Measurement Uncertainty in Decision-Making: How to Take Reliable Decisions under Uncertainty

Introduction

Typically measurements are used in order to get information on a phenomenon. Or, at times, merely it is required to know the behavior of a specific quantity. Such a task is not so simple. Often the available a priori information on the phenomenon is not sufficient. Therefore knowledge is not only by means of measurements. Sometimes measured data are subsequently put in comparison with fixed limits or warning thresholds (law limits or quality target) in order to verify the conformity with predetermined specifications or to control the occurrence of an alarm state. So decisions have to be taken about the possible limit overcoming. Generally when a parameter or value has to be compared with a limit or threshold, the simple mathematical comparison between the two values has to be avoided. In fact, the result of any measurement is affected by uncertainty. So, according to the Guide to the expression of Uncertainty in Measurement (JCGM, 2008), the result of a measurement is simply an approximation or estimate of the value of the measurand. The uncertainty characterizes the dispersion of the values that could reasonably be attributed to the measurand. Consequently the comparison does not concern two mere numerical values, but rather a value (limit or threshold) and an interval of values (measure). High values of uncertainty are cause of wide intervals and so the probability to take a mistaken decision is higher. Therefore the result of the comparison depends strongly on the measurement uncertainty, for that reason it cannot be disregarded. An underestimation of the uncertainty effects may be cause of wrong decisions. Consequently specific requirements on the reliability of measured data are required. Reliable data can be guaranteed not only by means of the choice of an appropriate measurement system. Although it should allow a decrease of the uncertainty to be got, an opportune analysis of the measurement process is needed too. In detail, in practice there are numerous possible sources of uncertainty in a measurement. The main contributions include: a) an incomplete definition of the measurand; b) an inadequate knowledge of the effects of environmental conditions; c) the used measurement method and procedure; d) the used measurement system. Therefore further information on measurement process is necessary in order to qualify the reliability of the decision making stage. In this sight, the comparison between measurement data and limits needs opportune decisional rules. In the state of art, several procedures and models have been proposed, but few ones keep into consideration the influence of uncertainty about the possibility to make a wrong decision. A methodical approach to such problems would require a suitable analysis of the possible risks and costs associated to the decisional alternatives. According to the number of decisional alternatives, it is possible to single out two categories of decision problems. In the present manuscript, attention is focused on Multi-Attribute problems where the number of alternatives is finite and determined, (Yuxun, 2010; Wang, 2010). With reference to the literature, the Standard EN ISO 14253-1, (International Organization for Standardization, 1998), provides a simple decisional rule for problems of conformity with specifications. Unfortunately in practice, it is a “rule of thumbs” because of some restrictions. In order to fix the problem, let us consider a generic process under control. Typically specific parameters or quantities are considered in order to monitor the process (Zingales, 1996; Catelani, 1998). So the occurrence of an out-control situation happens when the measured parameter or the quantity overcomes the specification limit or threshold. Such case includes several practical problems like environmental monitoring applications, the control of an industrial process, the check of compliance with specifications, or the observance of laws and regulations. The problem rises from putting in comparison the measured data with a fixed reference value. Often the overcoming of similar warning or threshold levels entails adopting appropriate corrective actions in order to avoid or to minimize the associated consequences. In this stage, the measurement uncertainty could have a significant influence. Moreover, if the inevitable cumulative effect of the uncertainty contributions is not considered, the whole decision-making stage could be invalidated. So the measured data could seem apparently lower or higher than the fixed limit, whereas the monitored parameter is really above or below the limit respectively. With regard to critical processes like environmental monitoring or health care applications, such an error during the decision making stage could put people safety at risk. Consequently reliable decisions need a suitable interpretation of available data by means of appropriate decisional rules. In order to optimize the decisional criteria, further information on the monitored process like statistical distributions, costs and risks have to be considered so to minimize the possibility to take erroneous decisions. In the present manuscript the authors report their experience in the field of decision making problems. By a detailed analysis of the literature, suitable decisional criteria have been developed and a Fuzzy Decision Making algorithm is described. Probability Theory and Fuzzy Logic have been used in order to improve the reliability of the proposed procedure and to manage the different features of the problem. The Probability Theory allows the stochastic aspects of the monitored process to be modeled. Whereas the Fuzzy Logic estimates the vague information on the measured process whose effects not are computed by the measurement uncertainty (Chen & Lee, 2010). The authors start from the guidelines of the Standard EN ISO 14253-1, and propose a procedure which guides the decision-maker to take a reliable decision. Two available alternatives: conformance or non-conformance of the considered parameter with specification limits. The algorithm is based on a Statistical Model that estimates the erroneous decision probabilities. The approach overcomes the limitations of the Standard EN ISO 14253-1. It provides practical criteria with an indication about the consistency of the final decision. The benefit of the procedure is to model the ‘fuzzy’ features of process under statistical control by means of a methodological approach to the decisional stage. Intuitive graphical tools guide the user about the most plausible decision. In the following Sections the Fuzzy Decision Making Algorithm is described and a case study is reported. An environmental monitoring application is considered in order to implement the procedure to a real case concerning a critical process.