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An Efficient Real Time Scheduling Scheme for Certain Flexible Manufacturing Systems

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.

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An Efficient Real Time Scheduling Scheme for Certain Flexible Manufacturing Systems

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  1. AnEfficient 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, • Type 2 parts are processed on the cell and on machine 2. • Type 3 parts are processed on the cell and on machine 3. 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= x1 +x2 +x3(Production per unit time) Subject to: 0.24 x1 +0.48x2 +0.72x3 <= 1 (Capacity of cell) 2 x1 <= 1 (Capacity of machine1) 3 x2 <= 1 (Capacity of machine1) 1 x3 <= 1 (Capacity of machine1) 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 Partof type a in cell Machines empty Decision States (? ; - - ; - -) (a;--;--) (b;--;--) (?;-a;--) (?;--;-b) (b;-a;--) (a;--;-b) (b;--;-b) (a;-a;--) (?;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 110 Parts/Hour OPTIMAL (No blocking) 109Parts/Hour OPTIMAL (Blocking) 95 Parts/Hour Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison

  16. 3. Evaluation • Example problem • 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 1 - 10 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 • 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

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