MODELING AND ANALYSIS OF MANUFACTURING SYSTEMS

1. MODELING AND ANALYSIS OFMANUFACTURING SYSTEMS Session 4 ASSEMBLY LINES February 2001

2. ASSEMBLY LINE • SET OF SEQUENTIAL WORKSTATIONS • CONNECTED BY A CONTINUOUS MATERIALS HANDLING SYSTEM • INPUT: RAW MATERIALS • OUTPUT: FINISHED PRODUCT

3. WORK ELEMENTS SMALLEST UNITS OF PRODUCTIVE (i.e. VALUE-ADDING) WORK

4. BACKBONES OF ASSEMBLY LINES • PRINCIPLE OF INTERCHANGEABILITY • DIVISION OF LABOR

5. ASSEMBLY LINE TYPES • SINGLE PRODUCT • MULTIPLE PRODUCT • MIXED LINES

6. ADVANTAGES easy work load balancing increasing scheduling flexibility job enrichment higher line availability more accountability DISSADVANTAGES higher setup costs higher equipment costs higher skill requirements slower learning complex supervision MULTIPLE PARALLEL LINES

7. WORKSTATION CYCLE TIME • PACED LINES • UNPACED LINES (ASYNCHRONOUS) • ROLE OF BUFFERS • PARALLEL WORKSTATIONS IN SERIAL SYSTEMS

8. BASIC LINE BALANCING PROBLEM TO ASSIGN WORK ELEMENTS TO WORKSTATIONS SUCH THAT ASSEMBLY COST IS MINIMIZED

9. TOTAL ASSEMBLY COST • LABOR COST (WHILE PERFORMING TASKS) • IDLE TIME COST • FOCUS: MINIMIZE IDLE TIME • LIMITS: PRODUCTION CONSTRAINTS

10. PROBLEM FORMULATION • PRODUCTION RATE P (UNITS/TIME) • NUMBER OF PARALLEL LINES m • TO MEET DEMAND: CYCLE TIME m/P • TIME TO PERFORM TASK i : ti • NO WORKER MUST BE ASSIGNED A SET OF TASKS OF DURATION LONGER THAN m/P = C !

11. SOME FEATURES OF TASKS • ORDER PARTIALLY DETERMINED • ASSEMBLY ORDER CONSTRAINTS IP • ZONING RESTRICTIONS • TASK PAIRS TO SAME STATION ZS • TASK PAIRS NOT PERFORMED IN SAME WORKSTATION ZD

12. DECISION VARIABLES • TASK i ASSIGNED TO STATION k ? • Xik = {1,0} • TOTAL NUMBER OF STATIONS K • COST COEFFICIENTS cik • TOTAL NUMBER OF TASKS N

13. PROBLEM FORMULATION • MINIMIZE  (cik Xik) • SUBJECT TO:  ti Xik < C (all stations k)  Xik = 1 (all tasks i) Xvh < Xuj (all k) & (u,v) in IP  (Xuk Xvk)=1 (all k) & (u,v) in ZS Xuh+Xvh < 1 (all k) & (u,v) in ZD

14. OBJECTIVE FUNCTION FEATURES • LOWERED NUMBER STATIONS FILL UP FIRST • ONLY STATIONS WITH AT LEAST ONE TASK ARE CONSTRUCTED • BECHMARKING GAGE: PROPORTION OF IDLE TIME • IDLE TIME = (PAID -PRODUCTIVE)

15. BALANCE DELAY(measures proportion of idle time) D = (K* C -  ti)/(K* C) = idle time/paid time where K* is the number of stations required by the solution

16. COMMMENTS • D IS IDLE TIME OVER PAID TIME • OBJECTIVE DOES NOT ALLOCATE IDLE TIME EQUALLY AMONG STNS • BEST SOLUTIONS: GOOD WORK LOAD BALANCING • TOTAL TASK TIME T =  ti • MINIMUM STATIONS (LOWER BOUND) Ko = | T/C |

17. LINE BALANCING APPROACHES • COMSOAL • RPWH • OPTIMAL SOLUTIONS • TREE GENERATION & EXPLORATION • PROBLEM STRUCTURE RULES • FATHOMING RULES

18. LINE BALANCING APPROACHES (contd) • Required cycle time, sequencing restrictions and tasks times are all known.

19. COMSOAL • Computer Method for Sequencing Operations for Assembly Lines • Simple record keeping to allow examination of many possible sequences • Sequences are generated by random picking a task and constructing subsequent tasks • New stations are opened when needed

20. COMSOAL (contd) • Sequences that exceed the best solution are discarded • Better sequences become upper bounds

21. COMSOAL (contd) • Array of number of Immediate Predecesors for each task i NIP(i) • Array of for which other tasks is i an immediate predecesor WIP(i) • Array of N tasks TK

22. COMSOAL (contd) • List of unassigned tasks A • List of tasks from A with all immediate predecesors assigned B • List of tasks from B with tasks times not exceeding remaining cycle time in the current workstation F

23. COMSOAL ALGORITHMFor generating X trial solutions 1.- SET x=0, UB=inf, c=C 2.- START NEW SEQUENCE: • SET x=x+1, A=TK, NIPW(i) = NIP(i) 3.- PRECEDENCE FEASIBILITY • FOR i IN A, IF NIPW(i) = 0 , ADD i TO B

24. COMSOAL ALGORITHM(contd) 4.- TIME FEASIBILITY • FOR i IN B, IF ti < c ADD i TO F . • If F empty , 5 , otherwise 6 5.- OPEN NEW STATION • IDLE=IDLE + c , c = C • If IDLE > UB , 2, otherwise 3

25. COMSOAL 6.- SELECT TASK: SET m = card{F} • RANDOM GENERATE RN in U(0,1) • LET i* = [m*RN]th TASK from F • REMOVE i* from A,B,F • c = c - ti • FOR ALL i in WIP(i*), NIPW=NIPW-1 • IF A EMPTY --> 7, OTHERWISE --> 3

26. COMSOAL 7.- SCHEDULE COMPLETION • IDLE = IDLE + c • IF IDLE < UB , UB = IDLE --> STORE SCHEDULE • IF x = X , STOP, OTHERWISE --> 2

27. Example 2.1 (pp. 40-42) • Assembly of a spring-activated toy car • Two 4-hr shifts w/ two 10 min breaks • Four days a week • Planned production rate 1500 units/week • Tasks, times and precedence constraints are shown in Table 2.2 and Fig. 2.5 • No zoning constraints • Cycle time C = 1.17 minutes/unit ~ 70 s

28. Example 2.1 (contd) • Four potential first tasks (a, d, e, or f) • Generate a random number R (=0.34) • Continue until schedule is completed. See Table 2.3 • Exercise: Develop a Table like Table 2.3 by doing your own random number generation.

29. RPWH • Ranked Positional Weight Heuristic • A single sequence is constructed • A task is prioritized by cummulative assembly time associated with itself and its succesors • Tasks are then assigned to the lowest numbered feasible workstation

30. RPWH (contd) • S(i) succesor tasks to task i • PW(i) = ti +  tj ; j in S(i)

31. RPWH (contd) 1.- TASK ORDERING • FOR ALL TASKS i , COMPUTE THE POSITIONAL WEIGHT PW(i) • RANK TASKS BY NONINCREASING PW 2.- TASK ASSIGNMENT • FOR RANKED TASKS i , ASSIGN TASK i TO FIRST FEASIBLE WORKSTATION

32. Example 2.2 (pp. 43-44) • RPWH applied to Example 2.1 • Starting at last task compute PW(l) • Compute backwards PW(k) = tk + PW(l) • See values in Table 2.4 • Iteratively assign tasks to first feasible station • See sequence in Table on p. 44

33. OPTIMAL SOLUTIONS • TREE GENERATION • Tree (Fig. 2.7, p. 46) • Backtracking (Fig. 2.8, p. 47) • Flowchart (Fig. 2.9, p. 49) • TREE EXPLORATION • PROBLEM STRUCTURE RULES • FATHOMING RULES

34. FATHOMING RULES 1.- TASK DOMINANCE 2.- STATION DOMINANCE 3.- SOLUTION DOMINANCE 4.- BOUND VIOLATION 5.- EXCESIVE IDLE TIME

35. Example 2.3 (pp. 52-54) • Same as Example 2.1 but using Optimal Solutions • Exercise: Work out Example 2.1

36. PRACTICAL ISSUES • Models are abstractions • Hard problem of stations with small number of tasks each (Parallel lines? Grouping?) • Is C cast in stone? • How about randomness? • Independence of task times? • Alternate “optimum”?

37. SEQUENCING MIXED MODELS 1.- INITIALIZATION: CREATE LIST OF ALL PRODUCTS TO BE ASSIGNED (A) 2.- ASSIGN A PRODUCT • FOR n from A, CREATE LIST B OF ALL PRODUCT TYPES ASSIGNABLE WITHOUT VIOLATING CONSTRAINTS • FROM LIST B SELECT PRODUCT WHICH MINIMIZES THE FUNCTION

38. MIXED MODELS sum n sum i ti,j - n Ck • ADD PRODUCT TYPE j* TO THE nth POSITION • REMOVE A PRODUCT TYPE j* FROM A IF n < N • GO TO 1

39. Example 2.4 (pp. 58-59) • Multiple toy car models. • Estimated sales by model (Table 2.6) • Exercise: Work out Example 2.4

40. UNPACED LINES • Paced line with K stations and cycle time C • Each time spends KC in system • Production rate is 1/C • In a deterministic unpaced line • Production rate is 1/C • Time in system is maybe not KC • WIP is smaller for unpaced lines