# Optimizing Asia-Europe container network: The Suez Canal and Cape of Good Hope routes in a changing world

Sadeque Hamdan, Dominique Feillet, Ali Cheaitou, Pierre Cariou, Nadjib Brahimi

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## ► To cite this version:

Sadeque Hamdan, Dominique Feillet, Ali Cheaitou, Pierre Cariou, Nadjib Brahimi. Optimizing Asia-Europe container network: The Suez Canal and Cape of Good Hope routes in a changing world. European Journal of Operational Research, 2025, 325, pp.167 - 188. 10.1016/j.ejor.2025.03.008. hal-05058856

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Submitted on 7 May 2025

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Innovative Applications of O.R.

# Optimizing Asia–Europe container network: The Suez Canal and Cape of Good Hope routes in a changing world

Sadeque Hamdan<sup>a,∗</sup>, Dominique Feillet<sup>b,†</sup>, Ali Cheaitou<sup>c</sup>, Pierre Cariou<sup>d,†</sup>, Nadjib Brahimi<sup>e,‡</sup>

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## ARTICLE INFO

**Keywords:**

Demand disruption

Freight rate

Suez Canal

Cape of Good Hope

Maritime transportation

## ABSTRACT

The aim of this paper is to develop an approach for evaluating various maritime transportation routes in light of ongoing disruptions and evolving global factors, including changes in demand, fluctuations in fuel prices, geopolitical shifts, and environmental considerations. For each alternative route, a profit maximization liner shipping problem with speed optimization is solved. Both edge-based speed optimization and maximum transit time of commodities are considered. We have proposed a mathematical programming formulation followed by an efficient hybrid approach for the optimization problem. The hybrid approach utilizes a population-based heuristic to optimize the route and an exact algorithm to optimize commodities and speed. A thorough analysis is made on the costs associated with the different alternative routes. Although the approach can be applied to different regions of the world and under the variation of several factors, we focus on the Asia–Europe trade route and on the evolution of the market demand. The comparison is made between a route going through Suez Canal and the alternative route of going through the Cape of Good Hope. Besides the high efficiency of the solution procedure, we have found that the Cape of Good Hope route can be economically interesting especially if there is demand in some African ports and if vessels of less than 20,000 TEU capacity are used.

## 1. Introduction

The transport of containerized goods by sea is a critical component of world trade. A key decision for shipping lines and shippers lies in choosing which route to use for their container vessels and containers. For Asia–Europe containerized trade, the second largest globally in terms of volume with 24.2 million twenty-foot equivalent units (TEUs) in 2022 (UNCTAD, 2023), the options are limited and the Suez Canal option is dominant. It offers shorter transit time and lower costs compared to the Arctic, rail, or via the Cape of Good Hope (Verny & Grigentin, 2009; Vukic & del Mar Cerban, 2022). Since its opening in 1869, the Suez Canal has become the most important shipping route in the world, carrying almost 80% of all seaborne trade between Asia and Europe, which is almost 20% of all global seaborne trade (Laih, Tsai, & Chen, 2015).

Although recent literature confirms that the Suez Canal is still the best economic and environmental option, statistics also show that the number of container ships opting to use the Cape of Good Hope route

has increased to a historic peacetime high, reaching 32 voyages in the first three months of 2020, and costing the Suez Canal Authority over 10 million USD in lost fees (Global-Maritime-Hub, 2021). However, the maritime industry has seen recently a phenomenon of shifting from Suez Canal to Cape of Good Hope or other alternative routes which has recently further accelerated due to major disruption risks for the Suez Canal such as the major piracy attacks on cargoes around the Horn of Africa in the 2010–2015 period (Statista, 2023) and the grounding of the container ship Ever Given in March 2021. More recently, amid the war on Gaza strip and due to attacks from Yemen Houthis on vessels crossing the strait Gate of Grief (Bab-el-Mandeb), major shipping firms, such as Maersk, Hapag-Lloyd, and CMA-CGM, announced in December 2023 that they are rerouting all their journeys that were supposed to pass through the red sea and suspending their operations in that region (BBC, 2023). The transit activity through Suez Canal has dropped by more than 41% between October 2023 and January 2024 (UNCTAD, 2024).

<footnote><sup>a</sup> Bangor Business School, Bangor University, Bangor, Gwynedd, LL57 2DG, United Kingdom</footnote>

<footnote><sup>b</sup> Mines Saint-Etienne, Univ Clermont Auvergne, INP Clermont Auvergne, CNRS, UMR 6158 LIMOS, Saint-Etienne, F-42023, France</footnote>

<footnote><sup>c</sup> Sustainable Engineering Asset Management (SEAM) Research Group and Department of Industrial Engineering and Engineering Management, University of Sharjah, Sharjah, P.O. Box 27272, United Arab Emirates</footnote>

<footnote><sup>d</sup> Kedge Business School, The Centre of Excellence in Supply Chain (CESIT), 680 Cours de la Libération, Talence, 33400, France</footnote>

<footnote><sup>e</sup> Rennes School of Business, Supply Chain Management and IS department, 2 Rue Robert d'Arbrissel, Rennes, 35065, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail addresses: s.hamdan@bangor.ac.uk (S. Hamdan), feillet@emse.fr (D. Feillet), acheaitou@sharjah.ac.ae (A. Cheaitou), pierre.cariou@kedgebs.com (P. Cariou), nadjib.brahimi@rennes-sb.com (N. Brahimi).</footnote>
In addition to the security-related issues and incidents, this shift has been motivated by a combination of other factors associated mainly with fuel prices, fluctuations in demands and international trade volumes between Europe and Asia, and other transport costs, especially those related to environmental regulations and canal fees.

Low shipping fuel prices and decreasing demand in the European market, affected by the coronavirus (COVID-19) pandemic, had a major impact on the shift from the Suez Canal to the Cape of Good Hope. Volumes of international trade between Europe and Asia have changed tremendously since the beginning of the COVID-19 pandemic. For instance, the demand for transport has decreased and the capacity dropped while at the same time, the fuel prices and freight rates have increased during the pandemic (Khan, Khurshid, & Köseoğlu, 2023; Loske, 2020; Xu, Yang, Chen, & Shi, 2021; Yang, Liu, & Chang, 2023). Currently, the freight rates have come back to values that are comparable to their levels before the crisis, but the transport demand is still low compared to the situation before the crisis (Pham Thi, Do, Paramaiah, Duong, Pham et al., 2023).

Another related factor affecting this change is new regional maritime environmental regulations and their impact on maritime transport costs. For instance, in December 2022, the 21 countries located around the Mediterranean Sea adopted at the International Maritime Organization, a new regulation to establish a Med Sea Sulfur Emission Control Area (Cullinane & Bergqvist, 2014; Sheng, Meng, & Li, 2019) The increase in transportation costs due to these regulations was estimated between 0.5 and 1% by EERA (2022). Such cost increase will be borne by shipping lines and customers when shipping cargoes through the Mediterranean Sea. These costs are related to the need for investments in new port infrastructure to adapt to the new regulation, such as shore power and liquefied natural gas (LNG) bunkering facilities, and additionally, to the cost of using cleaner fuels, such as low-sulfur marine diesel or LNG, that are higher than conventional fuels and will increase fuel costs.

Other factors, such as the Suez Canal Authority's decision to increase transit fees by 5%–15% (April 2021) to cover the increase in operating costs could also lead carriers to reconsider the Cape of Good Hope as an alternative route, in the near future, as this could save hundreds of thousand dollars per voyage in diverting ships towards the south of Africa (Wiesener, 2022). Hence, while the route around the Cape remains longer, it is generally considered to be safer, with fewer security risks and less congestion. Additionally, ships can avoid the tolls associated with using the Suez Canal, which can be a significant cost saving for shipping companies.

In light of the above-mentioned factors influencing maritime trade, there arises a pressing necessity for novel models to effectively evaluate the trade-off between utilizing the Suez Canal versus the route via the Cape of Good Hope. The paper aims to develop an approach to assess under which changes in market conditions the option of sailing through the Cape of Good Hope is more competitive than the Suez Canal. To do so, this paper addresses the single-liner shipping network design problem with speed optimization. It considers two subsets of ports, representing potential ports of call in two global regions such as Asia and Europe. Two possible splitting points separate the two sets of ports, namely the Suez Canal and the Cape of Good Hope leading to two potential routes for the vessels sailing between the two sets of ports. The problem consists in selecting the ports to be called at in the service, out of the candidate ports, and their sequence, to satisfy the deterministic weekly transport demand between any two considered ports, while determining the optimal sailing speed that can vary on every leg of the route. The studied problem considers a maximum allowable transit time for each commodity between every two ports, beyond which the corresponding transport demand cannot be satisfied, attempting to keep lead times at a reasonable level. The studied problem requires a weekly service, and the route that a vessel follows is referred to as a cycle. Given that a cycle can last several weeks and there is a weekly port call requirement, multiple vessels must be used. A mixed integer

programming model that maximizes the total profit generated by all the vessels is proposed. The profit consists of the difference between the generated revenue and the different incurred costs including port dues, handling costs, canal costs, chartering costs, and voyage costs. The resulting optimization problem is overly complex to solve. We propose a hybrid procedure exploiting a population-based heuristic and an exact procedure of a subproblem. A thorough analysis and several insights are provided through the consideration of three scenarios corresponding to the situation before COVID-19 pandemic (Case A), during the pandemic (Case B), and after the pandemic (Case C) respectively. The study focuses on these scenarios because of the availability of the corresponding data. It can be easily adapted to other situations, such as the current situation (by the time this paper was written) related to the war on Gaza.

It is worth noting that this paper addresses the problem of the design of single-liner shipping services, which can be seen as a subset of the broader network design problem and classified as a tactical issue (Dulebenets, Pasha, Abioye, & Kavoosi, 2021; Meng, Wang, Andersson, & Thun, 2014). Nevertheless, while theoretically significant, the overall network design in liner shipping has limited practical applicability (Psaraftis, 2019), primarily for two reasons. First, liner shipping operations typically involve joint services among multiple carriers (Merk, 2018), which link East–West and North–South routes, and each carrier participates in several individual agreements. This means that shipping companies rarely have full control over the entire network's design, and decisions are generally made at the level of single-liner shipping service design. Second, the choice of ports of call-in liner shipping is the outcome of negotiations between carriers and port authorities (Psaraftis 2019), who control the allocation of time slots to competing services. As a result, liner companies often rely on major hubs as buffer points, where containers typically stay between 2 to 5 days (dwell time) to coordinate with other interlinked services.

The remainder of this paper is structured as follows: Section 2 reviews the related literature. Section 3 describes the studied problem and develops the proposed model. Section 4 details the hybrid solution approach. In Section 5, we present numerical experiments including computational performance of the solution approach and several managerial insights. Finally, Section 6 provides concluding remarks and avenues for future research.

## 2. Literature review

Decisions related to the design, management, and operations of maritime liner services can be grouped into three categories (Christiansen, Fagerholt, & Ronen, 2004): strategic, focusing on the number of vessels in the fleet and the routes they have to sail on; tactical, focusing on fleet deployment decisions and vessel sailing speeds; and operational, such as selecting the cargoes to be accepted or rejected for each voyage. This study integrates strategic network design, tactical vessel sailing speed optimization, and operational commodity routing with maximum delivery dates into a unified framework. Moreover, it investigates alternative maritime routes to the Suez Canal, particularly in the current economic context, which features the importance of resilience in global liner shipping networks.

### 2.1. Liner shipping network design and routing

Liner shipping network design involves long term decisions to determine vessel routes, port call frequencies and service cycles (Cheaitou & Cariou, 2012) While network structures may remain fixed for years, changing economic and regulatory conditions require periodic adjustments. Network design and routing problems have their roots in the classical Traveling Salesman Problem (TSP) and the Vehicle Routing Problem (VRP), (Toth & Vigo, 2014). However, the adaptation of the TSP and the VRP to liner shipping introduces unique complexities, including vessel cyclic vessel schedules and port-to-port cargo flows.
Though these complexities might be similar in extensions of the VRP (such as the bus scheduling problem or the Pickup-and-Delivery problem), liner shipping has its own unique set of challenges due to the complex network of ports, the need for multi-ship coordination, and the long-term, global-scale nature of the operations, which make it different from traditional VRP extensions.

The initial works on ship routing and scheduling were mainly focused on tankers (Dantzig & Fulkserson, 1954; Flood, 1954) and then on cargo transportation (Olson, Sorenson, & Sullivan, 1969). Researchers have then proposed many routing models that are specific to the liner shipping problem (Christiansen et al., 2004). Foundational models by Chu, Kuo, and Shieh (2003) and Fagerholt (2004) prioritized route sequencing and cost minimization. Subsequent studies addressed real world challenges such as empty container repositioning (Shintani, Imai, Nishimura, & Papadimitriou, 2007), demand uncertainty (Chuang, Lin, Kung, & Lin, 2010), and multi-modal transportation (Tran, 2011). Models evolved further with the integration of transshipment options (Agarwal & Ergun, 2008; Dangaard Brouer, Desaulniers, & Pisinger, 2014), spot voyage (Moretti Branchini, Amaal Armentano, & Morabito, 2015), and time-constraints (Vad Karsten, Dangaard Brouer, & Desaulniers, 2016), and environmental strategies (Chen, Yi, Xin, & Zhang, 2023). Additionally, Pantuso, Fagerholt, and Hvattum (2014) conducted a review on maritime fleet size and mix problems with a slight focus on liner shipping problems. Wetzel and Tierney (2024) proposed a flexible cyclic liner shipping network design, allowing vessels to move between services to avoid inefficient connections without interruption to container flows.

Recent research has also explored network resilience in response to global disruptions such as the COVID-19 pandemic (Bai, Ma, & Zhou, 2023) and the impact of digital transformation (Raza, Woxenius, Vural, & Lind, 2023). Through these studies contributed considerably to a better understanding of liner shipping optimization, they fall short of quantitatively assessing major route changes. To address this, our study introduces a profit-maximization model to evaluate different maritime routes.

## 2.2. Speed models in maritime transportation

The tactical decision of vessel sailing speed optimization has been an important topic for researchers and practitioners since the global economic crisis of 2008, when slow steaming emerged as a cost saving strategy to reduce fuel consumption (Cariou, 2011; Degiuli, Martić, Farkas, & Gospić, 2021; Farkas, Degiuli, Martić, & Grlj, 2022; Farkas, Degiuli, Martić, & Mikulić, 2023). Studies on sailing speed optimization can be broadly classified into two categories: those optimizing economic performance (Alvarez, Longva, & Engebrethsen, 2010; Lagoudis, Lalwani, Naim, & King, 2002; Notteboom & Vernimmen, 2009; Ting & Tzeng, 2003) and those focusing on environmental performance (De, Kundu, Sheu, & Choi, 2023).

**Integrated Speed Optimization Models.** Ronen (2011) proposed a liner shipping model that finds a balance between the number of vessels to deploy in the service and the sailing speed. Meng and Wang (2011) proposed a mathematical model for liner shipping which identifies the optimal sailing speed, fleet deployment, and frequency of call. Cheaitou and Cariou (2012) built on the model of Ronen (2011) and accounted for auxiliary engine consumption, as well as a more accurate fuel consumption function. Using statistical analysis of historical data, Wang and Meng (2012) further validated the cubic correlation between sailing speed and main engine fuel consumption.

**Cargo Allocation and Service Levels.** More recent studies have explored speed optimization alongside cargo allocation decisions and service level constraints. For instance, Guericke and Tierney (2015) introduced a liner shipping model that jointly optimizes cargo allocation and vessel speed while considering service levels. Their model highlights the trade-off between cargo revenues, fuel costs, and customer satisfaction under varying speed decisions. See also Tierney, Ehmke,

Campbell, and Müller (2019) for an extension and application on real-world data from different liner shipping routes. Koza (2019) integrated the interdependency between liner shipping and cargo allocation into a model that considers a payload-dependent fuel consumption function.

**Buffer time consideration.** In transport services and maritime transport in particular, delays are inevitable. Hence, during schedule construction, companies include buffer or slack time. Mulder, van Jaarsveld, and Dekker (2019) proposed an approach integrating both the development and execution of the trip schedule with speed optimization. This approach results in considerable cost saving over the considered route. Mulder and Dekker (2019) further enhanced this by demonstrating a mixed-integer programming model that optimizes recovery policy actions and buffer time allocation, significantly reducing the total costs associated with delays and recovery actions.

**Environmental and Economic Considerations.** Many authors have focused on the liner shipping sailing speed from an environmental perspective. For example, Corbett, Wang, and Winebrake (2009) studied carbon dioxide (CO₂) reduction policies from maritime shipping calling at US ports, and more particularly fuel tax and vessel speed reduction mandates. Fagerholt, Laporte, and Norstad (2010) introduced a model for optimizing vessel speed with the goal of reducing fuel-related emissions in a liner service characterized by specific ports, their sequences, and associated time windows. Cariou and Cheaitou (2012) conducted a comparative study examining the impact of bunker taxes and regional speed limits on CO₂ emissions originating from liner services. Cariou (2011) concentrated on the influence of slow steaming strategies within liner shipping on CO₂ emissions. More recently, Dulebenets (2018) incorporated sailing speed as a decision variable when addressing the vessel scheduling problem within an alliance's liner service, with a specific focus on cost and emissions. De, Choudhary, Turkay, and Tiwari (2021) considered fuel bunkering policies under dynamic fuel price scenarios and fuel consumption in the liner shipping problem with speed deviations. Herrera Rodriguez, Agrell, Manrique-de Lara-Penate, and Trujillo (2022) integrated vessel sailing speed considerations into a bi-criterion model, striving to minimize shipping costs and transit time while adhering to environmental constraints. The study of Cariou, Parola, and Notteboom (2019) concluded that, recently, the emissions from global container shipping dropped mainly due to the reduction in the vessels' sailing speed and the changes in technology and network design.

**Other extensions.** Psaraftis and Kontovas (2014) discussed different important factors playing a role in ship speed optimization such as the used function for fuel consumption (i.e. cubic function or other forms), and the effect of weather conditions. Mulder and Dekker (2014) introduced an integrated problem encompassing fleet design, ship scheduling, and cargo routing, where determinations were made regarding sailing routes, the selection of vessel types for each route, cargo flows, and vessel sailing speeds. In the realm of RoRo shipping, Andersson, Fagerholt, and Hobbesland (2015) devised a modeling approach that fused speed optimization and ship routing. Wang and Meng (2015) proposed an additional optimization model that combines the selection of bunkering ports and sailing speed for a given liner service, accounting for variations in sailing speed. Aydin, Lee, and Mansouri (2016) formulated a dynamic stochastic speed optimization model for liner services, incorporating bunkering decisions within the context of predefined liner routes. Cheaitou and Cariou (2017) studied sailing speed optimization decisions and their effect on inventory management cost by considering two scenarios: centralized and decentralized supply chains. Mancini and Stecca (2018) developed a Large Neighborhood Search-based Matheuristic for planning tourist cruise itineraries, where vessel speeds and port visits were optimized to minimize total traveling and port costs. While the focus was on tourist cruises, their work highlights the applicability of speed optimization techniques across other maritime services. Asghari, Jaber, and e hashem (2023) proposed a model that allows rerouting and skipping ports in response to network disruptions while optimizing speed on arcs in the liner shipping
network design problem. They developed a Crowd-Learning Particle Swarm Optimization approach to solve large-scale instances.

While the above studies and other existing models incorporate speed optimization, they often lack maximum delivery time constraints or assume constant sailing speeds across network arcs. Our study addresses these limitations by allowing varying vessel speeds between network edges while respecting commodity-specific delivery deadlines.

## 2.3. Liner shipping models with commodity pickup and delivery

The operational level of decisions covers the commodity routing, i.e., the pickup and delivery of commodities while considering a maximum delivery date. This aspect relates to certain widely recognized extensions of the TSP and VRP models, such as the collection and delivery of goods (Hernandez-Perez, Rodriguez-Martin, & Salazar-Gonzalez, 2009; Hernandez-Perez & Salazar-Gonzalez, 2004), as well as the dial-a-ride problem (Cordeau & Laporte, 2007; Liu, Luo, & Lim, 2015).

Within maritime transportation, numerous research papers have tackled the pickup and delivery challenge particularly in the context of liner shipping. For instance, Karlaftis, Kepapsoglou, and Sambracos (2009) studied a containership routing problem involving pickups and deliveries between a central hub and multiple spoke ports. They employed a capacitated VRP formulation that accounted for both pickups, deliveries, and associated time constraints. Plum, Pisinger, Salazar-Gonzalez, and Sigurd (2014) introduced innovative formulations for pickup and delivery problems, utilizing both arc-flow and path-flow models. Zheng, Meng, and Sun (2014) proposed a two-phase hub-and-spoke liner shipping design problem. In phase 1, the hub location and feeder allocation problem is solved, and in phase 2, the ship route design with ship fleet deployment is formulated as a pickup-and-delivery routing problem. Vad Karsten, Pisinger, Ropke, and Dangaard Brouer (2015) presented two alternative formulations, namely arc-flow and path-flow models, to address the pickup and delivery challenge involving multiple commodities in the design of container liner shipping networks. Their formulations incorporated a critical time constraint, restricting the duration that a specific commodity could traverse the network. Wang, Zhang, and Wang (2016) devised a comprehensive model that encompassed three key decision areas: cargo allocation, optimization of ship capacity, and scheduling of ship routes. Zheng, Sun, and Zhang (2016) proposed a two-stage model, wherein the initial stage involved a liner shipping pickup and delivery model with multiple route options. The outcomes of this first stage informed the second stage, which focused on determining perceived container leasing prices at various ports. More specifically, Cuesta, Andersson, Fagerholt, and Laporte (2017) put forth solutions for single and multiple vessel routing problems, introducing selective pickups and deliveries tailored for maintenance equipment destined for offshore oil platforms.

Other authors have also explored routing and scheduling problems in the maritime sector, addressing pickup and delivery concerns for oil tankers (Hennig et al., 2012; Hennig, Nygreen, Furman, & Song, 2015), as well as the transportation of chemicals (Jevne Arnesen, Gjestvang, Wang, Fagerholt, Thun et al., 2017). More recently, Brahimi, Cheaitou, Cariou, and Feillet (2021) proposed a mathematical model aimed at designing service liner networks while incorporating speed optimization. In their study, they demonstrated that when delivery date constraints were absent for commodities, the optimal sailing speed remained constant within the cycle. Additionally, Yahyaoui, Dahmani, and Krichen (2021) developed a routing model for tankers, addressing split pickups and deliveries in the context of crude oil transportation.

Finally, Brouer, Vad Karsten, and Pisinger (2016, 2017), Christiansen, Fagerholt, Nygreen, and Ronen (2013), Christiansen et al. (2004), Christiansen, Hellsten, Pisinger, Sacramento, and Vilhelmsen (2020), Kjeldsen (2011), Meng et al. (2014), Psaraftis and Kontovas (2013), Tran and Haasis (2015) reviewed and categorized the works on liner shipping routing and scheduling and speed optimization in

maritime transportation. Related studies include the discussion of uncertainty in liner shipping design and scheduling. In particular, Berling and Martínez-de Albéniz (2016) solve a liner shipping problem with speed optimization and demand uncertainty. Later, Liu, Liu, Chu, Zhu, and Zheng (2020) proposed a two-stage stochastic and non-linear programming formulation to solve a liner shipping problem with speed optimization and uncertain demand. For a general survey of uncertainty in maritime ship routing and scheduling see Kciuk, Kuhlemann, Tierney, and Koberstein (2023)

We can observe that studies integrating speed optimization, time constraints on commodity delivery, and network design remain limited. For example, Brahimi et al. (2021) optimized vessel speeds but did not account for time constraints, while others simplified the problem by assuming constant speeds (Mulder & Dekker, 2014). Our study bridges this gap by introducing a comprehensive model that considers all three decision levels.

## 2.4. Solution approaches for liner shipping models

While the works mentioned above considered different decision levels in liner shipping, the solution approaches that have been considered for the proposed models can be categorized as exact, heuristic, or mixed approaches. For instance, Meng and Wang (2011) suggested linearization techniques to approximate their model with a mixed-integer linear program and they solve it then using branch-and-bound based e-optimal algorithm. Plum et al. (2014) applied a Branch-and-Cut-and-Price algorithm to achieve optimality in solving instances involving as many as 25 ports. Lin and Tsai (2014) developed a solution methodology using a Lagrangian relaxation technique and local search while Agarwal and Ergun (2008) solved their model using a combination of greedy heuristic, column generation, and Branchers decomposition. However, most of the other works use heuristic approaches (Andersson et al., 2015; Dangaard Brouer et al., 2014; Karlaftis et al., 2009). For instance, Mulder and Dekker (2014) proposed a decomposition approach that generates routes, and determines speed levels, fleet design, and routes cargo using an iterative way. For the ship routing problem, they used a genetic algorithm.

In this work, we propose a linear programming formulation for the liner shipping with speed optimization between edges and we propose a hybrid solution approach that uses a population-based heuristic and the exact approach to solve the problem.

## 2.5. Applications on alternative routes to the Suez Canal

Numerous studies have examined alternative routes to the Suez Canal route for Asia–Europe trade, particularly in light of changes in market conditions spurred by the Arctic ice melt, which has led to increased interest in the North Sea route (NSR), rail alternatives, and Cape of Good Hope route.

Theocharis, Pettit, Sanchez Rodrigues, and Haider (2018) noted that 20 out of 24 articles on the feasibility of the NSR since 2011 focused on liner shipping markets and included a comparison with the Suez Canal Route (Cariou, Cheaitou, Faury, & Hamdan, 2021; Erikstad & Ehlers, 2012; Lasserre, 2014; Liu & Kronbak, 2010; Meng, Zhang, & Xu, 2017; Verny & Grigentin, 2009; Vukic & del Mar Cerban, 2022). The main findings of these studies are that, despite shorter sailing distances (up to 40% compared with the Suez Canal route), this has not translated into significant economic benefits and the number of containerships transiting through the NSR has remained limited (Aksenov et al., 2017; Lee & Kim, 2015; Verny & Grigentin, 2009; Zhu, Fu, Ng, Luo, & Ge., 2018). Challenges such as navigating remote geographical locations and the lack of sufficient cargo to pick up along the NSR are probably the most important ones. Zhu et al. (2018) and Cariou et al. (2021) further suggest that environmental costs may also be higher, due to the need to use smaller vessel sizes, and to operate vessels at lower load factors. Meng et al. (2017) then argue, in line with Cariou et al.
(2021), that the NSR does not really offer an alternative, due to sea ice, weather, and geographical conditions, as the provision of regular east/west services requires a fixed schedule and frequency, conditions that are difficult to set when there is limited predictability of future ice conditions along the route. This finding is also in line with Vukic and del Mar Cerban (2022) who confirm that the NSR, despite having its high potential is less competitive than the conventional Suez transit.

On the other hand, Verny and Grigentin (2009) and Moon, Kim, and Lee (2015) concluded that the rail alternative could offer better competitiveness, especially for the most valuable cargoes (Psaraitis & Kontovas, 2010). As underlined by Yang, Jiang, and Ng (2018), this means that despite their limited capacity and higher transport costs (Lasserre, Huang, & Mottet, 2018), railways can offer lower transit times and have the main advantage of not being affected by oil price fluctuations or by the upcoming new environmental regulations of the International Maritime Organization (IMO), such as the global sulfur cap (January 2020). Railways transit time is always shorter than the NSR (Lasserre et al., 2018; Psaraitis & Kontovas, 2010), while the rail CO₂ emissions remain significantly higher than Suez Canal route due to limited capacity (Yang et al., 2018).

Comparisons with the Cape of Good Hope option, which is the focus of our study, are rare, although the uncertainty associated with the transit via the NSR and the limited transport capacity of the rail option mean that the Cape of Good Hope route has emerged as the main competitor of the Suez Canal passage (Chou, Hsu, & Chang, 2020; Vukic & del Mar Cerban, 2022) under certain market conditions. First, during a recession in the oil market, when lower bunker prices favor the Cape of Good Hope option, and second, when there is a significant increase in the Suez Canal transit toll. According to the authors, the Cape of Good Hope option would become viable at a bunker price of 70 USD per MT, if all other components remain stable. Gao and Lu (2019) also argue that the blockage of the Suez Canal route due to natural disasters, piracy, and war could lead to an increase in the use of alternative routes, although this competitiveness could remain limited for products with short life cycles (Herrera Rodriguez et al., 2022). Our study provides an economic analysis of this route in comparison with the Suez Canal under varying economic contexts.

## 2.6. Research gap, research questions and study contribution

Based on the previous discussions, we can conclude that despite extensive research on liner shipping optimization, research gaps persist. Firstly, existing studies often address speed optimization or commodity routing separately, without fully integrating these decisions into network design. Secondly, models that incorporate delivery time constraints typically assume constant vessel speeds across network edges. Finally, limited attention has been given to alternative routes, particularly the Cape of Good Hope, under dynamic economic conditions like the COVID-19 pandemic.

To address these gaps, this study answers the following questions. How can strategic network design, tactical speed optimization, and operational commodity routing decisions be integrated into a single model and be solved efficiently? What are the economic impacts of structural changes, such as the COVID-19 pandemic, on route selection? Under what conditions can alternative routes to the Suez Canal, such as the Cape of Good Hope, become competitive?

This study's main contributions include:

1. Developing a comprehensive model that integrates liner network design, variable vessel speeds, and commodity routing with time constraints.

2. Proposing an efficient hybrid solution approach combining a population-based heuristic with exact optimization techniques.

3. Conducting an economic analysis of maritime routes (Suez Canal vs. Cape of Good Hope) under varying economic contexts and considering demand uncertainty.

The positioning of our work and its contributions are illustrated on the layered framework diagram of Fig. 1

# 3. Problem description and formulation

In Section 3.1, we introduce the Single Liner Service Design problem with Edge-based Speed Optimization (SLSDES). In Section 3.2, we present a Mixed Integer Linear Programming formulation that serves as the foundation for the hybrid solution approach outlined in Section 4.

## 3.1. Problem description

We define a set $\mathcal{P}$ of ports, divided into two subsets $\mathcal{P}^w$ and $\mathcal{P}^e$, separated by what we call a splitting point $sp$. Vessels must traverse the splitting point $sp$ when traveling from a port in $\mathcal{P}^w$ to a port in $\mathcal{P}^e$ and vice versa. The splitting point in our data sets is either the Suez Canal or the Cape of Good Hope. Ports in $\mathcal{P}^w$ include those from the Atlantic Ocean and the Mediterranean Sea, whereas $\mathcal{P}^e$ consists of ports in Asia. The distance between two ports $i$ and $j$ within the same subset $\mathcal{P}^w$ or $\mathcal{P}^e$ is represented by $D_{i,j}$. The distance from any port $i$ in $\mathcal{P}$ to the splitting point $sp$ is denoted by $D_{i,sp}$ or $D_{sp,i}$, depending on the travel direction.

We assume a deterministic maximum weekly demand that the service can accommodate between any two ports in the network $i, j \in \mathcal{P}$, where $i$ and $j$ denote the source and destination ports. This demand is called a commodity. Accordingly, we define the set $\mathcal{K}$ of commodities, where each commodity $k \in \mathcal{K}$ is characterized by an originating port $s_k$, a destination port $d_k$, a maximum weekly containers $w_k > 0$ (in TEUs), and a revenue $\phi_k$ per container (in USD). Due to the large volume of containers transported per vessel, which could number in the thousands, we do not apply any integrality restrictions on the number of containers managed by the service. Each commodity $k \in \mathcal{K}$ is also constrained by a maximum allowable transit time $CT_k$.

The aim is to design a container ship liner service that maximizes the collected weekly profit where demand is met fully or partially. The service has specific characteristics: The shipping company provides a weekly service, and the route that a vessel follows is referred to as a cycle. Given that a cycle can last several weeks and there is a weekly port call requirement, multiple vessels must be used. Vessels within the same service are considered identical, with each ship visiting the same sequence of ports and each visit spaced one week apart from the next vessel. The vessels have a maximum load capacity defined as $Q$ (in TEUs). Moreover, each port included in the service may receive up to two visits from the same vessel within a single cycle. The time spent at a port $i \in \mathcal{P}$ for loading and unloading operations is called $PT_i$. To maximize the profit, the company focuses on three main strategies: (1) it specifies which ports of call to visit and their sequence, (2) it sets the sailing speed between any two ports while respecting the time restriction for each commodity, (3) it determines the quantities of containers to be transported to and from the selected ports.

The speed is a major decision variable of the problem. It can be different between every pair of ports. In order to take into account these facts, we assume that speed can be chosen in a discrete set $\mathcal{R}$ of values within a range $[r^{\min}, r^{\max}]$. This is a common choice when dealing with speed optimization models (Christiansen et al., 2020).

The profit is derived from revenues, which are offset by four categories of expenses. Revenue is accrued through the transportation of containers. A breakdown of these costs is provided as follows, with the financial calculations performed on a weekly basis due to the routine port calls. The first category encompasses costs associated with ports, which include port dues and handling fees. Port dues are calculated based on the duration the vessel spends at the port and the vessel's overall length. These dues are computed using a specific rate per meter per hour, expressed as $PT_i \times LOA \times CHM_i$, where LOA represents the vessel's total length and $CHM_i$ denotes the specific dockage cost at port $i$. This cost is represented as $\gamma_i$. Handling expenses are related to the volume of cargo loaded or unloaded, and vary according to the specific loading and unloading costs at each port, denoted as $LC_i$ and $UC_i$, respectively, for each TEU processed.
<figure><img src="image_5.png" /><figcaption>Fig. 1. Layered framework diagram for paper positioning and contributions.</figcaption></figure>

The second category of costs includes the canal fees incurred each time the vessel transits through the Suez Canal. These fees are calculated by multiplying the total number of containers transiting the canal by a unitary cost, denoted as $C_{sp}$. Notably, the model sets $C_{sp} = 0$ when the route bypasses the Suez Canal via the Cape of Good Hope. The third category of costs pertains to the chartering of the vessel, which encompasses expenses for investment or charter fees, crew salaries, insurance, maintenance, and other related costs. These are consolidated into a daily fixed cost $\Gamma$ (in USD), applicable to each vessel participating in the service.

The final category of costs is associated with the voyage itself, particularly fuel consumption, which involves both the main and auxiliary engines. The main engine, which propels the ship, incurs fuel costs that are influenced by the speed of the vessel and the distance covered. These costs are formulated as $C_M^b \times \mu_M \times D_{ij} \times r^2$, where $D_{ij}$ is the distance between ports $i$ and $j$, $r$ is the sailing speed between ports $i$ and $j$, $C_M^b$ is the fuel price for the main engine (USD per ton), and $\mu_M$ is a parameter specific to the vessel. For further details on these calculations, see Brahimi et al. (2021). Meanwhile, the cost for the auxiliary engine, which powers onboard systems such as lighting and air conditioning, is calculated based on the formula $C_A^b \times \mu_A \times (\text{total time})$, where $\mu_A$ is a vessel-specific parameter, $C_A^b$ is the auxiliary engine fuel price (USD per ton), and “total time” encompasses both the time spent in ports and the sailing duration.

In Table 1, we recapitulate the data involved in the definition of the SLSDS.

## 3.2. Mathematical formulation

Before presenting our mathematical formulation for the SLSDS, we introduce a few additional notation. In order to deal with the fact that ports can be visited twice, we duplicate ports. Every port $i \in P$ has a copy $i'$. The set of these copies is denoted $P'$. To simplify notation, we also consider the splitting point as a port and this port is also duplicated. The duplicated splitting point is $sp'$. The set of all ports is then $V$ with $V = P \cup P' \cup \{sp, sp'\}$. Ports in $P^w$ and their copies are denoted $V^w$. Ports in $P^e$ and their copies are denoted $V^e$.

Every notation involving a port is straightforwardly extended to the duplicate ports. Regarding the splitting points, their port time $PT_i$ is zero for the Cape of Good Hope and is equal to the traversal time for

<table><caption>Table 1
Input Data involved in the definition of the SLSDS.</caption><tr><td>P</td><td>set of ports</td></tr><tr><td>P<sup>w</sup></td><td>set of ports on one side of the splitting point (P<sup>w</sup> &sub; P)</td></tr><tr><td>P<sup>e</sup></td><td>set of ports on the other side of the splitting point (P<sup>e</sup> &sub; P)</td></tr><tr><td>R</td><td>set of discrete speed values</td></tr><tr><td>sp</td><td>splitting point</td></tr><tr><td>D<sub>i,j</sub></td><td>distance between two ports or between a port and the splitting point (in nautical miles)</td></tr><tr><td>K</td><td>Set of commodities</td></tr><tr><td>s<sub>k</sub></td><td>source port for commodity k &sub; K</td></tr><tr><td>d<sub>k</sub></td><td>destination port for commodity k &sub; K</td></tr><tr><td>w<sub>k</sub></td><td>maximal weekly number of containers for commodity k &sub; K (in TEUs)</td></tr><tr><td>&phi;<sub>k</sub></td><td>unitary revenue for commodity k &sub; K (in USD)</td></tr><tr><td>CT<sub>k</sub></td><td>maximum allowable transit time for commodity k &sub; K (in hours)</td></tr><tr><td>Q</td><td>vessel capacity (in TEUs)</td></tr><tr><td>PT<sub>i</sub></td><td>port time at port i &sub; P (in hours)</td></tr><tr><td>R</td><td>set of acceptable vessel speeds (in knots)</td></tr><tr><td>r<sub>i</sub></td><td>port dues for port i &sub; P (in USD)</td></tr><tr><td>LC<sub>i</sub></td><td>loading costs at port i &sub; P (in USD&times;TEU<sup>-1</sup>)</td></tr><tr><td>UC<sub>i</sub></td><td>unloading costs at port i &sub; P (in USD&times;TEU<sup>-1</sup>)</td></tr><tr><td>C<sub>sp</sub></td><td>unitary cost for the splitting point sp (in USD&times;TEU<sup>-1</sup>)</td></tr><tr><td>&Gamma;</td><td>daily chartering cost (in USD&times;day<sup>-1</sup>)</td></tr><tr><td>&mu;<sub>M</sub></td><td>cost coefficient for the main engine (in USD)</td></tr><tr><td>&mu;<sub>A</sub></td><td>cost coefficient for the auxiliary engine (in USD)</td></tr></table>

the Suez Canal; their port due $r_i$ is zero in all cases (but remembering that a unitary cost $C_{sp}$ is due for every TEU passing through the splitting point).

The following assumptions regarding the model parameters have been considered:

* Single problem network design without transshipment. However, any heterogeneous transshipment demand can be included in the demand values.

* In the same service, only one size of homogeneous vessels with the same propulsion system are used.

* A single fuel type is used.

* Demand does not change with transit time but a time limit is considered.
* The studied problem requires a weekly service. Given that a cycle can last several weeks and there is a weekly port call requirement, multiple vessels must be used.

* We assume that the sailing speed can be chosen in a discrete set of values within a range.

* Moreover, all the parameters defined in Table 1 are fixed for each solved instance of the problem. However, the model can handle any new values for these parameters that define new instances of the problem.

Decision variables are defined as follows:

* $X_{i,j} \in \{0,1\}$: binary variable equals to 1 if the vessel goes from port $i$ to port $j$, 0 otherwise.

* $X'_{i,j} \in \{0,1\}$: binary variable equals to 1 if the vessel goes from port $i$ to port $j$ at speed $r$, 0 otherwise.

* $X_{i,j}^k \in \{0,1\}$: binary variable equals to 1 if commodity $k$ is onboard when the vessel goes from port $i$ to port $j$, 0 otherwise.

* $Y_i \in \{0,1\}$: binary variable equals to 1 if port $i$ is included in the service, 0 otherwise.

* $T_{i,j} \ge 0$: travel time from port $i$ to port $j$.

* $\mathcal{T}_{i,j}^k \ge 0$: travel time for commodity $k$ from port $i$ to port $j$.

* $F_{i,j}^k \ge 0$: quantity of commodity $k$ onboard from port $i$ to port $j$.

* $L^k \ge 0$: total quantity loaded from both the original and duplicate origin (source) ports for commodity $k$.

* $U^k \ge 0$: total quantity unloaded at both the original and duplicate destination ports for commodity $k$.

* $N \in \mathbb{N}$: number of vessels used in the service.

* $Z_i \in \{0,1\}$: binary variable arbitrarily defining if port $i$ is the starting port for the service.

* $V_i \ge 0$: variable indicating the sequence of port $i$ in the service.

In order to avoid obtaining an empty tour, we force the cycle to contain at least two ports. Note that it is then possible to have a solution of negative profit. We finally introduce a large number $M$ to linearize logical constraints. The objective function of the model is given by:

$$ \max \text{ Profit} = \text{Revenue} - (\text{Port dues} + \text{Handling cost} + \text{Canal Cost} + \text{Chartering cost} + \text{Voyage cost}), \quad (1) $$

where

$$ \text{Revenue} = \sum_{k \in \mathcal{K}} \phi_k L^k, \quad (2) $$

$$ \text{Port dues} = \sum_{i \in \mathcal{V}} Y_i Y_i, \quad (3) $$

$$ \text{Handling cost} = \sum_{k \in \mathcal{K}} LC_{s_k} L^k + \sum_{k \in \mathcal{K}} UC_{d_k} U^k, \quad (4) $$

$$ \text{Canal cost} = \sum_{k \in \mathcal{K}} \sum_{j \in \mathcal{V}} \left( C_{sp} F_{j,sp}^k + C_{sp'} F_{j,sp'}^k \right), \quad (5) $$

$$ \text{Chartering cost} = 7 \times \Gamma \times N, \quad (6) $$

$$ \text{Voyage cost} = \sum_{j \in \mathcal{V}} \sum_{i \in \mathcal{V}} \sum_{r \in \mathcal{R}} \left( C_M^b \mu_M D_{i,j} r^2 + C_A^b \mu_A \frac{D_{i,j}}{r} \right) X_{i,j}^r + C_A^b \mu_A \sum_{i \in \mathcal{V}} PT_i Y_i, \quad (7) $$

The objective function (1) represents the total profit from the service. The total profit is calculated as the difference between the total revenue and the total costs, as detailed in Eqs. (2) through (7).

The model is subject to the following four sets of constraints:

**Service design constraints**

$$ \sum_{j \in \mathcal{V}} X_{i,j} = Y_i \quad \forall i \in \mathcal{V}, \quad (8) $$

$$ \sum_{i \in \mathcal{V}} X_{i,j} = Y_j \quad \forall j \in \mathcal{V}, \quad (9) $$

$$ X_{i,j} = 0 \qquad \forall i \in \mathcal{V}^w, j \in \mathcal{V}^e, \quad (10) $$

$$ X_{i,j} = 0 \qquad \forall i \in \mathcal{V}^e, j \in \mathcal{V}^w, \quad (11) $$

$$ \sum_{i \in \mathcal{V}} Y_i \ge 1, \qquad (12) $$

$$ Z_i \le Y_i \qquad \forall i \in \mathcal{V}, \quad (13) $$

$$ \sum_{i \in \mathcal{V}} Z_i = 1, \qquad (14) $$

$$ M Z_j + V_j \ge V_i + 1 - M(1 - X_{i,j}) \quad \forall i \in \mathcal{V}, j \in \mathcal{V}, \quad (15) $$

Constraints (8) and (9) link variables $X_{i,j}$ and $Y_i$. They also ensure flow balance at ports. Constraints (10) and (11) prevent vessels from going directly to a port that is not on the same side of the splitting point. Constraint (12) forces at least one port to be selected in the service. This constraint helps quantify losses if the service is not profitable; removing it results in zero profit in such cases. Constraints (13) force to select the (arbitrarily-defined) first port of the service among the ports that are visited. Constraint (14) guarantees that exactly one such port is defined. Inequalities (15) represent subtour elimination constraints.

**Commodity flow constraints**

$$ \sum_{k \in \mathcal{K}} F_{i,j}^k \le Q X_{i,j} \quad \forall i \in \mathcal{V}, j \in \mathcal{V}, \quad (16) $$

$$ \sum_{j \in \mathcal{V}} F_{i,j}^k = \sum_{j \in \mathcal{V}} F_{j,i}^k \qquad \forall i \in \mathcal{V} \setminus \{s_k, s'_k, d_k, d'_k\}, \quad k \in \mathcal{K}, \quad (17) $$

$$ L^k = \sum_{j \in \mathcal{V}} \left( F_{s_k,j}^k + F_{s'_k,j}^k \right) \quad \forall k \in \mathcal{K}, \quad (18) $$

$$ U^k = \sum_{j \in \mathcal{V}} \left( F_{j,d_k}^k + F_{j,d'_k}^k \right) \quad \forall k \in \mathcal{K}, \quad (19) $$

$$ L^k \le w_k \quad \forall k \in \mathcal{K}, \quad (20) $$

Constraints (16) ensure that the flow of commodities is supported by the service. In addition, they ensure that the total number of transported containers does not exceed the vessel capacity. Constraints (17) ensure flow conservation. Constraints (18) define the total loaded quantity for each commodity, factoring in the flows from both the original and duplicated source ports. In a similar vein, Constraints (19) establish the total unloaded quantity based on the flows at both the original and duplicated destination ports. Constraints (20) ensure that the loaded quantity does not exceed the maximum weekly containers for each commodity.

**Constraints for speed, time variables and number of vessels**

$$ \sum_{r \in \mathcal{R}} X_{i,j}^r = X_{i,j} \quad \forall i \in \mathcal{V}, j \in \mathcal{V}, \quad (21) $$

$$ T_{i,j} = \sum_{r \in \mathcal{R}} \frac{D_{i,j}}{r} X_{i,j}^r \quad \forall i \in \mathcal{V}, j \in \mathcal{V}, \quad (22) $$

$$ N \ge \frac{1}{168} \left( \sum_{j \in \mathcal{V}} \sum_{i \in \mathcal{V}} T_{i,j} + \sum_{i \in \mathcal{V}} PT_i Y_i \right), \quad (23) $$

Constraints (21) impose that a single speed $r$ is selected between two ports $i$ and $j$. Constraints (22) determine the time taken to travel from port $i$ to port $j$ based on the chosen speed value. Constraint (23) determines the number of vessels utilized in the service.

**Transit time constraints**

$$ F_{i,j}^k \le w_k X_{i,j}^k \quad \forall i \in \mathcal{V}, j \in \mathcal{V}, k \in \mathcal{K}, \quad (24) $$

$$ \mathcal{T}_{i,j}^k \ge T_{i,j} - M(1 - X_{i,j}^k) \quad \forall i \in \mathcal{V}, j \in \mathcal{V}, k \in \mathcal{K}, \quad (25) $$

$$ \mathcal{T}_{i,j}^k \le T_{i,j} \quad \forall i \in \mathcal{V}, j \in \mathcal{V}, k \in \mathcal{K}, \quad (26) $$

$$ \sum_{i \in \mathcal{V}} \sum_{j \in \mathcal{V}} \left( \mathcal{T}_{i,j}^k + PT_i X_{i,j}^k \right) + PT_{d_k} \le CT_k \quad \forall k \in \mathcal{K}, \quad (27) $$
Constraints (24) link binary variables $X_{i,j}^k$ with commodity flow variables $F_{i,j}^k$. Constraints (25) and (26) set the commodity time $T_{i,j}^k$ to the transportation time $T_{i,j}$. Constraints (27) ensure that the total travel time for a commodity does not exceed its maximum allowable duration $CT_k$.

### 3.3. Valid inequalities

In this section we propose valid inequalities that break symmetries.

$$ \sum_{j \in \mathcal{V}} (F_{j,s_k}^k + F_{j,s'_k}^k) = 0 \qquad \forall k \in \mathcal{K}, \quad (28) $$

$$ \sum_{j \in \mathcal{V}} (F_{d_k,j}^k + F_{d'_k,j}^k) = 0 \qquad \forall k \in \mathcal{K}, \quad (29) $$

$$ Y_{l'} \le Y_l \qquad \forall i \in \mathcal{P} \quad (30) $$

$$ X_{i,j} = 0 \qquad \forall i \in \mathcal{V}, \quad (31) $$

$$ X_{i,l'} = 0 \qquad \forall i \in \mathcal{P} \cup \{sp\}, \quad (32) $$

$$ X_{i',j} = 0 \qquad \forall i \in \mathcal{P} \cup \{sp\}, \quad (33) $$

$$ \sum_{0 \le j \le i} Z_j \ge Y_i \qquad \forall i \in \mathcal{V} \quad (34) $$

$$ \sum_{k \in \mathcal{K}} X_{i,j}^k \ge X_{i,j} \qquad \forall j \in \mathcal{V}, \quad (35) $$

$$ F_{i,j}^k \ge X_{i,j}^k \qquad \forall i \in \mathcal{V}, \ j \in \mathcal{V}, \ k \in \mathcal{K}, \quad (36) $$

$$ \sum_{k \in \mathcal{K}} F_{i,j}^k = 0 \qquad \forall i \in \mathcal{V}^e, \ j \in \mathcal{V}^w, \quad (37) $$

$$ \sum_{k \in \mathcal{K}} F_{j,j}^k = 0, \qquad \forall i \in \mathcal{V}^w, \ j \in \mathcal{V}^e, \quad (38) $$

$$ \sum_{j \in \mathcal{V}^w} X_{j,i}^k = \sum_{j \in \mathcal{V}^e} X_{i,j}^k \qquad \forall i \in \{sp, sp'\}, \ k \in \mathcal{K}, \quad (39) $$

$$ \sum_{j \in \mathcal{V}^e} X_{j,i}^k = \sum_{j \in \mathcal{V}^w} X_{i,j}^k \qquad \forall i \in \{sp, sp'\}, \ k \in \mathcal{K}, \quad (40) $$

$$ \sum_{j \in \mathcal{V}^w} F_{j,i}^k = \sum_{j \in \mathcal{V}^e} F_{i,j}^k \qquad \forall i \in \{sp, sp'\}, \ k \in \mathcal{K}, \quad (41) $$

$$ \sum_{j \in \mathcal{V}^e} F_{j,i}^k = \sum_{j \in \mathcal{V}^w} F_{i,j}^k \qquad \forall i \in \{sp, sp'\}, \ k \in \mathcal{K}, \quad (42) $$

$$ V_i \le V_{i'} \qquad \forall i \in \mathcal{P}, \quad (43) $$

$$ T_{i,j} = 0, \qquad \forall i \in \mathcal{V}, \quad (44) $$

$$ T_{i,j}^k = 0 \qquad \forall i \in \mathcal{V}, \ k \in \mathcal{K}, \quad (45) $$

$$ X_{i,j}^k = 0 \qquad \forall i \in \mathcal{V}, \ k \in \mathcal{K}, \quad (46) $$

$$ X_{i',j}^k = 0 \qquad \forall i \in \mathcal{P} \cup \{sp\}, \ k \in \mathcal{K}, \quad (47) $$

$$ X_{i,j}^k \le X_{i,j} \qquad \forall i \in \mathcal{V}, \ j \in \mathcal{V}, \ k \in \mathcal{K}, \quad (48) $$

Constraints (28) and (29) ensure that commodities neither are delivered to their source port nor depart from their destination port. Constraints (30) ensure that original ports are prioritized before their duplicated counterparts. Constraints (31) explicitly forbid self-loops. Constraints (32) and (33) forbid connecting a port and its duplicate. Constraints (34) force to set the first port of the service as the port with the smallest index among those that are visited. Constraint (35) ensures that a selected path, given by $X_{i,j}$, is used to transport at least one commodity. Constraint (36) mandates that if $X_{i,j}^k$ is equal to 1, a quantity $F_{i,j}^k$ is transported. Constraints (37) and (38) ensure that no quantities are transported directly between any two ports separated by the splitting point without passing through the splitting point itself. Constraints (39)–(42) are flow balance constraints at the splitting point. Constraints (43) confirm that the rank of an original port is lower than its duplicated version. Constraints (44)–(46) mandate the diagonal (going from and to the same port) to be zero. Constraints (47) prevent moving commodity $k$ from port $i$ to its duplicated port. Note that $i'$

is the duplicated (copy) port $i'$, $i' \in \mathcal{P}'$ as defined in Section 3.2. Constraints (48) define an upper bound for $X_{i,j}^k$ based on $X_{i,j}$, such that if $X_{i,j}$ is zero, no commodity flow is allowed on that arc.

## 4. Solution approach

The current problem is complex and challenging to address it using exact solution approaches. This complexity is supported by (i) the employment of a decision variable for the speed on each arc and (ii) the incorporation of the maximum commodity time constraint. For this reason, we have adopted a hybrid solution strategy. This strategy employs a population-based heuristic approach to optimize the route and an exact approach for speed, commodity flows, and other decisions when the route is known. Our method integrates a population-based search algorithm that produces feasible sequences, i.e., adhering to the design of the service constraints (Eqs. (8)–(15)). These sequences undergo continual modifications until the stipulated stopping criteria are satisfied. After route generation, a revised version of the model is invoked to optimize the residual decisions. This new model enables solving optimally these remaining decisions quickly. It benefits from the fact that service design variables are fixed. Also, it implements a Dantzig–Wolfe decomposition to replace flow variables with path-based variables for the commodities. Subsequent subsections explain the proposed search algorithm and detail the refined model.

### 4.1. General framework of the solution approach

We use a hybrid solution approach that combines a population-based heuristic and an exact approach. The population-based heuristic is used to solve the routing problem and the exact approach is used to optimize the speed and commodity flow. Population-based heuristics have demonstrated effectiveness in addressing several optimization problems such as the TSP, the VRP, and some maritime transportation problems (e.g. Cariou, Cheaitou, Larbi, & Hamdan, 2018; Cheaitou, Hamdan, & Larbi, 2021).

The solution procedure presented in Algorithm 1, initiates by generating a population of $\Psi$ feasible random sequences utilizing the ports and splitting points in $\mathcal{V}$ (see Section 4.2 for further details). These feasible sequences, denoted as $\mathcal{V}'_{\psi}$ for $\psi = 1, \dots, \Psi$, must adhere to the design of service constraints (Constraints (8)–(15)). After having established this initial population, the algorithm endeavors to enhance this population over $G$ iterations, denoted by $g$ where $g = 1, \dots, G$. In every iteration, Algorithm 1 executes the subsequent steps:

1. For each individual $\mathcal{V}'_{\psi}$, the algorithm calls a simplified version of the model presented in Section 3.2. This simplified model optimizes the speed in each arc in $\mathcal{V}'_{\psi}$, the commodities transported, and the arrival times. The process returns the optimal profit value for the individual, denoted as $\text{Obj}_{\psi}$.

2. The algorithm identifies the local best profit ($\text{Obj}_G$) and the best sequence ($\mathcal{V}_B'$) based on the profit values from the $\Psi$ individuals of the current iteration.

3. The global best profit ($\text{Obj}_G$) is compared with the local best profit ($\text{Obj}_B$). If the local best profit proves superior, the global best profit and sequence are updated accordingly.

4. Subsequently, the population is divided into $\Omega$ subgroups, with each subgroup containing $\eta$ individuals.

5. Within each subgroup $\omega$, the best individual ($\mathcal{V}_{\omega}''$) is identified. A set of $\eta$ operators are then applied independently to $\mathcal{V}_{\omega}''$ to create $\eta$ new individuals, in which the sequence is altered (see Section 4.3). The newly generated $\eta$ individuals replace the previous ones in subgroup $\omega$.

6. It is noteworthy that the algorithm monitors the number of iterations where no improvement in the global value $\text{Obj}_G$ is observed. If this count reaches $\Xi$, a new population is generated randomly.
<caption>**Algorithm 1:** The hybrid solution approach</caption>

**Data:** $\mathcal{P}, \mathcal{P}^w, \mathcal{P}^e, \text{sp}, D_{i,j}, s_k, d_k, w_k, \phi_k, CT_k, Q, PT_i, R, \gamma_i, LC_i, UC_i, C_{sp}, \Gamma, \mu_M, \mu_A, \mathcal{V}, \mathcal{W}, \mathcal{G}$

**Result:** $\mathcal{V}'_G$ and $\text{Obj}_G$

1. set $\text{Obj}_G = -\infty$:
2. for $\psi = 1$ to $\mathcal{W}$ do
3.   $\lfloor$ generate $\mathcal{V}'_\psi$ randomly (see Section 4.2);
4.   for $g = 1$ to $G$ do
5.     for $\psi = 1$ to $\mathcal{W}$ do
6.       use $\mathcal{V}'_\psi$ and call the optimization model in Section 4.4;
7.       set $\text{Obj}_\psi$ as the optimal objective value of $\mathcal{V}'_\psi$;
8.       set $\mathcal{V}'_B = \{\mathcal{V}'_\psi | \psi = \arg\max_{\psi'=1}^{\mathcal{W}} \text{Obj}_\psi'\}$ and
9.         $\text{Obj}_B = \max_{\psi'=1}^{\mathcal{W}} \text{Obj}_\psi$;
10.      if $\text{Obj}_B > \text{Obj}_G$ then
11.        set $\text{Obj}_G = \text{Obj}_B$;
12.        set $\mathcal{V}'_G$ as $\mathcal{V}'_B$;
13.        set $\xi = 0$;
14.      else
15.        $\lfloor$ set $\xi \leftarrow \xi + 1$;
16.        split $\{\mathcal{V}'_\psi : \psi = 1, \dots, \mathcal{W}\}$ randomly into $\Omega$ groups, each with $\eta$ individuals.
17.        for $\omega = 1$ to $\Omega$ do
18.          identify the best individual $\mathcal{V}''_\omega$ in subgroup $\omega$ based on the objective value;
19.          apply $\eta$ operators (see Section 4.3) independently to $\mathcal{V}''_\omega$ to generate $\eta$ new individuals;
20          replace all individuals in subgroup $\omega$ with the newly created individuals;
21          if $\xi > \Xi$ then
22            discard the current population;
23            for $\psi = 1$ to $\mathcal{W}$ do
24              $\lfloor$ generate $\mathcal{V}'_\psi$ randomly;

## 4.2. Initial route generation

Algorithm 1 is initialized with feasible route sequences generated randomly while respecting the service design constraints (Constraints (8)–(15)). We proceed as follows. We assume that each individual consists of three segments (in order $\mathcal{V}^e/\mathcal{V}^w/\mathcal{V}^e$ or $\mathcal{V}^w/\mathcal{V}^e/\mathcal{V}^w$). Each segment is assigned a random length between 0 and the number of ports in the set. Then, ports are selected randomly from their respective sets ($\mathcal{V}^e$ or $\mathcal{V}^w$) and assigned random locations within the segment. Ports already inserted cannot be used again, avoiding constraint violation. Unless it is empty, the intermediate segment is separated by the splitting points sp and sp'. Finally, we check that the total sequence contains at least two ports to avoid empty routes, otherwise it is discarded.

Fig. 2 presents six examples of feasible sequences. For instance $\mathcal{V}'_1 = \{5,7,6\}$ is translated to the following tour: Port 5 → Port 7 → Port 6 → Port 5. Given the randomness in the process, a sequence might comprise one group of ports ($\mathcal{V}^e$ or $\mathcal{V}^w$), as seen in $\mathcal{V}'_1$ and $\mathcal{V}'_2$, or both groups, as demonstrated in $\mathcal{V}'_3$ through $\mathcal{V}'_6$. The key requirement is that the transition between the ports in $\mathcal{V}^e$ and $\mathcal{V}^w$ must occur via a splitting point.

The route generation checks each generated sequence and ensures that ports in $\mathcal{P}^e \cup \mathcal{P}^{te}$ and $\mathcal{P}^w \cup \mathcal{P}^{tw}$ are separated by a splitting point $\{\text{sp}, \text{sp}'\}$.

<figure><img src="image_6.png" /><figcaption>**Fig. 2.** Illustrative examples of feasible sequences generated by the algorithm.</figcaption></figure>

## 4.3. Route modification

The subgroup best individuals generated in Algorithm 1 are modified in each iteration using the following 14 operators.

1. **Operator 1:** Randomly select a starting and an ending port in compliance with the splitting point constraints (10)–(11), and flip the sequence between these two ports.

2. **Operator 2:** Randomly select two ports, ensuring adherence to the splitting point constraints, and swap their positions.

3. **Operator 3:** Randomly select two ports in line with the splitting point constraints and slide their positions.

4. **Operator 4:** Select a port not currently in the route, and insert it at a random position, ensuring it aligns with the splitting point constraints.

5. **Operator 5:** Randomly select a port from the route and remove it.

6. **Operator 6:** Apply Operator 4 a random number of times.

7. **Operator 7:** Apply Operator 5 a random number of times.

8. **Operator 8:** Apply Operator 4 and subsequently Operator 1.

9. **Operator 9:** Apply Operator 4 followed by Operator 2.

10. **Operator 10:** Apply Operator 4, and then Operator 3.

11. **Operator 11:** Apply Operator 5 and then Operator 1.

12. **Operator 12:** Apply Operator 5 followed by Operator 2.

13. **Operator 13:** Apply Operator 5 and then Operator 3.

14. **Operator 14:** Keep the route without any changes.

Fig. 3 depicts the primary operators (Operator 1 through Operator 5). The left portion of the figure displays feasible scenarios, subsequently adjusted by the operators to ensure continued feasibility. Conversely, the right portion showcases scenarios that are bypassed due to feasibility constraints. Initially, these operators categorize ports in the sequence according to their respective sets ($\mathcal{V}^e, \mathcal{V}^w$, or $\{\text{sp}, \text{sp}'\}$). Subsequently, ports or positions from the identical set are selected at random. For example, in the context of Operator 1, both randomly chosen ports must belong to the same set and cannot be interspersed by a splitting point.

## 4.4. Speed and commodity flow optimization

Algorithm 1 employs a simplified version of the optimization model presented in Section 3.2, to determine the optimal speed and commodity flows based on a predetermined route $\mathcal{V}'_\psi$. This simplification significantly reduces the model complexity. The simplified model incorporates the subsequent adjustments (relating to the blocks of constraints as presented in Section 3.2).
<figure><img src="image_7.png" /><figcaption>Fig. 3. Sequence improvement operators.</figcaption></figure>

## Design of the service

Variables $X_{i,j}$, $Y_i$, $Z_i$ and $V_i$ are all set to 0 or 1 in compliance with sequence $V'_\psi$.

## Flow of commodities

Flow variables $F_{i,j}^k$ are transformed into path variables, mimicking a Dantzig-Wolfe decomposition. Given a commodity $k$, a set $P_k$ of at most four paths can be followed by the commodity in sequence $V'_\psi$: (1) from $s_k$ to $d_k$, (2) from $s_k$ to $d'_k$, (3) from $s'_k$ to $d_k$, and (4) from $s'_k$ to $d'_k$. Variables $F_{i,j}^k$ are replaced by variables $F_p^k$ ($p \in P_k$) and constraints are updated accordingly. In addition, Eq. (5) defining canal costs is also changed.

## Speed, time variables and number of vessels

This set of constraints is not modified, except that variables $X_{i,j}^r$, that define the speed on arcs, are made continuous. Clearly, this transformation relaxes the initial formulation and may result in overestimating the total profit computed for solution $V'_\psi$. The aim is to speed up the algorithm and allow more iterations. The real profit associated with the solution that is ultimately returned by the algorithm is recomputed with variables $X_{i,j}^r$ defined again as binary.

In addition, variables $X_{i,j}^r$ and $T_{i,j}$ associated with arcs that do not belong to solution $X_{i,j}^r$ are set to 0.

## Transit time constraint

Similarly to variables $F_{i,j}^k$, variables $X_{i,j}^k$ are changed to $X_p^k$, with $p \in P_k$. Variables $T_{i,j}^k$ are not needed anymore. Set of Constraints (24)–(27) then becomes:

$$F_p^k \le w_k X_p^k \quad \forall k \in \mathcal{K}, p \in P_k, \quad (49)$$

$$\sum_{(i,j) \in P} T_{i,j} + \sum_{i \in P} P T_i \le C T_k + M(1 - X_p^k) \quad \forall k \in \mathcal{K}, p \in P_k, \quad (50)$$

Compared to the previous formulation, transit time constraints (50) take advantage of the fact that the paths that can be followed by the commodities are known.

## Valid inequalities

Only Constraint (36) from the valid inequality set is used. Other valid inequalities are automatically implied in the linear relaxation of the new formulation.

# 5. Numerical experiments

This presents the datasets used in the numerical experiments followed by results on the performance of the proposed approach and several insights.

## 5.1. Input data

The proposed model and the hybrid solution approach have been implemented on a business case inspired by the trade between Asia and Europe. In this business case we considered four vessel sizes with capacities of 6000, 12000, 18000, and 22000 TEUs. The details of the corresponding parameters have been obtained mainly from Woo (2012) and are provided in Table 2.

Moreover, 27 different ports have been considered in the study, 15 in Asia, 10 in Europe, and two in Africa that have been considered for the Cape of Good Hope route option. More precisely, the ports have been used in three different scenarios. The first considered only the following seven ports: Shanghai, Ningbo, Xiamen, Tanjung Pelepas, Felixstowe, Rotterdam, Bremerhaven. The second scenario (11 ports) included the seven ports of the previous scenario in addition to the following four ports: Busan, Shenzhen, Le Havre, and Hamburg. The third scenario (25 ports) included in addition to the 11 previously mentioned ports, the following 14 ports: Tokyo, Dalian, Vung Tau, Port Klang, Colombo, Mundra, Karachi, Jabel Ali, Jeddah, Piraeus, Gioia Tauro,
<table><caption>**Table 2**
Vessel and economic data.</caption><thead><tr><th>Vessel capacity Q</th><th>6000</th><th>12000</th><th>18000</th><th>22000</th></tr></thead><tbody><tr><td>LOA</td><td>304</td><td>393</td><td>400</td><td>400</td></tr><tr><td>C<sup>b</sup><sub>A</sub></td><td>900</td><td>900</td><td>900</td><td>900</td></tr><tr><td>C<sup>b</sup><sub>M</sub></td><td>600</td><td>600</td><td>600</td><td>600</td></tr><tr><td>μ<sup>min</sup></td><td>12</td><td>12</td><td>12</td><td>12</td></tr><tr><td>μ<sup>max</sup></td><td>28</td><td>28</td><td>25</td><td>25</td></tr><tr><td>PT<sub>i</sub></td><td>12</td><td>12</td><td>15</td><td>15</td></tr><tr><td>C<sup>sp</sup></td><td>50</td><td>50</td><td>50</td><td>50</td></tr><tr><td>LC<sub>i</sub> and UC<sub>i</sub></td><td>45</td><td>45</td><td>45</td><td>45</td></tr><tr><td>CHM<sub>i</sub></td><td>2</td><td>2</td><td>2</td><td>2</td></tr><tr><td>Γ</td><td>10000</td><td>14000</td><td>16000</td><td>19000</td></tr><tr><td>μ<sub>M</sub> (ton × h<sup>2</sup>/nm<sup>3</sup>)</td><td>0.000020009</td><td>0.000025738</td><td>0.000041389</td><td>0.000040786</td></tr><tr><td>μ<sub>A</sub> (ton/h)</td><td>3</td><td>12</td><td>18</td><td>20</td></tr></tbody></table>

Malta, Algeciras, Southampton. Finally, for the Cape of Good Hope route option, two configurations have been considered. The first configuration in which only the three scenarios mentioned earlier (7, 11, and 25 ports) have been considered (and is noted hereafter CGHR), and the one two additional ports located in south and west Africa have been added, namely Durban and Lome (and is noted hereafter CGHR+).

In addition, two splitting points have been considered, Suez Canal and Cape of Good Hope (Capetown). The distances between the considered ports have been obtained from SEA-DISTANCES (2023). The demand between the ports have been inspired by Editor (2011) and some have been simulated. Moreover, the freight rates have been obtained mainly based on Sea-Rates (2023), World-Freight-Rates (2021) using apparel as shipped goods with a value of 10000 USD, high interval, and TEU as container size. All these data are reported in the Supplementary Material (Tables A1–A11). All these values correspond to the situation that is called hereafter “Before COVID-19” or “Case A”. In order to obtain the values of the situation that is called “During COVID-19” or “Case B”, the bunker prices $C_A^b$ and $C_M^b$ have been multiplied by 1.83, and the demand from Asia to Europe has been decreased by 1.4% and from Europe to Asia by 5.1%. Moreover, the freight rates that are reported in the Supplementary Material have been increased by 547% and 23% from Asia to Europe and Europe to Asia, respectively. Finally, to define the data for the situation that is called “After COVID-19” or “Case C”, all the data of Case A (“Before COVID-19”) have been used except the demand for which the values of Case B (“During COVID-19”) have been used.

## 5.2. Results

In this section, we discuss the obtained results. First, we investigate the impact of the valid inequalities on the computation time. Then, we investigate the performance of the proposed hybrid solution approach by comparing it with that of the exact approach and by including a time study. In the following section, we discuss the obtained profits in the different configurations considered. Finally, we discuss the vessel utilization and sailing speed results. It is worth noting that the exact and the hybrid approaches have been implemented in a computer equipped with 16 GB RAM, 12th Gen Intel(R) Core(TM) i5-1235U 1.30 GHZ processor, and a 64-bit Windows 10 operating system. We used the CPLEX 21 to solve the exact formulation.

### 5.2.1. Impact of valid inequalities

Tables 3 and 4 compare the performance of the exact model without valid inequalities, with some valid inequalities (up to (34)), and with all valid inequalities for the configurations with seven and 11 ports. These tables report the computation time in seconds and the MIP gap of the solver. When the transit time constraints are not considered, the model incorporating valid inequalities up to Constraint (34) achieves an optimal solution within 20 s for the 7 ports case. The model with all valid inequality constraints manages to find the optimal solution in

less than 95 s. In both scenarios, the computation time is considered acceptable. However, when the transit time constraints are included, employing all valid inequalities generally leads to solutions of better quality or to faster solutions in most cases.

### 5.2.2. Performance of the hybrid solution approach

In this section we compare the hybrid solution approach detailed in Section 4 with the exact solution approach. Fig. 4 shows the mean relative difference (RD) between the objective function values of the hybrid solution approach and the best solution found using the hybrid approach as a function of the number of iterations, for each number of ports under consideration (7, 11, and 25) and each route option (SCR, SCR TC, CGHR, CGHR TC, CGHR+, and CGHR+ TC), where TC stands for time constraint which means the model considering the time constraint of the commodities expressed in Eq. (24). Moreover, the instances with the sign “+”, i.e., CGHR+, correspond to the case in which the vessels taking the Cape of Good Hope route may call at two additional ports considered in West Africa, namely Lome in Togo and Durban in South Africa. Each of the resulting 18 cases has been solved using the hybrid approach five times for each of the demand and freight rate scenarios, i.e., for “Case A”, “Case B” and “Case C”, resulting in a total of 15 trials for each case. The relative difference has been calculated for each of the 15 cases and the average has been obtained and is shown in Fig. 4.

As can be seen from Fig. 4, the best solution is found for all the considered cases, after at most 250 iterations. However, it is much faster in the small problem sizes, i.e., with seven ports. Moreover, the impact of the time constraint of the commodities is clear on the convergence speed. All the problems with TC appear to require more iterations before the algorithm reaches the best solution than their counterparts without TC. Moreover, the impact of the TC on the convergence is magnified when the problem size, i.e., the number of ports, increases. Finally, the vessel size does not seem to have a clear impact on the convergence.

In addition, Fig. 5 shows the average computation time of the hybrid approach for the same cases considered in Fig. 4. The average computation time has been calculated considering the 15 trials detailed in the discussion of Fig. 4. As it can be seen in Fig. 5, the average computation time till convergence of the hybrid approach appears to be linear in function of the number of iterations. It also remains very reasonable and does not exceed 10 min in any of the cases except in two, namely the SCR TC cases with 11 and 25 ports. For these two cases, the computation time for 6000 TEUs and 12000 TEUs reached a value that is slightly above 100 min. The reasons may be related to the capacity constraints combined with the time limit constraints. It is worth noting that the hybrid approach computation time may be significantly decreased by reducing the precision in the discretization of the speed values, i.e., by increasing the speed step.

As shown in Tables 3 and 4, the optimal solution can be obtained for the configurations with 7 ports for all the cases without transit time constraint after using valid inequalities within the maximum computation time set to the solver, i.e., three hours. However, for some of the cases with transit time constraints, and for the configurations with 11 ports, in the majority of the cases, the optimality gap remained positive after the time limit set for the solver even with the strengthened formulations with valid inequalities. Table 5 shows the profit for instances with seven and 11 ports under the SCR obtained from the exact and hybrid solution approaches. Due to computational limitations, we conduct the exact experiments using $|R| = 11$, whereas, in the hybrid approach, which is not constrained by such limitations, we use $|R| = 160$. We calculate the relative difference (RD) between the exact and hybrid approach profits as $100 \times \frac{Profit_E - Profit_H}{Profit_H}$. A negative RD indicates that the hybrid approach provides a higher profit than the exact approach. The results show that in all instances where the exact approach reached the optimal solution (i.e., MIP = 0), the hybrid
<caption>**Table 3**</caption>

Computation time in Seconds (MIP gap) of the 7 ports scenario using the model without valid inequalities, with valid inequalities up to (34) and all valid inequalities. We use $\infty$ to indicate situations when the solver returned extremely large gap as it fails to find a feasible solution.

<table><caption>**Table 4**</caption><thead><tr><th colspan="10">SCR</th></tr><tr><th colspan="10">Without transit time constraints</th></tr><tr><th></th><th></th><th colspan="3">up to (27)</th><th colspan="3">up to (34)</th><th colspan="3">up to (48)</th></tr><tr><th></th><th></th><th colspan="3">up to (27)</th><th colspan="3">up to (34)</th><th colspan="3">up to (48)</th></tr></thead><tbody><tr><td rowspan="6">Case A</td><td>6000 TEU</td><td>10800 (40.48)</td><td>14 (0.00)</td><td>58 (0.00)</td><td>10800 (49.76)</td><td>10800 (10.70)</td><td>4674 (0.00)</td><td>10800 (49.76)</td><td>10800 (10.70)</td><td>4674 (0.00)</td></tr><tr><td>12000 TEU</td><td>10800 (100.03)</td><td>18 (0.00)</td><td>94 (0.00)</td><td>10800 (119.80)</td><td>10800 (21.81)</td><td>5490.73 (0.00)</td><td>10800 (119.80)</td><td>10800 (21.81)</td><td>5490.73 (0.00)</td></tr><tr><td>18000 TEU</td><td>10800 (310.15)</td><td>11 (0.00)</td><td>38 (0.00)</td><td>10800 (465.90)</td><td>10800 (510.48)</td><td>4798.30 (0.00)</td><td>10800 (465.90)</td><td>10800 (510.48)</td><td>4798.30 (0.00)</td></tr><tr><td>22000 TEU</td><td>10800 (496.46)</td><td>12 (0.00)</td><td>60 (0.00)</td><td>10800 (∞)</td><td>5784 (0.00)</td><td>608.21 (0.00)</td><td>10800 (∞)</td><td>5784 (0.00)</td><td>608.21 (0.00)</td></tr><tr><td>6000 TEU</td><td>10800 (8.46)</td><td>10 (0.00)</td><td>64 (0.00)</td><td>10800 (10.82)</td><td>10800 (5.25)</td><td>4522 (0.00)</td><td>10800 (10.82)</td><td>10800 (5.25)</td><td>4522 (0.00)</td></tr><tr><td>12000 TEU</td><td>10800 (15.31)</td><td>9 (0.00)</td><td>42 (0.00)</td><td>10800 (20.09)</td><td>10800 (5.56)</td><td>5267 (0.00)</td><td>10800 (20.09)</td><td>10800 (5.56)</td><td>5267 (0.00)</td></tr><tr><td rowspan="6">Case B</td><td>18000 TEU</td><td>10800 (20.62)</td><td>11 (0.00)</td><td>40 (0.00)</td><td>10800 (33.29)</td><td>10800 (15.84)</td><td>7257 (0.00)</td><td>10800 (33.29)</td><td>10800 (15.84)</td><td>7257 (0.00)</td></tr><tr><td>22000 TEU</td><td>10800 (25.39)</td><td>14 (0.00)</td><td>57 (0.00)</td><td>10800 (36.61)</td><td>10800 (13.01)</td><td>10800 (1.26)</td><td>10800 (36.61)</td><td>10800 (13.01)</td><td>10800 (1.26)</td></tr><tr><td>6000 TEU</td><td>10800 (41.35)</td><td>10 (0.00)</td><td>34 (0.00)</td><td>10800 (52.63)</td><td>10800 (10.56)</td><td>10800 (13.32)</td><td>10800 (52.63)</td><td>10800 (10.56)</td><td>10800 (13.32)</td></tr><tr><td>12000 TEU</td><td>10800 (102.89)</td><td>11 (0.00)</td><td>39 (0.00)</td><td>10800 (112.41)</td><td>10800 (21.02)</td><td>4921 (0.00)</td><td>10800 (112.41)</td><td>10800 (21.02)</td><td>4921 (0.00)</td></tr><tr><td>18000 TEU</td><td>10800 (348.00)</td><td>10 (0.00)</td><td>41 (0.00)</td><td>10800 (∞)</td><td>10800 (316.50)</td><td>5699 (0.00)</td><td>10800 (∞)</td><td>10800 (316.50)</td><td>5699 (0.00)</td></tr><tr><td>22000 TEU</td><td>10800 (573.60)</td><td>12 (0.00)</td><td>51 (0.00)</td><td>10800 (∞)</td><td>3791 (0.00)</td><td>398 (0.00)</td><td>10800 (∞)</td><td>3791 (0.00)</td><td>398 (0.00)</td></tr><tr><th colspan="10">CGHR</th></tr><tr><th colspan="10">Without transit time constraints</th></tr><tr><th></th><th></th><th colspan="3">up to (27)</th><th colspan="3">up to (34)</th><th colspan="3">up to (48)</th></tr><tr><th></th><th></th><th colspan="3">up to (27)</th><th colspan="3">up to (34)</th><th colspan="3">up to (48)</th></tr></tbody></table><table><thead><tr><th rowspan="6">Case C</th><td>6000 TEU</td><td>10800 (56.46)</td><td>8 (0.00)</td><td>23 (0.00)</td><td>10800 (83.78)</td><td>10800 (270.70)</td><td>10800 (163.93)</td><td>10800 (83.78)</td><td>10800 (270.70)</td><td>10800 (163.93)</td></tr><tr><td>12000 TEU</td><td>10800 (174.24)</td><td>6 (0.00)</td><td>21 (0.00)</td><td>10800 (360.38)</td><td>10800 (3083.61)</td><td>10800 (780.84)</td><td>10800 (360.38)</td><td>10800 (3083.61)</td><td>10800 (780.84)</td></tr><tr><td>18000 TEU</td><td>10800 (∞)</td><td>7 (0.00)</td><td>37 (0.00)</td><td>10800 (∞)</td><td>45 (0.00)</td><td>164 (0.00)</td><td>10800 (∞)</td><td>45 (0.00)</td><td>164 (0.00)</td></tr><tr><td>22000 TEU</td><td>10800 (∞)</td><td>4 (0.00)</td><td>32 (0.00)</td><td>10800 (∞)</td><td>32 (0.00)</td><td>36 (0.00)</td><td>10800 (∞)</td><td>32 (0.00)</td><td>36 (0.00)</td></tr><tr><td>6000 TEU</td><td>10800 (11.15)</td><td>7 (0.00)</td><td>8 (0.00)</td><td>10800 (13.25)</td><td>10800 (14.10)</td><td>3379.97 (0.00)</td><td>10800 (13.25)</td><td>10800 (14.10)</td><td>3379.97 (0.00)</td></tr><tr><td>12000 TEU</td><td>10800 (20.96)</td><td>6 (0.00)</td><td>24 (0.00)</td><td>10800 (27.70)</td><td>10800 (12.16)</td><td>3675.66 (0.00)</td><td>10800 (27.70)</td><td>10800 (12.16)</td><td>3675.66 (0.00)</td></tr><tr><td>18000 TEU</td><td>10800 (34.68)</td><td>7 (0.00)</td><td>61 (0.00)</td><td>10800 (48.88)</td><td>10800 (15.27)</td><td>4545 (0.00)</td><td>10800 (48.88)</td><td>10800 (15.27)</td><td>4545 (0.00)</td></tr><tr><td>22000 TEU</td><td>10800 (38.32)</td><td>7 (0.00)</td><td>10 (0.00)</td><td>10800 (53.64)</td><td>10800 (43.60)</td><td>4678 (0.00)</td><td>10800 (53.64)</td><td>10800 (43.60)</td><td>4678 (0.00)</td></tr><tr><td>6000 TEU</td><td>10800 (58.64)</td><td>7 (0.00)</td><td>69 (0.00)</td><td>10800 (90.37)</td><td>10800 (436.71)</td><td>10800 (177.21)</td><td>10800 (90.37)</td><td>10800 (436.71)</td><td>10800 (177.21)</td></tr><tr><td>12000 TEU</td><td>10800 (186.87)</td><td>7 (0.00)</td><td>17 (0.00)</td><td>10800 (270.67)</td><td>10800 (1889.45)</td><td>5951 (0.00)</td><td>10800 (270.67)</td><td>10800 (1889.45)</td><td>5951 (0.00)</td></tr><tr><td>18000 TEU</td><td>10800 (∞)</td><td>11 (0.00)</td><td>54 (0.00)</td><td>10800 (∞)</td><td>45 (0.00)</td><td>182 (0.00)</td><td>10800 (∞)</td><td>45 (0.00)</td><td>182 (0.00)</td></tr><tr><td>22000 TEU</td><td>10800 (∞)</td><td>3 (0.00)</td><td>14 (0.00)</td><td>10800 (∞)</td><td>14 (0.00)</td><td>9 (0.00)</td><td>10800 (∞)</td><td>14 (0.00)</td><td>9 (0.00)</td></tr></table>

Computation time in Seconds (MIP gap) of the 11 ports scenario using the model without valid inequalities, with valid inequalities up to (34) and all valid inequalities. We use "$\infty$" to indicate situations when the solver returned extremely large gap as it fails to find a feasible solution.

<table><thead><tr><th colspan="10">SCR</th></tr><tr><th colspan="10">Without transit time constraints</th></tr><tr><th></th><th></th><th colspan="3">up to (27)</th><th colspan="3">up to (34)</th><th colspan="3">up to (48)</th></tr><tr><th></th><th></th><th colspan="3">up to (27)</th><th colspan="3">up to (34)</th><th colspan="3">up to (48)</th></tr></thead><tbody><tr><td rowspan="6">Case A</td><td>6000 TEU</td><td>10800 (17.27)</td><td>10800 (1.42)</td><td>10800 (2.51)</td><td>10800 (∞)</td><td>10800 (64.40)</td><td>10800 (25.30)</td><td>10800 (∞)</td><td>10800 (64.40)</td><td>10800 (25.30)</td></tr><tr><td>12000 TEU</td><td>10800 (27.59)</td><td>3072 (0.00)</td><td>2619 (0.00)</td><td>10800 (326.80)</td><td>10800 (162.67)</td><td>10800 (∞)</td><td>10800 (326.80)</td><td>10800 (162.67)</td><td>10800 (∞)</td></tr><tr><td>18000 TEU</td><td>10800 (42.71)</td><td>3670 (0.00)</td><td>2683 (0.00)</td><td>10800 (191.11)</td><td>10800 (526.91)</td><td>10800 (∞)</td><td>10800 (191.11)</td><td>10800 (526.91)</td><td>10800 (∞)</td></tr><tr><td>22000 TEU</td><td>10800 (51.44)</td><td>6657 (0.00)</td><td>2103 (0.00)</td><td>10800 (∞)</td><td>10800 (199.19)</td><td>10800 (∞)</td><td>10800 (∞)</td><td>10800 (199.19)</td><td>10800 (∞)</td></tr><tr><td>6000 TEU</td><td>10800 (4.05)</td><td>10800 (0.23)</td><td>10800 (0.38)</td><td>10800 (∞)</td><td>10800 (14.30)</td><td>10800 (∞)</td><td>10800 (∞)</td><td>10800 (14.30)</td><td>10800 (∞)</td></tr><tr><td>12000 TEU</td><td>10800 (6.35)</td><td>2875 (0.00)</td><td>1102 (0.00)</td><td>10800 (138.91)</td><td>10800 (16.11)</td><td>10800 (19.32)</td><td>10800 (138.91)</td><td>10800 (16.11)</td><td>10800 (19.32)</td></tr><tr><td rowspan="6">Case B</td><td>18000 TEU</td><td>10800 (9.31)</td><td>3694 (0.00)</td><td>1415 (0.00)</td><td>10800 (∞)</td><td>10800 (40.45)</td><td>10800 (∞)</td><td>10800 (∞)</td><td>10800 (40.45)</td><td>10800 (∞)</td></tr><tr><td>22000 TEU</td><td>10800 (9.93)</td><td>2874 (0.00)</td><td>1475 (0.00)</td><td>10800 (∞)</td><td>10800 (18.73)</td><td>10800 (∞)</td><td>10800 (18.73)</td><td>10800 (18.73)</td><td>10800 (∞)</td></tr><tr><td>6000 TEU</td><td>10800 (19.10)</td><td>10800 (1.95)</td><td>10800 (2.77)</td><td>10800 (∞)</td><td>10800 (76.58)</td><td>10800 (∞)</td><td>10800 (76.58)</td><td>10800 (76.58)</td><td>10800 (∞)</td></tr><tr><td>12000 TEU</td><td>10800 (27.69)</td><td>3197 (0.00)</td><td>1664 (0.00)</td><td>10800 (110.49)</td><td>10800 (72.83)</td><td>10800 (∞)</td><td>10800 (110.49)</td><td>10800 (72.83)</td><td>10800 (∞)</td></tr><tr><td>18000 TEU</td><td>10800 (45.18)</td><td>7950 (0.00)</td><td>2264 (0.00)</td><td>10800 (137.13)</td><td>10800 (576.60)</td><td>10800 (∞)</td><td>10800 (137.13)</td><td>10800 (576.60)</td><td>10800 (∞)</td></tr><tr><td>22000 TEU</td><td>10800 (53.19)</td><td>6142 (0.00)</td><td>1549 (0.00)</td><td>10800 (∞)</td><td>10800 (∞)</td><td>10800 (∞)</td><td>10800 (∞)</td><td>10800 (∞)</td><td>10800 (∞)</td></tr><tr><th colspan="10">CGHR</th></tr><tr><th colspan="10">Without transit time constraints</th></tr><tr><th></th><th></th><th colspan="3">up to (27)</th><th colspan="3">up to (34)</th><th colspan="3">up to (48)</th></tr><tr><th></th><th></th><th colspan="3">up to (27)</th><th colspan="3">up to (34)</th><th colspan="3">up to (48)</th></tr></tbody></table><table><thead><tr><th rowspan="6">Case C</td><td>6000 TEU</td><td>10800 (20.31)</td><td>10800 (1.61)</td><td>10800 (3.00)</td><td>10800 (114.07)</td><td>10800 (397.88)</td><td>10800 (∞)</td><td>10800 (114.07)</td><td>10800 (397.88)</td><td>10800 (∞)</td></tr><tr><td>12000 TEU</td><td>10800 (33.36)</td><td>518 (0.00)</td><td>1430 (0.00)</td><td>10800 (∞)</td><td>10800 (∞)</td><td>10800 (∞)</td><td>10800 (∞)</td><td>10800 (∞)</td><td>10800 (∞)</td></tr><tr><td>18000 TEU</td><td>10800 (55.45)</td><td>772 (0.00)</td><td>791 (0.00)</td><td>10800 (240.71)</td><td>10800 (∞)</td><td>10800 (∞)</td><td>10800 (240.71)</td><td>10800 (∞)</td><td>10800 (∞)</td></tr><tr><td>22000 TEU</td><td>10800 (63.14)</td><td>460 (0.00)</td><td>2646 (0.00)</td><td>10800 (457.08)</td><td>10800 (∞)</td><td>10800 (∞)</td><td>10800 (457.08)</td><td>10800 (∞)</td><td>10800 (∞)</td></tr><tr><td>6000 TEU</td><td>10800 (4.78)</td><td>10800 (0.26)</td><td>10800 (0.43)</td><td>10800 (150.38)</td><td>10800 (47.77)</td><td>10800 (∞)</td><td>10800 (150.38)</td><td>10800 (47.77)</td><td>10800 (∞)</td></tr><tr><td>12000 TEU</td><td>10800 (7.59)</td><td>655 (0.00)</td><td>1761 (0.00)</td><td>10800 (76.27)</td><td>10800 (52.83)</td><td>10800 (86.41)</td><td>10800 (76.27)</td><td>10800 (52.83)</td><td>10800 (86.41)</td></tr><tr><td>18000 TEU</td><td>10800 (11.52)</td><td>806 (0.00)</td><td>1248 (0.00)</td><td>10800 (106.76)</td><td>10800 (262.71)</td><td>10800 (∞)</td><td>10800 (106.76)</td><td>10800 (262.71)</td><td>10800 (∞)</td></tr><tr><td>22000 TEU</td><td>10800 (12.47)</td><td>1461 (0.00)</td><td>1555 (0.00)</td><td>10800 (165.04)</td><td>10800 (179.59)</td><td>10800 (203.59)</td><td>10800 (165.04)</td><td>10800 (179.59)</td><td>10800 (203.59)</td></tr><tr><td rowspan="6">Case B</td><td>6000 TEU</td><td>10800 (22.78)</td><td>10800 (1.59)</td><td>10800 (1.99)</td><td>10800 (∞)</td><td>10800 (∞)</td><td>10800 (∞)</td><td>10800 (∞)</td><td>10800 (∞)</td><td>10800 (∞)</td></tr><tr><td>12000 TEU</td><td>10800 (34.10)</td><td>548 (0.00)</td><td>646 (0.00)</td><td>10800 (∞)</td><td>10800 (678.31)</td><td>10800 (∞)</td><td>10800 (678.31)</td><td>10800 (∞)</td><td>10800 (∞)</td></tr><tr><td>18000 TEU</td><td>10800 (57.09)</td><td>410 (0.00)</td><td>2597 (0.00)</td><td>10800 (107.22)</td><td>10800 (∞)</td><td>10800 (∞)</td><td>10800 (107.22)</td><td>10800 (∞)</td><td>10800 (∞)</td></tr><tr><td>22000 TEU</td><td>10800 (67.73)</td><td>450 (0.00)</td><td>2717 (0.00)</td><td>10800 (∞)</td><td>10800 (∞)</td><td>10800 (∞)</td><td>10800 (∞)</td><td>10800 (∞)</td><td>10800 (∞)</td></tr><tr><td>18000 TEU</td><td>10800 (34.10)</td><td>548 (0.00)</td><td>646 (0.00)</td><td>10800 (∞)</td><td>10800 (678.31)</td><td>10800 (∞)</td><td>10800 (678.31)</td><td>10800 (∞)</td><td>10800 (∞)</td></tr><tr><td>22000 TEU</td><td>10800 (57.09)</td><td>410 (0.00)</td><td>2597 (0.00)</td><td>10800 (107.22)</td><td>10800 (∞)</td><td>10800 (∞)</td><td>10800 (107.22)</td><td>10800 (∞)</td><td>10800 (∞)</td></tr></table>

approach also reached the optimal solution. In several cases, the hybrid approach achieved higher accuracy due to the use of smaller speed increments ($|R| = 160$), with an average RD of -0.34%. Additionally,

in instances where the exact approach did not reach the optimal solution (i.e., 11 ports with transit time constraints), the hybrid approach outperformed the exact approach with an average RD of -123.23%.
<table><caption>**Table 5**
Exact profit vs hybrid approach profit for the 7 and 11 ports instances under the SCR.</caption><thead><tr><th rowspan="2"></th><th colspan="5">Without transit time</th><th colspan="5">With transit time</th></tr><tr><th>Vessel</th><th>Profit<sub>E</sub></th><th>MIP (%)</th><th>Profit<sub>H</sub></th><th>RD</th><th>Profit<sub>E</sub></th><th>MIP (%)</th><th>Profit<sub>H</sub></th><th>RD</th></tr></thead><tbody><tr><td colspan="10" style="text-align:center;">7 ports</td></tr><tr><td>Case A</td><td>6000 TEU</td><td>3.71 × 10<sup>6</sup></td><td>0</td><td>3.72 × 10<sup>6</sup></td><td>-0.29%</td><td>3.01 × 10<sup>6</sup></td><td>0</td><td>3.02 × 10<sup>6</sup></td><td>-0.35%</td></tr><tr><td></td><td>12000 TEU</td><td>2.48 × 10<sup>6</sup></td><td>0</td><td>2.50 × 10<sup>6</sup></td><td>-0.54%</td><td>1.74 × 10<sup>6</sup></td><td>0</td><td>1.75 × 10<sup>6</sup></td><td>-0.95%</td></tr><tr><td></td><td>18000 TEU</td><td>1.11 × 10<sup>6</sup></td><td>0</td><td>1.11 × 10<sup>6</sup></td><td>-0.50%</td><td>-9.46 × 10<sup>4</sup></td><td>0</td><td>-9.46 × 10<sup>4</sup></td><td>0.00%</td></tr><tr><td></td><td>22000 TEU</td><td>7.57 × 10<sup>5</sup></td><td>0</td><td>7.74 × 10<sup>5</sup></td><td>-2.23%</td><td>-1.22 × 10<sup>5</sup></td><td>0</td><td>-1.22 × 10<sup>5</sup></td><td>0.00%</td></tr><tr><td></td><td>6000 TEU</td><td>2.65 × 10<sup>7</sup></td><td>0</td><td>2.65 × 10<sup>7</sup></td><td>0.00%</td><td>2.50 × 10<sup>7</sup></td><td>0</td><td>2.50 × 10<sup>7</sup></td><td>-0.07%</td></tr><tr><td>Case B</td><td>12000 TEU</td><td>2.45 × 10<sup>7</sup></td><td>0</td><td>2.45 × 10<sup>7</sup></td><td>-0.07%</td><td>2.29 × 10<sup>7</sup></td><td>0</td><td>2.29 × 10<sup>7</sup></td><td>-0.11%</td></tr><tr><td></td><td>18000 TEU</td><td>2.22 × 10<sup>7</sup></td><td>0</td><td>2.22 × 10<sup>7</sup></td><td>0.00%</td><td>1.91 × 10<sup>7</sup></td><td>0</td><td>1.91 × 10<sup>7</sup></td><td>-0.13%</td></tr><tr><td></td><td>22000 TEU</td><td>2.17 × 10<sup>7</sup></td><td>0</td><td>2.17 × 10<sup>7</sup></td><td>0.00%</td><td>1.88 × 10<sup>7</sup></td><td>1.26</td><td>1.88 × 10<sup>7</sup></td><td>-0.13%</td></tr><tr><td>Case C</td><td>6000 TEU</td><td>3.59 × 10<sup>6</sup></td><td>0</td><td>3.60 × 10<sup>6</sup></td><td>-0.24%</td><td>2.89 × 10<sup>6</sup></td><td>10.56</td><td>2.90 × 10<sup>6</sup></td><td>-0.36%</td></tr><tr><td></td><td>12000 TEU</td><td>2.37 × 10<sup>6</sup></td><td>0</td><td>2.38 × 10<sup>6</sup></td><td>-0.47%</td><td>1.64 × 10<sup>6</sup></td><td>0</td><td>1.64 × 10<sup>6</sup></td><td>-1.02%</td></tr><tr><td></td><td>18000 TEU</td><td>9.93 × 10<sup>5</sup></td><td>0</td><td>9.99 × 10<sup>5</sup></td><td>-0.56%</td><td>-9.46 × 10<sup>4</sup></td><td>0</td><td>-9.46 × 10<sup>4</sup></td><td>0.00%</td></tr><tr><td></td><td>22000 TEU</td><td>6.44 × 10<sup>5</sup></td><td>0</td><td>6.58 × 10<sup>5</sup></td><td>-2.23%</td><td>-1.22 × 10<sup>5</sup></td><td>0</td><td>-1.22 × 10<sup>5</sup></td><td>0.00%</td></tr><tr><td colspan="10" style="text-align:center;">11 ports</td></tr><tr><td>Case A</td><td>6000 TEU</td><td>9.25 × 10<sup>6</sup></td><td>1.42</td><td>9.25 × 10<sup>6</sup></td><td>0.00%</td><td>7.54 × 10<sup>6</sup></td><td>25.3</td><td>8.25 × 10<sup>6</sup></td><td>-9.37%</td></tr><tr><td></td><td>12000 TEU</td><td>1.07 × 10<sup>7</sup></td><td>0</td><td>1.07 × 10<sup>7</sup></td><td>-0.06%</td><td>4.10 × 10<sup>6</sup></td><td>162.67</td><td>9.30 × 10<sup>6</sup></td><td>-126.70%</td></tr><tr><td></td><td>18000 TEU</td><td>9.21 × 10<sup>6</sup></td><td>0</td><td>9.22 × 10<sup>6</sup></td><td>-0.16%</td><td>1.50 × 10<sup>6</sup></td><td>526.91</td><td>6.31 × 10<sup>6</sup></td><td>-321.75%</td></tr><tr><td></td><td>22000 TEU</td><td>8.83 × 10<sup>6</sup></td><td>0</td><td>8.85 × 10<sup>6</sup></td><td>-0.16%</td><td>2.99 × 10<sup>6</sup></td><td>199.19</td><td>6.09 × 10<sup>6</sup></td><td>-103.44%</td></tr><tr><td></td><td>6000 TEU</td><td>5.35 × 10<sup>7</sup></td><td>0.23</td><td>5.35 × 10<sup>7</sup></td><td>0.00%</td><td>4.70 × 10<sup>7</sup></td><td>14.3</td><td>5.14 × 10<sup>7</sup></td><td>-9.27%</td></tr><tr><td>Case B</td><td>12000 TEU</td><td>6.93 × 10<sup>7</sup></td><td>0</td><td>6.93 × 10<sup>7</sup></td><td>-0.02%</td><td>5.98 × 10<sup>7</sup></td><td>16.11</td><td>6.65 × 10<sup>7</sup></td><td>-11.26%</td></tr><tr><td></td><td>18000 TEU</td><td>6.68 × 10<sup>7</sup></td><td>0</td><td>6.68 × 10<sup>7</sup></td><td>0.00%</td><td>4.77 × 10<sup>7</sup></td><td>40.45</td><td>6.12 × 10<sup>7</sup></td><td>-28.34%</td></tr><tr><td></td><td>22000 TEU</td><td>6.63 × 10<sup>7</sup></td><td>0</td><td>6.63 × 10<sup>7</sup></td><td>0.00%</td><td>5.59 × 10<sup>7</sup></td><td>18.73</td><td>6.09 × 10<sup>7</sup></td><td>-8.99%</td></tr><tr><td>Case C</td><td>6000 TEU</td><td>9.09 × 10<sup>6</sup></td><td>1.95</td><td>9.09 × 10<sup>6</sup></td><td>0.00%</td><td>5.27 × 10<sup>6</sup></td><td>76.58</td><td>8.10 × 10<sup>6</sup></td><td>-53.79%</td></tr><tr><td></td><td>12000 TEU</td><td>1.04 × 10<sup>7</sup></td><td>0</td><td>1.04 × 10<sup>7</sup></td><td>-0.06%</td><td>6.10 × 10<sup>6</sup></td><td>72.83</td><td>9.03 × 10<sup>6</sup></td><td>-47.94%</td></tr><tr><td></td><td>18000 TEU</td><td>8.93 × 10<sup>6</sup></td><td>0</td><td>8.95 × 10<sup>6</sup></td><td>-0.16%</td><td>1.34 × 10<sup>6</sup></td><td>576.6</td><td>6.04 × 10<sup>6</sup></td><td>-351.01%</td></tr><tr><td></td><td>22000 TEU</td><td>8.56 × 10<sup>6</sup></td><td>0</td><td>8.57 × 10<sup>6</sup></td><td>-0.17%</td><td>5.19 × 10<sup>5</sup></td><td>∞</td><td>5.82 × 10<sup>6</sup></td><td>-1022.49%</td></tr></tbody></table>

These results show that the hybrid approach proposed in this paper appears to be a suitable alternative to the exact approach since it allows for achieving solutions that are close to the optimal/best exact ones within a reasonable computation time and for realistic instances of the considered problem. It will be therefore used in the following section of this numerical analysis.

The proposed hybrid solution approach has been compared with a traditional GA that includes capacity fix and splitting point insertion routines. We refer readers to Cariou et al. (2018) and Ali Cheaitou and Sadeque Hamdan and Rim Larbi and Imad Alsyouf (2021) for the basic algorithm details and to Appendix B of the Supplementary Material for the capacity fix and splitting point insertion routines, which ensure the algorithm fits the problem defined in this work. Table 6 shows the profit of the SCR under the three cases (A, B, and C) using the traditional GA with 5000 iterations. We then calculate the relative improvement (Imp) achieved by the proposed hybrid approach as $100 \times \frac{Profit_H - Profit_{GA}}{Profit_{GA}}$. Note that the profit values from the hybrid approach are reported in Table 7 to avoid redundancy. The analysis reveals that the traditional GA requires 5000 iterations to reach the same solution that the hybrid approach achieves in only 300 iterations (see Case B - 7 ports). The hybrid approach achieves an average improvement of 17% in solution quality, with improvements ranging from a minimum of 0% to a maximum of 91.27%, while requiring fewer iterations. Additionally, the GA requires an average of 33 min, compared to just 3 to 6 min for the hybrid approach. Note that the traditional GA performance deteriorates considerably when adding the transit time constraints.

### 5.2.3. Profits

Table 7 shows the profit defined in Eq. (1) for all the considered route options (SCR, SCR TC, CGHR, CGHR TC, CGHR+, and CGHR+ TC), vessel sizes, and number of ports. The first comment that can be mentioned is that some of the profit values are negative. This is due to the fact that the model forces the solution to include at least one port, as formulated in Constraints (13) and (14) which is not necessarily profitable in all cases, and in particular in the CGHR and CGHR+ cases with seven ports. The negative profit quantifies the loss in cases where a non-profitable service is established as part of the liner shipping network strategy, or to comply with regulatory or connectivity

requirements, creating a need for external monetary support to sustain operations. These losses may highlight the necessity of balancing unprofitable but strategically important routes with overall network profitability or exploring alternative designs to close the losses gap. The results show clearly the supremacy of the SCR route especially for the two cases with seven and eleven ports and without the time constraints. Indeed, for the cases with only seven ports, the SCR option is more profitable than the CGHR option. For the cases with 11 ports, the SCR option remains more profitable except in the cases with 6000 TEU vessels. However, with 25 ports, the CGHR becomes more profitable except when vessels with large capacities of 22000 TEUs are used. In addition, when the two ports in West and South Africa, Lome, and Durban, are potential ports of call in the service, then CGHR+ becomes more profitable than the SCR in all cases except two cases with seven ports. This can be explained by the additional revenue generated from the two additional African ports that compensates the additional costs related to longer sailing distances. However, for the cases with seven ports, adding these two ports to the service does not compensate the additional costs related to the long sailing distances, especially due to the small number of ports, and therefore the low total demand, with a large vessel capacity. When adding the time constraint, the SCR TC option becomes again more profitable than the CGHR TC in almost all the cases, which is expected given the shortcut it offers in terms of distance and therefore sailing time especially in contexts with constrained delivery lead times. Moreover, when the two African ports are added to the service, then the SCR TC remains more profitable when only seven or eleven ports are considered. However, CGHR+ TC comes back as the most profitable option in most of the cases with 25 ports especially with small to medium-sized vessels (8 out of 12 cases).

The other two conclusions that can be drawn from the results and that can be noticed in Figs. 6 and 7 are the following. First, the profit of Case B is the highest in all the configurations. This can be easily explained by the explosion of the demand and freight rates. In addition, for the same route and number of ports, the profit after COVID-19 pandemic (Case C) is lower than the profit before the pandemic. The reasons are also related to the dynamics of demand and freight rates. Second, for the small networks with only seven potential ports, whether the SCR or the CGHR is used, the smaller vessel capacities seem to
<figure><img src="image_8.png" /><figcaption>Fig. 4. Mean relative difference (RD) between the hybrid solution approach and the best solution.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 5. Mean computation time of the hybrid approach as a function of the number of iterations.</figcaption></figure>

generate more profit. However, for the large networks of ports, with 25 ports and without time constraints, the larger vessel sizes appear to be more profitable. However, when the time constraint is considered, vessels with large capacities are not recommended.

### 5.2.4. Utilization

Fig. 8 shows an example of the optimal utilization values over the arcs of the solution of the CGHR TC case with 11 ports for the 18,000 TEU vessel. It can be seen that the utilization values start increasing from the first port of call in East Asia, i.e., Busan, and keep increasing until the first port of call in Europe, i.e., Le Havre, and start decreasing after that until the last port of call in Europe, namely Bremerhaven. This result is expected because of the trade imbalance between Europe and Asia.

Table 8 provides the average and the maximum capacity utilization values for the cases with 11 ports based on the corresponding utilization values of the arcs of the optimal/best solution. The results show that, in general, the utilization average values decrease with the vessel capacity, which can be expected. However, there are some exceptions to that result in which the average utilization of the 12,000 TEU is higher than the other capacity values including the 6,000 TEU (e.g. Cases A, B and C under SCR model and CGHR+ TC). This can be explained by the solution that includes more ports and therefore more demand when the capacity increases, but that reaches a maximum potential demand with the capacity level of 12,000 TEUs and therefore the utilization starts to decrease when the capacity keeps increasing without additional ports and demand. The second conclusion from this table is that the maximum capacity utilization reaches 100% for the
<caption>**Table 6**</caption>

Relative improvement of the hybrid approach at 300 iterations compared to the traditional GA at 5000 iterations using the SCR without transit time constraints.

<table><caption>**Table 7**</caption><thead><tr><th rowspan="2"></th><th colspan="3">Case A</th><th colspan="3">Case B</th><th colspan="3">Case C</th></tr><tr><th>Profit<sub>GA</sub></th><th>Time (minutes)</th><th>Imp (%)</th><th>Profit<sub>GA</sub></th><th>Time (minutes)</th><th>Imp (%)</th><th>Profit<sub>GA</sub></th><th>Time (minutes)</th><th>Imp (%)</th></tr></thead><tbody><tr><td colspan="10" style="text-align:center; font-weight:bold;">7 ports</td></tr><tr><td>6000 TEU</td><td>3.72×10<sup>6</sup></td><td>27</td><td>0.00%</td><td>2.65×10<sup>7</sup></td><td>20</td><td>0.00%</td><td>3.60×10<sup>6</sup></td><td>32</td><td>0.00%</td></tr><tr><td>12000 TEU</td><td>2.47×10<sup>6</sup></td><td>31</td><td>1.01%</td><td>2.45×10<sup>7</sup></td><td>19</td><td>0.00%</td><td>2.34×10<sup>6</sup></td><td>28</td><td>1.56%</td></tr><tr><td>18000 TEU</td><td>1.11×10<sup>6</sup></td><td>27</td><td>0.00%</td><td>2.22×10<sup>7</sup></td><td>24</td><td>0.00%</td><td>9.99×10<sup>5</sup></td><td>24</td><td>0.00%</td></tr><tr><td>22000 TEU</td><td>7.74×10<sup>5</sup></td><td>32</td><td>0.00%</td><td>2.17×10<sup>7</sup></td><td>27</td><td>0.00%</td><td>6.21×10<sup>5</sup></td><td>33</td><td>5.90%</td></tr><tr><td colspan="10" style="text-align:center; font-weight:bold;">11 ports</td></tr><tr><td>6000 TEU</td><td>8.22×10<sup>6</sup></td><td>49</td><td>12.54%</td><td>5.34×10<sup>7</sup></td><td>21</td><td>0.15%</td><td>7.98×10<sup>6</sup></td><td>46</td><td>14.03%</td></tr><tr><td>12000 TEU</td><td>1.06×10<sup>7</sup></td><td>43</td><td>0.93%</td><td>6.85×10<sup>7</sup></td><td>8</td><td>1.16%</td><td>1.03×10<sup>7</sup></td><td>38</td><td>1.12%</td></tr><tr><td>18000 TEU</td><td>9.20×10<sup>6</sup></td><td>44</td><td>0.29%</td><td>6.62×10<sup>7</sup></td><td>9</td><td>0.94%</td><td>8.88×10<sup>6</sup></td><td>38</td><td>0.76%</td></tr><tr><td>22000 TEU</td><td>8.78×10<sup>6</sup></td><td>40</td><td>0.73%</td><td>6.61×10<sup>7</sup></td><td>9</td><td>0.27%</td><td>8.48×10<sup>6</sup></td><td>44</td><td>1.09%</td></tr><tr><td colspan="10" style="text-align:center; font-weight:bold;">25 ports</td></tr><tr><td>6000 TEU</td><td>1.03×10<sup>7</sup></td><td>56</td><td>91.27%</td><td>5.68×10<sup>7</sup></td><td>24</td><td>18.68%</td><td>1.08×10<sup>7</sup></td><td>52</td><td>82.51%</td></tr><tr><td>12000 TEU</td><td>1.73×10<sup>7</sup></td><td>60</td><td>64.75%</td><td>1.10×10<sup>8</sup></td><td>23</td><td>8.77%</td><td>1.58×10<sup>7</sup></td><td>56</td><td>77.07%</td></tr><tr><td>18000 TEU</td><td>2.23×10<sup>7</sup></td><td>56</td><td>48.89%</td><td>1.15×10<sup>8</sup></td><td>21</td><td>42.27%</td><td>2.10×10<sup>7</sup></td><td>48</td><td>55.73%</td></tr><tr><td>22000 TEU</td><td>2.70×10<sup>7</sup></td><td>35</td><td>28.42%</td><td>1.32×10<sup>8</sup></td><td>22</td><td>29.32%</td><td>2.78×10<sup>7</sup></td><td>52</td><td>22.43%</td></tr></tbody></table>

Profit for all the considered configurations from the hybrid solution approach.

<table><thead><tr><th></th><th></th><th>SCR</th><th>CGHR</th><th>CGHR+</th><th>SCR TC</th><th>CGHR TC</th><th>CGHR+ TC</th></tr></thead><tbody><tr><td colspan="8" style="text-align:center; font-weight:bold;">7 ports</td></tr><tr><td rowspan="4">Case A</td><td>6000 TEU</td><td>3.72 x 10<sup>6</sup></td><td>3.41 x 10<sup>6</sup></td><td>5.24 x 10<sup>6</sup></td><td>3.02 x 10<sup>6</sup></td><td>7.52 x 10<sup>5</sup></td><td>2.49 x 10<sup>6</sup></td></tr><tr><td>12000 TEU</td><td>2.50 x 10<sup>6</sup></td><td>1.83 x 10<sup>6</sup></td><td>3.61 x 10<sup>6</sup></td><td>1.75 x 10<sup>6</sup></td><td>-2.35 x 10<sup>4</sup></td><td>6.84 x 10<sup>5</sup></td></tr><tr><td>18000 TEU</td><td>1.11 x 10<sup>6</sup></td><td>1.05 x 10<sup>5</sup></td><td>1.83 x 10<sup>6</sup></td><td>-9.46 x 10<sup>4</sup></td><td>-9.46 x 10<sup>4</sup></td><td>-9.46 x 10<sup>4</sup></td></tr><tr><td>22000 TEU</td><td>7.74 x 10<sup>5</sup></td><td>-1.22 x 10<sup>5</sup></td><td>1.37 x 10<sup>6</sup></td><td>-1.22 x 10<sup>5</sup></td><td>-1.22 x 10<sup>5</sup></td><td>-1.22 x 10<sup>5</sup></td></tr><tr><td rowspan="4">Case B</td><td>6000 TEU</td><td>2.65 x 10<sup>7</sup></td><td>2.59 x 10<sup>7</sup></td><td>2.77 x 10<sup>7</sup></td><td>2.50 x 10<sup>7</sup></td><td>2.17 x 10<sup>7</sup></td><td>2.34 x 10<sup>7</sup></td></tr><tr><td>12000 TEU</td><td>2.45 x 10<sup>7</sup></td><td>2.33 x 10<sup>7</sup></td><td>2.51 x 10<sup>7</sup></td><td>2.29 x 10<sup>7</sup></td><td>1.88 x 10<sup>7</sup></td><td>2.04 x 10<sup>7</sup></td></tr><tr><td>18000 TEU</td><td>2.22 x 10<sup>7</sup></td><td>2.04 x 10<sup>7</sup></td><td>2.20 x 10<sup>7</sup></td><td>1.91 x 10<sup>7</sup></td><td>1.34 x 10<sup>7</sup></td><td>1.50 x 10<sup>7</sup></td></tr><tr><td>22000 TEU</td><td>2.17 x 10<sup>7</sup></td><td>1.98 x 10<sup>7</sup></td><td>2.14 x 10<sup>7</sup></td><td>1.88 x 10<sup>7</sup></td><td>1.31 x 10<sup>7</sup></td><td>1.47 x 10<sup>7</sup></td></tr><tr><td rowspan="4">Case C</td><td>6000 TEU</td><td>3.60 x 10<sup>6</sup></td><td>3.29 x 10<sup>6</sup></td><td>5.12 x 10<sup>6</sup></td><td>2.90 x 10<sup>6</sup></td><td>6.57 x 10<sup>5</sup></td><td>2.44 x 10<sup>6</sup></td></tr><tr><td>12000 TEU</td><td>2.38 x 10<sup>6</sup></td><td>1.71 x 10<sup>6</sup></td><td>3.49 x 10<sup>6</sup></td><td>1.64 x 10<sup>6</sup></td><td>-2.35 x 10<sup>4</sup></td><td>6.33 x 10<sup>5</sup></td></tr><tr><td>18000 TEU</td><td>9.99 x 10<sup>5</sup></td><td>-1.61 x 10<sup>4</sup></td><td>1.71 x 10<sup>6</sup></td><td>-9.46 x 10<sup>4</sup></td><td>-9.46 x 10<sup>4</sup></td><td>-9.46 x 10<sup>4</sup></td></tr><tr><td>22000 TEU</td><td>6.58 x 10<sup>5</sup></td><td>-1.22 x 10<sup>5</sup></td><td>1.24 x 10<sup>6</sup></td><td>-1.22 x 10<sup>5</sup></td><td>-1.22 x 10<sup>5</sup></td><td>-1.22 x 10<sup>5</sup></td></tr><tr><td colspan="8" style="text-align:center; font-weight:bold;">11 ports</td></tr><tr><td rowspan="4">Case A</td><td>6000 TEU</td><td>9.25 x 10<sup>6</sup></td><td>9.67 x 10<sup>6</sup></td><td>1.25 x 10<sup>7</sup></td><td>8.25 x 10<sup>6</sup></td><td>5.64 x 10<sup>6</sup></td><td>6.90 x 10<sup>6</sup></td></tr><tr><td>12000 TEU</td><td>1.07 x 10<sup>7</sup></td><td>1.05 x 10<sup>7</sup></td><td>1.32 x 10<sup>7</sup></td><td>9.30 x 10<sup>6</sup></td><td>5.81 x 10<sup>6</sup></td><td>7.91 x 10<sup>6</sup></td></tr><tr><td>18000 TEU</td><td>9.22 x 10<sup>6</sup></td><td>8.61 x 10<sup>6</sup></td><td>1.14 x 10<sup>7</sup></td><td>6.31 x 10<sup>6</sup></td><td>1.05 x 10<sup>6</sup></td><td>3.81 x 10<sup>6</sup></td></tr><tr><td>22000 TEU</td><td>8.85 x 10<sup>6</sup></td><td>8.11 x 10<sup>6</sup></td><td>1.09 x 10<sup>7</sup></td><td>6.09 x 10<sup>6</sup></td><td>7.52 x 10<sup>5</sup></td><td>3.51 x 10<sup>6</sup></td></tr><tr><td rowspan="4">Case B</td><td>6000 TEU</td><td>5.35 x 10<sup>7</sup></td><td>6.76 x 10<sup>7</sup></td><td>7.03 x 10<sup>7</sup></td><td>5.14 x 10<sup>7</sup></td><td>5.82 x 10<sup>7</sup></td><td>6.10 x 10<sup>7</sup></td></tr><tr><td>12000 TEU</td><td>6.93 x 10<sup>7</sup></td><td>6.85 x 10<sup>7</sup></td><td>7.13 x 10<sup>7</sup></td><td>6.65 x 10<sup>7</sup></td><td>6.05 x 10<sup>7</sup></td><td>6.33 x 10<sup>7</sup></td></tr><tr><td>18000 TEU</td><td>6.68 x 10<sup>7</sup></td><td>6.54 x 10<sup>7</sup></td><td>6.81 x 10<sup>7</sup></td><td>6.12 x 10<sup>7</sup></td><td>4.92 x 10<sup>7</sup></td><td>5.18 x 10<sup>7</sup></td></tr><tr><td>22000 TEU</td><td>6.63 x 10<sup>7</sup></td><td>6.48 x 10<sup>7</sup></td><td>6.75 x 10<sup>7</sup></td><td>6.09 x 10<sup>7</sup></td><td>4.88 x 10<sup>7</sup></td><td>5.15 x 10<sup>7</sup></td></tr><tr><td rowspan="4">Case C</td><td>6000 TEU</td><td>9.09 x 10<sup>6</sup></td><td>9.39 x 10<sup>6</sup></td><td>1.22 x 10<sup>7</sup></td><td>8.10 x 10<sup>6</sup></td><td>5.48 x 10<sup>6</sup></td><td>6.74 x 10<sup>6</sup></td></tr><tr><td>12000 TEU</td><td>1.04 x 10<sup>7</sup></td><td>1.02 x 10<sup>7</sup></td><td>1.30 x 10<sup>7</sup></td><td>9.03 x 10<sup>6</sup></td><td>5.53 x 10<sup>6</sup></td><td>7.78 x 10<sup>6</sup></td></tr><tr><td>18000 TEU</td><td>8.95 x 10<sup>6</sup></td><td>8.32 x 10<sup>6</sup></td><td>1.11 x 10<sup>7</sup></td><td>6.04 x 10<sup>6</sup></td><td>9.35 x 10<sup>5</sup></td><td>3.69 x 10<sup>6</sup></td></tr><tr><td>22000 TEU</td><td>8.57 x 10<sup>6</sup></td><td>7.83 x 10<sup>6</sup></td><td>1.06 x 10<sup>7</sup></td><td>5.82 x 10<sup>6</sup></td><td>6.33 x 10<sup>5</sup></td><td>3.39 x 10<sup>6</sup></td></tr><tr><td colspan="8" style="text-align:center; font-weight:bold;">25 ports</td></tr><tr><td rowspan="4">Case A</td><td>6000 TEU</td><td>1.98 x 10<sup>7</sup></td><td>2.97 x 10<sup>7</sup></td><td>3.58 x 10<sup>7</sup></td><td>1.61 x 10<sup>7</sup></td><td>1.33 x 10<sup>7</sup></td><td>2.22 x 10<sup>7</sup></td></tr><tr><td>12000 TEU</td><td>2.85 x 10<sup>7</sup></td><td>3.40 x 10<sup>7</sup></td><td>4.11 x 10<sup>7</sup></td><td>2.06 x 10<sup>7</sup></td><td>1.49 x 10<sup>7</sup></td><td>2.06 x 10<sup>7</sup></td></tr><tr><td>18000 TEU</td><td>3.32 x 10<sup>7</sup></td><td>3.40 x 10<sup>7</sup></td><td>3.84 x 10<sup>7</sup></td><td>1.49 x 10<sup>7</sup></td><td>1.39 x 10<sup>7</sup></td><td>1.71 x 10<sup>7</sup></td></tr><tr><td>22000 TEU</td><td>3.47 x 10<sup>7</sup></td><td>3.34 x 10<sup>7</sup></td><td>3.90 x 10<sup>7</sup></td><td>1.52 x 10<sup>7</sup></td><td>1.35 x 10<sup>7</sup></td><td>1.11 x 10<sup>7</sup></td></tr><tr><td rowspan="4">Case B</td><td>6000 TEU</td><td>6.75 x 10<sup>7</sup></td><td>1.64 x 10<sup>8</sup></td><td>1.70 x 10<sup>8</sup></td><td>5.93 x 10<sup>7</sup></td><td>1.00 x 10<sup>8</sup></td><td>6.54 x 10<sup>7</sup></td></tr><tr><td>12000 TEU</td><td>1.19 x 10<sup>8</sup></td><td>1.69 x 10<sup>8</sup></td><td>1.75 x 10<sup>8</sup></td><td>1.05 x 10<sup>8</sup></td><td>1.01 x 10<sup>8</sup></td><td>1.09 x 10<sup>8</sup></td></tr><tr><td>18000 TEU</td><td>1.63 x 10<sup>8</sup></td><td>1.68 x 10<sup>8</sup></td><td>1.71 x 10<sup>8</sup></td><td>1.10 x 10<sup>8</sup></td><td>1.13 x 10<sup>8</sup></td><td>1.20 x 10<sup>8</sup></td></tr><tr><td>22000 TEU</td><td>1.70 x 10<sup>8</sup></td><td>1.67 x 10<sup>8</sup></td><td>1.73 x 10<sup>8</sup></td><td>1.10 x 10<sup>8</sup></td><td>1.11 x 10<sup>8</sup></td><td>7.05 x 10<sup>7</sup></td></tr><tr><td rowspan="4">Case C</td><td>6000 TEU</td><td>1.97 x 10<sup>7</sup></td><td>2.91 x 10<sup>7</sup></td><td>3.52 x 10<sup>7</sup></td><td>1.58 x 10<sup>7</sup></td><td>1.31 x 10<sup>7</sup></td><td>2.19 x 10<sup>7</sup></td></tr><tr><td>12000 TEU</td><td>2.80 x 10<sup>7</sup></td><td>3.33 x 10<sup>7</sup></td><td>4.03 x 10<sup>7</sup></td><td>2.02 x 10<sup>7</sup></td><td>1.47 x 10<sup>7</sup></td><td>2.03 x 10<sup>7</sup></td></tr><tr><td>18000 TEU</td><td>3.27 x 10<sup>7</sup></td><td>3.33 x 10<sup>7</sup></td><td>3.76 x 10<sup>7</sup></td><td>1.45 x 10<sup>7</sup></td><td>1.37 x 10<sup>7</sup></td><td>1.70 x 10<sup>7</sup></td></tr><tr><td>22000 TEU</td><td>3.40 x 10<sup>7</sup></td><td>3.27 x 10<sup>7</sup></td><td>3.84 x 10<sup>7</sup></td><td>1.50 x 10<sup>7</sup></td><td>1.33 x 10<sup>7</sup></td><td>1.09 x 10<sup>7</sup></td></tr></tbody></table>

small vessel capacity values which is expected. The third result that can be drawn from Table 8 is related to the negative impact of time constraints on the utilization level. As it can be seen, the utilization level drops when the time constraints are activated due to the drop in the demand.

Moreover, the substantial difference between the utilization average and maximum values for each case is due to the imbalance in the trade

volumes between the arcs of the same service and in particular between European and Asian ports.

Similar conclusions can also be obtained from the utilization results provided in Tables B1 and B2 in the Supplementary Material.

In addition, the comparison of the results of the cases with seven, 11, and 25 ports shows that the vessel average and maximum capacity utilization increase with the number of potential ports in the service.
<figure><img src="image_10.png" /><figcaption>Fig. 6. Profit in the three cases (SCR, CGHR and CGHR+) for different vessel capacities before, during and after COVID-19.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 7. Profit in the three cases (SCR TC, CGHR TC and CGHR+ TC) for different vessel capacities before, during and after COVID-19 considering maximum time on commodity.</figcaption></figure>

For the cases with 25 ports, most of the maximum utilization values are equal to 100% (66 instances out of 72, Table B2).

### 5.2.5. Speed

Fig. 8 provides an example of the optimal speed values for the CGHR TC case with 11 ports. As it can be seen, the optimal speed on the blue arcs, corresponding to the sailing distance between the before last port of call in Europe, i.e., Hamburg, and the first port of call in East Asia, i.e., Busan, is equal to the minimum sailing speed, i.e., 12 kts. This result is justified by the low utilization rate and therefore the low sensitivity of the demand to the sailing time eastbound which justifies reducing the speed to save on the fuel cost. However, when

the utilization values start to increase in Busan, the demand sensitivity makes the sailing speed values increase until the speed reaches the maximum value of 25 kts between Shenzhen and Le Havre.

Table 9 provides the average and standard deviation values of the vessel sailing speed based on the corresponding speed values on the arcs included in the optimal/best service, for the cases with 11 ports.

As it can be seen, the average speed for most of the cases without the transit time constraint is very close to 12 kts with a standard deviation around 0 kt which means that the optimal speed on all the arcs of the service is equal to or very close to 12 kts which is the minimum possible technical speed. However, when the transit time constraints become active, the average speed increases values somehow midway
<table><caption>**Table 8**
Average (Maximum) vessel capacity utilization for the cases with 11 ports.</caption><thead><tr><th rowspan="2"></th><th rowspan="2"></th><th colspan="3">Without transit time</th><th colspan="3">With transit time</th></tr><tr><th>SCR</th><th>CGHR</th><th>CGHR+</th><th>SCR TC</th><th>CGHR TC</th><th>CGHR+ TC</th></tr></thead><tbody><tr><td rowspan="4">Case A</td><td>6000 TEU</td><td>54.38(100.00)</td><td>75.86(100.00)</td><td>76.22(100.00)</td><td>47.26(100.00)</td><td>58.20(100.00)</td><td>55.75(100.00)</td></tr><tr><td>12000 TEU</td><td>58.64(83.58)</td><td>64.27(92.58)</td><td>71.90(100.00)</td><td>35.50(74.25)</td><td>34.02(75.58)</td><td>64.37(100.00)</td></tr><tr><td>18000 TEU</td><td>39.06(55.72)</td><td>42.84(61.72)</td><td>52.79(67.83)</td><td>23.40(49.50)</td><td>32.35(55.11)</td><td>38.38(64.33)</td></tr><tr><td>22000 TEU</td><td>31.62(45.59)</td><td>35.05(50.50)</td><td>42.95(57.95)</td><td>19.15(40.50)</td><td>26.47(45.09)</td><td>31.41(52.64)</td></tr><tr><td rowspan="4">Case B</td><td>6000 TEU</td><td>56.05(100.00)</td><td>74.81(100.00)</td><td>75.17(100.00)</td><td>46.82(100.00)</td><td>59.45(100.00)</td><td>61.78(100.00)</td></tr><tr><td>12000 TEU</td><td>57.72(82.76)</td><td>65.38(87.04)</td><td>80.74(100.00)</td><td>34.49(73.42)</td><td>56.28(85.48)</td><td>63.96(99.15)</td></tr><tr><td>18000 TEU</td><td>38.00(55.17)</td><td>42.23(61.16)</td><td>52.11(67.14)</td><td>22.34(48.95)</td><td>31.90(54.69)</td><td>38.18(63.81)</td></tr><tr><td>22000 TEU</td><td>31.13(45.14)</td><td>34.55(50.04)</td><td>42.45(57.49)</td><td>18.27(40.05)</td><td>26.10(44.75)</td><td>31.24(52.20)</td></tr><tr><td rowspan="4">Case C</td><td>6000 TEU</td><td>53.67(100.00)</td><td>74.81(100.00)</td><td>75.17(100.00)</td><td>46.74(100.00)</td><td>56.49(100.00)</td><td>55.28(100.00)</td></tr><tr><td>12000 TEU</td><td>57.72(82.76)</td><td>63.34(91.73)</td><td>71.70(100.00)</td><td>34.87(73.42)</td><td>33.45(74.76)</td><td>63.96(99.15)</td></tr><tr><td>18000 TEU</td><td>38.43(55.17)</td><td>42.23(61.16)</td><td>52.11(67.14)</td><td>22.99(48.95)</td><td>32.11(54.59)</td><td>38.18(63.81)</td></tr><tr><td>22000 TEU</td><td>31.13(45.14)</td><td>34.55(50.04)</td><td>42.45(57.49)</td><td>18.81(40.05)</td><td>26.27(44.66)</td><td>31.24(52.20)</td></tr></tbody></table>

<figure><img src="image_12.png" /><figcaption>**Fig. 8.** Example of the speed and utilization values for the CGHR TC case with 11 ports for the 18,000 TEU vessel.</figcaption></figure>

between the minimum and maximum technical speeds, with a positive standard deviation and therefore a variability in the optimal speed values along the arcs of the route. It should be noted here that the average speed for the CGHR TC option is the highest which can be interpreted by the sensitivity of the main loaded demand from Asia to Europe to the long distance of the CGHR route which requires to speed up in the westbound direction to compensate for the longer distance. However, when the African ports are added, the average speed decreases a bit thanks to the demand between these two ports and the Asian and European ports. Moreover, for the option SCR TC, the average speed of the smaller vessels appears to be lower than that of the large vessels, although their maximum technical speed is higher. This can be explained by the smaller fuel consumption function coefficient of the smaller vessels compared to the larger vessels. However, for the CGHR TC and CGHR+ TC options, the average sailing speed of the small vessels appears to be larger with a smaller standard deviation. This can be interpreted mainly by the smaller revenue of these vessels due to their smaller capacity that requires capturing the maximum demand to compensate for the additional costs of the CGHR option and therefore that requires to speed up in order to reduce the demand sensitivity to transit time. Similar insights can be obtained from the cases with seven and 25 ports as shown in Tables B3 and B4 in the Supplementary Material.

### 5.3. Impact of demand uncertainty on the network decisions

In this section, we explore the impact of demand uncertainty on both strategic and tactical network decisions (route design) and operational decisions (sailing speed and quantities) by introducing a two-stage stochastic variant of our model. We define three demand (commodity) scenarios ($e$ = normal, $e$ = low, and $e$ = high), with each scenario having a probability of occurrence, denoted as $\zeta_e$. We then convert the deterministic model into a two-stage stochastic model, where $f$ is linked to the commodities transported in each scenario $e$. As a result, we update the mathematical model presented in 3.2 as follows. First, we revise the following elements of the objective function.

$$ \text{Revenue} = \sum_{e \in \mathcal{E}} \zeta_e \sum_{k \in \mathcal{K}^c} \phi_k L_e^k, \qquad (51) $$

$$ \text{Handling cost} = \sum_{e \in \mathcal{E}} \zeta_e \sum_{k \in \mathcal{K}^c} LC_{s_k} L_e^k + \sum_{k \in \mathcal{K}^c} UC_{d_k} U_e^k, \qquad (52) $$

$$ \text{Canal cost} = \sum_{e \in \mathcal{E}} \zeta_e \sum_{k \in \mathcal{K}^c} \sum_{j \in \mathcal{V}} (C_{sp} F_{j,sp}^{k,e} + C_{sp'} F_{j,sp'}^{k,e}), \qquad (53) $$

After that, we update the following commodity flow constraints and the transit time constraint. Valid inequalities involving the decision variable $F_{i,j}^k$ are updated accordingly. The remaining elements of the
<table><caption>**Table 9**
Average (Standard deviation) speed for the cases with 11 ports.</caption><thead><tr><th rowspan="2"></th><th rowspan="2"></th><th colspan="3">Without transit time</th><th colspan="3">With transit time</th></tr><tr><th>SCR</th><th>CGHR</th><th>CGHR+</th><th>SCR TC</th><th>CGHR TC</th><th>CGHR+ TC</th></tr></thead><tbody><tr><td rowspan="4">Case A</td><td>6000 TEU</td><td>12.00(0.00)</td><td>12.00(0.00)</td><td>12.13(0.07)</td><td>18.10(2.16)</td><td>25.33(1.82)</td><td>24.58(2.09)</td></tr><tr><td>12000 TEU</td><td>12.15(0.05)</td><td>12.23(0.19)</td><td>12.00(0.00)</td><td>19.62(1.01)</td><td>24.23(4.21)</td><td>22.09(5.75)</td></tr><tr><td>18000 TEU</td><td>12.37(0.04)</td><td>12.00(0.00)</td><td>12.01(0.03)</td><td>20.16(0.50)</td><td>21.59(5.41)</td><td>19.93(6.23)</td></tr><tr><td>22000 TEU</td><td>12.37(0.04)</td><td>12.00(0.00)</td><td>12.01(0.03)</td><td>20.15(0.51)</td><td>21.59(5.41)</td><td>19.93(6.24)</td></tr><tr><td rowspan="4">Case B</td><td>6000 TEU</td><td>12.00(0.00)</td><td>12.00(0.00)</td><td>12.00(0.00)</td><td>18.08(2.14)</td><td>20.49(4.92)</td><td>18.60(5.25)</td></tr><tr><td>12000 TEU</td><td>12.15(0.05)</td><td>12.00(0.00)</td><td>12.00(0.00)</td><td>19.12(0.54)</td><td>23.91(3.95)</td><td>22.08(5.75)</td></tr><tr><td>18000 TEU</td><td>12.00(0.00)</td><td>12.00(0.00)</td><td>12.01(0.03)</td><td>19.36(1.68)</td><td>21.20(5.91)</td><td>19.94(6.23)</td></tr><tr><td>22000 TEU</td><td>12.00(0.00)</td><td>12.00(0.00)</td><td>12.01(0.03)</td><td>19.39(1.67)</td><td>21.20(5.91)</td><td>19.94(6.22)</td></tr><tr><td rowspan="4">Case C</td><td>6000 TEU</td><td>12.00(0.00)</td><td>12.00(0.00)</td><td>12.13(0.07)</td><td>18.10(2.16)</td><td>24.87(1.33)</td><td>24.54(2.12)</td></tr><tr><td>12000 TEU</td><td>12.15(0.05)</td><td>12.23(0.19)</td><td>12.00(0.00)</td><td>19.64(1.00)</td><td>24.22(4.21)</td><td>22.07(5.74)</td></tr><tr><td>18000 TEU</td><td>12.37(0.04)</td><td>12.00(0.00)</td><td>12.01(0.03)</td><td>20.14(0.58)</td><td>21.58(5.40)</td><td>19.92(6.23)</td></tr><tr><td>22000 TEU</td><td>12.37(0.04)</td><td>12.00(0.00)</td><td>12.01(0.03)</td><td>20.14(0.51)</td><td>21.59(5.41)</td><td>19.96(6.20)</td></tr></tbody></table>

objective function and constraints remain unchanged.

$$ \sum_{k \in \mathcal{K}} F_{i,j}^{k,e} \le QX_{i,j} \qquad \forall i \in \mathcal{V}, j \in \mathcal{V}, e \in \mathcal{E}, \qquad (54) $$

$$ \sum_{j \in \mathcal{V}} F_{i,j}^{k,e} = \sum_{j \in \mathcal{V}} F_{j,i}^{k,e} \qquad \forall i \in \mathcal{V} \setminus \{s_k, s'_k, d_k, d'_k\}, k \in \mathcal{K}, e \in \mathcal{E}, \qquad (55) $$

$$ L^{k,e} = \sum_{j \in \mathcal{V}} \left( F_{s_k, j}^{k,e} + F_{s'_k, j}^{k,e} \right) \qquad \forall k \in \mathcal{K}, e \in \mathcal{E}, \qquad (56) $$

$$ U^{k,e} = \sum_{j \in \mathcal{V}} \left( F_{j, d_k}^{k,e} + F_{j, d'_k}^{k,e} \right) \qquad \forall k \in \mathcal{K}, e \in \mathcal{E}, \qquad (57) $$

$$ L^{k,e} \le w_k \qquad \forall k \in \mathcal{K}, e \in \mathcal{E}, \qquad (58) $$

$$ F_{i,j}^{k,e} \le w_k X_{i,j}^k \qquad \forall i \in \mathcal{V}, j \in \mathcal{V}, k \in \mathcal{K}, e \in \mathcal{E}, \qquad (59) $$

We conducted the experiments using $\zeta_e = 50\%$, 25% and 25% for the normal, low and high scenarios. We limit the analysis to the instances that were profitable in the deterministic model. Table 10 gives the cost components of the stochastic and deterministic solutions for all vessel sizes under Case B in the 7 ports problem. The optimal values of the strategic decisions under the stochastic setting remain the same as those of the deterministic case whereas the optimal quantities of the transported commodities are the only ones to change along with their corresponding cost elements. Fig. 9 shows the profit reduction due to commodity uncertainty, that is the relative difference between the deterministic profit and the weight average stochastic profit $100 \times \frac{Profit_D - Profit_S}{Profit_S}$, where $Profit_D$ is the profit from the deterministic model and $Profit_S$ is the weighted average profit from the stochastic model. In Case A, using CGHR the 18000 TEU vessel became not profitable under the stochastic scenario. This is due to the reduction in quantities in the low scenario that leads to higher costs than the collected revenue. Under Case B, profit reduction remains below 20% with an average reduction of 11%. Large vessels (18000 TEU and 22000 TEU) were impacted significantly by uncertainty in Cases A and C due to the limited commodities and high operational costs resulting from the vessel size.

## 6. Conclusion

We have presented a mathematical model that provides the ability to combine strategic network design aspects, tactical speed optimization and operational cargo pickup and delivery decisions. The aim has been to offer a model that can be used to account for structural economic changes in the liner shipping market, such as the general

<figure><img src="image_13.png" /><figcaption>**Fig. 9.** Profit reduction. Center represents the situation when the profit with uncertainty leads to non profitable solution. The first inner circle (0%) indicates non profitable service for both deterministic and stochastic cases.</figcaption></figure>

decline in demand for Asia–Europe trade and the increase in traffic in West African ports, and changes in maritime transport costs due variation in fuel costs or transit costs through the Suez Canal. More importantly, our model can be used to capture the shift that has happened in some of the maritime transport services from Suez Canal Route to the Cape of Good Hope route to avoid the recent risks and security issues in the Red Sea. This model helps us investigate whether the current elements could make the alternative liner shipping routes via the Cape of Good Hope viable. To answer this question, we developed a model allowing the sailing speed to vary between the edges with or without the consideration of a maximum transit time between loading at the source port and discharging at the destination ports. We also offered a hybrid solution approach to optimize the developed model, including a population-based heuristic method to optimize the route and an exact approach to optimize the sailing speed, commodity, and other decisions.

We validated the proposed approach using realistic data sets for container transport between Asia and Europe. In this business case, we considered that liner shipping companies are searching for the optimal solution to maximize their profit with four different services with vessel sizes of 6000, 12000, 18000 and 22000 TEUs and with the option of calling at different ports located in Asia (15 potential ports), Europe (10 ports) and South or West Africa (2 ports).
<table><caption>**Table 10**
Cost components of Case B under demand uncertainty for the two route options and the various vessels.</caption><thead><tr><th>Scenario</th><th>Normal</th><th>Low</th><th>High</th><th>Deterministic</th><th>normal</th><th>low</th><th>high</th><th>Deterministic</th></tr></thead><tbody><tr><td colspan="9">SCR</td></tr><tr><td colspan="9">6000 TEU</td></tr><tr><td>Revenue</td><td>3.02 x 10<sup>7</sup></td><td>2.11 x 10<sup>7</sup></td><td>3.02 x 10<sup>7</sup></td><td>3.02 x 10<sup>7</sup></td><td>3.02 x 10<sup>7</sup></td><td>2.11 x 10<sup>7</sup></td><td>3.02 x 10<sup>7</sup></td><td>3.02 x 10<sup>7</sup></td></tr><tr><td>Handling costs</td><td>5.68 x 10<sup>5</sup></td><td>3.97 x 10<sup>5</sup></td><td>5.68 x 10<sup>5</sup></td><td>5.68 x 10<sup>5</sup></td><td>5.68 x 10<sup>5</sup></td><td>3.97 x 10<sup>5</sup></td><td>5.68 x 10<sup>5</sup></td><td>5.68 x 10<sup>5</sup></td></tr><tr><td>Canal fees</td><td>2.36 x 10<sup>5</sup></td><td>1.66 x 10<sup>5</sup></td><td>2.36 x 10<sup>5</sup></td><td>2.36 x 10<sup>5</sup></td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>Profit</td><td>2.65 x 10<sup>7</sup></td><td>1.76 x 10<sup>7</sup></td><td>2.65 x 10<sup>7</sup></td><td>2.65 x 10<sup>7</sup></td><td>2.59 x 10<sup>7</sup></td><td>1.70 x 10<sup>7</sup></td><td>2.59 x 10<sup>7</sup></td><td>2.59 x 10<sup>7</sup></td></tr><tr><td>Weighted average profit</td><td></td><td>2.43 x 10<sup>7</sup></td><td></td><td>2.65 x 10<sup>7</sup></td><td></td><td></td><td>2.37 x 10<sup>7</sup></td><td>2.59 x 10<sup>7</sup></td></tr><tr><td colspan="9">12000 TEU</td></tr><tr><td>Revenue</td><td>3.02 x 10<sup>7</sup></td><td>2.11 x 10<sup>7</sup></td><td>3.02 x 10<sup>7</sup></td><td>3.02 x 10<sup>7</sup></td><td>3.02 x 10<sup>7</sup></td><td>2.11 x 10<sup>7</sup></td><td>3.02 x 10<sup>7</sup></td><td>3.02 x 10<sup>7</sup></td></tr><tr><td>Handling costs</td><td>5.68 x 10<sup>5</sup></td><td>3.97 x 10<sup>5</sup></td><td>5.68 x 10<sup>5</sup></td><td>5.68 x 10<sup>5</sup></td><td>5.68 x 10<sup>5</sup></td><td>3.97 x 10<sup>5</sup></td><td>5.68 x 10<sup>5</sup></td><td>5.68 x 10<sup>5</sup></td></tr><tr><td>Canal fees</td><td>2.52 x 10<sup>5</sup></td><td>1.76 x 10<sup>5</sup></td><td>2.52 x 10<sup>5</sup></td><td>2.52 x 10<sup>5</sup></td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>Profit</td><td>2.45 x 10<sup>7</sup></td><td>1.56 x 10<sup>7</sup></td><td>2.45 x 10<sup>7</sup></td><td>2.45 x 10<sup>7</sup></td><td>2.33 x 10<sup>7</sup></td><td>1.44 x 10<sup>7</sup></td><td>2.33 x 10<sup>7</sup></td><td>2.33 x 10<sup>7</sup></td></tr><tr><td>Weighted average profit</td><td></td><td>2.23 x 10<sup>7</sup></td><td></td><td>2.45 x 10<sup>7</sup></td><td></td><td></td><td>2.11 x 10<sup>7</sup></td><td>2.33 x 10<sup>7</sup></td></tr><tr><td colspan="9">18000 TEU</td></tr><tr><td>Revenue</td><td>3.02 x 10<sup>7</sup></td><td>2.11 x 10<sup>7</sup></td><td>3.02 x 10<sup>7</sup></td><td>3.02 x 10<sup>7</sup></td><td>3.02 x 10<sup>7</sup></td><td>2.11 x 10<sup>7</sup></td><td>3.02 x 10<sup>7</sup></td><td>3.02 x 10<sup>7</sup></td></tr><tr><td>Handling costs</td><td>5.68 x 10<sup>5</sup></td><td>3.97 x 10<sup>5</sup></td><td>5.68 x 10<sup>5</sup></td><td>5.68 x 10<sup>5</sup></td><td>5.68 x 10<sup>5</sup></td><td>3.97 x 10<sup>5</sup></td><td>5.68 x 10<sup>5</sup></td><td>5.68 x 10<sup>5</sup></td></tr><tr><td>Canal fees</td><td>2.52 x 10<sup>5</sup></td><td>1.76 x 10<sup>5</sup></td><td>2.52 x 10<sup>5</sup></td><td>2.52 x 10<sup>5</sup></td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>Profit</td><td>2.22 x 10<sup>7</sup></td><td>1.33 x 10<sup>7</sup></td><td>2.22 x 10<sup>7</sup></td><td>2.22 x 10<sup>7</sup></td><td>2.04 x 10<sup>7</sup></td><td>1.15 x 10<sup>7</sup></td><td>2.04 x 10<sup>7</sup></td><td>2.04 x 10<sup>7</sup></td></tr><tr><td>Weighted average profit</td><td></td><td>1.99 x 10<sup>7</sup></td><td></td><td>2.22 x 10<sup>7</sup></td><td></td><td></td><td>1.82 x 10<sup>7</sup></td><td>2.04 x 10<sup>7</sup></td></tr><tr><td colspan="9">22000 TEU</td></tr><tr><td>Revenue</td><td>3.02 x 10<sup>7</sup></td><td>2.11 x 10<sup>7</sup></td><td>3.02 x 10<sup>7</sup></td><td>3.02 x 10<sup>7</sup></td><td>3.02 x 10<sup>7</sup></td><td>2.11 x 10<sup>7</sup></td><td>3.02 x 10<sup>7</sup></td><td>3.02 x 10<sup>7</sup></td></tr><tr><td>Handling costs</td><td>5.68 x 10<sup>5</sup></td><td>3.97 x 10<sup>5</sup></td><td>5.68 x 10<sup>5</sup></td><td>5.68 x 10<sup>5</sup></td><td>5.68 x 10<sup>5</sup></td><td>3.97 x 10<sup>5</sup></td><td>5.68 x 10<sup>5</sup></td><td>5.68 x 10<sup>5</sup></td></tr><tr><td>Canal fees</td><td>2.52 x 10<sup>5</sup></td><td>1.76 x 10<sup>5</sup></td><td>2.52 x 10<sup>5</sup></td><td>2.52 x 10<sup>5</sup></td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>Profit</td><td>2.17 x 10<sup>7</sup></td><td>1.29 x 10<sup>7</sup></td><td>2.17 x 10<sup>7</sup></td><td>2.17 x 10<sup>7</sup></td><td>1.98 x 10<sup>7</sup></td><td>1.09 x 10<sup>7</sup></td><td>1.98 x 10<sup>7</sup></td><td>1.98 x 10<sup>7</sup></td></tr><tr><td>Weighted average profit</td><td></td><td>1.95 x 10<sup>7</sup></td><td></td><td>2.17 x 10<sup>7</sup></td><td></td><td></td><td>1.76 x 10<sup>7</sup></td><td>1.98 x 10<sup>7</sup></td></tr></tbody></table>

The obtained results show that the hybrid approach proposed in this paper seems to be a suitable alternative to the exact approach since it allows us to obtain solutions that are close to the optimal/best exact ones, within a reasonable computational time and for realistic instances of the considered problem. In addition, our practical results confirm the superiority of the SCR route, especially when we consider only services with 7 ports in Europe and Asia, and without the constraints of transit time between ports. The Cape of Good Hope can then become profitable for smaller vessels (6000 TEU) if we consider 11 Asian and European ports. If we add the 2 ports of call located in West (Port of Lome) and South Africa (Port of Durban), the additional revenue generated by the possibility of calling at these ports can be enough to offset the additional costs associated with the need to sail longer distances, except for larger units (22,000 TEU). In general, the rise of the African economy could then lead to the emergence of a viable alternative maritime route for some of the Asia-Europe services (e.g. services without restriction on the commodity transit time), that could then sail via the Cape of Good Hope.

The work presented in this paper can be extended in several directions. One interesting future aspect could involve the analysis of various maritime routes, such as those between Asia and America, as well as Arctic routes. Moreover, an extension to this work that may have a theoretical value is the consideration of the design problem of shipping networks including multiple liner services that may interconnect at certain port hubs which allows for the consideration of transshipment. Another avenue for exploration could be examining the sensitivity of demand to transit times and its implications for profit maximization and speed optimization. Additionally, considering vessel emissions

as a secondary objective function would be particularly relevant in the context of the IMO's strategy to reduce emissions from maritime shipping. Further research could also investigate the effects of both demand and price uncertainties on decision-making processes. Finally, the development of a multimodal transportation model could offer valuable insights into integrated transport solutions.

### CRediT authorship contribution statement

**Sadeque Hamdan:** Writing – review & editing, Writing – original draft, Visualization, Validation, Software, Resources, Methodology, Formal analysis, Conceptualization. **Dominique Feillet:** Writing – review & editing, Writing – original draft, Validation, Supervision, Resources, Methodology, Conceptualization. **Ali Cheaitou:** Writing – review & editing, Writing – original draft, Validation, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **Pierre Cariou:** Writing – review & editing, Writing – original draft, Validation, Resources, Investigation, Data curation, Conceptualization. **Nadjib Brahimi:** Writing – review & editing, Writing – original draft, Validation, Project administration, Methodology, Investigation, Formal analysis, Conceptualization.

### Appendix A. Supplementary data

Supplementary material related to this article can be found online at https://doi.org/10.1016/j.ejor.2025.03.008.
# Data availability

The data that support the findings of this study are available from the corresponding author, S.Hamdan, upon reasonable request.

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