Boxin Huang

Yong He

Li Wang

Jian Shi

Jie Hu

Elrade Rofaani

Ayako Yamada

Yong Chen

Microfluidic channel with embedded monolayer nanofibers for cell culture and co-culture

Keywords: Microfluidics, Organ on-a-chip, Basement membrane, Blood vessel

Mimicking the cellular microenvironment is important for organoids and organ on-a-chip studies. One of the current issues is to introduce vessel-like structures into the culture system to improve the cellular and tissue functions, which deserves particular efforts in design and systematic consideration. Based on a standard device configuration, we fabricated a vessel-like component which can be easily integrated for cell co-culture. This component consists of an embedded monolayer of gelatin nanofibers on the top of an open channel. It can then be enclosed with an upper plastic plate with molded chamber, channels and standard Luer connectors. Human umbilical vein endothelial cells (HUVECs) were firstly introduced into the vessel-like channel and cultivated threedimensionally with the help of a rotational device. Then, a flow was applied for cytoskeleton remolding, resulted in a dense and aligned HUVECs layer. Afterward, human glioblastoma cells (U87) were introduced in the upside of the fiber layer and a flow was also applied for the upper cell layer culture. Our results show adjunct formation of HUVEC and U87 cell layers on both sides of the monolayer of gelatin nanofibers, thereby providing a reliable support for a variety of co-culture assays.

Introduction

Blood vessels, which transport blood cells, nutrients, and oxygen to the tissues and organs, are omnipresent in the body. Any abnormal function of the vessels may cause diseases and, thus, it is important to develop new vessel models taking into account physiological and pathological specification. In the current context of organ on-a-chip development, introducing artificial vessels become a key step toward higher body functions. In this regard, one has to create hollow tubular structures covered by endothelial cells [START_REF] Koike | Creation of long-lasting blood vessels[END_REF][START_REF] Chwalek | Glycosaminoglycan-based hydrogels to modulate heterocellular communication in in vitro angiogenesis models[END_REF][START_REF] Ehsan | A three-dimensional in vitro model of tumor cell intravasation[END_REF] and to co-culture other types of cells around the model vessels [START_REF] Bhatia | Microfluidic organs-on-chips[END_REF][START_REF] Cho | Three-Dimensional Blood-Brain Barrier Model for in vitro Studies of Neurovascular Pathology[END_REF][START_REF] Bischel | The importance of being a lumen[END_REF] . By using different microfabrication techniques, various tubular structures have been proposed but plastic or elastomeric membranes were often used for co-culture, which differ significantly from the in vivo characteristics. In fact, the organ constituents such as alveolar and blood-brain-barrier are structurally formed with the help of basement membrane (BM), which is about 100 nm thick and made of specific extracellular matrix proteins. The challenge is then to mimic such a membrane and to make it integrative in a microfluidic device.

Previously, we have devised a so-called "patch method" by forming a monolayer of cross-linked gelatin nanofibers on a thin honeycomb frame [START_REF] Tang | Patch method for culture of primary hippocampal neurons[END_REF][START_REF] Tang | Induction and differentiation of human induced pluripotent stem cells into functional cardiomyocytes on a compartmented monolayer of gelatin nanofibers[END_REF][START_REF] Tang | Effective motor neuron differentiation of hiPSCs on a patch made of crosslinked monolayer gelatin nanofibers[END_REF] . Although this device has been initially developed for pluripotent stem cell differentiation, the fabricated device can also be used for co-culture. Here, we developed an alternative approach to embed a suspended monolayer of cross-linked nanofibers in a microfluidic channel for co-culture under dynamic flow conditions. To make this method flexible and versatile, we adopted a standardized chip configuration in which the device is composed of two plates. A microchannel was defined in the lower plate and the monolayer nanofibers were embedded on top of the channel. The upper plate with elastomeric interlayer and Luer connectors was clamped above the lower plate with a mechanical device for a tight and reversible sealing. Finally, the device was connected to a fluidic controller for endothelial cell culture and co-culture under flow conditions. With this system, we achieved the formation of continuous endothelial cell layer covering the whole microchannel wall and the embedded nanofiber layer. Co-culture of HUVECs and U87 cells separated by porous monolayer of gelatin nanofibers was also demonstrated and the shear flow-induced cytoskeleton remolding has been revealed. Overall, this approach is successful and valuable for further investigation.

Materials and methods

Reagents were purchased from Sigma-Aldrich (France) unless otherwise specified.

Fabrication of the device

We firstly fabricated the lower plate of the device (Fig. 1a). A 500 µm-wide, 4 cm-long rectangular channel with 4 mm-diameter reservoirs at its extremities was created in a 500 µm-thick layer of polydimethylsiloxane (PDMS, RTV 615, Eleco-EFD, France) by cutting plotter (Graphtec CE6000-40, France). After plasma treatment, the PDMS layer was bonded to a 1 mm-thick glass slide. Afterward, a thin layer of non-reticulated viscous PDMS (v-PDMS) was deposited on top of the PDMS layer by microcontact printing. Here, the v-PDMS was firstly prepared by mixing the PDMS pre-polymer and the crosslinker at the ratio of 10:1 (w/w), degassed, and spin-coated on a glass substrate. Then, the PDMS/glass assembly was gently pressed against the v-PDMS. After separation, a monolayer of gelatin nanofibers was electrospun on the v-PDMS-coated surface, followed by a thermal treatment at 100°C for 10 min. Thus, the gelatin nanofibers could be embedded in the PDMS, resulting in a free-standing monolayer of nanofibers. 10 wt% gelatin powder from porcine skin was dissolved in the mixed solution of acetic acid, ethyl acetate and DI water at the volume ratio of 21:14:10 prior to electrospinning. The gelatin solution was electrospun for 12 min from a syringe needle to the lower plate, which was placed on an aluminum foil at 10 cm distance from the needle tip. The solution was injected at 0.2 mL/h by syringe pump, while the voltage between the needle and the foil was kept at 11 kV. Nanofibers in the reservoir areas were carefully removed by a fine cotton stick wet with DI water and the remaining nanofibers were treated with 0.2 M N-(3-dimethylaminopropyl)-N′-ethylcarbodiimide hydrochloride and 0.2 M Nhydroxysuccinimide in ethanol for chemical crosslinking for 4 h at room temperature (RT). As a result, a robust and flat monolayer of cross-linked gelatin nanofibers was formed, being suspended on top of the microfluidic channel.

An upper plate with two inlets (Fig. 1b left) or four inlets with a chamber (Fig. 1e) provided by Mesobiotech, ltd, was assembled with the lower plate to finalize the microfluidic device for the formation of endothelial cell layer (Fig. 1d) or co-culture of HUVECs and U87 (Fig. 1e), respectively. The upper and the lower plates can be reversibly assembled with a hand-screw clamp device and the assembly could be easily mounted on a rotation system for 3D culture (Fig. 1c). Both the clamp device and the rotation system were also provided by Mesobiotech, ltd.

Characterization of the device

The microchannel with embedded gelatin nanofibers of the lower plate were observed by scanning electron microscope (SEM, Hitachi S800) operated at 10 kV. To observe the cross-section of the channel, the lower plate was cut perpendicularly to the channel by a glass cutter and a blade. A 5 nm-thick gold layer was sputtered on the samples prior to the SEM observation.

Cell culture

Human umbilical vein endothelial cells (HUVECs) and human glioblastoma cells (U87) purchased form Sigma-Aldrich, were cultured with endothelial cell growth medium (Cell Applications, Inc.) supplemented with 1% (v/v) penicillin streptomycin (Pen Strep, Thermo Fisher Scientific) (Medium 1) and Dulbecco's modified Eagle medium (DMEM) supplemented with 10% (v/v) fetal bovine serum and 1% (v/v) Pen Strep (Medium 2, all reagents from Thermo Fisher Scientific), respectively, at 37 °C with 5% CO2 in a humidified incubator.

After sterilization, the microfluidic channel was assembled with a two-inlet cover plate and reinforced by the clamp device. To promote cell adhesion, the channel was treated with 100 µg/mL poly-D-lysine (PDL) in Dulbecco's phosphate buffered saline (DPBS) at 4°C overnight and 1% (v/v) Geltrex hESC-Qualified matrix in DPBS at 37 °C for 1 h. HUVECs at the density of 1.5×10 7 /mL in 50 µL Medium 1 were seeded into the channel and incubated at 37 ℃ with 5% CO2 for 4 h (Fig. 1d).

During the first hour of incubation, the device was rotated 90° three times to allow cell attachment on the entire channel surface, including the exposed area of the nanofiber monolayer. 200 µL of Medium 1 was added into the channel and the device was incubated overnight in the CO2 incubator.

HUVECs were then cultured under a dynamic condition with a flow of the culture medium at 0.1 mL/h, controlled by a syringe pump, for 2 days.

For co-culture, the two-inlet upper plate was replaced by a four-inlet one equipped with a microfluidic channel and a central culture chamber (Fig. 1c,1e). 5 x 10 6 U87 cells suspended in 50 µL of Medium 2 supplemented with 10% (v/v) Geltrex were seeded on the other side of the nanofiber monolayer through the inlet connected to the upper channel. After 2 h incubation at 37 °C with 5% CO2, Medium 1 and Medium 2 were injected to fill up the lower channel and upper culture chamber, respectively. Both channels were perfused with a constant flow for further culture.

Control of microflow condition

After cell adhesion, a constant flow of the culture medium at the rate of 0.1 mL/h was applied using a syringe pump. The co-culture was also performed with the same flow rate but with different culture media above and below the nanofiber-supported cell layers. The shear stress (𝜏) in the channel is given by

𝜏 = 6μQ wh 2
where μ is the dynamic viscosity of the medium (0.012 dyn.s/cm 2 ), Q the flow rate (0.1 mL/h), w and h the width (500 μm) and the height (500 μm) of the channel, respectively. 𝜏 in our condition is 0.02 dyn/cm 2 .

Immunofluorescence imaging

HUVECs were fixed with 4% (w/v) paraformaldehyde in DPBS for 15 min at RT and rinsed with DPBS three times, prior to the incubation with 0.5% (w/v) Triton X-100 in DPBS for 10 min at RT for permeabilization. After rinsing with DPBS three times, the cells were further incubated with 3 % (w/v) bovine serum albumin, 0.1% (w/v) Triton X-100, and 0.1% (w/v) sodium azide in DPBS for 30 min at room temperature. Subsequently, the cells were incubated with Atto 488 Phalloidin and DAPI (Thermo Fisher Scientific) in DPBS for 20 min at RT to visualize actin filaments and nuclei, respectively. After three times rinsing with DPBS, cell morphologies and distribution were observed by confocal microscope (LSM 710, Zeiss).

Cell labeling

Cell Tracker TM fluorescent probes (Thermo Fisher Scientific) were used to distinguish the two cell types. 1 µM Cell Tracker™ Deep Red (C34565) and 25 µM Cell Tracker™ Green (2925) in serum free medium were used to label HUVECs and U87 cells, respectively. Prior to labeling, both cells were cultured separately in tissue culture dishes (Fisher Scientific) with the appropriate medium. After medium removal, pre-warmed Cell Tracker TM solutions were gently added to the corresponding dishes and the cells were incubated with the solutions for 30 min at 37°C . Then, the cells were trypsinized by TrypLE Express (Thermo Fisher Scientific), collected by centrifugation, and seeded into the device as described above. After 24 h of co-culture, the cells were imaged by fluorescence microscope (Observer Z1, Zeiss).

Results and discussion

The gelatin nanofibers embedded in the top PDMS layer can be easily observed bySEM (Fig. 2). In Fig. 2a, porous structures of the fiber layer can be clearly seen, indicating that all pores are small enough to capture all the cells during cell seeding. Although the exact thickness of the nanofibers layer cannot be determined from Fig. 2b and2c, the high porosity observed in Fig. 2a suggests the formation of a monolayer of nanofibers and a thickness of few hundred nanometers.

Such a monolayer of crosslinked nanofibers has many advantages for cell culture and co-culture in terms of nutrient diffusion, cell to cell interaction across the monolayer, and the structure closely mimicking the basement membrane. As seen in Fig. 2a-c, the monolayer is flat across the microchannel without deformation, proving that it is well integrated and supported by the thin PDMS layer on the surface of the lower plate.

Both the channel and the nanofiber embedding layers are in PDMS. Once the lower plate is clamped with the upper plate, a conformal contact is achieved to prevent leakage. Other methods such as PDMS casting can also be used to produce the lower plate structure. This device configuration might be versatile for many applications involving cell-cell, cell-organoid and organorgan co-culture.

After incubation for 24 h, the cell layers got confluent (Fig. 3a). The monolayer of endothelial cells with luminal structure created in the microchannel is similar to the small blood vessels in vivo (Fig. 3b) except the square channel shape. The fabrication of a rounded channel is possible with more sophisticated patterning techniques. The cell morphology can be more clearly seen in Fig. 3c from which the cell number can also be deduced by counting the nuclei of the cells with DAPI marker. Remarkably, we observed a higher cell density on the top wall than the bottom one (Fig. 3d), suggesting the suitability of the suspended monolayer nanofibers for endothelial cell culture.

It is known that endothelial cells in blood vessels are exposed to hemodynamic forces associated with blood flow, which cause shear stress and friction of blood with the luminal cell surface. [START_REF] Raasch | Microfluidically supported biochip design for culture of endothelial cell layers with improved perfusion conditions[END_REF][START_REF] Wevers | A perfused human blood-brain barrier on-a-chip for high-throughput assessment of barrier function and antibody transport[END_REF][START_REF] Partyka | Mechanical stress regulates transport in a compliant 3D model of the blood-brain barrier[END_REF] In our case, we observed remarkable changes of the HUVECs morphology on top and bottom walls of the channel after incubation for 48 h under dynamic flow conditions and cells were clearly aligned along the flow direction (Fig. 4). Under static conditions, cells were not aligned on both layers after incubation for 3 days. In addition, some cells at the bottom became round and tended to detach from the substrate, which means that the shear stress induced under dynamic culture conditions is necessary for long-term maintenance of HUVECs culture.

U87 cancer cells were chosen to demonstrate co-culture with HUVECs with suspended gelatin nanofibers above the channel. We seeded U87 directly on to the backside of the HUVEC layer separated by the monolayer of gelatin nanofibers. After incubation for 24 h, HUVECs and U87 cells were labeled with red and green markers, respectively. As can be seen in Fig. 5, U87 cells adhered well on the gelatin nanofibers and HUVECs remained healthy during co-culture. Here, the gelatin nanofibers served as a mechanic support for cell adhesion and cell layer formation. Such a device configuration should be useful for co-culture of other cells and more systematic studies are needed to understand more clearly how cells interact each other across a monolayer of nanofibers. More interestingly, tumor angiogenesis and tumor metastasis can be studied using such a device [START_REF] Magdeldin | Engineering a vascularised 3D in vitro model of cancer progression[END_REF][START_REF] Tsai | Tumour-on-a-chip: microfluidic models of tumour morphology, growth and microenvironment[END_REF][START_REF] Chung | Biomimetic Model of Tumor Microenvironment on Microfluidic Platform[END_REF][START_REF] Sobrino | 3D microtumors in vitro supported by perfused vascular networks[END_REF][START_REF] Avci | Investigating the Influence of HUVECs in the Formation of Glioblastoma Spheroids in High-Throughput Three-Dimensional Microwells[END_REF] . In such a case, it should be possible to form tumor spheroids close to a vessel and to observe the interplay of the two systems. More generally, our approach can be applied to generate different types of tissues and different organ constituents involving artificial vessels and/or artificial basement membranes.

Conclusion

We have developed a microfluidic device with an embedded monolayer of nanofibers in order to achieve a generalized device configuration for co-culture-based assays. This device has firstly been used for three-dimensional culture of endothelial cells (HUVECs) on the microfluidic channel wall and the fiber monolayer. We also examined the morphology changes of HUVECs as a function of shear stress. With the same device, co-culture of HUVECs and U87 cells has been achieved, showing a continuous layer of HUVECs on the downside and a U87 cell layer on the upside of the fiber layer, thereby proving the potential of this new approach with an artificial yet physiologically relevant basement membrane for a plethora of possible applications. Finally, the device presented in this work will soon be commercially available.

Figure 1 .

 1 Figure 1. Schematic of the device fabrication process and the cell culture and co-culture configuration: (a) Lower plate, (b) Device assembly, (c) Rotational culture device, (d) and (e) Side and cross-section views of culture and co-culture configurations, respectively.

Figure 2 .

 2 Figure 2. Scanning electron microscopy images of a microfluidic channel with embedded monolayer nanofibers. (a) Top view, (b) Cross-section view and (c) Enlarge cross-section view.

Figure 3 .

 3 Figure 3. Confocal images of HUVECs cultured in microfluidic channel for 24 h. 3D views (a) and cross-sectional views (b) of the HUVECs on all the surfaces. The morphology (c) and cell number (d) of HUVECs on gelatin nanofibers and at the bottom of channel are presented, respectively. Scale bars, 300 µm.

Figure 4 .

 4 Figure 4. Phase contrast images of HUVECs on the top (below the fiber layer) and bottom wall of the channel at day 1 and 3. Scale bars, 100 µm.

Figure 5 .

 5 Figure 5. Fluorescence images of U87 cells above the fiber layer, HUVECs below the fiber layer and HUVECs on the bottom wall. Staining: Cell Tracker TM . Scale bars, 100 µm.

Figure. 1 .

 1 Figure.1.

Figure. 2 13Figure. 3 .

 23 Figure.2

Figure. 4 .

 4 Figure.4.

Figure. 5 .

 5 Figure.5.

Acknowledgements

This work was supported by Agence de Recherche Nationale under contract ANR-17-CE09-0017 (AlveolusMimics), European Commission Cost Action BIONECA (CA 16122), DIM ELICIT program of Ile-de-France, and PSL Valorization through Pre-maturation project. Boxin Huang also thanks to the Oversea Study Program of Guangzhou Elite Project.


Fan Zhang

Cyril R R Adolf

Nicolas Zigon

Sylvie Ferlay

Mir Nathalie Kyritsakas

Wais Mir

Hosseini

Wais Hosseini

Molecular tectonics: hierarchical organization of heterobimetallic coordination networks into heterotrimetallic core-shell crystals

come   L'archive ouverte pluridisciplinaire

For the development of new molecular materials and devices, mastering of molecular organization in the solid state is of prime importance. Hierarchical construction of solid-state matter composed of several crystalline domains with continuity of short-and long-range orders is challenging. The generation of core-shell crystals is a pertinent example of such hierarchical construction. 1 Core-shell crystals are formed between different isostructural and almost isometric crystalline organizations by 3D epitaxial growth of crystalline layers on seed crystals. This construction strategy is based on principles developed by the molecular tectonics approach. 2 The latter deals with the formation of molecular networks 3 leading to extended crystalline architectures displaying translational symmetry. The formation of periodic assemblies results from mutual interconnection of complementary tectons, 4 by self-assembly processes. For the generation of core-shell crystals by 3D epitaxial growth, isostructurality and close unit cell metrics between crystals are the two prerequisites. So far, only few molecular core-shell crystals have been reported. Most of the documented examples are mainly based on H-bonded systems. 5,6 However, rare examples based on discrete coordination complexes 7 or coordination polymers 8 have been also reported. Examples of epitaxial growth using nanoscopic systems such as nanoparticles or films have been also described. 9

Here we report on the synthesis of the organometallic tecton 1 and on the formation of crystalline 1D heterobimetallic coordination compounds in the presence of MX2 (M = Co(II) or Zn(II)) metallic complexes. Furthermore, based on coordination bonding, we describe the generation of core-shell crystals by 3D epitaxial growth of crystalline layers on preformed seed crystals.

Dealing with the formation of core-shell crystals, we 1,5 and others 6a-c have previously designed H-bonded architectures.

One may argue that the formation of core-shell crystals may be due to the use of H-bonding which is rather tolerant both in terms of distance and angle and thus may compensate slight mismatches in metrics of unit cells between isostructural crystals. In order to further explore this issue, we have designed the tecton 1 (Figure 1) and investigated the formation of isostructural and almost isometric crystals upon its combinations with different neutral metal halide complexes MX2 (M = Co(II) and X = Cl -or Br -and M = Zn(II) and X = Cl -). It might be worth mentioning that, to the best of our knowledge, no example of molecular core-shell crystals based on heterobimetallic coordination polymers has been reported. The design of the organometallic tecton 1 (Figure 1) is inspired by our previous investigation on the analogous tecton 2. The latter, when combined with metallic complexes (M = Co(II), Ni(II) and Cd(II) and X = Cl -or Br -) leads to the formation of heterobimetallic 2D grid architectures. 10 Several neutral Pt(II) organometallic bis-alkyne complexes have been reported. 11 Tecton 2, when combined with 1,4diiodotetrafluorobenzene, leads to the formation of halogen bonded 1D compounds. 12 Compound 2 has been also incorporated into films for catalytic coupling reactions. 13 The tecton 1, a positional isomer of 2, is based on a Pt(II) core adopting a square planar geometry and surrounded by two P(Ph)3 ligands and two 3-pyridyl moieties in trans positions. Both compounds 1 and 2 are rigid tectons. However, 1, owing to free rotation around pyridyl-C bonds, the latter may adopt two distinct syn apolar and anti centric configurations (Figure 1).

Please do not adjust margins

Please do not adjust margins (C≡CPy)2(PPh3)2] 1 (syn andanti configuration) and2 as well as the crystal structure for 1 adopting the anti configuration. Platinum atoms are colored in yellow, nitrogen in blue and phosphosrus in pink. Hydrogen atoms have been omitted for clarity.

Tecton 1 was synthetized in 63% yield by condensation of cis-PtCl2(PPh3)2 with 3-ethynylpyridine (see ESI). In addition to common characterization techniques, the solid-state structure of tecton 1 was elucidated by X-ray diffraction on single crystal (XRD) (See ESI). Crystals of 1 have been obtained upon slow diffusion of pentane into a solution of 1 in C2H4Cl2. The latter crystallises in the space group Pbca, (see ESI, Crystallographic Table S1). As expected, the Pt(II) cation adopts a deformed square planar geometry (Figure 1) with P-Pt-C angle in the 86.79(6) -93.21(6)° range and P-Pt-P and C-Pt-C angles of 180.0°. The Pt-P and the Pt-C bond lengths of 2.3121(5) Å and 2.009(2) Å, respectively, are comparable to those previously observed for 2. 10 Pt-C14≡C15 and C14≡C15-C16 angles are equal to 172.8(2) and 172.8(3)°, respectively, indicating a slightly bent nature of the metallatecton. In order to explore the formation of heterometallic coordination networks, the neutral Pt(II) based tecton 1 was combined, under self-assembly conditions, with MX2 (M = Co(II) and X = Cl -or Br -and M = Zn(II) and X = Cl -). Upon slow diffusion of 1 dissolved in C2H4Cl2 with MX2 (M = Co(II) and X = Cl -or Br -and M = Zn(II) and X = Cl -) salts in MeOH (see ESI, experimental conditions), several crystals (1-CoCl2 (blue), 1-CoBr2 (blue) and 1-ZnCl2 (colourless)) were obtained and studied by XRD on single crystals (Figures 2a-c and5). In all cases, crystals ((C7H4N)2Pt(P(C6H5)3)2MX2, P-1) are isomorphous and almost isometric with less than 1.6 % difference in the unit cell parameters (see ESI, Crystallographic Table S1). For all architectures, the M/Pt stoechiometry is 1/1, the tecton 1 is in the trans centric configuration (Figure 1) and the metal M adopts a distorted tetrahedral coordination geometry. All crystals are composed of neutral 1D zig-zag (Figure 2d) heterobimetallic (Pt,M) coordination polymers resulting from the interconnection of consecutive tectons 1 by neutral MX2 units (Figure 2). The MX2 units, for which the metal M adopts a distorted tetrahedral geometry, behave as 2connecting V shape nodes. The N-M-N angle is equal to 95.11(13)°, 96.83(9)° and 97.56(13)° in 1-ZnCl2, 1-CoCl2 and 1-CoBr2 architectures respectively (see Table 1 for characteristic bond distances and angles). The purity of all three crystalline materials was established by X-Ray diffraction on polycrystalline powders (figure 4). The XRPD study confirmed the isomorphous nature of all three crystals. Indeed, all three powdered compounds (1-ZnCl2, 1-CoCl2 and 1-CoBr2) display analogous diffraction patterns with a good match with the one simulated using the XRD data for 1-ZnCl2. It is worth noting that, although crystals are reproducibly obtained, they are not stable for extended period of time outside the crystallisation solvent system. Taking advantage of the isostructurality of the three crystalline materials based on tecton 1 and the colour difference i.e. blue for 1-CoX2 (Figure 5a andb) and colourless for 1-ZnCl2 (Figure 5 c), the formation of core-shell crystals by 3D epitaxial growth was explored. Starting with either 1-CoCl2 or 1-CoBr2 as blue seed crystals, the epitaxial growth of colourless crystalline layer of 1-ZnCl2 leading to 1-ZnCl2@1-CoBr2 (Figure 5d) and 1-ZnCl2@1-CoCl2 (Figure 5e) core-shell crystals was achieved. Unfortunately, owing to low quality of the 1-ZnCl2 crystals (see ESI, Table S2), It was not possible achieve the reverse process, i.e. growth of crystalline layers of either 1-CoBr2 or 1-CoCl2 on 1-ZnCl2 seed crystal leading to 1-CoBr2@1-ZnCl2 and 1-CoCl2@1-ZnCl2. Both 1-ZnCl2@1-CoBr2 and 1-ZnCl2@1-CoCl2 core-shell crystals were analysed by XRD which revealed that the epitaxial growth takes place with conservation of both short and long range orders leading thus composite single crystals. Indeed, cell parameters of the core-shell crystal are almost identical to those of i) the pure 1-ZnCl2, ii) the 1-ZnCl2 crystalline zone obtained by cutting the core-shell crystal, iii) the pure 1-CoBr2 (or 1-CoCl2) crystal, iv) the 1-CoBr2 (or 1-CoCl2) crystalline zone obtained by cutting the core-shell crystal (Table 2). Cell parameters recorded at 173 K for the crystal of crystals 1-ZnCl2@1-CoBr2, and 1-ZnCl2@1-CoCl2, 1-ZnCl2 (from 1-ZnCl2@1-CoBr2 or 1-ZnCl2@1-CoCl2) and 1-CoBr2 or 1-CoCl2 (from 1-ZnCl2@1-CoBr2 or 1-ZnCl2@1-CoCl2).

Compound 1-ZnCl2@1- CoBr2 1-ZnCl2 (cut from 1-ZnCl2@1- CoBr2) 1-CoBr2 (cut from 1-ZnCl2@1- CoBr2)
It should be pointed out that the epitaxial growth process leading to core-shell crystals reported here leads to heterotrimetallic (Pt, M1 and M2) hierarchically organised crystalline materials. Indeed, combinations of the Pt(II)

Conclusions

Hierarchical organization of crystalline materials into core-shell crystals by epitaxial growth is a challenging matter and of current interest. This was achieved by designing the organometallic tecton 1 bearing two divergently oriented pyridyl coordinating sites. Combinations of 1 with MX2 (M = Co(II) or Zn(II) and X = Cl -or Br -) salts behaving as 2-connecting nodes lead to the formation of isostructural and almost isometric crystals composed of neutral 1D coordination compounds. Heterotrimetallic (Pt, Co and Zn) core-shell crystals of the type 1-ZnCl2@1-CoX2 (X = Cl or Br) have been generated by 3D epitaxial growth processes. Taking advantages of the difference in colour between 1-CoBr2 (blue) (or a 1-CoCl2) and 1-ZnCl2 (colourless) crystals, the crystalline growth processes could be visualized. The investigation also demonstrated that coordination bonding may be used for the generation of core-shell crystals. The interface between the two crystalline zones is currently under investigation using near field microscopy. Welding 4f of isostructural crystals mentioned above in solution and under mild condition is also being investigated.

Figure 1 :

 1 Figure 1: Metallatectons Pt(C≡CPy)2(PPh3)2] 1 (syn and anti configuration) and 2 as well as the crystal structure for 1 adopting the anti configuration. Platinum atoms are colored in yellow, nitrogen in blue and phosphosrus in pink. Hydrogen atoms have been omitted for clarity.

Figure 2 :

 2 Figure 2: Portions of the 1D zig-zag type compounds formed upon combining the bismonodentate metallatecton 1 with Co(II) salts: a) 1-ZnCl2, b) 1-CoCl2 c) 1-CoBr2 and d). For sake of clarity, hydrogen atoms and water molecules are omitted. Schematic representation of the connectivity between the metallatecton 1 and MX2 leading to a zigzag 1D chain.

  Please do not adjust marginsPlease do not adjust marginsThe chains are packed in a parallel fashion along the a and b axis (figure3).

Figure 3 :

 3 Figure 3: Portions of the packing of the 1-MX2 1D compounds (M = Co(II) and X = Cl -or Br -and M = Zn(II) and X = Cl -) a) along the b axis and b) along the a axis. Owing to the isostructural nature of all three crystals, on the packing of 1-CoCl2 is represented. For the sake of clarity hydrogen atoms are omitted.

Figure 4 :

 4 Figure 4: Simulated XRPD pattern for 1-CoCl2 (a) and recorded patterns for 1-CoCl2 (b), 1-CoBr2 (c) and 1-ZnCl2(d) showing the homogeneity of phases and their isostructurality.

  Please do not adjust marginsPlease do not adjust margins organometallic tecton 1 with MX2 metallic nodes lead to the formation of the seed heterobimetallic crystals (Pt and Co) which, upon epitaxial growth leads to heterotrimetallic architectures (Pt, Co and Zn).

Figure 5 :

 5 Figure 5: Photographs of crystals of: a) 1-CoBr2, b) 1-CoCl2, c) 1-ZnCl2 and 1 st generation core-shell crystals of d) 1-ZnCl2@1-CoBr2 and e) 1-ZnCl2@1-CoCl2.

Table 1 :

 1 Comparison

		1-ZnCl2	1-CoCl2	1-CoBr2
	Pt-C (Å)	1.995(4)	1.994(3)	2.005(4)
		2.013(4)	2.004(3)	2.007(3)
	Pt-P (Å)	2.3127(9)	2.3102(6)	2.3003(8)
		2.3003(9)	2.2962(7)	2.3116(8)
	P-Pt-C (°)	86.09(11) -	86.13(8) -	86.17(10) -
		93.91(11)	93.87(8)	93.83(10)
	Pt-C≡C (°)	176.6(3)	176.4(2)	176.7(3)
		178.8(3)	179.0(2)	178.2(3)
	C≡C-C (°)	174.1(4)	173.6(3)	174.1(4)
		176.7(4)	176.8(3)	176.4(4)
	M-X (Å)	2.2062(12)	2.2188(9)	2.3509(6)
		2.2139(12)	2.2275(10)	2.3708(6)
	M-N (Å)	2.058(3)	2.034(2)	2.034(3)
		2.076(3)	2.052(2)	2.048(3)
	N-M-N (°)	95.11(13)	96.83(9)	97.56(13)
	N-M-X (°)	107.09(10)-	107.72(7) -	108.43(9) -
		110.86(9)	111.33(7)	111.65(8)
	X-M-X (°)	122.77(5)	120.36(4)	118.84(3)

of bond distances and angles for 1-ZnCl2, 1-CoCl2 and 1-CoBr2.

ZnCl2@1- CoCl2 1-ZnCl2 (cut from 1-ZnCl2@1- CoCl2) 1-CoCl2 (cut from 1-ZnCl2@1-

	Crystal system	Triclinic	Triclinic	Triclinic
	Space group	P-1	P-1	P-1
	a(Å)	9.58	9.56	9.57
	b(Å)	9.64	9.65	9.63
	c(Å)	24.12	24.05	24.15
	α(deg)	95.07	95.14	95.15
	β(deg)	91.38	91.61	91.40
	γ(deg)	98.42	98.56	98.35
	V(Å 3 )	2191	2183	2206
	Compound	1-CoCl2)
	Crystal system	Triclinic	Triclinic	Triclinic
	Space group	P-1	P-1	P-1
	a(Å)	9.54	9.61	9.59
	b(Å)	9.62	9.70	9.70
	c(Å)	24.07	24.17	24.17
	α(deg)	95.24	95.42	95.13
	β(deg)	91.78	91.42	91.49
	γ(deg)	98.57	98.39	98.31
	V(Å 3 )	2172	2217	2213

Table 2 :

 2

Acknowledgments

We thank the University of Strasbourg, the C.N.R.S, the International centre for Frontier Research in Chemistry (icFRC), the Labex CSC (ANR-10-LABX-0026 CSC) within the Investissement d'Avenir program ANR-10-IDEX-0002-02, the Institut Universitaire de France and the Ministère de l'Enseignement Supérieur et de la Recherche for financial support.

COMMUNICATION

Please do not adjust margins Please do not adjust margins

Graphical Abstract

Core-shell heterotrimetallic crystals are generated by 3D epitaxial growth using isostructural and almost isometric crystals based on an organometallic tecton and MX2 complexes.


Charles Elegbede
email: charles.elegbede@launchers.eads.net

Kondo Adjallah
email: kondo.adjallah@utt.fr

Availability Allocation to Repairable Systems with Genetic Algorithms: A Multi-Objective Formulation

Keywords: Availability Allocation to Repairable Systems with Genetic Algorithms: A Multi-Objective Formulation availability allocation, multi-objective optimization, genetic algorithms mutation probability pc : crossover probability

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NOTATIONS : Availability target

Cmax

: maximum cost admissible for the system s : number of subsystems Let us recall first of all that if the reliability of a system decreases toward 0 when its operating time tends to infinity (i.e. becomes very large), then its availability decreases to a value called asymptotic availability [START_REF] Pagès | Fiabilité des systèmes[END_REF]. The main requirements for complex systems operating are usually specified in terms of cost and availability and/or of reliability, or equivalently in terms of mean operating time and/or of mean down time under cost constraint. These requirements have then to be taken into consideration in the system design stage in order to determine the appropriate reliability and availability of each of its components. The requirements, generally expressed by the owner of the system, correspond to a global dependability objective, which must be apportioned to each component of the system by the designer [START_REF] Elegbede | Allocation de fiabilité par les algorithmes génétiques et le recuit simulé[END_REF]. The result can then be used to conceive an appropriate logistic to support the system (determination of optimal stock of spare parts, development of an appropriate maintenance plan, etc.).

The problem of availability allocation can be formulated as a multi-objective optimization problem: minimize the cost and maximize the availability of the global system. As constraints relating to both objectives, there are the system asymptotic availability must be smaller than the target availability Aobj on the one hand, and the cost of the system must be lower than a given maximum cost Cmax on the other hand.

The decision variables of this problem are the number of components ki in each subsystem, the failure rate li and the repair rate µi of each component (i = 1, ..., s), which are also used to compute the cost and the availability of the system [START_REF] Villemeur | Sûreté de fonctionnement des systèmes industriels[END_REF]. The objective and constraint functions of the problem are non-linear, and in addition they depend on continuous and discrete variables (respectively the failure and repair rates, and the number of components). The considered optimization problem was proven NP-hard [START_REF] Chern | On the computational complexity of reliability redundancy allocation series system[END_REF]. Therefore, an heuristic or a meta-heuristic such as genetic algorithms can provide solutions close to the optimal solution with good efficiencies [START_REF] Kuo | An annotated overview of system-reliability optimization[END_REF]. Indeed, optimization based on GA is known to be an efficient approach to solve classical problems of multi-objectives optimization under constraints ( [START_REF] Michalewicz | Genetics algorithms + data structures = Evolution programs[END_REF], [START_REF] Goldberg | Les algorithmes génétiques[END_REF], [START_REF] Coello | An empirical study of evolutionary techniques for multiobjective optimization in engineering design[END_REF], [START_REF] Eckart | Multiobjective Algorithms: A comparative case study and the strength Pareto Approach[END_REF] , [START_REF] Eckart | Multiobjective Algorithms: A comparative case study and the strength Pareto Approach[END_REF]). Several papers relating to systems reliability optimization using GA was published these last few years ( [START_REF] Coit | Reliability optimization of series parallel systems using genetic algorithm[END_REF], [START_REF] Campbell | Genetic algorithm in optimization of system reliability[END_REF], [START_REF] Elegbede | Reliability allocation to components following Weibull law using genetics algorithms[END_REF], [START_REF] Elegbede | Contribution aux méthodes d'allocation d'exigences de fiabilité aux composants de systemes[END_REF], [START_REF] Elegbede | Contribution aux méthodes d'allocation d'exigences de fiabilité aux composants de systemes[END_REF], [START_REF] Elegbede | Contribution aux méthodes d'allocation d'exigences de fiabilité aux composants de systemes[END_REF]). For all those reasons, we chose genetic algorithms to solve the optimization problem stated in section 2.

In our approach, without loss of generality, we will consider only the case of parallel-series systems. Using appropriate objective functions and fitness functions, this approach can be generalized to other categories of system. More details on the use of GA for reliability optimization can be found in the state-of-the-art of Gen & Kim [START_REF] Gen | GA-based reliability design: state-of-the-art survey[END_REF] or the one proposed by Kuo & Prasad [START_REF] Kuo | An annotated overview of system-reliability optimization[END_REF]. To solve this problem, weights are used to transform the multi-objective functions into a single objective function. Then, we relax the constraints using an exterior penalty approach [START_REF] Minoux | Programmation mathématique[END_REF] to lead finally to a problem without constraint. The new objective function is then used as adaptation function (fitness function) of the GA for which we design an experiment plan to tune the parameters. Lastly, the methodology is applied to solve the availability allocation problem under cost constraint, for a repairable parallel-series system.

Section 2 of the paper states the multi-objective optimization problem. Section 3 presents the method used to relax the constraints, and gives details on the cost function and the new formulation of the problem. Then, details on the implemented GA are presented in section 4.

Next, we propose an experiment plan approach to tune the parameters of the algorithm in section 5. It enables to assess the sensitivity of each parameter on the GA effectiveness. Finally, section 6 summarizes the main results obtained with the approach proposed to solve the specific optimization problem.

STATEMENT AND FORMULATION OF THE PROBLEM

General model of the problem

Let us recall that a parallel-series system is a system made of parallel subsystems put in serial (figure 1). A parallel subsystem works when at least one of its components works and a series system fails when at least one of its components fails. We suppose in all the remainder of the paper that the subsystems are parallel, and that in each subsystem all the components are identical. Here, the "identical" term means that the components have the same reliability and availability. The number of components in each subsystem, the availability of the system and the system cost will be denoted respectively by ki, AS and CS. Assume now that during the design stage of a parallelseries system, the availability objective Aobj has to be apportioned to each component, with the aim of maximizing the availability and of minimizing simultaneously the total cost of the system. We then have a multi-objective optimization problem that can be formulated as follows:

( ) (1)
Figure 1: General structure of a parallel-series system.

( ) ( ) ( ) ( ) ï ï þ ï ï ý ü £ £ £ £ £ £ £ ³ ki i ki i i i i i i S obj S S S k and C C A A t s C Minimize and A Maximize u u u ! ! ! µ µ l l µ l ; ; . . max k µ, λ, k µ, λ, k µ, λ, k µ, λ, k µ, λ, k µ, λ, s i ,..., 1 =
For the parallel-series systems, the asymptotic availability and the cost are given by:

(2) [START_REF] Elegbede | Allocation de fiabilité par les algorithmes génétiques et le recuit simulé[END_REF] where ai, bi, pi, and qi are real numbers such that : ai >0 , bi >0, pi < 0 , qi > 0 (

).

The decision variables of the problem modeled by equation ( 1) are li, µi, ki (

).

Since maximizing the availability AS is equivalent to minimizing the unavailability (1-AS), the problem can be solved through a vector minimization. In addition, the following observations about the cost model could help to formulate the problem.

The cost model

Equation ( 3) describes the cost model of the system. The cost of the system is obtained by the sum of the costs of all the components, which are supposed to be identical for a given subsystem.

It then remains to choose the suitable models for the cost of components. The approach used in this paper consists in providing, for a given component, a cost model that depends on the main features enabling to describe the technical efficiency of the system. This technical efficiency is expressed in term of availability which depends on components failure and repair rates. Thus, in the formulation of a given component cost, there are two main contributions:

• the failure rate li contribution :

, where ai > 0, pi < 0, ( ) are real numbers, which means that the more the component failure rate is low the more the component will be expensive.

(

) Õ = ÷ ÷ ø ö ç ç è ae ÷ ø ö ç è ae + - = s 1 i k i i i S i 1 A µ l l k µ, λ, ( )
( ) å = + = s 1 i q i i p i i i S i i b a k C µ l k µ, λ, s i ,..., 1 = s i ,..., 1 = i p i i a l s i ,..., 1 =
• the repair rate µi contribution :

, where bi > 0, qi > 0, ( ) are real numbers, which means that the lower the repair rate of the component is, the less expensive the component will be.

In practice, the coefficients ai , bi, pi , qi ( ) may be obtained from a maintenance data base. If one assumes that (l (j) , µ (j) , k (j) , ) ( ) are known for a set of h configurations of the system to design (where is the cost of the jth configuration), the coefficients may be obtained by minimizing the quantity:

ie. by using the least square regression:

(i = 1, …, s) (4)
This set of 4s independent equations and 4s unknown have a solution [START_REF] Minoux | Programmation mathématique[END_REF]. There are various ways in engineering to model the cost of a component. For instance, Misra et al. [START_REF] Misra | Optimal reliability design of a system: a new look[END_REF] considered in their approach that the cost of a component may be modeled as a continuous function of the reliability. Recently, Giuggioli et al. [START_REF] Giuggioli Busacca | Multiobjective optimization by genetic algorithms : application to safety systems[END_REF] also used continuous functions to express the contribution of the reliability of the components to the cost of the system. In this cost model, they took into account the profit from the system operation, purchase and installation costs, the repair costs, and the penalty related to the downtime due to miss contract delivery of service.

The interest of their model is to highlight its bonds with the intention of the designer. For example, the downtime of a component may not cause a penalty if one considers that in case of redundancy the service continues to be delivered. On the contrary, Levitin et al. (1998) [START_REF] Levitin | Redundancy optimization for series-parallel multi-state systems[END_REF] consider that it is more realistic to suppose that there is no analytic function that expresses the dependence of cost on its nominal performance. They consider that for the nominal performance, the supplier offers reliability and price characterizing each system or component version. Many

i q i i b µ s i ,..., 1 = s i ,..., 1 = ) (j S C h 1 j ,..., = ) (j S C ( ) ( ) ( ) å = - = h j j S j j j S C C W 1 2 ) ( ) ( ) ( ) ( k , µ , λ q p, b, a, q p, b, a, ( ) ( ) ( ) ( )
ï ï ï ï ï þ ï ï ï ï ï ý ü = ¶ ¶ = ¶ ¶ = ¶ ¶ = ¶ ¶ 0 0 0 0 i i i i q W p W b W a W q p, b, a, q p, b, a, q p, b, a, q p, b, a,
others continuous functions are presented in [START_REF] Tillman | Optimization of systems reliability[END_REF]. But let us say, in general, that the cost model depends on the studied problem. Hence, only an analysis of the problem can lead to a relevant formulation of the cost model.

To our opinion, a solution could be found to the difficulties pointed out by Levitin et al. by building a cost function that depends on each version of component involved in the system. However, it is clear that there is no general solution to the problem and a tradeoff should be sought.

In our paper, we choose to model the cost function by equation ( 3) taking only into account the parameters (li and µi), which enables to calculate the performance of the system expressed in terms of asymptotic availability.

CONSTRAINTS RELAXATION AND NEW FORMULATION OF THE PROBLEM

Multi-objective optimization

In single-objective optimization problem, the optimum corresponds to the point that maximizes or minimizes the objective function in the feasible domain ( [START_REF] Minoux | Programmation mathématique[END_REF], [START_REF] Culioli | Introduction à l'optimization[END_REF], [START_REF] Cohon | Multi-objective programming and planning[END_REF]). But in multiobjective optimization, the notion of optimal solution is replaced by the notion of Pareto optimal solution, i.e. a non-dominated solution ( [START_REF] Cohon | Multi-objective programming and planning[END_REF], [START_REF] Eckart | Evolutionary algorithms for Multiobjective Optimization Methods and applications[END_REF], [START_REF] Deb | Multiobjective optimization using nondominated sorting in genetic algorithms[END_REF], [START_REF] Eckart | Multiobjective Algorithms: A comparative case study and the strength Pareto Approach[END_REF]).

Assume that a vector Z(x) has to be maximized i.e. Z(x) = (Z1(x),…,Zp(x)), where x IR n , with m constraints gi(x):

s.t. gi(x) 0 (i = 1, … m) (5)

a) Definition of a feasible solution

A solution x* is a feasible solution of the optimization problem (5) , if and only if it satisfies all the constraints of the problem. The set of feasible solutions is called "feasible domain".

Î ( ) x Z Maximize £ b) Definition [36]
A solution x* in the feasible domain is Pareto-optimal or non dominated solution, if there is no solution x** in the feasible domain such that Zj(x**) Zj(x*), j = 1,… , p , and Zl (x**) > Zl(x*) for at least one l [1, p].

A feasible solution x* is weakly Pareto-Optimal if there is no feasible solution x** such that

Zj(x**) > Zj(x*) j [1, p].

c) Generation of a Pareto optimal solution

There are several approaches to solve a multi-objective optimization problem to obtain a Pareto optimal solution ([14], [START_REF] Eckart | Evolutionary algorithms for Multiobjective Optimization Methods and applications[END_REF], [START_REF] Deb | Multiobjective optimization using nondominated sorting in genetic algorithms[END_REF], [START_REF] Eckart | Multiobjective Algorithms: A comparative case study and the strength Pareto Approach[END_REF], [START_REF] Fonseca | An overview of evolutionary algorithms in multiobjective optimization[END_REF]). The existing methods can be classified in two categories:

• the first one gathers the methods which are based on the preference of a decision maker.

It consists of approaches based on the use of weightings, the constraint-based approach, or distances based approach [START_REF] Fonseca | An overview of evolutionary algorithms in multiobjective optimization[END_REF]. The output of these methods is a single non-dominated solution, corresponding to the preference expressed by a decision maker for example. To obtain a set of Pareto-optimal solution, it is sufficient to consider a set of preferences, and to run the algorithm for each expressed preference.

• the second one gathers methods using genetic algorithms, the main one are the Vector Evaluation Genetic Algorithms developed by Shaffer [START_REF] Fonseca | An overview of evolutionary algorithms in multiobjective optimization[END_REF] and the Non dominated Sorting Genetic Algorithms (NSGA) developed by Srinivas et al. [START_REF] Deb | Multiobjective optimization using nondominated sorting in genetic algorithms[END_REF]. The output of these two methods is a set of Pareto-Optimal solutions.

Let us present briefly the first category of methods.

³ Î Î

Weighting-based approach [14][30][34]

Consider wj > 0, j = 1,… , p. The weighting approach consists in solving the following singleobjective optimization problem:

s.t. gi(x) 0 (i = 1, … m) (6)
Consequently, the solution of ( 6) is also a solution of [START_REF] Michalewicz | Genetics algorithms + data structures = Evolution programs[END_REF]. The coefficients wj are arbitrarily chosen for this method, and depend on the decision maker preference.

Constraints-based approach [30] [34]

To solve equation ( 5) by the constraints-based approach, one needs to choose one component of the vector to be optimized as objective and some appropriate coefficients hj (according to the decision maker preference) and solve the following system [START_REF] Coit | Reliability optimization of series parallel systems using genetic algorithm[END_REF], whose solution is also solution of ( 5):

(7)

Approach based on distance function [30][34]

This method consists of minimizing the objective chosen in the vector to optimize, compared to the specified target vector of the problem. The multi-objective problem scalarization is:

Where r Î N-{0}. The solution depends on the specified vector . The case r = 2 corresponds to the classical Euclidian metric.

( )

x å = p j j j Z Maximize 1 w £ þ ý ü ³ £ = = p 1 j j j m 1 i i k h x Z 0 x g t s Z , , , , ) ( ; )
( . . ) ( ! ! x Maximize z ~( ) ( ) r 1 p 1 i r i i z Z Minimise ú û ù ê ë é - å = x z ~

Other specific methods used for genetic algorithms

In the literature, there are some specific multi-objectives optimization methods based on genetic algorithms. The Vectors Optimization Genetic Algorithm (VEGA) for instance modifies the classical genetic algorithms by performing independent selection cycles according to each objective [START_REF] Fonseca | An overview of evolutionary algorithms in multiobjective optimization[END_REF], and provides a set of Pareto optimal solutions.

Equally, the Non-dominated Sorting Genetic Algorithms (NSGA) approach [START_REF] Deb | Multiobjective optimization using nondominated sorting in genetic algorithms[END_REF] provides a set of Pareto optimal solutions.

Multi-objective optimization and constraints relaxation

In this paper, we choose to use the weighting-based approach, more easy to implement.

a) Weighting coefficients

To a given set of weights (w1, w2) corresponds a Pareto optimal solution. Hence, if one wants to propose several choices to the decision maker, it will suffice to run the algorithm described in this paper the number of time it is necessary to obtain a set of Pareto optimal solutions. Indeed, the choice of weighting coefficients vector expresses a preference on the objective to be optimized. This approach may be time consumer with respect to the VEGA and NSGA [START_REF] Deb | Multiobjective optimization using nondominated sorting in genetic algorithms[END_REF] to perform the Pareto optimal search. But when the decision maker is sure on his preference (i.e. it does not need any other alternative), the simple weighting approach may be sufficient with a set of coefficients.

By choosing the appropriate values for w1 > 0 and w2 > 0 according to the preference of the decision maker, the optimization problem could be stated as follows: [START_REF] Elegbede | Reliability allocation to components following Weibull law using genetics algorithms[END_REF] where !li, !µi and !ki respectively denote the lower bound of the failure rate, of the repair rate and of the number of components in a subsystem, and uli, uµi and uki respectively denote the upper bound of the same variables.

On the basis of equation ( 9), the problem will be transformed into a single-objective optimization problem that can be solved by constraints relaxation.

b) Penalty coefficients for constraints relaxation

The exterior penalty methods are generally used to relax the constraints of a problem [START_REF] Coit | Reliability optimization of series parallel systems using genetic algorithm[END_REF][26] [START_REF] Michalewicz | Evolutionnary Algorithms for constrained Engineering Problems[END_REF], and enable to solve an equivalent optimization problem without constraints. The choice of penalty coefficients takes into account the preference of the decision maker with respect to the constraints violation. Penalty coefficients are chosen according to practical implication of constraints violation. For example, the penalty on the cost will be large if the cost violation is less acceptable than the availability violation.

Coit et al. 1996 [START_REF] Coit | Penalty guided genetic search for reliability design optimization[END_REF] used adaptive coefficients to relax the constraints which are difficult to satisfy. They introduced the notion of "non feasible threshold solutions penalty".

In order to compare a solution with the same evaluation function, the penalty coefficients are chosen before the beginning of the simulation and are kept constant throughout the simulation.

Thus, from a given generation i to i+1 (i = 1 to Ngene), the evaluation function which depends on weighting coefficients and penalty coefficients that will remain the same during all simulation enable to re-formulate the problem as follows:

( ) ( ) ( ) ( ) ( ) ï þ ï ý ü £ £ £ £ £ £ £ ³ + - ki i ki i i i i i i S obj S S S k and C C A A t s C A u u u ! ! ! µ µ l l µ l w w ; ; . . 1 inimize M max 2 1 k µ, λ, k µ, λ, k µ, λ, k µ, λ, µ,k λ, s i ,..., 1 =
13 [START_REF] Elegbede | Contribution aux méthodes d'allocation d'exigences de fiabilité aux composants de systemes[END_REF] where d1 and d2 are the penalty coefficients, and where the function F(l, µ , k) is defined as: [START_REF] Gen | GA-based reliability design: state-of-the-art survey[END_REF] A solution of ( 10) is also solution of ( 9) [START_REF] Cohon | Multi-objective programming and planning[END_REF]. Thus, the search will use equation [START_REF] Elegbede | Contribution aux méthodes d'allocation d'exigences de fiabilité aux composants de systemes[END_REF] which corresponds to a single-objective optimization problem.

c) The evaluation function used in the genetic algorithm

The new objective function F(l, µ , k) is bounded by a value Q such that for all li, µi, ki (i = 1, …, s), > 0. To prove this, it is sufficient to prove that the cost has an upper bound over the search space. Let us observe that if uk = , then the following relation holds:

In addition, according to Heine theorem [START_REF] Valiron | Théorie des fonctions[END_REF], as the function is continuous on a closed and bounded interval , it is bounded and reaches its upper and lower bounds on the interval. Here, we just need the fact that the function is upper-bounded. Let Mi Î IR + be the upper bound, then the relation holds. According to the continuity and the monotony of the function , the upper bound can be calculated as .

Similarly, it can be proved that there exits a value Pi Î IR + , , such that .

Therefore, one can conclude that .

( )

ï þ ï ý ü £ £ £ £ £ £ ki i ki i i i i i i k t s F Minimize u u u ! ! ! ; ; . . µ µ l l µ l k µ, λ, k µ, λ, s i , , 1 ! = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 ; max 0 ; max 1 max 2 1 2 1 C C A A C A F S S obj S S - + - + + - = k µ, λ, k µ, λ, k µ, λ, k µ, λ, k µ, λ, d d w w ( ) k µ, λ, F Q- ) ( sup ,.., 1 ki s i u = ( ) ( ) å = + £ s i q i i p i i k S i i b a C 1 µ l u k µ, λ, l l i i i p a i ! [ ] i i l l u , ! ) ( sup ,..., 1 1 i s i p i s i i M s a i = = × £ å l l l i i i p a i ! i p i i i a M ) ( l ! = i q i i i b P ) ( µ ! = ) ( sup ,..., 1 1 i s i q i s i i P s b i = = × £ å µ ( ) ( ) ( ) ÷ ø ö ç è ae + × £ = = i s i i s i k S P M s C ,..., 1 ,..., 1 sup sup u k µ, λ,
Let us denote: [START_REF] Culioli | Introduction à l'optimization[END_REF] Given that , , ,

Taking into account these properties, we can determine an upper bound of the objective function in equation ( 11) as: .

Hence, for any positive real e, there exists a value Q such that , defined by . We chose e = 0 in this paper.

Let

, then the optimization problem can be stated as follows:

By construction, the solutions of ( 14) satisfy the constraints related to the search domain [START_REF] Cohon | Multi-objective programming and planning[END_REF]. Note that H is the scaling form of the objective function also called evaluation function because through this function, the decision maker preferences are taken into account from two points of view: 1) to obtain a single objective optimization problem, 2) to relax the constraints of the problem.

The quality of the results of GA depends strongly on its adaptation function. The function H defined in this section is the adaptation function used in the proposed GA. H is also called fitness function or evaluation function.

( )

( )÷ ø ö ç è ae + × = = = s i i s i i k P M s Q ,..., 1 ,..., 1 1 sup sup u 1 0 < < obj A ( ) 1 0 < < k µ, λ, S A C max > 0 ( ) ( ) 1 0 ; , , max < - i i i S obj k A A µ l ( ) 1 max 0 ; ) ( max Q C C S < - k µ, λ, C max > 0 ( ) ( ) 1 0 ; max < - k µ, λ, S obj A A ( ) 2 1 1 1 2 1 d d w w + + + £ Q Q F k µ, λ, ( ) 0 > - k µ, λ, F Q ( ) 1 1 2 2 1 ) ( ) 1 ( Q Q d w d w e + + + × + = ( ) ( ) k µ, λ, k µ, λ, F Q H - = ( ) ï ï þ ï ï ý ü = £ £ £ £ £ £ s i k t s H Maximize ki i ki i i i i i i , , 1 ; ; . . ! " " " u u u µ µ l l µ l k µ, λ, k µ, λ,

PRINCIPLE OF GENETIC ALGORITHMS

Genetic algorithms are introduced by John Holland at Michigan University in the late sixties [START_REF] Goldberg | Les algorithmes génétiques[END_REF].

Genetic algorithms belong to evolutionary algorithms and are devoted to simulate animals reproduction by using the Darwinian theory [START_REF] Darwin | L'origine des espèces[END_REF] which stipulates that only individuals that are strong may survive and will be able to reproduce. Then naturally the weak individuals will be eliminated.

There are many papers that give details on this family of algorithm. The reader may refer to [START_REF] Goldberg | Les algorithmes génétiques[END_REF][18] [START_REF] Schoenauer | Evolutionary Computation[END_REF] for more details on the topic. A set of recent PhD dissertations [START_REF] Coello | An empirical study of evolutionary techniques for multiobjective optimization in engineering design[END_REF], [START_REF] Eckart | Evolutionary algorithms for Multiobjective Optimization Methods and applications[END_REF] on the topic shows that genetic algorithms are very efficient to solve NP-hard problems.

The general schema of genetic algorithms may be summed up as follows (figure 2). First of all, an initial population (generation) of potential solutions (individuals) is generated randomly.

One makes also evolve this population by recombination of individuals by the mean of crossover operator and/or mutation operator. A selection procedure based on a fitness function (also called evaluation function or adaptation function) enables to choose the individuals candidate for reproduction. The reproduction consists in recombining two individuals by the crossover operator, possibly followed by a mutation of the offspring. Therefore, from the initial population a new generation is obtained. From this new generation, a second new generation is produced by the same process and so on. The stop criterion is generally based on the number of generations. In the remainder of this section, the details of the items of the genetic algorithm we have implemented are presented with appropriate comments.

Encoding of the solution

A solution of equation ( 1) represents a chromosome which can be coded as a matrix of dimension 3´s. The first row of the matrix contains the failure rates li Î of the components of the system. The second row contains the repair rates µi Î and the third row contains the necessary number ki Î of components in each subsystem such that the availability of the system reaches the target Aobj. This encoding appears to be simpler, in comparison to the binary encoding where the computing times are large without significant improvement in the search process [START_REF] Michalewicz | Genetics algorithms + data structures = Evolution programs[END_REF].

The encoding proposed is phenotypic because the genes of the chromosomes are directly chosen in the variables space. One could have used a binary string to encode the chromosome of

i i l l u ! ] , [ i i µ µ u ! ] , [ ki ki u !
solutions; in this case the gene would be a string of bits that would be decoded to obtain the solution in the variables space. This latter approach is a genotypic one.

Generating the initial population

The generation of an initial population is necessary to start solving the optimization problem with a GA. This process is based on the construction of initial solutions or individuals chromosomes by sampling randomly values of li, µi and ki in their appropriated interval. The size of any population is given and remains the same at each generation. The main difficulty in the initial population is that the individuals may not satisfy all or part of the constraints of the problem. A GA must improve the individuals of the populations throughout the successive generations, so as to tend towards the optimal solution.

Selection of the individuals

The selection principle is based on Darwin theory [START_REF] Darwin | L'origine des espèces[END_REF], which postulates that in a natural system, only strong individuals will survive. Selecting individuals who are prone to reproduction improves the new successive generations of individuals. Thus, the fitness of each individual of the population is evaluated in order to select the best individual who is in fact the individual that has the highest fitness. The mainly known selection processes are the roulette technique and the tournament technique. The first one may be consulted in [START_REF] Michalewicz | Genetics algorithms + data structures = Evolution programs[END_REF] [START_REF] Bäck | Evolutionnary algorithms in theory and practice[END_REF]. We shall present only the tournament technique used in this paper.

The tournament technique

The tournament technique consists in choosing M individuals at random (M < n), according to uniform law in (0,1) in a population and in retaining the m best individuals among M (m < M), who will be inserted in the new generation that is being formed ( [START_REF] Michalewicz | Genetics algorithms + data structures = Evolution programs[END_REF], [START_REF] Sado | Les plans d'expériences[END_REF]) for the crossover.

Generally, one chooses m = 1 ( [START_REF] Michalewicz | Genetics algorithms + data structures = Evolution programs[END_REF], [START_REF] Sado | Les plans d'expériences[END_REF]).

In this paper, we use the tournament technique for selecting the individuals in the current population. The fitness of the individuals is measured by means of the adaptation function H defined in section 3.2, which corresponds to the relaxed form of the penalized objective function.

Crossover operators

There are several crossover operators in the literature [START_REF] Bäck | Evolutionnary algorithms in theory and practice[END_REF]. The crossover must enable to obtain offspring whose chromosomes represent children who have kept the good properties of the parents. But it is not always possible to satisfy the latter crossover properties, and one defines crossover just to obtain new individuals that will be evaluated by the fitness function. Crossover operators must be adapted to the studied problem. The most current operators are the arithmetic crossover [START_REF] Michalewicz | Genetics algorithms + data structures = Evolution programs[END_REF] [START_REF] Bäck | Evolutionnary algorithms in theory and practice[END_REF]) and the crossover with one or several crossing points [START_REF] Sado | Les plans d'expériences[END_REF]. We shall present only the one point crossover, which is used in our paper.

Crossover with one point

Let [f1 f2 … fn] and [t1 t2 … tn] be two strings encoding two potential solutions of a problem.

Let us choose randomly a value k in the set {1,…,n} following an uniform law. The crossover with one crossing point consists in creating two offspring as follows: f1f2...fktk+1...tn and t1t2...tk fk+1...fn. This crossover is the one used in our approach and simple to implement. It will be applied to each row of the chromosomes at the same position randomly chosen in each chromosome.

Mutation operators

This operator is very relevant, for it enables to insert new genes in the chromosomes [6][18][27]. This can be seen as local reparation on the chromosome and thus brings new genes into the chromosomes of the individuals in order to increase diversity in the population. The mutation occurrence probability should be very low if one wants to ensure stability in the neighborhood of the current solution found by the algorithm. In our approach, the mutation consists in reducing the failure and the repair rates of 5% on 20% of the genes of the chromosomes. In addition, the number of components is increased by 1 in the corresponding genes. The values 5% and 20% are chosen because they appeared to be appropriate, but they could also be adjusted for instance by an experiment plan. But according to the importance of these parameters in the algorithm, we consider that it is not necessary to implement such a technique.

Immigration technique

In absence of sufficient diversity in the population, the algorithm can quickly converge towards a weak solution. This behavior is due to the heuristic character of the GA. To avoid such situation, we decide to change all or part of individuals of the current population when the best individual remains the same after a given number of successive generations. This process is called immigration [START_REF] Hajela | Genetic algorithms optimization problems with discrete and integer design variables[END_REF]. In our case, it consists in removing randomly 50% of the individuals from the current population and in replacing them by new individuals generated randomly in the search space. This is performed when during successive generations the algorithm has not improved the quality of the best individual, i.e. a new best individual is not obtained.

ë û 4 / Ngene optimal solution. Therefore, it is suggested to consider small size of populations, according to the computing time necessary to obtain such a solution, when the initial population is made up of feasible solutions. In the contrary case where the initial population is not made up of feasible solutions, it seems better to consider large size of populations. For our concern, we have generated non-feasible solutions and have relaxed the constraints.

b) Number of generations

The current population is called generation in GA. The number Ngene of generations must be fixed at the beginning of the algorithm by taking into account its convergence. This number may be large or not according to the problem under consideration. It is also one of the most used stop criteria. Let us observe that it is the computing time which will give a limitation to Ngene.

Anyways, it is better to have largest size possible for Ngene. A good tradeoff must be found between the computing time and Ngene.

c) Crossover probability

The crossover probability (pc) is used to decide which couples of individuals constituting the current population to be crossed, according to the random variable u sampled in the interval (0, 1): if u < pc, then the couple of individuals corresponding to this pulling is crossed. The value of pc is generally chosen bigger than 0.5, and is often considered equal to 0.6 [START_REF] Goldberg | Les algorithmes génétiques[END_REF]. In order to increase the possibility of obtaining good individuals among the populations generated during the successive improvements, we choose these two values for the crossover probability: 0.5 and 0.8 for the experiment plan.

EXPERIMENT PLAN AND APPLICATION

Generally, researchers use arbitrary parameters values in the algorithms they implemented. For example, some authors consider that a crossover probability value may range from 0.6 to 0.8 and a mutation probability may range from 0.0 to 0.05. The size of population and number of generation are parameters that authors used to choose arbitrary without any particular investigation. It seems important to find the set of parameters that would provide the best results in terms of search efficiency. This set of (optimal) parameters may be found by using experiments plan to tune the parameters of the GA before running it with the set of optimal parameters. In this section, we present first of all the principle of experiments plan. Then, we present the application of this technique to help the tuning of the parameters of the GA. For that, we present the main elements from which the tuning is performed: the coding of the concerned parameters, the experimental matrix, the sensitivity measurement function of the efficiency of the algorithm and the result in term of choice of the parameters. The selected parameters are those used in the experiment that gives the best value of adaptation function. Finally, some comments related to the results are given.

The experiment plan

The tuning of a complex process controlled by several parameters is a very hard time consuming task in practice. Achieving an optimal tuning of the parameters of the process may be ensured through an experiment plan ( [START_REF] Goupy | La méthode des plans d'expériences[END_REF], [START_REF] Sado | Les plans d'expériences[END_REF]), which is an optimization technique easy to implement. This technique, which principle is the following, is based on effectiveness evaluation and sensitivity analysis of the process compared to the parameters. It aims at finding the right parameters through a small number of experiments. Let k be the number of parameters whose variation can affect the effectiveness of the process, i.e. k is the dimension of the vector of parameters also called factors. To reduce the number of experiments during the tuning of the parameters, let us consider discrete values for each parameter and let m be the possible number of discrete values for each parameter. Hence, there are m k possible combinations of the values of, parameters enabling to evaluate the effectiveness of the process. This process can then be evaluated through m k experiments. The evaluations will be based, in this experiment plan, on the input and output data of the process and on given criteria functions. Let us note that an experiment represents the running of the algorithm with a combination of the values of the parameters. For example, according to the experiment plan used, the experiment #1 will consist in running the algorithm with parameters (Pop, pm, pc, Ngene) = (100, 0.02, 0.5, 200).

Tuning the parameters of the GA

The process under consideration here is the GA, and we define two levels (m = 2) for each of its factors or parameters (-1 for a low value, +1 for a high value), as in table 1. Four parameters (k = 4) are concerned {pc, pm, Pop, Ngene}. The optimal combination of parameters will be investigated in a factorial plan of dimension 2 4 , i.e. 16 experiments will be performed to determine the best values combination of the parameters. The experiment plan can be modeled by a matrix M(i, j) in which the index of the rows corresponds to the experiment index, and index of the column corresponds to the index of the parameters. Let y be a parameter, y Î ( pc, pm, Pop Ngene). The matrix M defined as follows is summarized in table 2.

ï î ï í ì + - = value high has if 1 value low has if 1 ) , ( y y j i M
With this model, the evaluation of the sensitivity of the algorithm will be based on the following measurement functions with respect to each parameter:

, j = 1, …, k
The aim is to identify the best experiment, which is defined by the one that gives statistically the best results. The criteria that measure the best experiment will be the score Si of each experiment after a given number of runs of the algorithm with the experiment plan. To obtain the score of an experiment, it must be run during a given number of times with a change of the seed of the random generator. The score of each experiment corresponds to the number of times it finds the best solution. Thus, the sensitivity analysis will be done by the use of the score Si. The ad hoc sensitivity analysis is then given by: , j = 1, …, k

The input data of the optimization problem are the following: The efficiency criterion according to experiment of index i is denoted Hi, which corresponds to the adaptation function defined in section 3.2. In this paper, the relevant criterion for the efficiency of the algorithm is the score Si defined above. The sensitivity measurement increases with the influence of the corresponding parameter.

å = = Q k m 1 i i k j H j i M m 1 ) , ( å = = Q k m 1 i i k j S j i M m 1 ) , (
Table 1 describes the encoding of the parameters and Table 2 shows the matrix of the experiment plan for the efficiency evaluation of the GA.

Results and comments

The adaptation function defined in section 3.2 is considered as an efficiency measurement function for the individuals of the GA. After 250 runs of the GA, the experiment #9 gives the best results according to the experiment plan (see Table 2). Indeed, this experiment has obtained the best score (number of maximum values of the adaptation function). The parameters of the experiment #9 are selected for implementing the GA to find the solution of the optimization problem. This helps to choose the parameters of the GA which enable to reach an optimal solution. In addition, the sensitivity measurement of this experiment plan enables to identify the most sensitive parameters of the algorithm: Pop and pm (factors 3 and 2). In addition, it is interesting to notice that the factor 1 (Pc) is also important (see Table 3).

Optimal search with GA

The stability of the algorithm with respect to the parameters and even to the structure of the cost function constitutes a problem of thorough investigation. But, with regard to our optimization problem, the method which is suggested for tuning the parameter of the algorithm remains statistically acceptable. By using the retained parameters in the case studied and referring to the experiment plan technique, experiment # 9 corresponding to factors level (-1, -1, -1, +1) gives the solution depicted in table 4 for our problem. For a sufficiently exhaustive search of solution and to ensure that the solution is close to the optimal solution, it is essential to increase the size of the population (300 for example) on the one hand. It can be also necessary to increase the number of factors considered in the implementation of the experiment plan on the other hand.

But that orientation may be expensive in computing times and it thus becomes important to find very quickly a good compromise.

The system considered consists of 5 parallel subsystems in serial (cf. figure 1). The Table 4 summarizes the optimization results, with the failure and repair rate and the number of components allocated to each subsystem. One notes that the cost of the system is also less than CMax.

CONCLUSION

In this paper, a GA-based approach is proposed to solve a multi-objective optimization problem. It simultaneously aims at maximizing the availability and minimizing cost of repairable parallel-series systems. In a first step, we used the weighting technique to transform the problem into a problem of single objective optimization. Next we relaxed the constraints by mean of the technique of exact penalty to reformulate the problem and to find a solution. The objective function obtained is then used for adaptation in a GA. In a second step, we developed the GA for which we have tried to provide guidance to adjust the parameters and control the fitness. For that purpose, we designed an experiment plan that shows its relevance for the GA implementation.

The use of experiment plans could also be suitable to solve the optimization problems with simulated annealing, where there are several parameters to adjust. The methodology can be structured in two steps as follows: 1) working out of an experiment plan for the tuning of the parameter of the algorithm; 2) selection of the best parameters and implementation of the GA for

i : index of a subsystem li :

 li failure rate of components in subsystems i l : l = (l1, l2,...,ls) µi : repair rate of components in subsystems i µ : µ = (µ1, µ2,...,µs) ki : number of components in subsystem i k : k = (k1, k2,..., ks) lk, uk : vector of minimum and maximum number of components allowable in each subsystem ll, ul : vector of lower and upper bound of failure rate of components in each subsystem lµ, uµ : vector of lower and upper bound of repair rate of components in each subsystem ai, bi, pi, qi : coefficients of the cost function a, b, p, q : vector of ai, bi, pi, qi CS(l,µ,k) : cost of the system AS(l,µ,k) : asymptotic availability of the system w1, w2 : weighting factors respectively for the cost and the availability d1, d2 : relaxation coefficients respectively for the cost and the availability Qi : sensitivity factors (i = 1,...,4) : integer part of x such that Ngene : number of generation Tpop

Figure 2 :

 2 Figure 2: Flowchart representing the GA's principle.

s= 5 ;

 5 Aobj = 0.999 Cmax= 600 units ; uk= 9x(1 1 1 1 1) w1 = 0.75; w2 =0.25 d1 = 100; d2 =10000 p =-0.8x(0.4, 0.2, 1, 0.8, 1.2) q = 0.85 x (0.4, 0.2, 1, 0.8, 1.2) a = 0.01 x (4, 2, 5, 8, 12); b = 0.1 x (0.4, 0.2, 1, 0.8, 1.2)ll=10-3 x(0.4, 0.4, 0.5, 0.3, 0.5) ul= 2.10 -3 x (1, 1, 1, 1, 1) lµ =0.5 x (0.4, 0.7, 0.9, 0.8, 0.7) uµ= 0.85 x (0.4, 0.7, 0.9, 0.8, 0.7)

Figure 3 :

 3 Figure 3: Experiments results.

Table 4 :Figure 4 :

 44 Figure 4: Max Fitness from the first generation

Table 1 :

 1 Factors of the experiment plan.

Table 2 :

 2 Experiments matrix and statistics on 250 runs.

	N	F1	F2	F3	F4	250 runs
	1	-1	-1	-1	-1	38
	2	+1	-1	-1	-1	20
	3	-1	+1	-1	-1	16
	4	+1	+1	-1	-1	9
	5	-1	-1	+1	-1	6
	6	+1	-1	+1	-1	6
	7	-1	+1	+1	-1	12
	8	+1	+1	+1	-1	11
	9	-1	-1	-1	+1	41*
	10	+1	-1	-1	+1	28
	11	-1	+1	-1	+1	20
	12	+1	+1	-1	+1	10
	13	-1	-1	+1	+1	11
	14	+1	-1	+1	+1	9
	15	-1	+1	+1	+1	6
	16	+1	+1	+1	+1	7

Table 3 :

 3 sensitivity measurement for 250 runs.

	Qj	Q1	Q2	Q3	Q4
	250 runs 3,125	4,25	7,125	0,875

1,1 1,j 1,k1

2,1 2,j 2,k2 i i,1 i,j i,ki s s,1 s,j s,ks

Acknowledgements:

The authors indebted to anonymous referees for their invaluable comments and suggestions.

This work has been done when (*)

Tuning of the parameters of the algorithm

The described GA needs the tuning of several parameters. The main parameters are the size of populations, the number of generations, the crossover probability, the mutation probability, and the stop criterion. The last one consists in stopping the algorithm after a given number of successive generations without any improvement of the best solution. But tuning the other parameters, remains a difficult task. To overcome this difficulty, we propose to study the sensitivity of each parameter on the results and the algorithm efficiencies, by using the experiment plan described in section 5. This approach gives two indications for the parameters algorithm adjustment:

• the sensitivity of each parameter on the solution,

• the optimal set of parameters that will be used for the algorithm.

In 1986, Grefenstette [START_REF] Grefenstette | Optimization of control parameters for genetics algorithms[END_REF] proposed to find the set of optimal parameters of genetic algorithms by means of efficiency criteria based on the objective function to be optimized (for unconstrained problems). He determined the set of optimal parameters by means of a meta GA to optimize the parameters of the GA to be designed. In our concerns, we not only take into account the objective to be optimized, but also the fact that the random process (seed) may have an effect on the simulation.

a) Population size

Usually, the number of individuals in the population which stands for the population size (Pop) depends on the problem to be solved. In the case where the initial population consists of feasible solutions, generally difficult to generate, we suggest to consider populations of small size. Indeed, in an optimization problem, find a feasible solution may be as difficult as to find the

d) Mutation probability

While referring to the natural systems, the mutation is a rare phenomenon, and occurs with a small probability. Thus, we adopted the usual values of the probability of mutation pm [START_REF] Goldberg | Les algorithmes génétiques[END_REF]: pm is generally chosen in the interval (0.03, 0.05).

General structure of the algorithm

1. Encode the solution; the search of an optimal solution. Numerical experiments enable to illustrate the effectiveness of the approach. Even if GA do not give the optimal solution, it remains a practical way to search a solution complying at best with the problem constraints.

AUTHORS:

Charles