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