## Introducing impact and cost into conformity assessment risks

### Risks and costs

In general, the impact of a wrong decision in conformity assessment is expressed as a risk Risk, defined as the probability p of the wrong decision occurring multiplied by the cost (utility) C of the consequences of the in-correct decision:

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:

Overall costs - equation

where RPV 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:

1. 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
2. a range of test dispersion, σ, for a given quantity value η, the so-called “optimised uncertainty curve

3D costs

Figure Overall costs versus Uncertainty and Entity value in the vicinity of a specification limit USL (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]

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### Examples

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### 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 attributeAccred. 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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