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ISO9001:2000 and TS 16949 have become the major quality system management models in present traditional and Hi-tech industries. The Measurement System Analysis (MSA) Reference Manual, on the other hand, is one of the core tools in ISO/TS 16949. MSA aims to evaluate Gauge Repeatability and Reproducibility (GR&R) where the control, monitoring, and maintenance of the measurement process are required in measurement systems so that the measurement capability could be ensured under statistical control. An ideal measurement system should present the statistical characteristic of zero error on any measured product. Nevertheless, such an ideal measurement system hardly exists. Managers therefore have to adopt such measurement systems with unsatisfactory statistical characteristics. Traditional MSA indexes are constructed with definite observed values. Nevertheless, measurements with observed values are not entirely error-free. For this reason, this study proposes to research three cases in a case company and apply the integration of Fuzzy Theory and GR&R to discuss the differences in the evaluation index GR&R and the Number of Distinct Categories (NDC). Substituting fuzzy numbers for definite numbers found that the data of %GR&R were increased and NDC was decreased after fuzzification. Such results verify that the fuzzified %GR&R and NDC become stricter in the determination criterion. The research outcomes could assist the case company in improving the reference data of measurement systems and promoting the measurement quality.

In the 1980s, Motorola in the United States created a set of techniques known as Six Sigma to improve quality management systems. Subsequently, this technique was successfully adopted by major enterprises such as General Electric, producing benefits including reduced costs, and gradually leading a new wave of thinking about quality across the globe [

In quality management, measurement techniques can be used to determine whether a product meets certain specifications and safety requirements; thus, measurement techniques are effective tools for increasing product quality [

As the precision and accuracy of different types of measurement equipment have increased, companies have developed various methods for measuring quality. The objective of the analysis of measurement methods is to find the total variation between measurement methods, widely referred to as the measurement uncertainty [

The introduction of QS 9000/TS 16949 certification and Six Sigma has ensured that quality control personnel now pay more attention to measurement systems and GR&R [

Mandel’s [

Research by Montgomery and Runger [

ganizations improve quality. Therefore, a successful measurement system should provide effective techniques for dealing with process variation (PV) and be able to establish the causes of variation. Measurement systems use GR&R to analyze the sources of measurement variation and quantify the variation. Tsai [

Montgomery and Runger [

Reilly [

Research by Dolezal et al. [

Assuming inspectors and parts both have fixed effects, Wang and Eldon [

Fang and Wang [

Fang et al. [

TS 16949 provides internationally universal and proven technical specifications for the automobile and precision industries, covering the operating requirements for the quality systems of both manufacturers and suppliers. Before the establishment of TS 16949, all suppliers of parts and services in the automobile industry had to obtain quality system certification for each country or region they supplied, for instance EAQF in France, AVSQ in Italy, QS 9000 in the United States, and VDA-6 in Germany.

Reid [

all suppliers involved in the manufacture or maintenance of parts in the automotive industry were required to use the ISO/TS 16949 automobile industry quality management system. Aside from the automotive industry standards in the ISO 9001quality management system, ISO/TS 16949 also contains five core tools: 1) advanced product quality planning (APQP); 2) production parts approval process (PPAP); 3) measurement system analysis (MSA); 4) failure mode and effects analysis (FMEA); and 5) statistical process control (SPC). These five core tools make the ISO/TS 16949quality management system more rigorous.

The basic framework of TS 16949:2002 is based on ISO9001:2000. The process is used as the basic framework, looking at process management models formed by a series of interactions and relationships during the conversion of inputs to outputs. This management model is the process management from receiving an input to producing an output, and continued maintenance of the PDCA cycle during process management. Continued reductions in variation are achieved to ensure that outputs achieve customer satisfaction, while also strengthening the organization’s quality control system [

MSA uses statistical and diagrammatic methods for experimental design and statistical analysis of measurement system error to assess variance in measurement systems, allowing an assessment of the variance in individual measurement equipment and inspectors, and providing a basis for the management of measurement equipment systems [

Equipment qualification, equipment calibration, and MSA are all within the scope of measurement system evaluation. Each of these methods aims to evaluate the reliability of the measurement system and establish the stability of the manufacturing process and benefits from improvement. This study summarizes frequently used evaluation methods for measurement systems according to type and function, as shown in

If the errors produced by inspectors or measurement equipment in measurement systems are large, the reliability of the data recorded in the manufacturing process needs to be reconsidered. An ideal measurement system should have an error rate that is statistically zero [

Item | Calibration | MSA | Correlation |
---|---|---|---|

Sample selection | Standard item could refer to international standard | Sampling the actual product | Customer specified or customer and manufacturer agreed standard items |

Measured result | International standard is available | Acceptable deviation and variation | Measurement consistency of the same sample with different measuring instruments |

Measuring environment | Controlled laboratory | Manufacturing environment | Manufacturing environment |

Method of judgment | Errors within 1/10 of measurement tolerance | Based on the AIAG MSA manual | Customer specified error range or 1/10 of measurement tolerance |

peatability and reproducibility: the range method, the average and range method, and the ANOVA method.

The AIAG [

Pan [

Fuzzy theory refers to the fuzzy conceptualization of certain data. Uncertain information is made more precise through an approximate reasoning process [

A crisp set is a very clearly defined set, without any fuzzy areas. Therefore, a characteristic function of a crisp set can be expressed as, if not 0, then 1. Conversely, in order to express the fuzziness of fuzzy sets, a membership function of between 0 and 1 must be used to represent the degree of membership to the set [

In order to distinguish between crisp and fuzzy sets, this study sets fuzzy sets as Ã. For example, if the crisp set A is defined as {x is equal to b}, when x is equal to b, its membership degree of set A is 1. However, when x is not equal to b, its membership degree of set A is 0, as shown in Equation (1):

Conversely, if the fuzzy set Ã is defined as {x is approximately equal to b}, when x is equal to b, its membership degree of set Ã is 1. However, when x is a value close to b, its membership degree of set Ã is between 0 and 1, as shown in

Commonly found membership functions of fuzzy sets include trapezoidal membership functions, triangular membership functions, and bell membership functions. This study uses the triangle membership function (a,b,c) shown in

In practical application in industry and academia, since there are many possible shapes for membership func-

tion, the choice of membership function will depend on the practical considerations of decision-makers.

However, the fuzzy concepts in mathematics proposes a fuzzification process of turning numbers into functions of fuzzy numbers and carrying out expansion of fuzzy relations on crisp sets. Therefore, the use of fuzzy control involves determining the operating range based on the size of the measurement values. Therefore, the semantic value is an expression of the membership function of the fuzzy set. When a crisp value is observed, this will correspond to the membership function of the domain and obtain the membership function of the inputs.

After carrying out MSA, we first calculate the X-bar-R values and make these values independent. Construction of the MSA uses crisp numbers, but errors are still inevitable in the measurement process. These errors are non- sampling errors and cannot be avoided in statistical theory. Therefore, this study regards measurement values as a fuzzy concept where real numbers are used to represent a domain. These measurement values are therefore fuzzy numbers, with fuzzy theory used in MSA. When it is determined that the measurement values are not precise values, fuzzy mathematics is used to replace crisp numbers with fuzzy numbers, with the values also being independent. Then, these two sets of data are compared using GR&R number of distinct categories (NDC).

In real life, many problems are uncertain or imprecise. Conventional binary logic may be unable to solve these problems [

Membership function is the basis for the application of fuzzy sets to actual problems. However, there is still no objective means of determining the membership function [^{th} factor u_{i} in the sample set is evaluated as having the j^{th} element v_{j} degree of membership r_{ij} for the common set. Therefore, the evaluation results can be expressed using the fuzzy set R_{i}, with R_{i} the fuzzy subset of evaluation set V, which can also be simply expressed as a vector:^{th} factor, r_{ij} must satisfy the normality condition_{i} for each of the factors as the rows, we can compose the fuzzy matrix.

1) In this paper, we use linear membership functions since they are more commonly used, and produce a more stable 45-degree angle, making them more acceptable to ordinary readers. An appropriate universal set that contains all data domains is divided into a number of short segments (normally five short segments), and the corresponding data segment for each data item is identified, with the triangle membership function fuzzifying the original data as fuzzy data values [

2) The sample specification tolerance is divided into five intervals, with a positive or negative specification tolerance of 1 mm divided into intervals of 0.02, with a positive or negative interval of 0.02 as v_{1} = extremely precise; a positive or negative interval of 0.04 as v_{2} = very precise; a positive or negative interval of 0.06 as v_{3} = precise; a positive or negative interval of 0.08 as v_{4} = less precise; and a positive or negative interval of 0.08 as v_{5} = imprecise. This produces the fuzzy subset for each sample.

The evaluation set represents all the possible evaluations of an object made by the evaluator and is normally represented as V; therefore, _{1} = extremely precise, V_{2} = very precise, V_{3} = precise, V_{4} = less precise, and V_{5} = imprecise.

Interval I = [a,b] is a special fuzzy number called an interval number. Li [

The measurement record shows three inspectors coded A, B, and C measuring different samples, as shown in _{1} = the interval median for specification tolerance A; v_{2} = the interval median for specification tolerance B; v_{3} = the interval median for specification tolerance C; v_{4} = the interval median for specification tolerance D; v_{5} = the interval median for specification tolerance E.

The MSA average and range method is applied to measurement equipment MX-01 hardness tester and MX-02 micrometer. Three measures for each of the ten samples are taken by the three inspectors, producing 90 data points for XBar and R calculation. Pen [

Research steps:

Step 1: Select ten sample parts and code them to 1 - 10. These sample parts must fully reflect the range of variation in the manufacturing process.

Step 2: Select three inspectors coded A, B, and C.

Step 3: Each inspector carries out random repeated measurement of each sample part three times. Inspectors

Sample | A | B | C | ||||||
---|---|---|---|---|---|---|---|---|---|

1^{st} | 2^{nd} | 3^{rd} | 1^{st} | 2^{nd} | 3^{rd} | 1^{st} | 2^{nd} | 3^{rd} | |

Sample No. 01 | |||||||||

Sample No. 02 | |||||||||

Sample No. 03 | |||||||||

Sample No. 04 | |||||||||

Sample No. 05 | |||||||||

Sample No. 06 | |||||||||

Sample No. 07 | |||||||||

Sample No. 08 | |||||||||

Sample No. 09 | |||||||||

Sample No. 10 |

Note: A1^{st}: First measurement of inspector; A2^{nd}: Second measurement of inspector A; A3^{rd}: Third measurement of inspector A; B1^{st}: First measurement of inspector; B2^{nd}: Second measurement of inspector B; B3^{rd}: Third measurement of inspector B; C1^{st}: First measurement of inspector C; C2^{nd}: Second measurement of inspector C; C3^{rd}: Third measurement of inspector C.

are not permitted to compare their results. Inspectors A, B, and C will each produce thirty sets of data, giving a total of ninety sets of data.

Step 4: The average and range for the three measurements of each sample part produce a total of ten averages and ten ranges for each inspector.

Step 5: Calculate the sample average for each part. The average and range of each sample part is represented as Rp.

Step 6: Inspectors A, B, and C each produce ten averages, which produce an overall average of

Step 7: Calculate the

Step 8: Repeatability measures EV, with average range multiplied by constant K_{1}, dependent on the number of tests and equal to the reciprocal of_{1} = 0.5908. The reproducibility is measured as the variation between appraisers, AV, calculated as themaximum variation in average between inspectors _{2}. K_{2} is determined by the number of inspectors used in the research, and is the reciprocal of_{2} = 0.5231, the number of samples is n = 10, and the number of repeat tests is r = 3.

Step 9: The measurement system GR&R is calculatedas shown in Equation (7).

Step 10: Calculating PV. The average range of the sample parts multiplied by constant K_{3}; K_{3} is determined by the number of sample parts used in the study and is reciprocal of_{3} = 0. 3146,

Step 11: Total variation can be obtained from the square of GR&R plus the square of the part variation (PV), as shown in Equation (9).

Step 12: %GR&R is GR&R as a percentage of total variation, as shown in Equation (10).

Step 13: NDC. Wheeler and Lyday [

Step 14: Repeat steps 4 to 13 according to data from the fuzzy model.

Step 15: Comparison of fuzzification and standard deviation for GR&R and NDC.

This study looks at the case of a machine tool factory in Taiwan, using the test methods and steps to analyze measurement results and calculate EV, appraiser variation (AV), and GR&R. We carry out tests on the measurement systems and use the results of the analysis to put forward appropriate recommendations, as a basis for industry to determine the %GR&R and as a reference for future research.

In order to obtain variance in a real measurement system, the data gathered in this study were measured during actual processes, as explained below: 1) Using actual production equipment; 2) Personnel were quality personnel or operators who regularly use this equipment; 3) Measurement was carried out using typical measuring tools; 4) Data was collected according to plan; and 5) Three case studies were carried out, each with three inspectors, ten sample parts, and each with three tests.

In case study 1, the measurement instrument was a micrometer codenamed MX-02 with an accuracy of obtained values of 0.001 mm. Three inspectors coded A, B, and C were selected, then ten production line guide rod machines were chosen, and the thickness at a certain position was measured. The drawing specification states a thickness of 4.62 mm at this point, with a tolerance range of +0 - −0.035 mm. Each of the three inspectors measured the sample part three times, and the measurement results are recorded in

After the data were collected, we used the average and range method in the MSA Reference Manual to calculate GR&R, giving a result of 0.0006, and a %GR&R for the measurement system of 6.73%. The NDC of 20.904 was rounded to the nearest integer, producing a value of 21.

In case study 2, the measuring instrument was a hardness tester coded MX-01, with an accuracy of obtained values of 1 degree. Three inspectors coded A, B, and C were selected, ten production line hand tool blades were chosen, and the hardness at a certain position was measured. The drawing specification states a thickness of HRC55 at this point, with a tolerance range of +5 degrees - −0 degrees. Each of the three inspectors measured the sample part three times, and the measurement results are recorded in

After the data were collected, we used the average and range method in the MSA reference manual to calculate GR&R, giving a result of 0.327, and a %GR&R for the measurement system of 28.89%. The NDC of 4.672 was rounded to the nearest integer, producing a value of 5.

In case study 3, the measuring instrument was a hardness tester coded MX-01, with an accuracy of obtained values of 1 degree. Three inspectors coded A, B, and C were selected, ten production line tension spindles were chosen, and the hardness at a certain position was measured. The drawing specification states a thickness of HRC48 at this point, with a tolerance range of +5 degrees - −0 degrees. Each of the three inspectors measured the sample part three times, and the measurement results are recorded in

Sample | A | B | C | ||||||
---|---|---|---|---|---|---|---|---|---|

1^{st} | 2^{nd} | 3^{rd} | 1^{st} | 2^{nd} | 3^{rd} | 1^{st} | 2^{nd} | 3^{rd} | |

Sample No. 01 | 4.620 | 4.620 | 4.620 | 4.620 | 4.619 | 4.619 | 4.620 | 4.620 | 4.619 |

Sample No. 02 | 4.618 | 4.619 | 4.618 | 4.620 | 4.619 | 4.620 | 4.619 | 4.618 | 4.619 |

Sample No. 03 | 4.618 | 4.619 | 4.619 | 4.618 | 4.619 | 4.619 | 4.618 | 4.617 | 4.618 |

Sample No. 04 | 4.612 | 4.613 | 4.613 | 4.619 | 4.619 | 4.620 | 4.618 | 4.619 | 4.619 |

Sample No. 05 | 4.619 | 4.619 | 4.618 | 4.618 | 4.617 | 4.617 | 4.619 | 4.618 | 4.618 |

Sample No. 06 | 4.617 | 4.618 | 4.617 | 4.619 | 4.619 | 4.618 | 4.620 | 4.619 | 4.620 |

Sample No. 07 | 4.619 | 4.618 | 4.619 | 4.620 | 4.620 | 4.620 | 4.620 | 4.620 | 4.620 |

Sample No. 08 | 4.620 | 4.619 | 4.620 | 4.619 | 4.618 | 4.619 | 4.619 | 4.620 | 4.620 |

Sample No. 09 | 4.619 | 4.618 | 4.618 | 4.619 | 4.619 | 4.618 | 4.620 | 4.620 | 4.619 |

Sample No. 10 | 4.595 | 4.595 | 4.600 | 4.585 | 4.585 | 4.586 | 4.585 | 4.586 | 4.587 |

Sample | A | B | C | ||||||
---|---|---|---|---|---|---|---|---|---|

1^{st} | 2^{nd} | 3^{rd} | 1^{st} | 2^{nd} | 3^{rd} | 1^{st} | 2^{nd} | 3^{rd} | |

Sample No. 01 | 57 | 57 | 57 | 58 | 57 | 58 | 57 | 57 | 57 |

Sample No. 02 | 56 | 56 | 57 | 57 | 57 | 56 | 57 | 57 | 57 |

Sample No. 03 | 55 | 55 | 55 | 55 | 56 | 55 | 55 | 55 | 56 |

Sample No. 04 | 57 | 57 | 57 | 58 | 58 | 58 | 58 | 58 | 58 |

Sample No. 05 | 58 | 58 | 58 | 57 | 58 | 57 | 57 | 57 | 58 |

Sample No. 06 | 58 | 58 | 58 | 57 | 57 | 57 | 59 | 58 | 58 |

Sample No. 07 | 57 | 58 | 58 | 58 | 58 | 58 | 58 | 58 | 59 |

Sample No. 08 | 58 | 58 | 57 | 56 | 56 | 56 | 58 | 58 | 57 |

Sample No. 09 | 59 | 59 | 58 | 59 | 58 | 59 | 59 | 59 | 58 |

Sample No. 10 | 59 | 58 | 59 | 59 | 59 | 59 | 58 | 58 | 58 |

Sample | A | B | C | ||||||
---|---|---|---|---|---|---|---|---|---|

1^{st} | 2^{nd} | 3^{rd} | 1^{st} | 2^{nd} | 3^{rd} | 1^{st} | 2^{nd} | 3^{rd} | |

Sample No. 01 | 48 | 48 | 48 | 48 | 48 | 48 | 49 | 50 | 49 |

Sample No. 02 | 50 | 51 | 50 | 50 | 49 | 49 | 50 | 50 | 50 |

Sample No. 03 | 49 | 48 | 49 | 49 | 49 | 50 | 50 | 50 | 50 |

Sample No. 04 | 52 | 52 | 52 | 52 | 52 | 52 | 51 | 52 | 51 |

Sample No. 05 | 52 | 51 | 51 | 51 | 51 | 52 | 52 | 52 | 52 |

Sample No. 06 | 51 | 50 | 50 | 51 | 51 | 50 | 49 | 50 | 49 |

Sample No. 07 | 50 | 50 | 49 | 50 | 49 | 50 | 51 | 51 | 51 |

Sample No. 08 | 48 | 48 | 48 | 49 | 49 | 49 | 48 | 48 | 48 |

Sample No. 09 | 49 | 50 | 50 | 50 | 50 | 50 | 49 | 49 | 50 |

Sample No. 10 | 48 | 48 | 48 | 48 | 49 | 49 | 48 | 48 | 48 |

After the data were collected, we used the average and range method in the MSA Reference Manual to calculate GR&R, giving a result of 0.333, and a %GR&R for the measurement system of 28.53%. The NDC of 4.736 was rounded to the nearest integer, producing a value of 5.

In case study 1, each of the three inspectors measured the sample parts three times. Following fuzzification, the measurement results for the fuzzy case are recorded in

After the data were collected, we used the average and range method in the MSA Reference Manual to calculate GR&R, giving a result of 0.0006, and a %GR&R for the measurement system of 8.14%. The NDC of 17.266 was rounded to the nearest integer to produce a value of 17.

In fuzzy case study 2, each of the three inspectors measured the sample parts three times as shown in

After the data were collected, we used the average and range method in the MSA Reference Manual to calculate GR&R, giving a result of 0.290, and a %GR&R for the measurement system of 32.73%. The NDC of 4.071 was rounded to the nearest integer, producing a value of 4.

Sample | A | B | C | ||||||
---|---|---|---|---|---|---|---|---|---|

1^{st} | 2^{nd} | 3^{rd} | 1^{st} | 2^{nd} | 3^{rd} | 1^{st} | 2^{nd} | 3^{rd} | |

Sample No. 01 | 4.617 | 4.617 | 4.617 | 4.617 | 4.617 | 4.617 | 4.617 | 4.617 | 4.617 |

Sample No. 02 | 4.617 | 4.617 | 4.617 | 4.617 | 4.617 | 4.617 | 4.617 | 4.617 | 4.617 |

Sample No. 03 | 4.617 | 4.617 | 4.617 | 4.617 | 4.617 | 4.617 | 4.617 | 4.617 | 4.617 |

Sample No. 04 | 4.607 | 4.613 | 4.613 | 4.617 | 4.617 | 4.617 | 4.617 | 4.617 | 4.617 |

Sample No. 05 | 4.617 | 4.617 | 4.617 | 4.617 | 4.617 | 4.617 | 4.617 | 4.617 | 4.617 |

Sample No. 06 | 4.617 | 4.617 | 4.617 | 4.617 | 4.617 | 4.617 | 4.617 | 4.617 | 4.617 |

Sample No. 07 | 4.610 | 4.617 | 4.617 | 4.617 | 4.617 | 4.617 | 4.617 | 4.617 | 4.617 |

Sample No. 08 | 4.610 | 4.617 | 4.617 | 4.617 | 4.617 | 4.617 | 4.617 | 4.617 | 4.617 |

Sample No. 09 | 4.610 | 4.617 | 4.617 | 4.617 | 4.617 | 4.617 | 4.617 | 4.617 | 4.617 |

Sample No. 10 | 4.596 | 4.596 | 4.600 | 4.589 | 4.589 | 4.589 | 4.589 | 4.589 | 4.589 |

Sample | A | B | C | ||||||
---|---|---|---|---|---|---|---|---|---|

1^{st} | 2^{nd} | 3^{rd} | 1^{st} | 2^{nd} | 3^{rd} | 1^{st} | 2^{nd} | 3^{rd} | |

Sample No. 01 | 57 | 57 | 57 | 58 | 57 | 58 | 57 | 57 | 57 |

Sample No. 02 | 56 | 56 | 57 | 57 | 57 | 56 | 57 | 57 | 57 |

Sample No. 03 | 56 | 56 | 56 | 56 | 56 | 56 | 56 | 56 | 56 |

Sample No. 04 | 57 | 57 | 57 | 58 | 58 | 58 | 58 | 58 | 58 |

Sample No. 05 | 58 | 58 | 58 | 57 | 58 | 57 | 57 | 57 | 58 |

Sample No. 06 | 58 | 58 | 58 | 57 | 57 | 57 | 59 | 58 | 58 |

Sample No. 07 | 57 | 58 | 58 | 58 | 58 | 58 | 58 | 58 | 59 |

Sample No. 08 | 58 | 58 | 57 | 56 | 56 | 56 | 58 | 58 | 57 |

Sample No. 09 | 59 | 59 | 58 | 59 | 58 | 59 | 59 | 59 | 58 |

Sample No. 10 | 59 | 58 | 59 | 59 | 59 | 59 | 58 | 58 | 58 |

In fuzzy case study 3, each of the three inspectors measured the sample parts three times as shown in

After the data were collected, we used the average and range method in the MSA Reference Manual to calculate GR&R, giving a result of 0.276, and a %GR&R for the measurement system of 30.12%. The NDC of 4.463 was rounded to the nearest integer to produce a value of 4.

If MSA produces acceptable results, the measurement system is considered reliable. Conversely, unacceptable results indicate that the measurement system has room for improvement. For example, following MSA analysis, if it is discovered that unstable measurement instruments cause instability in the production process, it is not necessary to waste resources changing the production process. Similarly, when a high defect rate for a product is observed, and it is discovered that the reason is excessive EV, if the use of more sophisticated equipment reduces the defect rate or if MSA identifies product design flaws, design changes can be carried out [

Observation of the three cases in the study shows that %GR&R after fuzzification is higher than the original data (see

Sample | A | B | C | ||||||
---|---|---|---|---|---|---|---|---|---|

1^{st} | 2^{nd} | 3^{rd} | 1^{st} | 2^{nd} | 3^{rd} | 1^{st} | 2^{nd} | 3^{rd} | |

Sample No. 01 | 49 | 49 | 49 | 49 | 49 | 49 | 49 | 50 | 49 |

Sample No. 02 | 50 | 51 | 50 | 50 | 49 | 49 | 50 | 50 | 50 |

Sample No. 03 | 49 | 49 | 49 | 49 | 49 | 50 | 50 | 50 | 50 |

Sample No. 04 | 52 | 52 | 52 | 52 | 52 | 52 | 51 | 52 | 51 |

Sample No. 05 | 52 | 51 | 51 | 51 | 51 | 52 | 52 | 52 | 52 |

Sample No. 06 | 51 | 50 | 50 | 51 | 51 | 50 | 49 | 50 | 49 |

Sample No. 07 | 50 | 50 | 49 | 50 | 49 | 50 | 51 | 51 | 51 |

Sample No. 08 | 49 | 49 | 49 | 49 | 49 | 49 | 49 | 49 | 49 |

Sample No. 09 | 49 | 50 | 50 | 50 | 50 | 50 | 49 | 49 | 50 |

Sample No. 10 | 49 | 49 | 49 | 49 | 49 | 49 | 49 | 49 | 49 |

Case No. | Original % GR&R | Fuzzy % GR&R | Original NDC | Fuzzy NDC |
---|---|---|---|---|

Case 1 | 6.73% | 8.14% | 21 | 17 |

Case 2 | 28.89% | 32.73% | 5 | 4 |

Case 3 | 28.53% | 30.12% | 5 | 4 |

However, following fuzzification of the measurement values, %GR&R for the three cases was 8.14%, 32.73%, and 30.12%, respectively.

In practice, if %GR&R is less than 10%, it means that the measurement system is acceptable. If %GR&R is between 10% and 30%, it means that the measurement system is acceptable depending on the application, the cost of the measuring device, cost of repair, or other factors. If %GR%R is greater than 30%, it means that the measurement system is unacceptable and should be improved. In the first case, since the sample part was a key precision part, (as poor tolerance would cause downtime, and make it unusable to the operator), the company stipulates that %GR&R must be less than 10%. Although the value was higher following fuzzification, it was still within the specified range, 10%. The measurement system is acceptable.

In the second and third cases, the original %GR&R of 28.89% and 28.53%, respectively, fell within the edge 30% range, but after fuzzification, exceeded the original 30% range. This diametrically opposite result can act as a reference for assessors. The measurement system should be improved.

The original NDC values for the three cases were larger than 5 and within the specifications. However, fuzzification produced a fall in the NDC values, indicating that the analysis was unable to distinguish between categories with maximum variation. The NDC for the three cases was 21, 5, and 5, respectively. However, following fuzzification of the measurement values, the NDC for the three cases fell to 17, 4, and 4, respectively. In the first case, since the sample part was a key precision part, and while the NDC value was lower after fuzzification, it was still within the specified range. The NDC of 5 for both the second and third cases fell on the edge of the range of 5 or larger; however, following fuzzification it fell to outside of this range, producing a diametrically opposite result. This finding fits with customer complaints of insufficient hardness, and is an important reference for the company.

Finally, this study used the concept of fuzzy theory to test %GR&R and NDC for measurement systems. The results of MSA following fuzzification can help the companies to assess in more depth any uncertainty in measurement systems. In order to ensure that their products achieve good quality, companies need to have accurate measurement systems in place. At present, the industry mostly follows TS 16949 criteria to assess measurement systems and determine whether they are acceptable. Following fuzzification, the measurement systems in the second and third cases are not within an acceptable range and require improvement.

This study applies the existing literature to gather measurement data and uses fuzzy theory to process the results. The standards set out in the long form method in the MSA Reference Manual [

1) %GR&R comparison of measurement systems using the standards set out in the long form method in the MSA Reference Manual [

2) The measurement capability of the measurement system should be tested regularly. In order to ensure that the analysis results from a measurement system can be used, relevant criteria should be put in place. At the same time, these criterions can be used to grade the measurement capability of the measurement system, providing a reference to the production department when dispatching workers or introducing new products, and preventing the misuse of measuring equipment with poor measurement capability.

3) GR&R comparison of measurement systems using the standards set out in the long form method in the MSA Reference Manual [

4) This study believes that using the %GR&R set out in the MSA Reference Manual as the only criterion for the acceptability of the measurement systems is not an appropriate approach. Assessment of the measurement system should not be based on a single indicator or criterion. NDC for measurement tests under different conditions should also be applied to avoid false positives.