Saif Aati

Ewelina Rupnik

Samir Nejim

COMPARATIVE STUDY OF PHOTOGRAMMETRY SOFTWARE IN INDUSTRIAL FIELD

Keywords: photogrammétrie industrielle, logiciel de photogrammétrie, nuage de points, vision par ordinateur Industrial photogrammetry, photogrammetry software, point cloud, computer vision

Cet article présente une étude comparative entre différent logiciels de photogrammétrie capables de traiter automatiquement plusieurs images pour générer un nuage de points dense dans le domaine industriel. Dans le cadre d'un projet mené dans le l'entreprise Urbica 3D Laser Scanning Survey, quatre jeux de données industrielles ont été réalisés en utilisant les solutions photogrammétriques suivantes : PhotoScan, Pix4DMapper Pro, RealityCapture, MicMac, ReCap Photo et ContextCapture. Chaque jeu de données présente un nombre varié d'images, de résolutions et d'objets d'intérêt de différents niveaux de complexité industrielle. L'évaluation était basée sur trois critères principaux : la précision de l'orientation de l'image, la qualité visuelle et la précision géométrique du nuage de points dense 3D et les différentes caractéristiques du logiciel telles que le temps de traitement et la variabilité des produits de sortie.

Introduction

Nowadays 3D geometry products are used in many fields such as heritage, archaeology, medicine, and industry, to permanently record the form of complex objects and sites. As a result, many advanced survey techniques are evolving in order to obtain accurate 3D models.

Photogrammetry and laser scanning are the most used techniques in the industrial field as they can exhibit three-dimensional point clouds in a fast and accurate mode. Furthermore, they are considered as complementary technologies and no longer defined as two separate methodologies [START_REF] Kadobayashi | Comparaison and evaluation of laser scanning and photogrammetry and their combined use for digital recording of culutral heritage[END_REF], [START_REF] Graussenmeyer | Comparaison methods for terrestrial laser scanning, photogrammetry and tacheometry data for recording of cultural heritage building[END_REF], [START_REF] Pepe | 3D Point Cloud Model Color Adjustement by Combining Terrestrail Laser Scanner and Close Range Photogrammetry Datasets[END_REF], [START_REF] Heno | Numérisation 3D de Bâtiments Cas Des Edifices Remarquables[END_REF].

Industrial field such as, oil & gas, nuclear and refinery factories are characterized by the complexity of their facilities and the perpetual equipment change. Hence, photogrammetry has a significant role in those cases either by completing a complex part that cannot be acquired by laser scanning or by updating a point cloud after field modification (e.g. adding or removing equipment).

In the book entitled "Close Range Photogrammetry" photogrammetry was classified into several categories with different fields of application using the following criteria [START_REF] Luhman | Close Range Photogrammetry: Principles, Techniques and Applications[END_REF]:

-camera position and object distance, -methods of recording and processing, -availability of measurement results, -number of measurement images, -applications For instance, camera position and object distance criteria has been used in aerial photogrammetry, terrestrial photogrammetry, close-range photogrammetry and macro photogrammetry. In this paper, we will focus mainly on close-range photogrammetry and its valuable application in industrial field. The ongoing spread of this technique has led to more diversified software solution. Nearly a hundred-software are regularly updated [START_REF] Heno | Numérisation 3D de Bâtiments Cas Des Edifices Remarquables[END_REF]. However, choosing the right software to meet the required endpoints remains a challenge. Among these varieties of solutions, it is significantly important to determine the optimal ones for the generation of point cloud in industrial sites.

The performance of six photogrammetry solutions (Agisoft PhotoScan, Pix4D Pix4DMapper Pro, Capturing Reality RealityCapture, MicMac, ReCap Photo and ContextCapture) which are commonly used in archaeology and aerial photogrammetry were evaluated in industrial environment. Four different industrial data sets were carried out as part of a project conducted in Urbica 3D Laser Scanning Survey Company

1 .

An overview of the organization of the rest of this paper is given as follow: section 2 reviews the standard photogrammetry workflow for generating dense point clouds. Then without being exhaustive, a brief presentation of the selected software solutions are given in section 3. Afterwards, section 4 displays the different data sets and equipment used. Section 5 presents the main criteria chosen for the comparative study. Section 6 summarizes the results and section 7 concludes the paper.

Photogrammetry workflow

The pipeline for photogrammetry image acquisition and processing has a more or less standardized workflow and so does all software. Acquired images have to be oriented by determining both interior and exterior orientation parameters of: the camera and the object space points coordinates measured on photos [START_REF] Mcglone | Analytic data-reduction schemes in non-topographic photogrammetry[END_REF].

During the internal orientation, also known as "Internal calibration", two sets of parameters have to be considered. The first covers the geometric parameters of the camera: the focal length (i.e. the principal distance) and the coordinates of the principal point.

The second set includes the estimation of the camera distortion model, which describes the systematic errors of the camera. With regards to the external orientation, known as "Pose estimation" problem, it aims to define the position and attitude of the camera at the instant of exposure.

The alignment phase can be processed in steps (as relative and absolute orientation). Currently simultaneous methods such "Bundle Adjustments (BA)" are available in many software packages. BA is the task of estimating the 3D location of points using multi-images as well as the camera orientation (the orientation include the exterior parameters and may include the intrinsic parameters if the same camera is used under alike setup and configuration). In general, two approaches to perform BA may be followed [START_REF] Remondino | Low-Cost and Open-Source Solutions for Automated Image Orientation -A Critical Overview[END_REF], [START_REF] Granshaw | Bundle Adjustment Methods in Engineering Photogrammetry[END_REF]:

 Free-network BA: this approach includes the estimation of the interior (optional) and exterior camera parameters in an arbitrary coordinate system (relative orientation), followed by a 3D similarity transformation to align the network to absolute coordinate system (absolute orientation).

 Block Bundle Adjustment (BBA): this approach involves a simultaneous estimation of the 3D point coordinates, the internal (optional) camera parameters and external camera parameters in the absolute coordinates system. This is done by introducing in the observation matrix at least 3 Ground Control Points (GCPs). Formally, the orientation phase consists of the following steps. The first step requires extracting tie 1 http://www.urbica.fr/ points (i.e. image correspondences), which represent details extracted from all images relaying on outperforming detector and descriptor algorithms such as SIFT, SURF, BRIEF, FAST, etc. Reader may refer to [START_REF] Hartmann | Recent Developments in Large-scale Tie-point matching[END_REF] [START_REF] Salahat | Recent Advances in Features Extraction and Description Algorithms: A Comprehensive Survey[END_REF], [START_REF] Salti | Learning a Descriptor-specific 3D Keypoint Detector[END_REF] for a comprehensive overview of the state of the art and recent advances in various detector and descriptor algorithms. As for the second step, matching algorithm is set up, where the descriptor of one feature (Keypoint) in the first set is matched with all other features from other images. The keypoints matching is normally performed with brute force method based on the Hamming distance [START_REF] Remondino | A critical review of automated photogrammetric processing ff large datasets[END_REF]. As a last step the bundle adjustment is used iteratively to estimate all unknown parameters (interior and exterior parameters) and, to optimize the reprojection error between the image locations of observed and predicted image points [START_REF] Lourakis | SBA: A software package for generic sparse bundle adjustment[END_REF].

Once alignment phase is accomplished, the dense image matching step takes place in order to generate dense and colored 3D point cloud. Different reconstruction methods are available, a critical review and analysis of various dense image-matching algorithms are available in the literature, reader may refer to [START_REF] Remondino | State of the art in high density image matching[END_REF].

Software solutions

Photogrammetry solutions can be grouped into three main categories: web solutions, open access solutions and commercial solutions. The first group targets, in most cases, beginner users with modest photogrammetric knowledge. One of the main downfalls of this type of solution is the limited number of images that may be uploaded on the server. The second category is the open access solutions where public organizations, universities or communities display their software to the public for free. Most often, the code is accessible on open source, which allows more discussions and further understanding of used methods. The last category is the commercial solutions where many companies have developed their own packages to meet the growing demand of the market. Commercial software are often very ergonomic and offer a fully integrated processing chain.

Numerous software evaluation studies are available in the literature [START_REF] Nex | Benchmark for Multi-Platform Photogrammetry[END_REF], [START_REF] Knapitsh | Tanks and Temples: Benchmarking Large-Scale Scene Reconstruction[END_REF], [START_REF] Gini | UAV Photogrammetry: Block triangulation comparaison[END_REF]), (Pierrot-desseilligny, et al., 2011), [START_REF] Mendes | Photogrammetry with UAV's: Quality Assessment of Open-Source Software for Generation of Orthophotos and Digital Surface Models[END_REF], [START_REF] Alidoost | Comparison of UASbased photogrammetry software for 3D point cloud generation: a survey over a historical site[END_REF], [START_REF] Murtiyoso | OpenSource and Independent Methods for Bundle Adjustement Assessment in Close-Range UAV Photogrammetry[END_REF]. In most cases, the main objective of these evaluation papers was to compare software which are commonly used in archaeology field. Therefore, in this paper, the analysis was focused on close-range photogrammetry software in industrial field.

The evaluation was carried out using the following 6 software: ContextCapture: is a photogrammetry software developed by the technological software company Acute3D which became part of the end-to-end Reality Modeling solutions of the american software company Bentley Systems since 2015. It is available as a standalone desktop or cluster software solution and includes tilling mechanism which allows to handle terabytes of input imagery. This 3D photogrammetry solution is characterized by its capacity to automatically generate 3D meshes with level-of-detail and paging directly with several leading 3D GIS software. RealityCapture: the software RealityCapture (also known as RC) of the Slovakian company Capturing Reality is a stand-alone commercial photogrammetry program based on Structure-from-Motion (SfM), which creates 3D models out of terrestrial/aerial images or laser scans. It is used in different fields such as cultural heritage, gaming, virtual reality and surveying. Characterized by its parallel data processing and the possibility of direct integration of laser scanning point clouds. PhotoScan (new: Metashape): is a stand-alone commercial software developed by the russian company Agisoft LLC, which has many different fields of application. It is one of the widely used SfM program, which performs 3D reconstruction of objects from images and employs the entire photogrammetric workflow.



Data sets and image sensors description

Two Digital Single-Lens Reflex (DSLRs) cameras (Nikon D610 and D810) were used to acquire 4 data sets (overview in Table 1). Both cameras have full frame CMOS sensor with a 5.984-µm pixel size and 35-mm focal length for Nikon D610, and 4.8859-µm pixel size with 24-mm focal length for Nikon D810. An overview of the technical characteristics of used cameras can be found in Table 3.

Nikon D610 was used to acquire images of 3 data sets. Dataset-1 is a test bench established during my internship with Urbica 3D Laser Scanning Survey Company in Paris. It is a set of pipes with different shapes, sizes and color of paintings that are commonly used in industrial sites. A total of 90 images were taken with a Ground Sampling Distance (GSD) up to 0.3 mm/pix. Dataset-2 and Dataset-3, were acquired as part of a project aiming to generate 3D model of a ship engine overhaul factory at Saint-Nazaire bay area in France. Dataset-2 presents a complex set of indoor industrial pumps, 157 photos were taken to cover this area with a GSD up to 0.5 mm/pix. Whereas, Dataset-3 presents a set of electrical boxes where 73 photos were taken with a GSD up 0.8 mm/pix. The images of 3 data sets were saved in JPEG format with low compression.

Regarding Dataset-4, it was recorded using Nikon D810 DSLR camera. It was the similar test bench used in Dataset-1; however, we added a set of 12 calibrated scale bars and a set of 10-coded targets. Scale bars were designed by Cultural Heritage Imaging2 (CHI) Corporation. They were provided in several sizes with printed targets separated by known distance and calibrated to 1/10 th mm. The acquired images were saved in TIFF format.

All recorded objects were marked with coded targets which were generated using the methodology presented in the following paper [START_REF] Chen | An accurate and reliable circular coded target detection algoruithm for vision measurement[END_REF]. Moreover, no topographical survey was conducted around the object of interest, hence targets were used as tie points.

Evaluation criteria

Each software offers a slightly different set of parameters, different terminology, as well as different approaches for image orientation and dense point clouds generation. To assess the differences, the strengths and the weaknesses of the considered software, we focused on three main components:  Image orientation: number of aligned images, re-projection error and error on scale bars (to assess the estimation of the intrinsic parameters and the results after the BA);  Dense point clouds: visual quality, density, edge sharpness and geometric accuracy;  Software features: time processing, documentation and variability in output products. Furthermore, it is important to underline that images of the different data sets have been acquired in such a way that basic photogrammetric rules and shooting strategy are applied by : ensuring high overlapping of 80-90% between adjacent images, combining vertical and oblique images for optimum intersection conditions, small aperture to achieve a sufficiently large depth of field, and numerous tilted (90° rotated) images for reliable camera calibration (Pierrot-Deseilligny, et al., 2011). Additionally, for unbiased comparison and for the sake of consistency, all datasets have been processed with the same computer. An overview of computer technical characteristics are summarized in Table 1.

Processor : Intel® Core™ i7-3770 CPU @ 3.40GHz RAM : 32 Go Operating system : Windows © 7 64bits Graphic Card : Nvidia Quadro K4000 -4GB [START_REF] Remondino | Digital Camera Calibration Methods: Consideration and Comparaisons[END_REF]. Most of software use the auto calibration (i.e. self-calibration) approach in order to estimate the intrinsic camera parameters. In this paper, the same parameters were computed using different software.

The calibration parameters could be used as variables in the comparison study, however due to the large number of possible correlations between them performing a direct comparison is complex. Thus, it is necessary to compare the effect of the aforementioned parameters on the accuracy of the 3D restitution.

Dataset-4 was designed to assess the orientation step of each software package; a graphical illustration of the conducted experiment is shown in Table 2. In this data set, 12 calibrated bars were used where the distance between the target centers is well defined 1/10 th mm (see section 4). Three bars were employed for scaling, and the remaining ones as control bars. The alignment of images was performed by each software and the internal orientation parameters were calculated. The tie point extraction phase was performed forcing the same number of extracted Keypoint (40,000) and the same image resolution (full size). In addition, Free-network BA approach was adopted. For Free-network BA we used "Tapas" with the classical Brown's parameters lens camera model "RadialBasic".

Data set information

Software

Orientation results

First re-projection errors of automatic extracted tie points were examined. Whereby each software calculate the projection of tie points from image space towards field space at first, then re-project the pseudo intersection towards the image and compute the error. Formally, mean re-projection errors of less than one pixel is acceptable, whereas above 1 pix results in a noisier dense reconstruction. Subsequently, the accuracy of the alignment was assessed based on scale bars. The mean re-projection errors of most software was 0.56 ± 0.1 pix, as shown in Fig. 2. The best result was recorded with RealityCapture 0.3 pix. Furthermore, ReCap Photo software did not reveal the necessary information about the internal orientation settings and the evaluation of outcomes was not possible. Thus, this software was excluded from the study. Afterward, the accuracy of the orientation was evaluated by estimating measured distance of the control bars. Results are summarized in Table 4 and Fig. 2. PhotoScan software displayed a root mean square error of 0.07 mm, which represents the best accuracy value. The highest error was recorded with Pix4DMapper software.

It is important to underline that the coded targets exhibited on the control bars were fully detected automatically only with PhotoScan and RealityCapture whereas for the other software, manual method was adopted. Hence, presented results introduce a new parameter which is the precision of targets marking.

Dense point clouds assessment

In order to apply same settings, the resulting dense point clouds was considered as the final output of each software. Visual assessment between different generated point cloud using various photogrammetric software was carried out on different benchmark data set (Table 2). Then, to assess the edge sharpness and the geometric accuracy, 3D-templates (surfaces, cylinders) were adjusted to the point cloud. The distance between point cloud and geometric shapes was evaluated. While Dataset 3 electrical boxes' were used as test element for surface adjustment, Dataset 2 cylinders' were used for cylinder adjustment. A graphical representation of these templates is shown in Fig. 1.

It is worth mentioning that tests using ReCap Photo and ContextCapture did not provide access to the results of the digital image matching, being it fused with the meshing step. Accordingly, as previously mentioned, textured mesh rendering was not considered as evaluation criteria.

Parameters of Digital Image Matching

The same settings were used to generate dense point clouds. With PhtoScan, both "high" and "Mild filtering depth map" options were chosen. With RealityCapture we used the "high model" parameter with image downscale factor 1. Regarding Pix4DMapper we set up the image scale to original image size and the point density to "High". As for MicMac (Pierrotdeseilligny & Paparodits, 2006), two modules were used: C3DC Statue and C3DC BigMac.

The evaluated point clouds were raw data without any filtering or manual post-processing.

Dense point cloud evaluation results

First, we conducted visual quality assessment. The point cloud generated by Pix4DMapper led to locally noisy point clouds especially on uniform surfaces and homogenous textured feature as illustrated in Fig. 3. Regarding the Cylinder adjustment, the overall root mean square error was 0.85 mm for PhotoScan and 1.09 mm for RealityCapture. In contrast, the highest error was registered with Pix4DMapper 2.73 mm for Plan adjustment and 3.37 mm for Cylinder adjustment.

Furthermore, a significant difference in the generated point densities that fits plan and cylinder was registered (Fig. 67). The highest level of points was recorded with PhotoScan and the lowest with MicMac and Pix4D. Likewise, the total number of generated points by PhotoScan in different data sets was higher than other software, the histogram in Figure 8 reports the density of generated point cloud per data set.

Software features' comparison

The processing stage in both orientation and image dense matching was resource and time consuming. We used the same computer to process all data sets. In addition, we enabled the use of GPU option with PhotoScan, RealityCapture and Pix4DMapper. We noticed that commercial software were able to optimize the processing time. A significant processing speed was registered with RealityCapture, controversy to MicMac where the highest processing time was recorded. Histogram in Figure 9 The web solution ReCap Photo is limited in image input format (only JPEG). Moreover, it does not provide parameters for users. Therefore, the main drawback of this solution is the low reliability of the procedure and the lack of accuracy and metrics in results.

Note on orientation and reconstruction algorithms

Due to commercial consideration, scarce information was provided about commercial software. With PhotoScan, the first step consists of image alignment regrouping the interior and exterior parameters estimation. The interior orientation covers the estimation of the following parameters: focal length, principal point location, three radial and two tangential distortion coefficients. To do this, PhotoScan use an improved version of SIFT to detect image feature points then perform the BA [START_REF] Chiabrando | SfM for Orthophoto Generation: A Winning Approach for Cultural Heritage Knowledge[END_REF]. The resulting data is a sparse 3D point cloud corresponding to the locations of the estimated feature points. In the second step, a dense multi-view stereo reconstruction on the aligned images is applied. According to Remondino's paper [START_REF] Remondino | State of the art in high density image matching[END_REF], the implemented imagematching algorithm appears to be a Semi-Global Matching reconstruction method (SGM) [START_REF] Hirschmüller | Stereo processing by semiglobal matching and mutual information[END_REF], [START_REF] Hirschmüller | Accurate and efficient stereo processing by semi-global matching and mutual informatio[END_REF], [START_REF] James | Optimising UAV topographic surveys processed with structure-from-motion: Ground control quality, quantity and bundle adjustement[END_REF]. Like PhotoScan, there is reference limitation about the matching algorithms used by RealityCapture. Nevertheless, the implemented image-matching algorithm seems to be a local reconstruction method, a variant of patch match belief propagation approach [START_REF] Besse | PMBP: PathMatch Beleif Propagation for Correspondence Field Estimation[END_REF]. For the orientation phase, MicMac offers two modules Pastis and Apero. The former is performed under Tapioca command in order to generate and match tie points. This module use an improved version of SIFT called SIFT ++ (Pierrot Deseilligny & Clery, 2011), [START_REF] Chiabrando | SfM for Orthophoto Generation: A Winning Approach for Cultural Heritage Knowledge[END_REF]. "Apero" module which is found under "Tapas" command, performs the camera calibration and the Free-network BA. MicMac also offer the possibility to perform the BBA under Campari module.

Subsequently, a dense image matching for surface reconstruction is performed with MicMac module (Pierrot Deseilligny & Paparoditis, 2006), under C3DC interfaces. Micmac performs different approaches of image matching where the user can choose between two different processing strategies, called "image" and "terrain geometries". With the former strategy, the user selects a set of master images for the correlation procedure, then for each 3D point candidate a patch in the master image is identified and projected to all the neighboring images and a global similarity is derived. On the other hand, with "terrain" strategy, a voxel is defined according to the block size and the camera-to-object distance then every 3D point candidate is back-projected onto images and global similarity is derived. Regarding the remaining tested solutions, no information was found in literature about the used algorithms in orientation and reconstruction.

Conclusion

This study was part of a project conducted in Urbica 3D Laser Scanning Survey Company, to evaluate different photogrammetric solution in industrial filed, with the aim of updating or giving more completeness to 3D point clouds generated by Terrestrial Laser Scanner (TLS). We assessed six photogrammetry solutions based on four data sets acquired in industrial field with two DSLRs cameras. The evaluated solutions are commonly used in the cultural and heritage community. It covers the main categories in photogrammetric solutions e.g. commercial, open source and web solutions. The evaluation was based on three main criteria, the accuracy of image orientation, the visual quality and the geometric accuracy of 3D dense point cloud and, the different software features such as time processing and output variability. Additionally, in each data set the same parameters were adopted for all software.

The assessment of image orientation was based on the re-projection error and the root mean square error (rmse) of 12 calibrated scale bars estimated distances. According to the achieved results, the mean re-projection errors of most software was 0.5 ± 0.2 pix. The best performance was recorded with RealityCapture: 0.3 pix. However, there was differences in the rmse of estimated scale bars' distances. PhotoScan and Reality Capture software displayed a root mean square error of 0.1 mm, which represents the best accuracy value. The highest error was recorded with Pix4DMapper software. ReCap Photo, which is a web solution, did not reveal the necessary information about the internal orientation settings. The evaluation of outcomes was not possible, thus this software was excluded from the study.

Based on the results of dense point cloud assessment a significant difference was registered. The total number of generated points by PhotoScan was higher than any other software. In addition, the visual qualities and the accuracies of point clouds generated by RealityCapture and PhotoScan were very close to each other in different types of data set and higher than the accuracy of point cloud generated by other software. However, RealityCapture provided better geometric details. Additionally, point clouds generated by Pix4DMapper and MicMac led to locally noisy results especially on uniform surfaces. ContextCapture software does not provide access to the results of the digital image matching, being it fused with the meshing step. Concerning software features, commercial software had a fast processing speed. For instance, RealityCapture recorded the shortest processing time opposing to MicMac. The latest has a large number of parameters and no graphical interface unlike commercial software which does not display many parameters for the user to choose. This can be seen as advantage (it is easy to use) but also as a drawback since the user does not have a lot of options. The preselected data sets allowed us to evaluate the performance of different software solution in industrial filed. The aim was not to declare a winner, but to suggest a solution to the company in order to update and complete the 3D point cloud generated by TLS. At the present time, we suggest RealityCapture solution as it showed optimal results in image orientation accuracy and the quality of generated dense point clouds in term of sharpness and exhaustiveness. It also has a high parallel optimized implementation with simultaneous TLS data fusion.

Due to the lack of access to the full version of some software, further investigation are planned, where more photogrammetry solution will be tested.

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  Number of images : 90/JPEG -Ground resolution: 0.342 mm/pix -Camera : Nikon D 610 Area size : ca 4.77 x 1.969 x 2.67 m 3 Number of images : 157 /JPEG -Ground resolution : ca 0.513 mm/pix -Camera :Nikon D 610 -Area size: ca 10.6 x 0.5 x 3.85 m 3 Number of images : 73 /JPEG -Ground resolution : ca 0.854 mm/pix -Area size: ca 1,2 x1 x 0.5 m 3 Number of images : 51/TIFF -Ground resolution: ca 0.102 mm/pix -

Figure 1 :

 1 Figure 1 : Views of shape templates used for geometry evaluation, (a): view of Dataset 3 electrical boxes' used for plan-fitting, (b): view of Dataset 2 cylinders' used for cylinderfitting.

Figure 2 :

 2 Figure 2 : Mean, Max re-projection errors and accuracy on control bars obtained using different software

Figure 3 :

 3 Figure 3 : Generated dense point clouds with Pix4DMapper software, (a), (c) front view of respectively Dataset-3 and Dataset-1 point cloud. (b), (d) bottom view of a cropped uniform area As for MicMac, with both modules (C3DC Statue and C3DC BigMac), uniform areas were characterized by the absence of points, typically on electrical boxes of Dataset-3, Fig. 4 stipulates the results. In contrast with both PhotoScan and RealityCapture, the generated point cloud was exhaustive and less noisy (Fig.5). With respect to the geometric aspects, 3DReshaper software was employed to allow a local adjustment to geometric shapes such as plans and cylinders. The smallest deviation in both adjustment tests was recorded with Photoscan and RealityCapture

Figure 4 :Figure 5 :

 45 Figure 4 : Generated dense point cloud with MicMac software. (a) Dataset 3 point cloud using C3DC BigMac module. (b) Dataset 2 point cloud using C3DC Statue module

Figure 6 :Figure 7 :Figure 8 :

 678 Figure 6 : Combined graph: histogram distribution of adjusted points to plans and the respective standard deviation

  reports the time duration of each software to process the different data sets. Unlike MicMac, all tested commercial solution support GPU computing, which enables the acceleration of the processing time especially during the generation of dense point cloud. Furthermore, PhotoScan and Pix4DMapper supports non-Nvidia GPUs, which is not the case with RelaityCapture. The tested commercial software used in this study are available with a complete trial version through different period span. PhotoScan has a 30-day trial period, Pix4DMapper 14 days and both ContextCapture and RealityCapture have 7 days. Evaluated commercial software, i.e. PhotoScan, RealityCapture, Pix4dMapper and ContextCapture, have a user-friendly interface with several simplified functionalities and parameters. ContextCapture and Pix4DMapper provide fewer parameters to users than RealityCapture and PhotoScan. Regarding open source solution, MicMac has an extended number of parameter.It can handle a large number of input image format and give a variety of output products. In addition, MicMac, PhotoSan, Pix4DMapper and RealityCapture have a good documentation and provide the user with detailed processing report.

Figure 9 :

 9 Figure 9 : Time processing of tested software

  is the photogrammetry solution of the American multinational software company Autodesk. RecCap Photo is featured under the main 3D scan program ReCap. It is a web-based interface photogrammetry solution, which supports terrestrial and areal photogrammetry, with only cloud processing support (i.e. no local processing support). MicMac: is a free open-source photogrammetric suite operating as command-line based tool and developed by the National Institute of Geographic and Forest Information in France. Its high degree of versatility, as well as its various fields of application such as cartography, environment, industry, forestry, heritage, archaeology, characterize MicMac. Pix4DMapper: is part of a suite of software products developed by the Swiss company Pix4D. This software solution operate on desktop, cloud and mobile platforms. The major industries where Pix4DMapped software is used are aerial survey, cultural heritage, surveying and construction.

	-ContextCapture 4.10	-RealityCapture
	1.0.3
	-Pix4DMapper Pro 4.3	-PhotoScan Pro
	1.43
	ReCap Photo:
		Web solution : ReCap photo
		 Open access solutions: MicMac
		 Commercial solution:

Table 1 :

 1 Computer's characteristics

		6. Results and analysis
	6.1 Evaluation of Image orientation accuracy
	Accurate camera calibration and orientation are
	prerequisites for the extraction of precise and reliable
	3D metric information. Numerous studies describing
	different approaches of camera calibration are
	available in the literature. An overview of camera
	calibration	approaches	in	close-range
	photogrammetry and computer vision was presented
	by

Table 2 :

 2 Different data sets

Table 3 :

 3 Cameras

' specification

6.1.1 Orientation processing parameters

With PhotoScan, "Align Images" module was employed using "High" parameter as accuracy (i.e. image full resolution), 40,000 and 4,000 respectively as Key Point and Tie point limit and, "Auto" was set up for the camera calibration. As for RealityCapture "Alignment" module was used with the following parameters: Image downscale factor 1, Max feature per image 40,000, Preselector features 4,000, Distortion Model Browen3 with tangential2 and the rest of parameters remained as default. With regard to Pix4DMapper, advance "Initial Processing" module was used alongside the following parameters: Image scale 1, Number of KeyPoints 40,000, Calibration automatic. Concerning the parameters used with ContextCapture, Key point Density was set as "Normal" and Focal Length, Principal Point, Radial Distortion and Tangential distortion were set as "Adjust". Although MicMac provides a rich library to estimate camera intrinsic parameters, we employed the classical options to be consistent with other software. For tie point extraction, we used 'Tapioca' command with ALL and -1, which corresponds to finding out all possible matching in a full size image.

Table 4

 4 Estimated distances and errors relative to each software

http://culturalheritageimaging.org/

Acknowledgment

This work was supported by National School of Geographic Sciences of France and Urbica 3D Laser Scanning Survey Company.


Julie Tottey

Gilles Lalmanach

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Cryptosporidium parvum is an apicomplexan parasite responsible for a zoonotic disease affecting both human health and livestock. The parasite is transmitted through the fecal-oral route and develops in ileal epithelial cells. The resulting enteric disease leads to watery diarrhoea that can be life-threatening in individuals with an immature or a compromised immune system. The only approved treatments against cryptosporidiosis have limited efficacy in the total clearance of the parasite. Improving our understanding of the parasite biology and host-pathogen interactions, as well as identifying new drug targets, are key steps towards the development of much needed effective control strategies.

Cysteine proteases play key roles in the pathogenesis of protozoan parasites, including cell or tissue penetration, hydrolysis of host or parasite proteins, and evasion or modulation of the host immune response, making them attractive chemotherapeutic and vaccine targets. For instance, papain-like proteases (clan CA, family C1) from protozoan parasites responsible for human or bovine trypanosomiasis (cruzipain, congopain 1 ) and malaria (falcipains 2 ) are already well characterised as druggable targets based on in vitro and in vivo models.

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Evaluating the scalability of reconfigurable manufacturing systems at the design phase

Keywords: Scalability, Reconfigurable manufacturing systems, Multi-objective indicator, Combinatorial analysis, Line Balancing

To face market volatility, reconfigurable manufacturing systems (RMS) aim to efficiently and cost-effectively react to changes. We focus on one characteristic of RMS: the scalability (ability to adapt the volume of throughput). In the literature, the only few indicators for scalability are not always formally defined and usually only consider a partial view of scalability. Moreover, most of them are actually more suited for the configuration planning rather than for the design. However, the design of the RMS has a high impact on its scalability. We propose the first combinatorial definition of this problem and a new measure to fully assess the scalability of a system at the design phase. This measure, based on a multi-objective approach, can assess the scalability of single-product manufacturing systems, analyzing all configurations that it can implement. We present numerical experiments to compare this indicator with a state-of-the-art scalability indicator and with some classical production line design indicators, and we show that future research should focus on scalability as a specific criterion to optimize during the design of a RMS. In addition, the results obtained allow us to infer some managerial insights on the best levers to use when performing a reconfiguration for scalability purpose.

Introduction

Nowadays, one of the main challenge industrial companies are facing is to design manufacturing systems that can deal with the volatile market and uncertain future (see e.g., [START_REF] Aheleroff | Mass personalisation as a service in industry 4.0: A resilient response case study[END_REF][START_REF] Dolgui | Expected trends in production networks for mass personalization in the cloud technology era[END_REF]. They call for reduced time-tomarket as well as the ability to deal efficiently with numerous ramp-up phases. [START_REF] Koren | Impact of manufacturing system configuration on performance[END_REF] have introduced the Reconfigurable Manufacturing Systems (RMS) as an answer to this need of reactivity. Basically, RMS aim at managing shorter product lifecycles while keeping longer production system lifecycles. To achieve this goal, RMS relies on six key features [START_REF] Koren | Reconfigurable manufacturing systems: Principles, design, and future trends[END_REF]:

• Modularity, i.e. compartmentalization of operational functions into modules • Integrability to rapidly and efficiently integrate modules • Diagnosability allowing the automatic identification of problems in the production system and real-time monitoring the product quality • Customization to produce different parts in a family, • Convertibility to enable the system to change for new products • Scalability which is the ability to adapt the volume of production The recent emergence of Industry 4.0 enables the implementation of such systems by providing new solutions to implement these features [START_REF] Zheng | Smart manufacturing systems for industry 4.0: Conceptual framework, scenarios, and future perspectives[END_REF][START_REF] Ivanov | The impact of digital technology and Industry 4.0 on the ripple effect and supply chain risk analytics[END_REF][START_REF] Napoleone | How the technologies underlying cyber-physical systems support the reconfigurability capability in manufacturing: a literature review[END_REF]. For example, Plug & Produce Cyber-physical systems bring modularity and integrability, Internet of Things (IoT) and Digital Twins work together to improve diagnosability, Collaborative robot (Cobot) and additive manufacturing can provide the flexibility needed for the customization. . . Beside the use of new digital technologies, the decision support systems used by companies working with RMS have to be thought accordingly (see e.g., [START_REF] El | Scalable, reconfigurable simulation models in industry 4.0-oriented entreprise modeling[END_REF]. According to [START_REF] Andersen | Towards a generic design method for reconfigurable manufacturing systems: Analysis and synthesis of current design methods and evaluation of supportive tools[END_REF] and [START_REF] Dolgui | Expected trends in production networks for mass personalization in the cloud technology era[END_REF], due to the multiple variants and generations of product over the system life cycle, the design of RMS is a challenge compared to the design of traditional manufacturing systems. [START_REF] Koren | Value creation through design for scalability of reconfigurable manufacturing systems[END_REF] states that scalability might be the most important feature to deal with the uncertain demand or ramp-up phase. According to [START_REF] Putnik | Scalability in manufacturing systems design and operation: State-of-the-art and future developments roadmap[END_REF], the scalability in manufacturing systems can be defined as "the capacity for adding or removing the resources in a cost-effective manner, in order to adjust the production capacity on a system in steps or stages". Contrary to convertibility, it concerns only the throughput volume and not a change in the specifications of the product. The scalability assessment of a system must integrate components measuring the ability to quickly adapt the system, with incremental steps, and cost-effective changes, to answer to the need to provide at any time an efficient configuration with the exact capacity needed. Such an adaptation is particularly useful during ramp-up and product end-of-life phases.

RMS are typically composed of serial stages or stations with several identical parallel resources, joined by a gantry and a conveyor to move the parts in this grid system. The resources are generally Computer Numerical Control (CNC) or Reconfigurable Manufacturing Tools (RMT) but can also be of other types (for instance workers or cobots). Figure 1 schematizes this typical organization of RMS. Such layouts are sometimes denoted in the literature as parallel-serial lines with crossover (see e.g., [START_REF] Freiheit | Productivity of paced parallel-serial manufacturing lines with and without crossover[END_REF].

A configuration is defined as the state of the system, determined by the resources used, in terms of functionality, performance and number. Two levers can be involved in a reconfiguration: adding or removing parallel resources on the stations, or processing a full reconfiguration of the system by changing the tools used in RMT or CNC machines [START_REF] Putnik | Scalability in manufacturing systems design and operation: State-of-the-art and future developments roadmap[END_REF]. In the former approach, the assignment of operations to stages stays the same, thus the only organizational change in the system is the addition of resources and their integration on the gantry. The latter enables us to change the balancing of the system, implying a longer setup time, as many resources can be impacted and the system must be purged of semi-finished products. In both cases,

Main conveyor

Production resource (CNC, RMT, worker, cobot, . . . )

Available slot for scalability Station

Figure 1. RMS layout as described by [START_REF] Koren | Reconfigurable manufacturing systems: Principles, design, and future trends[END_REF] the choice of equipment is a tactical decision that should be made when designing the system to enable operational reconfigurations.

In this article, we investigate the scalability of RMS by considering the two aforementioned levers. In particular, we consider two research questions:

(1) What is the relationship between the balancing of an RMS and its scalability, both for the choice of the initial configuration and for further reconfigurations? (2) How to compare the scalability of different possible initial configurations of manufacturing system taking into account the policy to be applied for future reconfigurations and considering an unknown demand evolution?

In order to address these questions, two main contributions are provided. First, we give a formalization of the combinatorial optimization problem associated with the balancing of an RMS such as defined by [START_REF] Koren | Impact of manufacturing system configuration on performance[END_REF], and we analyze its complexity, which appear to be polynomial. Second, we propose a new measure to assess the scalability of a single-product system since the design phase, based on a multi-objective approach. In addition, the experiments performed to validate these contributions allow us to infer some interesting managerial insights for practitioners dealing with the design or the reconfiguration of an RMS.

The remainder of the paper is organized as the following: Section 2 presents the related literature, Section 3 formalizes the combinatorial definition of the problem, the proposed scalability measure is explained in Section 4 and experimental results are analyzed in Section 5. Section 6 concludes this paper and discusses few perspectives.

Related literature

The scalability of manufacturing systems has been studied in different research communities (optimization, model-based systems engineering,. . . ). Since this paper aims to formalize the problem optimizing the scalability of a single-product manufacturing system from the design phase, we choose to focus this literature review on papers dealing with optimization approaches and scalability evaluation.

If the design of manufacturing systems has generated a lot of publications on various balancing and facility layout problems, the specific characteristics of RMS make it an original optimization problem requiring new methods. The first methods proposed were heuristics [START_REF] Youssef | Modelling and optimization of multipleaspect RMS configurations[END_REF][START_REF] Dou | A GA-based approach for optimizing single-part flow-line configurations of RMS[END_REF][START_REF] Essafi | A reactive GRASP and path relinking for balancing reconfigurable transfer lines[END_REF][START_REF] Lahrichi | A new split-based hybrid metaheuristic for the reconfigurable transfer line balancing problem[END_REF] but exact models and algorithms have also been presented [START_REF] Essafi | A mip approach for balancing transfer line with complex industrial constraints[END_REF][START_REF] Borisovsky | Balancing reconfigurable machining lines via a set partitioning model[END_REF]. A recent review on the main optimization problem associated with RMS has been provided by [START_REF] Yelles-Chaouche | Reconfigurable manufacturing systems from an optimisation perspective: a focused review of literature[END_REF]. In addition, works on robotic assembly line balancing (see e.g., [START_REF] Borba | Exact and heuristic methods for solving the robotic assembly line balancing problem[END_REF] or parallel stations (see [START_REF] Aguilar | A survey of the parallel assembly lines balancing problem[END_REF], for a recent survey) present some similar features with the case of RMS.

However, all these works only focus on the generation of a single configuration minimizing the cost and do not take into account the possibility to further scale the system.

Scalability in the context of RMS has been studied on two different levels: the management of the scalability (on an operational level) and the design of a scalable system (on a tactic level).

Regarding the management of the scalability, [START_REF] Deif | A control approach to explore the dynamics of capacity scalability in reconfigurable manufacturing systems[END_REF]; [START_REF] Deif | Assessing capacity scalability policies in RMS using system dynamics[END_REF] were among the first to consider several reconfigurations, assessing different reconfiguration policies on various scenarios of demand evolution. [START_REF] Wang | Scalability planning for reconfigurable manufacturing systems[END_REF] started to consider successive reconfigurations with a heuristic method minimizing the number of machines required. An extension of this method has been presented in [START_REF] Koren | Value creation through design for scalability of reconfigurable manufacturing systems[END_REF], integrating buffers in RMS. Several optimization methods [START_REF] Hees | A production planning system to continuously integrate the characteristics of reconfigurable manufacturing systems[END_REF]Hu et al., 2017, e.g.) and simulation-based methods [START_REF] Hsieh | Design of scalable agent-based reconfigurable manufacturing systems with petri nets[END_REF]Gola et al., 2021, e.g.) have been developed to handle the scalability during the planning process. It is interesting to note that, even if some authors working on this question do not accurately explain the nature of the changes performed during a reconfiguration, most of them consider reconfigurations based on a new line balancing.

Some works have focused on the scalability since the design phase by analysing the characteristics allowing an RMS to be scalable. [START_REF] Koren | Impact of manufacturing system configuration on performance[END_REF] and [START_REF] Spicer | Design principles for machining system configurations[END_REF] started to study the impact of various system configurations on throughput capacity and scalability. [START_REF] Koren | Design of reconfigurable manufacturing systems[END_REF] went further and discussed the relationship between the structure of RMS and the number of configurations available. A similar idea was considered by [START_REF] Delorme | A multi-objective approach for design of reconfigurable transfer lines[END_REF] who suggested to evaluate the solution of their heuristic by generating the set of trade-off configurations between capacity throughput and number of resources needed. [START_REF] Son | An approach to scalability and line balancing for reconfigurable manufacturing systems[END_REF] questioned the link between the balancing of a production system and its productivity and scalability. They highlighted that unbalanced RMS can sometimes allow us to obtain smaller steps of capacity changes. More recently, [START_REF] Napoleone | Reconfigurable manufacturing: A classification of elements enabling convertibility and scalability[END_REF] introduced a classification of the main root causes leading to scalability and convertibility, [START_REF] Rösiö | Towards an assessment criterion of reconfigurable manufacturing systems within the automotive industry[END_REF] provided a list of key enablers for scalability including the possibility to change the line balancing and to add machines, and Beauville Dit [START_REF] Dit Eynaud | Framework for the design and evaluation of a reconfigurable production system based on movable robot integration[END_REF] presented a framework for the design of RMS and a reconfiguration strategy to answer to a specific demand evolution. Finally, some works tried to integrate the design and the planning problems. In [START_REF] Moghaddam | Configuration design of scalable reconfigurable manufacturing systems for part family[END_REF], the problem of designing multiproduct and scalable RMS for multiple production periods has been addressed, minimizing design and reconfiguration costs while fulfilling demand. The authors presented two approaches: an up-and downgrading method based on approximate demand in each period; and RMT selections and reconfigurations based on long-term demand forecast. A similar problem of design and configuration planning was considered in [START_REF] Cerqueus | Balancing and configuration planning of RMS to minimize energy cost[END_REF] where the authors used the scalability of a mono-product RMS to minimize the energy cost in a time-of-use pricing scheme.

Finally, the evaluation of the scalability of an RMS has received little attention. In [START_REF] Wang | Scalability planning for reconfigurable manufacturing systems[END_REF], a first metric based on the smallest possible incremental capacity change has been presented to evaluate a priori the scalability on the first reconfiguration. However this metric is dependent on the current state of the system and does not allow to take into account the subsequent reconfigurations. In addition, the definition of the proposed metric can be confusing because incremental capacity can be defined either as the gap in terms of resources used or in terms of capacity throughput. Both definitions are equivalent in the examples presented in the paper but it is not true in the general case. In addition, the authors showed that the metric can be coupled with a heuristic simulating the future reconfigurations (i.e. rebalancing of the line and addition of resources) to forecast the production throughput gain of different configurations. Based on the experimental results obtained, the authors observed that a lower number of stages leads to higher throughput and gain, mainly because it leads to a more reliable system. [START_REF] Nishith | Flexibility measurement criteria with respect to reconfigurable system properties[END_REF] has also proposed a slightly different metric based on the same principle of potential capacity increase but they provided very few details on the calculation. Indeed, [START_REF] Putnik | Scalability in manufacturing systems design and operation: State-of-the-art and future developments roadmap[END_REF] stated that new performance measures of scalability were still needed. The first evaluation of the scalability based on the set of possible configurations identified has been proposed by [START_REF] Wang | Reconfiguration schemes evaluation based on preference ranking of key characteristics of reconfigurable manufacturing systems[END_REF]. Their function takes into account the gap between the minimum and maximum throughput among all the configurations, as well as the average gap between the configuration and the effort needed to go from one configuration to another (time, cost, . . . ). However, the authors do not indicate how to identify the set of available configurations and the numerous parameters used in the function. More recently, some authors have proposed to extend the concept of scalability either to reconfigurable supply chain [START_REF] Zidi | New metrics for measuring supply chain reconfigurability[END_REF] or to process manufacturing [START_REF] Accorsi | Scalability analysis in industry 4.0 manufacturing[END_REF]. Important to note is that the metrics proposed in these papers are similar to those proposed for RMS, the former being defined as a combination of throughput capacity amplitude and of latency, and the latter being calculated by the simulation of a given curve of demand.

Overall, we can observe that scalability is rarely considered alone and most authors consider it together with convertibility. Even if both features do not have the same purpose, this choice may be explained by the fact that most authors use the same levers (i.e. adding resources and rebalancing operations) to deal with them. This observation is confirmed by [START_REF] Maganha | Understanding reconfigurability of manufacturing systems: An empirical analysis[END_REF] which report that there is a confusion among industrials between these two features.

Combinatorial definition of the problem

We are considering the design of a paced reconfigurable manufacturing system. A set N of operations to process is given and each operation n ∈ N has a processing time t n . The set of operations have to be partitioned into a sequence of stations (i.e., we have a line balancing problem), with n 0 and m 0 the upper bound on the number of operations which can be assigned to the same station and on the number of stations, respectively. The assignment of operations to stations should respect a set P rec ⊂ N × N of precedence relations. The workload of a station is defined as the sum of the processing times of the operations assigned and the production system is paced at a takt time defined according to the station with the highest operating time. In order to increase the production level, each station can be equipped with at most r 0 resources working in parallel and the whole sequence of stations can be duplicated while the total number of resources used does not exceed r. The data are summarized in Table 1.

All definitions presented in this section will be illustrated on the same didactic example composed of 7 operations (N = {1, . . . , 7}), given by the precedence graph in Figure 2 on which the processing times are given above the node representing the operation. We take n 0 = 4, m 0 = 4, r 0 = 3 and r = 7.

We call a sequence B = (S 1 , . . . , S m ), S k ⊆ N, ∀k ∈ 1, . . . , m a feasible balancing (see Figure 3) if all operations are assigned once ( k S k = N ), the precedence constraints are respected ((i, j) ∈ P rec ∧ i ∈ S k ⇒ j ∈ k ≥k S k ), and the number of operations

N

The set of operations t n

Processing time of the operation n n 0

Maximum number of operations assigned to a same station m 0

Maximum number of stations P rec Set of precedence constraints r 0

Maximum number of resources assigned to a same station r Maximum total number of resources assigned in the system The performance of each line L is a 2-dimensional vector z(L) = (τ (L), ρ(L)) which can be calculated according to (1), where τ (L) (resp. ρ(L)) corresponds to the takt time (resp. the total number of resources used).

               τ (L) = max k∈1,...,L.m n∈L.Sk t n /L.r k ρ(L) = L.m k=1 L.r k (1)
When comparing two lines from an economical perspective, the lower number of resources and the lower takt time are preferred by decision-maker, so it is a bi-objective problem and we can use the Pareto dominance [START_REF] Pareto | Manuel d'économie politique[END_REF]

: a line L dominates another line L if τ (L) ≤ τ (L ), ρ(L) ≤ ρ(L ) and we have τ (L) < τ (L ) or ρ(L) < ρ(L ), denoted z(L) z(L ).
We can focus only on the non-dominated lines. Let L * (B) ⊆ L(B) denote the subset of feasible lines using the balancing B which are not dominated by another line using B. For any given B, the set L * (B) can be constructed by starting from the line L ∈ L(B) with L.r k = 1, ∀k ∈ 1, . . . , L.m and then increasing iteratively of one unit the number of resources of the bottleneck station (i.e., argmax k∈1,...,L.m n∈L.Sk t n /L.r k )2 until either the upper bound r 0 or r is attained. Since the number of steps of this process is bounded by r and each step can be computed in O(n 0 m 0 ), we have |L * (B)| ≤ r and the process can be done in constant time since n 0 , m 0 and r are given constants.

We call a λ-tuple C = (L 1 , . . . , L λ ) a feasible configuration (see Figure 5) if all L q , q ∈ 1, . . . , λ are feasible lines which share the same balancing (i.e., ∀q, q ∈ 1, . . . , λ, L q .m = L q .m and ∀k ∈ 1, . . . , L q .m, L q .S k = L q .S k ) and ). Reciprocally, let the singleton B(C) denote the balancing used by the configuration C.

q k L q .r k ≤ r. Let C
Similarly to the lines, the performance of a configuration C is a 2-dimensional vector z(C) = (τ (C), ρ(C)) which can be calculated according to (2), where τ (C) (resp. ρ(C)) corresponds to the takt time (resp. total number of resources used). We can naturally

Line C.L1 {1, 3, 5} {1, 3, 5} {1, 3, 5} {2, 6} {4, 7} {4
                             τ (C) = 1 C.λ q=1 1 τ (L q ) = C.λ q=1 τ (L q ) C.λ q=1   u∈{1,...,C.λ}\q τ (L u )   ρ(C) = C.λ q=1 ρ(L q )
(2)

Changing the system state from one configuration to another in order to answer to an evolution of market demand is called a reconfiguration. Such changes generate costs and require a reconfiguration time, which are usually non-decreasing functions of the magnitude of the change, i.e. the gap between the old configuration Ĉ and the new one C. The minimal change occurs when the same balancing is kept (B(C) = B( Ĉ)) since such a reconfiguration does not impact the tools or modules used. On the contrary, whenever there is a change in the balancing, some tools or modules must be replaced. We call N p the set of neighborhoods (3) based on the number of operations movements from one initial balancing. Starting from an initial configuration C ∈ C these neighborhoods can thus define the subset B∈Np(B(C)) C(B) of possible future configurations reachable over the lifecycle of the RMS for a given threshold p of changes allowed.

                                   N 0 (B) = {B} N 1 (B) = {B ∈ B|B .m = B.m and ∃n ∈ N, k 1 , k 2 ∈ 1, . . . , B.m, k 1 = k 2 , B .S k1 = B.S k1 ∪ {n}, B .S k2 = B.S k2 \ {n}, B .S k = B.S k ∀k = k 1 , k 2 } ∪ {B} N p (B) = {B ∈ B|B .m = B.m and ∃((n 1 , k 1 1 , k 1 2 ), . . . , (n p , k p 1 , k p 2 )), ∀i ∈ 1, . . . , p, n i ∈ N, k i 1 , k i 2 ∈ 1, . . . , B.m, ∀k ∈ 1, . . . , B.m, B .S k = B.S k ∪ {n i |k i 1 = k} \ {n i |k i 2 = k}} N |N | (B) = {B ∈ B|B .m = B.m} (3)
Given the balancing B 1 in Figure 3, the neighborhood N 1 (B 1 ) is composed of seven balancing:

B 2 : B 2 .S 1 = {3, 5}, B 2 .S 2 = {1, 2, 6}, B 2 .S 3 = {4, 7}, B 3 : B 3 .S 1 = {1, 2, 3, 5}, B 3 .S 2 = {6}, B 3 .S 3 = {4, 7}, B 4 : B 4 .S 1 = {1, 3, 5}, B 4 .S 2 = {6}, B 4 .S 3 = {2, 4, 7}, B 5 : B 5 .S 1 = {1, 3, 5}, B 5 .S 2 = {2, 4, 6}, B 5 .S 3 = {7}, B 6 : B 6 .S 1 = {1, 3}, B 6 .S 2 = {2, 5, 6}, B 6 .S 3 = {4, 7}, B 7 : B 7 .S 1 = {1, 3, 5, 6}, B 7 .S 2 = {2}, B 7 .S 3 = {4, 7}, B 8 : B 8 .S 1 = {1, 3, 5}, B 8 .S 2 = {2}, B 8 .S 3 = {4, 6, 7}.
According to these definitions, the optimization problem of maximizing the scalability of an RMS at the design stage can be expressed as (4), where f s refers to a scalability metric evaluating the set of configurations available for future reconfiguration and p is a parameter defining the maximal level of changes allowed. Considering that checking the feasibility of each balancing can be done in polynomial time, and as long as the function f s can also be calculated in polynomial time, an approach based on the enumeration of all feasible balancing and configurations will be polynomial, i.e. theoretically efficient [START_REF] Garey | Computers and intractability: a guide to the theory of NP-completeness[END_REF]), since we have

|B| ≤ O(|N | n0m0 ), |C| ≤ O(|N | n0m0 r m0 r 0
) and |C * | ≤ O(|N | n0m0 rr ) with n 0 , m 0 and r being given constants. However, in practice these constants usually take large values and such an enumeration approach will only be applicable for small problems.

max B∈B f s   B ∈Np(B) C(B )   (4)
Note that the design of a dedicated manufacturing system (DMS) would correspond to a specific case where the set of configurations available would be restricted within a subset of C DM S ⊂ C such that only one resource is assigned per station (C.L q .r k = 1, ∀q ∈ 1, . . . , C.λ, k ∈ 1, . . . C.L q .m).

After the design phase, each reconfiguration will correspond to another optimization problem (5), where Ĉ is the initial configuration of the RMS and T is the new takt time deduced from current market demand. When choosing a new configuration from the available set N p (B( Ĉ)), there can be several configurations with the same minimal value of p but a rational decision-maker will prefer to select the non-dominated one. As a consequence, we can discard the dominated configurations from the search space. min

C∈C(B) p s.t. B ∈ N p (B( Ĉ)) τ (C) ≤ T p ∈ 0, . . . , |N | (5)
In the next section, we will focus on the definition of the scalability measure f s which both take into account the whole set of reachable configurations and is calculable in polynomial time.

Hypervolume based indicator for scalability

Based on the literature review, we extracted three main properties to evaluate the scalability level of a set of reachable configurations:

(1) all the configurations should be as economically efficient as possible (i.e., we are seeking configurations with the lowest possible values for both the number of resources and the takt time). ( 2) they should cover a large range of takt time to be able to answer both small and large demands.

(3) the configurations should be well-spread on the two-dimensional performance space (i.e., we prefer to have regular steps to increase the capacity throughput incrementally).

The first one is omnipresent as soon as we consider the design of a manufacturing system. The second one is linked to the work of [START_REF] Wang | Reconfiguration schemes evaluation based on preference ranking of key characteristics of reconfigurable manufacturing systems[END_REF] taking into account the gap between the minimum and the maximum throughput. The third one comes from [START_REF] Wang | Scalability planning for reconfigurable manufacturing systems[END_REF] searching for the smallest incremental capacity change as well as the adaptation "in steps" in the definition of scalability of [START_REF] Putnik | Scalability in manufacturing systems design and operation: State-of-the-art and future developments roadmap[END_REF]. These three characteristics are actually very similar to those considered in the evaluation of a multi-objective approximate method, where sets of potentially non-dominated points have to be evaluated and compared. We can thus use multiobjective indicators to assess the quality of a set of reachable configurations (and the associated values of z).

Among the numerous multi-objective indicators, the most commonly used is probably the hypervolume [START_REF] Zitzler | Performance assessment of multiobjective optimizers: an analysis and review[END_REF]. For any subset of configurations C ⊆ C, let us consider F ( C) = {(τ (C), ρ(C)), C ∈ C}, the set of points which are the images by z of each configuration from C. The hypervolume of F ( C) is the area between the points (τ (C), ρ(C)) and a reference point P . Its formula is given by Equation ( 6).

H( C) = A     C∈ C(τ (C), ρ(C)) -R 2   (P + R 2 )   (6)
A is the geometrical area. R 2 denotes the dominance cone {y ∈ R 2 : y (0, 0)}.y + R 2 stands for the vector addition of y ∈ R 2 and any vector of R 2 , it defines the area whose points weakly dominate y. y -R 2 is defined similarly as the area whose points are weakly dominated by y.

Of course, the reference point P have to be such all (τ (C), ρ(C)) have lower values on both τ and ρ. For our application, we can take

P =   max K⊆N,|K|≤n0    j∈K t j    + ε τ , r + ε ρ   with ε τ , ε ρ ∈ R + .
Indeed, the takt time cannot be greater than the sum of the n 0 highest processing times, since there are no more than n 0 operations per stations, and the number of resources used is bounded by r. Since the number of resources used is necessarily an integer, we can take ε ρ = 1. To have a proportional gap for τ , we take 6 graphically illustrates the notion y + R 2 and the hypervolume of a set of points F ( C). Then we can define the scalability function f s ( C) = H( C). The hypervolume based scalability indicator f s , in this bi-objective context, is the sum of | C| rectangles. Thus it is computed in O(| C|). Considering that the dominated configurations can be discarded from the set C, the cardinality of the set is bounded by r since the performance on the number of resources used ρ is integral and bounded by r by definition. Thus f s can be computed in constant time, O(r), and used to solve the problem (4).

ε τ = maxK⊆N,|K|≤n 0 { j∈K tj} r . Figure
τ ρ × × × z(C 1 ) z(C 2 ) z(C 3 ) P z(C 2 ) + R 2 H(F )
However, the values taken by the function f s are difficult to interpret by the decisionmaker and do not really allow him to have a precise idea of the quality of a set of configurations. Moreover, the values of f s can vary significantly from one instance to another which makes it difficult to compare the results obtained for various instances. As a consequence, it can be useful to have a relative value to compare the set of reachable configurations with a reference set for the instance considered. By definition, the subset of non-dominated configurations considering any possible balancing C * has the highest value for H. We can thus evaluate the quality of a set of reachable configurations by comparing H( C) with the value of the hypervolume of the ideal set H(C * ). We define the relative hypervolume based indicator f s in Equation (7). f s takes values in [0,1] and a low value indicates that the set of reachable configurations are close from those of the ideal set of configurations, indicating its good quality. Since H(C * ) is constant for a given instance, maximizing f s and minimizing f s are equivalent optimization problems.

f s = 1 - f s ( C) H(C * ) (7)

Experimental results

The goals of this section are to: (i) to compare the hypervolume based indicator for scalability to classical line balancing indicators and to the average of the scalability measure proposed by [START_REF] Wang | Scalability planning for reconfigurable manufacturing systems[END_REF], to see if the same RMS design are considered scalable by both metrics and what makes the difference for other designs; (ii) to determine some characteristics of scalable systems, in order to help decision maker in the design of their systems, and (iii) to analyse of the impact of the length of the neighborhoods on the scalability of the system, to determine the gain in reconfiguring a system by re-balancing it.

For the instances considered, we enumerated all balancing satisfying the precedence constraints, under the following assumptions:

• There are no more than three resources on each station.

The total number of resources is bounded by the number of operations in the system. • There are no more than 50% of the operations assigned to the same station.

The benchmark of instances used has been extracted from [START_REF] Scholl | Balancing and Sequencing of Assembly Lines[END_REF] and [START_REF] Otto | Systematic data generation and test design for solution algorithms on the example of salbpgen for assembly line balancing[END_REF]. Since our experimental process is based on a total enumeration of all feasible balancing, we chose to focus on a restricted set of instances: the ten smallest instances of [START_REF] Scholl | Balancing and Sequencing of Assembly Lines[END_REF] and ten from the instances with 20 operations of [START_REF] Otto | Systematic data generation and test design for solution algorithms on the example of salbpgen for assembly line balancing[END_REF]. For the latter, we arbitrarily chose the instances numbered {50, 100, . . . , 500}, and checked that they are representative of the whole set of instances, in terms of values for the parameters on the structure of the graph, the order of strength (density of the precedence graph) and time distribution. All experiments were implemented in C++ and ran on an Ubuntu system using an Intel Xeon 6230 chip and 5 MO of RAM. For the numerical experiments, we limited the computational time allocated to the balancing enumeration to an hour. Only the five smallest instances completed the process under the timeout. For the others, we randomly deleted operations until the resulting instance completes the enumeration phase within the time limit. The initial and reduced size of the instances are given in Table 2.

In Table 3, the correlations between the scalability indicator f s and the average incremental measure of [START_REF] Wang | Scalability planning for reconfigurable manufacturing systems[END_REF] are presented for all considered instances. These correlations are computed on the sets of reachable configurations extracted from the same balancing (i.e. N 0 ).

Since the scalability indicator presented in Wang and Koren ( 2012) is a local evaluation of the scalability, we adapted it by computing the average of the smallest incremental capacity in percentage. Sorting the non-dominated configurations by number of resources, for two successive configurations, the increment is the difference between the inverse of the takt time of two configurations. In the following, this indicator is denoted avgScal. The two indicators avgScal and f s are to be minimized.

Table 3 shows that there is almost no correlation between the two scalability indicators. This reflects a major difference in the computation of the two indicators regarding what is a good increment. By definition, the measure of Wang and Koren 3. Correlation of the hypervolume with the average scalability from [START_REF] Wang | Scalability planning for reconfigurable manufacturing systems[END_REF] (2012) gives better values when the difference between the takt time of two consecutive configurations is small. The increments being averaged, avgScal is lower if the spread of the τ values for the whole set of reachable configuration is tighter. On the contrary, the f s indicator favors the biggest increment it is possible to obtain when adding a resource, i.e. the largest possible capacity increment for the lowest possible resource cost. Moreover, the hypervolume computation gives better values if the values of ρ and τ are spread on a large range. Thus it is not surprising to see that these two indicators are not correlated.

In Table 4, we present the correlations between these scalability indicators and the classical line balancing indicators (takt time, number of stations, idle time, smoothness). We still suppose that the sets of reachable configurations are N 0 , i.e. all configurations of a reachable set are based on the same balancing. All indicators are to be minimized, thus a value close to 1 indicates a strong positive correlation. Table 4 shows that there are positive correlations between avgScal and all the classical line balancing indicators, except for the number of stations. On the contrary, there is no correlation between the classical line balancing indicators and the indicator f s , except for the number of stations.

As mentioned earlier, the avgScal favors small capacity increments. When the system is well-balanced, adding one resource will lead to a small increment. Indeed, by adding a resource on the bottleneck station, the takt time will be reduced only to achieve the workload of the second bottleneck station, which is most likely to be close to the previous takt, since the stations are well-balanced. Thus the correlation of avgScal with the smoothness ratio was expected. Even if there is a correlation between avgScal and the smoothness, this correlation is not complete, which is coherent with [START_REF] Son | An approach to scalability and line balancing for reconfigurable manufacturing systems[END_REF] stating that there can be unbalanced scalable systems.

Moreover, a well-balanced system has a low idle time, thus the correlation is similar for this indicator. The indicator avgScal is also correlated with the takt time since the well-balanced systems offering the smallest capacity increments are those with a low takt time, leading to a long sequence in the sequential process of adding resources on the same line. The absence of correlation with the number of stations is due to the fact that the number of stations alone is not sufficient to obtain well-balanced systems, and for the unbalanced balancing, some increments are high. In [START_REF] Koren | Value creation through design for scalability of reconfigurable manufacturing systems[END_REF], the authors stated that the scalability is correlated to the number of stations, which does not reflect with the indicator avgScal.

However this assertion seems correct for the indicator f s , since it presents a strong positive correlation with the number of stations. This correlation can be explained by the fact that having a low number of stations allows to derive a larger set of configurations. Indeed, generally from a balancing, we can derive a configuration for any number of resources between the number of stations of the balancing and the maximum number of resources. The hypervolume being highly impacted by the number of configurations, it tends to have higher values when the number of stations in the balancing is low.

f s is slightly negatively correlated with the takt time. It can be seen as a consequence of the strong positive correlation with the number of stations, since a low number of stations implies a high takt time, also explaining that the correlation is negative. However, it is mitigated by the fact that not all balancing with a high number of stations has a low takt time. For balancing with a high takt time and a high number of stations, the value of f s is very variable, some can have highly productive configurations for the number of resources but others not.

The idle time being the product of the number of stations and the takt time, a similar explanation can be given for the positive correlation between the idle time and f s .

It is interesting to note that f s is not correlated with the smoothness, which means that it is as likely to have a scalable system with a well-balanced system as it is for an unbalanced one. This is a stronger statement than the one made in [START_REF] Son | An approach to scalability and line balancing for reconfigurable manufacturing systems[END_REF] saying that some unbalanced systems could be scalable.

Indeed, Table 4 shows that the classical line balancing indicators cannot be used to evaluate the scalability of a RMS, save for the number of stations. However, even if there is a strong correlation between the number of stations and f s , we looked more in-depth at the relationship between these two indicators and we statistically analyzed the repartition of the f s values for the balancing with the lowest number of stations and the neighborhood N 0 (Table 5).

This table shows a significant difference between the minimum and the maximum value of f s , since the maximum is 2 to 4 times more than the minimum value. The first quartile represents on average an increase, regarding the minimum, of 1 percentage point, 2 percentage points for the median, 4 percentage points for the third quartile and 10 percentage points for the maximum. This shows that not all balancing with the lowest number of stations has a good value for f s . Thus minimizing the number of stations offers no guarantee regarding the scalability of the system, which means that having a low number of stations is a necessary condition to have good scalability but not a sufficient one.

All the previous experiments have focused on N 0 , i.e. deriving configurations without re-assigning the operations to the stations. In the literature, most authors have often considered reasonable to allow some operations to be reassigned during a reconfiguration of RMS. In Table 6, we aim to evaluate the gain that could come from the movement of one operation. It contains the minimum and average values of f s of all balancing considering N 0 and N 1 .

First of all, it can be observed that the average minimum value for N 0 is 5.76%. This value represents the gap between the scalability level achieved with an optimal balancing when future reconfigurations will not allow to change the balancing and the ideal scalability level obtained if any balancing could be considered for the reconfigurations (i.e., reconfigurations can fully redesign the system without considering its initial or current configuration). It corresponds to the cost of imposing that all the configurations of a reachable set must share the same balancing. This cost seems really low by comparison with the additional costs of reconfiguration times associated with a change of tools. By comparing the values with the minimum values of f s when focusing only on the balancing with the lowest number of stations (Table 6), it can be noted that the minimum always corresponds to a balancing with the fewest stations. This emphasizes that to obtain scalable systems, the number of stations must be minimal.

When allowing one movement of operations (neighborhood N 1 ), this average drops to 0.93%, meaning that reachable configurations are an approximation of good quality of the ideal set. The cost of reconfiguration compared to the ideal set is expected to be largely lower, since the ideal set could imply moving a large number of operations.

The average value of f s over all balancing is around 37 % for all instances when considering N 0 and is only reduced to 33 % when considering N 1 . It highlights that the potential gain of optimizing this indicator during the design phase w.r.t. picking a random solution for scalability (i.e. optimizing other criteria only), remains very close, above 30 percentage points, whatever the neighborhood considered. Moreover, this potential gain is much higher than the one obtained by switching from neighborhood N 0 to N 1 (around 4 percentage points).

Figure 7 shows the evolution of the minimum value of f s with respect to the different N i , for the 5 smallest instances of our benchmark. We can notice that the minimum f s quickly decreases when i increases and stabilises after i = 2 or 3. This shows that allowing only of small number of movements is already enough to achieve the best configurations.

Finally, we can note that for Jackson and Mansoor instances, the curve stabilises to a value close but not equal to 0 (respectively 0.07% and 0.01%). This reveals that the ideal set of configurations for those instances cannot be obtained entirely with configurations based on a balancing with the same number of stations.

Conclusion

Despite scalability being considered to be one of the main characteristics of RMS, its evaluation at the design phase has not yet received a lot of attention. The existing measures in the literature are actually difficult to use, due to some imprecisions in the definitions, to the multitude of parameters, or to the dependence to the initial state of the system. In this paper, we propose a new measure based on a classical multi-objective metric to assess the scalability level of a balancing by taking into account all the configurations which can be achieved. An important theoretical result is that, due to the nature of the problem defined, we proposed a method to enumerate the whole set of balancing in polynomial time.

The experimental results show that this hypervolume based indicator allows a different evaluation than the smallest increment measure proposed by [START_REF] Wang | Scalability planning for reconfigurable manufacturing systems[END_REF]. They also reveal that the usual line balancing criteria are mainly unrelated with the scalability, enhancing the need for dedicated metrics which could be used to optimize the scalability at the design step. From a managerial point of view, to obtain a scalable system, the decision maker does not need a balanced system, as it is as likely to have a scalable system when implementing a balanced system as with an unbalanced one. This offers a stronger statement than in [START_REF] Son | An approach to scalability and line balancing for reconfigurable manufacturing systems[END_REF]. It is also interesting to note that, similarly to the conclusions reported in [START_REF] Koren | Value creation through design for scalability of reconfigurable manufacturing systems[END_REF], in our setting, the designer should prefer a lower number of stages, as it is strongly and positively correlated with the scalability, even without taking the reliability into account.

Moreover, the experimental results demonstrate that the cost of keeping the same balancing for the reachable configurations of a design is low if the scalability measure is taken into account during the design. In addition, this cost rapidly decreases when allowing few operations to be moved to a different machine when reconfiguring. Indeed, allowing the move of a single operation is enough to achieve a scalability level with a gap lower than 1% from the optimum. From a managerial point of view, we can conclude that, contrary to a reconfiguration for convertibility, when implementing a reconfiguration for scalability, decision-makers should focus on the first reconfiguration lever consisting in the addition of resources rather than on rebalancing the operations. Even if they want to use the second lever and change some of the tools used, they could restrict their search at one operation moved (i.e. one or two tools impacted) which will make the reconfiguration problem easier to solve and the reconfiguration itself easier to process.

Thus further works should seek to develop methods to optimize the scalability indicator when designing RMS, along with some classic economic performance measures. The proposed measure can also be used to determine further the characteristics of scalable systems and identify the parameters affecting scalability. As the scalability characteristic may not be totally independent from the five other key features of RMS, studying these interactions would be another research axis.

Figure 2 .Figure 3 .

 23 Figure 2. Precedence graph and processing times of the illustrative example

Figure 4 .

 4 Figure 4. Example of a line L ∈ L for a problem with 7 operations and r 0 = 3. Here we have, τ (L) = max {(7 + 4 + 8)/2; (5 + 3)/1; (4 + 8)/2} = 9.5 and ρ(L) = 5.

  denote the set of all feasible configurations and C(B) ⊆ C denote the subset of configurations using the balancing B, i.e. C(B) = {C ∈ C|C.L 1 .m = B.m and C.L 1 .S k = B.S k , ∀k ∈ 1, . . . , C.L 1 .m}. Since all the components of C(B) are elements from the set L(B) and λ ≤ r, we can deduce that |C(B)| ≤ O(r m0 r 0

Figure 5 .

 5 Figure 5. Example of a configuration C ∈ C with 2 lines for a problem with 7 operations and r 0 = 3. Here we have τ (C) = 8 × 12 8 + 12 = 4.8 and ρ(C) = 10.

Figure 6 .

 6 Figure 6. Illustration of the notion y + R 2 and the hypervolume of a set of points F ( C) for a set of 3 configurations C = {C 1 , C 2 , C 3 } (depicted by ×).

Figure 7 .

 7 Figure 7. Evaluation of the minimum value of f s regarding the length i of the neighborhood N i

Table 1 .

 1 Problem data

Table 2 .

 2 Description of the benchmark instances original and reduced size (number of operations), "Otto-i" denotes the instances from[START_REF] Otto | Systematic data generation and test design for solution algorithms on the example of salbpgen for assembly line balancing[END_REF] named "instance n=20-i"

	Inst. name Orig. size Red. size	Inst. name Orig. size Red. size
	Mertens	7	-	Otto-50	20	11
	Bowman8 8	-	Otto-100	20	12
	Jaeschke	9	-	Otto-150	20	10
	Jackson	11	-	Otto-200	20	11
	Mansoor	11	-	Otto-250	20	12
	Mitchell	21	13	Otto-300	20	10
	Roszieg	25	13	Otto-350	20	9
	Heskia	28	9	Otto-400	20	10
	Buxey	29	12	Otto-450	20	15
	Sawyer30	30	12	Otto-500	20	14
		Instance	avgScal	Instance avgScal
		Mertens	0.289	Otto-50	0.297
		Bowman8 0.039	Otto-100 0.209
		Jaeschke	0.251	Otto-150 0.167
		Jackson	0.154	Otto-200 0.210
		Mansoor	0.138	Otto-250 0.194
		Mitchell	0.090	Otto-300 0.137
		Roszieg	0.201	Otto-350 0.070
		Heskia	0.105	Otto-400 0.224
		Buxey	0.024	Otto-450 0.060
		Sawyer30 0.071	Otto-500 -0.107

Table

Table 4 .

 4 Correlation between the scalability indicators and classical line balancing indicators.

		takt time nb stations idle time smoothness
	avgScal	0.648	0.010	0.783	0.793
	f s	-0.403	0.979	0.393	0.160

Table 5 .

 5 Statistical repartition of the value of f s (in percentage) restricted to the balancing with the lowest number of stations for each instance

	Instance	min 1st quartile median 3rd quartile	max average
	Mertens	9.49	11.65	12.95	15.54 28.57	14.36
	Bowman8 6.57	8.01	9.69	12.00 17.16	10.26
	Jaeschke	7.73	8.80	9.32	10.64 19.05	9.99
	Jackson	5.63	6.13	6.88	7.73 14.09	7.30
	Mansoor	4.93	5.48	6.37	8.07 12.85	7.00
	Mitchell	4.32	5.26	5.85	7.48 11.31	6.42
	Roszieg	4.29	4.98	5.76	6.85 12.23	6.31
	Heskia	5.33	6.39	7.25	10.97 17.30	8.61
	Buxey	4.32	5.17	5.85	7.83 14.36	6.61
	Sawyer30	3.93	5.89	6.98	9.38 17.30	7.86
	Otto-50	5.51	6.23	6.87	8.10 15.01	7.55
	Otto-100	6.38	7.81	8.36	9.33 13.80	8.94
	Otto-150	6.91	8.12	8.88	10.29 15.64	9.42
	Otto-200	5.53	6.64	7.84	10.21 15.57	8.58
	Otto-250	5.68	6.45	7.47	9.12 14.43	8.05
	Otto-300	6.69	7.70	8.67	10.55 17.80	9.33
	Otto-350	6.70	7.92	9.19	11.80 18.59	10.11
	Otto-400	7.01	8.03	8.91	9.87 16.54	9.40
	Otto-450	4.52	5.32	5.75	7.01 12.22	6.37
	Otto-500	3.76	4.63	4.97	6.13 10.01	5.51
	Average 5.76	6.83	7.69	9.44 15.69	8.40

Table 6 .

 6 Instancemin f s on N 0 avg f s on N 0 min f s on N 1 avg f s on N 1 Minimum and average value of f s (in percentage) among all balancings, for N 0 and N 1

	Mertens	9.49	34.98	1.73	27.62
	Bowman8 6.57	34.87	1.45	30.36
	Jaeschke	7.73	36.55	2.71	32.04
	Jackson	5.63	36.89	0.96	33.45
	Mansoor	4.93	38.60	1.63	35.81
	Mitchell	4.32	39.84	0.74	37.39
	Roszieg	4.29	39.74	1.22	37.00
	Heskia	5.33	32.21	0.67	28.37
	Buxey	4.32	37.81	0.69	35.24
	Sawyer30 3.93	37.83	0.41	35.11
	Otto-50	5.51	37.30	0.51	33.80
	Otto-100	6.38	37.29	0.41	33.74
	Otto-150	6.91	34.65	0.70	30.58
	Otto-200	5.53	37.78	0.52	34.15
	Otto-250	5.68	37.59	0.81	34.16
	Otto-300	6.69	32.62	0.67	28.54
	Otto-350	6.70	34.32	0.41	29.90
	Otto-400	7.01	34.98	0.45	30.83
	Otto-450	4.52	40.72	0.69	38.28
	Otto-500	3.76	40.06	1.14	38.15
	Average 5.76	36.83	0.93	33.23

Whenever there is not a single bottleneck station, we can arbitrarily pick one of them since the next steps will consider the others. In this case, the line's takt time will only decrease when the number of resources will have been increased in all of the aforementioned stations. The set of solutions generated during these iterations will differ according to the sequence of bottleneck stations considered (they would be denoted as equivalent solutions in multi-objective optimization) but all these solutions will be dominated since they have the same takt time but a higher number of resources.

Data availability statement

The data that support the findings of this study are available from the corresponding author, [AC], upon reasonable request.