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An Efficient Real Time Scheduling Scheme for Certain Flexible Manufacturing Systems. Arne Thesen Department of Industrial Engineering University of Wisconsin-Madison Madison, WI USA. 1 This talk. - PowerPoint PPT Presentation
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Professor Arne Thesen, University of Wisconsin-Madison
1
An Efficient Real Time Scheduling Scheme for Certain Flexible Manufacturing Systems
Arne ThesenDepartment of Industrial EngineeringUniversity of Wisconsin-MadisonMadison, WI USA
Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 2
1 This talk Problem is to develop simple but efficient control scheme for a given
class of production systems Pre-production analysis
– Rule independent bounds on performance– Introduce three state-independent schemes– Optimal state-dependent scheduling
Evaluation– Both analytic, and simulation results
Conclusion– Very good simple scheduling rules can be found
Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 3
1.1 Example Type 1 parts are processed on the cell and on machine 1, Type 2 parts are processed on the cell and on machine 2. Type 3 parts are processed on the cell and on machine 3.
Mean proces-sing time is 2Cell
Meanprocessing
time ism1=0.24m2=0.48m3=0.72
Additional stations
Buffer, Capacity = 3
Machine 1
Machine 2 Additional stations
Buffer, Capacity = 3Mean proces-sing time is 3
Machine 3 Additional stations
Buffer, Capacity = 3Mean proces-sing time is 1
Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 4
1.2 The Real-Time Scheduling Problem Determine in real time what part should be processed next at a cell
– A number of different parts are available for processing– Processing times are not known with certainty
The cell feeds a number of machines– Information about current and future states is limited
The expected production rate for the overall system should be maximized.
The best control system is no control system
Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 5
1.3 Four Real Time Decision Rules Random (Push)
– Next part is selected at random, probabilities reflect product mix– Cell often blocked
Rotation (Push)– Parts produced in fixed sequence– Sequences for some mixes may be difficult to develop
Circulating tokens (Pull)– A fixed number of part-specific tokens rotate in a FIFO manner– Token mix established from part routing and product mix
SMDP (Push)– Optimal state-dependent rule if Markov assumptions hold
Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 6
1.4 The Problem: More Details
Parts Parts of M different types are produced.There is an unlimited supply of raw materials for all parts.All parts produced by the system can be sold.
Facilities All parts are first processed at a cellThey then continue to separate assembly lines for each part type.The first station in each line is the bottleneck; hence subsequent stationsneed not be analyzed.Buffer capacity (bi) in front of each line i is restricted.
Processing Processing times are exponentially a distributed random variables with the following parameters:mi = Mean processing time at the machining center for parts of type i.ai = Mean processing time at station i for parts of type i.
Control Cell: Parts may be processed in any sequence. The identity of the part tobe processed next is determined at the time processing start.Assembly lines: Any part in the input buffer may be selected for processing.
AvailableInformation
Three types of information is available:Processing times (actual or expected)System states (Buffer full/ not full, expected completion times time)Product mix (Target and/or actual)
Objective Maximize parts produced per hour.
Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 7
1.5. The Problem: Previous Work1984 Yao and Buzacott PSQ heuristic: predictable performance
1988 Seidmann Introduced Semi-Markov Decision Processes(SMDPs) for control policies for manufacturingsystems.
1991 Yih & Thesen Extend SMDP to applications where transitionprobabilities were difficult to establish analytically
1992 Chen Extended Seidmann’s work to non-exponentials
1991 Chen and Thesen Reviews other policies for real time control.1993 Stidham and
WeberReviews other policies for real time control
1996 Thesen & Chen Unequal buffer allocations
1996 Thesen State independent rules
Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 8
2. Pre-Production Analysis Bound on Performance Three state independent schemes Optimization
Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 9
2.1 A Bound on PerformanceIgnoring issues of blocking and queuing delays, linear programming can be used to establish
an upper bound on expected profit:Maximize z= x1 + x2 + x3 (Production per unit time)
Subject to: 0.24 x1 + 0.48 x2 + 0.72 x3 <= 1 (Capacity of cell)
2 x1 <= 1 (Capacity of machine 1) 3 x2 <= 1 (Capacity of machine 1) 1 x3 <= 1 (Capacity of machine 1)
Where:xi = Parts of type i produced per unit time
The optimal production rate is: z= 110 parts per hour, and x1 = 30 pph, x2 = 20 pph, x3 = 60 pph
The corresponding product mix is: x1 = 27.3%, x2 = 18.2%, x3 = 54.5%
Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 10
2.2 The Rotation schedule The bound suggests that we produce parts in the following proportions
– 30/110 of Part 1, 20/110 of Part 2 and 60/110 of Part 3. Thus the following sequence is feasible;
– 1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3,… However, to avoid blocking parts should be evenly spaced:
– 3, 1, 3, 2, 3, 1, 3, 2, 3, 1, 3, … The resulting product mix is
– 27.3%, 18.2% and 54.2% of parts 1, 2 and 3.
Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 11
2.3 Circulating tokens Must arrive at Cell at a rate equal to the desired production rate Round trip times depends on token count and processing times Queuing theory man be used to estimate proper # of tokens Optimal initial token sequence is : 3, 1, 2, 3 , 1 , 2
Cell
Machine 1
Machine 2
Machine 3
Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 12
2.4 A heuristic for allocation of tokens
Step Tokens Throughput Add1 Add 2 Add 3 Maxstates
0 0, 0, 0 0 29.6 19.6 46.1 3
1 0, 0, 1 46.1 72.6 62.4 51.4 13
2 1, 0, 1 72.6 74.0 87.3 76.7 79
3 1, 1, 1 87.3 89.1 87.8 90.5 168
4 1, 1, 2 90.5 93.1 91.2 91.7 427
5 2, 1, 2 93.1 94.0 94.0 94.9 1282
6 2, 1, 3 94.9 96.1 95.9 89.3 3057
7 3, 1, 3 96.1 78.6 91.1 91.8 8207
Throughput estimated from steady-state Markov balance equations
Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 13
2.5 Optimization: Semi-Markov Decision Processes
Assuming that– All system states can be enumerated (next slide)– Decisions in a given state are always made the same way , and,– Processing times are exponentially distributed.
Then we can compute steady state probabilities for– being in each state,– making any state transition.
If rewards are given for some transitions (e.g. “make part”),– expected profit for given set of decisions can be computed,– dynamic programming can be used to find optimal set of decisions.
Resulting decisions can form “rule-base” for optimal state dependent scheduling system
Optimal decisions for state space with 50,000 states easily obtained
Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 14
2.5 State transition diagram for case with two machines, each with one buffer space
Probabilistic states
(?;--;bb)
Decision States
Blocked State
(? ; - - ; - -)(a;--;--)
(?;-a;--)
(a;-a;--) (b;-a;--) (a;--;-b) (b;--;-b)
(?;aa;--)
(b;aa;--)
(?;-a;-b)
(?;--;-b)
(b;--;--)
(a;--;bb)(b;-a;-b)(a;-a;-b)
(?;aa;-b)
(?;aa;bb)
(b;aa;-b) (a;-a;bb)
(?;-a;bb)
Cell Buffer 1 Machine 1 Buffer 2 Machine 2Part of type a in cellMachines empty
Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 15
2. 5. The optimal rule-base Thesen and Chen found the following optimal policy
L.P. BOUND 110 Parts/Hour OPTIMAL (No blocking) 109 Parts/Hour OPTIMAL (Blocking) 95 Parts/Hour
Space Available In Start Production of Part of
Line 1? Line 2? Line 3? Type 1 Type 2 Type 3
Y I I Y
N Y I Y
N N I Y
Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 16
3. Evaluation Example problem
– Simulation– Markov process
Other scenarios Blocking avoidance
Simulation results are averages for 10,000,000 partsAnalytic results are obtained for statespaces of up to 100,000 states
Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 17
3.1 The example proble
Rule Parameters Throughput Product mix
Random 0.273, 0.182, 0.545 69.6 pph 27.3, 18.2, 54.5
Rotation 3, 1, 3, 2, 3, 1, 3, 2,3, 1, 3, …
80.1 pph 27.3, 18.2, 54.5
Tokens t1=3, t2=1, t3=3 96.1 pph 29.3, 17.0, 53.7
SMDP if space then 1, else if space then 2 else 3
94.9 pph 30.6, 20.5, 48.9
Bound 110.0 pph 27.3, 18.2, 54.5
Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 18
3.2 Additional Scenarios
Mean ProcessingTimes at Cell
Ideal ProductMix (%)
Expected MachineUtilization (%)
m1 m2 m3 p1 p2 p3 Cell m1 m 2 m 3
1 0.12 .24 .36 27 18 55 50 100 100 1002 0.24 .48 .72 27 18 55 100 100 100 1003 .375 .75 1.13 37 25 38 100 100 100 504 0.60 1.20 1.80 50 33 17 100 100 100 175 1.00 2.00 3.00 67 33 0 100 100 75 06 .36 .24 0.12 27 18 55 38 100 100 1007 .72 .48 0.24 27 18 55 76 100 100 1008 1.13 .75 .375 20 20 60 100 67 100 1009 1.8 1.20 0.6 0 25 75 100 0 100 100
10 3.00 2.00 1.00 0 0 100 100 0 0 100
Mean processing times at machines 1, 2 and 3 are 2, 3, and 1 minutes
Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 19
3. 2 Rules for scenarios 1 - 10Scenario
Ideal Mix Rotation Sequence Tokens
12
27.3, 18.2, 54.5 3, 1, 3, 2, 3, 1, 3, 2, 3, 1, 3, 2-2-33-1-3
3 37.5,25,37.5 1, 3, 2, 1, 3, 1, 3, 2, . 3-3-24 50, 33.3, 16.7 1, 2, 1, 2, 1, 3, ... 3-3-15 67, 33, 0 1, 1, 2, ... 3-2-067
27.3, 18.2, 54.5 3, 1, 3, 2, 3, 1, 3, 2, 3, 1, 3, 2-3-32-2-3
8 20, 20, 60 3, 1, 3, 1, 3, ... 1-3-39 0, 25, 75 3, 3, 3, 2 0-1-310 0, 0, 100 3 0-0-3
Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 20
3.2 Simulation Analysis: Simple rules
0.0
20.0
40.0
60.0
80.0
100.0
120.01 2 3 4 5 6 7 8 9 10
Scenario
Thro
ughp
ut (p
ph)
randomrotationtokensSMDP
Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 21
3.3 Blocking avoidance
0.0
20.0
40.0
60.0
80.0
100.0
120.0
1 2 3 4 5 6 7 8 9 10
Scenario
Thro
ughp
ut (p
ph)
randomABRotationABtokensSMDP
Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 22
3.4 Observations Good rules must
– Produce parts in proper mix– Avoid delays due to blocking
The token rule achieves this by– Using a small number of tokens– Using proper combination of tokens
The optimality of the token assignment heuristic must be proven Extensions to other distributions and unequal buffer sizes yield similar
results
Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison Slide 23
4. Conclusion Our goal was to find a simple control scheme for a production system Three state-independent schemes were developed Their performance was compared to an optimal control scheme The token based scheme was found to give near optimal performance A benefit of this scheme is its lack of need for real-time information Future work include
– Analytic estimators of expected throughput for this rule– Proof of optimality for token allocation heuristic