OM Session2

transcript

Variability BasicsOperations Management - WS 2014/2015

Univ.Prof. Werner Jammernegg / Dr. Emel Arikan, M.Sc.

W. Jammernegg: Variability Basics 1 – 46

Overview

1. Process Variability

2. Quality Control: Statistical process control

3. Quality Improvement: Process capability

Variability

DefinitionVariability is anything that causes the system to depart from regular, predictablebehavior.

Sources of Variability

I setupsI machine failuresI materials shortagesI yield lossI reworkI operator unavailability

I workpace variationI differential skill levelsI engineering change ordersI customer ordersI product differentiationI material handling

Process Variability

Overview

Process Variability

Measuring Process Variability

coefficient of variation of effective process times (CVe)

ce = σete

σe . . . standard deviation of process timete . . . mean process time of a job

Note: we often use the "‘squared coefficient of variation"’ (SCV), c2e

Process Variability

Variability Classes

Effective Process TimesI actual process times are generally LVI effective process times include setups, failure outages, etc.I HV, LV, and MV are all possible in effective process times

Relation to Performance Cases: For balanced systemsI MV - Practical Worst CaseI LV - between Best Case and Practical Worst CaseI HV - between Practical Worst Case and Worst Case

Process Variability

Measuring Process Variablity - Example

Process Variability

Causes of Variability

Natural VariabilityVariability without explicitly analyzed cause

Preemptive Outages (Breakdowns)Variability due to outages right in the middle of a job (e.g. due to power outages,operators being called away on emergencies, running out of consumables)

Nonpreemptive OutagesVariability due to outages for which we have some control as to exactly when theyoccur (e.g. due to setups, replacement of tools, shift changes)

ReworkVariability due to quality problems

Process Variability

Natural Variability

DefinitionVariability without explicitly analyzed cause

SourcesI operator paceI material fluctuationsI product type (if not explicitly considered)I product quality

ObservationNatural process variability is usually in the LV category

Process Variability

Variability from Preemptive Outages (Breakdowns)

Definitionst0 = base mean process timeσ0 = base standard deviation of process timec0 = base process coefficient of variability (σ0/t0)r0 = 1

t0 = base capacity (rate, e.g. parts/hr)

mf = mean time to failuremr = mean time to repaircr = coefficient of variability of repair times (σr/mr )

Process Variability

Variability from Preemptive Outages (cont.)

Availability - Fraction of time machine is up

A = mfmf +mr

Effective Processing Time and Ratere = A ∗ r0

te = t0/A

Process Variability

Example: Tortoise and Hare

Two machinesI subject to same workload: 69 jobs/day (2.875 jobs/hr)I subject to unpredictable outages (availability = 75%)

Hare X19I long, but infrequent outages

Tortoise 2000I short, but more frequent outages

PerformanceHare X19 is substantially worse on all measures than Tortoise 2000. Why?Variability!

Process Variability

Example: Tortoise and Hare (cont.)

Hare X19t0 = 15minσ0 = 3.33minc0 = σ0/t0 = 3.33/15 = 0.22mf = 12.4hrs(744min)mr = 4.133hrs(248min)cr = 1.0

Tortoiset0 = 15minσ0 = 3.33minc0 = σ0/t0 = 3.33/15 = 0.22mf = 1.9hrs(114min)mr = 0.633hrs(38min)cr = 1.0

AvailabilityA = A =

Process Variability

Variability from Preemptive Outages (cont.)

Effective Variabilityte = t0/A

σ2e = ( σ0A )2 + (m2

r +σ2r )(1−A)t0Amr

c2e = σ2e

t2e= c20 + (1 + c2r )A(1− A) mr

Note: Variability depends on repair times in addition to availability!

ConclusionsI Failures inflate mean, variance, and CV of effective process timeI Mean (te) increases proportionally with 1/AI SCV (c2e ) increases proportionally with mr

I SCV (c2e ) increases proportionally with c2rI For constant availability (A), long infrequent outages increase SCV more

than short frequent onesW. Jammernegg: Variability Basics 14 – 46

Process Variability

Example: Tortoise and Hare (cont.)

Hare X19te =

Tortoise 2000te =

Process Variability

Variability from Non-Preemptive Outages (Setups)

DefinitionsNs = number of parts (or jobs) between setupsts = mean duration of setup timescs = CV of setup times (σs/ts)

Effective Variabilityte = t0 + ts

σ2e = σ20 + σ2s

Ns+ Ns −1

c2e = σ2e

Process Variability

Variability from Rework

Definitionsp = probability that a given part is defective

Effective Variabilityte = E [Te ] = t0

σ2e = Var(Te) = σ20

1−p + pt20(1−p)2

c2e = σ2e

t2e= (1−p)σ2

0+pt20t20

= c20 + p(1− c20 )

Process Variability

Quality and the Supply Chain

Effect of Variability on Purchasing Lead Times

Single Component SystemsI Required Service: 95% service levelI Consequences: supplier 1 has 14 days and supplier 2 has 23 days lead time

Process Variability

Quality and the Supply Chain (cont.)

Effect of Variability on Purchasing Lead Times

Multiple Component SystemsI Required Service: 10 component assembly

I each component with 95% service level: p = 0.95 → p10 = (0.95)10 = 0.5987I 95% service on the assembly: p10 = 0.95 → p = 0.951/10 = 0.9949

I Consequences: supplier 1 has 16.3 days and supplier 2 has 33.6 days lead time

Process Variability

Other Process Variability Inflators

SourcesI operator unavailabilityI recycleI batchingI material unavailabilityI et cetera, et cetera, et cetera

EffectsI inflate te

I inflate ce

ConsequencesEffective process variability can be LV, MV, or HV.

Process Variability

Total Productive Maintenance (TPM)

cf. Nakajima S (1988) Introduction to TPM: Total Productive Maintenance. Productivity Press, Inc.

Process Variability

Overall Equipment Effectiveness (OEE)

cf. Nakajima S (1988) Introduction to TPM: Total Productive Maintenance. Productivity Press, Inc.

Quality Control: Statistical process control

Overview

Quality Control

Sample testing

I Acceptance sampling

I Sampling inspection(acceptance sampling, sampling inspection)

I OK/defect detection(qualitative attributes)(sampling by ATTRIBUTES)

Acceptance check (Approval check)

Incoming goods inspection, Finished goods inspection

I Initial situation: A defective product primary damages the image of theproducer and only secondary affects the supplier

I Consequence: Sampling inspections with inspection plans(e.g. Standards with MIL STD 105 D)

I Best practice: Define the Acceptable Quality Level AQL and the rejectionlevel LTPD (Lot Tolerance Percent Defective)

I AQL is defined as the accepted defect rate of a shipment by the customer

I LTPD is the defect rate of a shipment with inadequate quality

I AQL and LTPD are mainly used for the calculation of the required lot size

I Dynamical sampling plans: Change the lot size according to the history of thesupplier and the delivered products:

I Full inspection – Sampling inspection – Skip Lot

SPC (Statistical Process Control)

Process Control: Monitoring of the production process:

I Quality control chart (Control chart)I Samples are taken at fixed time intervalsI Given sample size

I Control chart for important KPIs of the process parameterI Mean, variance, range,...

I SHEWHART–CHARTS (Classical control charts)I The decision (e.g. Stop production) will be taken according to ONE KPI.

Statistical process control: Control–chart

I Process parameters are tracedI Mean valueI Percentage of defects

I Differentiation:I Common causes for variation

(within the control limits)I Classifiable reasons for

variation (out of the controllimits)

I Measurement of the processperformance:What is the natural variation ofthe process while it is undercontrol?

Control chart

Classical control chart: Mean-chart(X̄ -chart)

UCL = ¯̄x + 3σ√n

LCL = ¯̄x − 3σ√n

UWL = ¯̄x + 2σ√n

LWL = ¯̄x − 2σ√n

UCL Upper Control LimitUWL Upper Warning LimitLWL Lower Warning LimitLCL Lower Control Limit¯̄x Process meanσ Process standard deviationn Number of observations per sample

Range (R) - Chart

UCL = R̄ + 3× σR

LCL = R̄ − 3× σR

UWL = R̄ + 2× σR

LWL = R̄ − 2× σR

UCL Upper Control LimitLCL Lower Control LimitUWL Upper Warning LimitLWL Lower Warning LimitR̄ Range meanσR Range standard deviation

p-chart

p̄ = No. of defective goodsNo. of observations

Sp =√

p̄(1− p̄)n̄

UCL = p̄ + 3× Sp UWL = p̄ + 2× Sp

LCL = p̄ − 3× Sp LWL = p̄ − 2× Sp

Classical control chart: KPI development

A process is under control if

I all values are within the control limits.I not more than one value out of 40 is out of the warning limits.I two consecutive values are not out of the same warning limit.I no trend exists containing five or more values, which exceeds a warning limit.I no more than six consecutive values are either above or below the process

mean.I no trend of more than six values exists.

Quality Improvement: Process capability

Overview

Process capability

Goal: Reduction of the defect rate and thereby continuous process improvement

I Process capability is defined by the production tolerance.

I Production tolerance = Upper tolerance limit – Lower tolerance limit= [UTL–LTL]

I The product is defective if the measurement is out of the productiontolerance.

Process capability indices

cp = (UTL− LTL)/6σ

I If the target m = (LTL + UTL)/2 and the mean of the process ¯̄x do notcoincide, the usage of cp would lead to erroneous results.

I Therefore:

cpk = Min{UTL− ¯̄x , ¯̄x − LTL}3σ

Measurements of process capability

I Process capability studies are only be meaningful if the process is “undercontrol”.

I Calculation of ¯̄x , σ by using process date, where the mean is within thecontrol limits of the control chart.

Normal distributed parameter

Process capability indices

I A process is capable if cpk ≥ 1.

I Otherwise the process is called not capable.

Example

Process capability

I Quality design: UTL=85kg, LTL=75kg, m=80kg

I Quality control: ¯̄x=82,5kg, σ=4,2kg

I cp =

I cpk =

Improvement: Process mean=Target value

Improvement: Reduction of process variance

Quality control / Quality improvement

OM Session2

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