# Application of Multi-Criteria Optimization Methods in the Calibration Process of Digital Measuring Instruments

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## Abstract

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## 1. Introduction

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- Single-criterion—defining one function that describes a specific problem and finds its extreme.
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- Multi-criteria—finding the optimal solution that is appropriate from the point of view of each criterion.

## 2. Materials and Methods

#### 2.1. Calibration

#### 2.2. Optimization

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- Performance criteria (functional, aesthetic);
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- Technical criteria (general technical, manufacturing, material);
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- Economic criteria (production costs and time, operating costs) [35].

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- The criterion related to the measurement error $\left[f1\right(x\left)\right]$;
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- The criterion related to the measurement uncertainty $\left[f2\right(x\left)\right]$;
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- The criterion of the total time of execution of the calibration $\left[f3\right(x\left)\right]$.

#### 2.2.1. Criterion Related to the Measurement Error [${f}_{1}\left(x\right)$]

- E—measurement error;
- ${y}_{i}$—an indication of the measuring instrument (i symbolizes successive measuring points);
- ${y}_{k}$—value generated by the calibrator;
- $\delta {y}_{i}$—instrument reading correction due to finite resolution;
- $\delta {y}_{k}$—correction of the value generated by the calibrator, including the following factors:

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- Drift since the last calibrator calibration;
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- Changes caused by the effect of offset, non-linearity, and changes in the gain factor;
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- Changes in ambient temperature;
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- Supply voltage changes;
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- Load effect resulting from the finite input resistance of the calibrated multimeter.

- $\overline{{y}_{i}}\left(x\right)$—mean value from a series of x measurements.

- x—the number of measurements in series;
- ${y}_{i}$—the result of the i-th measurement;
- W—the mean value of the result from 30 measurements.

#### 2.2.2. Criterion Related to the Measurement Uncertainty $\left[{f}_{2}\left(x\right)\right]$

- A—those that have been calculated by statistical methods;
- B—those that have been calculated by other methods.

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- Data from previous measurements;
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- Experience and general knowledge of the behavior and properties of appropriate measuring instruments;
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- Manufacturer specifications;
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- Data obtained from calibration certificates or other certificates;
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- Uncertainties related to reference data obtained from the literature.

- $u\left({y}_{k}\right)$—the uncertainty related to the calibration of the standard, estimated on the basis of the records of the last calibration certificate;
- $u\left(\delta {y}_{i}\right)$—uncertainty related to the resolution of the calibrated multimeter;
- $u\left(\delta {y}_{k}\right)$—the uncertainty related to the factors affecting the quantity generated by the calibrator, estimated from the manufacturer’s data;
- $u\left(\overline{{y}_{i}}\left(x\right)\right)$—uncertainty related to the dispersion of the series of measurements.

- j—successive component of uncertainty;
- N—number of uncertainty components;
- c—sensitivity coefficient.

- p—confidence level.

#### 2.2.3. Criterion of the Total Time of Execution of the Calibration $\left[{f}_{3}\left(x\right)\right]$

- ${t}_{0}$—automatic calibration time with one measurement;
- ${t}_{s}$—time of the next measurement in the series;
- ${p}_{p}$—total number of measurement points.

#### 2.2.4. Multi-Criteria Optimization Methods

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- Weight objectives method;
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- Hierarchical optimization method;
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- Trade-off method;
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- Global criterion method;
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- Method of distance functions;
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- Min–max method;
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- Goal programming method.

## 3. Result

#### 3.1. Weight Objectives Method

- k—number of objective criteria;
- x—number of measurements in series;
- ${w}_{i}$—weights such that:

- ${f}_{i}^{0}=-min{f}_{i}\left(x\right),\phantom{\rule{0.222222em}{0ex}}when-min{f}_{i}\left(x\right)\u2a7emax{f}_{i}\left(x\right)$;
- ${f}_{i}^{0}=max{f}_{i}\left(x\right),\phantom{\rule{0.222222em}{0ex}}when-min{f}_{i}\left(x\right)<max{f}_{i}\left(x\right)$;

#### 3.2. Method of Distance Functions, Min–Max Method

- $x=6$—for the weight objectives method;
- $x=6$—for the min–max method;
- $x=7$—for the distance function method.

## 4. Practice

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- Weight objectives method;
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- Min–max method;
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- Distance function method.

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- Shortening the calibration time;
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- Improving the quality of the results obtained.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Measurement with a digital caliper [34].

**Figure 2.**Three measurement results of the same value [34].

**Figure 4.**Optimal solutions in the weighted criteria method for a simulated series of 1000 sets of input data.

**Figure 7.**Optimal solutions with the distance function method for a simulated series of 1000 sets of input data.

**Table 1.**Series of 10 measurements performed with the FLUKE 27 multimeter connected to the FLUKE 5500 calibrator for 40 V DC.

Measurement | Series 1 | Series 2 | Series 3 | Series 4 | Series 5 |
---|---|---|---|---|---|

1 | 40.1 | 40.1 | 40.1 | 40.1 | 40.0 |

2 | 40.0 | 40.1 | 40.1 | 40.0 | 40.1 |

3 | 40.1 | 40.1 | 40.0 | 40.1 | 40.0 |

4 | 40.0 | 40.1 | 40.0 | 40.0 | 40.1 |

5 | 40.1 | 40.1 | 40.1 | 40.1 | 40.0 |

6 | 40.1 | 40.1 | 40.1 | 40.1 | 40.1 |

7 | 40.0 | 40.0 | 40.1 | 40.0 | 40.0 |

8 | 40.0 | 40.1 | 40.0 | 40.0 | 40.1 |

9 | 40.1 | 40.1 | 40.1 | 40.0 | 40.0 |

10 | 40.1 | 40.0 | 40.1 | 40.0 | 40.1 |

Average | 40.06 | 40.08 | 40.07 | 40.04 | 40.05 |

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**MDPI and ACS Style**

Klebba, M.; Adamczyk, A.; Wąż, M.; Iwen, D.
Application of Multi-Criteria Optimization Methods in the Calibration Process of Digital Measuring Instruments. *Sensors* **2023**, *23*, 2984.
https://doi.org/10.3390/s23062984

**AMA Style**

Klebba M, Adamczyk A, Wąż M, Iwen D.
Application of Multi-Criteria Optimization Methods in the Calibration Process of Digital Measuring Instruments. *Sensors*. 2023; 23(6):2984.
https://doi.org/10.3390/s23062984

**Chicago/Turabian Style**

Klebba, Maciej, Arkadiusz Adamczyk, Mariusz Wąż, and Dominik Iwen.
2023. "Application of Multi-Criteria Optimization Methods in the Calibration Process of Digital Measuring Instruments" *Sensors* 23, no. 6: 2984.
https://doi.org/10.3390/s23062984