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Variability

Variability. Example. A single machine performs an operation for a unit of product. The mean operation time is 30 seconds. Units arrive at the station with an average time between arrivals of 40 seconds. There is room for three waiting units. Arrival and service processes are constant.

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Variability

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  1. Variability Paul A. Jensen Operations Research Models and Methods Copyright 2004 - All rights reserved

  2. Example • A single machine performs an operation for a unit of product. • The mean operation time is 30 seconds. • Units arrive at the station with an average time between arrivals of 40 seconds. • There is room for three waiting units.

  3. Arrival and service processes are constant • Arrival rate: 1.5/minute • Service rate: 2/minute • Percent utilization • Average delay • Average WIP

  4. Arrival process is randomService process is constant • Arrival rate: 1.5/minute • Average Service rate: 2/minute • Percent utilization • Average delay • Average WIP

  5. Arrival and service processes are random • Average Arrival rate: 1.5/minute • Average Service rate: 2/minute • Percent utilization • Average delay • Average WIP

  6. Conclusion:

  7. u Unit flow t operation time U = ut Unit time V Production volume Analytical Determination of System Characteristics s Number of machines

  8. For analytical purposes • Assume service and interarrival times have exponential distributions

  9. We want to compute: • State Probabilities • Average Number and Time in the Queue • Average Number and Time in the System • Percent Utilization

  10. The State Probabilities • The probability that the system is empty • The probability of n in the system for n ≤ s • The probability of n in the system for n > s

  11. Average Number and Time • In the queue • In the system • Utilization

  12. Queuing Networks

  13. Equivalence Property • Assume: All stations have exponential service times and unlimited queues and all inputs to the system are Poisson processes. • Then: Each station can be analyzed independently with queuing analysis. • Then: System characteristics can be determined by summing station characteristics.

  14. Consider the Job Shop

  15. Queuing Analysis

  16. System Characteristics

  17. Stations Performing More than One Operation

  18. Items Processed in Lots

  19. The effects of producing in lots

  20. Individual vs. lot production (neglecting setup time) • The minimum number of stations is the same • The traffic intensity is the same • The average number in the queue is the same • But: For individual production Lq is in units • For lot production Lq is in lots • For lot production, WIP is Q times greater • For lot production, W and Wq is Q times greater

  21. Effects of setup time • Setup time increases the minimum number of machines • Setup time increases the traffic intensity • The effects are reduced by increasing the lot size • But, increasing the lot size increases WIP and throughput time by a factor of Q

  22. Oper. 1 2 3 4 5 6 7 8 9 10 Time A 5 4 18 5 min. Time B 10 12 10 15 min. Unit 1.272 1.272 1.272 1.272 1.272 1.111 1.111 1.111 1.111 1 Flow WIP A 2.65 2.12 8.332 2.315 WIP B 10.6 12.72 9.258 13.89 Example • Production of A is 0.417 per minute, and production of B is 0.8333 per minute.

  23. Analyze Station 3 for flows in individual units

  24. Determining the number of machines

  25. Analyze Station 3 for flows in lots of 10

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