1. 1 Chapter 7 Dynamic Job Shops Advantages/Disadvantages Planning, Control and Scheduling Open Queuing Network Model

2. 2 Definition of a Job Shop Several types of machines –Similar machines grouped, focused on particular operations Storage for work in process (no blocking) Material handling between machine groups Different job types have different routings among machine groups; may revisit the same machine group Manufacturing process in evolution “Job” is typically a batch of identical pieces May have to set up machines between different job types

3. 3 Advantages & Disadvantages Advantages (machine groups) +Ease of supervision/operation by skilled workers +High utilization of expensive machines +Flexibility – broad scope of products Disadvantages (flexibility) –High WIP and long flow times –Conflict between resource utilization and customer service –Difficult to meet promise dates –Cost of variety: reduced learning, difficult scheduling

4. 4 Planning, Control, Scheduling Focus on produce-to-order (if produce-to-stock, treat PA as a “customer order”) Design and layout –evolutionary –based on real or perceived bottlenecks Order acceptance and resource planning –Quoted delivery date: need distribution of flow time –Required resource levels: complicated by random order arrivals

5. 5 Planning, Control, Scheduling (cont.) Loading and resource commitment –Schedule a new job on critical machine groups first; this schedule determines sequencing on noncritical groups and timing of release to the shop –Forward load jobs on all machines, looking ahead from present –Backward load jobs on all machines, working back from due dates –Production schedule based on deterministic processing and transport times; slack time built in to handle uncertainty Allocation and resource assignment –Assign jobs at a machine group to worker/machine –Often from resource perspective, ignoring promise dates

6. 6 Why Models? Estimate flow times for use in quoting delivery dates Identify policies –order release –scheduling –sequencing to reduce unnecessary flow time and work in process Model a particular policy, then observe numbers of jobs in system, at each machine group, and in transition between machine groups

7. 7 Job Assumptions 1.A job released to the system goes directly to a machine center 2.Job characteristics are statistically independent of other jobs 3.Each job visits a specified sequence of machine centers 4.Finite processing times, identically distributed for each job of specific type at a given machine center 5.Jobs may wait between machine centers

8. 8 Machine Assumptions 1.A machine center consists of a number (perhaps one) of identical machines 2.Each machine operates independently of other machines; can operate at its own maximum rate 3.Each machine is continuously available for processing jobs (in practice, rate in #2 is adjusted down to account for breakdowns, maintenance, etc.)

9. 9 Operation Assumptions 1.Each job is indivisible 2.No cancellation or interruption of jobs 3.No preemption 4.Each job is processed on no more than one machine at a time 5.Each machine center has adequate space for jobs awaiting processing there 6.Each machine center has adequate output space for completed jobs to wait

10. 10 Aggregation of Job Flow Follow Section 7.3.2 in the text to determine:

11. 11 Example 10.15312--641-- 20.101231286127 30.052313-4942-

12. 12 Job Shop Capacity Capacity is the maximum tolerable total arrival rate, * Job arrival rates to machine centers satisfy Let v i be the average number of times an aggregate job visits machine center i during its stay in the system. unaffected by variability!

13. 13 Jackson Open Queuing Network Assume A single class (aggregated) of jobs arrives from outside according to a Poisson process with rate The service times of jobs at machine center i are iid exponential random variables with mean 1/  i If there are n jobs present at machine center i, then the total processing rate is  i r i (n) (e.g., if there are c i identical machines then r i (n) = min{n, c i }) Service protocol (queue discipline) is independent of job service time requirements, e.g., FCFS or random

14. 14 Jackson Network If N i (t) is the number of jobs at machine center i at time t, and then that is, the network has a product-form solution. If each m/c center has a single machine, then if The network can be decomposed into independent M/M/1 queues (M/M/c i if multiple m/c’s at each center)

15. 15 Single Machine at Each Station Average number of jobs in m/c center i: Total number of jobs in the system: Average flow time of a job: Variance of the number of jobs in the system:

16. 16 Assigning Tasks to Machine Centers Minimize E[T] or equivalently E[N] s.t. average number of tasks for an arbitrary job = K Then task allocation determines values of v i with Assume that if a given task is assigned to m/c center i then it will require an exponential (  i ) amount of time; Optimal:

17. 17 Achieving the Optimal Visit Rates Total set of tasks for all job types: Let w i be the average number of times task i needs to be performed on an arbitrary job: Ideally, find a partition More practically, for each k=1,…,m, find l(k) as the largest task index that satisfies Then if

18. 18 Generalized Jackson Networks (Approximate Decomposition Methods) The arrival and departure processes are approximated by renewal process and each station is analyzed individually as GI/G/m queue. The performance measures (E[N], E[W]) are approximated by 4 parameters. (Don’t need the exact distribution) References –Whitt, W. (1983a), “The Queuing Network Analyzer,” Bell System Technical Journal, 62, 2779-2815. –Whitt, W. (1983b), “Performance of Queuing Network Analyzer,” Bell System Technical Journal, 62, 2817-2843. –Bitran, G. and R. Morabito (1996), “Open queuing networks: optimization and performance evaluation models for discrete manufacturing systems,” Production and Operations Management, 5, 2, 163-193.

19. 19 Single-class GI/G/1 OQNS Notation: n = number of internal stations in the network. For each j, j = 1,…,n: = expected external arrival rate at station j [ ], = squared coefficient of variation ( scv) or variability of external interarrival time at station j { = [Var(A 0j )]/[E(A 0j ) 2 ]}, = expected service rate at station i [  j =1/E(S j )] = scv or variability of service time at station j { = [Var(S j )]/[E(S j ) 2 ]}.

20. 20 Notation (cont) -For each pair (i,j), i = 1,…,n, j = 1,…,n: = probability of a job going to station j after completing service at station i. *We assume no immediate feedback, -For n nodes, the input consists of n 2 + 4n numbers The complete decomposition is essentially described in 3 steps oStep 1. Analysis of interaction between stations of the networks, oStep 2. Evaluation of performance measures at each station, oStep 3. Evaluation of performance measures for the whole network.

21. 21 Step 1: Determine two parameters for each station j: (i) the expected arrival rate j = 1/E[A j ] where A j is the interarrival time at the station j. (ii) the squared coefficient of variation (scv) or variability of interarrival time, = Var(A j )/E(A j ) 2. Consists of 3 sub-steps: (i)Superposition of arrival processes (ii)Overall departure process (iii)Departure splitting according to destinations

22. 22 Step 1: Superposed arrival rate can be obtained from the traffic rate equations: for j = 1,…,n where ij = p ij i is the expected arrival rate at station j from station i. We also get = j /  j, 0  < 1 (the utilization or offered load) The expected (external) departure rate to station 0 from station j is given by. Throughput = or The expected number of visits v j = E(K j ) = j / 0.

23. 23 Step 1(i): (cont.) The s.c.v. of arrival time can be approximated by Traffic variability equation (Whitt(1983b)) (1) where and

24. 24 Step 1(ii): Departure process is interdeparture time variability from station j.(2) Note that if the arrival time and service processes are Poisson (i.e., = = 1), then is exact and leads to = 1. Note also that if  j  1, then we obtain  On the other hand if  j  0, then we obtain 

25. 25 Step 1(iii): Departure splitting = the interarrival time variability at station i from station j. = the departure time variability from station j to i. (3) Equations (1), (2), and (3) form a linear system – the traffic variability equations – in

26. 26 Step 2: Find the expected waiting time by using Kraemer & Lagenbach-Belz formula [modified by Whitt(1983a)] whereif < 1 if  1 Step 3: Expected lead time for an arbitrary job (waiting times + service times)

27. 27 Single-class GI/G/m OQNS with probabilistic routing More general case where there are m identical machines at each station i. for j = 1,…,n where

28. 28 Single-class GI/G/m OQNS with probabilistic routing (cont.) Then the approximation of expected waiting time is similar to GI/G/1 case:

29. 29 Symmetric Job Shop A job leaving m/c center i is equally likely to go to any other m/c next: Then So we can approximate each m/c center as M/G/1. Then, can be replaced by mean waiting time in M/G/1 queue.

30. 30 Symmetric Job Shop

31. 31 Uniform Flow Job Shop All jobs visit the same sequence of machine centers. Then In this case, M/G/1 is a bad approximation, but GI/G/1 works fine. The waiting time of GI/G/1 queue can be approximated by using equation 3.142, 3.143 or 3.144 above and the lower and upper bound is

32. 32 Uniform-Flow Shop

33. 33 Job Routing Diversity Assume high work load: and machine centers each with same number of machines, same service time distribution and same utilization. Also, Q: What job routing minimizes mean flow time? A1: Ifthen uniform-flow job routing minimizes mean flow time; A2: Ifthen symmetric job routing minimizes mean flow time.

34. 34 Variance of Flow Time The mean flow time is useful information but not sufficient for setting due dates. The two distributions below have the same mean, but if the due date is set as shown, tardy jobs are a lot more likely under distribution B! E[T]E[T] A B Due

35. 35 Flow Time of an Aggregate Job Pretend we are following the progress of an arbitrary tagged job. If K i is the number of times it visits m/c center i, then its flow time is (K i is the r.v. for which v i is the mean) Given the values of each K i, the expected flow time is and then, “unconditioning”,

36. 36 Variance In a similar way, we can find the variance of T conditional upon K i and then uncondition. Assume {T ij, i = 1,…,m; j = 1, …} and {K i, i = 1,…,m} are all independent and that, for each i, {T ij, j = 1, …} are identically distributed. Similar to conditional expectation, there is a relation for conditional variance:

37. 37 Using Conditional Variance Since T depends on {K 1, K 2, …, K m }, we can say The first term equals

38. 38 Using Conditional Variance The second term is

39. 39 Variance The resulting formula for the variance is: If arrivals to each m/c center are approximately Poisson, we can find Var[T i ] from the M/G/1 transform equation (3.73), p. 61. But we still need Cov(K i, K j ).

40. 40 Markov Chain Model Think of the tagged job as following a Markov chain with states 1,…,m for the machine centers and absorbing state 0 representing having left the job shop. The transition matrix is K i is the number of times this M.C. visits state i before being absorbed; let K ji be the number of visits given X 0 = j.

41. 41 with probability with probability, l = 1,…,m Where if j = i and 0 otherwise. with probability with probability, l = 1,…,m

42. 42 Expectations Take expectation of (7.83), we get and take expectation of (7.84), and

43. 43 Because and, we obtain and where Therefore

44. 44 Job Routing Extremes Symmetric job shop E[K i ] = 1, If all m/c centers are identical and m is large, and we can use an exponential dist’n to approximate T. Uniform-flow job shop E[K i ] = 1, Then

45. 45 Setting Due Dates Use mean and variance of T to obtain approximate distribution F T (t) = P[T < t], e.g., if Var[T] < E[T] 2, fit an Erlang distribution; if Var[T] = E[T] 2, use an exponential dist’n with mean E[T]; if Var[T] > E[T] 2, use a hyperexponential distribution. If t d is the due date, what matters is

46. 46 Costs Related to Due Dates Setting the due date too far away: Completing job after its due date: Completing job before its due date: Total expected cost has derivative

47. 47 Optimal Due Date