### Risks and costs

In general, the impact of a wrong decision in conformity assessment is expressed as a risk ** Risk**, defined as the probability

**of the wrong decision occurring multiplied by the cost (utility)**

*p***of the consequences of the in-correct decision:**

*C**Risk = p.C *

In this section, new expressions for decision-making risks including costs are presented, together with a novel tool – the operating cost characteristic curve – as an extension of traditional statistical tools, with the addition of an economic decision-theory approach. Complementarity with the optimised uncertainty methodology [Fearn *et al *2002] is emphasised in the concluding remarks.

### Cost model

An in-correct accept on inspection of a non-conforming object will lead to customer costs associated with out-of-tolerance product. Overall costs, consisting of a sum of testing costs, *D*, and the costs, *C*, associated with customer risk can be calculated with the expression:

where *R _{PV}* denotes the region of permissible entity values, η – where test costs,

*D*, are modelled as varying inversely to the squared dispersion σ² . The expression can be applied to both specific and global conformity assessment [Pendrill 2007].

### Plots of cost against (i) quantity value and (ii) uncertainty

Overall costs, according to the above equation, can be either plotted over:

- a range of quantity values of η for a given test dispersion, σ, and ‘guard-band’ factor
*h*– yielding an “operating cost characteristic” analogous to the traditional, probability-based operating characteristic - a range of test dispersion, σ, for a given quantity value η, the so-called “optimised uncertainty curve”

*Figure *** ***Overall costs ** versus Uncertainty and Entity value in the vicinity of a specification limit **U _{SL} (adapted from [Pendrill 2008])*

It is possible to view the two tools – the operating cost characteristic [I] and optimised uncertainty [II] methodologies – as together providing a complete basis for risk-assessment in conformity assessment: Overall costs are plotted in three dimensions – shown in figure* ** *** **– where at each entity value on the operating characteristic curve, the corresponding optimized uncertainty curve would cross in the orthogonal direction. The familiar U-shaped optimized uncertainty curve – where the costs of testing are balance against the costs of incorrect decision-making – is clearly visible along the ‘uncertainty’ axis of the 3D-plot of figure* **.*** **In this way, the optimum uncertainty required at specific conformity assessment points, such as those for customer and supplier risk, could be identified across the full range of entity values.

### Criticism of approach

A common criticism is that it is “very difficult to quantify the total cost in all possible applicative scenarios. On the other hand, it is known that, by setting a suitable amount of guard-banding, the risk of accepting an out-of-tolerance item or rejecting an in-tolerance item can be kept under control with good accuracy” [Macii and Petri 2009]

Answers to such criticism include:

- any estimate of cost (or ‘impact’) in any scenario is better than none – however uncertain!
- how do you decide what ‘good accuracy’ in guard-banding is?

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### Examples

- Measurement instruments
- Geometric characteristics of products
- Vehicle panel closure gap *(For password, contact leslie.pendrill@sp.se )

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- Operating (cost) characteristics
- Optimised uncertainty
- Introducing impact and cost into conformity assessment risks
- Specifications of Process and Measurement Capabilities

### References:

G Beges, J Drnovsek and L R Pendrill 2009 Optimising **calibration and measurement capabilities** in terms of economics in conformity assessment”, *Accred Qual Assur*, DOI http://dx.doi.org/10.1007/s00769-009-0599-3

Hake R 2010 “The Cult of Statistical Significance”, http://bit.ly/dkTyXP

T. Fearn, S. Fisher, M. Thompson and S. Ellison 2002, “A decision-theory approach to fitness for purpose in analytical measurement,” *Analyst*, **vol. 127**, pp. 818 – 824

Macii D and Petri D 2009 “Guidelines to manage measurement uncertainty in conformance testing procedures”, *IEEE Trans Instrum Meas* **58**, 33 – 40

H Källgren and L R Pendrill 2006, “**Exhaust gas analysers** and optimised sampling, uncertainties and costs*”, **Accreditation and Quality Assurance – Journal for Quality, Reliability and Comparability in Chemical Measurement.** Vol 11, 496- 505*

*http://dx.doi.org/10.1007/s00769-006-0163-3 (2006)*

Källgren H, Pendrill L R, Lindlov K (2006) ”Uncertainty in conformity assessment in legal metrology (related to the MID)”, *OIML Bull* **XLVII**(3):15–21

Pendrill L R 2007 “Optimised Measurement Uncertainty and Decision-Making in Conformity Assessment”, *NCSLi Measure*, Vol**. 2**, no. 2, pp 76 – 86 Balancing the costs of testing against the consequence costs of in-correct decision-making

Pendrill L R 2008, “Operating ‘cost’ characteristics in sampling by variable and attribute” *Accred. Qual. Assur*., **13**, 619 – 631, DOI: 10.1007/s00769-008-0438-y Extending classical statistical significance tools to include measures of impact

L.R. Pendrill and H. Källgren 2008, “Optimised measurement uncertainty and decision-making in the **metering of energy, fuel and exhaust gases**,” *Izmerite’lnaya Technika (Measurement Techniques), *Vol **51**,* *No. 4, pp. 370 – 7, April, 2008* DOI: *http://dx.doi.org/10.1007/s11018-008-9047-8

L. R. Pendrill 2009, **”**An Optimised Uncertainty Approach to Guard-banding in Global Conformity Assessment”, *Advanced Mathematical and Computational Tools in Metrology VIII* in Data Modeling for Metrology and Testing in Measurement Science Series: Modeling and Simulation in Science, Engineering and Technology, Birkhauswer (Boston), 2009, ISBN: 978-0-8176-4592-2 http://www.worldscibooks.com/mathematics/7212.html

L R Pendrill 2010 “Optimised uncertainty and cost operating characteristics: new tools for conformity assessment. Application to **geometrical product control in automobile industry**”, *Int. J. Metrol. Qual. Eng.***1***,* 105 – 110, DOI: http://dx.doi.org/10.1051/ijmqe/2010020

Ziliak, S.T. & D. McCloskey. 2008. “The Cult of Statistical Significance: How the Standard Error Costs Us Jobs, Justice, and Lives (Economics, Cognition, and Society).” University of Michigan Press; publisher’s information at <http://bit.ly/aWQtbX>. Amazon.com information at <http://amzn.to/a4n2yE>. Note the searchable “Look Inside” feature.

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