Assessing the Cost of providing Quality in Inventory Systems

Georg-August-Universität Göttingen

Institut für Statistik und Ökonometrie

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Dirk Lehnick

Six Sigma in the context of logistic processes

Measuring the quality of a process based on

continuous attributes


Institut für Statistik und Ökonometrie______________________________________________________________________________________________________________

Dirk Lehnick

The Concept ofSix Sigma

Breakthrough Strategy

Measuring ConceptMetric

DPO, DPMO, Sigma Level

Operational ConceptSchemes and Tools for Improving

and Developing ProcessesDMAIC, DMADV

Strategic ConceptScheme for Introducing Six Sigma

Educational Programme

Six SigmaConceptional Frame



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Six Sigma Metric



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Dirk Lehnick

Converting DPO/DPMO to Sigma Level

Source: Breyfolge, F. W. (1999), Implementing Six Sigma



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Philosophy behind Six Sigma Metric:Do not loose too much information

but keep it simple!

Disadvantage of the Six Sigma Metric:• Loss of information (quality is a multidimensional,

complex phenomenon)

Advantages of the Six Sigma Metric:• simple, one-dimensional measure• easy to interpret• easy to calculate• suited for all kinds of processes and products• combinable to overall measures (e. g. for a whole

department or company)



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Critical-to-quality-characteristic (CTQ)

discrete mixed continuous

CTQ



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Specification limits

represent customer requirements

• Mimimum requirements are often given by normsand standardizations (ISO, DIN, ASTM, ...).

• The assumption of deterministic specification limitsimplies the idea of homogeneous customer requirements.



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Specification limits

For continuous attributes specification limits are used to distinguish between defects and non-defects.

specification limit

lower spec. limit

upper spec. limit

non-defect defect non-defect defectdefect



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Measuring DPO/DPMO/Sigma level in case of continuous CTQs

Three ways of measuring:

1. Counting the number of defects

2. Fit a normal distribution

3. Fit an appropriate model

Remember: DPOs/DPMOs/Sigma levels calculated from samplesare estimators and, thereby, random variables



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1. Simply count the number of defectsand calculate the ratio

Example: n = 100, real proportion of defects: 0.05

[2.78; 3.82]3.14Sigma level[0.01; 0.10]0.05DPO95% IntervalReal Value Measure

Inprecise, large loss of information



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2. Fit a normal distribution and calculateDPO as probability to fall outside the

specification limits

Example: n = 100, real proportion of defects: 0.05, CTQ normally distributed

[2.95; 3.42]3.14Sigma level[0.027; 0.073]0.05DPO95% IntervalReal Value Measure



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What happens if the CTQ fails to have a normal distribution?

Example: CTQ exponentially distributed (λ = 1/3)

2.400.1852.540.15

2.800.0962.780.1

3.500.0233.140.05

5.110.0001563.830.010.00000000173

0.271E(estim. DPO)

7.414.590.001

2.112.340.2E(estim. Sigma Level)Real Sigma Level Real DPO

Not appropriate if assumption ofnormal distribution fails

3. Fit an appropriate model and calculateDPO as probability to fall outside the

specification limits



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Better solution but takes a lot of time and money



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Measuring DPO/DPMO/Sigma level in case of continuous CTQs

1. Counting the number of defectsInprecise, large loss of information

2. Fit a normal distributionFails if CTQ is not normally distributed

3. Fit an appropriate modelToo costly to do for every single CTQ



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Common disadvantage of all three methods

Specification limits are assumedas single, deterministic values

Homogeneous customer requirementsimplies

But, in general, customer requirements are

heterogeneous.



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Deterministic vs. random specification limits

Borderline representsaverage customer

(heter. requirements)

Borderline representsall customers

(homog. requirements)specification

limitspecification

limit

non-defect defect



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Defect evaluation functionHomogeneous

customers requirementsHeterogeneous

customers requirements

specification limit

specification limit



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Logistic defect evaluation function

Other defect evaluation functions could be apllied as well, e. g. transformations of the normal distribution function



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Logistic defect evaluation function

lower spec. limit

upper spec. limit

target value

specification limit

target value



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Parameter β in the logistic d. e. functionSmall λ (e.g. β = 5): wide spread of customer requirements,

small slope of defect evaluation functionLarge λ (e.g. β = 9): narrow spread of customer requirements,

large slope of defect evaluation function



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Calculating DPO with the logistic defect evaluation function



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Influence of β on the calculation of DPOExample: n = 100, CTQ exponentially distributed (λ = 1/3),

expected DPO under homogeneous customer requirements: 0.05

97,5% quantile

2,5% quantile

50% quantile

DPO

β



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Dirk Lehnick

DPO ist bigger (and Sigma level lower) than you think!

In general, one has:

DPO under heterogeneous customer requirements

is bigger than

DPO under homogeneous customer requirements(the probability to fall outside the specification limits)



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DPO ist bigger than you think!



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What is the optimal value for β?Example: n = 100, CTQ exponentially distributed (λ = 1/3),

expected DPO under homogeneous customer requirements: 0.05

97,5% quantile

2,5% quantile

50% quantile

DPO

β



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How should one select the value for β?

Don't know yet ...



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Dirk Lehnick

How should one selct the value for β?

Don't know yet ...

The value of β depends on the spreadof the customer requirements.

Thus, it would be nice to define β as a function of thespread (variance) of the customer requirements.

Still to do ...



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Dirk Lehnick

Still to do ...

Defining β as a function of the spread (variance) of the customer requirements.

Constructing confidence intervals for DPO/DPMO/Sigma level under the assumption of heterogeneous customer requirements.

Customer requirements (deterministic as well as random)are dynamic!They change (increase) with time and the specification limits tend to move towards the target value.



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Dirk Lehnick

Summary

Calculating DPO/DPMO/Sigma Level with a defect evaluation function

uses the information of the CTQ values themselves (instead of just using a discrete defect/non-defect classification)

does not need to assume or to fit CTQ distribution models

is able to measure quality under the assumption of heterogeneous customer requirements



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Dirk Lehnick

Questions to the participants from business and industry

Central topic: customer requirements

How are customer requirements are observed or measured in practice?

What kind of information about customer requirements is available in Six Sigma applying firms?

How often is information about customer requirements updated?



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Dirk Lehnick

Thank you very much for your attention!

Assessing the Cost of providing Quality in Inventory Systems

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