IE 469 Manufacturing Systems 694 II Modeling Systems

IE 469 Manufacturing Systems ﺼﻨﻊ ﻨﻈﻢ ﺍﻟﺘﺼﻨﻴﻊ 694 II - Modeling Systems Tutorial

Problem #1 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 (S 1, S 2, S 3). The processing time of stations (S 1, S 2, S 4, 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 (S 2). 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. Question Graph (NOT GIVEN!): 400 S 1 S 2 S 3 I 4% 6%

Problem #1 Solution Arrival Rate, λ unit/hr Service Rate, m unit/hr S 1 10 7. 5 S 2 10 6 S 3 10 4. 615385 I 10 7. 5 Arrival Rate, λ’ unit/hr Utilization Ratio, r=λ’/ m 10 1. 333333 10. 6383 1. 77305 10. 6383 2. 304965 10. 6383 1. 41844 Number of machines, n 2 2 3 2 At this stage a decision to be taken; 1 st alternative to have single production line with multi station at all staion, OR 2 nd two have two production lines [each produce half the weekly production at 200. in this case the solution will be: Arrival Rate, λ = Rp, unit/hr 5 5 Arrival Rate, λ’ = λ/(1 -p), unit/hr Service Rate, m =1/Tp, unit/hr Utilization Ratio, r=λ’/ m Number of machines, n =int(ρ)+1 Utilization Ratio, r=λ’/n m 5 7. 5 0. 666667 1 0. 666667 5. 319149 6 0. 886525 1 0. 886525 5. 319149 4. 615385 1. 152482 2 0. 576241 5. 319149 7. 5 0. 70922 1 0. 70922 Po =probability 0. 333333 0. 113475 0. 268841 0. 29078

Problem #1 Solution v= λ’/ l Lq = length of queue L =System Queue , unit Total Queue in system =ΣL, units Wq =queue throughput, W = System Throughput, hr v*w W=Snw, hr 1 1. 333333 2 1. 06383 6. 925975 0. 57293 7. 8125 1. 7254 13. 98 1. 06383 1. 729805 2. 439024 0. 27 0. 4 1. 30 0. 11 1. 46875 0. 324377 1. 5625 0. 345082 2. 7954 0. 33 0. 458537 0. 487805

Problem #2 Question 2 Five parts are processed using four machining centers (C 1, C 2, C 3, and C 4) 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. Product 1 2 3 4 5 Demand / Week 15 10 6 10 8 Routing Data [Centers, Process Time in hr] 1 2 3 C 3, 1 C 4, 3 C 4, 4 C 3, 2 C 3, 3 C 2, 2 C 1, 3 C 3, 2 C 2, 1 C 1, 2 C 2, 2 C 1, 2 4 C 4, 2 C 3, 1 C 1, 1

Problem #2 Solution a- Calculate Arrival Rate Product, (i) 1 2 3 4 5 Sum Product, (i) 1 2 3 4 5 λ’ C 1 15 10 6 10 8 49 Effective arrival Rate λ’ C 2 15 0 0 10 8 33 λ’ C 3 15 10 6 10 8 49 λ’ C 4 15 10 6 0 0 31 Process time data, Tp , Hr Machine centers C 1 C 2 C 3 C 4 1 3 2 1 2 2 0 0 2 1 1 1 2 2 3 4 0 0

Problem #2 Solution b- Calculate Process Times (λ’i x Tp)/(Sum (λ’i) x 40) Average Process Time Product, (i) m-1 C 1 m-1 C 2 m-1 C 3 1 15 30 15 2 30 0 10 3 12 0 12 4 10 20 20 5 16 8 24 Sum m-1 0. 0423 0. 043939 0. 0413 m 23. 614 22. 75862 24. 1975 m-1 C 4 30 30 24 0 0 0. 0677 14. 762 c) Number of machines Station C 1 C 2 C 3 C 4 λ 49 33 49 31 m 23. 61 22. 76 24. 2 14. 76 M(c)=l/μ 3 2 3 3 r 0. 692 0. 725 0. 675 0. 7 Po 0. 099 0. 159 0. 107 0. 096 L 3. 152 3. 057 2. 973 3. 249 Lq 1. 077 1. 607 0. 948 1. 149 W 0. 064 0. 093 0. 061 0. 105 Wq 0. 022 0. 049 0. 019 0. 037

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

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) Weekly Part Demand 1 2 3 4 5 6 7 3 5 4 Ʃ 25 Process Sequence A>C>B>D>A A>B>D>E>A A>C>D>E>A A > C >E > B > A A>B>E>C>A Load A 0. 2 0. 2 Operation Time, hr Process Station B C D 2 1. 5 2. 1 2. 8 0 2. 2 0 1. 5 1. 8 1. 5 0 0 2. 5 2 0 E 0 1. 5 2 2. 2 1. 8 Unload A 0. 2 0. 2

Problem #3 Solution a) Arrival rate calculation demand of part (i) at department (k), λ’ A B C D E P 1 6 6 --- P 2 7 7 --7 7 P 3 3 --3 3 3 P 4 5 5 ----5 P 5 4 4 4 --4 ld=Σ(lki) 25 22 13 16 19 operation time of part (i) at department (k) , Tki A B C D E ΣTo P 1 0. 4 2 1. 5 2. 1 --6 P 2 0. 4 2. 8 --2. 2 1. 5 6. 9 P 3 0. 4 --1. 5 1. 8 2 5. 7 P 4 0. 4 1. 5 ----2. 2 P 5 0. 4 2. 5 2 --1. 8 4. 1 6. 7

Problem #3 Solution b) Service rate calculation Calculate the operation time (service time) for a department, md-1 =S(λki/λd)*(Tki/40), P 1 2. 4 12 9 12. 6 0 A B C D E P 2 2. 8 19. 6 0 15. 4 10. 5 P 3 1. 2 0 4. 5 5. 4 6 P 4 2 7. 5 0 0 11 P 5 1. 6 10 8 0 7. 2 Ʃ(OT&AR) 10 49. 1 21. 5 33. 4 34. 7 μd-1 0. 0100 0. 0558 0. 0413 0. 0522 0. 0457 μd 100. 0 17. 923 24. 186 19. 162 21. 902 λ μ C ρ Po L Lq W Wq 25 22 13 16 19 100. 0 17. 923 24. 186 19. 162 21. 902 1 1 1 0. 250 0. 614 0. 538 0. 835 0. 868 0. 750 0. 239 0. 463 0. 165 0. 133 0. 33333 1. 96932 1. 16216 5. 06061 6. 54717 0. 0833 0. 7418 0. 6247 4. 2256 5. 6797 0. 013 0. 090 0. 089 0. 316 0. 345 0. 0033 0. 0337 0. 0481 0. 2641 0. 2989

Problem #3 Solution Average work in process L ƩL 0. 33333 1. 96932 15. 0726 1. 16216 5. 06061 6. 54717 Throughput for each product W 1 W 2 W 3 W 4 W 5 TP= ΣTo/40 Tp+ΣWq 0. 150 0. 173 0. 143 0. 103 0. 168 0. 499 0. 773 0. 757 0. 438 0. 552