.037 Problem #2 Solution b- Calculate Process Times (λ’ i x T p )/(Sum (λ’i) x 40) Product,(i) Average Process Time m -1 C1 m -1 C2 m -1 C3 m -1 C4 115301530 2 01030 3120 24 41020 0 5168240 Sum m -1 0.04230.0439390.04130.0677 m23.61422.7586224.197514.762

8. e) Throughput time for each product (Troughput time, Wp = Σprocess time+ΣWq) d) Number of products in the system Problem #2 Solution

9. Problem #3 Question 3 A five departments manufacturing system is used to produce five parts according the data given in table (2). Find (a) Determine the effective arrival rate at each department. (b) Determine the number of machines at each station. (c) Find the average work in process. (d) Find the throughput time. TABLE (2) Part Weekly Demand Process Sequence Operation Time,hr LoadProcess StationUnload ABCDEA 16A > C > B > D > A0.221.52.100.2 27A > B > D > E > A0.22.802.21.50.2 33A > C > D > E > A0.201.51.820.2 45A > C >E > B > A0.21.5002.20.2 54A > B > E > C > A0.22.5201.80.2 Ʃ 25

10. a) Arrival rate calculation demand of part (i) at department (k), λ’ P1P2P3P4P5l d =Σ(l ki ) A 67354 25 B 67 --- 54 22 C 6 --- 3 4 13 D 67 3 --- 16 E --- 7 3 54 19 Problem #3 Solution operation time of part (i) at department (k), T ki P1P2P3P4P5 A0.4 B2 2.8 --- 1.52.5 C1.5 --- 1.5 --- 2 D2.1 2.2 1.8 --- E 1.5 2 2.21.8 ΣToΣTo 66.95.74.16.7

11. b) Service rate calculation Calculate the operation time (service time) for a department, m d -1 =S(λ ki /λ d )*(T ki /40), P1P2P3P4P5 Ʃ (OT&AR) μ d -1 μdμd A2.42.81.221.6100.0100100.0 B1219.607.51049.10.055817.923 C904.50821.50.041324.186 D12.615.45.40033.40.052219.162 E010.56117.234.70.045721.902 Problem #3 Solution λμCρPoPo LLqLq WWqWq A 25100.010.2500.7500.333330.08330.0130.0033 B 2217.92320.6140.2391.969320.74180.0900.0337 C 1324.18610.5380.4631.162160.62470.0890.0481 D 1619.16210.8350.1655.060614.22560.3160.2641 E 1921.90210.8680.1336.547175.67970.3450.2989

12. Throughput for each product T P = ΣT o /40Tp+ΣWqTp+ΣWq W10.1500.499 W20.1730.773 W30.1430.757 W40.1030.438 W50.1680.552 Problem #3 Solution Average work in process L ƩLƩL 0.33333 15.0726 1.96932 1.16216 5.06061 6.54717