Approaches for Measurement System Analysis Considering Randomness and Fuzziness

Introduction

Quality control and management of products are very important during production to ensure customer satisfaction. However, their effectiveness greatly depends on the data collected from the production line. The collected data are generally examined through a series of measurement processes by which the actual product quality level should be reflected. If the data collected from the production line cannot reflect the actual variability of the production process, incorrect decisions may be made. For this purpose, measurement system analysis (MSA), which is a design of experimentation approach for assessing the quality of measurements, is usually applied in applications. Such a system should be reliable in terms of low variability in the measurement process; however, some variability usually exists in real applications, due to measurement instruments, procedures, work-pieces, operations, personnel, and environment, which may distort observation data. Therefore, the examination of variability of a measurement system is an important work to make sure that the variability of collected data is as close to the actual process (product) variability as possible.

For MSA, variable and attribute measurements are two kinds of data. However, their applications are somehow limited in existing approaches. For example, most approaches for variable measurements only consider exact measurements. Imprecise measurements due to uncertainty, such as the appraiser’s subjective judgement, unstable measurement instruments, and change of measurement environment, are not considered. For attribute measurements, visual inspections are usually adopted in MSA by using the cross-tab method to compare one appraiser’s inspection outcomes to that of another for determining consistency among the appraisers. This application is only useful when inspection outcomes are limited to a very finite number of categories. In practice, the visual inspection process is frequently carried out in industries to subjectively judge the quality characteristic’s size or its ratio to an acceptable size to determine whether a part is nonconforming. The judgment includes appraisers’ cognitive processes based on their experience, and thus the measurements are imprecise. In other words, those subjective judgements include randomness and fuzziness. Existing approaches for MSA are limited to deal with observations that include randomness and fuzziness. In addition, the variability of the measurement system due to randomness and fuzziness cannot be quantified by existing approaches to determine how good a measurement system is.

Two studies have applied MSA in a fuzzy environment. Hajipour et al. (2013) proposed MSA with the index of gauge repeatability and reproducibility (GR&R) based on the average and range method, in which observations are represented as triangular fuzzy numbers (TFNs). A fuzzy expert system was developed to make final decisions. This study fuzzified the collected precise data as TFNs by using the deviation value, which may not reflect the actual fuzziness. Based on analysis of variance (ANOVA) and Zadeh’s extension principle, Moheb-Alizadeh (2014) formulated a pair of nonlinear programming problems to obtain fuzzy percent gauge variability (GRR%) and C_gk indices as assessment criteria, where a ranking method is applied to determine whether the measurement system is reliable. The formulated nonlinear models in this approach do not consider the relationship of the sum of squares of factors so that the fuzziness of GRR% may be over-emphasized, resulting in unrealistic outcomes. This is because the relationship of the total sum of squares of factors should be satisfied before investigating the reliability of a measurement system based on its variability. However, this important and essential point in data analyses has not been considered in the literature. No previous studies have investigated the reliability of a measurement system based on its variability.