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The stringent customer demands and competitive market emphasize the importance of efficient and effective inspection in industrial metrology. Therefore, the implementation of an appropriate sampling strategy, i.e., the number of points and their distribution, has become very important in the inspection process using a coordinate measuring machine. Moreover, the quality of inspection results has frequently been influenced by sampling plan as well as workpiece size and surface characteristics. It has been an indispensable problem in the present-day measurement processes. Thus, this paper investigates various sample sizes and different point distribution algorithms that can be employed in the evaluation of form error. The effect of specimen size and surface quality on the sampling strategy has also been investigated. Furthermore, this work employs a fuzzy based Technique for Order Performance by Similarity to Ideal Solution approach to realize the best sampling strategy. The results have demonstrated the significance of robust optimization techniques as well as the importance of a suitable sampling strategy in coordinate metrology. This study has also established that Poisson point distribution achieved the best accuracy and the Grid point distribution had taken the least measurement time.

Coordinate Measuring Machine (CMM) has turned up as a leading technology to accomplish inspection in manufacturing industries. This can be attributed to its high precision and accuracy. Indeed, the large-scale benefits of CMM inspection are evident only in the presence of an explicit and well-planned inspection strategy. The irregular market requirements, as well as the needs of higher productivity, have emphasized the importance of efficient and effective inspection plan in CMM [

The sampling strategy involves not only the selection of sample size or the number of measurement points but also their appropriate distribution on the measuring surface [

The principal objective of this work focused on producing a user-friendly, manageable, and responsive sampling plan. For this purpose, different accessible sampling strategies based on existing sequences and distributions were investigated in the assessment of the flatness error. The different sample sizes, as well as diverse point distribution algorithms, were explored. The algorithms considered in this study were Hammersley, Halton-Zaremba, Poisson, Random, and Grid. The test specimens with variable surface characteristics were fabricated to determine the effect of different manufacturing signatures on the sampling strategy. Moreover, the parts with the different areas were utilized to examine the influence of part size on the sampling strategy. Finally, the Fuzzy Technique for Order Performance by Similarity to Ideal Solution (FTOPSIS) was applied to derive a robust and compliant sampling plan for the surface of the given size and quality. This problem was modeled as a multiresponse optimization problem with two responses, the accuracy and the measurement time.

The sampling strategy can be defined as a plan designed to achieve appropriate sample size and precise point locations on the geometric feature being inspected using CMM [

The implementation of a relevant sampling strategy for an enhanced CMM performance is a demanding job. In spite of the existence of abundant approaches for specifying sampling strategy, their application is constrained owing to part intricacy, the shape of the object, etc. Correspondingly, more work is needed to devise sampling strategies for the measurement of manufactured parts adequately and efficiently. According to Bosch [

The blind sampling strategies can be defined as the methods which identify the point locations based on standard sequences, canonical distributions, user’s experience or knowledge, etc. For instance, the point allocation may include strategies based on Hammersley sequence, Halton-Zaremba series, Stratified based allocation, Poisson distribution, Random sampling, etc. Although they are not as effective as adaptive strategies, but they can be valuable in the inspection of prismatic and low curvature parts. It is because they are computationally less expensive and can save significant processing time. If the manufacturing process is known to fabricate components with precision, it is always convenient to implement a sampling plan based on existing distributions. However, the execution of the blind sampling strategies necessitates a proper combination of the number of points and the corresponding distribution algorithm. The workpieces with well-known manufacturing signatures can also be inspected effectively, provided appropriate sample size is combined with a suitable point allocation algorithm. Certainly, the choice for the point distribution algorithm is also critical because they are based on different assumptions and behave diversely in different applications. Therefore, prior knowledge and understanding of these algorithms are crucial to manage their performance. The correct assessment of the sample size for the corresponding allocation on the part surface has always been a mighty assignment to metrologists. Furthermore, the paucity of standard methods and systematic studies make this task more vigorous for the engineers [

Based on the preceding discussion, there is a need to explore the applications of different standard sequences, existing distributions, etc., in sampling strategies. It is because they are easy to use and interpret, relatively simple, highly accessible, etc. However, the performance of these available sequences or distributions needs to be enhanced using an appropriate sample size. A large number of points would unnecessarily increase the inspection time and insufficient points would degrade the measurement accuracy. It mandates the need for an optimum combination of sample size and points allocation algorithm depending on the quality and size of the surface being measured. It can be realized that a befitting consolidation of the number of points and their distribution has the competency to provide economic and accurate results. It advocates the need for an optimization approach that must consider the associated uncertainties and determine the relevant sampling plan. Henceforth, the FTOPSIS has been employed in this research to acknowledge its pertinence for acquiring a suitable sampling scheme in coordinate metrology.

In this section, different sampling methods are briefly described. The simplest methods to distribute sample points on the measuring surface are Random and Grid point distribution methods. The random point distribution strategy haphazardly allocates points on the inspection feature. Although it is easy to implement, most often, it is inconsistent and inadequate as it neglects many portions on the measuring surface, resulting in a cluster of points in some regions and negligible points in the other. The inconsistency in the random sampling method may have been induced due to the random point generation process. On the other hand, Grid point distribution is systematic because it distributes the sampling points on the measuring surface in a regular pattern and is one of the most accepted methods. It comprises equally spaced straightness contours in two orthogonal directions to generate a rectangular grid [

where

Note that the value of

These sequences can analyze the measuring surface more efficiently than the Random and Grid point distributions. They are especially useful to determine the set of points, which are more significant in the accurate evaluation of the form tolerance. The limitations of the Grid and Random sampling strategies can also be overcome by the Poisson point distribution method. It results in a more uniform distribution than the Random point distribution and is independent of any periodic behavior like Grid point distribution. It allocates the sampling points in such a way that each of the two points is at least at a specified minimum distance apart [

Initialize an

Choose the initial sample,

If the active list is not empty, select a random index from it (say

The implementation of a proper sampling strategy in CMM inspection is a complicated task due to the presence of numerous uncontrollable factors. These factors instill ambiguity, uncertainty, and vagueness as well as making the determination of appropriate sampling strategy a cumbersome task. Generally, the metrologists decide the sampling strategy depending on intuition or experience, which is not a convenient approach if a trade-off between inspection accuracy and measurement time is desired. Therefore, the concept of the fuzzy set theory developed by Zadeh, 1965 [

A simple and an effective vertex method was developed by Chen [

The implementation of a suitable sampling strategy is a complicated problem involving numerous uncertain factors and imprecise information. Therefore, the consideration of linguistic terms is quite helpful [

In this paper, the linguistic variables are represented using trapezoidal fuzzy numbers owing to their effectiveness, simplicity, ease of application, and general acceptance. In other words, it can be asserted that the decision matrix (or the response matrix) has been described by utilizing trapezoidal fuzzy numbers. The membership function of trapezoidal fuzzy number designated by

The normalized fuzzy decision matrix can be represented as

Positive response: the higher the value of the response, the better will be the performance,

Negative response: the lower the value of the response, the better will be the performance,

The weighted normalized decision matrix assigns importance to different responses (or attributes) depending on the requirements. This matrix is calculated by multiplying the normalized decision matrix with the weights designated to various responses as follows.

The FPIS and FNIS are computed as follows.

The alternative with highest closeness coefficient represents the best alternative.

If

The primary goal of this work was the investigation of different sampling strategies as well as the determination of a suitable sampling plan using FTOPSIS for CMM. Five different types of point distribution algorithms, including Hammersley, Halton-Zaremba, Poisson, Random, and Grid, were employed in this study. These algorithms were chosen because of their simplicity, accessibility, convenience, and ease of application. The sample sizes preferred for the experiments were 22, 45, 90, 180, 360, and 720, depending on the time and cost involved in the experiments. The manufacturing signatures play a significant role in the selection of a sampling strategy because they can be traced back to the manufacturing method [^{2}.

The flatness was measured using a commercial CMM (Zeiss ACCURA, 900x1200x700 mm^{3}) with a tactile probe. The part setup and the CMM can be seen in Figure

Measurement of test specimen using a CMM.

Consequently, the coordinates of measurement points were generated using different algorithms in Matlab program. The coordinates were independently generated for each region, i.e., 90x90, 45x45, and 22.5x22.5 mm^{2}. Figure ^{2}.

Coordinates generated using different algorithms for the measurement region of 90x90 mm^{2}.

The generated ^{2} respectively. The inspection time was the time recorded for the inspection path obtained through the coordinates generated from the given sampling strategy.

Subsequently, the FTOPSIS was implemented to obtain the appropriate sampling policy. Henceforth, the different intervals (or the range) (as shown in Table

Linguistic variable and corresponding fuzzy numbers for inspection regions: (a) 90x90 mm^{2}; (b) 45x45 mm^{2}; and (c) 22.5x22.5 mm^{2}.

Variable | Linguistic values | Range | Fuzzy numbers |
---|---|---|---|

CA | Very Low | ≤ 0.001 | (0.0001, 0.0003, 0.0006, 0.001) |

Low | 0.0011 - 0.010 | (0.0011, 0.00407, 0.00704, 0.01) | |

Middle | 0.011-0.025 | (0.011, 0.0157, 0.0203, 0.025) | |

High | 0.0251-0.05 | (0.0251, 0.0334, 0.0417, 0.05) | |

Very High | > 0.05 | (0.051, 0.367, 0.683, 1) | |

| |||

Measurement time | Very Low | ≤ 1 | (0.1, 0.3, 0.7, 1) |

Low | 1.1-8 | (1.01, 3.33, 5.66, 8) | |

Middle | 8.1-15 | (8.01, 10.4, 12.7, 15) | |

High | 15.1-30 | (15.01, 20.07, 25.04, 30) | |

Very High | > 30 | (30.01, 36.73, 43.36, 50) |

Variable | Linguistic values | Range | Fuzzy numbers |
---|---|---|---|

CA | Very Low | ≤ 0.001 | (0.0001, 0.0003, 0.0006, 0.001) |

Low | 0.0011 - 0.01 | (0.0011, 0.00407, 0.00704, 0.01) | |

Middle | 0.011-0.015 | (0.011, 0.0123, 0.0136, 0.015) | |

High | 0.0151-0.0200 | (0.0151, 0.0167, 0.0184, 0.02) | |

Very High | > 0.020 | (0.021, 0.347, 0.673, 1) | |

| |||

Measurement time | Very Low | ≤ 1 | (0.1, 0.3, 0.7, 1) |

Low | 1.01-5 | (1.01, 2.34, 3.67, 5) | |

Middle | 5.01-10 | (5.01, 6.673, 8.336, 10) | |

High | 10.01-20 | (10.01, 13.34, 16.67, 20) | |

Very High | > 20 | (20.01, 30, 39.99, 50) |

Variable | Linguistic values | Range | Fuzzy numbers |
---|---|---|---|

CA | Very Low | ≤ 0.001 | (0.0001, 0.0003, 0.0006, 0.001) |

Low | 0.0011 - 0.0050 | (0.0011, 0.0024, 0.0037, 0.005) | |

Middle | 0.0051-0.015 | (0.0051, 0.0084, 0.0117, 0.015) | |

High | 0.0151-0.0200 | (0.0151, 0.0167, 0.0184, 0.02) | |

Very High | > 0.020 | (0.021, 0.347, 0.673, 1) | |

| |||

Measurement time | Very Low | ≤ 1 | (0.1, 0.3, 0.7, 1) |

Low | 1.01-3 | (1.01, 1.67, 2.33, 3) | |

Middle | 3.01-8 | (3.01, 4.673, 6.336, 8) | |

High | 8.01-15 | (8.01, 10.34, 12.67, 15) | |

Very High | > 15 | (15.01, 26.673, 38.336, 50) |

Figures

CA of algorithms for different surfaces on inspection region 90x90 mm^{2}.

CA of algorithms for different surfaces on inspection region 45x45 mm^{2}.

CA of algorithms for different surfaces on inspection region 22.5x22.5 mm^{2}.

It can be observed in these trends that Poisson point distribution has achieved the minimum CA. It means that it has achieved better accuracy as compared to other algorithms. The better performance of Poisson point distribution can be attributed to its characteristics of both randomness and systemization. It was also realized that Poisson point distribution possessed a higher CA at a lower sample size, but it decreased steeply as the number of points increased.

The Poisson point distribution algorithm was followed by the Hammersley and Halton-Zaremba point distributions in terms of CA. The CA for Hammersley algorithm improved with an increase in the sample size; however, at lower sample sizes Halton-Zaremba algorithm performed better than the Hammersley point distribution. Although the Grid point distribution has also shown significant improvement, its CA was inferior to other considered algorithms.

The subsequent step was the determination of inspection time using the coordinates obtained from different algorithms. As shown in Figure

Measurement time for different inspection regions: (a) 90x90 mm^{2}; (b) 45x45 mm^{2}; and (c) 22.5x22.5 mm^{2}.

Table ^{2}. In contrast, enhancement of 55.34%, 31.64%, 39.92%, and 51.48% was noticed with Hammersley, Halton-Zaremba, Random, and Grid point distributions, respectively, when the number of points increased from 22 to 720.

Relationship between CA and sample size.

Size (mm^{2}) | Surface | Percentage improvement in CA | ||||
---|---|---|---|---|---|---|

Hammersley | Halton-Zaremba | Poisson | Random | Grid | ||

90x90 | 1 | 61.50 | 54.85 | 85.30 | 96.18 | 53.55 |

2 | 55.34 | 31.64 | 65.25 | 39.92 | 51.48 | |

3 | 96.47 | 92.59 | 99.88 | 96.17 | 95.14 | |

| ||||||

45x45 | 1 | 77.23 | 71.98 | 74.82 | 66.10 | 73.39 |

2 | 80.88 | 79.76 | 90.06 | 74.11 | 79.05 | |

3 | 94.48 | 88.67 | 96.71 | 97.93 | 90.11 | |

| ||||||

22.5x22.5 | 1 | 79.06 | 71.03 | 82.51 | 83.63 | 65.45 |

2 | 92.59 | 80.51 | 96.20 | 81.64 | 80.95 | |

3 | 89.07 | 85.44 | 86.86 | 79.07 | 74.43 |

The investigation also revealed a significant effect of specimen size on the flatness error. As observed in Figure

Effect of specimen size on the form error.

Moreover, it is shown in Figure

Influence of different surfaces on the form error.

After analyzing the performance of different algorithms, the FTOPSIS was implemented and the following results were obtained for surface 1 with inspection region 90x90 mm^{2}. This problem was considered as a multiresponse optimization problem with CA and measurement time as the two responses. Tables

Conversion of gathered data into trapezoidal fuzzy numbers.

Parameters | CA (mm) | Time taken (min) | Fuzzy numbers | ||
---|---|---|---|---|---|

Sample size | Point distribution | CA | Measurement time | ||

22 | Hammersley | 0.0387 | 0.9500 | (0.0251, 0.0334, 0.0417, 0.05) | (0.1, 0.3, 0.7, 1) |

45 | 0.0371 | 2.0833 | (0.0251, 0.0334, 0.0417, 0.05) | (1.01, 3.33, 5.66, 8) | |

90 | 0.0300 | 4.1500 | (0.0251, 0.0334, 0.0417, 0.05) | (1.01, 3.33, 5.66, 8) | |

180 | 0.0253 | 8.3167 | (0.0251, 0.0334, 0.0417, 0.05) | (8.01, 10.4, 12.7, 15) | |

360 | 0.0228 | 16.6500 | (0.011, 0.0157, 0.0203, 0.025) | (15.01, 20.07, 25.04, 30) | |

720 | 0.0149 | 33.1667 | (0.011, 0.0157, 0.0203, 0.025) | (30.01, 36.73, 43.36, 50) | |

| |||||

22 | Halton-Zaremba | 0.0392 | 1.3000 | (0.0251, 0.0334, 0.0417, 0.05) | (1.01, 3.33, 5.66, 8) |

45 | 0.0334 | 2.0333 | (0.0251, 0.0334, 0.0417, 0.05) | (1.01, 3.33, 5.66, 8) | |

90 | 0.0282 | 4.0667 | (0.0251, 0.0334, 0.0417, 0.05) | (1.01, 3.33, 5.66, 8) | |

180 | 0.0256 | 8.1967 | (0.0251, 0.0334, 0.0417, 0.05) | (8.01, 10.4, 12.7, 15) | |

360 | 0.0248 | 16.4333 | (0.011, 0.0157, 0.0203, 0.025) | (15.01, 20.07, 25.04, 30) | |

720 | 0.0177 | 33.3667 | (0.011, 0.0157, 0.0203, 0.025) | (30.01, 36.73, 43.36, 50) | |

| |||||

22 | Poisson | 0.0415 | 1.2000 | (0.0251, 0.0334, 0.0417, 0.05) | (1.01, 3.33, 5.66, 8) |

45 | 0.0303 | 1.9667 | (0.0251, 0.0334, 0.0417, 0.05) | (1.01, 3.33, 5.66, 8) | |

90 | 0.0292 | 3.6667 | (0.0251, 0.0334, 0.0417, 0.05) | (1.01, 3.33, 5.66, 8) | |

180 | 0.0244 | 6.9667 | (0.011, 0.0157, 0.0203, 0.025) | (1.01, 3.33, 5.66, 8) | |

360 | 0.0182 | 15.1167 | (0.011, 0.0157, 0.0203, 0.025) | (15.01, 20.07, 25.04, 30) | |

720 | 0.0061 | 30.0500 | (0.0011, 0.00407, 0.00704, 0.01) | (30.01, 36.73, 43.36, 50) | |

| |||||

22 | Random | 0.0419 | 1.2667 | (0.0251, 0.0334, 0.0417, 0.05) | (1.01, 3.33, 5.66, 8) |

45 | 0.0370 | 2.0667 | (0.0251, 0.0334, 0.0417, 0.05) | (1.01, 3.33, 5.66, 8) | |

90 | 0.0349 | 4.0500 | (0.0251, 0.0334, 0.0417, 0.05) | (1.01, 3.33, 5.66, 8) | |

180 | 0.0276 | 8.1333 | (0.0251, 0.0334, 0.0417, 0.05) | (8.01, 10.4, 12.7, 15) | |

360 | 0.0211 | 16.1333 | (0.011, 0.0157, 0.0203, 0.025) | (15.01, 20.07, 25.04, 30) | |

720 | 0.0016 | 32.8833 | (0.0011, 0.00407, 0.00704, 0.01) | (30.01, 36.73, 43.36, 50) | |

| |||||

22 | Grid | 0.0394 | 1.1667 | (0.0251, 0.0334, 0.0417, 0.05) | (1.01, 3.33, 5.66, 8) |

45 | 0.0352 | 1.8167 | (0.0251, 0.0334, 0.0417, 0.05) | (1.01, 3.33, 5.66, 8) | |

90 | 0.0325 | 3.3333 | (0.0251, 0.0334, 0.0417, 0.05) | (1.01, 3.33, 5.66, 8) | |

180 | 0.0290 | 6.5333 | (0.0251, 0.0334, 0.0417, 0.05) | (1.01, 3.33, 5.66, 8) | |

360 | 0.0272 | 15.0167 | (0.0251, 0.0334, 0.0417, 0.05) | (15.01, 20.07, 25.04, 30) | |

720 | 0.0183 | 26.0833 | (0.011, 0.0157, 0.0203, 0.025) | (15.01, 20.07, 25.04, 30) |

Weighted normalized decision matrix.

Weighted normalized fuzzy decision matrix | |||||||
---|---|---|---|---|---|---|---|

CA | Measurement time | ||||||

a_{1} | b_{1} | c_{1} | d_{1} | a_{2} | b_{2} | c_{2} | d_{2} |

0.011 | 0.01319 | 0.01647 | 0.02191 | 0.05 | 0.07496 | 0.15015 | 0.5 |

0.011 | 0.01319 | 0.01647 | 0.02191 | 0.00625 | 0.00883 | 0.01502 | 0.0495 |

0.011 | 0.01319 | 0.01647 | 0.02191 | 0.00625 | 0.00883 | 0.01502 | 0.0495 |

0.011 | 0.01319 | 0.01647 | 0.02191 | 0.00333 | 0.00394 | 0.00481 | 0.00624 |

0.022 | 0.02709 | 0.03503 | 0.05 | 0.00167 | 0.002 | 0.00249 | 0.00333 |

0.022 | 0.02709 | 0.03503 | 0.05 | 0.001 | 0.00115 | 0.00136 | 0.00167 |

0.011 | 0.01319 | 0.01647 | 0.02191 | 0.00625 | 0.00883 | 0.01502 | 0.0495 |

0.011 | 0.01319 | 0.01647 | 0.02191 | 0.00625 | 0.00883 | 0.01502 | 0.0495 |

0.011 | 0.01319 | 0.01647 | 0.02191 | 0.00625 | 0.00883 | 0.01502 | 0.0495 |

0.011 | 0.01319 | 0.01647 | 0.02191 | 0.00333 | 0.00394 | 0.00481 | 0.00624 |

0.022 | 0.02709 | 0.03503 | 0.05 | 0.00167 | 0.002 | 0.00249 | 0.00333 |

0.022 | 0.02709 | 0.03503 | 0.05 | 0.001 | 0.00115 | 0.00136 | 0.00167 |

0.011 | 0.01319 | 0.01647 | 0.02191 | 0.00625 | 0.00883 | 0.01502 | 0.0495 |

0.011 | 0.01319 | 0.01647 | 0.02191 | 0.00625 | 0.00883 | 0.01502 | 0.0495 |

0.011 | 0.01319 | 0.01647 | 0.02191 | 0.00625 | 0.00883 | 0.01502 | 0.0495 |

0.022 | 0.02709 | 0.03503 | 0.05 | 0.00625 | 0.00883 | 0.01502 | 0.0495 |

0.022 | 0.02709 | 0.03503 | 0.04583 | 0.00167 | 0.002 | 0.00249 | 0.00333 |

0.055 | 0.07813 | 0.13514 | 0.5 | 0.001 | 0.00115 | 0.00136 | 0.00167 |

0.011 | 0.01319 | 0.01647 | 0.02191 | 0.00625 | 0.00883 | 0.01502 | 0.0495 |

0.011 | 0.01319 | 0.01647 | 0.02191 | 0.00625 | 0.00883 | 0.01502 | 0.0495 |

0.011 | 0.01319 | 0.01647 | 0.02191 | 0.00625 | 0.00883 | 0.01502 | 0.0495 |

0.011 | 0.01319 | 0.01647 | 0.02191 | 0.00333 | 0.00394 | 0.00481 | 0.00624 |

0.022 | 0.02709 | 0.03503 | 0.05 | 0.00167 | 0.002 | 0.00249 | 0.00333 |

0.055 | 0.07813 | 0.13514 | 0.5 | 0.001 | 0.00115 | 0.00136 | 0.00167 |

0.011 | 0.01319 | 0.01647 | 0.02191 | 0.00625 | 0.00883 | 0.01502 | 0.0495 |

0.011 | 0.01319 | 0.01647 | 0.02183 | 0.00625 | 0.00883 | 0.01502 | 0.0495 |

0.011 | 0.01319 | 0.01647 | 0.02174 | 0.00625 | 0.00883 | 0.01502 | 0.0495 |

0.011 | 0.01319 | 0.01647 | 0.02165 | 0.00625 | 0.00883 | 0.01502 | 0.0495 |

0.011 | 0.01319 | 0.01647 | 0.02157 | 0.00167 | 0.002 | 0.00249 | 0.00333 |

0.022 | 0.02709 | 0.03503 | 0.05 | 0.00167 | 0.002 | 0.00249 | 0.00333 |

FPIS and FNIS.

FPIS | FNIS |
---|---|

| |

(0.5, 0.5, 0.5, 0.5) | (0.011, 0.011, 0.011, 0.011) |

(0.0495, 0.0495, 0.0495, 0.0495) | (0.00625, 0.00625, 0.00625, 0.00625) |

(0.0495, 0.0495, 0.0495, 0.0495) | (0.00625, 0.00625, 0.00625, 0.00625) |

(0.02191, 0.02191, 0.02191, 0.02191) | (0.0801, 0.0801, 0.0801, 0.0801) |

(0.05, 0.05, 0.05, 0.05) | (0.00167, 0.00167, 0.00167, 0.00167) |

(0.05, 0.05, 0.05, 0.05) | (0.001, 0.001, 0.001, 0.001) |

(0.0495, 0.0495, 0.0495, 0.0495) | (0.00625, 0.00625, 0.00625, 0.00625) |

(0.0495, 0.0495, 0.0495, 0.0495) | (0.00625, 0.00625, 0.00625, 0.00625) |

(0.0495, 0.0495, 0.0495, 0.0495) | (0.00625, 0.00625, 0.00625, 0.00625) |

(0.02191, 0.02191, 0.02191, 0.02191) | (0.00333, 0.00333, 0.00333, 0.00333) |

(0.05, 0.05, 0.05, 0.05) | (0.00167, 0.00167, 0.00167, 0.00167) |

(0.05, 0.05, 0.05, 0.05) | (0.001, 0.001, 0.001, 0.001) |

(0.0495, 0.0495, 0.0495, 0.0495) | (0.00625, 0.00625, 0.00625, 0.00625) |

(0.0495, 0.0495, 0.0495, 0.0495) | (0.00625, 0.00625, 0.00625, 0.00625) |

(0.0495, 0.0495, 0.0495, 0.0495) | (0.00625, 0.00625, 0.00625, 0.00625) |

(0.05, 0.05, 0.05, 0.05) | (0.00625, 0.00625, 0.00625, 0.00625) |

(0.04583, 0.04583, 0.04583, 0.04583) | (0.00167, 0.00167, 0.00167, 0.00167) |

(0.5, 0.5, 0.5, 0.5) | (0.001, 0.001, 0.001, 0.001) |

(0.0495, 0.0495, 0.0495, 0.0495) | (0.00625, 0.00625, 0.00625, 0.00625) |

(0.0495, 0.0495, 0.0495, 0.0495) | (0.00625, 0.00625, 0.00625, 0.00625) |

(0.0495, 0.0495, 0.0495, 0.0495) | (0.00625, 0.00625, 0.00625, 0.00625) |

(0.02191, 0.02191, 0.02191, 0.02191) | (0.00333, 0.00333, 0.00333, 0.00333) |

(0.05, 0.05, 0.05, 0.05) | (0.00167, 0.00167, 0.00167, 0.00167) |

(0.5, 0.5, 0.5, 0.5) | (0.001, 0.001, 0.001, 0.001) |

(0.0495, 0.0495, 0.0495, 0.0495) | (0.00625, 0.00625, 0.00625, 0.00625) |

(0.0495, 0.0495, 0.0495, 0.0495) | (0.00625, 0.00625, 0.00625, 0.00625) |

(0.0495, 0.0495, 0.0495, 0.0495) | (0.00625, 0.00625, 0.00625, 0.00625) |

(0.0495, 0.0495, 0.0495, 0.0495) | (0.00625, 0.00625, 0.00625, 0.00625) |

(0.02157, 0.02157, 0.02157, 0.02157) | (0.00167, 0.00167, 0.00167, 0.00167) |

(0.05, 0.05, 0.05, 0.05) | (0.00167, 0.00167, 0.00167, 0.00167) |

Computation of distances and closeness coefficients.

| | | |
---|---|---|---|

0.8399 | 0.2632 | 1.1030 | 0.2386 |

0.0684 | 0.0324 | 0.1008 | 0.3210 |

0.0684 | 0.0324 | 0.1008 | 0.3210 |

0.0249 | 0.0146 | 0.0395 | 0.3705 |

0.0672 | 0.0345 | 0.1017 | 0.3393 |

0.0683 | 0.0346 | 0.1029 | 0.3363 |

0.0684 | 0.0324 | 0.1008 | 0.3210 |

0.0684 | 0.0324 | 0.1008 | 0.3210 |

0.0684 | 0.0324 | 0.1008 | 0.3210 |

0.0249 | 0.0146 | 0.0395 | 0.3705 |

0.0672 | 0.0345 | 0.1017 | 0.3393 |

0.0683 | 0.0346 | 0.1029 | 0.3363 |

0.0684 | 0.0324 | 0.1008 | 0.3210 |

0.0684 | 0.0324 | 0.1008 | 0.3210 |

0.0684 | 0.0324 | 0.1008 | 0.3210 |

0.0543 | 0.0514 | 0.1057 | 0.4860 |

0.0596 | 0.0331 | 0.0926 | 0.3569 |

0.8555 | 0.2630 | 1.1185 | 0.2351 |

0.0684 | 0.0324 | 0.1008 | 0.3210 |

0.0684 | 0.0324 | 0.1008 | 0.3210 |

0.0684 | 0.0324 | 0.1008 | 0.3210 |

0.0249 | 0.0146 | 0.0395 | 0.3705 |

0.0672 | 0.0345 | 0.1017 | 0.3393 |

0.8555 | 0.2630 | 1.1185 | 0.2351 |

0.0684 | 0.0324 | 0.1008 | 0.3210 |

0.0685 | 0.0323 | 0.1008 | 0.3207 |

0.0685 | 0.0323 | 0.1008 | 0.3204 |

0.0685 | 0.0323 | 0.1008 | 0.3202 |

0.0264 | 0.0154 | 0.0418 | 0.3681 |

0.0672 | 0.0345 | 0.1017 | 0.3393 |

As shown in the above computations, the combination of sample size 180 and a Poisson point distribution algorithm provided superior results for surface 1 with a size 90 x 90 mm^{2}. Table

Sampling schemes obtained using FTOPSIS.

Surface | Size (mm^{2}) | Sampling strategy | CA (mm) | Measurement time (min) | |
---|---|---|---|---|---|

Sample size | Point distribution algorithm | ||||

1 | 90 x 90 | 180 | Poisson | 0.0244 | 6.9667 |

45 x 45 | 90 | Poisson | 0.0198 | 3.6167 | |

90 | Random | 0.0194 | 3.4167 | ||

22.5 x 22.5 | 45 | Poisson | 0.0177 | 1.6500 | |

| |||||

2 | 90 x 90 | 360 | Poisson | 0.0493 | 14.1667 |

45 x 45 | 360 | Poisson | 0.0198 | 14.0376 | |

22.5 x 22.5 | 180 | Poisson | 0.0170 | 6.6904 | |

180 | Random | 0.0187 | 6.9167 | ||

| |||||

3 | 90 x 90 | 720 | Poisson | 0.0008 | 32.1236 |

45 x 45 | 360 | Poisson | 0.0197 | 14.2667 | |

22.5 x 22.5 | 360 | Poisson | 0.0191 | 13.1333 | |

360 | Random | 0.0200 | 13.7833 |

A quintessential sampling strategy in CMM inspection should produce meticulous measurements in lesser time. In this work, different point distribution algorithms, varying number of inspection points, and distinctive specimen sizes and surfaces have been investigated to accomplish the highest accuracy in lesser inspection time. There can be uncertainty sources which cause unreliable selection of sampling scheme. Henceforth, the appropriate selection of a sampling strategy requires a technique such as FTOPSIS which assumes fuzziness and uncertainty existed in the system. Correspondingly, this work discusses the applicability of FTOPSIS in the optimization of the sampling strategy. The FTOPSIS optimization technique certainly systemized the optimization of sampling plan with multiple responses. For instance, it has recommended a higher sample size and a Poisson point distribution algorithm (predominantly) for large and poor quality surfaces. However, the optimum results may differ in case the effects of the process generating the surface are considered. It means that the FTOPSIS will adapt the sampling plan, i.e., modify the sample size and point distribution algorithm depending on the degree of surface curvature and the condition of the surface being measured.

From the exploratory analysis, it was revealed that different algorithms have shown varying performance in terms of accuracy. The Poisson point distribution has demonstrated superior performance among the different studied algorithms. Nevertheless, the accuracy in the case of all algorithms improved with an increase in the number of inspection points. Moreover, it was noticed that the Halton-Zaremba algorithm had performed better than the Hammersley point distribution at a lower number of inspection points. However, with an increase in the sample size, the performance of Hammersley was better than the Halton-Zaremba algorithm. Besides, the percentage improvement in accuracy with sample size was also explored. For example, a minimum of 65% improvement in accuracy can be expected with the Poisson point distribution algorithm, when the sample size is increased from 22 to 720. Also, the measurement time was also considered to establish the efficiency of different algorithms. The coordinates obtained using the Grid point distribution had taken the lowest inspection time owing to its entirely systematic pattern. The Poisson point distribution was efficient but was on the higher side as compared to the Grid point distribution because of the presence of randomness element. The Hammersley and Halton-Zaremba algorithms had taken the highest inspection time.

The best performance of Poisson point distribution in terms of both accuracy and inspection time can be credited to its properties of volatility as well as systemization. It can be established that a Poisson point distribution with appropriate sample size is generally useful in components with prismatic features and low curvatures as well as in surfaces characterized by a regular pattern of form errors. Its significance can be attributed to its many benefits, such as being easily accessible and economical both in terms of cost and time, as well as involving lesser computation. Furthermore, its performance which may deteriorate due to the influence of part manufacturing method can be sustained through various ingenious means. For example, the grid-wise Poisson point distribution can be utilized, especially in higher degree curvature surfaces or surfaces with distinct manufacturing defects or patterns. In this approach, the surface would first be divided into grids depending on the curvature profile or surface conditions and then individual grids would be filled with sample points according to Poisson point distribution. In the future, this work will be extended to utilize and adapt the Poisson point distribution for complex free form surfaces. Finally, it can be asserted that the outcomes from this analysis can act as a guide for metrology experts or operators who require a sampling strategy for effective and efficient CMM inspection.

The data used to support the findings of this study are included within the article.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The authors are grateful to the Raytheon Chair for Systems Engineering for funding.