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© Wallace J. Hopp, Mark L. Spearman, 1996, 2000 1 Variability Basics.

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1 © Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 1 Variability Basics

2 © Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 2 Variability Makes a Difference! Little’s Law: TH = WIP/CT, so same throughput can be obtained with large WIP, long CT or small WIP, short CT. The difference? Penny Fab One: achieves full TH (0.5 j/hr) at WIP=W 0 =4 jobs if it behaves like Best Case, but requires WIP=27 jobs to achieve 95% of capacity if it behaves like the Practical Worst Case. Why?

3 © Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 3 Tortise and Hare Example Two machines: subject to same workload: 69 jobs/day (2.875 jobs/hr) subject to unpredictable outages (availability = 75%) Hare X19: long, but infrequent outages Tortoise 2000: short, but more frequent outages Performance: Hare X19 is substantially worse on all measures than Tortoise 2000. Why?

4 © Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 4 Variability Views Variability: Any departure from uniformity Random versus controllable variation Randomness: Essential reality? Artifact of incomplete knowledge? Management implications: robustness is key

5 © Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 5 Probabilistic Intuition Uses of Intuition: driving a car throwing a ball mastering the stock market First Moment Effects: Throughput increases with machine speed Throughput increases with availability Inventory increases with lot size Our intuition is good for first moments g

6 © Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 6 Probabilistic Intuition (cont.) Second Moment Effects: Which is more variable – processing times of parts or batches? Which are more disruptive – long, infrequent failures or short frequent ones? Our intuition is less secure for second moments Misinterpretation – e.g., regression to the mean

7 © Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 7 Variability Definition: Variability is anything that causes the system to depart from regular, predictable behavior. Sources of Variability: setups workpace variation machine failures differential skill levels materials shortages engineering change orders yield loss customer orders rework product differentiation operator unavailability material handling

8 © Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 8 Measuring Process Variability Note: we often use the “squared coefficient of variation” (SCV), c e 2

9 © Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 9 Variability Classes in Factory Physics Effective Process Times: actual process times are generally LV effective process times include setups, failure outages, etc. HV, LV, and MV are all possible in effective process times Relation to Performance Cases: For balanced systems MV – Practical Worst Case LV – between Best Case and Practical Worst Case HV – between Practical Worst Case and Worst Case 0.75 High variability (HV) Moderate variability (MV) Low variability (LV) 01.33 cece

10 © Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 10 Natural Variability Definition: variability without explicitly analyzed cause Sources: operator pace material fluctuations product type (if not explicitly considered) product quality Observation: natural process variability is usually in the LV category.

11 © Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 11 Down Time – Mean Effects Definitions:

12 © Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 12 Down Time – Mean Effects (cont.) Availability: Fraction of time machine is up Effective Processing Time and Rate:

13 © Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 13 Totoise and Hare - Availability Hare X19: t 0 = 15 min  0 = 3.35 min c 0 =  0 /t 0 = 3.35/15 = 0.05 m f = 12.4 hrs (744 min) m r = 4.133 hrs (248 min) c r = 1.0 Availability: Tortoise: t 0 = 15 min  0 = 3.35 min c 0 =  0 /t 0 = 3.35/15 = 0.05 m f = 1.9 hrs (114 min) m r = 0.633 hrs (38 min) c r = 1.0 A =

14 © Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 14 Down Time – Variability Effects Effective Variability: Conclusions: Failures inflate mean, variance, and CV of effective process time Mean (t e ) increases proportionally with 1/A SCV (c e 2 ) increases proportionally with m r SCV (c e 2 ) increases proportionally in c r 2 For constant availability (A), long infrequent outages increase SCV more than short frequent ones Variability depends on repair times in addition to availability

15 © Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 15 Tortoise and Hare - Variability Hare X19: t e = c e 2 = Tortoise 2000 t e = c e 2 =

16 © Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 16 Setups – Mean and Variability Effects Analysis:

17 © Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 17 Setups – Mean and Variability Effects (cont.) Observations: Setups increase mean and variance of processing times. Variability reduction is one benefit of flexible machines. However, the interaction is complex.

18 © Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 18 Setup – Example Data: Fast, inflexible machine – 2 hr setup every 10 jobs Slower, flexible machine – no setups Traditional Analysis?

19 © Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 19 Setup – Example (cont.) Factory Physics Approach: Compare mean and variance Fast, inflexible machine – 2 hr setup every 10 jobs

20 © Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 20 Setup – Example (cont.) Slower, flexible machine – no setups Conclusion:

21 © Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 21 Setup – Example (cont.) New Machine: Consider a third machine same as previous machine with setups, but with shorter, more frequent setups Analysis: Conclusion:

22 © Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 22 Other Process Variability Inflators Sources: operator unavailability recycle batching material unavailability et cetera, et cetera, et cetera Effects: inflate t e inflate c e Consequences:

23 © Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 23 Illustrating Flow Variability t Low variability arrivals t High variability arrivals

24 © Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 24 Measuring Flow Variability

25 © Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 25 Propagation of Variability Single Machine Station: where u is the station utilization given by u = r a t e Multi-Machine Station: where m is the number of (identical) machines and c d 2 (i) = c a 2 (i+1) ii+1 departure var depends on arrival var and process var ce2(i)ce2(i) ca2(i)ca2(i)

26 © Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 26 Propagation of Variability – High Utilization Station HV LV HV LV HV Conclusion: flow variability out of a high utilization station is determined primarily by process variability at that station.

27 © Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 27 Propagation of Variability – Low Utilization Station HV LV Conclusion: flow variability out of a low utilization station is determined primarily by flow variability into that station. HV LV HV

28 © Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 28 Variability Interactions Importance of Queueing: manufacturing plants are queueing networks queueing and waiting time comprise majority of cycle time System Characteristics: Arrival process Service process Number of servers Maximum queue size (blocking) Service discipline (FCFS, LCFS, EDD, SPT, etc.) Balking Routing Many more

29 © Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 29 Kendall's Classification A/B/C A: arrival process B: service process C: number of machines M: exponential (Markovian) distribution G: completely general distribution D: constant (deterministic) distribution. A B C Queue Server

30 © Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 30 Queueing Parameters r a = the rate of arrivals in customers (jobs) per unit time (t a = 1/r a = the average time between arrivals). c a = the CV of inter-arrival times. m = the number of machines. r e = the rate of the station in jobs per unit time = m/t e. c e = the CV of effective process times. u = utilization of station = r a /r e. Note: a station can be described with 5 parameters.

31 © Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 31 Queueing Measures Measures: CT q = the expected waiting time spent in queue. CT = the expected time spent at the process center, i.e., queue time plus process time. WIP = the average WIP level (in jobs) at the station. WIP q = the expected WIP (in jobs) in queue. Relationships: CT = CT q + t e WIP = r a  CT WIP q = r a  CT q Result: If we know CT q, we can compute WIP, WIP q, CT.

32 © Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 32 The G/G/1 Queue Formula: Observations: Useful model of single machine workstations Separate terms for variability, utilization, process time. CT q (and other measures) increase with c a 2 and c e 2 Flow variability, process variability, or both can combine to inflate queue time. Variability causes congestion!

33 © Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 33 The G/G/m Queue Formula: Observations: Useful model of multi-machine workstations Extremely general. Fast and accurate. Easily implemented in a spreadsheet (or packages like MPX).

34 © Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 34 basic data failures setups yield measures VUT Spreadsheet

35 © Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 35 Effects of Blocking VUT Equation: characterizes stations with infinite space for queueing useful for seeing what will happen to WIP, CT without restrictions But real world systems often constrain WIP: physical constraints (e.g., space or spoilage) logical constraints (e.g., kanbans) Blocking Models: estimate WIP and TH for given set of rates, buffer sizes much more complex than non-blocking (open) models, often require simulation to evaluate realistic systems

36 © Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 36 The M/M/1/b Queue B buffer spaces Infinite raw materials Model of Station 2 1 2 Note: there is room for b=B+2 jobs in system, B in the buffer and one at each station. Goes to u/(1-u) as b  Always less than WIP(M/M/1) Goes to r a as b  Always less than TH(M/M/1) Little’s law Note: u>1 is possible; formulas valid for u  1

37 © Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 37 Blocking Example B=2 t e (1)=21t e (2)=20 M/M/1/b system has less WIP and less TH than M/M/1 system 18% less TH 90% less WIP

38 © Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 38 Seeking Out Variability General Strategies: look for long queues (Little's law) look for blocking focus on high utilization resources consider both flow and process variability ask “why” five times Specific Targets: equipment failures setups rework operator pacing anything that prevents regular arrivals and process times

39 © Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 39 Variability Pooling Basic Idea: the CV of a sum of independent random variables decreases with the number of random variables. Example (Time to process a batch of parts):

40 © Wallace J. Hopp, Mark L. Spearman, 1996, 2000 http://www.factory-physics.com 40 Basic Variability Takeaways Variability Measures: CV of effective process times CV of interarrival times Components of Process Variability failures setups many others - deflate capacity and inflate variability long infrequent disruptions worse than short frequent ones Consequences of Variability: variability causes congestion (i.e., WIP/CT inflation) variability propagates variability and utilization interact pooled variability less destructive than individual variability

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