Operating (cost) characteristics

In some cases, the supplier and customer may wish to come to an agreement about how the risks of incorrect decisions might be shared between them by examining the respective levels of risk – in the first case, in percent risk terms, but also in economic terms.

Operating characteristic (power) curves

operating USL

Operating characteristic
Operating characteristic

Classical statistical hypothesis testing [Montgomery 1996] uses percent risk tools such as the so-called ‘operating characteristic’ or ‘power curve’. This is obtained by calculating the risk of incorrect decisions associated with either measurement uncertainty or entity dispersion in terms of the area of the probability distribution function extending ‘beyond’ the specification limit, for instance, in the case of consumer risk, the tail (figure (a)) above the upper specification limit. This risk is calculated as the location of the uncertainty interval – of fixed width – is swept across the specification limit, USL, of interest, as illustrated.

Customer and supplier can use such curves to agree on:

  • a maximum level, β, of consumer risk – say, 10% – where the uncertainty interval is located at the value LQL (limiting quality level) of the quality characteristic. Characteristic entity values further away from (below) the (upper) specification limit, will have probabilities of in-correctly accepting a non-conforming entity less than β.
  • a minimum level, α, of supplier risk – say, 95% – where the uncertainty interval is located at the value AQL (acceptable quality level) of the quality characteristic. Characteristic entity values further away from (above) the (upper) specification limit, will have probabilities of correctly rejecting a non-conforming entity greater than α.

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Operating ‘cost’ characteristic (power) curves

It may be more meaningful and easier to communicate between customer and supplier in economic terms rather than just percentage terms. Additionally, economics terms can capture additional impact factors, such as increasing costs as product deviates increasingly from specification, as well as whether it is a profit or loss in terms of the sign for either party.

Overall costs - equationOverall costs – equation

In a new kind of plot [Pendrill 2008] – the operating ‘cost’ characteristic – overall costs, according to the above equation, can be plotted over:

(I) a range of quantity values of  ym for a given test dispersion, σ, and ‘guard-band’ factor h

– yielding an “operating cost characteristic” analogous to the traditional, probability-based operating characteristic.

Cost operating characteristic

Cost operating characteristic

in the case of a linear cost model.

Examples:

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References:

L R Pendrill 2008, “Operating ‘cost’ characteristics in sampling by variable and attribute” Accred. Qual. Assur., 13, pp. 619-31, DOI: 10.1007/s00769-008-0438-y

L. R. Pendrill 2009, ”An Optimised Uncertainty Approach to Guard-banding in Global Conformity Assessment”, in Advanced Mathematical and Computational Tools in Metrology VIII, 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

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