Patrick A Pintus
email: pintus@univ-aix.fr.

Local Determinacy with Non-separable Utility *

Keywords: real business cycles, intertemporal substitution, capital-labor substitution, externalities, indeterminacy, sunspots. Journal of Economic Literature Classification Numbers: C62, D58, D91, E32

This paper introduces general formulations for both technology (with input substitution) and non-separable utility (compatible with balanced growth and stationary worked hours) into a benchmark RBC model. It is shown that intertemporal substitution and input substitutability lead to local determinacy and rule out stationary sunspot equilibria when labor demand is downward-sloping, in contrast with recent results obtained under the assumption of separable utility. The main intuition behind this result is shown to work as follows: in contrast with separable preferences, increasing the elasticity of intertemporal substitution in consumption necessarily implies decreasing the elasticity of constant-consumption labor supply, when utility is non-separable and concave, which affects unfavorably the occurrence of local indeterminacy.

Introduction

A major challenge of the literature exploring expectation-driven business cycles is to provide models that pass the empirical tests, e.g. calibration. Bennett and Farmer [5] have argued that non-separable utility is a key element to achieve this objective. However, Hintermaier [6] has recently shown that assuming Cobb-Douglas technology and a downward-sloping labor demand rules out local indeterminacy, in contrast with the examples by Bennett and Farmer [5] which turn out to violate concavity. On the other hand, Pintus

[9] has stressed that local indeterminacy may occur with small externalities (and downward-sloping labor demand): this requires both risk aversion to be small enough (very close to zero, indeed, for reasonable parameter values) and input substitutability to be large enough (greater than two). However, he assumes that utility is separable in consumption and leisure. Taking all recent results together, one wonders whether or not departing from Cobb-Douglas technology (which is key in Pintus [9]) may overturn the result by Hintermaier [6] and reconcile local indeterminacy with small externalities, in the spirit of the examples with low risk aversion studied in Bennett and Farmer [5].

In this paper, I address this open question and show that input substitution and non-separable utility compatible with both balanced growth and stationary worked hours do not help to improve the plausibility of local indeterminacy and sunspots. More precisely, I demonstrate that the result by Hintermaier [6] generalizes to any technology with constant (private) returns to scale, when utility belongs to the class of utility functions considered in King et al. [7] (see also King and Rebelo [8]). Under the maintained assumption of a downward-sloping labor demand, I show, first, that local indeterminacy requires both the elasticity of capital-labor substitution to be less than one (complementary inputs) and the coefficient of risk aversion to be less than the elasticity of input substitution (low risk aversion). Then I demonstrate that local indeterminacy is ruled out when the value of capital share is smaller than some upper bound (for example, less than a half for reasonable parameter values). In other words, complementary inputs do not lead to indeterminacy with negatively sloped labor demand when utility is non-separable, which puts some light on a conjecture stated in footnote 6 of Bennett and Farmer [5,p. 130]. As a corollary, I derive the expression of a minimal level of externalities below which indeterminacy is ruled out and illustrate that, for reasonable parameter values, this critical level of increasing returns is large and well outside the range of recent estimates (for instance, Basu and Fernald [2], based on constant shares of input costs).

The analysis also delivers an intuitive description that may help to understand the effects of non-separable utility on local indeterminacy, in terms of two key parameters stressed by Bennett and Farmer [5] (the elasticity of intertemporal substitution -the inverse of risk aversion -and the slope of the constant-consumption labor supply curve). In that context, the intuition behind the main result may be presented as follows. As already known from the literature (e.g. Benhabib and Farmer [3]), a greater elasticity of labor supply helps to get local indeterminacy for lower externalities. When utility is non-separable (and of the King et al.

[7] form) and when labor demand has a negative slope, local indeterminacy requires both inputs to be complementary (or, equivalently, a very elastic rate of return on capital investment) and risk aversion to be small enough.

Moreover, when capital externalities are small enough, indeterminacy requires the coefficient of (relative) risk aversion to be less than the elasticity of capital-labor substitution. Therefore, when technology, starting from the Cobb-Douglas configuration, is getting closer to the Leontief case, risk aversion has therefore to decrease to zero. However, imposing that utility is concave in both consumption and labor implies that the lower risk aversion, the lower the (constant-consumption) labor supply elasticity to the real wage, which in fact goes to zero when risk aversion goes to zero. In other words, although intuition may suggest that decreasing risk aversion should increase the plausibility of indeterminacy (indeed, this is the case when separable utility is almost linear; see Pintus [9]), this necessarily reduces labor supply elasticity when utility is non-separable and concave, which makes indeterminacy less likely. In contrast, separable utility allows to treat labor supply elasticity as a parameter that is independent of risk aversion. This also explains why complementary inputs are needed to get indeterminacy. As noted above, labor supply is close to inelastic when risk aversion is small so that wage and interest rate have to be elastic enough to move labor hours, i.e. technology has to be close to the Leontief case (see also the discussion in Bennett and Farmer [5,).

It turns out that, in our model with input substitution and non-separable utility, there is no net gain in terms of indeterminacy being more plausible. It is worth noting that our formulation of non-separability does not nest the separable case studied by Pintus [9]: equation (3) below shows that separability implies logarithmic consumption utility, which is less general than separable utility as assumed in Pintus [9].

The rest of the paper is organized as follows. Section 2 presents the extended Ramsey model, following Bennett and Farmer [5] and Pintus [9], and derives its perfect-foresight competitive equilibria. Section 3 establishes some necessary conditions for indeterminacy to occur and establishes the main result. Finally, Section 4 concludes, while the proofs are gathered in two appendices.


Arthur Bontemps
email: arthur.bontemps@uca.fr

Gaël Godi
email: gael.godi@uca.fr

Eric Fournely
email: eric.fournely@uca.fr

Rostand Moutou Pitti
email: rostand.moutoupitti@uca.fr

Joseph Gril
email: joseph.gril@cnrs.fr

Implementation of an Optical Measurement Method for Monitoring Mechanical Behaviour

Keywords: Marker tracking, crack monitoring, timber, homography, fiducial markers

come   L'archive ouverte pluridisciplinaire

Introduction

In experimental mechanics, measurements by optical methods are increasingly common. The main optical methods are: marker tracking; digital image correlation (DIC); grid method, photoelasticimetry; moiré; interferometry [START_REF] Brémand | Mesures en mécanique par méthodes optiques[END_REF]. These methods are very accurate and are among the only ones that can be used to perform full-field measurements [START_REF] Bornert | Assessment of Digital Image Correlation Measurement Errors: Methodology and Results[END_REF]. Obviously, DIC or grid method are more powerful than the marker tracking. Most of the time, both the marker tracking and the DIC are realised under controlled and restrictive conditions. The lightening conditions, the temperature of the room and the camera positions are controlled, allowing a high accuracy for the measurement of mechanical parameters. But theses conditions are rarely found in the monitoring of actual structures, making these methods unsuitable.

On the other hand, the computer vision community develops increasingly efficient methods for camera pose estimation [START_REF] Muñoz-Salinas | Mapping and Localization from Planar Markers[END_REF]. Augmented reality or robot navigation are examples among others. Efforts have recently been made by some authors to introduce the development of computer vision into photomechanics, such as planar markers for the monitoring of crack patterns in historic structures with a camera of UAVs [START_REF] Germanese | Long-Term Monitoring of Crack Patterns in Historic Structures Using UAVs and Planar Markers: A Preliminary Study[END_REF], or an image processing technique for automatic crack monitoring using homography and circular grid onto concrete object [START_REF] Valença | Automatic crack monitoring using photogrammetry and image processing[END_REF]. This paper is another attempt to use fiducial markers in mechanical measurements. With a need for a flexible and polyvalent method for monitoring crack propagation in wooden beams, an adapted marker tracking technique has been developed. This method uses the development of computer vision algorithms to increase marker location accuracy. Moreover, fiducial markers are unique and differentiable, allowing to measure large displacements. A homography process ensures reliable measurements while the camera is moving. Settings and application examples of the method in creep tests on wood will be described. This paper does not aim at analysing creep tests results, but at showing how this method may be used.

2 Material and methods

Preparation

This tracking technique uses ArUco (Augmented Reality University of COrdoba) markers, developed in Spain in 2013 and common in computer vision applications but employed here as a tool to measure relative displacements.

It consists in sticking markers on the surface of an object (e.g., a beam) and taking pictures of a portion of that object containing the marker. The marker follows the point where it is glued and by measuring the position of this marker in a photo, the relative position of the considered point can be calculated. The marker detection in a picture is illustrated in Figure 1.

ArUco markers and OpenCV library

Fiducial markers are reference objects placed in the field of view of the camera when an image or video frame is captured. ArUco markers are a specific type of fiducial markers, consisting of a black square with a white foreground that has been generated in a particular pattern (Figure 1). The black border surrounding the marker makes it easier for computer vision and image processing algorithms to detect it in a variety of scenarios, including variations in rotation, scale and lightening conditions. It looks like a QR code: a 2D binary pattern that can be detected using computer vision algorithms. ArUco markers were chosen since they are reliable, versatile and their recognition is very fast and easy [START_REF] Garrido-Jurado | Automatic generation and detection of highly reliable fiducial markers under occlusion[END_REF]. Throughout this study the markers are 18 × 18 mm, printed on standard paper with a printer Sharp MX-3051 and protected by matt lamination in order to avoid reflection and humidity degradation. A comparison between markers location on a regular paper and on a laminated paper showed that the lamination process did not affect the marker position.

The ArUco markers are integrated in OpenCV (Open source Computer Vision), an open-source library of computer vision algorithms, written in python and C++. The ArUco module that was affected here contains functions allowing to import and print the markers, detect them in a picture and many other actions. More information about the available functions can be obtained in OpenCV website.

In the present study the following configuration was used:

• Python 3.8.5;

• Anaconda environment;

• OpenCV 4.5.2.52.

The ArUco markers were detected using the detectMarkers function of OpenCV, using default parameters; the detection process is documented in [START_REF] Garrido-Jurado | Automatic generation and detection of highly reliable fiducial markers under occlusion[END_REF]. Each marker has its own pattern so that it can be identified separately from the others.

To improve the detection precision, the cornerSubPix method of OpenCV was used. The supbix function process is described in [START_REF] Forstner | A fast operator for detection and precise location of distinct points, corners and centres of circular features[END_REF]. The centers of the markers are interpolated from the corners returned by the previous functions. This detection procedure ensures a much better precision than one pixel. Many other functions are available in OpenCV, e.g., calibrateCamera, findHomography.

The homography process and the calibration targets

When a photo is captured, it is necessary to compute the real distance between markers. Therefore, a reference is needed to convert pixel distance into metric distance. For this purpose, the homography process is used, as explained in Appendix A. It consists in projecting the image plane into the intended metric plane, through the homography matrix H that converts distances and allows taking pictures with a view angle.

H =   h 11 h 12 h 13 h 21 h 22 h 23 h 31 h 32 h 33   (1)
This homography matrix is computed using a calibration target, which is a reference object composed here of 4 ArUco markers stuck onto an aluminium object and with known relative distances. Every photo has to contain this calibration target in order to perform the projection and compute the distance between the markers. The thickness of the calibration target and that of the rigid body where the markers are stuck are the same, in order to have all markers placed in the same plane. Two calibration targets were created: a horseshoe target and a window target (Figure 2). The input dimensions for the printer is presented in Figure 2 but their exact dimensions, checked by taking a zoomed-picture of the printed target next to a ruler, was : window target, 211.3 mm × 181.5 mm; horseshoe target, 74 mm × 73.5 mm.

These aluminium calibration targets are only subjected to temperature variations due to outdoor conditions. As the linear thermal expansion coefficient of aluminium is α al = 23 × 10 -6 K -1 , a temperature variation ∆T = 60 K (the maximum expected in a whole year), for the length L 0 = 210 mm of the window target, induced a length variation:

∆L = α al L 0 ∆T = 0.29 mm (2)
and for the 74 mm of the horseshoe target: ∆L = 0.102 mm. As the markers of the calibration targets are stuck onto the aluminium with superglue, it was worth checking if the adhesive expands with the aluminium. A climate chamber was used to apply a ∆T = 70 K on an aluminium calibration target and its dimensions was measured with another calibration target of control.

The experimental thermal coefficient determined with this setup was 22 × 10 -6 K -1 . This value, close to the theoretical one, proved that the markers follow the material subjected to thermal expansion. As a consequence, the thermal expansion has a significant effect on the calibration target's dimensions and requires a correction. The two calibration targets were adapted to different ranges. The horseshoe and the window measured distance variations between markers initially 4 cm and 17.5 cm apart, respectively. It was considered that the closer the dimension to the target distance, the better the precision. Note that when a picture is taken, the top-left marker of the calibration target used is taken as origin. Therefore, each picture has its own local coordinate system relative to the position of the calibration target.

Feasibility tests

The main camera used in this study is a Canon EOS 1D X Mark II: 5472×3648 pixels. It is a Single Lens Reflex (SLR) camera with CMOS (Complementary Metal Oxide Semiconductor) sensor. For all the photos taken in the context of this study, the parameters used were:

• Fixed focal length 50 mm;

• ISO varying from 1000 to 4000, depending on the outside light;

• Aperture F6.3;

• Autofocus activated;

• Shutter speed varying depending on the outside light.

As autofocus is activated and ISO and shutter speed are varying, no distortion correction is applied. Indeed this needs a camera calibration that is only efficient when the last mentioned parameters are constant. It was not conceivable to compute camera calibration and correct distortion at each picture because the method aims to be flexible and simple. Besides, for a 50 mm fixed-focal length distortions remain very low and only affect the border of a picture. In every case, the gray scale between the markers and the surrounding white border is estimated. The goal is to check that the gray scale gradient is as large as possible but without reaching saturation (i.e., a gray scale of 255 or 0 for the pixels around the markers). In order to provide this information for the analysis, all graphic captions contain three data:

• Diffgray, average gray scale gradient between black and white of all pictures of the running test; it should be as large as possible;

• Mingray, minimum gray scale level reached among all pictures of the running test; it should be above 0; • Maxgray, maximum gray scale level reached among all pictures of the running test; it should be below 255.

Test on a paper sheet

To validate the method in laboratory conditions, pictures of a paper sheet containing ArUco markers were taken (Figure 3). The distance between the markers was known. Some markers were used for the homography and some were used as subject to check if the distance between them was correctly computed.

In Figure 3, two calibration targets were used: markers 10, 11, 12, 13 (square of 74 mm) stand for the horseshoe and markers 14, 15, 16, 17 (rectangle of 210 × 180 mm) for the window. The distance between markers 1 and 2 (40 mm), as well as that between markers 3 and 4 (175 mm) were computed using both the horseshoe and the window target. These distances were chosen because they correspond to the distance measured during the application of this method. This paper sheet was photographed with two cameras: the Canon camera (20 MPx) and a smartphone Fairphone 3 (12 MPx). For both cameras 20 photos were taken at the same distance from the paper sheet: 120 cm. No tripod was used. The first results show a quite good accuracy, Table 1. As expected, the Canon camera is more reliable than the Fairphone 3 camera regarding the standard deviation. The dispersion is around 10 times lower and the result is closer to the true value for 40 mm. This difference may comes from multiple parameters such as the photo resolution or the manufacturing quality.

Test in real conditions

The next step was to check the feasibility in real conditions. A 4-point bending test was conducted until failure on a notched wooden beam of silver fir (Abies alba). The method was used for measuring the crack opening, by computing the positions of the 2 markers stuck onto the notch where the crack was expected to occur. The position of the two markers was converted into a distance with a homography using the horseshoe target. The variation of x distance between the two markers yields the crack opening in mode II (sliding) and the y distance that in mode I (separation). The curvature of the beam does not impact the distances because the local coordinate system rotates with the beam. The thickness of the calibration targets being 3 mm, the markers were also glued on a 3 mm thick PLA (Poly-Lactic Acid) support so that they were in the same plane.

The bending test was carried out respecting the norm NF EN 408+A1 [START_REF]AFNOR: NF EN 408+A1 -Structures en bois. Bois de structure et bois lamellé-collé[END_REF] and controlled in displacement. Every 2 mm imposed, or each time a crack was heard or observed, a series of 5 to 10 photos were taken to estimate the crack opening at the imposed displacement. Computing the mean of the 10 photos decreases the random uncertainties by reducing the dispersion. The photos were taken with two cameras: the Canon camera and a smartphone Asus X00TD Zenfone Max (12 MPx), Figure 4.

Figure 4 shows the measurements of the crack opening at each stage of the test. Each point represents the result of a photo. The graph clearly shows that the Canon camera gives much less dispersed results than the Asus camera.

The beam cracked at the 6th step, when the central deflection exceeded 20 mm.

Precision and application example

Effect of the angle and of the distance between camera and markers

As the method was planed to be used from different shooting positions, the impact of the observation angle and of the distance between camera and markers was estimated.

The effect of the distance, affecting the marker size in a picture (i.e., in pixels), was estimated using the same paper sheet (Figure 3). It was placed in front of the camera, which was moved back from 40 cm to 200 cm by steps of 10 cm, Figure 5. Five pictures were taken at each step. The effect of the observation angle was estimated with the camera placed on a tripod at 90 cm from the paper sheet (Figure 3). To simulate different view angles, the target was rotated around a vertical axis, then turned by 90°and rotated again, Figure 6. The paper sheet was rotated from 0 to 45°by steps of 5°. Ten pictures were taken at each step.

Application to creep tests

The method was then employed for the monitoring of long-term tests. The experimental campaign aimed at gaining a better understanding of long-term mechanical evolution, the delayed mechanical response of green or partially dried silver fir wood, coupled with crack propagation. Creep tests were performed in 4-points bending on notched wooden beams in outdoor sheltered conditions. The beams were 4000 × 175 × 63 mm and were notched at both ends like in the previous test, in order to trigger fracture initiation during the creep. Six beams were loaded at a time, by series of three spaced 50 cm from each other (Figure 7a). Three campaigns of multi-step creep tests were conducted, each lasting 4 to 6 months, with an initial load of around 350 kg subsequently increased by steps using additional weights. The first started in May 2021, the second in October 2021, the third in April 2022. Global movements of the beam, such as central or lateral deflections, were recorded using linear variable differential transducers (LVDT) placed under the beams. However, a large number of local data such as crack propagation or height variations due to moisture-induced expansion were also needed and not so easily obtained using conventional sensors (e.g., strain gauges, digital image correlation, linear transducers), due to outdoor conditions, limited available space and limited available devices. ArUco method has been tried as an alternative, owing to its flexibility. In order to resist to outdoor conditions, the markers were stuck using superglue onto a PLA support, 3 mm thick as well so that the calibration targets and the markers on the PLA are placed in the same plane.

As the calibration targets are subjected to outdoors conditions, a temperature correction was applied using the thermal expansion of aluminium 23 × 10 -6 K -1 and the outside temperature measured by the weather station close to the experiment.

During the creep tests the shrinkage-swelling, crack opening, length and height were measured with the method (Figure 7b). The crack opening was measured with the same scheme as the 4-points bending tests, i.e. the distance between two markers placed on the notch. The crack length and height (CL, CH) refer to the crack tip position in comparison to the markers placed at the notch, the one used for crack opening measurements. The crack tip position is measured by positioning a marker at the eyed-observed end of the crack. Hence CL and CH refer respectively to the horizontal and vertical distances between the crack tip marker and the marker on the notch. Considering that the markers on the notch are only subjected to transverse translation (i.e. neglecting the longitudinal dilatation and the crack opening in mode 2), they were a static reference for the crack length. The tip marker was placed with magnetic tape to a removable slit in steel. Both the marker and the slit could be moved. The horseshoe target was used for this measurement because the distance between the markers is of the same order of magnitude as the distance between the markers used for the crack opening measurement.

The shrinkage-swelling (or transverse expansion) of wood was measured with two markers placed at the extremity of the beam's height. Indeed, the wood is a hygroscopic material whose dimensions are in balance with its environment.

If the beam shrinks, the transverse expansion is negative and if it swells, the transverse expansion is positive. During the first campaign, twelve pairs of markers per beam were used for this measurement: on each of the four sides (face 1 left, face 1 right, face 2 left and face 2 right), three couples spaced 30, 60 and 90 cm from the notch. During the second and third campaign, six pairs of markers were used: at the left, the middle and the right of the two faces of each beam. The window target was used for this measurement because the distance between the markers is of the same order of magnitude as the distance between the markers used for this measurement.

Effect of the out-of-plane displacement

Wood is a hygroscopic material that may shrink and warp if subjected to meteorological variations, such as during the presented creep tests. Among the various warping modes, only the ones that locally distort the beam surface may involve an out-of-plane displacement. It is mostly the case of cupping, that happens in plain sawn lumber, and its effect should be given particular attention. Moreover, a calibration target misplacement may happen due to operator's error and also create an out-of-plane motion.

The effect of wood cupping on shrinkage-swelling measurement is represented in Figure 8. The 4 markers of the calibration targets always remain in the same plane because they are stuck onto a rigid aluminium object, but there is no possibility to have them in the same plane as all the beam's markers. However, the homography process projects all markers into the real world distances as if they were in the same plane as the calibration target.

The error due to such out-of-plane displacement can be estimated geometrically. Let define d as the out-of-plane displacement, D 1 and D 2 the distances between the camera and two markers on the beam, H p1 their distance projected in the window target plane, H p2 their real distance. According to the intercept theorem (Figure 8b):

D 1 H p1 = D 1 + d H p2 (3)
where H p1 = H; H p2 = H + er, D 1 = D 2 = D and er is the location error coming from the out-of-plane motion. As the relative error is the error over the theoretical value, the result is divided by H. It leads to :

er H = d D (4)
This result remains valid for any view angle. The out-of-plane displacement due to wood cupping is a slow process depend- ing on drying speed. If the calibration target is always placed in the same position, which was the case, then this displacement becomes a systematic error. This error is discarded when focusing on shrinkage-swelling variations. It would matter when analysing long-term variations, but as the studied beams started with high moisture content, the long-term shrinkage-swelling overwhelmed this systematic error.

Nevertheless the out-of-plane displacement induces also a random error when positioning the calibration target. A realistic upper bound for operator's target placement error during the creep test is d = ±1 mm, leading to a 0.2% error for D = 1000 mm.

Guidance for success and accuracy

It is very important to keep a large white space around the markers. A black line too close to the edge of the ArUco pattern would disturb the marker detection.

During the first campaign there were problems during the markers detection process because of black cut lines close to the pattern. The problem was overcome by reducing the photos quality to 1000 × 667 pixels, only where it was needed. It is supposed that reducing the photo quality decreases the impact of the black lines relative to the whole pattern.

Obviously, using a tripod decreases significantly the random dispersion of a measurement. Its impact was estimated by taking 60 photos with a tripod and 60 photos without a tripod of the same paper sheet Figure 3 with the Canon camera. The results showed a 4 times lower standard deviation for the tripod measurements.

3 Results and discussion

Precision

Effect of distance between camera and markers

Figure 9 shows the evolution of the error, the relative error and the standard deviation with the distance between camera and markers. The error corresponds to the difference between real and measured values, where a real value is the distance between markers taken as input for the printer and checked with a ruler after the printing, and a measured value the distance computed using the photos of the paper sheet. The relative error corresponds to the error divided by the real value, i.e. 40 mm for dy12 and 175 mm for dy34, and the standard deviation refers to the measured distance obtained from five pictures. Obviously, the relative error of the x distance cannot be calculated because the real value is 0. The distance between the camera and the markers influenced both the size of the marker and its position in a photo.

It may be noted that:

• The distance had no observable impact on the error and very little on the relative error; • The standard deviation increased with the camera-markers distance. This may be explained by the marker size reduction. For the 60 cm distance the The error associated with the distance between the camera and the markers was at most 0.08 mm.

Effect of angle

Figure 10 shows the evolution of the error, the relative error and of the standard deviation of the ten measurements, for an increasing horizontal angle. Similar results were obtained for the vertical angle. It may be noted that the error associated with the 90 cm distance camera markers of these pictures has been corrected for the values (Figure 10). The view angle had no impact on the standard deviation of a series of photos. However, it increased the error especially in the considered direction: error on dy increased with the horizontal angle and error on dx increased with the vertical angle. This was particularly true for the large distances (dy34), as if the measured distance of 175 mm was reduced with the augmentation of the view angle. The depth of field could have been the reason, as some markers were less sharp than others, but in this case an error would have appeared for the x distance measurements, and the standard deviation would have increased. Possibly the homography result was imperfect due to the low number of markers used for the matrix calculation. New trials with more than 4 markers for the homography is a perspective to solve that problem. The relative error can reach 0.3% in inappropriate conditions, such as a large distance measured with a small calibration target (dy34 ht). Otherwise (dy12 ht, dy34 wt, dy12 wt), it remains below 0.2% and even below 0.08% as long as the view angle is kept lower than 25°.

Discussion on the precision tests

The precision tests confirm that choosing a Single Lens Reflex camera instead of a cheap smartphone camera does decrease the dispersion of the photo measurement. When measuring a distance between markers 175 mm apart, the results suggest to use the window target (Figure 10). Both the view angle and the camera-marker distance are random errors. The view angle has more impact on the precision and should be as close to 0 as possible. Only dy12 ht, dy34 wt and dy12 wt results are relevant to the situation encountered in the creep tests. The conditions of the creep tests photos were usually close to 90 cm for the camera-marker distance and the angle between 0 and 25°, that is a relative error lower than 0.08%; at which must be added the out-of-plane error presented in section 2.3.3, that is a relative error lower than 0.2%. This leads to an estimated total error majored by 0.28%. It corresponds to an error of 0.112 mm for the 40 mm distance and of 0.5 mm for the 175 mm distance. Nevertheless, other potential errors like the lightening conditions and camera parameters are not considered. As a comparison, Malbezin et al. found an increase of the camera pose estimation error when the angle increases in the considered direction [START_REF] Malbezin | Measuring ARTootKit accuracy in long distance tracking experiments[END_REF]. Moreover, Bremand et al. showed that during a marker tracking process, the smaller the marker, the bigger the standard deviation of its location [START_REF] Brémand | Mesures en mécanique par méthodes optiques[END_REF]. Finally, Abawi et al. found that the distance between the camera and the target increases the standard deviation and the error of a pose estimation [START_REF] Abawi | Accuracy in optical tracking with fiducial markers: an accuracy function for ARToolKit[END_REF]. But they also found that the angle reduces the error especially between 30 and 40°, contrary to the observations of this study.

Results on creep tests

Measurement of transverse expansion

Figure 11 shows, for two beams of the second campaign, the correlation between the local transverse expansion near the notch and in the core of the beam where the whole height was measured (Figure 7). The expansion near the notch was computed using the markers of the crack opening in-between crack propagation events. The results are given as a fraction of the initial distance, so that they can be compared.

The results show a good correlation between the two measurements of trans- verse expansion, with the value at the notch slightly higher than that measured in the middle of the beam. The transverse shrinkage-swelling rate of wood is about 0.2 to 0.3 %/% [START_REF] Guitard | Mécanique du Matériau Bois et Composites[END_REF], which at least corresponds to 0.35 mm of variation for 1% of moisture content change and with an initial distance of 175 mm. As the precision estimated is 0.28%, a variation of 1% of moisture content may not be identified. Nevertheless, a variation of more than 1%, such as the longterm shrinkage-swelling observed during the creep tests, should be identified. The correlation between both transverse expansions in Figure 11 is very good and the slope slightly lower than 1 may be explained by variation of wood properties along the beam combined with a delay resulting from the diffusion process.

Measurement of crack parameters

Figure 12a shows the crack opening of a beam subjected to crack propagation during the first campaign. The evolution of wood equilibrium moisture content (EMC) corresponding to the external climate is added in order to indicate whether the beam tends to shrink or to swell, as this should have an impact on the crack opening measurement. The EMC was computed using the Merakeb-Pedersen model ( [START_REF] Pedersen | Combined heat and moisture transfer in building constructions[END_REF], [START_REF] Merakeb | Modélisation des hystérésis de sorption dans les matériaux hygroscopiques[END_REF]) that takes as inputs the previous moisture content, the outside temperature and relative air humidity measured with a weather station close to the experiment. A rapid increase of crack opening is observed at the beginning of the test, after the initial loading, then again at the end of the 4-months period when additional loads are applied. Figure 12c focuses on an intermediate 2-months period, were the crack opening seems to be triggered by drying phases, since a significant increase is observed each time EMC falls below 10%. Note that the rupture eventually occurred on the left side of the beam, while the data shown are those of the right side.

Figure 12b shows the relationship between the crack opening and the crack length.

Three types of propagation are identified:

• Initiation and propagation, both CO and CL increase; • Shrinkage with propagation, CO decreases because of the shrinkage but CL increases due to crack propagation. Maybe the crack propagation is only on the surface, which would explain why CO does not increase with CL; • Swelling or apparent threshold effect, CL is constant but CO increases. It could be either due to the swelling of the beam or due to a threshold effect before crack propagation as identified by [START_REF] Pambou Nziengui | Notched-beam creep of Douglas fir and white fir in outdoor conditions: Experimental study[END_REF]. The threshold effect may be explained by the heterogeneity of the wood, if a knot is in the crack trajectory it would stop its propagation while the crack opening continues to grow.

The final CL after the beam cracked is not represented in the graph but reached 1290 mm. The results showing that the shrinkage helps the crack to propagate are in line with another similar experimental study ( [START_REF] Pambou Nziengui | Notched-beam creep of Douglas fir and white fir in outdoor conditions: Experimental study[END_REF]) that identified the same phenomenon. Moreover, Figure 12b shows a crack opening varying from 0 to 3 mm and a crack length varying from 0 to 1290 mm, to be compared with those of that previous study, a crack opening between 0 and 3 mm and a crack length between 0 and 800 mm. Finally, a crack propagation of silver fir wooden beams is characterised by an increase of the crack opening of 0.3 mm minimum and an increase of the crack length of at least a few centimeters. 0.3 mm corresponds to 0.75% of the 40 mm markers distance, and the precision was majored by 0.28%. Therefore, for both the crack opening and the crack length the method is sufficiently reliable to identify the phenomenon as long as a correction is made for the thermal expansion of the calibration target.

Conclusion

The ArUco marker tracking technique presented in this article is flexible.

Owing to the homography procedure, it enables to take photo with view angles and with different camera locations while maintaining a low error.

This advantage was useful in the context of the creep test where the experimental conditions were not suitable for other optical methods. The method also allows to compute distances between markers in a wide variety of ranges without additional effort. For example, the markers may be stuck 20 mm apart as well as 500 mm apart. The limitations is that all markers have to be in the same picture, the precision then comes from the number of pixels. On this subject, the marker tracking is more flexible than the LVDT sensors or transducers that have a fixed measuring range, so that it is necessary to consider what the LVDT sensor will be used for before purchasing it. The financial cost of the method is attractive. The most expensive necessary device is the camera. A much better accuracy is obtained with a good SLR camera rather than with a smartphone. The markers just need to be printed on regular paper or cardboard and protected, for example with a matt lamination. A piece of metal is a good solution to have a reference calibration target for the homography. A computer is obviously necessary as well. Once the camera has been acquired, the number of measurements can be increased at no additional costs. The method is quite simple to use as all the important functions are already programmed by the OpenCV community. Basic skills on python are necessary to adapt the use of the function to a specific case. The algorithms can be self-taught and for free with OpenCV and their community. The precision of the method was estimated considering the out-of-plane displacement, the impact of a view angle and of the distance camera-markers.

The most significant relative error may come from the out-of-plane displacement, it was geometrically determined that it can reach 0.2% due to calibration target misplacement. Then experimental results showed that the view angle had more impact on the measurements than the camera-markers distance. Under suitable conditions, i.e. a view angle of less than 25°, the relative error was below 0.08%. The total relative error is hence majored by 0.28%. The printed markers must have a large white space around the corners in order to ensure a good detection. Furthermore, they should be protected and stuck to a rigid body. As an application examples, the ArUco markers tracking method have been used to measure the crack parameters and the shrinkage-swelling of wooden beams subjected to multi-steps creep tests. The results of the measurements made with the method are consistent with the physical phenomenon involved.

Firstly, an increase of the crack propagation due to wood shrinkage of the beam was identified. Secondly, a good correlation between the transverse expansion of the beams in two different locations was obtained.

Other parameters probably impact the precision of the method but have not been estimated, for example picture brightness and camera parameters. The out-of-plane error analysis may be developed in case of wood application by computing the theoretical cupping of wood based on its growth rings. Moreover, it would be interesting to estimate the impact of the view angle when the homography is computed with more than 4 markers.

• x = (x, y, 1) are the homogeneous 2D image coordinates, related to the projective space. When passing from the Euclidean coordinates to the homogeneous coordinates, a dimension is added. This distance is fixed to 1 and allows to get mathematical coherence; The transformation needed to get the homogeneous 2D real-world coordinates is: x.( h31 x + h32 y + 1) = h11 x + h12 y + h13 ŷ.( h31 x + h32 y + 1) = h21 x + h22 y + h23 By transforming H into a vector, the following equation can be written:

X =   X/Z Y /Z 1   =   x ŷ 1   = X Z (
x y 1 0 0 0 -xx -yx -1 0 0 0 x y 1 -xŷ -ŷy -1

.( h11 , h12 , h13 , h21 , h22 , h23 , h31 , h32 , 1) T = 0 0 (A3) Basically:

A. H = 0

Solving this problem, avoiding H = 0, is the homography computation. It is a minimization problem as the solution is never perfect for images obtained experimentally. It is worth noting that 1 point in a picture with known realworld positions gives 2 equations, and the homography matrix consists of 8 unknowns. As a consequence, at least 4 points with known physical positions are necessary to compute the homography matrix. These 4 points may be the 4 corners of a single marker, or as in the case of this study 4 centers of 4 markers. There is at least two reasons for using 4 centers of 4 markers instead of 4 corners of a single marker:

• The centers are more precisely located than the corners; • The projection of the pixels in the physical real world plane is globally better with 4 centers rather than 4 corners. Indeed, the projection with 4 corners is good mainly around that reference marker but not for remote pixels.

Obviously, 3 collinear points give a useless redundant equation, so that no more than two should be alined. If more than 4 points are available, the homography solution should gives a better minimization.

The homography used to get real world coordinates of a given picture is the same for all markers. Thus, they are all projected on the same plane. As a consequence, the homography has to be computed one time for each photo so that the distance between two markers is correctly computed even with a view angle involving a perspective.

Fig. 1

 1 Fig. 1 Detection of ArUco markers stuck on a notched beam. An ArUco marker is a 6 × 6 matrix of black or white squares.

Fig. 2

 2 Fig. 2 Drawing of the calibration target: (a) the horseshoe target and (b) the window target. Numerical values are in millimeters.

Fig. 3

 3 Fig. 3 Photo of a paper sheet containing ArUco markers taken with the Canon camera. Markers 1 and 2 are 40 mm apart, markers 3 and 4 are 175 mm apart. Markers 10, 11, 12, 13 form a square of 74 mm and markers 14, 15, 16, 17 form a rectangle of 210 × 180 mm.

Fig. 4

 4 Fig. 4 Crack opening measurement: (a) photo taken with the smartphone Asus; (b) photo taken with the Canon camera; (c) results of crack opening as a function of the central deflection imposed, left side of the beam. Gray scale: diffgray = 171, mingray = 37, maxgray = 248.

Fig. 5

 5 Fig. 5 Photos taken for the estimation of the impact of distance between camera and target. (a) camera 60 cm from the target; (b) 130 cm; (c) 200 cm.

Fig. 6

 6 Fig. 6 Photos taken to estimate the impact of an angle. (a) camera straight in front of the target; (b) vertical angle of 45°; (c) camera straight in front of the target turned 90°; (d) horizontal angle of 45°, simulated by the 90°turn of the target.

Fig. 7

 7 Fig. 7 Configuration of the creep tests: (a) Illustration of creep tests of three notched beams, (b) ArUco markers measurements.

Fig. 8

 8 Fig. 8 Effect of wood cupping on the shrinkage-swelling measurement of a beam: (a) illustration of the out-of-plane displacement ; (b) application of the intercept theorem.

Fig. 9

 9 Fig.9The impact of the distance between camera and markers: (a) error: difference between real and measured distance, relative error: error divided by real distance, and standard deviation of measured distance on ten photos for the y distances; (b) error, standard deviation for the x distance. dy12 and dy34 are the y distance between markers 1-2 (40 mm) and 3-4 (175 mm), respectively; dx12 and dx34 are the x distance between markers 1-2 (0 mm) and 3-4 (0 mm), respectively. Calibration target used for the homography: ht, horseshoe target; wt, window target. Gray scale: diffgray = 156, mingray = 29, maxgray = 201.

Fig. 10

 10 Fig. 10 Impact of a horizontal angle: (a) error, relative error and the standard deviation on the y distance; (b) error and standard deviation on the x distance. The caption is the same as for Figure 9. Gray scale: diffgray = 164, mingray = 35, maxgray = 216.

Fig. 11

 11 Fig. 11 Correlation between transverse expansion at different locations. Beam S25 and S26 are considered on face 1 on the left. Gray scale: diffgray = 142, mingray = 13, maxgray = 253

Fig. 12

 12 Fig. 12 Evolution of the crack parameters for beam S14: (a) simultaneous evolution of crack opening right side and daily mean wood equilibrium moisture content (EMC) during 4 months, where zone 1 is the end of the primary creep, zone 2 the 2 months concerned by (c) and zone 3 is the start of the daily load increase; (b) crack opening as a function of crack length, both left side; (c) time evolution of crack opening right side focused on 2 months. Gray scale: diffgray = 140, mingray = 9, maxgray = 252.

  = h11 x + h12 y + h13 ŷ.Z = h21 x + h22 y + h23 Z = h31 x + h32 y + 1 Thus:

Table 1

 1 Results for 20 photos taken with two cameras: a smartphone camera and a Canon SLR camera. Gray scale: diffgray = 155, mingray = 18, maxgray = 190

	Markers 1-2	Markers 3-4

  • H is the 3 × 3 homography matrix: 11 h 12 h 13 h 21 h 22 h 23 h 31 h 32 h 33• ω is a scale factor. In homogeneous coordinates, (x, y, 1) and (ωx, ωy, ω)represent the same point for ω a non-zero scalar. It may be fixed at ω = 1 h33 and injected into H. In this case let's note:

					
			H =			(A1)
	hij =	h ij h 33	,	(i, j) = ((1, 2, 3), (1, 2, 3))

h

Compliance with Ethical StandardsOn behalf of all authors, the corresponding author states that there is no conflict of interest.

Acknowledgements

The authors would like to thank the Auvergne Rhône-Alpes region for the Ph.D. scholarship as well as the funding of the Hub-Innovergne program of the Clermont Auvergne University. Moreover, the method could only be developed thanks to the significant contributions of the OpenCV community. Finally the authors thank Benoît Blaysat for lending equipment and fruitful discussions.

Appendix A Homography

The aim of the homography is to project the coordinates in a given plane into another, as illustrated in Figure A1 coming from [START_REF] Hartley | Multiple View Geometry in Computer Vision[END_REF]. In the left picture the photo is taken with a view angle and on the right picture it is projected onto the plane parallel to the wall. In this study, the goal is to go from the virtual image plane to the physical real world plane, where the real distance between two markers is defined by their coordinates. The detectMarkers function of OpenCV returns the position of the markers into the 2D image coordinates (i.e., in pixels). In order to get real world coordinates (i.e., in meters), a homography must be computed:

Where:

• X = (X, Y, Z) are the real world coordinates;


Onrawee Laguerre

Steven Duret

N Chaomuang

A Denis

Evelyne Derens-Bertheau

P Mawilai

F T Ndoye

T Pongsuttiyakorn

S Rakmae

K Srisawat

P Sirisomboon

P Pornchaloempong

Influence of long

supply chain were developed. The numerical and the experimental values were in good agreement.

These models were able to predict the product mass loss and peel color evolution as a function of a supply chain scenario. Temperature and relative humidity are the key factors that determine the preservation of quality and shelf life of mangoes (Mangifera indica L., [START_REF] Yasunaga | Quality changes in fresh mango fruits (Mangifera indica L.'Nam Dok Mai') under actual distribution temperature profile from Thailand to Japan[END_REF]). Mango is a time-and-temperature sensitive fruit with an optimal storage temperature of 13 °C [START_REF] Kader | Mango quality attributes and grade standards: A review of available information and identification of future research needs[END_REF] and an optimal relative humidity of 85-95 %RH [START_REF] Hussen | Impact of temperature and relative humidity in quality and shelf life of mango fruit[END_REF]. To maintain the product quality, mangoes should be properly handled during harvest, preparation, packing and transportation throughout the entire chain [START_REF] Sivakumar | Maintaining mango (Mangifera indica L.) fruit quality during the export chain[END_REF].

Keywords

The main quality attributes of mango are peel color, firmness, sugar content, mass and visual quality [START_REF] Penchaiya | Modelling quality and maturity of 'Namdokmai Sithong'mango and their variation during storage[END_REF]. [START_REF] Yasunaga | Quality changes in fresh mango fruits (Mangifera indica L.'Nam Dok Mai') under actual distribution temperature profile from Thailand to Japan[END_REF] investigated the effect of storage temperature and relative humidity on quality changes in fresh mango (Mangifera indica L. cv. "Nam Dok Mai")

shipped from Thailand to Japan by marine transport. The ambient temperature and relative humidity were recorded throughout the supply chain over the 18-day shipment period. The authors reported that non-refrigerated (~25 °C) transport conditions and the duration of transport between the farm and the packing house exerted a significant impact on product quality. In other words, fruit should be cooled down and stored at an appropriate temperature as soon as possible after harvest. Numerous experimental studies on mango quality can be found in the literature, but a few modeling studies have been conducted. [START_REF] Fukuda | Modelling the relationship between peel colour and the quality of fresh mango fruit using Random Forests[END_REF] modeled the relationship between the peel color and some quality attributes (firmness, total soluble solids -TSS, and ascorbic acid) of mangoes (cv. "Nam Dok Mai" and "Irwin") using an artificial intelligence approach (random forests). [START_REF] Penchaiya | Modelling quality and maturity of 'Namdokmai Sithong'mango and their variation during storage[END_REF] carried out a laboratory study and proposed first-order exponential models to predict the color and firmness evolution of mango (cv. "Nam Dok Mai") as a function of time during constant storage temperature, and a first-order logistics model for variations in the storage temperature. [START_REF] Schouten | Mango firmness modeling as affected by transport and ethylene treatments[END_REF] have developed a kinetic model to predict the mangoes softening (cv. 'Keitt' and 'Kent'), which depends on the ethylene releasing rate. This rate is related to the storage temperature according to the Arrhenius' law.

For years, the air cargo industry has played an important role in global food trade. As distribution by air typically costs 12-16 times more than marine transport (The World Bank, 2009), the key drivers for utilizing air transport are the short product shelf life and high product value [START_REF] Pelletier | Analysis of air cargo temperature variations during transport operations[END_REF]. The short duration of air transport in comparison with marine transport allows minimization of product loss throughout the supply chain. For most air shipments, the time spent in flight corresponds to approximately 50 % of the total travel time, including storage and transfers at airports [START_REF] Sharp | Air transport of perishables[END_REF].

Maintaining suitable temperatures during air transport operations is challenging since the cargo may spend a long period on the tarmac. Variation of the ambient temperature due to the logistic at the airport has a significant impact on product temperature fluctuations [START_REF] Thompson | Air transport of perishable products[END_REF]. Many authors revealed that high temperature variation was influenced by solar radiation [START_REF] Pelletier | Analysis of air cargo temperature variations during transport operations[END_REF], material used to manufacture the walls of the container [START_REF] Heap | Cold chain performance issues now and in the future[END_REF][START_REF] Oskam | Validated models of the thermodynamic behaviour of perishables during flight[END_REF][START_REF] Villeneuve | Effect of environmental conditions on air shipment of perishables during ground operations[END_REF], and airflow pattern depending on the ventilation system during the flight [START_REF] Emond | In-flight temperature conditions in the hold of a widebody aircraft[END_REF].

In spite of the temperature control issues, few studies on in-flight environmental conditions and their impacts on fresh fruit and vegetables have been undertaken. Such studies are essential in order to develop new handling methods that preserve product quality throughout a supply chain. Pelletier (2010) conducted a thermal analysis on horticultural products in an individual box placed in an aircraft container (loaded with several boxes) exposed to detrimental temperatures in the laboratory. In this study, different fruit sizes and packing arrangements were tested, and it was observed that there were small temperature differences between the pulp and air temperatures within the individual boxes (~1.5 °C) and relatively fast rates of temperature change were observed even in the core region of the boxes i.e. the temperature increased from 3 °C to 12 °C within 7 h. For the aircraft container, similar thermal behavior was observed in the fruit near the external environment, particularly in the case of the boxes located in the top row.

In Thailand, high-quality mango (Mangifera indica L. cv. "Nam Dok Mai") is one of major fruits exports. The export value of fresh mango increased about 6 folds in a decade from 500 million Thai Baht in 2011 to almost 3,000 million Thai Baht in 2021 (Thai Information and Communication Technology Center, 2021). This figure reflects the great demand of Thai mango from international markets. However, around 15-50% of this mango cultivar harvested are lost during transportation [START_REF] Sardsud | Postharvest loss assessment of mango cv[END_REF]. To ensure fresh mango quality and minimize loss throughout the air transport supply chain, this study was conducted with the following objectives:

(i) To carry out a preliminary study to highlight the time-temperature and fruit quality evolution throughout the supply chain from postharvest of mangoes in Thailand to France including direct and indirect flights.

(ii) To develop kinetic models to predict product mass loss and color evolution as a function of temperature.

(iii) To study numerically the influence of supply chain scenarios on product quality.

As a first approach, a simple configuration was studied in this work. It involved the placing of mangoes in corrugated boxes under blind conditions during air shipment, i.e. no control of the box position in the aircraft hold which was not temperature-controlled.

Materials and methods

Mango characteristics

The mango investigated was Mangifera indica L. cv. "Nam Dok Mai". Based on measurements of 20 mangoes, the average mass was 0.415 ± 0.037 kg, the volume was 4.02×10 -4 ± 3.75×10 -5 m 3 , the surface area was 2.60×10 -2 ± 1.28×10 -3 m 2 , and the density was 1030 ± 19 kg•m -3 . Other thermal properties of mangoes (cultivar not specified) were assumed from ASHRAE (2018) as a moisture content of 81.71 %, a thermal conductivity of 0.418 W•m -1 •K -1 , a specific heat capacity of 3740 J•kg - 1 •K -1 (above freezing), and a heat of respiration of 356.0 mW•kg -1 at 25 °C.

Supply chain

This study was carried out in March 2022. Mangoes were harvested 90-93 days after anthesis (80-85 % maturity) from an orchard in Chachoengsao province (13.606977 °N, 101.298767 °E), then transported to a packing house in Nakhon Pathom province (13.80253 °N, 100.06714 °E) by a non-controlled temperature truck (157 km, 2.5 h travel duration from the orchard). At the packinghouse, the fruit were sorted and graded according to the Thai Agriculture Standard for exported fresh mangoes (MOAC, 2015), which judged its weight (> 350 g in this study) and external appearance i.e., absence of defects. To control fruit flies, the fruits were immersed in hot water until their core temperature reached 46 °C, then were held at this temperature for 10 min (total duration of 1 h). They were dried under room temperature conditions using industrial fans before being wrapped in foam net (made of polystyrene) to minimize product damage by vibration. Twelve fruits were arranged in a corrugated box (Fig. 1) with 4 vent holes covered by net for insect prevention, corresponding to 21 % of occupied volume. In a box, 8 recorders (iButton®, precision ±0.2 °C after calibration, ±5 %RH according to the manufacturer) were installed: 4 recorders near the side walls for internal air temperature monitoring (denoted "Ai" in Fig. 1b), 2 recorders in front of the vent holes for external air and relative humidity monitoring ("Ae", Fig. 1b) and 2 recorders inserted under the foam net, sensors facing the product, and stick on the mango peel for the product surface temperature monitoring ("M", Fig. 1c). These temperatures were recorded every 5 min along the supply chain from Bangkok, Thailand to Paris, France.

To ensure the repeatability, four instrumented boxes of mangoes were transported from the packing house to the airport cargo terminal for customs control, transfer to aircraft, loading and lift-off. Two boxes were sent by a direct flight and the other two boxes by an indirect flight on the same day.

Tables 1 and2 present the time schedule of these 2 supply chains.

To highlight the influence of air transport conditions, two boxes of the same batch of mangoes were transported to the Department of Food Engineering, School of Engineering, King Mongkut's Institute of Technology Ladkrabang (KMITL), Thailand. These boxes were stored in a temperature-controlled room (average temperature 20.5 ± 0.6 °C and relative humidity 68 ± 5.0 %RH) for a duration of 11 days.

Fig. 2a shows the upstream stages of the studied supply chain in Thailand from harvest to Suvarnabhumi Airport in Bangkok (BKK).

Quality assessment

Individual mangoes to be shipped to France were weighed and labelled before packing in the boxes while the temperature recorders were triggered to start the acquisition. This moment was considered as the initial time and the initial mass for the mangoes from both flights. As it was too complicated to conduct color measurement at the packing house, the initial color value of all mangoes shipped to France was considered to be the same value as those measured at the laboratory in Thailand for 20 °C storage treatment. This consideration could be justified because of the same fruit batch. These two quality attributes were successively measured at different time points during 20 °C storage at INRAE for both direct and indirect flights: day 3 (the first evaluation day in France), day 7, day 9, day 11 and day 14. The measurement was ended after day 14 because the mangoes were visually too deteriorated and considered as having reached the end of their saleable lifetime. Several quality attributes were assessed: mass loss, peel color, pH, TSS, firmness and visual quality based on score ranging (peel color, shrivelling, chilling injury, and decay). Almost the same quality attributes were measured for the mangoes stored in the laboratory in Thailand.

To avoid data redundancy, only the product mass loss and peel color evolution are presented in this article since they are considered as the main criteria of product saleability.

Mass loss

The product mass loss is related to water evaporation from the surface which leads to product shrinkage and texture modification.

Nine mangoes from each supply chain (direct and indirect flights) were randomly selected and weighed at different time points (0, 3, 7, 9, 11 and 14 days) using a digital balance (Sartorius CPA34001P, accuracy 0.1 g). Referencing the initial mass of the fruit (𝑚 0 ) measured in the packing house in Thailand and the mass on different days (𝑚 𝑡 ) at INRAE in France, the % mass loss was calculated using Eq. (1):

𝑀𝑎𝑠𝑠 𝑙𝑜𝑠𝑠 (%) = 𝑚 0 -𝑚 𝑡 𝑚 0 × 100 (1)

Color

The peel color of each individual mango was assessed by a non-destructive spectroscopic method using a chroma meter because it is handy and practical. The value of L*, a* and b* were recorded by placing the sensor at 2 positions on the mango peel (1 measurement/mango side) of 9 mangoes. It is to be emphasized that L* represents the product whiteness, a* the product redness (positive value) and greenness (negative value), and b* the product yellowness (positive value) and blueness (negative value). The b* represents the product visual quality to the greatest extent.

Model development

Thermal model

Product temperature is considered uniform in a box and its temperature varies exponentially with the ambient temperature throughout the supply chain as expressed in Eq. (2).

𝑑𝑇 p 𝑑𝑡 = (𝑇 𝑎 -𝑇 𝑝 ) 𝜏 𝑇 (2)
where 𝑇 𝑝 is the predicted product temperature, 𝑇 𝑎 is the measured ambient temperature, and 𝜏 𝑇 is the characteristic time of the temperature evolution (h) which depends on the product properties and geometry (heat capacity, mass, and surface area) and the heat transfer coefficient in the box. The 𝜏 𝑇 value was identified by fitting the product temperature to that measured throughout the supply chain.

Mass loss model

As in other horticultural products, mango mass loss is determined by the difference in water vapour concentration between the mango peel and the surrounding air. A simple model describing the evolution of water loss (𝑚 𝑤 ) due to evaporation was used in our study (Eq. (3)):

𝑑𝑚 𝑤 𝑑𝑡 = 𝑘 𝑡𝑎 𝐴(𝐶 𝑤 -𝐶 𝑤,∞ ) (3)
where 𝑘 𝑡𝑎 is the moisture transfer coefficient between the product and the air (including peel resistance) (m•s -1 ), where 𝐴 is product surface area (= 2.60 × 10 -2 m 2 , average of 20 mangoes), 𝑚 𝑤 is the water loss due to transpiration (kg), 𝐶 𝑤 is the water vapour concentration in the mango peel (kg water•m -3 humid air). This value is related to the saturated vapour pressure at the product surface temperature 𝑃 𝑠𝑎𝑡 (𝑇 𝑠 ), which can be represented by Eq. (4):

𝐶 𝑤 = 𝑀 𝐻 2 𝑂 𝑃 𝑠𝑎𝑡 (𝑇 𝑠 )𝑎 𝑤 𝑅𝑇 𝑠 (4)
𝑀 𝐻 2 𝑂 is the molecular weight of water (18 kg•mol -1 ) and R is the gas constant = 8.314 J.mol -1 .K -1 . The water activity (𝑎 𝑤 ) is considered constant and equal to 1 throughout the supply chain. 𝐶 𝑤,∞ is the water vapour concentration in the surrounding air (kg water•m -3 humid air), and it can be calculated using Eq. ( 5):

𝐶 𝑤,∞ = 𝑀 𝐻 2 𝑂 𝑃 𝑠𝑎𝑡 (𝑇 𝑎 ) 𝑅𝑇 𝑎 • 𝑅𝐻 100 (5)
The saturation pressure of water vapour (𝑃 𝑠𝑎𝑡 (𝑇) in Pa) can be determined from Eq. ( 6) [START_REF] Batina | Water vapour condensation in a partly closed structure. Comparison between results obtained with an inside wet or dry bottom wall[END_REF]:

𝑃 𝑠𝑎𝑡 (𝑇 𝑎 ) = 10 (2.7877+ 7.625𝑇 𝑎 241.6+𝑇 𝑎 ) (6)
The value of the moisture transfer coefficient (𝑘 𝑡𝑎 ) was obtained by fitting the predicted mass loss (obtained using Eq. ( 3)) with the experimental values reported by [START_REF] Penchaiya | Assessing of postharvest quality of mango fruit with change of texture characteristics[END_REF] for the same mango cultivar (cv. "Nam Dok Mai"). In that study, mangoes were stored under 4 different constant ambient conditions (constant temperature and relative humidity). For each storage condition, the mass loss of 30 mangoes was measured every day until day 8 and then every 2 days until day 14. The fitted 𝑘 𝑡𝑎 value (3.01×10 -4 m.s -1 ) was then used in Eq. (3) to predict the product mass loss under variable conditions such as those investigated in the experimental study conducted by [START_REF] Penchaiya | Assessing of postharvest quality of mango fruit with change of texture characteristics[END_REF] i.e., mangoes stored at a temperature and relative humidity of 13.4 °C/91.6 %RH respectively for 12 days then under 25-28 °C/75.5 %RH conditions for an additional 8 days (20 days in total). Good agreement between the predicted and experimental values (Fig. 3) was obtained.

After having validated the model as described previously, the temperature and relative humidity of the air and the surface temperature measured on the peel of the mango and recorded every 5 min.

throughout the 14-day supply chain we investigated, then were applied to Eq. ( 2) to predict the product mass loss evolution.

Color

As mentioned in Section 2.3.2, 𝑏 * is a parameter representing the product yellowness, which is a commercial quality criterion for mango.

where 𝑏 𝑚𝑖𝑛 *

and 𝑏 𝑚𝑎𝑥 * are the lowest and highest asymptotic values of the measured 𝑏 * , respectively.

The characteristic period of time during which color evolves, 𝜏′, is defined as:

𝜏′ = ∫ 𝑘 𝑟𝑒𝑓 𝑒 𝐸 𝑎 𝑅 ( 1 𝑇𝑟𝑒𝑓 - 1 𝑇(𝑡 ′ ) ) 𝑑𝑡′ 𝑡 0 (8)
where ∆𝑡 * is a dimensionless time shift factor. ∆𝑡 * depends on the initial value of 𝑏 * (𝑏 𝑖𝑛𝑖 * ), which can be estimated using Eq. ( 9):

∆𝑡 * = 𝑙𝑛 ( 𝑏 𝑚𝑎𝑥 * -𝑏 𝑚𝑖𝑛 * 𝑏 𝑖𝑛𝑖 * -𝑏 𝑚𝑖𝑛 * -1) (9)
In the case of the mango cv. "Nam Dok Mai" stored at temperatures ranging from 13 to 30 °C, [START_REF] Penchaiya | Modelling quality and maturity of 'Namdokmai Sithong'mango and their variation during storage[END_REF] values can also be estimated by an expert.

The simulation was undertaken with time intervals 𝑑𝑡 of 5 min, (being equivalent of the data acquisition during air transport). It should be borne in mind that the position of the boxes on the aircraft was unknown, and [START_REF] Emond | In-flight temperature conditions in the hold of a widebody aircraft[END_REF] reported large spatial temperature variations on aircraft. In general, the air and surface temperatures were very close: the temperature difference between the air and the surface temperature of the mangoes was 0.5 °C on average for the direct flight and 0.7 °C on average for the indirect flight.

Results and discussion

Experimental results

Air and product temperatures throughout the supply chain

The difference between the air and product temperatures of the two replications (Box 1 and Box 2 on the same flight) was not significant (0.5 °C on average). This led us to assume that there is low air renewal in the boxes, and thus the air was stagnant. This might be due to the fact that during air transport, the studied boxes were surrounded by other boxes or they were located in a poorly ventilated position within the aircraft. Hence, conduction is the dominant heat transfer mode inside the box. This observation is in agreement with that reported by [START_REF] Pelletier | Air transport of horticultural products: A thermal analysis[END_REF] who conducted thermal analysis in a laboratory using a similar configuration. It needs to be noted that small differences between the air and the fruit surface temperatures could also be observed under high airflow rate conditions. However, high airflow was not our case since the boxes were closed and held in low airflow environments along the supply chains i.e., airflow rates are typically 0.02-0.06 L•kg -1 •s -1 in transport and 0.001-0.002 L•kg -1 •s -1 in cold storage [START_REF] Wu | Virtual cold chain method to model the postharvest temperature history and quality evolution of fresh fruit-A case study for citrus fruit packed in a single carton[END_REF]. The flow rate would be even lower during air transport because of no ventilation system in the aircraft container (ULD). Moreover, the vent holes on the walls of the boxes are covered by net, which in turn resists air infiltration.

There were instances where the difference between the air and product temperatures varied by up to 5

°C during periods of rapid cooling (beginning in cold storage) and warming (beginning of nonrefrigerated transport to INRAE). In the indirect supply chain (between activities 9 and 10, Fig. 5), the mango temperature decreased to about 8 °C, and this can be explained by the fact that the boxes were stored in a cold room used by the logistics company.

Product mass loss

Fig. 6 presents the percentage of product mass loss from harvest to storage in a room maintained at 20 °C for 12 days after the direct and indirect flights and that observed following storage in a laboratory test room maintained at a temperature of 20 °C. Each point represents the average value of 9 mangoes. The linear relationship with time is obtained for all conditions. The mass loss seen in the case of the laboratory test room maintained at a temperature of 20 °C is the lowest, while that observed for the indirect flight supply chain was the highest. This result can be explained by the higher amplitude of temperature fluctuations and the longer duration of air transport and transit periods.

These temperature fluctuations induce relative humidity variations along the supply chain, both of them have an impact on the product mass losses. Knowledge of the mass loss threshold value for saleability of the studied mangoes (Mangifera indica L. cv. "Nam Dok Mai") and the product shelf life could be determined. Taking into account this finding, the supply chain should be well-managed to ensure fruit quality up until consumption. [START_REF] Nunes | Relationship between weight loss and visual quality of fruits and vegetables[END_REF] have reported that the maximum postharvest life of mangoes (cv. 'Tommy Atkins' and 'Palmer') stored at 20 °C is 5 days.

Color parameter b*

The value 𝑏 * measured at day 0 and on different days during storage following air transport is shown in Fig. 7, increasing until day 7 for the direct flight and day 11 for the indirect flight, then the value stabilized until day 14 for both flights. It is noteworthy that the peel color 𝑏 * value seemed to reach an asymptote following a number of days that depended on the supply chain. The differences in mango peel color for the direct and indirect supply chains could be explained by the variation in time-temperature history (or accumulated time × temperature unit) of these two chains. Moreover, this difference could be explained by the fact that for the indirect flight supply chain, the product temperature decreased and reached 8 °C during storage overnight in the cold room of the logistics company prior to delivery to INRAE. It should be borne in mind that each point is an average value of 18 measurements (one measurement on each side of 9 mangoes). The measurement of the 𝑏 * value of mangoes stored in a laboratory test room at a temperature of 20 °C in Thailand was not undertaken.

Identification of characteristic time of temperature evolution

The characteristic duration of temperature evolution 𝜏 𝑇 was identified by fitting the predicted product temperature evolution throughout the supply chain (using Eq. ( 2)) with the measured values. The result is shown in Table 3.

The value of 𝜏 𝑇 was about 1-2 h longer with the direct flight in comparison to that obtained with the indirect flight. This result means that the product was less responsive to the ambient temperature (that is, the rate of temperature change was slower) in the case of the direct flight supply chain than that of the indirect flight. This led us to suppose that the boxes may have been located at the center of a container during the direct flight, and more so on the edge of a container during the indirect flight. The 𝜏 𝑇 value difference between the 2 boxes on the same flight could be explained by the difference in box positions within the same container (e.g., box 2 on top of box 1). These fitted 𝜏 𝑇 values can be used to predict the product temperature evolution in different supply chain scenarios given solely ambient conditions. Virtual scenarios were developed as an illustration of the influence of the supply chain on the product quality (mass loss and color) using a modeling approach presented in Section 3.4.

Comparison between measured and predicted results

Product mass loss

A comparison between the experimental and predicted mass loss is shown in Fig. 8. There is good agreement between the predicted and measured values for the direct flight (RMSE = 1.4 % mass loss).

However, for the indirect flight, the predicted values are slightly underestimated (RMSE = 2.9 % mass loss). The difference is observed from day 3 until the end of the experiment. This discrepancy may be explained by the uncertainty of the humidity measurement in the box due to the accuracy of the hygrometer. In fact, the temperature recorders were previously calibrated; however, the calibration of hygrometer was not possible in our study. In the storage cold room at a temperature of 20 °C, in which the temperature and the relative humidity were controlled and homogeneous, the model shows the best performance with experiments (RMSE = 0.8 % mass loss), in comparison to the field studies. It should be emphasized that the prediction of mass loss is very sensitive to the measured relative humidity, a difference of a few percent may lead to a large difference in mass loss. Hence, robust, accurate probes should be used for humidity measurements, and this is not always possible when conducting field experiments.

b* value

Fig. 9 shows a comparison between the measured and predicted 𝑏 * values. Good agreement was obtained. It needs to be noted that the predicted initial value of 𝑏 * were determined from Eq. ( 7) in which 𝑏 𝑚𝑖𝑛 * and 𝑏 𝑚𝑎𝑥 *

were taken from the values estimated with more than hundreds of mangoes (cv.

"Nam Dok Mai") by [START_REF] Penchaiya | Modelling quality and maturity of 'Namdokmai Sithong'mango and their variation during storage[END_REF]. The evolution of measured and predicted 𝑏 * values exhibited the same tendency i.e., the value increases with time and approaches a stable value after 8-10 days. The discrepancy observed at the beginning (< 3 days) could be explained by the difference of the initial 𝑏 * value. As explained previously, the high number of fruits used for the model development by [START_REF] Penchaiya | Modelling quality and maturity of 'Namdokmai Sithong'mango and their variation during storage[END_REF] would assure the prediction precision other fruit batches.

Influence of the supply chain scenario

Three virtual scenarios were developed for a 6-day supply chain: scenario 1 considered as a reference consists of a constant ambient temperature of 12 °C and 80 %RH, scenario 2 takes into account potential temperature abuse that can occur on airport tarmacs during departure and arrival, and scenario 3 consists of storage under non-optimal conditions following air transport (18 °C, 70 %RH) (Fig. 10). The ambient temperature and relative humidity evolution of these scenarios are the input parameters of the models.

The predicted % mass loss and 𝑏 * value corresponding to these 3 scenarios are presented in Fig. 11.

As expected, it was shown that the constant conditions (scenario 1) led to less product mass loss and better preserved peel color (a lower 𝑏 * value). The rate of mass loss increased during the period of high vapour pressure deficit at the fruit surface in comparison with the surrounding air, i.e., high ambient temperature and low relative humidity. The mass loss is co-incident with the peel color change of fruit. Thus, 𝑏 * increases. In scenario 2, the quality attributes deteriorate more rapidly, but the impact of the temperature abuse and humidity changes are limited by their short duration (4 h). For scenario 3, although temperature and humidity abuse is less severe than in scenario 2, the deterioration of the fruit quality is more significant due to its longer duration. These results emphasize that both the intensity and the duration of temperature abuse are of major importance in the deterioration of products. The storage conditions used for mangoes should be optimal up until consumption in order to ensure quality and to avoid product waste.

Overall discussion

Although the present study used experimental data generated by an independent study [START_REF] Penchaiya | Modelling quality and maturity of 'Namdokmai Sithong'mango and their variation during storage[END_REF] to develop the model of mass loss and color evolution along the logistic chain of mango.

The main limitation of this study concerns the reproducibility and representativeness of the data because only one batch along 2 supply chains was considered. To address these limitations, the solution would be to replicate the field study as many times as possible as proposed by [START_REF] Derens-Bertheau | Cold chain of chilled food in France[END_REF]. In that study, more than a hundred food products were instrumented and dispatched through various ground logistic chains so that the variability of the logistics chains was thoroughly represented.

The implementation of the field study presented in this paper is complicated to execute as it requires excellent logistics coordination among the professionals in different sectors (e.g., agriculture, packing house, carrier, phytosanitary control at custom service, airport logistics in Bangkok and in Paris, onfield instrumentation by KMITL, temperature analysis by INRAE and quality assessment by both institutions). For this reason, only a few measurements of air and product temperatures during international air transport were conducted and few fruit samples were used to evaluate the quality evolution.

An additional limitation of this study is the validity of the results with respects to the biological maturity and variability of the products as only one batch was studied. The presented data represents only the intra-batch fruit variability of a single batch. The presented results might be different from the products harvested at other times of the same season. To account for the likely variability, the solution would be to reproduce such experiments at different times of the season. The results of this study demonstrated the potential for quality models to be used to predict quality evolution through a complex supply chain.

Conclusions and perspectives

An air shipment of mangoes in instrumented corrugated boxes was studied for the first time without control of the position of the boxes aboard the aircraft. This fruit is temperature-sensitive and requires an optimal temperature of 13 °C throughout the supply chain. Four instrumented boxes of mangoes The results of this study will be used to design another field study to be conducted in 2023. This future study will consist of an aircraft container fully loaded with boxes of mangoes (about 160 boxes in total), which will allow the reproducibility of analysis data and the study of the heterogeneity of temperature distribution according to the box position. This knowledge is useful for the understanding of airflow, heat transfer phenomena and model development, which are original since few data are available in literature in comparison with land or sea transportation. where 𝑛 = number of temperature recordings, 𝑇 𝑝,𝑚𝑒𝑎𝑠 = measured temperature, and 𝑇 ̂𝑝 = predicted temperature.

  : air transport; temperature; tropical fruit; mass loss; color in the mango peel (kg water•m -3 humid air) 𝐶 𝑤,∞ Water vapor concentration in air (kg water•m -3 humid air) 𝐸 𝑎 Activation energy = 87,600 J•mol -1 𝑘 𝑟𝑒𝑓 Constant rate at reference temperature (d -1 ) 𝑘 𝑡𝑎 Moisture transfer coefficient between the product and the air (m•s -1 ) 𝑚 Product mass (kg) 𝑚 𝑤 Water loss due to transpiration (kg) 𝑀 𝐻 2 𝑂 Water molecular weight = 18 kg•mol -1 𝑃 𝑠𝑎𝑡 Saturated vapor pressure (Pa) 𝑅 Gas constant = 8.314 J•mol -1 •K -1 𝑅𝐻 Relative humidity of air (%RH)

  Fig. 2b shows the consecutive stages of the supply chain of the direct flight and Fig. 2c shows those of the indirect flight to Charles de Gaulle Airport (CDG) in Paris. After arrival in Paris, all boxes were received by a logistics company for delivery to the FRISE laboratory, National Research Institute for Agriculture, Food and Environment (INRAE) located in Antony, a suburb of Paris.

  [START_REF] Penchaiya | Modelling quality and maturity of 'Namdokmai Sithong'mango and their variation during storage[END_REF] developed a logistic model for 𝑏 * prediction of mango (cv. "Nam Dok Mai") exposed to variable temperatures such as those in a supply chain as below:

Figs 4

 4 Figs 4 and 5 present the air and product surface temperature evolution measured at different positions in Box 1 for direct and indirect flights, respectively.

  were sent from Bangkok, Thailand, to Paris, France: two boxes were shipped on a direct flight and two boxes were shipped on an indirect flight. The product quality was assessed at the packing house in Thailand and in a controlled-temperature room in France on different storage days until the end of life of 14 days. A maximum fruit temperature of 33 °C (during transport from the orchard to the packing house) and a minimum fruit temperature of 8 °C (in the cold room of the logistics company following arrival at Charles de Gaulle Airport in Paris via an indirect flight) were observed. The slight temperature difference between the air and the fruit surface (<1 °C on average) for both the direct and indirect flights led us to suppose that the air was stagnant in the box and, thus conduction was the main heat transfer mode. Models of product mass loss and peel color change throughout the supply chain were developed. The numerical and the experimental values were in good agreement. These models were able to predict the product quality evolution as a function of a supply chain scenario.

Fig. 1 Fig. 2

 12 Fig. 1 Corrugated box loaded by 12 mangoes: (a) box and vent hole dimensions, (b) 6 positions of air temperature and relative humidity measurements, (c) 2 positions of product surface temperature measurement.

Fig. 3

 3 Fig. 3 Model validation by comparing the predicted product mass loss (%) with the experimental values for 2 storage conditions: 13.4 ± 0.5 °C/91.6 ± 3.0 %RH for 12 days then in 25-28 °C/75.5 ± 5.2 %RH for 8 days (experimental data from Penchaiya (2018)). CI = Confidence Interval due to uncertainty of humidity measurement.

Fig. 4

 4 Fig. 4 Time-temperature profile in Box 1 direct flight with activity numbers.

Fig. 5

 5 Fig. 5 Time-temperature profile in Box 1 indirect flight with activity numbers.

Fig. 6

 6 Fig. 6 Experimental mass loss variation (%) during storage for several days after 2 supply chain scenarios (direct and indirect flights) in comparison to the one stored in a 20 °C laboratory test room. Error bars represent the standard deviation of 9 samples.

Fig. 7

 7 Fig. 7 Experimental values of color parameter 𝑏 * variation with storage time for 2 supply chain scenarios including direct and indirect flights. Error bars represent the standard deviation of 9 samples.

Fig. 8

 8 Fig. 8 Comparison between measured and predicted values of mass loss during storage for (a) supply chain with direct flight, (b) supply chain with indirect flight, and (c) storage in 20 °C room. CI = Confidence Interval due to uncertainty of humidity measurement.

Fig. 9

 9 Fig. 9 Comparison between measured and predicted values of color parameter 𝑏 * variation with storage time (a) supply chain with direct flight and (b) supply chain with indirect flight.

Fig. 10

 10 Fig. 10 Three virtual supply chain scenarios: (a) evolution of ambient temperature and (b) evolution of relative humidity. Note: scenario 1 (blue line) is a reference scenario (constant temperature of 12 °C and constant relative humidity of 80%), scenario 2 (red dash line) represents temperature break in airport tarmacs, scenario 3 (green dash-dot line) represents high ambient temperature and low relative humidity in storage room after air transport.

Fig. 11 :

 11 Fig. 11: Predicted evolution of product quality according to 3 virtual supply chain scenarios (a) product mass loss and (b) color (𝑏 * value).

  = 2.61 (estimated by Eq. (9) taking into account the measured initial value of 𝑏 𝑖𝑛𝑖 * (32.34).It is to be emphasized that the 𝑏 * value increases with time. Where there are no experimental data,

	𝑏 𝑚𝑖𝑛 *	and 𝑏 𝑚𝑎𝑥 *

proposed: 𝑏 𝑚𝑖𝑛 * = 27.85, 𝑏 𝑚𝑎𝑥 * = 45.78, 𝑘 𝑟𝑒𝑓 = -0.468 d -1 , and 𝐸 𝑎 = 87,600 J•mol -1 . 𝑡 *

Table 1

 1 Direct flight schedule

				Duration
	Activity number	Date	Local time of Thailand	between activities	Source of data
				n and n-1
	0. Packing house	21/03/2022	12:01	0	KMITL experimenter
	1. Transfer of all boxes to	21/03/2022	15:31	3.50 h	Temperature profile from
	the storage room of the				temperature recorders
	packing house
	2. Departure from the	22/03/2022	9:00	17.50 h	Packing house manager
	packing house
	3. Arrival at the airport	22/03/2022	15:00	6.00 h	Packing house manager
	cargo terminal
	4. Departure from the	23/03/2022	00:05	9.08 h	Boarding pass
	airport (BKK)
					TG930 Boeing 787
	5. Arrival at Paris CDG	23/03/2022	12:50	12.75 h	Boarding pass
	Airport
					TG930 Boeing 787
	6. Arrival at INRAE	23/03/2022	20:20	7.50 h	INRAE laboratory report

Table 2

 2 Indirect flight schedule

				Duration
	Activity number	Date	Local time of Thailand	between activities n	Source of data
				and n-1
	0. Packing house	21/03/2022	12:01	0	KMITL experimenter
	1. Transfer of all boxes to	21/03/2022			Temperature profile from
	the storage room of the		15:31	3.50 h	temperature recorders
	packing house
	2. Departure from the packing house	22/03/2022	9:00	17.50 h	Packing house manager
	3. Arrival at the airport cargo terminal	22/03/2022	15:00	6 h	Packing house manager
	4. Departure from the	22/03/2022			Boarding pass
	airport (BKK)		21:10	6.17 h
					EK0379 Boeing 777
	5. Arrival at Phuket Airport	22/03/2022			Boarding pass
	(HKT)		22:40	1.50 h
					EK0379 Boeing 777
	6. Departure from Phuket	23/03/2022			Boarding pass
	Airport (HKT)		00:10	1.50 h
					EK0379 Boeing 777
	7. Arrival at Dubai Airport	23/03/2022			Boarding pass
	(DXB)		06:46	6.60 h
					EK0379 Boeing 777
	8. Departure from Dubai	23/03/2022			Boarding pass
	Airport (DXB)		10:50	4.06 h
					EK0073 Airbus A380
	9. Arrival at Paris CDG	23/03/2022			Boarding pass
	Airport		18:25	7.58 h
					EK0073 Airbus A380

Table 3

 3 Characteristic time of temperature evolution 𝜏 𝑇 (h) of mango boxes in the supply chain from Thailand to France via direct and direct flights Root Mean Square Error) is defined as √ ∑ (𝑇 𝑝,𝑚𝑒𝑎𝑠 -𝑇 ̂𝑝) 2

			Direct flight	Indirect flight
	Box number
		𝝉 𝑻 (h)	RMSE (°C)*	𝝉 𝑻 (h)	RMSE (°C)
	1	3.2	0.09	2.2	0.35
	2	3.5	0.42	1.3	0.54
	𝑖=1 *RMSE (𝑛	𝑛

Acknowledgements

This study was financially supported by Program Management Unit Competitiveness (PMUC), Office of National Higher Education Science Research and Innovation Policy Council, Thailand (Contract no. C10F640316).

Declaration of Interest Statement

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