Nikola Kutin

Marie-Sabine Saget

Thomas Vallee

CONTAINER PORT HIERARCHY AND CONNECTIVITY BASED ON NETWORK ANALYSIS

Keywords: Network, ASEAN, Trade, Shipping, Connectivity, Centrality

This study aims to analyze port hierarchy between 153 container ports. For this purpose, a network analysis was conducted. Particular attention was paid to 68 ports from the ASEAN+3 1 community. We have created five director weighted networks at both port and country levels. Results reveal that the prevailing structure of the global maritime network is hub and spoke, and that the port rankings change according to different centrality measures.

Regarding the intra-ASEAN+3 connectivity, ASEAN member states form a cluster of interconnected ports. A comparative analysis shows that both the Export and Maritime connectivity networks have similar patterns, which indicates that the containerized trade within ASEAN+3 has the same features as the intra-regional exports.

Introduction

Maritime transport is the cheapest and most efficient way to transport goods in large quantities. According to data from Clarkson Research, containerized trade has increased by more than 600% since 1990. This has dramatically changed the character of ports' infrastructure. Due to the expansion of ports' hinterlands, shipping lines have been able to streamline their itineraries by reducing the number of ports of call, creating load-centre ports (hub ports). It also induced a further increase in vessel size, along with a growth in terminal area [START_REF] Midoro | Maritime liner shipping and the stevedoring industry: market structure and competition strategies[END_REF]. Indeed, [START_REF] Notteboom | Concentration and the formation of multi-port gateway regions in the European container port system: an update[END_REF] stated that 'the success of the port is strongly affected by the ability of the port community to fully exploit synergies with other transport nodes.' Furthermore, the maritime network is a complex subject due to the fact that its prevailing structure is 'hub and spoke'. [START_REF] Asgari | Network design approach for hub ports-shipping companies competition and cooperation[END_REF] define a hub port as a port that connects regional ports (feeder lines) to other hub ports (global network). Such hub ports might exhibit different purposes or importance within a network. Some ports such as Singapore and Hong Kong have a strategic geographical position and play an important intermediate role in connecting other regional ports to major sea trade routes, while others like Shenzhen are a gateway to large manufacturing centres situated in the hinterland [START_REF] Wang | Determinants of port centrality in maritime container transportation[END_REF]. Therefore, analysing the position and the purpose of a port within a network is essential to understanding the nature of the liner shipping network.

According to the International Maritime Organization, over 90% of the world's trade is carried out via sea. Hence, it is interesting to analyze whether a network based on the maritime connectivity has similar features to the network related to the value of countries' total exports. An interesting case study is the ASEAN+3 2 community which aims to promote regional economic integration in Eastern and South-Eastern Asia. According to [START_REF] Tongzon | The challenges of economic integration: the case of shipping in ASEAN countries[END_REF] this integration may result in lower shipping costs and improved quality of shipping throughout the region, and thus contribute to an improvement in the ASEAN members' trade performance and international competitiveness.

This research aims to determine the characteristics of the maritime and trade networks at port and country levels by using the graph theory and centrality measures. For this purpose, a dataset of 153 ports from 50 countries, including statistics for 6,410 maritime routes in the year 2014, was used. Particular attention was paid to a subset of 68 ASEAN+3 ports, including 38 designated ports of the ASEAN Single Shipping Market (ASSM). This paper attempts to answer the following research questions: What are the characteristics of the global maritime shipping network? What are the key ports in the network and what is their main purpose (transhipment port, gateway port, regional port, etc.)? What is the position of the ASEAN community in the global shipping network? How are the regional maritime connectivity networks at port and country levels organized within the ASEAN+3 community? Are there any fundamental differences between maritime connectivity and countries' exports of all goods?

The remainder of this paper is as follows: Section 2 discusses the literature related to ports' connectivity and maritime trade. Section 3 explains the methodology related to the Network Analysis and the dataset used in the study. All results are depicted in Section 4. Lastly, a discussion on the outcomes of the study is provided in the final section.

2 ASEAN+3 community includes all member states of the Association of Southeast Asian Nations (ASEAN), the People's Republic of China, Japan, and the Republic of Korea.

Review of the literature

Network Analysis in the context of maritime transport

While the main ports and shipping routes have been described in a number of studies, the structure and evolution of the global maritime network, as well as some crucial issues such as unequal traffic distribution and port hierarchies, has not been fully documented [START_REF] Ducruet | The worldwide maritime network of container shipping: spatial structure and regional dynamics[END_REF][START_REF] Ducruet | Network diversity and maritime flows[END_REF]. It is important to analyze the ports' connectivity because even a small change in the network can cause shipping carriers to modify their shipping routes in order to meet the demands of the shipping markets [START_REF] Peng | A stochastic seaport network retrofit management problem considering shipping routing design[END_REF]. A suitable method to measure the connectivity of container ports is Network Analysis. It allows for the identification of port clusters, which may result in agglomeration economies and a strengthened bond between individual and organizational linkage networks [START_REF] Yang | Port choice strategies for container carriers in China: a case study of the Bohai Bay Rim port cluster[END_REF]. Such an analysis is a good complement to traditional measures of individual throughput [START_REF] Ducruet | Ports in multi-level maritime networks: evidence from the Atlantic (1996-2006)[END_REF].

The cost effectiveness and production efficiency of ports are closely linked to the optimization of their networks. [START_REF] John | A risk assessment approach to improve the resilience of a seaport system using Bayesian networks[END_REF] applied Bayesian belief networks to identify various influencing variables related to different risk factors in a seaport system. [START_REF] Peng | A stochastic seaport network retrofit management problem considering shipping routing design[END_REF] used a mixed-integer nonlinear stochastic programming model and showed that the deployment of the containers handled at each seaport was not directly related to its throughput capacity. In a study conducted by [START_REF] Wang | Determinants of port centrality in maritime container transportation[END_REF], Freeman's centrality measures were estimated in order to determine the importance of a given port at both regional and global levels. The degree centrality measures revealed that the key port in the network was Shenzhen. In terms of closeness centrality, which is related to the number of connections with diverse ports, Shanghai, Hong Kong and Singapore had the highest scores. In addition, the betweenness centrality measures pointed out that Hong Kong and Singapore are also hubs connecting different clusters of tightly connected ports. [START_REF] Notteboom | Concentration and the formation of multi-port gateway regions in the European container port system: an update[END_REF] conducted a detailed container traffic analysis for the period for 78 ports. His study showed that the performance of European ports was related to their position within supply chains and networks. There is also a gradual deconcentration process in the container handling market, though it remains more concentrated than other cargo handling segments. [START_REF] Ducruet | Network diversity and maritime flows[END_REF] applied network simplification techniques based on linkage analysis which removes from each port all its links except the largest one. His study revealed that the most diversified ports have a high level of centrality and dominance in the network. They are also connected with more ports situated at a longer physical distance. On the other hand, the most specialized ports capture the majority of global traffic, although their average connection distance is lower due to their distribution role at the intra-regional level.

Maritime and Trade networks in ASEAN+3 community

Since the establishment of ASEAN maritime transport cooperation, ASEAN Member States have been working to promote the progressive liberalization of maritime transport to improve the region's maritime performance and international competitiveness (ASEAN 2017).

Regarding the development of the ASEAN transport network, [START_REF] Banomyong | Trade Facilitation & Logistics Development: Bridging the GMS and beyond[END_REF] applied the Geographical Simulation Model (GSM) to analyze the effects of the implementation of The Master Plan on ASEAN Connectivity (MPAC) by the member states. The study shows that port development is mostly concentrated in specific locations and in countries with better trade facilitation, and that the positive effect it entails could spread to non-ASEAN member states such as Japan, China and India which have a better trade facilitation environment. According to the Liner Shipping Index, computed by UNCTAD, the ASEAN community has had steady progress in terms of integration into the global network. However, the index varies across countries and additional efforts are needed to strengthen the maritime connectivity in order to realize the ASEAN Single Shipping Market (ASEAN 2017). Therefore it is necessary to analyze ASEAN port integration within the regional and global networks, and the relations with ASEAN's closest trade partners: People's Republic of China, Japan, and the Republic of Korea.

There have been a few case studies regarding the countries in the ASEAN+3 community. [START_REF] Sutomo | Assessment Model of the Port Effectiveness and Efficiency (Case Study: Western Indonesia Region)[END_REF] applied Multi-criteria Analysis, Analytical Hierarchy Process and Graph Theory to detect the most efficient ports in Indonesia. Their analysis was conducted with a sample of 18 ports with respect of the density of the transportation infrastructure, the capacity of the port, and the speed of goods movement to ports. [START_REF] Zaman | Connectivity Analysis of Port in Eastern Indonesia[END_REF] used the Gravity Model to analyze the connectivity of 12 Indonesian ports. By taking into account the ports' facilities, the size of the regional economies, supporting infrastructure (highways, roads, import and export zones, etc.) and connectivity (global, regional and national), the study identified the ports which could play the role of hub under various scenarios. A recent study from [START_REF] Fahmiasari | Container Shipping Network Efficiency Comparison in Indonesia: Nusantara Pendulum and Sea Tollway[END_REF] assessed the efficiency of two plans used for the creation of an integrated container shipping network in Indonesia. They used a few network efficiency indices by taking into account the demand (TEU per year) at the port's origin and destination, ocean freight rates, and the port's connectivity. They concluded that the government should develop the least important ports in the network in order to improve the efficiency of intra-national trade and avoid empty return containers. [START_REF] Lam | Dynamics of liner shipping network and port connectivity in supply chain systems: analysis on East Asia[END_REF] analyzed slot capacity in terms of TEUsi.e. vessel capacity based on 3000 container shipping services from 1995 to 2006 for the Eastern Asian ports of Shanghai, Ningbo, Busan and Kaohsiung. The study found that inter-container port complementarity accounted for a significant share of changes to shipping capacity and that the ports periodically restructured their networks to adjust to the demands of the market.

Similar to shipping networks where we see flows of containers between ports of origin and destination, trade flows of goods and services between countries can be represented by directed graphs where all edges are potentially bidirected, as long as exports and imports between two countries are of nonzero values [START_REF] Nguyen | Economic integration in ASEAN+ 3: A network analysis[END_REF]). An analysis of both trade flows and FDI flows within ASEAN+3 countries was recently conducted by [START_REF] Nguyen | Similarity in trade structure: Evidence from ASEAN + 3.' Review of[END_REF]. They concluded that during the 1990-2012 period, the degree of trade and FDI integration varied among the ASEAN+3 member states. Furthermore, ASEAN+3's intra-regional trade network seemed to be more densely connected than its intra-regional FDI network, and large or advanced countries tend to be better linked and form a sub-regional bloc of tightly connected economies.

Adding value to the literature

We have not found any studies in the literature comparing the trade and maritime networks.

We expect some similarities between the two networks since 90% of exchanged goods are carried via sea [START_REF] Ducruet | Network diversity and maritime flows[END_REF]. The position of the ASEAN community within the global shipping network, as well as the connectivity of the designated ASEAN container ports, are topics which also require additional research. Furthermore, most of the studies analysing the importance of various ports use a very limited sample size. This research adds value to the literature by providing a comprehensive analysis of different types of networks at port and country levels using a sample of 6,410 maritime routes connecting 153 major ports around the world. Since we have almost all the major ports in the ASEAN+3 community, we can conduct an analysis on their intra-regional maritime and trade connectivity. The Network Analysis involves the computation of different centrality measures, allowing us to identify different clusters of tightly linked ports/countries. The method also detects nodes having the highest connectivity, good strategic location, and important intermediate connections. We also identify hub ports, which concentrate most of the maritime traffic or trade of goods between highly connected ports in the network. The findings of this study provide a foundation for understanding the typology of the global shipping industry, as well as the role of different ports/countries and communities within the network.

Methodology and data

Network analysis is the process of investigating complex structures through the use of graph theory [START_REF] Otte | Social network analysis: a powerful strategy, also for the information sciences[END_REF]. It has been used in a variety of studies, with the nodes representing individuals, organizations, or even countries, and the ties can refer to communication, cooperation, friendship, or trade. Ties can be differentiated in most settings [START_REF] Opsahl | Node centrality in weighted networks: Generalizing degree and shortest paths[END_REF]. [START_REF] Freeman | Centrality in social networks conceptual clarification[END_REF] developed the well-known centrality measures such as Degree, Closeness and Betweenness for binary directed and undirected networks. Degree centrality assesses the number of connections of a node with the other individuals in the network.

Closeness centrality shows how easily a node can be reached by other nodes, and betweenness centrality describes how important a node is in terms of connecting other clusters of individuals (Jackson 2010). These measures have been conceptualized for weighted networks [START_REF] Newman | The structure of scientific collaboration networks[END_REF][START_REF] Borgatti | Centrality and network flow[END_REF][START_REF] Brandes | A faster algorithm for betweenness centrality[END_REF][START_REF] Barrat | Weighted evolving networks: coupling topology and weight dynamics[END_REF][START_REF] Opsahl | Node centrality in weighted networks: Generalizing degree and shortest paths[END_REF].

Network Analysis based on centrality measures in the context of maritime trade

Maritime transport, and containerized trade in particular, consists of carriers transporting goods from port of origin to a given destination. Network analysis has been used by [START_REF] Wang | Determinants of port centrality in maritime container transportation[END_REF] and [START_REF] Ducruet | Network diversity and maritime flows[END_REF] to analyze port hierarchy and connectivity. We use directed weighted graphs, where the importance of the flow between the nodes depends on the edges' weights which can be represented by the value of exports, or, the average size of ships multiplied by the number of trips. In this study, the nodes are either a port or a country.

Variables and data

Data

Networks

Recall that each maritime route connects a port of origin and a port of destination. In order to analyze the global shipping network as well as the regional networks of the ASEAN+3 community, we built five directed weighted networks named the following: 'Maritime Port Network', 'Maritime Country Network', 'ASEAN+3 Maritime Port Network', 'ASEAN+3 Maritime Country Network' and 'ASEAN+3 Export Country Network'. 12 In the case of maritime networks at port and country levels, the links between each pair of ports/countries are weighted according to the number of trips multiplied by the average size of the vessels operating on a maritime route. This indicator not only shows the frequency of the services (𝑁𝑏𝑇𝑟𝑖𝑝𝑠) but also the relative importance of the container trade flows on the maritime routes 𝐴𝑣𝑔𝑇𝐸𝑈. 1 depicts some basic features of the networks at port and at country levels. In the maritime network 'ASEAN+3 Export Country Network', the edges are weighted according to the value of exports in USD (𝐸𝑋𝑃). This indicator allows us to assess the importance of the trade between countries for all goods. Appendix 1 shows additional descriptive statistics about the variables that we used in this study.

Type of network

Type of nodes

No of nodes

No of edges

Weight of edges

Centrality measures used in this study

In order to analyze the different types of networks, and the hierarchy of ports or countries within them, we have computed a set of centrality measures for weighted directed graphs. The weight of the edges are according to the type of network and nodes (Table 1).

Weighted degree Centralities

In the case of directed graphs, there are two types of degree centralities, Indegree and Outdegree. The former shows the sum of all edges that are directed (pointing in) to node 𝑖 while the latter depicts all edges that are going out from the node 𝑖. Mathematically, both measures can be represented by equations 1 and 2.

𝑤𝑠 𝑖 𝑜𝑢𝑡 = ∑ 𝑤 𝑖𝑗 𝑁 𝑗 (1) 𝑤𝑠 𝑖 𝑖𝑛 = ∑ 𝑤 𝑖𝑗 𝑁 𝑗 (2)
𝑤𝑠 𝑖 𝑜𝑢𝑡 and 𝑤𝑠 𝑖 𝑖𝑛 refers to the weighted Outdegree and Indegree centralities of node 𝑖, which are the sum of the weights of the edges 𝑤 𝑖𝑗 of all nodes 𝑁 which have a direct incoming or outgoing connection with node 𝑖. Nodes 𝑖 and 𝑗 are connected when 𝑤 𝑖𝑗 > 0. Ports having many connections with others and accommodating large ships have high Indegree coefficients, while ports which are a point of departure of a considerable number of vessels have high Outdegree scores. The same logic applies to the analysis of the interaction at a country level as well as in the case of the countries' exports where the weights of the edges are according to the value of exported goods.

Weighted Betweenness and Closeness Centralities

Betweenness and closeness centralities in weighted networks are based on the Dijkstra's algorithm [START_REF] Dijkstra | A note on two problems in connexion with graphs[END_REF]). The [START_REF] Newman | The structure of scientific collaboration networks[END_REF] and [START_REF] Brandes | A faster algorithm for betweenness centrality[END_REF] algorithm to estimate the shortest path between two nodes in a weighted network is as follows:

𝑑 𝑤 (𝑖, 𝑗) = 𝑚𝑖𝑛 ( 1 𝑤 𝑖ℎ + ⋯ + 1 𝑥 ℎ𝑗 ) (4)
𝑑 𝑤 (𝑖, 𝑗) refers to the weighted distance between node 𝑖 and node 𝑗, while ℎ is an intermediate node situated on the shortest path between 𝑖 and 𝑗. Therefore, weighted closeness centrality of node 𝑖 (𝑤𝐶 𝑖 ) can be depicted by equation 5.

𝑤𝐶 𝑖 = [∑ 𝑑 𝑤 (𝑖, 𝑗) 𝑁 𝑗 ] (5)
In the context of containerized trade, a port with high weighed closeness centrality refers to a port with the possibility of reaching all other ports in the network by passing through not so many intermediate nodes.

Weighted Betweenness centrality (𝑤𝐵 𝑖 ) is based on a combination of the number of intermediary nodes and tie weights [START_REF] Opsahl | Node centrality in weighted networks: Generalizing degree and shortest paths[END_REF]. It can be represented by the following equation:

𝑤𝐵 𝑖 = 𝑏 𝑗𝑘 𝑤 (𝑖) 𝑏 𝑗𝑘 𝑤 (6)
𝑏 𝑗𝑘 𝑤 (𝑖) refers to the number of times that node 𝑖 is an intermediate passage for node 𝑗 to reach all 𝑘 nodes in the network, as well as the associated weighs to the links. Ports with high weighted betweenness have a strategic position in the network, and might play a role as an important hub port or an obligatory passage to reach other ports.

Hub Coefficient

There are two main types of network configurations in the maritime trade: the hub and spoke network and the multi-port network (Figure 3). The first one is a system of connections where all traffic moves along spokes connected to the hub at the centre. By contrast, the multiport calling configuration is characterized by direct trajectories of any given vessel between the ports. Such structure is used in the case of relatively small container port networks. We used the algorithm of [START_REF] Kleinberg | Authoritative sources in a hyperlinked environment[END_REF] to detect hub ports, which was applied in the context of online queries and analysis of the hyperlink structure among 'www' pages. This algorithm identifies authoritative 'www' pages (authorities) for broad search topics. Pages that connect a number of authorities are called hubs.

Regarding maritime transport, ports having high Hub coefficients are connected to ports with high Indegree centrality. This algorithm allows us to identify ports which are mainly connected to other major ports in the network.

Identification of clusters

A related question of interest is to know if one port belongs to a particular community of ports, and if yes, how this community is defined? As stated by [START_REF] Pons | Détection de structures de communautés dans les grands réseaux d'interactions[END_REF] and [START_REF] Pons | Computing communities in large networks using random walks[END_REF], the concept of community in graph theory is not clearly defined. However, it is possible to define a community as a set of vertices (nodes) where the density of internal connections is greater than the density of connections to the outside [START_REF] Pons | Détection de structures de communautés dans les grands réseaux d'interactions[END_REF]. Accordingly, to calculate the potential communities within a given network, one should split the network into a given number of groups.

In order to determine different groups of ports or countries, we use the modularity algorithm for weighted direct networks applied by [START_REF] Blondel | Fast unfolding of communities in large networks[END_REF].

In summary, by applying Network Analysis and Graph Theory we can identify the ports/countries (i) with the most outgoing and incoming connections (degree centrality), (ii) which have diverse connections and are easily reachable by other nodes in the network (closeness centrality), (iii) which are intermediate links 'bridges' connecting different clusters or peripheral nodes (betweenness centrality) and (iv) which play the role of hub in regional and global shipping networks. In addition, this methodology allows us to discern tightly connected clusters of ports/countries.

Results

The networks analyses were conducted using Gephi software (version 0.9.2) and R software (Package 'tnet'4 ).

Maritime Port Network (153 nodes)

Typology of the global 'Maritime Port Network'

As shown in Appendix 1, the descriptive statistics of the nodes and the link attributes reveal a significant dispersion around the mean, as the studied sample consists of highly heterogeneous ports and maritime routes. The network has a high degree of connectivity ( The clustering coefficient reflects the probability of being connected to other nodes in the network. It is relatively significant (0.69), as should be expected, considering the average number of neighbours of each node. It is also observed that the network density is small (0.276). The Graph Density measures how complete the network is [START_REF] Bastian | Gephi: an open source software for exploring and manipulating networks[END_REF]. A complete network contains all possible edges and has a density of 1.

These results indicate that the network has a scale-free configuration. Such networks, introduced by [START_REF] Barabási | Emergence of scaling in random networks[END_REF], are mainly characterized by a degree of distribution that follows a power law, with a few highly-connected nodes and a majority of nodes with lower degrees of centralities.

Ranking of ports in the Global Maritime network based on centrality measures

When we look at the rankings of the 153 ports in the 'Maritime Port Network' (Appendix 2), we observe that ports change their position according to different centrality measures. Recall that the weight of the links between the ports is according to the Number of Trips multiplied by the average size of the vessels operating between each pair of ports. annual throughput of 600,000 TEUs is ranked 10 th . This means that the port is not a hub, but is compulsory passage for many smaller regional ports (feeder ports). Other Indonesian ports such as Port Klang, Tanjung Priok and Tanjung Perak (Surabaya) also play a key role in connecting the ASEAN community to the global shipping network.

In order to detect major hub ports in the global 'Maritime Port Network' we look at the Hub Coefficient. Unsurprisingly, the most important hub ports are identified as Singapore, Hong

Kong, Ningbo-Zhoushan, Shanghai, Shenzhen, Port Klang and Rotterdam. These ports have very strong links between them as well as with other major ports in the network.

When we look at the correlation coefficients of the centrality measures and the annual throughput in 2014 (Table 4), there is very strong correlation between Outdegree and Indegree centralities (0.99) not surprisingly. It means that ports which are major points of departure for many ships are also major destinations. In addition, indegree and outdegree are both positively correlated to Hub, Betweenness and Container Throughput. Thus, ports with high hub coefficients are intermediate links with a capacity to handle many containers and, due to their geographical locations, are easily reachable by other ports in the network. The existence of hub ports in the 'Maritime Port Network' reveals that any dysfunction or removal of one hub can lead to the separation of different clusters or communities, and to a larger extent, the destabilization of the logistics chain [START_REF] Xu | Modeling transportation network redundancy[END_REF].

Appendix 3 shows a graphical representation of the Maritime Port Network where we see that five communities have been identified. The first community consists of ports belonging to eleven countries from North and South America (see last column of the table in Appendix 2). In the second cluster there are 10 European, 1 Canadian and 1 Moroccan ports. The third community is formed by 51 ports from Europe, Asia, the Middle East and Africa. The fourth group of closely connected ports is dominated by ports from China, Taiwan, the U.S. and the Philippines. In this group we also have a few ports from Mexico. The last community only contains ports from Indonesia. These results reveal that the geographical location of a port is not enough by itself to explain the connections with other ports.

Maritime Country Network

At a country level, the maritime network is a polarized or scale-free structure with a high level of connectivity. It is a core-periphery structure where the most important countries are the world's main importers and exporters. Japan, Egypt and Spain have very high closeness coefficients, which means that they have strategic locations in the network. In terms of intermediate nodes, Spain, Japan, Germany and Malaysia are found to be the most important nodes. We have not estimated the hub coefficient due to the high level of density of the network and the fact that almost all countries in the sample are interconnected. Unsurprisingly, at the country level the clustering coefficient and the graph density are higher than in the Maritime Port Network (Table 5).

When we compare the countries in the sample, we observe that China and the U.S. are the most connected nodes according to all centrality measures (Table 6). In terms of Indegree and Malaysia, Cambodia, the Philippines and Brunei. We did not include Myanmar due to the lack of data, and we did not include Laos since it is a land locked country. The low graph density (0.383) and the average path length of 1.677 suggest that the ports are well connected and that on average they interact with 25.7 nodes in the sample (Table 7). The good connectivity between the ASEAN+3 ports is confirmed by the high clustering coefficient (0.741).

In contrast with the global 'Maritime Port Network', within 'The ASEAN+3 Ports Network' Singapore is ranked 5 th and 7 th in terms of Outdegree and indegree respectively. In terms of intra-ASEAN outgoing and incoming traffic, the Japanese ports and Honk Kong are the most important ones (Table 8). However, Singapore is the gateway port of ASEAN+3 to the global shipping market. It has many direct connections with smaller (feeder) ports and its role as a major transhipment centre for Eastern and South-Eastern Asia is confirmed by its high closeness and betweenness coefficients. Shanghai, Hong Kong and Shenzhen are also important intermediate ports. The high hub coefficients of Hong Kong, Singapore, Shanghai, Shenzhen and Kaohsiung

show that these ports are mainly connected with other hubs of the 'ASEAN+3 Maritime Port Network'. Appendix 6 shows detailed centrality measures for all ports in the sample. The correlation matrix between the centrality measures of the 'ASEAN+3 Maritime Port Network' is similar to that of the maritime port network. However, it should be noted that the correlation between Outdegree and hub, as well as indegree and hub, is lower in this network with 0.68 and 0.69 respectively. This is normal since we observe a regional network of well interconnected ports. Kong, Taiwan and the Philippines. These results suggest that the ASEAN ports have a strong maritime intra-regional integration. As a corollary, it means that the ASEAN Single Shipping Market and its designated ports maintain close ties. In addition, there is also a high level of connectivity between Chinese ports and Hong Kong.

Comparison between ASEAN+3 Maritime Country Network and ASEAN+3 Export Country Network

Since the dataset includes almost all major ports situated in ASEAN+3 nations, we focus our analysis on the particular case of the maritime and export networks at a country level in this The other states belong to the second community which is dominated by Japan and South Korea.

29

In both observed networks Japan, China and Hong Kong have the highest scores. In addition, the rankings of Cambodia, Brunei, Philippines and Thailand are very similar. Vietnam has low Outdegree and Indegree centralities while its betweenness scores are higher in the two networks. It should be highlighted that in the Maritime as well as the Export network, Singapore plays an important role as a bridging organization (high betweenness centrality). These results

suggest that in the context of ASEAN+3, the development of the port infrastructure and the connectivity has a substantial impact on the overall trade performance. When we look at the structure of the Trade and Maritime Networks of ASEAN+3, we observe similar features not only at the country level but also at port level.

Conclusion

Network Analysis is a suitable method to investigate the structure of the maritime transport system as well as the hierarchy of the ports/countries at global and regional levels according to different centrality measures. We have analyzed five types of directed weighted networks. The weights of the links between nodes are according to the number of trips and the size of the containership vessels operating between the origin and destination ports, or, the value of countries' exports.

The first network consists of 153 ports from all around the world. Our findings confirm the hub and spoke structure of the Global Maritime Network. As major hubs have been identified

Singapore, Hong Kong, Ningbo-Zhoushan, Shanghai and Shenzhen. Ports with relatively low Indegree and Outdegree centralities, such as Port Said, Balboa and Belawan, have very high betweenness centralities which indicates their strategic geographical position and connections with ports which are situated at the periphery of the global network. When we aggregate the data at the country level, the 'Maritime Country Network' shows a similar pattern. The European states as well as North and South America form linked communities. China, the U.S., the Netherlands, and Japan are intermediate passages.

We have also analyzed the intra-maritime connectivity between ports situated in the ASEAN+3 countries. The Japanese ports and Hong Kong play the most important roles in terms of outgoing and incoming traffic. However, Shanghai and Singapore are found to have the highest betweenness scores which make them important intermediate points connecting many ports in the network. Furthermore, modularity analysis identifies a community of ports from exclusively ASEAN member states except the Philippines. This finding reveals that ASEAN states had a good intra-regional connectivity in 2014.

A comparative analysis was conducted between the maritime and trade integration of ASEAN+3 countries. In the case of maritime connectivity, all ASEAN member states form the first cluster, while China, Taiwan, Hong Kong, Republic of Korea and Japan are associated with the second cluster. When we weight the links between countries according to the exports, the 'ASEAN+3 Export Country Network' has similar features as the 'ASEAN+3 Maritime Country Network', even though the rankings for some countries was slightly different. Similar to the Ports Network, China and Hong Kong form one community, while Singapore plays an important intermediate role in both networks. These findings confirm the comparable pattern of countries' exports and their maritime connectivities.

Potential future research could be conducted by analysing competition in the liner shipping sector by taking into account the number of carriers operating on different maritime routes. In the context of ongoing investments in port infrastructure and Climate Change, the sustainability of the supply chain network is a hot topic. Network analysis could be a useful tool to detect ports which are important in the network and to assess their resilience to different disturbances such as natural and man-made disasters. A similar analysis for a longer period of time with dynamic graphs could be also conducted in order to observe how containerized transport has evolved throughout the years. In the near future, data availability should increase and similar analyses might be possible for a bigger sample.

Appendices:

Appendix 1: Descriptive statistics of different type of networks

  on port connectivity was obtained from Lloyd's Marine Intelligence Unit. It contains information about port to port connectivity in 2014, and consists of 153 ports from 51 countries, including 79 ports in Asia, 27 in North America, 20 in Europe, 19 in Africa and 8 in Latin America and the Caribbean. There are 6,410 direct maritime routes connecting the ports in the sample: Container port of origin (Source, port i) Container port of destination (Target, port j) Average size of the ships in TEU going from port i to port j (AvgTEUij) Number of trips from port i to port j (NbTripsij) Data were also aggregated at a country level in order to analyze the interactions between the states in the sample: Country of departure, Source (country a) Country of arrival, Target (country b) Average size of the ships in TEU going from country a to port b (AvgTEUab) Number of trips from port A to port B (NbTripspab) The export dataset was collected from UN ComTrade 3 commodity databases in the Standard International Trade Classification (SITC) Revision 3. It contains the exports in USD for ASEAN + 3 states (EXPab).

Figure 1

 1 Figure 1 Hub and Spoke versus Multi-Port Calling configuration. Source: (Imai et al., 2009)

Table 1 .

 1 Type of networks and weightingTable

	Maritime Port Network	Ports	153	6410	𝑤 𝑖𝑗 = 𝐴𝑣𝑔𝑇𝐸𝑈 𝑖𝑗 * 𝑁𝑏𝑇𝑟𝑖𝑝𝑠 𝑖𝑗
	Maritime
	Country	Countries	51	2490	𝑤 𝑎𝑏 = 𝐴𝑣𝑔𝑇𝐸𝑈 𝑎𝑏 * 𝑁𝑏𝑇𝑟𝑖𝑝𝑠 𝑎𝑏
	network
	ASEAN+3
	Maritime Port	Ports	68	1745	𝑤 𝑖𝑗 = 𝐴𝑣𝑔𝑇𝐸𝑈 𝑖𝑗 * 𝑁𝑏𝑇𝑟𝑖𝑝𝑠 𝑖𝑗
	Network
	ASEAN+3
	Maritime Country	Countries	13	135	𝑤 𝑎𝑏 = 𝐴𝑣𝑔𝑇𝐸𝑈 𝑎𝑏 * 𝑁𝑏𝑇𝑟𝑖𝑝𝑠 𝑎𝑏
	Network
	ASEAN+3
	Export Country	Countries	13	156	𝑤 𝑎𝑏 = 𝐸𝑋𝑃 𝑎𝑏
	Network

Table 2

 2

	Clustering Coefficient	0.695
	Diameter	4
	Average Path Length	1.809
	Average bivariate Degree	47.81699346
	Average Weighted degree	16,409,626.386
	Node Count	153
	Density	0.276

). On average, the bivariate distance between any two connected nodes is 1.809 edges and each interacts with approximately 48 other ports in the sample.

Table 2 .

 2 Characteristics of the Maritime Port Network

Table 3

 3 depicts the top 20 ports ranked in terms of the selected network indicators. The most important port in Europe is the port of Rotterdam, as it has the highest Outdegree score. This means that the frequency and the size of the ships departing from this port of origin are substantial.In terms of Indegree centrality, Singapore has the highest score and Rotterdam ranks second.

	No	Rank Outdegree (weighted)	Rank Indegree (weighted)	Rank Closeness (weighted)	Rank Betweenness (weighted)	Rank Hub
	1	Rotterdam	Singapore	Port Said	Singapore	Singapore
	2	Singapore	Rotterdam	Singapore	Port Said	Hong Kong
						Ningbo-
	3	Shanghai	Port Said	Shanghai	Shanghai	Zhoushan
	4	Hong Kong	Shanghai	Rotterdam	Rotterdam	Shanghai
		Ningbo-		Ningbo-
	5	Zhoushan	Hong Kong	Zhoushan	Hong Kong	Shenzhen
	6	Port Said	Antwerp	Hamburg	Balboa	Port Klang
	7	Hamburg	Hamburg	Hong Kong	Valencia	Rotterdam
			Ningbo-
	8	Shenzhen	Zhoushan	Shenzhen	New York	Tanjung Pelepas
			Bremen/Bremer
	9	Antwerp	haven	Antwerp	Tokyo	Kaohsiung
		Bremen/Breme		Bremen/Bremer
	10	rhaven	Shenzhen	haven	Belawan	Port Said
	11	Tokyo	Le Havre	Valencia	Port Klang	Hamburg
	12	Le Havre	Valencia	Felixstowe	Marsaxlokk	Antwerp
					Tanjung Priok
	13	Valencia	Tokyo	Yokohama	(Jakarta)	Algeciras
	14	Felixstowe	Yokohama	Tokyo	Antwerp	Jebel Ali (Dubai)
	15	Yokohama	New York	Algeciras	Yokohama	Le Havre
					Tanjung Perak
	16	New York	Felixstowe	Tanjung Pelepas	(Surabaya)	Valencia
	17	Algeciras	Nagoya	Port Klang	Shenzhen	New York
	18	Nagoya	Algeciras	Le Havre	Ningbo-Zhoushan	Norfolk (USA)
	19	Kaohsiung	Jebel Ali (Dubai)	Southampton	Port Everglades	Balboa
						Bremen/Bremer
	20	Port Klang	Kaohsiung	Keelung	Algeciras	haven

Table 3 .

 3 Ranking of top 20 ports according to different centrality measuresAlthough Port Said had an annual throughput of only 3.96 million TEU in 2014, it has the highest closeness coefficient due to its unique geographical location. The port is located at the north gate of the Suez Canal, a 200 km-long canal that connects the Mediterranean with the Red Sea[START_REF] El-Bastawisy | Integrated Socio-economic Development for Accelerating the Regional Role of Port-Said in Tourism Development of Egypt[END_REF], and is at a strategic position easily reachable by many ports in the sample. Singapore is ranked second due to its role as a major hub port connecting Southeast Asia with Western Asia, Europe and Africa. Similar to the findings of[START_REF] Wang | Determinants of port centrality in maritime container transportation[END_REF], the Chinese ports of Shanghai, Shenzhen and Hong Kong have very high weighted closeness scores and rank 3 rd , 7 th and 8 th respectively. Ningbo-Zhoushan is another Chinese container port which is ranked 5 th and is situated in the Yangtze River Delta, a large industrial and logistics area.Similar to closeness centrality, the betweenness measures show that the ports serving as the most important intermediate passages are Singapore, Port Said and Shanghai. It is interesting to highlight that the port of Balboa is in 6 th place. It is an important hub connecting Far East and North America to the West Coast of South America, Central America and the Caribbean. Like Port Said in Egypt, the Panamanian port has a strategic location on the Panama Canal which makes it

a compulsory intermediate node for the majority of traffic in Latin and Central America. Although weighted betweenness centrality identifies major hubs of the global network, Belawan with an

Table 4 .

 4 Correlation matrix between weighted centrality measures for the Maritime Port Network

		Outdegree Indegree Closeness Betweenness	Hub	Container Throughput
	Outdegree	1
	Indegree	0.99	1
	Closeness	0.34	0.35	1
	Betweenness	0.74	0.74	0.18	1
	Hub	0.80	0.81	0.61	0.47	1
	Container Throughput	0.72	0.68	0.22	0.63	0.62	1

Table 5 .

 5 Characteristics of the Maritime Country Network

	Clustering Coefficient	0.792
	Diameter	3
	Average Path Length	1.408
	Average Bivariate Degree	29.784
	Average Weighted degree	42,509,324.093
	Node Count	51
	Graph Density	0.596

Table 7 .

 7 Characteristics of the ASEAN+3 Maritime Port Network

	Clustering Coefficient	0.741
	Diameter	4
	Average Path Length	1.677
	Average Bivariate Degree	25.662
	Average Weighted degree	7,332,918
	Node Count	68
	Graph Density	0.383

Table 8 .

 8 Ranking of top 20 ASEAN+3 ports according to different centrality measures

		Outdegree	Indegree	Closeness	Betweenness	Hub
	1	Tokyo	Tokyo	Singapore	Singapore	Hong Kong
	2	Yokohama	Yokohama	Tokyo	Shanghai	Singapore
	3	Nagoya	Nagoya	Yokohama	Hong Kong	Shanghai
	4	Hong Kong	Hong Kong	Hong Kong	Shenzhen	Shenzhen
	5	Singapore	Shenzhen	Shenzhen	Tokyo	Kaohsiung
	6	Shenzhen	Shanghai	Shanghai	Yokohama	Port Klang
	7	Shanghai	Singapore	Nagoya	Tanjung Priok (Jakarta)	Ningbo-Zhoushan
	8	Keelung	Ningbo-Zhoushan	Ningbo-Zhoushan	Port Klang	Nagoya
	9	Ningbo-Zhoushan	Keelung	Keelung	Tanjung Perak (Surabaya)	Laem Chabang
	10	Kaohsiung	Kaohsiung	Hanshin Ports	Belawan	Tanjung Priok (Jakarta)
	11	Hanshin Ports	Hanshin Ports	Kaohsiung	Bangkok	Hanshin Ports
	12	Bangkok	Bangkok	Penang	Manila	Tokyo
	13	Port Klang	Busan	Port Klang	Ningbo-Zhoushan	Yokohama
	14	Laem Chabang	Port Klang	Bangkok	Kota Kinabalu	Penang
	15	Busan	Laem Chabang	Laem Chabang	Kaohsiung	Tanjung Pelepas
	16	Penang	Tanjung Pelepas	Busan	Tanjung Mas (Semarang)	Xiamen
	17	Lianyungang	Penang	Belawan	Laem Chabang	Busan
	18	Xiamen	Lianyungang	Lianyungan g	Zhanjiang	Manila
	19	Tanjung Priok (Jakarta)	Xiamen	Xiamen	Ho Chi Minh City	Keelung
				Tanjung
	20		Tanjung Priok	Mas
		Tokyo	(Jakarta)	(Semarang)	Nantong	Bangkok

Table 9 .

 9 Correlation matrix between weighted centrality measures for the 'ASEAN+3 MaritimeAs we can see in Appendix 7, three communities have been identified in this network. The first one includes 8 ports from Japan, Korea, and China. The second one includes 29 ports exclusively from ASEAN. The only ASEAN country we studied which does not have any ports in this cluster is the Philippines. The last cluster is the biggest one and has 31 ports from China, Hong

		Outdegree	Indegree	Closeness	Betweenness Hub	Container Throughput
	Outdegree	1
	Indegree	0.9971	1
	Closeness	0.3418	0.3446	1
	Betweenness	0.7032	0.7	0.2583	1
	Hub	0.6798	0.6875	0.6614	0.5346	1
	Container Throughput	0.5678	0.5839	0.2286	0.7652	0.57	1
				Port Network'
	Container throughput is highly correlated with betweenness centrality. Therefore, ports
	with high betweenness in the ASEAN+3 network also constitute an important part of the intra-
	ASEAN+3 maritime traffic.

Table 11 )

 11 community. Recall that in the former network weights of edges are based on the Number of Trips multiplied by the average size of the containerships in TEUs, while in the latter, weights correspond to country Exports in USD. . In the case of the ASEAN+3 Maritime Country Network, we have identified two clusters. The first one includes all ASEAN states except the Philippines, while the second is constituted of China, Philippines, Japan, South Korea, Hong Kong and Taiwan. It confirms our previous finding that ASEAN countries form a well-connected community where the dominant port is Singapore (being both a port and a country).

		ASEAN+3 Maritime Country	ASEAN+3 Export Country
		Network	Network
	Clustering Coefficient	0.909	1
	Diameter	2	1
	Average Path Length	1.135	1
	Average Bivariate Degree	10.385	12
	Average Weighted degree	25,181,747	224,695,259
	Node Count	13	13
	Graph Density	0.865	1
	Table 10. Characteristics of ASEAN+3 Maritime Country Network and ASEAN+3 Export
		Country Networks
	Unsurprisingly, the 'Export Country Network' is denser, since it takes into account all
	exported goods. On the other hand, ASEAN+3 countries have 10.38 maritime country connections
	on average.
	At first glance we can see that major economies such as China, Japan and Hong Kong,
	which act as bridges toward Europe and North America, play a substantial role in both networks
	(

Table 11 .

 11 Rankings of the ASEAN+3 countries in the Export and Maritime Network according to different weighted centrality measures In the case of the Export Country Network (Appendix 11), we have identified two clusters of countries. The most dominant nodes are China and Hong Kong, and they form the first community. These results are consistent with the outcomes of the modularity analysis of the third network, 'ASEAN+3 Port Network', where Chinese ports and Hong Kong form the third cluster.

https://comtrade.un.org/

https://cran.r-project.org/web/packages/tnet/tnet.pdf

Acknowledgments

We would like to thank Ms. Kerry Slattery, International Programme Director, National University of Management, Phnom Penh, Cambodia, for reviewing this paper and providing us with valuable feedback.

This work was supported by the European Union's Erasmus+ Programme, Key Action 2, Capacity building in the field of higher education under DOCKSIDE Project (www.dockside-kh.eu), Grant number 573790-EPP-1-2016-1-FR-EPPKA2-CBHE-SP.

Outdegree centralities, Germany has the highest connectivity in Europe, followed by Spain and the Netherlands.

Outdegree

Ranking Indegree (weighed) ). The first one is constituted by only European countries. The second cluster includes 21 nodes mainly from Asia and the Middle East. The third cluster is the most diverse with countries from Europe, Asia, South America and Africa. Finally, similar to the port network, North and South American countries form a community.

ASEAN+3 Maritime Port Network

In this section we analyze the ports belonging to the ASEAN+3 community, which includes China, Hong Kong, Taiwan, South Korea, Japan, Singapore, Thailand, Vietnam, Indonesia,

Appendix 2: Main weighted centralities measures for the 'Maritime Port Network'

Weighted centrality measures for the 'Maritime Port Network' as well as the modularity class to which every port belongs. Note: Ports for which we do not have data for the container throughput, in the first column is written 'N/A'.

Appendix 4. Main centralities measures for the 'Maritime

Appendix 5 Graphical Representation of the Maritime Country Network

Maritime Network of 51 countries in the sample. The size of the nodes corresponds to their outdegree weighted centrality. The weight of the links is 𝑤 𝑎𝑏 = 𝐴𝑣𝑔𝑇𝐸𝑈 𝑎𝑏 * 𝑁𝑏𝑇𝑟𝑖𝑝𝑠 𝑎𝑏 . The different colours of edges and nodes correspond to the modularity classes (closely linked nodes).

Appendix 6 Main centralities measures for the 'ASEAN+3 Maritime


Pierre-Jean Barjhoux
email: pierre-jean.barjhoux@irt-saintexupery.com

Youssef Diouane
email: youssef.diouane@isae.fr

Stéphane Grihon
email: stephane.grihon@airbus.com

Dimitri Bettebghor
email: dimitri.bettebghor@onera.fr

Joseph Morlier
email: joseph.morlier@isae.fr

Mixed Variable Structural Optimization: toward an Efficient Hybrid Algorithm

Keywords: mixed optimization, structural optimization, branch and bound, categorical variable, multimaterial

Designing a structure implies a selection of optimal concept and sizing, with the aim of minimizing the weight and/or production cost. In general, a structural optimization problem involves both continuous variables (e.g., geometrical variables, …) and categorical ones (e.g., materials, stiffener types, …). Such a problem belongs to the class of mixed-integer nonlinear programming (MINLP) problems. In this paper, we specifically consider a subclass of structural optimization problems where the categorical variables set is non-ordered. To facilitate categorical variables handling, design catalogs are introduced as a generalization of the stacking guide used for composite optimization. From these catalogs, a decomposition of the MINLP problem is proposed, and solved through a branch and bound method. This methodology is tested on a 10 bars truss optimization inspired from an aircraft design problem, consistent with the level of complexity faced in the industry. 2.

Introduction

In the field of aircraft design, structural optimization involves different kinds of variables. Continuous variables describe the size of aircraft structural parts: in case of thin-sheet stiffened sizing, they represent panel thicknesses and stiffening cross-sectional areas. The set of discrete variables encompasses two kinds of variables. First, variables that are ordered, e.g. a list of cross-sectional areas. Second, technological choices or even computation assumptions (e.g. material choices, buckling margin policy) are depicted by categorical variables.

Many industrial problem formulations natively involve a mixture of continuous and discrete variables. Mixedvariable optimization algorithms have been proposed, such as Genetic Algorithms [START_REF] Goldberg | Genetic Algorithms in Search, Optimization, and Machine Learning[END_REF], Differential Evolution [START_REF] Storn | Differential Evolution -A Simple and Efficient Heuristic for global Optimization over Continuous Spaces[END_REF], Particle Swarm Optimization [START_REF] Kennedy | Swarm Intelligence[END_REF] and Pattern Search [START_REF] Torczon | On the convergence of pattern search algorithms[END_REF]. In many cases, discrete variables are considered as ordered through a continuous relaxation approach, for example in [START_REF] Lampinen | Mixed Integer-Discrete-Continuous Optimization, by Differential Evolution, Part 1: the optimization method[END_REF][6] [START_REF] Dimopoulos | Mixed-variable engineering optimization based on evolutionary and social metaphors[END_REF]. In other cases, categorical variables are handled through a native mixed-variable optimization approach for both categorical and continuous variables without relaxation [START_REF] Deb | A Flexible Optimization Procedure for Mechanical Component Design Based on Genetic Adaptive Search[END_REF] [START_REF] Audet | Pattern Search Algorithms for Mixed Variable Programming[END_REF]. Other strategies consist in turning categorical variables into continuous ones, allowing the use of continuous optimization algorithms. For example, continuous indices containing shape and material information are built [START_REF] Rakshit | Simultaneous material selection and geometry design of statically determinate trusses using continuous optimization[END_REF], meta-models are constructed through Multiple Kernel Regression [START_REF] Herrera | Metamodel-assisted optimization based on multiple kernel regression for mixed variables[END_REF].

In industry, most structural optimization software solutions mainly handle continuous variables and ignore ordinal and categorical variables. This is due to the difficulty of managing all variable types simultaneously while maintaining the optimization performance: discrete variables always introduce a combinatorial behavior leading to an exponential number of configurations to be explored. For instance, at Airbus, two structural optimization software were developed to split up the problem. A first one is inspired by classical continuous optimization approaches using gradients. The second one focuses on discrete and categorical variables in a preliminary phase of the design process where trade-offs are evaluated [START_REF] Grihon | Structure Sizing Optimization Capabilities at AIRBUS[END_REF]. On this basis, major decisions governing the structural definition are made.

In this article, the work related to mixed variable optimization will be presented in section 4, and an industrial formulation is proposed in section 5. Then, the formulation of a test case is introduced in section 6. The section 7 is dedicated to the description of the proposed methodology, followed by an analysis of numerical results.

Related work

To our knowledge, an open problem is to define an efficient and generic algorithm combining both categorical and continuous variables for solving multi-concept structural optimization problems. In industry, attempts have been made to tackle this kind of optimization problems. One can cite the PRESTO approach, developed at Airbus [START_REF] Grihon | PRESTO: A rapid sizing tool for airframe conceptual design studies[END_REF], that is a pure discrete approach to manage both sizing and categorical variables. However, sizing variables (e.g., thicknesses, areas) are treated as discrete ones. In this setting, a value of a categorical variable points to a choice among catalogs for a single element or group of elements. In the literature, a catalog for an element can be considered as a guide to a given composite panel [START_REF] Adams | Genetic algorithm optimization and blending of composite laminates by locally reducing laminate thickness[END_REF][15] [START_REF] Carpentier | Buckling optimisation of composite panels via layup tables[END_REF]. Composite guide approach has been extended to stiffening profiles as illustrated in Figure 1. On top of stacking sequences, the profile dimensions can be varied according to the area. For metal, a skin catalog reduces to a list of thicknesses and a stiffener catalog to a list of profiles.

Figure 1: catalogs principle, where the links between the thickness and stacking sequences, area and detailed variables, are depicted.

Indeed, for PRESTO, the catalogs allow for mapping detailed variables values with a given couple of thickness and area values. This means that a choice of thickness is equivalent to a choice of a stacking, and a choice of area corresponds to a choice of scaled profile (including stacking). Furthermore, a change in stacking sequence (symmetric, balanced, oriented, iso) leads to a change of catalog choice. Using the same approach, information on material, stiffening profile or even buckling margin policy choices have been added to catalogs. Indeed, results of ISAMI, the reference strength analysis package of Airbus used for sizing and airframes certification, have been stored into the PRESTO database. So in addition to the sizing variables link mentioned above, a catalog contains a table of Reserve Factors (RF) corresponding to predefined internal forces. This RF aims at measuring the strength of an element: a RF greater than 1 illustrates that the element does hold the given load.

For PRESTO, the sizing is performed separately for each element by finding, through the catalog database, the skin and stiffener parameters (i.e.; super-stiffener) that minimize the weight with respect to internal loads (IL). Considering that the catalog set describes a finite set of choices, this local (spatially speaking, in the structural model) optimization is straightforward and consists of a simple loop that carries out a numerical sorting to find the weight minimum feasible super-stiffener as illustrated in Figure 2. Since the choice of catalog is performed separately for each element, it does not take into account overall effects of optimal internal forces redistribution, or rigidity constraints. Both sizing and categorical variables are concerned by this drawback. The main objective of this work consists in improving the existing industrial PRESTO approach by setting up an efficient and performing methodology that tackles directly both categorical and sizing variables (as continuous variables) using a new formulation of the problem.

In order to implement a generic approach as far as possible, no distinction is made between material and stiffening principle selection. To that end, a particular attention has been paid to avoid methods that are focused Thickness Stacking Area Profile on specific categorical variables (e.g. dedicated to material, stacking sequences). Our methodology relies on PRESTO catalogs to ensure a unified approach handling all the categorical variables.

Industrial problem formulation

Let us name t the thicknesses and A the areas where (𝒕, 𝑨) ∈ ℝ 2𝑁 , N being the number of elements in the structural model. The set 𝛤 is an enumerated set which allows to list choices 𝒄, for example the material or the stiffener type of an element. If 𝑚 is the number of catalogs, we can note 𝛤 = ⟦1, 𝑚⟧.

All these variables are vectors for which each component is associated to an element. In our industrial case, a finite element model involves about N=100 elements leading to 200 continuous variables, and 10 up to 100 categorical choices per element. The categorical variable can take a value among 𝑚 𝑁 (number of categorical choices per element at the power of the number of elements) values of the set 𝛤 𝑁 . This high combinatory dimension demonstrates the need for a methodology to solve efficiently such problems. Within the framework of stiffened panels, as key concept of aeronautics structures, the optimization problem to solve is: min 𝑊(𝒄, 𝒕, 𝑨)

(1) (𝒕, 𝑨) ∈ ℝ 2𝑁 𝒄 ∈ 𝛤 𝑁 Subjected to: 𝑹𝑭(𝒄, 𝒕, 𝑨, 𝐼𝐿(𝒄, 𝒕, 𝑨)) ≥ 1 𝐺(𝒄, 𝒕, 𝑨) ≤ 0 𝑡(𝒄) ≤ 𝒕 ≤ 𝑡(𝒄)

𝛼(𝒄) ≤ 𝑨 𝑏𝒕 ≤ 𝛼(𝒄)
In this formulation, the constraints on RF ensure the structural strength element per element. 𝐺 represents any constraint, especially stiffness constraint that depends on all thicknesses and areas through the stiffness matrix (e.g. displacement). Bounds on sizing variables, in particular on the stiffening ratio 𝛼 (of areas over thicknesses), prevent over-stiffened stiffeners supporting thin skin, or the reverse (design rule). Note that both continuous variables (i.e.; thickness 𝒕, area 𝑨) and categorical variables (i.e.; 𝒄, hiding material or stiffening principle choices) have a significant role in this problem. The categorical variables affect the weight 𝑊, internal loads IL, rigidity constraints 𝐺, reserve factors RF, bounds on thicknesses and stiffening ratio. On the other hand, continuous variables affect the weight, internal loads, reserve factors and rigidity constraints. It is worth noting that a change in a categorical variable, thickness or area will modify 𝐼𝐿 distribution along the structure. Since the stresses and then RF require the value of 𝐼𝐿, each component of RF vector depends on the whole structure description. Internal loads 𝐼𝐿 are computed using a finite element model (FEM).

Test case formulation

Before tackling the targeted industrial case that deals with aeronautics box-section structures, a preliminary step has consisted in the implementation of a test-case that can be handled within short execution-times. This makes possible the exploration of new algorithms and allows for identifying promising leads.

To that end, the well-known 10 bars truss optimization test-case, introduced as a continuous optimization problem in [START_REF]L'Optimisation Multiniveaux D'une Structure[END_REF][18] [START_REF] Samuelides | Modèles réduits en optimisation multiniveau de structures aéronautiques, 9e Colloque national en calcul des structures[END_REF], has been adapted to match the needs of the mixed optimization problem. The continuous optimization part of the problem remains unchanged, while categorical variables have been introduced. As described in the whole problem in Eq.( 1), a choice of categorical value corresponds to a choice of material or stiffening principle. Thus, the problem becomes:

min 𝑊(𝒄, 𝑨) (2)
𝑨 ∈ ℝ 10 𝒄 ∈ 𝛤 10 Subjected to: 𝑹𝑭 𝒊 (𝑐 𝑖 , 𝐴 𝑖 , 𝐼𝐿(𝒄, 𝑨)) ≥ 1, ∀𝑖 ∈ ⟦1, 10⟧ 𝐺(𝒄, 𝑨) ≤ 0 𝑨(𝒄) ≤ 𝑨 ≤ 𝑨(𝒄)

Many dummy variables are hidden by categorical variables. For the sake of clarity, Table 1 gives the dependencies between them.

      Constraints        Internal Loads (IL)  
As in the whole problem in Eq.( 1), the i th component of the vector c corresponds to a choice of material and shape for the i th bar. For each section, and regarding cross-section sizing variables, catalogs establish the link between areas and detailed variables (geometrical description of the cross sections). The PRESTO approach is replicated here, except that the sizing variables (areas, detailed variables) are considered as continuous.

Based on the hypothesis that for each bar, the vector of detailed variables 𝒙 𝑖 of the 𝑖 𝑡ℎ bar varies linearly with respect to the detailed variables 𝒙 0 of the reference profile:

𝒙 𝑖 = 𝜆 𝑖 𝒙 0 , 𝑖 ∈ ⟦1, 10⟧ (3)
The corresponding cross-sectional area can then be written as:

𝐴 𝑖 = 𝜆 𝑖 2 𝐴 0 , 𝑖 ∈ ⟦1, 10⟧ (4)
From ( 3) and ( 4) can be deduced an expression of 𝜆 𝑖 , the scaling ratio of the profiles:

𝜆 𝑖 = √𝐴 𝑖 𝐴 0 ⁄ , 𝑖 ∈ ⟦1, 10⟧ (5)
Thus, the simple mathematical function detailed below can be used compute the detailed variables with respect to areas 𝐴 𝑖 coming from the optimizer, and the reference profile described by 𝐴 0 and 𝒙 0 :

𝒙 𝑖 (𝐴 𝑖 ) = √𝐴 𝑖 𝐴 0 ⁄ 𝒙 0 , 𝑖 ∈ ⟦1, 10⟧ (6)
, 𝑖 ∈ ⟦1, 10⟧ (7)
with:

𝜎 𝑖 𝐿𝑜𝑐𝑎𝑙.𝑏𝑢𝑐𝑘𝑙 (𝑐 𝑖 , 𝐴 𝑖 ) = 4𝜋 2 𝐸 𝑐 𝑖 12(1 -𝜈 𝑐 𝑖 2 ) ( 𝑥 𝑖 (1) (𝐴 𝑖 ) 𝑥 𝑖 (2) (𝐴 𝑖 ) ) 2 (8)
𝜎 𝑖 𝐸𝑢𝑙𝑒𝑟.𝑏𝑢𝑐𝑘𝑙 (𝑐 𝑖 , 𝐴 𝑖 ) = 𝜋 2 𝐸 𝑐 𝑖 𝐼 𝑐 𝑖 (𝒙 𝒊 (𝐴 𝑖 )) 𝐴 𝑖 𝐿 2 (9)
In [START_REF] Dimopoulos | Mixed-variable engineering optimization based on evolutionary and social metaphors[END_REF], 𝜎 𝑖 𝑎𝑙𝑙𝑜𝑤.𝑡𝑒𝑛𝑠𝑖𝑜𝑛 and 𝜎 𝑖 𝑎𝑙𝑙𝑜𝑤.𝑐𝑜𝑚𝑝𝑟 are the allowable values in terms of tension and compression respectively.

They are directly linked to the choice of material, whereas 𝜎 𝑖 𝐿𝑜𝑐𝑎𝑙.𝑏𝑢𝑐𝑘𝑙 and 𝜎 𝑖 𝐸𝑢𝑙𝑒𝑟.𝑏𝑢𝑐𝑘𝑙 depend on both shape (through detailed variables 𝑥 𝑖 (1) (𝐴 𝑖 ) , 𝑥 𝑖 (2) (𝐴 𝑖 ) and quadratic moment 𝐼 𝑐 𝑖 (𝒙 𝒊 (𝐴 𝑖 )) ) and material choices. The structural constraints computed by FEM are noted 𝜎 𝑖 . All allowable constraints value are defined as positive.

In this simplified test-case, the catalog is made of two aluminum alloys with different physical properties: one has a higher young modulus, the other higher allowable values. This is typically the case of a metallic wing, where the extrados working under compression requires a higher young modulus, whereas the intrados part needs higher fatigue allowable summarizing other properties than stiffness.

Methodology

As a first approach it was necessary to build a baseline methodology. For that purpose, we chose the branch and bound algorithm, whose guiding principle is mainly due to [START_REF] Land | An Automatic Method of Solving Discrete Programming Problems[END_REF], [START_REF] Dakin | A tree-search algorithm for mixed integer programming problems[END_REF]. Many authors improved this method afterwards, for example in the field of structural optimization [START_REF] Tseng | Enhancing Branch-and-Bound Method for Structural Optimization[END_REF].

Generic branch and bound process

Based on an enumerative divide-and-conquer technique, this generic approach searches for the best solution into the whole space of solutions for a given problem. The algorithm repeatedly splits the original design space into several design spaces, this is called "branching". A diagram of the branches can be seen as a "tree", while the subspaces and their associated sub-problem are illustrated by "nodes" of this tree. A sub-problem is then generated for each subspace. For each sub-problem, a lower bound (if the original objective is to get a minimum) of a feasible solution is computed. This corresponds to the "bound" step. Relying on this evaluation, all active sub-problems that meet the following two criteria are deleted. First, a design sub-space that does not contain allowable values has no reason to be further explored. Secondly, the subspace may not contain a solution better than the best known solution (called upper bound). Since an exploration of this subspace would not lead to the optimal solution, this part of the design space is discarded.

Here are the main steps of this algorithm: 0. Initialization : let be 𝑃 0 the initial problem, 𝑏 ̅ (𝑃) the upper bound of the full problem 𝑃 and equal to the best known solution of 𝑃 (or +∞ if no best solution known), Q the queue containing active nodes and equal to {𝑃 0 } 1. Branch: select the best problem 𝑃 𝑖 (𝑗) according to a given strategy among all active nodes from Q, and delete it from Q. Split the design space of problem 𝑃 𝑖 (𝑗) into 𝑚 subspaces. Build all associated subproblems 𝑃 𝑖 (𝑗) and add them to Q, with a priority following a given strategy.

2. Bound: compute lower bound 𝑏(𝑃 𝑖+1 (𝑗) ) of 𝑃 𝑖+1 (𝑗) through continuous relaxation of 𝑃 𝑖+1 (𝑗) noted

𝑃 𝑖+1 (𝑗) ̂. If the problem is infeasible, 𝑏(𝑃 𝑖+1 (𝑗) ) ≔ -∞. a. If 𝑏(𝑃 𝑖+1 (𝑗) ) > 𝑏 ̅ (𝑃), 𝑃 𝑖+1 (𝑗) node is removed from Q b. If 𝑃 𝑖+1 (𝑗) not feasible, 𝑃 𝑖+1 (𝑗) node is removed from Q c. If 𝑏(𝑃 𝑖+1 (𝑗) ) ≤ 𝑏 ̅ (𝑃) and 𝑏(𝑃 𝑖+1 (𝑗) ) is an integral feasible solution of 𝑃, update 𝑏 ̅ (𝑃): 𝑏 ̅ (𝑃) ≔ 𝑏(𝑃 𝑖+1 (𝑗) ). If Q is not empty, go to step 1,
The index 𝑗 of 𝑃 𝑖 (𝑗) reminds that at each branching step, we have to manage 𝑚 sub-problems, each of them being associated with a choice among 𝑚 catalogs (index 𝑗 ∈ ⟦1, 𝑚⟧). 𝑃 𝑖 (𝑗) is an instance of a problem 𝑃 𝑖 where the catalog 𝑗 has been chosen for the current bar 𝑖 . For simplification purposes, 𝑃 𝑖 (𝑗) and 𝑃 𝑖 will be used interchangeably later in this article.

The developed methodology

In the context of structural optimization, at each level of the tree, the categorical variable corresponding to a bar is chosen. Thus at level 𝑖 of the tree, 𝑖 -1 categorical variables have been fixed by the previous 𝑖 categorical variables associated to the first 𝑖 bars are fixed. The branch and bound process generates, at each branching step, as many subspaces as the number of available catalog choices (i.e.; 𝑚). This formulation is inspired from the generalized Benders [START_REF] Geoffrion | Generalized Benders decomposition[END_REF] decomposition, that allows to treat independently complicating variables (in our case, categorical variables). The problem formulation in Eq.( 2) at a level 𝑖 of the branch and bound tree is the following:

(Problem 𝑃 𝑖 ) min 𝑊(𝒄, 𝑨) = ∑ 𝑊 𝑘 (𝑐 𝑘 ̿ , 𝐴 𝑘 ) 𝑖 𝑘=1 + ∑ 𝑊 𝑘 (𝑐 𝑘 , 𝐴 𝑘 ) 𝑛 𝑘=𝑖+1 (10)
𝑨 ∈ ℝ 10 𝒄 ∈ ⟦1, 𝑚⟧ All fixed variables from upper levels are noted with double over-bar accent: for example, 𝒄 ̿ is the 𝑖-dimensional vector of fixed categorical values. As in the 10 bars truss problem description, 𝑨 is the vector of continuous design variable describing the areas, invariably of dimension 10 during the branch and bound process: the areas of all the bars are design variables regardless of the position in the tree. A breakdown of the objective function shows that two terms are involved. The first one gives the weight of the truss made up of the 𝑖 first bars. It only depends on the continuous variables: the areas. The second corresponds to the weight of the remaining bars of the truss: from 𝑖 + 1 to 10. This last term depends on two variables: the categorical variables that have not been fixed already, and the areas corresponding to the same undetermined bars from 𝑖 + 1 to 10. Since the evaluation of internal loads depends on all the categorical variables and areas, both fixed and undetermined categorical variable computation are required (𝒄 ̿ and 𝒄 respectively) in addition to all the areas.

Although branching is performed on categorical variables associated to the 𝑖 first bars, there is still a dependency to the categorical variables linked to the remaining bars. Even if the categorical variable dimension of the generated subspaces decreases when branches depth increases, the computation of the associated subproblems still requires to handle the categorical variable. Indeed, computation of constraints 𝑹𝑭 𝒌 , even for 𝑘 ∈ ⟦1, 𝑖⟧, requires categorical choices of the whole structure through internal load computation. One major advantage of the separation and evaluation process is that no exact solution of the sub-problems is required. Actually, only a lower bound of the optimum has to be provided. Since the convergence speed of the branch and bound algorithm is directly driven by its accuracy, a specific attention has to be paid to the lower bound computation process. Now that the formulation of a sub-problem has been described, the computation of a lower bound 𝑏(𝑃 𝑖 ) of such problem can be detailed. A lower bound of an optimization problem could, for example, be provided by removing one or several constraints from the problem. In the context of the branch and bound algorithm, it would consist in adding progressively the constraints, from one level of the tree to another, that can be computed when one more categorical variable is fixed. This would ensure to get a lower bound of the problem, while avoiding the complexity linked to categorical variables. However in the case of problem 𝑃 𝑖 in Eq.( 10), each 𝑹𝑭 𝒌 computation depends on internal loads, and thus to the whole vector of categorical variables (including the undetermined components). In this paper, two different optimization problems formulation that allow to compute a lower bound are detailed.

lower bound computation problem #1

The optimization constraints linked to buckling have been removed from the optimization problem, as a first approach. Indeed, the removal of the buckling 𝑹𝑭 𝒌 makes the optimization problem independent from crosssectional shape choices. On the other hand, whereas the internal loads computation requires categorical variables, two properties of the material information are still needed: the Young modulus, and the allowable constraint in tension and compression. Taking the highest allowable constraint value in Eq.( 12) leads to a relaxed problem, where the resulting optimum weight is a lower bound of the sub-problem. Concerning the objective function, only density is required through material component of the categorical choices. Also, applying the lowest density of the available materials to the undetermined bars ensures then to get a lower bound of the truss weight. Finally, the lower bound process comes to a weight optimization with respect to all areas and the young modulus of the remaining bars. This optimization problem is the following one:

(
, ∀𝑘 ∈ ⟦𝑖 + 1, 10⟧ (12)
where 𝜎 𝑚𝑎𝑥 𝑎𝑙𝑙𝑜𝑤.𝑐𝑜𝑚𝑝𝑟 and 𝜎 𝑚𝑎𝑥 𝑎𝑙𝑙𝑜𝑤.𝑡𝑒𝑛𝑠𝑖𝑜𝑛 are the highest allowable constraints available into the set of catalogs.

lower bound computation problem #2

In this second approach, the buckling constraints are not removed from the optimization problem: as in 7.2.1 with the constraints on allowable stresses, the material values and shape are chosen so that the limit stress for buckling described in ( 8) and ( 9) is maximized. Thus, the limit stress for buckling becomes, for all bars that are not defined:

𝜎 𝑚𝑎𝑥 𝐿𝑜𝑐𝑎𝑙.𝑏𝑢𝑐𝑘𝑙 (𝐴 𝑘 ) = 4𝜋 2 𝐸 𝑚𝑎𝑥 12(1 -𝜈 𝑚𝑎𝑥 2 ) ( 𝑥 𝑘 (1) (𝐴 𝑘 ) 𝑥 𝑘 (2) (𝐴 𝑘 ) ) 2 (13)
𝜎 𝑚𝑎𝑥 𝐸𝑢𝑙𝑒𝑟.𝑏𝑢𝑐𝑘𝑙 (𝐴 𝑘 ) = 𝜋 2 𝐸 𝑚𝑎𝑥 𝐼 𝑚𝑎𝑥 (𝒙 𝒌 (𝐴 𝑘 )) 𝐴 𝑘 𝐿 2 (14)
with 𝐼 𝑚𝑎𝑥 (𝒙 𝒌 (𝐴 𝑘 )) the function that returns the highest quadratic moment values, for a given shape (the highest function plotted in Figure 4 corresponds to a 'I' shape). The expression of 𝑹𝑭 𝒌 constraints are then given by: (𝜎 𝑚𝑎𝑥 𝐿𝑜𝑐𝑎𝑙.𝑏𝑢𝑐𝑘𝑙 (𝐴 𝑘 ) -𝜎 𝑘 (𝑬 𝒄 ̿ , 𝑬 𝒄 , 𝑨))

𝜎 𝑚𝑎𝑥 𝐿𝑜𝑐𝑎𝑙.𝑏𝑢𝑐𝑘𝑙 (𝐴 𝑘 ) ⁄ ≤ 0 ] (15)
The overall formulation of the optimization problem (11) remains unchanged, but there is a better approximation of the constraints because buckling is now taken into account (15).

Numerical examples and discussion

As explained previously, the combinatorial complexity is directly linked to the number of available catalogs. This is why, to start evaluating our methodology, first runs have been launched with only 2 catalogs based on two different materials and an identical cross-section shape I. The first material named 'Material 0' has higher density and allowable values than the second one 'Material 1', which is stiffer. A load of 10 5 N has been applied downward on the 4 th node of the truss. In order to highlight the changes in optimal structural concepts, several bounds have been imposed on vertical displacement of the 4 th node. The sub-problems 𝑃 𝑖 ̂ are solved using the Method of Moving Asymptotes (MMA) [START_REF] Svanberg | A Class of Globally Convergent Optimization Methods Based on Conservative Convex Separable Approximations[END_REF] algorithm available in NLOPT library.

The implemented node selection methodology (or queue data structure) during branching step consists in a mix of two strategies: depth-first search and best-first search. Indeed, the branching is performed on the node having the lowest lower bound, with a higher priority assigned to the nodes of the current generation (i.e.; current level in the tree). The advantage of this method is that the process returns a choice for the whole truss as quickly as possible. It provides a solution that is possibly better than the best current upper bound. In this case, the upper bound is updated with this solution, and all the active nodes that have a higher lower bound than this new upper bound can be deleted from the queue. In this way, a number of nodes are not explored further, leading to an improvement of the whole process convergence. Table 2 shows the results obtained, with the following convention: a '0' in position k of categorical variable vector c corresponds to a choice of 'Material 0' for the bar k, a '1' corresponds to a choice of 'Material 1'. A comparison of the results with those obtained with a factorial design of experiments shows that for these cases, the branch and bound approach returns the global solutions w.r.t. displacements bound changes. In addition, it can be seen that the optimal weight increases with the displacement bound: the more the structure is constrained, the higher is the weight. We can note that if the whole tree was to be explored, it would require ∑ 2 𝑘 10 𝑘=1

computations. This is twice more than the 2 10 optimizations required to solve the problem using a full combinatorial approach.

The results in Table 2 shows that the first approach is not efficient: all the tree volume (twice the combinatorial search cost) is explored in cases 3, 4 and 5. This means that the branch and bound needs more optimizations compared to those required by a full combinatorial approach. This is because our minoring problems 𝑃 𝑖 ̂ in Eq.( 11) are not close enough to the original problem in Eq.( 2). However, the results obtained with the second approach shows that the number of optimizations performed during the branch and bound run is equal to 16.2% of the optimizations required by a combinatorial approach. This means that despite the fact that the whole design space has been considered (thanks to branch and bound theory) during the optimization process, a lot of nodes have been pruned and thus a lot of optimizations were not required to find the optimal solution.

The computation history of case 3 is depicted on Figure 5 and Figure 6. On Figure 5, the lower bound estimation of the current node is presented in blue, the best upper bound in red. An iteration is defined by a branching and bounding step. We can note that the upper bound is not updated until the end of the computation: the branching route obtained by our strategy leads to the first upper bound after 10 optimizations, that appears to be the optimal solution. In Figure 6 we can follow for example the evolution of the number of pruned nodes (line in yellow) during the optimization. We can see for example a gap between the iterations 50 and 60: a node placed at the top of the tree has been deleted because the lower bound was higher than the upper bound. Thus, the remaining nodes (line in blue) decreases significantly: all the nodes behind the deleted one do not need to be explored. Figure 7 shows the truss with stress values, optimal areas, and reserve factors at the optimum for the case 3. This illustrates the trades that are made by the optimizer between the materials and the areas in this case. First, it can be noted that the solution is non-uniform in terms of categorical variables: each of the two available materials have been chosen at least once to build the optimal truss. Compared to case 4, the optimal material of bars 5 and 9 changed to the first material ('Material 0'), with higher density and allowable values. nodes. As it has proven its efficiency, the second methodology to compute the lower bound. The optimal solution is c*= '2220023000', the Figure 8 shows the convergence history. 7040 continuous optimizations have been performed to solve this problem, corresponding to less than 0.7% of the optimizations required by a combinatorial approach. In Figure 8, the bar diagram shows that a few nodes at level 3, 4 and 5 of the tree have been removed during the computation. This is very interesting in terms of algorithm efficiency, because the design space associated to these node has been eliminated, including all depending nodes. It can be seen that most of the deleted nodes are located between the 6 th and 10 th level of the tree. The number of explored nodes, exposed in Table 2, demonstrates that the number of optimizations during the branch and bound process using the first approach exceeds the number of optimizations of a full design of experiment on categorical variables (because the whole tree counts twice more optimizations than the 2 10 optimizations of the whole combinatorial problem). Indeed, as explained before, the branch and bound algorithm efficiency relies on the pruning rate of the tree. Nodes are discarded because either the solution is infeasible, or the lower bound exceeds the best upper bound. In our cases, independently of the quality of categorical choices, the optimizer can balance with the areas values. Thus, considering the available material quality and the bounds of the continuous optimization subspaces, there is no unfeasible solution. The pruning relies on lower bounds and upper bound comparison too. In this sense, the second approach offers a better approximation of the constraints, and thus a better lower bound. This allows to significantly decrease the number of optimizations: less than 16.2% of the number of optimizations required by a combinatorial approach are now performed to find a global solution of the mixed problem. In the case of 4 catalogs, only 0.7% of the optimizations are required, but it still needs 7040 optimizations. Knowing that an industrial finite element of a pylon counts around 100 elements, with 10 up to 100 choices available per element, this methodology based on the branch and bound approach remains costly.

The efficiency could be improved for example by choosing the branching order of the categorical variables, or adapting the numerical parameters (initial point, tolerances, …) of the continuous optimization during the branch and bound process. One promising approach consists in replacing categorical variables by continuous ones belonging to [0,1], using the work done for multi-material and topology optimization [START_REF] Stegmann | Discrete material optimization of general composite shell structures[END_REF]. This would keep the consistency between all the optimal physical features, and thus increase the lower bound accuracy.

Conclusion

Relying on Airbus experience and previous work on this topic, a formulation of a mixed optimization problem that tackles both continuous and non-ordered categorical variables has been proposed in this article. Specific attention has been paid to keep this formulation as generic as possible, without exhibiting any physical feature directly. A toy case involving a simplified physical model, that illustrates the industrial optimization problem complexity, has been presented. A hybrid approach based on branch and bound generic formulation, and taking advantage of the generalized Benders decomposition, has been implemented. This method allows a trade-off between categorical and continuous variables that leads to the global optimum. However, numerical examples have highlighted the limitations of this approach. The performance in terms of computation time prevents from scaling up the toy problem for industrial purpose. Several improvements are under investigation, mainly based on a better handling of categorical variables transformation into continuous ones.

Figure 2 :

 2 Figure 2: sizing engine of PRESTO

Figure 3 :

 3 Figure 3: 10 bars truss structure working in tension/compression, with 6 nodes

Figure 4 :

 4 Figure 4: continuous profile scaling

  Subjected to: 𝑹𝑭 𝒌 (𝑐 𝑘 ̿ , 𝐴 𝑘 , 𝐼𝐿(𝒄 ̿, 𝒄, 𝑨)) ≥ 1, ∀𝑘 ∈ ⟦1, 𝑖⟧ 𝑹𝑭 𝒌 (𝑐 𝑘 , 𝐴 𝑘 , 𝐼𝐿(𝒄 ̿, 𝒄, 𝑨)) ≥ 1, ∀𝑘 ∈ ⟦𝑖 + 1, 10⟧ 𝐺(𝒄 ̿, 𝒄, 𝑨) ≤ 0 𝑨(𝒄 ̿) ≤ 𝑨 ≤ 𝑨(𝒄 ̿) 𝑨(𝒄) ≤ 𝑨 ≤ 𝑨(𝒄)

  𝑹𝑭 𝑘 (𝐼𝐿(𝑬 𝒄 ̿ , 𝑬 𝒄 , 𝑨)) = [ (𝜎 𝑚𝑎𝑥 𝑎𝑙𝑙𝑜𝑤.𝑐𝑜𝑚𝑝𝑟 -𝜎 𝑘 (𝑬 𝒄 ̿ , 𝑬 𝒄 , 𝑨)) 𝐸𝑢𝑙𝑒𝑟.𝑏𝑢𝑐𝑘𝑙 (𝐴 𝑘 ) -𝜎 𝑘 (𝑬 𝒄 ̿ , 𝑬 𝒄 , 𝑨)) 𝜎 𝑚𝑎𝑥 𝐸𝑢𝑙𝑒𝑟.𝑏𝑢𝑐𝑘𝑙 (𝐴 𝑘 ) ⁄ ≤ 0

Figure 5 :

 5 Figure 5: Lower bound w.r.t. iterations in case 3.Figure 6: Number of nodes computed, remaining, and pruned w.r.t iterations in case 3.

Figure 6 :

 6 Figure 5: Lower bound w.r.t. iterations in case 3.Figure 6: Number of nodes computed, remaining, and pruned w.r.t iterations in case 3.

Figure 7 :

 7 Figure 7: results of case 3, with c*= '0001100111'The computation case 3 has been launched again, at the exception that two stiffening principles have been added to the two material choices: one U shape and one I shape. Since a categorical value corresponds to a choice of material and shape, the combination raises the number of catalogs from 2 to 4: 𝑚 = 4. This means that 4 10 optimizations could solve the problem, and the total volume of the tree counts ∑ 4 𝑘 10 𝑘=1

Figure 8 :

 8 Figure 8: Computation history of case 3 with 4 catalogs

Table 1 :

 1 variables description and links with the functions of the optimization problem 𝑐

	Shape		Material		Cont. variables
		𝜌	𝐸 𝜎 𝑎𝑙𝑙𝑜𝑤	𝜈	𝐴	𝑥	IL
	Weight				
	Constraints (compr/tension)						
	Constraints (loc/buckling)

  Since the buckling constraints have been removed from the problem 𝑃 𝑖 in Eq.(10), the constraints 𝑹𝑭 𝒌 are 2 dimensional: only constraints on allowable in tension and compression are included. Thus, in the context of Eq.(11), Eq.(7) becomes, for all bars that are currently not defined:

	Problem 𝑃 𝑖 ̂) 𝐴 𝑹𝑭 𝑘 (𝐼𝐿(𝑬 𝒄 ̿ , 𝑬 𝒄 , 𝑨)) = 𝑨 ∈ ℝ 10 min 𝑊(𝑬 𝒄 , 𝑨) = ∑ 𝑬 𝒄 ∈ ℝ 10-𝑖 Subjected to: 𝑹𝑭 𝒌 ( 𝐸 𝑐 𝑘 ̿̿̿ , 𝐴 𝑘 , 𝐼𝐿(𝑬 𝒄 ̿ , 𝑬 𝒄 , 𝑨)) ≥ 1, ∀𝑘 ∈ ⟦1, 𝑖⟧ 𝑊 𝑘 (𝜌 𝑐 𝑘 ̿̿̿ , 𝐴 𝑘 ) 𝑖 𝑘=1 + ∑ 𝑊 𝑘 (𝜌 𝑚𝑖𝑛 , 𝐴 𝑘 ) 𝑛 𝑘=𝑖+1 𝑹𝑭 𝒌 (𝐸 𝑐 𝑘 , 𝐴 𝑘 , 𝐼𝐿(𝑬 𝒄 ̿ , 𝑬 𝒄 , 𝑨)) ≥ 1, ∀𝑘 ∈ ⟦𝑖 + 1, 10⟧ 𝐺(𝑬 𝒄 ̿ , 𝑬 𝒄 , 𝑨) ≤ 0 𝐴 𝑘 (𝑐 𝑘 ̿ ) ≤ 𝐴 𝑘 ≤ 𝐴 𝑘 (𝑐 𝑘 ̿ ), ∀𝑘 ∈ ⟦1, 𝑖⟧ 𝐴 𝑚𝑖𝑛 ≤ 𝐴 𝑘 ≤ 𝐴 𝑚𝑎𝑥 ̅̅̅̅̅̅̅ , ∀𝑘 ∈ ⟦𝑖 + 1, 10⟧ 𝐸 𝑚𝑖𝑛 ≤ 𝐸 𝑐 𝑘 ≤ 𝐸 𝑚𝑎𝑥 , ∀𝑘 ∈ ⟦𝑖 + 1, 10⟧ (𝜎 𝑚𝑎𝑥 𝑎𝑙𝑙𝑜𝑤.𝑐𝑜𝑚𝑝𝑟 -𝜎 𝑘 (𝑬 𝒄 ̿ , 𝑬 𝒄 , 𝑨)) 𝑎𝑙𝑙𝑜𝑤.𝑐𝑜𝑚𝑝𝑟 ⁄ ≤ 0 𝜎 𝑚𝑎𝑥 [ (𝜎 𝑘 (𝑬 𝒄 ̿ , 𝑬 𝒄 , 𝑨) -𝜎 𝑚𝑎𝑥 𝑎𝑙𝑙𝑜𝑤.𝑡𝑒𝑛𝑠𝑖𝑜𝑛 ) 𝜎 𝑚𝑎𝑥 𝑎𝑙𝑙𝑜𝑤.𝑡𝑒𝑛𝑠𝑖𝑜𝑛 ⁄ ≤ 0 ]	(11)

𝑚𝑖𝑛 and 𝐴 𝑚𝑎𝑥 ̅̅̅̅̅̅̅ are the lowest lower bound and maximum upper bound on area of all catalogs respectively: 𝐴 𝑚𝑖𝑛 = min ∀𝑘 (𝐴 𝑘 (𝑐 𝑘 ̿ )), 𝐴 𝑚𝑎𝑥 ̅̅̅̅̅̅̅ = max ∀𝑘 (𝐴 𝑘 (𝑐 𝑘 ̿ )). The bounds on Young modulus 𝐸 𝑚𝑖𝑛 and 𝐸 𝑚𝑎𝑥 are defined similarly: 𝐸 𝑚𝑖𝑛 = min ∀𝑘 (𝐸 𝑐 𝑘 (𝑐 𝑘 ̿ )), 𝐸 𝑚𝑎𝑥 ̅̅̅̅̅̅̅ = max ∀𝑘 (𝐸 𝑐 𝑘 (𝑐 𝑘 ̿ )).

Table 2 :

 2 optimal weight and categorical values w.r.t. bounds on displacements

	cases	Bounds on	Optimal Weight	Optimal Categorical	Lower bound #1	Lower bound #2
		displacements (mm)	W*(kg)	variable c*	Combinatory explored (%)	Combinatory explored (%)
	1	displ < -17.	14.592	0011100111	153%	2.9%
	2	displ < -18.	14.053	1011100111	197%	11.3%
	3	displ < -19.0	13.722	0001100111	200%	16.2%
	4	displ < -19.96	13.633	0001000101	200%	11.1%
	5	displ < -25.	13.633	0001000101	200%	11.3%

Acknowledgements

This work is part of the MDA-MDO project of the French Institute of Technology IRT Saint Exupery. We wish to acknowledge the PIA framework (CGI, ANR) and the project industrial members for their support, financial backing and/or own knowledge: Airbus, Airbus Group Innovations, SOGETI High Tech, Altran Technologies, CERFACS. We would like to thank Charlie Vanaret for all his advices, specifically for the branch and bound formulation.