MODELING AND ANALYSIS OF MANUFACTURING SYSTEMS

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MODELING AND ANALYSIS OFMANUFACTURING SYSTEMS

• Session 4
• ASSEMBLY LINES
• February 2001

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ASSEMBLY LINE

• SET OF SEQUENTIAL WORKSTATIONS
• CONNECTED BY A CONTINUOUS MATERIALS HANDLING
  SYSTEM
• INPUT RAW MATERIALS
• OUTPUT FINISHED PRODUCT

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WORK ELEMENTS

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

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BACKBONES OF ASSEMBLY LINES

• PRINCIPLE OF INTERCHANGEABILITY
• DIVISION OF LABOR

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ASSEMBLY LINE TYPES

• SINGLE PRODUCT
• MULTIPLE PRODUCT
• MIXED LINES

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MULTIPLE PARALLEL LINES

• 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

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WORKSTATION CYCLE TIME

• PACED LINES
• UNPACED LINES (ASYNCHRONOUS)
• ROLE OF BUFFERS
• PARALLEL WORKSTATIONS IN SERIAL SYSTEMS

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BASIC LINE BALANCING PROBLEM

• TO ASSIGN WORK ELEMENTS TO WORKSTATIONS SUCH THAT
  ASSEMBLY COST IS MINIMIZED

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TOTAL ASSEMBLY COST

• LABOR COST (WHILE PERFORMING TASKS)
• IDLE TIME COST
• FOCUS MINIMIZE IDLE TIME
• LIMITS PRODUCTION CONSTRAINTS

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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 !

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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

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DECISION VARIABLES

• TASK i ASSIGNED TO STATION k ?
• Xik 1,0
• TOTAL NUMBER OF STATIONS K
• COST COEFFICIENTS cik
• TOTAL NUMBER OF TASKS N

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PROBLEM FORMULATION

• MINIMIZE ?? (cik Xik)
• SUBJECT TO
• ? ti Xik lt C (all stations k)
• ? Xik 1 (all tasks i)
• Xvh lt??? Xuj (all k) (u,v) in IP
• ? (Xuk Xvk)1 (all k) (u,v) in ZS
• XuhXvh lt 1 (all k) (u,v) in ZD

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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)

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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

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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

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LINE BALANCING APPROACHES

• COMSOAL
• RPWH
• OPTIMAL SOLUTIONS
• TREE GENERATION EXPLORATION
• PROBLEM STRUCTURE RULES
• FATHOMING RULES

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LINE BALANCING APPROACHES (contd)

• Required cycle time, sequencing restrictions and
  tasks times are all known.

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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

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COMSOAL (contd)

• Sequences that exceed the best solution are
  discarded
• Better sequences become upper bounds

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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

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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

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COMSOAL ALGORITHMFor generating X trial
solutions

• 1.- SET x0, UBinf, cC
• 2.- START NEW SEQUENCE
• SET xx1, ATK, NIPW(i) NIP(i)
• 3.- PRECEDENCE FEASIBILITY
• FOR i IN A, IF NIPW(i) 0 , ADD i TO B

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COMSOAL ALGORITHM(contd)

• 4.- TIME FEASIBILITY
• FOR i IN B, IF ti lt c ADD i TO F .
• If F empty , 5 , otherwise 6
• 5.- OPEN NEW STATION
• IDLEIDLE c , c C
• If IDLE gt UB , 2, otherwise 3

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COMSOAL

• 6.- SELECT TASK SET m cardF
• RANDOM GENERATE RN in U(0,1)
• LET i mRNth TASK from F
• REMOVE i from A,B,F
• c c - ti
• FOR ALL i in WIP(i), NIPWNIPW-1
• IF A EMPTY --gt 7, OTHERWISE --gt 3

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COMSOAL

• 7.- SCHEDULE COMPLETION
• IDLE IDLE c
• IF IDLE lt UB , UB IDLE --gt STORE SCHEDULE
• IF x X , STOP, OTHERWISE --gt 2

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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

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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.

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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

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RPWH (contd)

• S(i) succesor tasks to task i
• PW(i) ti ? tj j in S(i)

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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

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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

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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

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FATHOMING RULES

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

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Example 2.3 (pp. 52-54)

• Same as Example 2.1 but using Optimal Solutions
• Exercise Work out Example 2.1

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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?

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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

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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 lt N
• GO TO 1

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Example 2.4 (pp. 58-59)

• Multiple toy car models.
• Estimated sales by model (Table 2.6)
• Exercise Work out Example 2.4

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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