Transcript Professor's Slides: Transfer Lines

MODELING AND ANALYSIS OF
MANUFACTURING SYSTEMS
Session 5
TRANSFER LINES
February 2001
DEFINITIONS
• TRANSFER LINE
– FLOW-LINE SYSTEM (SET OF SERIAL
MACHINES AND/OR INSPECTION
STATIONS LINKED BY A COMMON
TRANSFER & CONTROL SYSTEM)
SUBJECT TO BREAKDOWNS AND
PROCESS TIME VARIABILITY
• BREAKDOWN
DEFINITIONS
• CYCLE TIME
– PERIOD OF TIME AFTER WHICH THE
HANDLING SYSTEM INDEXES PARTS ON
THE LINE AHEAD TO NEXT STATION
• HARD AUTOMATION
– PRODUCT DEDICATED TRANSFER LINES
• AUTOMATED FLEXIBLE LINES
DEFINITIONS
• BUFFERS
• WORKSTATION UP-TIME
• WORKSTATION DOWN-TIME
–
–
–
–
STATION FAILURE
TOTAL LINE FAILURE
STATION BLOCKED
STATION STARVED
GOAL OF ANALYSIS
• TO DETERMINE THE EFFECTIVENESS
OF A FLOW-LINE GIVEN :
– BUFFER CAPACITIES
– FAILURE RATE FOR EACH STATION
– REPAIR RATE FOR EACH STATION
CLASSIFICATION OF
FAILURES
• TIME DEPENDENT FAILURES
– TAKE PLACE WITH CHRONOLOGICAL
FREQUENCY
– EXAMPLE: DAILY MAINTENANCE
• OPERATION DEPENDENT FAILURES
– TAKE PLACE WHEN SYSTEM IS
RUNNING (DURING A CYCLE)
– EXAMPLE: TOOL WEAR
PERFORMANCE OF A LINE
• EFFECTIVENESS (or AVAILABILITY)
E = q /Q
q = actual capacity (output)
Q = theoretical capacity
E = E(up)/(E(up)+E(down))
PACED LINES WITHOUT
BUFFERS
• LIKE AN ASSEMBLY LINE
EXCEPT FAILURES CAN
HAPPEN AND THEN THE
LINE MUST STOP
ASSUMPTIONS
• LINE HAS M STATIONS
• FOR STATION i THE NUMBER OF
CYCLES TO FAILURE AND THE
NUMBER OF CYCLES FOR REPAIR
ARE GEOMETRIC RANDOM
VARIABLES
• UP & DOWNTIME R.V.S ARE
INDEPENDENT
ASSUMPTIONS
• IDLE STATIONS DO NOT FAIL
• FAILURES TAKE PLACE AT END OF
CYCLE AND DO NOT DESTROY THE
PRODUCT
• AT MOST ONE STATION FAILS ON ANY
CYCLE
PARAMETERS
• STATION FAILURE RATE
•
•
•
•
i
MEAN CYCLES TO FAILURE 1/ i
STATION REPAIR RATE
bi = b
MEAN CYCLES FOR REPAIR 1/bi
LINE FAILURE PARAMETER
 ~  i
COMMENTS
• ASSUMPTION OF G.R.V. ALLOWS USE
OF DISCRETE TIME, DISCRETE STATE
MARKOV CHAIN MODEL
• ASSUMPTION OF AT MOST ONE
STATION FAILING ALLOWS NEGLECT
OF SECOND ORDER TERMS
• ASSUMPTION OF OPERATING ONLY
FAILURE DEFINES FAILURE TYPE
EFFECTIVENESS OF THE
PACED LINE WITHOUT
BUFFERS AND OPERATION
DEPENDENT FAILURES
E = 1/(1 + /b)
• Example 3.1 p. 71
EFFECTIVENESS OF THE PACED
LINE WITHOUT BUFFERS AND
TIME DEPENDENT FAILURES
INDIVIDUAL WORKSTATION
Ei = 1/(1 + i /bi)
FULL LINE (Independent Stations)
E = E1*E2*E3*....
• Example 3.2, p. 73
TWO-STAGE PACED LINES
WITH BUFFER
• TWO SERIAL STAGES SEPARATED
BY AN INVENTORY BUFFER
• THE BUFFER REDUCES
DEPENDENCE BETWEEN
STATIONS
• BUFFER CAPACITY Z
ASSUMPTIONS
• SAME AS BEFORE, EXCEPT NOW
BUFFERS ARE ALLOWED
• STATUS OF STATIONS
– UP (W) OR DOWN (R)
– S1 AND S2
• NUMBER OF ITEMS IN BUFFER z
• MARKOV CHAIN STATES (S1,S2,z)
CONVENTIONS
• FAILURES AND REPAIRS OCCUR AT
THE END OF A CYCLE
• WHEN A CYCLE STARTS, IF BOTH
STATIONS ARE WORKING, STATION 2
RECEIVES ITS NEXT PART FROM
STATION 1
• IF STATION 1 IS DOWN, STATION 2
GRABS A PART FROM THE BUFFER
CONVENTIONS
• IF BUFFER IS EMPTY, STATION 2
BECOMES STARVED
• IF STATION 2 IS DOWN AND 1 IS UP,
THE PART FROM 1 IS SENT TO THE
BUFFER
• IF THE BUFFER IS FULL, STATION 1
BECOMES BLOCKED
TRANSITIONS (Table 3.1)
• FROM WWx TO
– WWx
– WRx
– RWx
p = 1 - 1 - 2
p = 2
p = 1
• FROM RW0 TO
– WW0
– RW0
p = b1
p = 1 - b1
TRANSITIONS
• FROM RWx TO
– WWx-1
– RWx-1
– RRx-1
p = b1
p = 1 - b1 - 2
p = 2
• FROM WRx TO
– WRx+1
– WWx+1
– RRx+1
p = 1 - 1 - b2
p = b2
p = 1
TRANSITIONS
• FROM WRZ TO
– WRZ
– WWZ
p = 1 - b2
p = b2
• FROM RRx TO
– RRx
– RWx
– WRx
p = 1 - b1 - b2
p = b2
p = b1
CHAPMAN-KOLMOGOROV
• LET S = SET OF STATES OF THE
SYSTEM
• LET p(u,v) = TRANSITION
PROBABILITY FROM STATE u TO v
• PROBABILITY THAT THE SYSTEM BE
IN STATE s1
P(s1) =  P(s)*p(s,s1)
CHAPMAN-KOLMOGOROV
• P(WW0) = P(WW0)*p(WW0,WW0) +
P(RW0)*p(RW0,WW0) +
P(RW1)*p(RW1,WW0)
• P(WW0) = P(WW0)*(1-1-2) +
P(RW0)*b1 + P(RW1)*b1
• SIMILARLY FOR ALL OTHER
STATES OF THE SYSTEM
PARAMETERS
 = ROW VECTOR OF STEADY-STATE
PROBABILITIES
• P = [p(i,j)] TRANSITION PROB.
VECTOR
• BALANCE EQUATIONS
*P = 
• CONSTRAINT
OR
*(P - I) = 0
 P(s) = 1
SYSTEM EFFECTIVENESS
• EFFECTIVENESS FOR SYSTEM WITH
BUFFER OF MAXIMUM SIZE Z
E(Z) =  P(WWx) + P(RWx)
BUZACOTT’S RESULT
• LET xi = i/bi (= repair time/up time)
• LET s = x2/x1 AND r = 2/1
• LET
C = [(1+2)(b1+b2) - 1*b2(1+2+b1+b2)]/
[ (1+2)(b1+b2) - 2*b1(1+2+b1+b2) ]
BUZACOTT
• SYSTEM EFFECTIVENESS Ez
– IF s > 1 OR s < 1
Ez = [1 - s CZ]/[1 + x1 - (1+x2)s CZ ]
– IF s = 1
Ez = [1 + r - b2(1+x) + Z b2 (1+x) ] /
[ (1+2x)( 1 + r - b2(1+x)) + Z b2(1+x)2 ]
BUZACOTT (contd)
• GAIN DUE TO BUFFER OF SIZE Z
GZ = EZ - E0
• WHAT HAPPENS AS Z --> Inf ?
– AS THE BUFFER CAPACITY IS
INCREASED, EFFECTIVENESS
APPROACHES THE CAPACITY OF THE
LEAST EFFECTIVE STATION
• Examples 3.3 and 3.4 , pp. 76-78
Deterministic Failures & Repairs
• For failures alternating between stations
(deterministically) with Z > 1/b
–
–
–
–
No starving nor blocking
Ez = 1/(1+x)
See Fig. 3.4
Rule: “Buffers should be large enough to
accomodate at least average repair time
production”
SYSTEM REDUCTION
• A SET OF STATIONS THAT MUST BE
JOINTLY ACTIVE OR IDLE CAN BE
AGGREGATED INTO A SINGLE
WORKSTATION PROVIDING THEY
HAVE A COMMON REPAIR RATE
SYSTEM REDUCTION RULES
• EQUIVALENCY RULE
– ANY SET OF STATIONS WITHOUT
BUFFERS CAN BE REPLACED BY A
SINGLE STATION
• MEDIAN BUFFER LOCATION
– BUFFERS SHOULD BE IN THE MIDDLE
• REVERSIBILITY
– INDEPENDENCE OF FLOW DIRECTION
EQUIVALENCY RULE
(Fig 3.5, p. 80)
1-> 2 -> B -> 3 -> 4 ->5
IS EQUIVALENT TO
(1 + 2) -> B -> (3 + 4 + 5)
• Example 3.5, p. 79
UNPACED LINES WITH
FAILURES (asynchronous line)
• EACH WORKSTATION ACTS
INDEPENDENTLY
• ONCE STATION FINISHES JOB, IT
ATTEMPTS TO PASS IT TO THE NEXT
STATION OR BUFFER
• IF STATION 2 IS DOWN AND BUFFER IS
FULL, 1 BECOMES BLOCKED
• Examples 3.7, 3.8 3.9, pp. 86-89
BUFFERS AND PRODUCTION
CONTROL
• IF LINE IN FRONT OF BUFFER HAS
HIGHER AVAILABILITY THAN LINE
BEHIND BUFFER --> BUFFER FULL
• IF LINE IN FRONT OF BUFFER HAS
LOWER AVAILABILITY THAN LINE
BEHIND BUFFER --> BUFFER EMPTY
• FOR SIMILAR AVAILABILITIES -->
BUFFER HALF FULL
BUFFERS AND PRODUCTION
CONTROL
• BALANCED VS UNBALANCED LINES
• PUSH VS PULL SYSTEMS
– MATERIALS REQUIREMENTS
PLANNING (MRP)
– JUST IN TIME (JIT)
Homework
• Prepare for class presentation and for
you personal binder, your choice of
– A solution to an end-of-chapter problem,
– An explanation/elaboration of an example in
the chapter not sufficiently discussed in class
– An explanation/elaboration of a section in
the chapter not sufficiently discussed in
class.