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

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

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

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

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

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

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

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

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 + 0.48 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

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

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

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

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

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

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

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

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

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

3. 2 Rules for scenarios 1 - 10 Professor Arne Thesen, University of Wisconsin-MadisonProfessor Arne Thesen, University of Wisconsin-Madison

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

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

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

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