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