IE 368: FACILITY DESIGN AND OPERATIONS MANAGEMENT

1. IE 368: FACILITY DESIGN AND OPERATIONS MANAGEMENT Lecture Notes #3 Production System Design Part #2

2. Performance Evaluation So far: Identified the general type of production system flow For each workstation in the system we have calculated the equipment fraction based on throughput, reliability, efficiency, etc. Calculations are based on averages

3. Performance Evaluation (cont.) Will the design meet performance requirements? Throughput Quantity over time System responsiveness Time-In-System WIP Levels Inventory on the floor

4. Performance Evaluation (cont.) Throughput After computing the equipment fraction, can the system meet long-run throughput requirements? Over the short-run many things are possible Can workstations block each other? If so, it is not simple to determine if throughput requirements can be met If workstations do not block each other, throughput requirements can be met

5. Performance Evaluation (cont.) Blocking in a production system WS 1 WS 2 WS 3 Jobs

6. Performance Evaluation (cont.) System responsiveness/WIP Unless all workstations work with complete predictability and reliability, this is generally not known The lack of complete predictability and reliability creates variability in system operations

7. Performance Evaluation (cont.) How do you evaluate performance? Computer simulation Needed for evaluating throughput when blocking occurs Can be detailed and time consuming Queuing (waiting line) models Mathematical formulas Applied at an early design stage Can be used to evaluate TIS & WIP

8. Queuing Models Mathematical models of waiting line systems e.g., a workstation receiving jobs for processing Their use requires some background in probability and statistics

9. Probability/Statistics Concepts Random variable A measurable quantity whose value is unpredictable Equipment fractions were calculated assuming fixed average values In reality many of the quantities in the equipment fraction equation vary e.g., the time per job, reliability, scrap, etc.

10. Probability/Statistics Concepts (cont.) Random variables are characterized by distribution functions Distribution functions describe the probability of observing various outcomes for the random variable Examples of distributions Normal Uniform Binomial …

11. Probability/Statistics Concepts (cont.) To utilize the queuing models, an understanding of the following concepts is needed Averages Variance Coefficient of variation The concepts of the true and estimated values for these quantities are also needed

12. Probability/Statistics Concepts (cont.) Expected Value The true average of a random variable It is a measure of the central tendency of X Denoted E(X) for the random variable X E(X) is estimated using the sample average E(X) is a constant and is a random variable

13. Probability/Statistics Concepts (cont.) Variance A measure of the true spread or predictability of a random variable Denoted as Var(X) or V(X) Var(X) is estimated by the sample variance Var(X) is a constant and s2 is a random variable

14. Probability/Statistics Concepts (cont.) Example x 110 115 120 125 130 135 140 Normal distribution E(X) = 125, Var(X) = 25 Exponential distribution E(X) = 1/2, Var(X) = 1/4

15. Probability/Statistics Concepts (cont.) The coefficient of variation (or CV) for a random variable X is a measure of relative variability Denoted CV(X) CV(X) is dimensionless

16. Probability/Statistics Concepts (cont.) For waiting line systems (e.g., a workstation receiving jobs) it is relative variability that affects performance instead of absolute variability x x 39.5 110 115 120 125 130 135 140 N(125,25) N(39.5,2.5)

17. Random Outages Different types of downtimes have different impacts on variability The most important distinction is between preemptive and non-preemptive downtimes Preemptive outages occur right in the middle of a process Typically, these are outages for which there is no control as to when they happen (e.g., failures) In contrast, non-preemptive outages require the tool to be idle before they can happen This means that we have some control as to exactly when they occur. This is usually the case for planned maintenance activities or setup times Schmidt, K., Rose, O. (2007). Queue time and x-factor characteristics for semiconductor manufacturing with small lot sizes. Proceedings of the 3rd Annual IEEE Conference on Automation Science and Engineering, 1069-1074.

18. Probability/Statistics Concepts (cont.) Calculating and estimating CV(X) From data From a known or assumed distribution Example 1 – Calculating CV(X) from data

19. Probability/Statistics Concepts (cont.) Example 2 – Calculating CV(X) from known distribution X is assumed to follow a triangular distribution with Min (a) = 2 Mode (b) = 5 Max (c) = 10

20. In-class Exercise Compute the estimated CV for the following data: 1, 1, 2, 1, 1, 1, 1, 25, 1, 1 If X is normally distributed with E(X) =10, and Var(X) = 100

21. Performance Evaluation To evaluate the performance of a workstation (TIS and WIP), the concept of workstation utilization is needed Utilization for a workstation is the ratio of: The average time between job departures (if each machine in the workstation always has work), and average time between job arrivals The average rate of job arrivals, and the average rate of job departures (if each machine in the workstation always has work) To compute either one of these ratios the concept of effective process time of a job at a workstation is needed

22. Performance Evaluation (cont.) Effective Process Time The total time seen by a job at a workstation Effective process time is a random variable From the perspective of the output side of a workstation, if a job is being processed at a workstation and is delayed, it does not matter if the delay is due to Product type, Machine down time, Operator break time, Human variability,… Effective process time does NOT include idle time

23. Performance Evaluation (cont.) Effective Process Time Examples Manual operations – Process time of each job will vary. Effective process time is the average. Automated operations – Identical process times, interrupted by down time. Combination – Varying process times interrupted by equipment down times. Eff. Proc. Time = Avg. Time Between Jobs Observer Jobs WS Eff. Proc. Time = 1/(Avg. Rate of Jobs Seen)

24. Performance Evaluation (cont.) Effective Process Time Down time or other delays occurring during the processing of a job are included as part of the effective process times Idle time Time when the WS is not working on a job, is NOT included in the effective process time e.g., a workstation is starved for jobs because its upstream workstation is experiencing a long down time. This idle time is taken out of calculations of effective process time.

25. Performance Evaluation (cont.)

26. Performance Evaluation (cont.)

27. Performance Evaluation (cont.) Job arrival rate If each machine in the workstation always has work, utilization for a workstation is the ratio of: The average time between job departures (if each machine in the workstation always has work), and average time between job arrivals The average rate of job arrivals, and the average rate of job departures (if each machine in the workstation always has work) Avg Intearrival Time = Avg. Time Between Jobs Observer Jobs WS Arrival Rate = 1/(Avg. Intearrival Time)

28. Performance Evaluation (cont.) Job arrival rate Time X X X X X X X 0 x1 x2 x3 x4 x5 x6 x7 Xi= Time between job arrivals (a random variable)

29. Performance Evaluation (cont.) Workstation utilization te = Average effective process time for a job at a workstation (on a single machine) ta= Average inter-arrival (time between job arrivals) time of jobs to the workstation. Note that ta is the inverse of the arrival rate of jobs to a workstation. u = Utilization = = The percent of time machines in a workstation are busy m = The number of machines working in parallel at a workstation

30. In-class Exercise – Estimate u

31. Performance Evaluation (cont.) To evaluate the performance of a workstation (TIS and WIP) using queuing models, the concept of workstation relative variability is needed The measure for relative variability is the coefficient of variation For a random variable X, the CV of X is

32. Performance Evaluation (cont.) Variance of the effective process times at a workstation  Variance of the job interarrival times to a workstation  Then Coefficient of variation of the effective process times Coefficient of variation of the job interarrival times

33. Classification of CVs

34. In-class Exercise – Estimate the CVs

35. Queuing Models for Performance Evaluation So far, we have learned about (and calculated) seven different parameters to describe a production system made up of workstations What are these seven parameters? In the next few slides, the basic theory (i.e., formulations and assumptions) to evaluate the performance of a production system made up of workstations is presented

36. Queuing Models for Performance Evaluation Will evaluate long-run average Throughput (with no blocking) Time-In-System WIP Consider the simplest case A single machine workstation WS Job Departures Job Arrivals

37. Evaluating Average Throughput When a workstation (WS) is never blocked Why? 1/ta , if utilization < 1 WS throughput = m/te , otherwise

38. Queuing Models for Performance Evaluation TIQ= Average time in queue Time in line before the start of processing

39. Example

40. Example

41. Example

42. Evaluating Average WIP Apply a result from queuing theory called Little’s Law Where TP = Throughput Assumes no limit on storage space

43. Example

44. In-class Exercise Suppose option 2 in the prior example will be adopted but there is uncertainty in the exact throughput (job arrival rate) Plot the average TIS as a function of throughput for the following throughput values (jobs per hour) 2.5, 2.6, 2.7, 2.8, 2.9, and 2.95

45. In-class Exercise

46. Examination of the Queuing Model TIQ/TIS and WIP depend linearly on CVa2 and CVe2 TIQ/TIS and WIP depend non-linearly on u As CVa2 and CVe2 get smaller it is possible to Operate at a higher utilization (throughput) with the same TIQ/TIS and WIP Have less TIQ/TIS and WIP for the same throughput This is the fundamental idea behind many Just-In-Time production systems