Stochastic Models of Manufacturing Systems

1. Stochastic Models of Manufacturing Systems Jan-Kees van Ommeren Website: wwwhome.math.utwente.nl/~ommerenjcw/LNMB-SMMS/ Notes, sheets, exercises and assignments: on website Prior knowledge: probability theory, Markov processes Exam: assignments

2. Manufacturing system Products to customers Customer orders MTO manufacturing system

3. Make and Assemble to Order manufacturing systems Manufacturing Assembly

4. Basic Definitions • Workstation: a collection of one or more identical machines. • Part: a component, sub-assembly, or an assembly that moves through the workstations. • End Item: part sold directly to customers; relationship to constituent parts defined in bill of material. • Consumables: bits, chemicals, gasses, etc., used in process but do not become part of the product that is sold. • Routing: sequence of workstations needed to make a part. • Order: request from customer. • Job: transfer quantity on the line.

5. Probability

6. 2 1 3 0 Markov chains Countable State Space Matrix with transition rates Q Steady state probabilitiesπQ=0 Special non stationary MC: Poisson process

7. 0 1 2 Work Arrival Departure Time The M/M/1 queue

8. 0 1 2 The M/M/1 queue (continued) ….. n-1 n n+1 n = number of jobs in system (waiting + in process) with Since we find and therefore

9. The M/M/1 queue (continued) Work in process (average amount of jobs in system) = From Little’s Law we find Average time in the system (flow time) = Average waiting time in queue = Applying Little’s Law to the jobs in queue yields:

10. 0 1 3 2 4 Parallel machines workstations: the M/M/c queue

11. Multiple parallel machines: the M/M/c queue

12. Multiple parallel machines: the M/M/c queue (continued) Average number of jobs in queue = From Little’s Law applied to the queue, we find Hence Applying Little’s Law to the system yields hence the number of jobs in service = A good approximation for the number of jobs in queue is

13. 25 20 15 10 5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Work load per employee

14. General service times: the M/G/1 queue With we find Define the squared coefficient of variation as then Example: M/D/1

15. General arrival and service times: G/G/1 and G/G/c queues with and Define A good approximation for the waiting time in queue is Note that Similar from which and

16. Ample servers with Poisson arrivals: M/M/ and M/G/ queues Consider the M/M/c queue with c , then hence Similar for the M/G/ queue: (Palm’s theorem)

17. Real time Service time Departure Start of service Machines subject to failure

18. Machines subject to failure Let the “up-time” be exponentially distributed with parameter is the mean time between failures (MTTF) Let D be the downtime (repair time): Availability A is defined as: Let and let

19. Machines subject to failure (continued) If failure and repair times are both exponential it is possible to show that the variance of the effective service time satisfies: we now find With Machines with setup times and batch size Ns Jobs that need rework on the same machine with probability fr

20. Analysis of job flow time when processed in a batch of size N Jobs arrival with interarrival times A, and have a process time S0 . Batches are of size N and need a batch setup time Ss . Average time until batch completion: Batch arrivals: Batch service: Batch workload: Batch time waiting in queue:

21. Analysis of job flow time when processed in a batch (continued) When jobs leave individually after being processed: When jobs leave in a batch after being processed: The total flow time (time in the system) of a tagged job is then:

22. 13 12 11 10 9 8 7 6 5 4 3 30 40 50 60 70 80 90 100 110 120 Batch size N Job flow time as a function of the batch size

23. S order-up-to level A production to stock model Note that: Hence, n describes the complete state space, and the system is completely equivalent to an M/M/1 queue with Poisson arrival rate  and exponential service rate . In particular, the backlog k satisfies:

24. Transition diagram of k 0 1 2 3 A production to stock model (continued) S order-up-to level Let q denote the probability that an arriving request has to wait, then

25. Exit Station 1 Station 2 0,0 0,1 0,2 1,1 2,1 2,2 1,0 2,0 1,2 Manufacturing flow line, modeled as tandem queues

26. Manufacturing system, modeled as Open Queuing Network (product-form solution, single-class case) Visit ratio , where where

27. Performance criteria for open queueing networks Throughput of the system (Little’s Law at system level)

28. A network production to stock model S order-up-to level System behaves as an open queueing network, with The backlog k satisfies

29. Non-product-form Open Queuing Networks I: coupling arrival and departure processes or, even better,

30. Non-product-form Open Queuing Networks II: Traffic equations for split and merge configurations Given: Traffic rates: Splitting: (exact for renewal processes) Merging: where and

31. Example: serial production lines

32. General multi-class manufacturing systems where Take Performance measures

33. 3 2.5 2 1.5 1 0.5 0 1 2 3 4 5 6 I I I EWQ,1 EWQ,2 EW

34. Transport or move batches between workstations Waiting time until batch completion at station i: Transportation time between stations i and j: (independent of batch size) Batch utilization at station j: Waiting time in the queue at station j: Waiting time in batch until start of processing:

35. Example of a closed queuing network (single-class manufacturing system)

36. 1 2/3 1 1 4/5 1 1/3 1/5 a b c e d f Routing matrix Example of a closed queuing network system

37. Mill M M Drill M M M M A Single-Stage Example: Machines and Operators Mill A Drill B Lathe Lathe Lathe Lathe C Average response times?

38. Manufacturing system, modeled as Closed Queuing Network (product-form solution, single-class case)

39. Mean Value Analysis (single class, single server)

40. MVA example: a four stage, single machine per stage, production line

41. Throughput 4 4 3.5 3.5 3 3 2.5 2.5 2 2 1.5 1.5 1 1 0.5 0.5 0 0 Number of jobs in the network 0 10 20 30 40 50 I I I I EL1 EL2 EL1 EL2

42. Marginal Distribution Analysis (single class, multi-servers) (Arrival Theorem) (Little’s Law) (Arrival Theorem)

43. MDA for product form CQN’s with multiple classes (fixed number N(r) for each class r)

44. MDA for non-product form Closed Queuing Networks (single class case) 1. where 2. 3.

45. MDA for non-product form Closed Queuing Networks (multiple class case; fixed number N(r) for each class r) Let Then (all other equations being equal)

46. PAC (Production Authorization Card) systems Routing probabilities (i,j = 1,2, … , M) (j = 1,2, … , M) (j = 1,2, … , M)

47. 0,0,3 1,0,3 2,0,3 0,0,2 0,1,2 1,1,2 2,1,2 0,0,1 0,1,1 0,2,1 1,2,1 2,2,1 0,0,0 0,1,0 0,2,0 0,3,0 1,3,0 2,3,0 Exact analysis by Matrix-Geometric approach

48. Open manufacturing systems with workload control Analyze as M/M(n)/1-queue Avi-Itzhak and Heyman

49. Aggregation method for multi-class systems (Flow-equivalent server) Analysis as multi-dimensional random walk

50. M+1 card feedback loop PAC systems: from an open to a closed system view i,j = 1,2, … , M i,j = 1,2, … , M i = 1,2, … , M j = 1,2, … , M j = 0,1,2, … , M