Alexandre Quemet
email: alexandre.quemet@cea.fr

Melody Maloubier

Vincent Dalier

Alexandre Ruas

Development of an analysis method of minor uranium isotope ratio measurements using electron multipliers in Thermal Ionization Mass Spectrometry

Keywords: TIMS, uranium, isotope ratio, electron multipliers

A simple analytical procedure was developed to measure with high accuracy the isotope ratio of minor isotope of natural uranium present in small quantities using a thermal ionization mass spectrometer (TIMS). The reduction of quantities used for analysis and the measurement of non-abundant isotopes are of prime interest in the nuclear industry. Indeed it is necessary to reduce the analyst received dose and the effluent released, as well as realizing measurement at trace level. The new generation of TIMS is equipped with a multicollection system of electron multipliers: discrete dynode electron multiplier (SEM) and continuous dynode electron multiplier (MIC), that improve the sensitivity compared to faraday cups. The procedure developed was verified using Certified Reference Material IRMM 052. Results were evaluated relying on NF T 90-210 norm regarding method validation. First, the isotope ratio 234 U/ 238 U was examined by total evaporation using the SEM and MIC to measure 234 U and the faraday cup to measure 238 U. In a second approach, the isotope ratio 235 U/ 238 U was studied by total evaporation using the SEM to measure 235 U and the faraday cup to measure 238 U. The classical method with peak-jumping SEM measurement was also used. Total evaporation method employing only the faraday cup was used to confront the results obtained. The analyzable quantity was reduced from 250 ng to 50 ng for the 235 U/ 238 U isotope ratio and from 1270 ng to 50 ng for the 234 U/ 238 U isotope ratio with acceptable uncertainties thanks to the use of electron multipliers. For all experiments were the accuracy was achieved, the calculated uncertainties were below to 0.28 % for the 235 U/ 238 U isotope ratio and 5 % for the 234 U/ 238 U isotope ratio. I.

Introduction

The determination of isotope ratio with high accuracy is very important for different applications such as: environmental, geological, forensics or in the nuclear industries. The natural isotopic composition of an element is affected by the geological history, the age and the water permeability of the mineral. The knowledge of such characteristic is useful to identify and understand the processes affecting an element, and determine the ages of the minerals. [START_REF] Keegan | The provenance of Australian uranium ore concentrates by elemental and isotopic analysis[END_REF][START_REF] Wieser | The development of multiple collector mass spectrometry for isotope ratio measurements[END_REF] The interpretation of the isotopic composition of uranium (both natural and enriched) is of prime interest for the International Atomic Energy Agency (IAEA) for monitoring nuclear installations in order to verify the material conformity to the nuclear grade specification, to identify material origins and to control the nuclear activities. [START_REF] Keegan | The provenance of Australian uranium ore concentrates by elemental and isotopic analysis[END_REF][START_REF] Bürger | Uranium and plutonium analysis of nuclear material samples by multi-collector thermal ionisation mass spectrometry: Quality control, measurement uncertainty, and metrological traceability[END_REF][START_REF] Mayer | Nuclear forensics--a methodology providing clues on the origin of illicitly trafficked nuclear materials[END_REF][START_REF] Donohue | Strengthening IAEA safeguards through environmental sampling and analysis[END_REF][START_REF] Wallenius | Nuclear forensic investigations: Two case studies[END_REF][START_REF] Varga | Age determination of uranium samples by inductively coupled plasma mass spectrometry using direct measurement and spectral deconvolution[END_REF] Uranium mineral has an isotope ratio 235 U/ 238 U relatively constant whatever its origin. [START_REF] Keegan | The provenance of Australian uranium ore concentrates by elemental and isotopic analysis[END_REF] In the nuclear field, the modification of this isotope ratio gives information about the enrichment level of nuclear material. The safeguard authorities requirement for 235 U abundance measurement uncertainty by TIMS is set to 0.28 %. [START_REF]Internaltional Atomic Energy Agnecy, International Target Values 2010 for Measurement Uncertainties in Safeguarding Nuclear Materials -STR368[END_REF] The isotopic abundance of 234 U provides information about the type of activities conducted in the nuclear installations: enrichment or reprocessing. [START_REF] Bürger | Uranium and plutonium analysis of nuclear material samples by multi-collector thermal ionisation mass spectrometry: Quality control, measurement uncertainty, and metrological traceability[END_REF][START_REF] Kraiem | Development of an improved method to perform single particle analysis by TIMS for nuclear safeguards[END_REF] However, 234 U stays a minor isotope with an isotopic abundance below 0.1 % whatever its origins: enriched uranium, spent nuclear fuel, reprocessed uranium or depleted uranium. Moreover, the reduction of the analyzable quantities is important to realize environmental measurement where uranium is present at trace level. Finally in the nuclear industry the reduction of the analyzable quantities reduces the radioactive effluent generated and the exposure to the sample radiation by the analyst.

Thermal ionization mass spectrometry (TIMS) is a reference technique for uranium isotope measurement. [START_REF] Kraiem | Development of an improved method to perform single particle analysis by TIMS for nuclear safeguards[END_REF][START_REF] Callis | High-precision isotopic analyses of uranium and plutonium by total sample volatilization and signal integration[END_REF][START_REF] De Oliveira | Evaluation of the n(235U)/n(238U) isotope ratio measurements in a set of uranium samples by thermal ionization mass spectrometry[END_REF][START_REF] Suzuki | A new method for isotope ratio measurement of uranium in trace amount by thermal ionization mass spectrometry: The continuous heating method[END_REF][START_REF] Goldberg | Improved environmental and forensics measurements using multiple ion counters in isotope ratio mass spectrometry[END_REF][START_REF] Richter | Improvements in routine uranium isotope ratio measurements using the modified total evaporation method for multicollector thermal ionization mass spectrometry[END_REF][START_REF] Mathew | Total evaporation method for uranium isotopeamount ratio measurements[END_REF] TIMS allows measurement of isotope ratio with high trueness and precision, allowing the knowledge of isotope ratio with an uncertainty sometimes lower than 0.1 %. [START_REF] Callis | High-precision isotopic analyses of uranium and plutonium by total sample volatilization and signal integration[END_REF][START_REF] De Oliveira | Evaluation of the n(235U)/n(238U) isotope ratio measurements in a set of uranium samples by thermal ionization mass spectrometry[END_REF][START_REF] Goldberg | Improved environmental and forensics measurements using multiple ion counters in isotope ratio mass spectrometry[END_REF][START_REF] Wakaki | Isotope ratio measurements of trace Nd by the total evaporation normalization (TEN) method in thermal ionization mass spectrometry[END_REF][START_REF] Bürger | Isotope ratio analysis of actinides, fission products, and geolocators by high-efficiency multi-collector thermal ionization mass spectrometry[END_REF][START_REF] Richter | Improved techniques for high accuracy isotope ratio measurements of nuclear materials using thermal ionization mass spectrometry[END_REF] The sample measurement with a low quantity of analyte or the isotope detection with low abundance (less than 0.1 %) is very challenging due to the low intensity of the ion bean affecting the measurement trueness and precision. This is a reason why some methods and technologies have been developed in order to improve sensitivity, accuracy or detection.

The major development for the low quantities measurements is the use the electron multipliers. [START_REF] Kraiem | Development of an improved method to perform single particle analysis by TIMS for nuclear safeguards[END_REF][START_REF] Goldberg | Improved environmental and forensics measurements using multiple ion counters in isotope ratio mass spectrometry[END_REF][START_REF] Richter | Improvements in routine uranium isotope ratio measurements using the modified total evaporation method for multicollector thermal ionization mass spectrometry[END_REF][START_REF] Mathew | Total evaporation method for uranium isotopeamount ratio measurements[END_REF][START_REF] Richter | Improved techniques for high accuracy isotope ratio measurements of nuclear materials using thermal ionization mass spectrometry[END_REF][START_REF] Mathew | Characterization of uranium isotopic abundances in depleted uranium metal assay standard 115[END_REF] They are used to further extend the sensitivity of the instrument by several orders of magnitude. It is useful for the minor isotope ratio determination or for low quantities of analyte. These types of detector operate by converting the ions hitting the detector into secondary electrons, which are amplified then detected. Multiple ion counting can be done with discrete dynode electron multipliers (secondary electron multiplier) or with continuous dynode electron multipliers (channel electron multipliers or Channeltron) or a combination of both. Due to their small size, these detectors can be moved, which improves the isotope measurement. Indeed, it is possible to perform static measurements. [START_REF] Bürger | Isotope ratio analysis of actinides, fission products, and geolocators by high-efficiency multi-collector thermal ionization mass spectrometry[END_REF][START_REF] Mialle | The use of total evaporation method using Channeltron electron multipliers by thermal ionization mass spectrometry for europium isotope ratio measurements on picogram sample amounts[END_REF] After a brief description of the TIMS's detector optimization, the results obtained for the 235 U/ 238 U and 234 U/ 238 U isotope ratio will be discussed in term of measurement trueness and precision. Different configurations using discrete and continuous dynode electron multipliers were studied in order to determine the best and most simple analytical method. Several quantities of uranium were investigated to challenge the limits of the different configurations. Each configuration was compared to the use of the faraday cups (FC) only.

II. Experimental

Reagent

All solutions were prepared in 8 M nitric acid (Merck suprapur). This 8 M nitric acid solution was prepared by diluting high purity nitric acid with deionized water (resistivity 18.2 M .cm).

Certified reference materials

To calibrate the electron detectors, uranium CRM U500 standard solution obtained from the National Institute of Standard and Technology (NIST) at 0.96 µg/µL (± 5%, k = 2) was used. This standard solution has a 234 U isotopic abundance (0.5181 % ± 0.0008) higher than natural uranium, necessary to obtain of an adequate intensity for the 234 U isotope.

The assessment of the investigated procedure was carried out on the certified reference material IRMM 052. This solution is certified for the following isotope ratios: 235 U/ 238 U (0.0072718 ± 0.0000030, k = 2), 234 U/ 238 U (0.00005548 ± 0.00000022, k =2) and 236 U/ 238 U (0.000000151 ± 0.000000045, k = 2). The initial solution concentration was 1.2711 g.kg -1 (± 0.05%, k = 2). Solutions of 250, 100, 50, 30, 20, 10, 3 and 1 ng/µL (± 5%, k = 2) were prepared from the IRMM 052 standard in 8 M nitric acid. Dilutions were performed gravimetrically with a high precision balance.

Instrumentation

Isotope measurements were performed on a Thermo Scientific Triton Thermal Ionization Mass Spectrometer, equipped with a glove box. The analyzer is a low resolution magnetic field (resolution around 400). The multicollector is equipped with 7 movable Faraday cups, a fixed discrete dynode Secondary Electron Multiplier (hereafter referred to as SEM), located behind the central faraday cup, and 3 movable continuous dynode electron multipliers operated in ion counting mode (hereafter referred to as MIC, abbreviation of Multi-Ion Counting). Faraday cups are equipped with high-ohmic resistors (10 11 ). The SEM is combined with a RPQ retardation filter to improve the abundance sensitivity to about 10 ppb at mass M compared to mass M+1 or M-1.

Uranium samples were loaded onto the outgassed zone of a refined Re filament. A double Re-filament configuration was used to control independently the evaporation and the ionization temperature. These filaments, provided by ATES, are obtained from a 99.995% pure Re metal. 1 µL of diluted solutions (described in section 2.2) was deposited onto the filament. After deposition, the sample preparation was dried with a 0.5 A current.

Then the current was progressively increased to 2 A over 10 s.

Detectors optimization

Intercalibration gains of the faraday cup detectors were measured daily by an automatic process. [START_REF] Ramakumar | Calibration procedures for a multicollector mass spectrometer for cup efficiency, detector amplifier linearity, and isotope fractionation to evaluate the accuracy in the total evaporation method[END_REF] The gains were extremely stable with a repeatability lower than 0.001%. Baselines were measured before each intercalibration gain and before each analysis.

For the SEM and the MIC, one of the major sources of uncertainty is the dead time, which is the time interval required between the impact of two ions to identify each ion as an independent event. [START_REF] Wieser | The development of multiple collector mass spectrometry for isotope ratio measurements[END_REF] The dead time of each detector was implemented by the manufacturer. It was set up at 24 ns for the SEM and 70 ns for the MIC.

For the SEM and the MIC, two parameters have to be measured before starting an analysis: the dark noise and the intercalibration gain. The process of the calibration is schematized in Figure 1.

In the first place, dark noise measurements were performed with the isolation valve closed for the SEM and the MIC. Each measurement takes about 10 minutes with an integration time set at 1 sec. A dark noise level lower than 7 cps was observed for the SEM and the MIC. In practice the dark noise level measurement was also realized before each measurement, analysis or intercalibration.

In order to measure SEM/FC or MIC/FC intercalibration gains, 400 ng of the certified reference material U500 was deposited in order to obtain adequate intensity on 234 U + : 1.5 mV for the MIC and 5 mV for the SEM, and to measure the corresponding yield, given by the following formula:

100 10 . 6 , 1 / (%) 8 / FC MIC SEM I I Yield (1)
With I SEM/MIC the ion beam intensity obtained with the SEM or the MIC, and I FC the ion beam intensity obtained with the faraday cup.

Electron multipliers have a limited lifespan which is an important performance parameter. It is common for the yield to decrease after several utilizations, requiring periodic voltage adjustments to obtain the adequate performance. Varying this voltage across the device alter the number of secondary electrons produced per strike.

A minimum yield of 90 % for the SEM and 85 % for the MIC must be obtained prior to the beginning of the analysis. When the yield was too small, the voltage of the detector was increased in order to reach a yield higher than 90 % for the SEM and 85 % for the MIC.

For the SEM/FC intercalibration, a method implemented within the TIMS software was used. The ion beam was measured alternatively 10 times on the central faraday cup and the SEM with an integration time of 8.389 seconds and 2.097 seconds, respectively. Several intercalibration gains were performed for the SEM (Figure 2). It was relatively stable over time: the average of all intercalibration gains was 97.2 ± 1.1 %. The detector voltage was unchanged during this study. However, the yield could change up to 2 % during a same day without any trend observable. To minimize the impact of the detector yield variation on the analysis, the intercalibration gain was realized before and after each measurement. Then, the measurement was corrected by the average of these two intercalibration gains.

In opposition to the SEM/FC intercalibration, no automatic procedure was available on the TIMS software to evaluate the MIC/FC yield. A procedure was developed to obtain the MIC/FC intercalibration gain. Data were acquired in peak-jumping collection mode with two measurements sequences. During the first sequence, 234 U isotope was measured over 2.097 s on the MIC. In the second sequence, the 234 U isotope was then measured over 4.194 s on the central faraday cup. The measurement sequences were repeated 10 times. Gains were calculated with equation [START_REF] Keegan | The provenance of Australian uranium ore concentrates by elemental and isotopic analysis[END_REF]. A yield diminution from 97 to 87 % was observed during the experiments (Figure 2) after approximately twenty analyses. The detector voltage was then increased in order to obtain a yield up to 90%. Finally, the MIC/FC intercalibration gain was realized before and after each measurement, and the measurement was corrected by the average of these two intercalibration gains. The yield of the MIC was less stable than the one of the SEM (Figure 2): the average of all the intercalibration gains before the change of voltage was 92.1 ± 3.7 % and after the change of voltage was 92.0 ± 1.7 %.

2.5. Measurement method 2.5.1. Classical method

In the classical or traditional method, the different isotopes are collected in a limited period of the sample evaporation. The isotope ratios are then corrected of the mass fractionation, the main cause of the measurement bias, [START_REF] Dubois | Isotopic analysis of rare earth elements by total vaporization of samples in thermal ionization mass spectrometry[END_REF][START_REF] Fiedler | Total evaporation measurements: experience with multi-collector instruments and a thermal ionization quadrupole mass spectrometer[END_REF] by using an internal or external normalization. This method is available in single or multi collection mode. [START_REF] Richter | Improved techniques for high accuracy isotope ratio measurements of nuclear materials using thermal ionization mass spectrometry[END_REF] In single collection or peak-jumping measurement, the different isotopes are successively introduced into the same detector by a variation of the magnetic field. This procedure eliminates the dependence of the current amplifier gains and the Faraday cup efficiencies. In multi collection or static measurement, the ions beams reach simultaneously the different detectors that overcome the fluctuation of the ions beams. The knowledge of the gain of each detector is required before the measurement.

Measurements in peak-jumping mode with the classical method were performed for the 235 U/ 238 U ratio with the SEM. The mass cycle for uranium isotopes in a peak-jumping SEM detector measurement was performed in two sequences. During the first sequence, the 235 U isotope was measured for 4.194 s on the SEM and in the second sequence the 238 U isotope was measured for 4.194 s on the SEM as well. An idle time was fixed at 1 s. Examples of target intensities are shown in supplementary materials. Each measurement corresponded to 8 blocks of 10 cycles. Mass fractionation was corrected using the mass fractionation factor determined with the certified reference material U500 analyzed in the same condition.

Total evaporation

The total evaporation method is based on the evaporation and ionization of the entire sample. [START_REF] Dubois | Isotopic analysis of rare earth elements by total vaporization of samples in thermal ionization mass spectrometry[END_REF] Therefore the ion beam of the element is totally collected by a multi-collection system. This method was developed in order to overcome the mass fractionation. Other significant advantages can be noted: sample quantities reduction and improvement of the accuracy. [START_REF] Callis | High-precision isotopic analyses of uranium and plutonium by total sample volatilization and signal integration[END_REF][START_REF] Wakaki | Isotope ratio measurements of trace Nd by the total evaporation normalization (TEN) method in thermal ionization mass spectrometry[END_REF] The total evaporation method is composed of three steps: adjustment, acquisition and shutdown. [START_REF] Callis | High-precision isotopic analyses of uranium and plutonium by total sample volatilization and signal integration[END_REF][START_REF] Wakaki | Isotope ratio measurements of trace Nd by the total evaporation normalization (TEN) method in thermal ionization mass spectrometry[END_REF][START_REF] Fiedler | Total evaporation measurements: experience with multi-collector instruments and a thermal ionization quadrupole mass spectrometer[END_REF] The adjustment phase was the same for all measurements using the total evaporation method. First, detectors were moved in order to collect uranium isotopes. Then the ionization filament was heated to a filament current of 4.5 A in 10 min. The ionization filament temperature was adjusted in order to obtain a 20 mV signal on 187 Re + ion beam. The 187 Re + ion beam from the ionization filament was then used to realize a "peak center" (mass calibration and ions beam centering in the detector) and to optimize the lens of the ion source: "focus setting" and "Z-focus". The ionization filament temperature was then adjusted to a 80 mV signal on the 187 Re + ion beam. Next, the evaporation filament was heated to obtain a 238 U + ion beam of 1 mV. Peak center, focus setting and Z-focus were optimized using uranium isotope 238 U + . Electronic baselines were then measured prior to data acquisition. When the data acquisition started, the evaporation filament current started to increase until the ion beam intensity of uranium isotopes reached the target intensity. This intensity was set by the operator. The target isotope and the optimized target intensities, depending on the uranium quantity and the type of detector, are presented in supplementary materials. The evaporation filament was then controlled to keep the ions beam intensity constant, by increasing the evaporation current when necessary. When the evaporation filament current reached a maximum value of 5 A and the ion beam decreased down to a 6 mV signal, the data acquisition was then terminated.

Methods evaluation

The results were evaluated relying to the NF T 90-210 norm regarding method validation. [START_REF] Afnor | 210 norm : Water quality -Protocol for the intial method performance assesment in a laboratory[END_REF] This evaluation aims at studying the method accuracy by evaluating measurement trueness and precision. A maximal bias (MB) was defined by the Laboratory of Analysis and Materials Metrology (LAMM) according to its requirements and the IAEA requirements. [START_REF]Internaltional Atomic Energy Agnecy, International Target Values 2010 for Measurement Uncertainties in Safeguarding Nuclear Materials -STR368[END_REF] This maximal bias (MB) was fixed to 0.28 % when isotope ratio involved 2 major isotopes ( 238 U and 235 U) and 5 % when isotope ratios involved a minor isotope ( 234 U).

For each method and each uranium quantity, five independent determinations were realized.

Bias, or measurement trueness, of the method was calculated with the following formula:

100 (%) ref ref Z Bias (2)
Where Z is the average of the series and ref is the certified value of the reference.

The precision control consisted in checking the repeatability, given by the relative standard deviation (RSD).

Finally, the accuracy control (i.e. measurement trueness and precision) consisted in checking that the value obtained was within the limits defined by the laboratory. The method is validated in term of accuracy when the following inequality is verified:

MB s ref Z 2 (3)
Where s is the standard deviation of the series.

For the determination of measurement uncertainty, the normalized bias (NB) was calculated:

2 2 ref u s ref Z NB (4)
Where u ref is the uncertainty at k = 1 of the reference value. When the normalized bias is below to 2, the observed bias is considered statistically negligible compared to and the uncertainty of the reference value u ref must be taken into account in the final uncertainty calculation. [START_REF] Afnor | 210 norm : Water quality -Protocol for the intial method performance assesment in a laboratory[END_REF] In this case, the following formula is used for the determination of the uncertainty (U) at k = 2:

2 2 2 ref u s U (5)
If the normalized bias is beyond to 2, a correction of the bias and the integration of this error in the final uncertainty is necessary. [START_REF] Afnor | 210 norm : Water quality -Protocol for the intial method performance assesment in a laboratory[END_REF] In other words, equation ( 5) cannot be applied.

III. Results and discussion

Uranium isotope measurements have been performed both by the total evaporation method and the classical method with quantities varying from 1270 ng down to 1 ng. Five measurements were performed for the different quantities and methods. Table 1, Table 2 and Figure 3 report the results obtained for the 234 U/ 238 U and 235 U/ 238 U isotope ratio for each deposited quantity.

3.1. 234 U/ 238 U isotope ratio measurement 3.1.1. Total evaporation with faraday cup 234 U/ 238 U isotope measurements have been performed by total evaporation method using the faraday cup to measure both 234 U and 238 U. The measurement trueness for a quantity of 1270 ng (1.1 %) and 250 ng (1.3 %) was lower than 5 %. However a degradation of the precision was observed for a deposit of 250 ng (RSD = 4.5 %) compared to a deposit of 1270 ng (RSD = 0.7 %). For a quantity of 250 ng, the signal obtained at m = 234 uma was in the limit of detection range of faraday cups, which is consistent with the observation of the precision deterioration. Finally, the control of the accuracy (inequality (3)) showed that the method was only validated for 1270 ng.

3.1.2. Total evaporation using SEM ( 234 U) and faraday cup ( 238 U)

In order to improve the accuracy on the 234 U/ 238 U ratio, measurements have been performed by the total evaporation method using the SEM to measure 234 U and the faraday cup to measure 238 U. For uranium quantities from 1270 ng down to 10 ng, the measurement trueness and RSD were lower than 5 %. A degradation of the measurement trueness (1.5 %) and precision (RSD = 2.1 %) was observed for a 10 ng uranium deposit compared to larger quantities: for instance a measurement trueness of 0.2 % and a precision of 0.3 % were measured for a 50 ng deposit. As a consequence, the method accuracy was validated for uranium quantities between 50 and 1270 ng.

3.1.3. Total evaporation using MIC ( 234 U) and faraday cup ( 238 U)

In order to compare the performance of the SEM and the MIC, the same experiment as in section 3.1.2 was performed using the MIC to measure 234 U. Results were similar to the ones obtained with the SEM. The measurement trueness and RSD were lower than 5 % for uranium quantities from 1270 ng down to 10 ng. The method was only validated in term of accuracy for uranium quantities from 1270 ng down to 50 ng. Indeed, a degradation of the measurement trueness and precision was observed for a uranium quantity of 10 ng which rendered the method inaccurate for such quantity.

235 U/ 238 U isotope ratio measurement

The measurement obtained with the SEM and the MIC in section 3.1 showed comparable results. However, contrary to the MIC, an automatic procedure for the intercalibration gain performance was incorporated in the software for the SEM, making the use of the latter detector more straightforward. Therefore, in this section, the isotope measurement using electron multipliers were only realized with the SEM.

Total evaporation with faraday cups

235 U/ 238 U isotope measurements have been performed using the total evaporation method with faraday cups for uranium quantities varying from 1270 ng down to 10 ng (five measurements for each quantity). The measurement trueness for a quantity of 1270 ng (0.10 %), 250 ng (0.01 %), 100 ng (0.10 %) and 50 ng (0.22 %) was less than 0.28 %. For a 10 ng quantity the measurement trueness increased to 1.38 %. A degradation of the repeatability was observed for the lowest amounts of sample. The residual standard deviation of 1270 ng (0.06 %), 250 ng (0.10 %), 100 ng (0.14 %) and 50 ng (0.45%) was calculated. Due to the degradation of the precision for the quantity of 100 ng, the method was inaccurate for this quantity. The method was validated for uranium quantities from 1270 down to 250 ng.

3.2.2. Total evaporation using SEM ( 235 U) and faraday cup ( 238 U)

235 U/ 238 U isotope measurements have been performed by total evaporation method using the SEM to measure 235 U and the faraday cup to measure 238 U. Uranium quantities ranged from 50 down to 10 ng. For a deposit quantity of 50 ng, the measurement trueness of the method was lower than 0.28 % and increased for lower quantities of uranium (> 0.5 %). However, the repeatability, 0.92 % for 50 ng, was important. Thereby, for a maximal bias equal to 0.28 %, the method accuracy was not validated for all the experiments to measure the 235 U/ 238 U ratio using SEM and faraday cup. In fact, the intensity detected on the faraday cup for the 238 U isotope was too small (< 300 mV), deteriorating the stability of the signal, especially the repeatability of the measurement. To conclude, the simultaneous use of faraday cup and SEM for the detection of the major ratio 235 U/ 238 U was not suitable to significantly improve the results.

Classical method with peak-jumping SEM measurement

235 U/ 238 U isotope measurements have been performed by the classical method using the peak-jumping measurements and the SEM to measure 235 U and 238 U. The measurement trueness and the repeatability was lower than 0.28 % for uranium quantities of 50 ng (measurement trueness = 0.03 %, RSD = 0.11 %). For smaller uranium quantities (10 ng, 3 ng and 1 ng), the measurement trueness (0.03 %, 1.08 % and 42 %) and the repeatability (0.34 %, 2.05 % and 95 %) increased too much, more than 0.28 %, to validate the method. Finally, the control of the accuracy showed that the method was only validated for 50 ng.

Uncertainty

For all experiments, the normalized bias was lower than 2 (see supplementary materials). In this case, the different enlarged uncertainties (k = 2) were calculated using formula 5 (Figure 3 and supplementary materials).

For the deposit quantities and cup configurations which conducted to the validated method, the 234 U/ 238 U isotope ratio repeatability was between 0.2 % and 0.7 %. The IRMM 052 certified reference material had a relative uncertainty (k = 2) of 0.4 % for the 234 U/ 238 U ratio. Therefore the repeatability and the reference value uncertainty had an equivalent contribution to the final uncertainty.

The 235 U/ 238 U isotope ratio of the material is certified with a low uncertainty (relative uncertainty of 0.04 % at k = 2). Therefore, in the present case, the reference value uncertainty represented a minor part in the final uncertainty, and the repeatability represented the major component in the final uncertainty. The repeatability was between 0.06 % and 0.11 % for the different configurations where the accuracy was achieved (Table 2)

The relative uncertainties of the 235 U/ 238 U isotope ratio for the different configurations were between 0.12 % and 0.23 %. These values were in agreement with the international target value recommended by the IAEA (0.28 %). [START_REF]Internaltional Atomic Energy Agnecy, International Target Values 2010 for Measurement Uncertainties in Safeguarding Nuclear Materials -STR368[END_REF].

IV. Conclusion

This study presents an analytical method to measure major and minor isotopes of natural uranium by TIMS using faraday cups, SEM and MIC. The 235 U/ 238 U and 234 U/ 238 U isotope ratios were studied in terms of measurement trueness, precision and accuracy according to the NF T 90-210 norm. The most effective methods, in term of accuracy and reduction of the analyzable quantities of uranium, to measure the 235 U/ 238 U and 234 U/ 238 U isotope ratios are summarized in the Table 3.

The minor isotope ratio 234 U/ 238 U was studied by total evaporation method using 3 different detector configurations: only faraday cups, SEM ( 234 U) / faraday cup ( 238 U) and MIC ( 234 U) / faraday cup ( 238 U). The analyzable quantity was reduced from 1270 ng down to 50 ng with an acceptable accuracy, thanks to the use of electron multipliers. Finally, the results obtained with the SEM were close to the ones obtained with the MIC. However, the straightforward use of the SEM and its stability conferred it an advantage when compared to the MIC.

The 235 U/ 238 U isotope ratio was studied by total evaporation method using 3 different detector configurations: only faraday cups, SEM ( 235 U) / faraday cup ( 238 U) and by the classical method using peak-jumping SEM measurement. The lowest analyzable quantity (50 ng) was achieved by peak-jumping measurement. Indeed, with the two others method, the intensity detected by the faraday cup was too small, lower than 300 mV and deteriorated the repeatability of the measurement. The method using only the faraday cups was accurate for uranium quantities between 250 ng and 1270 ng: the calculated uncertainties were below to 0.28 %, in agreement with the international target value recommended by the IAEA. [START_REF]Internaltional Atomic Energy Agnecy, International Target Values 2010 for Measurement Uncertainties in Safeguarding Nuclear Materials -STR368[END_REF] The use of electron multipliers (SEM) allowed to significantly decrease the analyzable quantities to 50 ng and respect the international target value as well. 3: Summary of the most effective method for the 235 U/ 238 U and 234 U/ 238 U isotope ratio determination and the associated reduction of the uranium quantities in comparison to the method using total evaporation with the faraday cups exclusively U isotope ratio measurements by total evaporation method using the faraday cups (a), the SEM (b) and the MIC (c) and results of 235 U/ 238 U isotope ratio measurements by total evaporation method using the faraday cups (d), the SEM (e) and by the SEM measurement (f). Each point is represented with an uncertainty at k=2 (calculated according to equation 5. The dark line (-) corresponds to the reference value and the dotted line (…) represents its uncertainty at k = 2 (not represented for better clarity for the isotope ratio 235 U/ 238 U): 0.0072718 ± 0.00000030 for isotope ratio 235 U/ 238 U and 0.00005548 ± 0.00000022 for isotope ratio 234 U/ 238 U. The prolonged dotted line (---) represent the maximal bias fixed by the laboratory

FigureFigure 1 :Figure 2 :

 12 Figure

Figure 3 :

 3 Figure3: Results for 234 U/ 238 U isotope ratio measurements by total evaporation method using the faraday cups (a), the SEM (b) and the MIC (c) and results of 235 U/ 238 U isotope ratio measurements by total evaporation method using the faraday cups (d), the SEM (e) and by the SEM measurement (f). Each point is represented with an uncertainty at k=2 (calculated according to equation 5. The dark line (-) corresponds to the reference value and the dotted line (…) represents its uncertainty at k = 2 (not represented for better clarity for the isotope ratio 235 U/ 238 U): 0.0072718 ± 0.00000030 for isotope ratio 235 U/ 238 U and 0.00005548 ± 0.00000022 for isotope ratio 234 U/ 238 U. The prolonged dotted line (---) represent the maximal bias fixed by the laboratory

Table Quantity

 Quantity

	(ng)	234 U/ 238 U	Measurement trueness (%)	RSD (%)	Accuracy
	Total evaporation with faraday cup ( 234 U and 238 U)
	1270	0.00005610	1.1	0.7	Valid
	250	0.00005618	1.3	4.5	Not valid
	Total evaporation using SEM ( 234 U) and faraday cup ( 238 U)
	1270	0.00005498	0.9	0.7	Valid
	250	0.00005515	0.6	0.5	Valid
	100	0.00005552	0.1	0.3	Valid
	50	0.00005535	0.2	0.3	Valid
	10	0.00005463	1.5	2.1	Not valid
	Total evaporation using MIC ( 234 U) and faraday cup ( 238 U)
	1270	0.00005542	0.1	0.3	Valid
	250	0.00005572	0.4	0.4	Valid
	100	0.00005575	0.5	0.3	Valid
	50	0.00005535	0.2	0.2	Valid
	10	0.00005466	1.5	1.9	Not valid

Table 1 :

 1 234 U/ 238 U isotope ratio results obtained with different methods and deposited amounts. For each determination five measurements have been performed (certified value: 0.00005548 ± 0.00000011). The method is validated in term of accuracy when the inequality 3 is verified.

	Quantity (ng)	235 U/ 238 U	Measurement trueness (%)	RSD (%)	Accuracy
	Total evaporation with faraday cup ( 235 U and 238 U)
	1270	0.0072644	0.10	0.06	Valid
	250	0.0072714	0.01	0.10	Valid
	100	0.0072645	0.10	0.14	Not valid
	50	0.0072878	0.22	0.45	Not valid
	10	0.0073718	1.38	2.79	Not valid
	Total evaporation using SEM ( 235 U) and faraday cup ( 238 U)
	50	0.0072671	0.06	0.92	Not valid
	30	0.0073285	0.78	1.15	Not valid
	20	0.0072356	0.50	1.08	Not valid
	10	0.0074211	2.05	2.60	Not valid
	Classical method with peak-jumping SEM measurement ( 235 U and 238 U)
	50	0.0072695	0.03	0.11	Valid
	10	0.0072696	0.03	0.34	Not valid
	3	0.0073506	1.08	2.05	Not valid
	1	0.0041864	42	95	Not valid

Table 2 :

 2 235 U/ 238 U isotope ratio results obtained with different methods and deposited amounts. For each determination five measurements have been performed (certified value: 0.0072718 ± 0.0000015). The method is validated in term of accuracy when the inequality 3 is verified.

	Isotope ratio	Method	Reduction of uranium quantities
	235 U/ 238 U	Classical method with SEM	250 ng 50 ng
	234 U/ 238 U	Total evaporation with SEM/FC	1270 ng 50 ng
	Table

Acknowledgements: We are grateful to S. Baghdadi (IRSN/PRP-Hom/SDI/LRC) for her precious advice on the present paper.


Ben Malek

Jérémie Mechlia

Sofiene Schutz

Dellagi

Anis Chelbi
email: anis.chelbi@planet.tn

Agadir -Morocco "New advances and challenges for sustainable and smart industries" OPTIMAL DESIGN OF A LEASED VEHICLE FLEET WITH CONSIDERATION OF MAINTENANCE AND ENVIRONMENTAL CONSTRAINTS

Keywords: Leasing, logistics, vehicle fleet, maintenance, environmental constrains

In this paper, we are interested in designing a fleet of vehicles made of two types (fuel and electric), which will be leased to achieve a set of transportation missions during a maximum period of time. A mathematical model is developed in order to determine the number of vehicles of each type to be leased, the number of months of use of each vehicle and the preventive maintenance period for each one. The objective is to minimize the expected total cost including leasing, operation, preventive and corrective maintenance, and environmental impact. An illustrative example is provided and numerical results are presented and discussed.

INTRODUCTION AND LITTERATURE REVIEWS

This paper deals with the selection of the types and the number of vehicles to be leased by logistics departments to achieve transportation missions. This decision is not an easy one given the diversity of vehicles on the market, each one involving specific parameters such as the type (fuel, electric, or hybrid), reliability, costs related to leasing, operation, maintenance, and environmental impact. Due to significant changes in the environmental impacts legislation, the logistics service providers have to take these incremental changes into account. Consequently, it is important to think about the right choice of transportation means and take them into consideration when designing a logistics network, which provides a comprehensive visibility and control. Logistics costs are usually computed according to the type of means of transport as well as the quantity transported, the distance traveled, and the penalty costs caused by delays. [START_REF] Benoist | Recherche locale pour un problème d'optimisation de tournées de véhicules avec gestion des stocks[END_REF] deal with the vehicle routing problem combined with inventory management in order to minimize logistics costs. The authors demonstrated that their model could reduce costs by 20 % compared to what was done by experts in the field of logistics. [START_REF] Mtalaa | Un modèle bi-objectif pour la conception de chaînes logistiques vertes[END_REF] proposed a mathematical model that minimizes the effects of carbon dioxide. They then proposed an alternative model that minimizes the overall economic costs of the supply chain i.e. the direct traditional logistics costs, in closed loop and the costs of reverse logistics.

These costs are considered in the paper as external transport costs. Similarly, [START_REF] Sawadogo | Modèle de plus court chemin multi objectif pour le transport intermodal au sein de la chaîne logistique verte. 8e Conférence Internationale de modélisation et Simulation[END_REF] focused on the integration of environmental and societal impacts in an intermodal transport network. The aim of their work is to determine the most efficient path to minimize transport costs, transport time and the various environmental and societal impacts. The criteria used for decision making are the following: economic cost, transport time, air pollution, energy consumption, noise pollution, damage due to transshipments and risk of accidents. They presented each criterion using a mathematical model. The specificity of this approach lies in allowing the link between the environmental impacts emanating from the means of transport and their estimated costs.

To address the problem of integrating maintenance into transport logistics, [START_REF] Mechlia | Integrated maintenance policy for transportation system according to a supply chain[END_REF] presented a motivational and bibliographic study on the selection of the best transportation means in order to ensure the supply of customers depending on the type of merchandises. Moreover, they presented works that deal the reliability and maintenance strategy for several types of transportation means, while pointing out that little work addressed the integration of maintenance and logistics. Further [START_REF] Troudi | An optimal maintenance policy for transport vehicles in a supply chain under infrastructure / environment constraints[END_REF] worked on the influence of road type and the weather conditions on vehicle failure rates. In fact, they sequentially established the optimal number of preventive maintenance actions to be performed on the vehicle as well as the optimal distance to be travelled. Furthermore, [START_REF] Rezg | Maintenance Optimization for Transport Vehicles in a Supply Chain[END_REF] MOSIM '20 -November 12-14, 2020 -Agadir -Morocco considered one type of vehicle required to carry out a set of tours during a deterministic period. The different circuits to be carried out are characterized by the time required to complete a tour as well as their road types. Each tour generates a process of vehicle degradation, which reduces the profit generated by the tour, which can cause failures where the maintenance cost increases rapidly. They developed a mathematical model that minimizes the cost of transport between different elements of the logistics network, as well as the road cost, and the cost of maintenance actions.

Regarding the choice of vehicles by type, [START_REF] Turki | Study of a manufacturing system with transport activities in urban area[END_REF]) studied a manufacturing system with transport activities in urban area. The authors considered two cases for the transportation of the products from the warehouse to the customer. In the first case, the transport is performed by an electric vehicle and in the second case by a fuel vehicle. They presented a model that allows the manufacturer to decide to use fuel or electric vehicle based on transportation, maintenance costs and the cost of carbon penalty. Two simulation algorithms based on discrete events are used for each type of vehicle to compare total costs and then provide a decision. Furthermore, (Prud'homme and Koning, 2015) performed a comparative study between an electric and a fuel vehicle taking into account the negative environmental impacts generated by the vehicles. They have a model to estimate the costs of an electric vehicle, compared to a similar fuel vehicle, as well as the CO 2 emission. These magnitudes are a function of a dozen parameters, such as purchase cost, electrical efficiency, the cost of the battery, the price of electricity, CO 2 content of electricity (for the electric vehicle), the purchase cost, the efficiency and price of fuel, the costs of local pollution, CO 2 emission (for the fuel vehicle), as well as the lifetime of the vehicle and the mileage travelled. The methodology adopted is a comparative study between an electric vehicle and a fuel vehicle providing approximately the same level of service during the same period. It does so from three important points of view: costs to the consumer, socio-economic costs and CO 2 emissions.

Management of vehicles leasing has attracted the attention of several researchers. For example, [START_REF] Balcik | Centralized vehicle leasing in humanitarian fleet management: the UNHCR case[END_REF] proved in their study how the UN High Commissioner for Refugees (UNHCR) has successfully improved its vehicle fleet management through the introduction of an Internal Leasing Program (ILP). The authors compared the fleet performance indicators before and after implementation of the (ILP). They indicated that the introduction of the (ILP) has positively impacted the fleet management by reducing procurement costs, fleet size and the average age of the fleet. Thanks to the use of the (ILP), a decrease of the fleet size by 11% has been realized between April 2013 and October 2015.

Dealing electric vehicles leasing, (Miao et al., 2018) proposed a comprehensive pricing for a Chinese electric vehicle lease company in order to optimize the annual operations profit for vehicle lease service based on mileage-based pricing (MBP). The profit introduced by authors integrated the revenue and the maintenance costs.

In order to reach this objective, the authors developed a differential pricing strategy. In addition, an enumerative algorithm is proposed in order to determine the optimal pricing and examine the influence of some parameters on the optimal profit. The authors proved through the use of the differential pricing strategies, a significant increased profit (between 2.3% and 69.8%).

Looking at the literature related to fleet vehicles sizing or comparison between several types of vehicles in logistic problems, the simultaneous consideration of a vehicle fleet size while considering maintenance and environmental impacts in the context of leasing, has not been studied in depth to highlight the strong interaction between these aspects.

The specificity of our study consists in designing a fleet of vehicles made of two types (fuel and electric), which will be leased to achieve a set of transportation missions during a maximum period of time. This is done taking into account the environmental impact, the maintenance of the vehicles and their availability.

The paper is organized as follows. The description of the problem is presented in section 2. In section 3 we provide the mathematical model with the working assumptions and the used notation. In section 4 we propose a numerical example, an analytical result is presented in the same section to understand and ensure the performance of the model developed and solved with MATLAB software. Finally, the last section concludes the paper and gives some perspectives to our work.

PROBLEM DESCRIPTION

Generally, deliveries have to be made to customers scattered over a very large area (different cities on a continent) and the vehicles must therefore be available for distribution.

In such situations, breakdowns are likely to occur at any time on the way between two recipients and repairs can last more or less time causing sometimes important delays and consequently significant penalties. Today, the impacts of transportation mean on the environment are well known and include global warming, deterioration of the ozone layer, dispersion of organic and inorganic toxic substances, and the landscapes degradation. In fact, the gaseous emissions are responsible for respiratory diseases, environmental damage and visibility problems, such as the fog. Therefore, to protect the environment, all companies must find solutions to control gas emissions as well as those of other toxic products. In addition, the logistics sector has experienced a significant growth. This growth has been marked by a negative impact on the environment, which has led governments to establish laws and requirements such as the (Kyoto protocol, 1998) and the (ISO 14001, 2015). This standard sets an efficient framework for the company to move towards an environmentally efficient system.

MOSIM '20 -November 12-14, 2020 -Agadir -Morocco This study focuses on the case of a logistics company that wants to lease a commercial vehicle fleet in order to satisfy a well-defined mission. More precisely, the present study was particularly motivated by the determination of the optimal number of two types of vehicles to lease (fuel and electric), in order to satisfy customers and meet delivery schedules. This objective will be reached by minimizing the expected total cost integrating leasing, operation, environmental impact, and maintenance costs.

THE MATHEMATICAL MODEL

Notation

The following notations are used in the mathematical formulation of the model:



Working assumptions

The following assumptions are considered:

 Two types of vehicles are considered (fuel and electric vehicles).  Each type of vehicle will travel a (𝑌 𝑖 . 𝑣 𝑖 ) number of kilometers.  The renter of vehicles will ensure the maintenance of each vehicle.  The leasing cost of vehicles depends on the type, agency and duration.  The renter is in charge of renewing the batteries

The objective function and constraints

Our objective consists in minimizing an objective function of cost including the leasing, the operating, the maintenance, and the environmental impact costs according to variable decisions; which are the number of vehicles to lease by type i (𝑋 𝑖 ), the number of months of use for one leased vehicle of type i (𝑌 𝑖 ) and the periodicity of a preventive maintenance action associated to the leased vehicle (𝑚 𝑖 ) in order to ensure 𝐾 kilometers during a specified period (𝑇 𝑚𝑎𝑥 ) with a fixed mean usage rate (𝑣 𝑖 ) for each type. We start by developing analytically the objective function which will be optimized in order to determine the optimal decision variables. We also presented the constraints which will be respected.

The expected total costs are as follows:

𝑚𝑖𝑛 ∑ 𝑋 𝑖 . (𝐶 𝐿𝑖 . 𝑇 𝑖 + 𝐶 𝑂𝑖 + 𝐶 𝑀𝑖 + 𝐶 𝐸𝑖 ) 𝑁 𝑖=1

(1)

Detailed constraints

The constraints of the objective function are defined as follows:

∑ 𝑋 𝑖 . 𝑌 𝑖 . 𝑣 𝑖 ≥ 𝐾 𝑁 𝑖=1 (2) 𝑇 𝑖 ≤ 𝑇 𝑚𝑎𝑥 ∀ 𝑖 = 1, … , 𝑁 (3)
𝑇 𝑖 ≥ 𝑇 𝑚𝑖𝑛 ∀ 𝑖 = 1, … , 𝑁 (4) 𝑌 𝑖 ≤↿ 𝑋 . min (𝑇 𝑚𝑎𝑥 , 𝐾 𝑣 𝑖 ) ∀ 𝑖 = 1, … , 𝑁 (5)
We recall that the objective function (1) minimizes the sum of leasing, operating, maintenance and environmental impact costs; whereas, constraint (2) ensures that the number of kilometers traveled by the all vehicles must be greater than or equal to the minimum total number of kilometers (𝐾) to be traveled by all types of vehicles. Constraint (3) ensures that the possession duration of each vehicle of type i (𝑇 𝑖 ) is shorter than or equal to the period during which the missions have to be performed (𝑇 𝑚𝑎𝑥 ). Constraint (4) denotes that the possession duration of leased vehicles of type i (𝑇 𝑖 ) must MOSIM'20 -November 12-14, 2020 -Agadir -Morocco be greater than or equal to the minimum period for leasing vehicles of type i (𝑇 𝑚𝑖𝑛 ). This constraint ensures the longterm lease of vehicles as well as a minimum leasing cost. Constraint (5) ensures that the number of months of use for one leased vehicle of type i (𝑌 𝑖 ), will be 0 if no vehicle of this type is leased.

The leasing cost

The leasing cost represents an important parameter in our work. It is obtained by the product of the unit leasing cost (proposed by the lessor) and the possession duration of each leased vehicle of type i (𝐶 𝐿𝑖 . 𝑇 𝑖 ). In fact,(𝑇 𝑖 ) includes the number of months of use of each leased vehicle of type i (𝑌 𝑖 ) and the average duration of all corrective and preventive maintenance actions performed on the vehicles. It is expressed as follows:

𝑇 𝑖 = ⌈𝑌 𝑖 . ∅(𝑌 𝑖 , 𝑚 𝑖 ). 𝜇 𝑐𝑖 + ⌊ 𝑌 𝑖 𝑚 𝑖 ⌋ . 𝜇 𝑃𝑖 ⌉ (6)
Moreover, we consider in our study two types of vehicles to lease (fuel (𝑖 = 1) and electric vehicle (𝑖 = 2)). Therefore, the leasing cost is different between the two types.

We consider that the leasing cost of vehicles depends on the type of the vehicle, the leasing agency and the lease duration. It is negotiable. Based on data from rental agencies, we propose in Table 2 the leasing costs of vehicle for each type i. Indeed, these costs are proposals for a long leasing duration and they do not include the operating, maintenance and environmental impact.

The expected maintenance cost

We adopt a strategy of an imperfect periodic preventive maintenance with minimal repair at failures; known in the literature [START_REF] Gertsbakh | Reliability theory: with applications to preventive maintenance[END_REF] and used by [START_REF] Gouiaa-Mtibaa | Integrated Maintenance-Quality policy with rework process under improved imperfect preventive maintenance[END_REF]. More precisely a PM action is performed for vehicles of type i after each 𝑚 𝑖 kilometers. The model assumes that, after each PM action, the failure rate is decreased to a certain extent and the failure rate function is then increased by a certain coefficient that can be determined empirically. Minimal repairs take place whenever failures are detected between PM actions to restore the system into the operating state without changing its failure rate function.

In our work, the number of PM actions is the ratio of the number of months of use of each leased vehicle of type i (𝑌 𝑖 ) and the periodicity of a PM action associated to the leased vehicle of type i;

⌊ 𝑌 𝑖 𝑚 𝑖 ⌋.
Adapting the model proposed by [START_REF] Gertsbakh | Reliability theory: with applications to preventive maintenance[END_REF] to our study, the average number of breakdowns (minimal repairs) ∅(𝑌 𝑖 , 𝑚 𝑖 ) is expressed in the following way:

∅(𝑌 𝑖 , 𝑚 𝑖 ) = ∑ ∫ 𝑒 (𝑗-1).𝛼 𝑚 𝑖 0 . 𝜆 𝑖 (𝑥)𝑑𝑥 ⌊ 𝑌 𝑖 𝑚 𝑖 ⌋ 𝑗=1 + ∫ 𝑒 ⌊ 𝑌 𝑖 𝑚 𝑖 ⌋ . 𝛼 𝑌 𝑖 -⌊ 𝑌 𝑖 𝑚 𝑖 ⌋.𝑚 𝑖 0 . 𝜆 𝑖 (𝑥)𝑑𝑥 (7)
It is easy to see in equation ( 7), that PM actions induce a reduction in the failure rate. However, the average number of breakdowns is increased, between PM actions, by a "degradation" factor (e  ) where  represents a positive parameter to be determined empirically.

The total maintenance cost is expressed in the following way:

𝐶 𝑀𝑖 = 𝐶 𝑀𝐶𝑖 . ∅(𝑌 𝑖 , 𝑚 𝑖 ) + 𝐶 𝑀𝑃𝑖 . ⌊ 𝑌 𝑖 𝑚 𝑖 ⌋ (8)

The expected operating cost

The operating cost is obtained according to the use of leased vehicles of type i during their operation duration.

We refer to the literature in order to formulate the operating cost for each type:

Fuel type. The fuel vehicle is characterized by its fuel efficiency, which is the number of kilometers driven per unit of fuel consumed (km/liter). Efficiency and productivity are generally defined as an (output/input) ratio and increases in productivity or efficiency are considered desirable. This is in conflict with the European standards of measuring the performance of a fuel vehicle in liter/100 km or in kWh/100 km for electric vehicle (but not with the US standards of measuring it in miles/gallon).

Based on the formulation developed by (Prud'homme and Koning, 2012), the efficiency is therefore considered as the inverse of the definition commonly used in Europe.

The following equation ( 9) presents the operating cost for fuel vehicle (𝑖 = 1):

𝐶 𝑂1 = 𝐶 𝑢_𝑑𝑖𝑒𝑠𝑒𝑙 . ( 𝑌 1 . 𝑣 1 𝑒𝑓𝑓 1 ) (9)
With :

 𝐶 𝑢_𝑑𝑖𝑒𝑠𝑒𝑙 : Unit cost of fuel (m.u/liter)  𝑌 1 : Number of months of use of each leased fuel vehicle (months)  𝑣 1 : the usage rate associated to the leased fuel vehicle (kms/months)  𝑒𝑓𝑓 1 : Fuel efficiency (km/liter) Electric type. The operating of an electric vehicle is characterized by the electric consumption (the recharge cost and the battery autonomy). Usually, the driver of this vehicle must lease a battery or buy a new battery just at the end of its service life. Thus, the lessor of this vehicle type supports of the battery leasing or purchase. The user pays only the leasing cost. Therefore, we have developed the operating cost of electric vehicle.

MOSIM '20 -November 12-14, 2020 -Agadir -Morocco The following equation ( 10

The expected environmental impact cost

We consider the emission of CO 2 is an important factor in the proper use of vehicles and the respect of environment, therefore a mathematical model on the cost of CO 2 emission is considered in our objective function as essential negative environmental impact. Based on the work of (Prud'homme and Koning, 2012), we propose the following equation that represents the cost of impact environmental calculated from the quantity of CO 2 emitted by the vehicle of type i multiplied by its unit cost. In fact, the unit cost of CO 2 is not the same in the world but can change between countries. Therefore, we used in our work the unit cost defined in the work of [START_REF] Ba | Joint optimization of preventive maintenance and spare parts inventory for an optimal production plan with consideration of CO2 emission[END_REF].

The impact environmental impact costs are given by:

𝐶 𝐸𝑖 = 𝐶𝑈 𝐶𝑂2

NUMERICAL EXAMPLE

We conducted many computational experiments using the numerical resolution with MATLAB software (Version 2019a). We present in this section an illustrative example and show the obtained results.

Experiments input

For our numerical example, we used the values given in the following Tables 1 and2. Table 2 shows the input parameters used by type of vehicle (fuel and electric). In our study, we suppose to realize at least 𝐾 kilometers with a same usage rate associated with each vehicle of type i (𝑣 𝑖 ) during 60 months (𝑇 𝑚𝑎𝑥 ). During this period, the missions have to be performed while obviously taking into account the environmental impact for which the nuisance caused by 1 kg of CO 2 emitted into the air is estimated to 0.05 m.u/kg by [START_REF] Ba | Joint optimization of preventive maintenance and spare parts inventory for an optimal production plan with consideration of CO2 emission[END_REF].

Parameters

Point of view reliability, we assume that the type of vehicle has an increased failure rate following Weibull distribution with (shape parameter = 2) and (scale parameter = 25) for fuel vehicle and (scale parameter = 30) for electric vehicle. The durations of maintenance actions follow an exponential distribution with an average of 1 day for repairs (𝜇 𝑐𝑖 ) and a mean of 0.5 day for PM (𝜇 𝑃𝑖 ), regardless of vehicle type.

MOSIM '20 -November 12-14, 2020 -Agadir -Morocco Regarding the energetic efficiency and the CO2 emissions, we use the same parameters used by (Prud'homme and Koning, 2012).

Obtained results

Based on data from Tables 1 and2, the numerical resolution of our problem provided the results shown in Table 3.

We recall that the decision variables (𝑋 𝑖 ), (𝑌 𝑖 ) and (𝑚 𝑖 ) represent respectively the total number of vehicles of type i to lease, the number of months of use of each leased vehicle, the PM period to be adopted for a vehicle of type i.

Fuel vehicle (i=1) Electric vehicle (i=2)

𝑋 𝑖 6 5

𝑌 𝑖 52 57

𝑚 𝑖 14 15

Table 3: The obtained results

The obtained results show that it is recommended to lease 6 fuel vehicles and 5 electric vehicles. The fuel vehicles will be used for 52 months, each one of them is expected to travel a total distance of (𝑌 1 . 𝑣 1 ) (873 600 kms) and be submitted to PM every 14 months. The electric vehicles will be used 57 months, each one of them is expected to travel a total distance of (𝑌 2 . 𝑣 2 ) (957 600 kms) with a PM period of 15 months.

CONCLUSIONS AND PROSPECTS

In this paper, we considered optimizing the typology and size of a vehicle fleet to be leased for a given maximum period of time taking into account maintenance and environmental constraints. Two types of leased vehicles (fuel and electric) were considered. The decision variables are the number of vehicles to lease, the number of months of use of each vehicle and the periodicity of preventive maintenance actions to be performed on each vehicle. The derived solution must satisfy the fact that at least K kilometers must be travelled during a specified period (𝑇 𝑚𝑎𝑥 ) with an average usage rate (νi).

A mathematical cost minimization model has been developed. Furthermore, the problem is then solved using MATLAB language. A numerical experimentation has been performed and the obtained results have been analysed.

The average usage rate of every type of vehicles has been considered as constant and predefined. As an extension of this work, it is worth to consider it as an additional decision variable. In fact, the combination between the number of vehicles and the average usage rate for every type of vehicle impacts maintenance cost and environmental costs.

  𝑇 𝑀𝑎𝑥 : Period during which the mission has to be performed (months)  N: Number of vehicle types, with i is considered the type of vehicle 𝑖 𝜖 {1, … , 𝑁}  𝐾 : The minimum total number of kilometers to be traveled by all types of vehicles (kms)  𝐶 𝐿𝑖 : The leasing cost of a vehicle of type i (cost fixed by the lessor) (m.u/month)  𝐶 𝑂𝑖 : Operating cost of a vehicle of type i (m.u)  𝐶 𝑀𝑖 : Expected total maintenance cost for a vehicle of type i (m.u)  𝐶 𝐸𝑖 : Expected cost of environmental impact for a vehicle of type i (m.u)  𝐶 𝑀𝐶𝑖 : Cost of corrective maintenance action for vehicles of type i (m.u)  𝜇 𝑐𝑖 : The average duration of a corrective maintenance action for a vehicle of type i (months)  𝐶 𝑀𝑃𝑖 : Expected cost of preventive maintenance (PM) for a vehicle of type i (m.u)  𝜇 𝑝𝑖 : The average duration of a preventive maintenance for a vehicle of type i (months)  𝑣 𝑖 : The average usage rate associated with a vehicle of type i (kms/month)  𝑇 𝑖 : The possession duration of each leased vehicle of type i (months)  𝑇 𝑚𝑖𝑛 : The minimum period for leasing vehicles of type i (months)  ∅(𝑌 𝑖 , 𝑚 𝑖 ) : The average number of breakdowns (minimal repairs) over a period of use of a vehicle of type i for a given PM period 𝜇 𝑝𝑖  𝜆 𝑖 (•): The failure rate function of a vehicle of type i  ⌊. ⌋: Rounds to the nearest inferior integer  ⌈. ⌉: Rounds to the nearest superior integer The decision variables are defined as follows:  𝑋 𝑖 : Number of vehicles to lease of type i, 𝑖 𝜖 {1, … , 𝑁}, with 𝑋 𝑖 𝜖 ℕ  𝑌 𝑖 : Number of months of use of each leased vehicle of type i, 𝑖 𝜖 {1, … , 𝑁}, with, 𝑌 𝑖 𝜖 ℕ  𝑚 𝒊 : Preventive maintenance period to be adopted for a vehicle of type i (months) with 𝑚 𝒊 𝜖 ℕ *

  . 𝑌 𝑖 . 𝑣 𝑖 . 𝑔 𝑖 . (

	1 𝑒𝑓𝑓 𝑖	)	(11)
	With:
	 𝐶𝑈 𝐶𝑂2 : The unit cost of CO 2 (m.u/kg)
	 𝑌 𝑖 : Number of months of use for one leased vehicle of
	type i
	 𝑣 𝑖 : The usage rate associated to the vehicle of type i
	(kms/month)
	 𝑔 𝑖 : CO 2 emission in (kg/liter) for fuel vehicle (𝑖 = 1)
	and in (kg/kWh) for electric vehicle (𝑖 = 2)
	 𝑒𝑓𝑓 𝑖 : The energetic efficiency (number of kilometers
	traveled per liter for fuel type (km/liter) or per
	kilowatt-hour for electric type (km/kWh)

Table 2 :

 2 Numerical input data for fuel and electric vehicle

		Numerical input data
	𝐾 (kms)		10 000 000
	𝑇 𝑚𝑎𝑥 (months)		60
	𝑇 𝑚𝑖𝑛 (months)		24
	𝐶𝑈 𝐶𝑂2 (m.u/kg)		0.05
			1.05
	Table 1: Numerical input data
		Fuel vehicle	Electric
		(i=1)	vehicle (i=2)
	𝐶 𝑙𝑖 (m.u/month)	250	260
	𝐶 𝑀𝐶𝑖 (m.u)	250	300
	𝐶 𝑀𝑝𝑖 (m.u)	60	80
	𝑣 𝑖 (kms/month)	16 800	16 800
	𝑒𝑓𝑓 𝑖 (kms/liter)	20	5
	𝐶 𝑢_𝑑𝑖𝑒𝑠𝑒𝑙 (m.u/liter)	1.5	-
	𝑔 1 (kg/liter)	2.6	-
	𝑔 2 (kg/kWh)	0.09	-
	𝐶 𝑐ℎ (m.u)	12.50	-
	𝑐 𝑢𝑛𝑖𝑡 (m.u)	-	6000
	𝑁𝑏 𝑀𝑎𝑥𝐶ℎ	-	2000
	𝜃 (kms)	-	160