1. Professor Arne Thesen, University of Wisconsin-Madison
An Efficient Real Time Scheduling Scheme for Certain Flexible Manufacturing Systems
Arne Thesen
Department of Industrial Engineering
University of Wisconsin-Madison
Madison, WI USA
Professor Arne Thesen, University of Wisconsin-Madison

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

3. 1.1 Example Type 1 parts are processed on the cell and on machine 1,
Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison

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

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

6. 1.4 The Problem: More Details
Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison

7. 1.5. The Problem: Previous Work
Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison

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

9. 2.1 A Bound on Performance
Ignoring issues of blocking and queuing delays, linear programming can be used to establish an upper bound on expected profit:
Maximize z= x x x (Production per unit time)
Subject to: 0.24 x x x3 <= 1 (Capacity of cell)
2 x <= 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

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

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
Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison

12. 2.4 A heuristic for allocation of tokens
Throughput estimated from steady-state Markov balance equations
Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison

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

14. 2.5 State transition diagram for case with two machines, each with one buffer space
Cell Buffer 1 Machine 1 Buffer 2 Machine 2
Part of type a in cell
Machines empty
Decision States
(? ; - - ; - -)
(a;--;--)
(b;--;--)
(?;-a;--)
(?;--;-b)
(a;-a;--)
(b;-a;--)
(a;--;-b)
(b;--;-b)
(?;aa;--)
(?;-a;-b)
(?;--;bb)
(b;aa;--)
(a;-a;-b)
(b;-a;-b)
(a;--;bb)
Probabilistic
states
(?;aa;-b)
(?;-a;bb)
(b;aa;-b)
(a;-a;bb)
(?;aa;bb)
Blocked State
Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison

15. 2. 5. The optimal rule-base
Thesen and Chen found the following optimal policy
L.P. BOUND Parts/Hour
OPTIMAL (No blocking) Parts/Hour
OPTIMAL (Blocking) Parts/Hour
Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison

16. 3. Evaluation Example problem Other scenarios Blocking avoidance
Simulation
Markov process
Other scenarios
Blocking avoidance
Simulation results are averages for 10,000,000 parts
Analytic results are obtained for statespaces of up to 100,000 states
Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison

17. 3.1 The example proble
Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison

18. 3.2 Additional Scenarios
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

19. 3. 2 Rules for scenarios
Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison

20. 3.2 Simulation Analysis: Simple rules
Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison

21. 3.3 Blocking avoidance
Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison

22. 3.4 Observations Good rules must The token rule achieves this by
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

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
Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison