On the optimization of green multimodal transportation: a case study of the West German canal system

Binsfeld, Tom; Hamdan, Sadeque; Jouini, Oualid; Gast, Johannes Article — Published Version On the optimization of green multimodal transportation: a case study of the West German canal system Annals of Operations Research Provided in Cooperation with: Springer Nature Suggested Citation: Binsfeld, Tom; Hamdan, Sadeque; Jouini, Oualid; Gast, Johannes (2024) : On the optimization of green multimodal transportation: a case study of the West German canal system, Annals of Operations Research, ISSN 1572-9338, Springer US, New York, NY, Vol. 351, Iss. 1, pp. 667-726, https://doi.org/10.1007/s10479-024-06075-5 This Version is available at: https://hdl.handle.net/10419/330543 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Annals of Operations Research (2025) 351:667–726 https://doi.org/10.1007/s10479-024-06075-5 ORIGINAL RESEARCH On the optimization of green multimodal transportation: a case study of the West German canal system Tom Binsfeld1,3 ·Sadeque Hamdan2·Oualid Jouini1·Johannes Gast3 Received: 30 May 2023 / Accepted: 22 May 2024 / Published online: 4 June 2024 © The Author(s) 2024 Abstract In this study, we address a biobjective multimodal routing problem that consists of selecting transportation modes and their respective quantities, optimizing transshipment locations, and allocating port orders. In the objective functions, we minimize total transportation costs and use the EcoTransit methodology to minimize total greenhouse gas emissions. The optimization model selects the transportation mode and transshipment port where quantities are transshipped from one mode to another. We compare inland waterway transportation and trucks encountering infrastructure failures that require rerouting or modal shifting in a reallife case study on the supply of goods for the chemical industry in the West German canal system. We propose a population-based heuristic to solve large instances in a reasonable computation time. A sensitivity analysis of demand, of varying lock times, and of infrastructure failure scenarios was conducted. We show that compared with inland waterway transportation, multimodal transportation reduces costs by 23% because of longer lock times. Our analysis shows that the use of inland waterway transportation only during infrastructure failures imposes nearly 28% higher costs per day depending on the failure location compared to that of the case of no failures. We also show that the use of a multimodal transportation system helps to reduce this cost increase in lock failure scenarios. Keywords Multimodal transportation ·Inland waterway transport ·Greenhouse gas emissions ·Sustainability ·Vehicle routing ·Modal shift ·Optimization BSadeque Hamdan [email protected] Tom Binsfeld [email protected] Oualid Jouini [email protected] Johannes Gast [email protected] 1Laboratoire Genie Industriel, Université Paris-Saclay, CentraleSupélec, 3 Rue Joliot-Curie, Gif-sur-Yvette 91190, Île-de-France, France 2Bangor Business School, Bangor University, College Rd, Bangor, Gwynedd LL57 2DG, UK 34flow Research, 4flow, Hallerstraße 1, 10587 Berlin, Germany 123 668 Annals of Operations Research (2025) 351:667–726 1 Introduction While the need for energy security has induced pressure on economies and societies in 2024, inland waterways ensure a scalable supply of energy feedstock. Under typical operating conditions, waterways are reliable and flexible transport systems based on an efficient infrastructure (Federal Ministry of Transport and Digital Infrastructure, 2019). However, extreme weather and dilapidated infrastructure threaten the availability of waterways for freight transport in Germany. Approximately 18 million tons of goods are transported monthly on German inland waterways depending on their availability. This volume equates to more than two million long-haul truckloads (Federal Office of Statistics, 2019). In general, public authorities aspire to further utilize existing capacity reserves of this environmentally friendly mode of transport; the plan is to shift traffic from roads to inland waterways: The European Union is pursuing the target of doubling their modal shift share up to 9% in 2030, as aligned with the German "Masterplan Binnenschiff" from the German Federal Ministry of Transport and Digital Infrastructure (Sims et al., 2014). Overall, inland waterway transport represents an elementary component of the German andEuropeanlogisticalsupplychains.Nonetheless,inlandbargescannotserveandsatisfythe logistical requirements of every industry. Other transportation modes, such as trucks, promise greater flexibility and availability while not requiring dedicated infrastructure (i.e., ports and canals). Evolving risks, such as infrastructure failures or climate change, among others, also impact transportation mode choice. Hence, multimodal transport is often established to exploit the advantages of each mode. Infrastructure failures, such as lock failures or bridge damage, are the main reasons for the nonavailability of whole canal sections, resulting in specific ports becoming temporarily unavailable (Gast et al., 2020). This unavailability has caused significant (economic) damage to companies. For example, the low amount of water on the Rhine River in 2018 induced a cost of e245 million for a chemical company in Germany (Reuters, 2019). Moreover, these risks lead to higher costs, higher emissions, and missing the time schedule; such issues can incur additional costs in the downstream supply chain. These observations highlight the need for a novel multimodal concept to ensure good flow along the supply chain, regardless of the availability of primary transportation infrastructure at the time. In this study, we formulate an optimization problem that addresses these issues. In this formulation, different transportation modes can be used either directly from the depot or at any port that acts as a transshipment port, where quantities unloaded by one mode are shifted to another mode of transportation. The problem is formulated as a biobjective mathematical model in which we minimize the total transportation cost and the greenhouse gas emissions of the network system. The proposed mixed-integer linear model helps decision-makers to identify the optimal route for each transportation mode and each vehicle, the quantity transported by each vehicle and each transportation mode, the location of the transshipment port, and the quantity shifted from one mode to another. We demonstrate how this multimodal consideration can be reduced to a single-mode optimization model. We combine the advantages of different transportation modes and develop an optimization tool to determine the optimal transportation mode. The problem is NP-hard, as we show in Sect. Aof the Appendix. Therefore, we propose a population-based heuristic to solve this problem. The heuristics generate a set of feasible individuals, and the model attempts to improve these individuals in each iteration by using 18 operators. We also compare different scenarios by using single and multiple modes. Furthermore, we analyze the impact of different scenarios on the transport of chemical goods in the West German canal system in a case study. 123 Annals of Operations Research (2025) 351:667–726 669 This study provides the following contributions. We optimized the selection of transshipment ports that can be used for both loading and unloading in this setting, which is inspired by a real-world problem, given the multimodal choice of either truck or inland waterway transportation. This type of problem has received limited attention in the literature. Furthermore, we allow for different types of vehicles to be chosen; in the literature, the focus has been on a single-vehicle type in multimodal problems. By adding a virtual port to act as a depot, our approach maintains model linearity; this approach avoids further complexity in this NP-hard problem. We use a biobjective formulation to model the potentially conflicting objectives of cost reduction and reduction of greenhouse gas emissions, with emissions calculated by using the EcoTransit methodology. Although this problem can still be solved in an acceptable runtime for small to medium instances, solving large instances optimally becomes infeasible. To address this issue, we developed a particular population-based heuristic that performs well, with less than 5% error from the best exact solution found and more than 83% time savings in most cases. We derive several managerial insights by using a practical case study from the research project Preview. In this project, we assess the vulnerability and the resilience of supply chains that depend on the infrastructure of the West German canal systems by mainly requiring inland waterway transport. We highlight the benefits of multimodal transportation over single-mode transportation in terms of cost and emissions under various scenarios. The scenarios vary in terms of lock time, demand, and whether infrastructure failure occurs. Interestingly, it appears that the use of multimodal transportation reduces the impact of increasing lock times; moreover, multimodal transportation allows for savings in the case of infrastructure failure. This demonstrates the economic effectiveness of rerouting and modal shifts as risk-mitigation strategies for supply chains. We calculated the cost of reducing emissions by using each transportation model, and we showed that the multimodal transportation mode provides a faster reduction rate of 1.12% of emissions for each 1% increase in the costs. The remainder of this study is organizedas follows. In Sect.2, we review the relevant work related to this study. In Sect.3, we describe the methodology used in this study. Section4 contains the solution approach. In Sect.5, we present a case study, and in Sect.6, we describe the numerical experiments related to the proposed heuristic. Section7provides managerial insights. Finally, Sect.8concludes the paper, and we highlight future research avenues. 2 Literature review The multimodal transportation model is a routing problem variant that uses multiple vehicles and different transportation modes. The use of multiple vehicles is similar to the multiple traveling salesman problem (m-TSP), which is a generalization of the TSP problem, which originally includes only one vehicle (salesperson). One important feature of the multimodal problem is the use of a transshipment point, where quantities are transferred from one mode to another. Therefore, in this section we review the related literature. First, we discuss different routing model variants. Then, we review works that focus on multimodal transportation and provide context for our work within the literature. 2.1 Routing model variants and solution algorithms The TSP is a fundamental routing challenge to find the shortest optimal routes to minimize travel costs. First introduced in the 1930s and extensively analyzed since then, the TSP requires finding a sequence for a salesperson to visit a set of nodes exactly once, starting 123 670 Annals of Operations Research (2025) 351:667–726 and ending at a depot (Miller et al., 1960). Gutin and Punnen (2006) offered an overview up to 2006, highlighting the broad applicability of the TSP and the emergence of many variants. The m-TSP, involving multiple salespeople, is determine the optimal sequence for multiple vehicles, with each salesman visiting a subset of nodes exactly once (Miller et al., 1960). Rao (1980) explored both symmetric and asymmetric m-TSPs and compared their transformations.TheVRP,whichiscloselyrelatedtotheTSP,differsinthatitseekssequences for vehicles while considering their capacity. Variants include simultaneous pick-up and delivery (Min, 1989), split pick-up m-VRP (Lee et al., 2006), and multidepot m-VRP with fuel constraints (Sundar et al., 2016). A two-echelon multivehicle location-routing problem with time windows was introduced by Govindan et al. (2014), and a hybrid multiobjective multidepot VRP was developed by Londoño et al. (2024) to minimize the distance and the control route length standard deviation. Other interesting variants include the railway TSP, where salespeople utilize railways to minimize travel time, considering trains’ schedules and nonstop routes (Hadjicharalambous et al., 2007), and the colored TSP and colored bottleneck TSP,which address multimachine engineeringsystem planning problems (Dong& Cai, 2019; Dong et al., 2023). The traveling purchaser problem (TPP), a routing and purchasing challenge, involves visiting suppliers to buy products at varying prices to satisfy demand at the lowest cost. The TPP is distinguished by the need to minimize both traveling and purchasing costs, making the problem more complex (Cheaitou et al., 2021b). Variants of the TPP include deterministic, biobjective, and budget constraints or total quantity discounts (Manerba et al., 2017;Ravi &Salman,1999; Riera-Ledesma & Salazar-González, 2005; Manerba & Mansini, 2012). Finally, the Family TSP is focused on minimizing costs to visit a predetermined number of cities, requiring decisions on which cities to visit within each group (Bernardino & Paias, 2018). For further exploration of variants such as quota, profit-based, and time window TSPs, readers can refer to Ilavarasi and Joseph (2014), Pop et al. (2024). The TSP and its variants, which are strongly NP-hard, have prompted researchers to develop various heuristics for efficient solution finding. Xing and Tu (2020)employeda Monte Carlo tree search to offer an alternative to traditional exact methods for the TSP. For clustered cities, Jafarzadeh et al. (2017) introduced an enhanced genetic algorithm that was based on nearest neighbor search, while Smith and Imeson (2017) developed a competitive large neighborhoodsearch heuristic forhard-instance andnonclustered problems. Mahmoudinazlou and Kwon (2024) proposed a hybrid genetic algorithm for the m-TSP to shorten the longest tour length. In addressing the TPP, strategies such as tabu search (El-Dean, 2008; Mansini et al., 2005), simulated annealing (Voß, 1996), and genetic algorithms (Almeida et al., 2012; Goldbarg et al., 2009) have been utilized. Roy et al. (2023) studied a multivehicle clustered TPP with a variable-length genetic algorithm to minimize system costs by optimizing cluster selection, market visits, procurement quantities, and routing. For the family TSP, Bernardino and Paias (2021) applied population-based heuristics combined with local search methods. An overview of exact and heuristic algorithms for TPP variants is given by Manerba et al. (2017); they highlight the diverse solution approaches in the field. 2.2 Multimodal transportation The use of different modes of transportation, such as ships, trains or trucks, can lead to a tradeoff between transit time and cost. Every combination is possible in theory, but only some of the possible combinations are common in logistics systems of multimodal transport, such as truck/vessel-train-truck and truck/vessel-ship-truck, depending on the leg considered in the 123 Annals of Operations Research (2025) 351:667–726 671 supply chain. ViaDonau (2012) introduced loading units to accelerate unloading processes that occur in multimodal transportation networks. Currently, there are standardized load units suchas containers, swap bodies, and semitrailers.When transshipping from inlandwaterways to trucks, for example, products are transferred to containers (Ghiani et al., 2004). In recent studies, researchers have combined trucks and drones in multimodal models (Jeong et al., 2019). An example of a truck-air model is a multimodal hub location and hub network design problem, such as that developed by Alumur et al. (2012). SteadieSeifi et al. (2014) mentioned the different definitions of terminologies that circulate in the literature, such as multimodal, intermodal, comodal, and, more recently, synchromodal transportation. After revising these definitions, we refer to this model as a multimodal transportation model. Infante et al. (2009) developed a ship-truck TPP where the aim is to define the sequence with a focus on where trucks should distribute goods to final warehouses. Sun et al. (2018) proposed a biobjective nonlinear truck-rail routing model that minimizes the total cost and total CO2 emitted. In contrast to their work, our linear model does not require specific service sets for each mode of transportation because mode exchange can occur at any node. Fazayeli et al. (2018) developed a multimodal routing-location model in which mode changes are allowed at predefined nodes. Again, the model presented in this manuscript optimizes transshipment locations and does not require specifying transshipment locations, unlike their work. HaoandYue(2016) used dynamic programming to solve a multimodal transportation model. They did not consider the quantities transported in their model. Instead, they assumed that the same model delivers all the quantities. Compared to their model, our model considers thenumberofvehicles,deliveryquantity,andpossibilityofdeliveryviatwomodestothesame node. Zhang et al. (2011) solved a multimodal uncapacitated routing problem. The model selected one mode for each city pair. However, it did not consider the transport quantities or the possibility of having multiple vehicles with different capacities. Xiong and Wang (2014) studied a biobjective multimodal routing problem with a time window. The model minimizes the total transportation cost and total traveling time. However, they do not consider the capacity of each transportation mode. They integrated the k-shortest path and a genetic algorithm to address this problem. Moccia et al. (2011) studied the multimodal routing problem with shipment consolidation options and time windows. Zameni and Razmi (2015) proposed an uncapacitated mixed-integer multimodal hub location-routing problem with simultaneous pickups and deliveries. The proposed model allocates a transportation mode to each route between the hubs and minimizes the total network cost. Riessen et al. (2015) considered barge and rail transportation modes in the container transportation problem in northwest Europe. In their model, they did consider the penalty for overdue deliveries. Demir et al. (2016) developed an intermodal service network design problem that reduces costs and emissions while using inland waterway transportation, rail, and trucks. Assadipour et al. (2016) proposed a bilevel biobjective mathematical model to regulate hazmat shipments by using road and rail transportation modes. The model was solved by using a particle swarm algorithm. Qu et al. (2016) considered emission and transfer costs in service network design in a multimodal transportation problem. Tawfik and Limbourg (2019) proposed a bilevel path-based intermodal mathematical model that maximizes profit and minimizes disutility. Wang and Qi (2019) studied the design of a time-dependent service network with multiple service types. More recently, Nitsenko et al. (2020) studied the risks of multimodal transportation by using fuzzy logic. Kaewfak et al. (2021) developed a decision support model to determine optimal multimodal routes. The proposed multiobjective model considered transportation costs, transportation time, and seven transportation risks to improve logistics and transportation performance. In addition, the analytic hierarchy process and zero–one goal programming 123 672 Annals of Operations Research (2025) 351:667–726 methods were used to determine multimodal route selection. He et al. (2021) proposed an approach to a multimodal network model and robustness assessment for freight transport networks. They analyzed the interdependencies between transport modes and considered the disruptions of single nodes. The robustness of the network was considered based on the travel time resulting from perturbations in the network. Their model helps to schedule maintenance operations by prioritizing the critical elements in the network. The approach differs based on the proposed algorithm and the multiobjective functions. Przystupa et al. (2021) developed a multiobjective optimization model to solve a multicriteria transport problem. Their algorithm is for large-data case studies with any number of types of transport and optimization criteria. The case study considered minimizes the transportation costs and transportation risk levels. Real et al. (2021) proposed a mixed integer programming model to solve multimodal hub design problems with flexible routes, and they developed metaheuristics based on an adaptive large neighborhood search. Ye et al. (2021) proposed a bilevel mathematical model to determine the transfer location and the infrastructure capacity for a multimodal transportation network design with elastic demand. Readers may refer to Elbert et al. (2020)for a systematic review of the topic. To the best of our knowledge, none of the models in the reviewed literature optimize the selection of the transshipment port in a capacity-constrained linear problem formulation. In addition, we formulated a biobjective model by using the EcoTransit emission calculator, applied the model to a real case study, and analyzed the effect of infrastructure failure on the multimodal formulation. Real-life analysis allows for a better understanding of the optimization problem, as well as more insightful results. 3 Methodology Our main target is to optimally determine the transportation mode and the route by considering both the total cost and the emissions. Additionally, we study the impact of considering different modes on the total cost and emissions. To achieve these goals, we compared three transportation modes, namely, inland waterway transportation, trucks, and multimodal transportation (Fig.1). The multimodal mode allows switching from one mode to another, where quantities are unloaded at a transshipment port and loaded in another mode. We consider the scenario of delivering chemical products from a depot to different ports by using three transportation modes. We list all the sets, parameters, and decision variables in Sect.3.1. Then, we describe the problem, and we present the modeling concept in Sect.3.2.Thecost and emission calculations required for the objective function formulation are presented in Sects.3.3 and 3.4. Finally, we present the biobjective model and its solution approach in Sects.3.5 and 3.6, respectively. 3.1 Notations The sets used are as follows: •L: Set of ports included in the network indexed by iand j. The indices i0,i1and i1 represent the virtual, actual and duplicated actual depots, respectively. •M: Set of transportation modes indexed by m. •Km: Set of vehicles belonging to transportation mode m, indexed by k. Table 1lists the vehicle parameters used; they were classified as inland waterway transportation and trucks. In addition, we have: 123 Annals of Operations Research (2025) 351:667–726 673 Fig. 1 The considered transportation modes and objectives •Di: Demand at port i[t/day]. •Sis a small number. The decision variables for the optimization model defined in Sect.3.5 are: •yk,m i: Decision variable that equals 1 if port iis visited by vehicle kof mode m. •Xk,m i,j: Decision variable that equals 1 if vehicle kof mode mtravels from port ito jby using vehicle kof mode m. •Qk,m i: Quantity for port iloaded at vehicle kof mode m. •uk,m i: Integer variable representing the sequencing of visits for port iand vehicle kof mode m. •fk,m i,j: Continuous (nonnegative) variable that gives the total quantity loaded in vehicle k of mode mfrom ito j. •UQk.m i: Transshipment quantities or transfer cargo unloaded at port iby using vehicle k in mode m. These quantities are then transported by another vehicle(s) and mode mto other ports. •Tk,m i: Decision variable (binary) equals 1 if vehicle kof mode mis used to transport quantities from transshipment port i. 3.2 Problem description and modeling idea Goods are to be transported from a depot port (i1=Port 1, as shown in Fig.2)toother ports by using inland waterway transportation, trucks, or both modes to fulfill the demand at each port i. Trucks are available at all ports. Thus, they can be used for either the main service of transporting goods from the depot to other ports or the secondary service of transporting goods from a transshipment port to other ports. Figure2shows one main service starting from the depot (Port 1), represented by solid arrows, and visiting Ports 2, 3 and 4, after which it returns to the depot. Figure2also shows one secondary service (transshipment) denoted by dashed arrows starting from Port 3 and visiting Ports 5 and 6. Because inland waterway transportation is only available at the depot port, it can be used only for the main service. 123 674 Annals of Operations Research (2025) 351:667–726 Table 1 Description of the parameters for the various transportation modes Notation Description Unit Inland Waterway transportation Truck αk,mVehicle unit cost to cover maintenance and depreciation [e/h]†,[e/km]†† xx βk,mHourly personnel cost per worker [e/h] x x μk,mUnit fuel cost [e/km] x x pPort charges paid by inland waterways upon unloading [e/t] x γk,mCost of handling transshipment (cost of loading cargo at another vehicle when changing the transportation mode) [e/t] x x k,mCost of handling deliveries (unloading costs at destination port) [e/t] x x k,mRent cost for container [e/h] x k,mhourly truck fixed cost to cover insurance, parking and capital return [e/h] x Tk,mToll rate on highways [e/km] x nk,mNumber of workers needed to operate the vehicle −x∗ dk,m i,jDistance between ports iand jusing transportation mode mof vehiclek [km] x x qi,jNumber of locks between ports iandj −x hDocking time at the ports [h] x τLock time at the locks [h] x Ck,mCapacity of transportation mode mof vehiclek [t] x x ηHandling performance of port during loading and unloading activities [t/h] x sk,mSpeed of transportation mode [km/h] x x κk,m i,jEmpty percentage factor for transportation mode −xx zconversion factor from MJ to kWh x rk,mLoad factor (utilization of transport mode) −xx ρEnergy factor (tank-to-wheel) [g/MJ] x x 123 Annals of Operations Research (2025) 351:667–726 681 startsfromthisport to deliverto other ports doesnotincludeadelivery tothisport.Constraints (25)and(26) are Miller–Trucker–Zemlin (MTZ) subtour elimination constraints that ensure that a vehicle does not subtour on a route. Constraint (27) defines the flow from ito jas the flow that enters ifrom all possible jvalues minus the quantities. Constraint (28) defines the first flow as the total loaded quantity. Constraint (29) links fk,m i,jwith Xk,m i,jso that fk,m i,jis zero if the corresponding Xk,m i,jis zero. Constraint (30) ensures that the total quantity unloaded by multiple vehicles at one port is equal to the demand. Constraint (31) ensures that the total delivered and unloaded quantities of vehicle kin mode mdo not exceed the vehicle capacity if the vehicle is used. Constraint (32) links Tk,m jwith Xk,m i,jsuch that if Tk,m j=1, the vehicle moves directly from the virtual depot to the transshipment port j. Constraint (33) states that if vehicle kof transportation mode mis used from the virtual depot, the vehicle must either use the arc from the virtual depot to the actual depot or be used at a transshipment port. Constraint (34) ensures that the quantity delivered to the ports from transshipment port jdoes not exceed the total quantity unloaded by other vehicles from other modes at port j. Constraint (35) ensures that unloaded quantities are allowed if and only if the vehicle moves from the virtual depot to the actual depot (i.e., the vehicle starts from the actual depot). Constraint (36) ensures that the number of vehicles and their capacities are sufficient to transport the unloaded quantity at port i. Constraint (37) ensures that all the unloaded quantities are delivered. Constraint (38) ensures that if a vehicle is used at the exchange port, then it must deliver some quantities; that is, the initial flow should be positive. Constraint (39) states that the vehicle must carry some nonzero quantities if it travels from the virtual port to the actual port. Constraint (40) ensures that if a vehicle moves from the virtual depot to the original depot, it moves from the original depot to the virtual depot. Consequently, the tour is complete. Constraint (41) prevents unloading at the actual and virtual depots. Sect. Bof the Appendix shows the steps required to convert the multimodal model into a single-mode model. 3.6 Biobjective optimization approach Dealing simultaneously with cost minimization and emission reduction leads to a biobjective optimization with two conflicting goals. Improving cost efficiency may worsen emissions and vice versa. For instance, in inland waterway transportation, prioritizing cost minimization might emphasize total docking and freight-related costs over total distance and fuel costs. This approach can result in better cost performance but can produce higher emissions due to a slight increase in fuel consumption and distance traveled. Conversely, focusing on emission reduction tends to reduce travel distance and, consequently, fuel consumption, leading to lower emissions while inducing higher costs associated with docking and freight handling. Several studies in the literature have explored these conflicting costs and emission objectives across different modes of transportation, such as ground transportation (Cheaitou et al., 2021b; Demir et al., 2016; Molina et al., 2014) and maritime transportation (Dulebenets, 2018;Zhaoetal.,2019). As a result, there is a set of nondominated, efficient, optimal solutions that are Pareto-optimal (Fonseca & Fleming, 1998). Pareto points can be obtained by using several methods, such as the utility function method, lexicographic method, goal programming, normal boundary intersection method and evolutionary algorithms (GhaneKanafi & Khorram, 2015; Singh Yadav, 2023). The method choice depends on the preference availability from decision-makers (no preference method, a priori, a posteriori and interactive methods). In this work, an a priori scalarization method is used because the decision-maker preference is known in advance. The weighted comprehensive criterion method is used to 123 682 Annals of Operations Research (2025) 351:667–726 solve the biobjective problem because of its simplicity and suitability for heuristics because it does not introduce additional constraints. In this method, the multiobjective function must be divided into two single functions that dispose of different units and orders of magnitude. On the one hand, the cost function is in e, and on the other hand, the emission function is in g. Single objective functions are defined as objective functions and are subject to the same constraints in the respective sections. Both single subproblems are solved to obtain the optimal solutions, which we call cmin and emin. After receiving both single optimal solutions, we merge them into a single normalized objective function (Dehghani et al., 2013). The equations used to normalize the objective functions are as follows: c=c−cmin cmin ,(42) e=e−emin emin .(43) Equations (42)and(43) are the relative variations between the single objective functions of the transportation costs and emission costs and their respective optimal values. We then multiply each normalized single-objective function by a relative weight, depending on the decision-maker. This multiplication leads to the following objective function. min δ=a1×c+a2×e,(44) where a1and a2are the weights that depend on the decision-maker, and a1+a2=1. This objective function is minimized and is subject to the same constraints as the initial problem. For a given set of a1and a2, there is only one Pareto-optimal solution. However, if these weights are changed, different Pareto-optimal solutions may result (Marler & Arora, 2004). The use of the scalarization technique may not result in all Pareto optimal points, as this depends on convexity, among other factors (Ghane-Kanafi & Khorram, 2015). 4 Solution approach This problem is a generalization of the traveling purchaser problem, which is an NP-hard problem (see Sect. Ain the appendix). Therefore, the problem cannot be solved by using an exact approach for large instances. We use a population-based heuristic to overcome this issue by designing search operators to align with the studied problem. An interesting advantage of population-based heuristics is the possibility of adapting them through the design of search operators, number of iterations and population size to control solution quality and computation time. Population-based heuristics have been used successfully in the literature to solve similar problems, such as TSP with processing time (Bo˙zejko & Wodecki, 2009), heterogeneous VRP (Liu, 2013), TPP with speed optimization (Cheaitou et al., 2021b), dynamic VRP with unknown customers (Créput et al., 2012; Sabar et al., 2021), capacitated electric VRP (Wang et al., 2023), periodic VRP (Vidal et al., 2012; Borthen et al., 2018) and liner shipping (Cariou et al., 2018; Cheaitou et al., 2021a). In addition, variants of this heuristic have been successfully used in other domains, such as supplier selection (Hamdan et al., 2023), maintenance strategy optimization (Alsharqawi et al., 2021), machine scheduling (Nearchou, 2010) and aircraft motion planning (Wu, 2021). The heuristic used to solve the problem is presented in Algorithm 1. It starts by calling a population generation subroutine (see Sect. Cin the appendix) to create a population (P) of feasible individuals (potential solutions) that should be multiples of the total number 123 Annals of Operations Research (2025) 351:667–726 683 of operators (=18). The randomly generated feasible individuals satisfy all the model constraints(20)–(41).Thestructureofeachindividualis detailedinSect.D.1in theappendix. The global cost, the emission, and the variation values are initialized as follows: cglobal =∞, eglobal =∞and δglobal =∞. The algorithm attempts to enhance the quality of individuals in each iteration under a predefined total number of iterations or generations (G). In each iteration, g=1, ..., G, the algorithm first calculates the cost and emission objective function (cψand eψ) by using Equations (18)and(19) for each individual (Algorithm 2). Based on the optimization setting (ObjType), either a single objective or a biobjective optimization is executed. In the case of single objective setting, the algorithm compares the best cost (emission) value for all individuals with the global cost (emission) value and updates cglobal (eglobal) accordingly. The algorithm returns cglobal as the best objective value and the corresponding solution details (i.e., route sequence, quantity allocation and vehicle assignment). In addition, it returns the corresponding total emission (cost) value and updates relevant heuristic parameters, such as . In the case of the biobjective setting and before calculating the variations, the global variation is recalculated if the global cost (cglobal)orglobal emissions (eglobal) are updated. This procedure ensures the use of the most recent reference points for the variation calculation. The variation for each individual (δψ) is calculated by using Equation (44). Then, the algorithm identifies the best objective function value in the current iteration (minψ∈{1,...,}δψ) and compares it with the current global objective value δglobal. The algorithm then updates the global objective value if the best-found objective value is better than the current global objective value. The algorithm returns the best objective value δglobal, which corresponds to the solution details, the corresponding total cost and emission of the best individual and the updated heuristic parameters, such as ,cglobal and eglobal,as shown in Algorithm 2. Finally, the algorithm randomly divides all individuals () into subsets, called ϕυ, each of which is individuals. These subsets are constructed randomly. In each subset, the algorithm identifies and selects the best individual in terms of the value of Equation (18), (19), or (44) depending on the optimization setting (ObjType) in the subset then uses one operator at a time to produce a new individual from each operator, which results in a total of individuals stored in a temporary population Ptmp.Theoperators used in the algorithm are as follows: •Operator 1: Keep the best individual in the subgroup unchanged “Do nothing”. Route operators (refer to Sect. D.2): •Operator 2: Randomly select one vehicle and flip its route. •Operator 3: Randomly select multiple vehicles then flip the route of each vehicle. •Operator 4: Randomly select one vehicle and swap its route. •Operator 5: Randomly select multiple vehicles then swap the route of each vehicle. •Operator 6: Randomly select one vehicle and slide its route. •Operator 7: Randomly select multiple vehicles then slide routes for each vehicle. Starting point operator (refer to Sect. D.3): •Operator 8: Randomly select a vehicle that starts from a transshipment port and force it to start from the depot. Vehicle type operators (refer to Sect. D.4): •Operator 9: Change the vehicle type of a random vehicle to a lower capacity. •Operator 10: Randomly change the vehicle type of a random vehicle. The quantity exchange operators are as follows (refer to Sect. D.5): •Operator 11: Randomly exchange two ports between two random vehicles. •Operator 12: Conduct multiple random port exchanges between two random vehicles. 123 684 Annals of Operations Research (2025) 351:667–726 •Operator 13: Randomly select two vehicles then exchange unique ports with similar quantities. •Operator 14: Select multiple random vehicles and exchange unique ports with similar quantities. The quantity transfer operators are as follows (refer to Sect. D.6): •Operator 15: Randomly transfer quantities from the vehicle with the lowest utilization to the vehicle with the highest utilization. •Operator 16: Randomly transfer quantities from multiple vehicles with low utilization to the vehicle with high utilization until it is fully utilized. •Operator 17: Randomly transfer quantities from one port to another between random vehicles. •Operator 18: Randomly transfer quantities from multiple ports served by one random vehicle to another random vehicle. Algorithm 1 Main heuristic 1: Input: Context-related: (L,M,Km,Di), vehicle-related: (αk,m,Ck,m,βk,m,nk,m,dk,m i,j,qi,jh,τ,η, μk,m,p,k,m,γk,m,sk,m,k,mk,m,Tk,m,κk,m i,j,z,rk,m,ρ,φ,ξ,ωk,m,χk,m,Ek,m), and heuristicrelated: (G,,,, ObjType). 2: Output: The best total cost, the best total emission, the best route for each vehicle, quantities delivered by each vehicle. 3: set cglobal ←∞,eglobal ←−∞,δglobal ←∞and =0 4: P←Population Generation(Context-related, vehicle-relate and heuristic-related parameters, ) 5: for g←1to Gdo 6: {Best value and corresponding details, ,cglobal,eglobal,δglobal }←Get Objective Value(Contextrelated, vehicle-relate and heuristic-related parameters, ,cglobal,eglobal,δglobal) 7: divide Prandomly and equally into ϒsub groups (ϕυ)ofIndividuals. 8: for υ←1to ϒdo 9: select the best individual from ϕυ: 10: =⎧ ⎪ ⎨ ⎪ ⎩ argminψ∈ϕυcψ,if ObjType =‘Cost’ argminψ∈ϕυeψ,if ObjType =‘Emission’ argminψ∈ϕυδψ,if ObjType =‘Bi’ 11: perform operations on the best individual (ϕ υ) 12: store new individuals in Ptmp 13: end for 14: if >then 15: discard individuals in Pand Ptmp 16: P←Population Generation(Context-related, vehicle-relate and heuristic-related parameters,) 17: else 18: P←Ptmp 19: end if 20: end for 21: return the heuristic best solution. The operators are designed such that the feasibility of the individual operator is not affected. Since the initial individuals are feasible, the operators maintain this feasibility, where any quantity movement is permitted only after ensuring that enough capacity is available (capacity is never violated in this case). In addition, since the initial chromosomes satisfy the demand requirements and none of the operators create or generate additional quantities, demand constraints are not impacted. Route and other constraints are respected through chromosome design. The algorithm then replaces all old individuals () with newly produced individuals in the temporary population Ptmp created by the operators. Notably, one of the 123 Annals of Operations Research (2025) 351:667–726 685 operators performs no changes on the solution. Thus, the best individual in each subset is kept unchanged and moved to the next iteration. Moreover, the heuristic discards all individuals and produces a new generation if the number of iterations without improvement in the solution counter () reaches a predefined limit . The process continues until the maximum number of iterations (G) is reached. Algorithm 2 Get objective value 1: Input: Context-related, vehicle-related and heuristic-related parameters, cglobal,eglobal,δglobal,and 2: Output: The best objective value and the corresponding solution details. 3: for ψ←1to do 4: evaluate Equations (18)and(19)andstore the values in cψand eψ, respectively 5: end for 6: if TypeOPT = ’Cost’ then 7: if minψ∈{1,...,}cψ<cglobal then 8: store the solution of (argminψ∈{1,...,}cψ) individual 9: ←0 10: else 11: ←+1 12: end if 13: return the updated cglobal and its corresponding solution details and total emission value, and the updated . 14: end if 15: if TypeOPT = ’Emission’ then 16: if minψ∈{1,...,}eψ<eglobal then 17: store the solution of (argminψ∈{1,...,}eψ) individual 18: ←0 19: else 20: ←+1 21: end if 22: return the updated eglobal and its corresponding solution details and total cost value, and the updated . 23: end if 24: if TypeOPT = ’Bi’ then 25: if minψ∈{1,...,}cψ<cglobal and g= 1then 26: recalculate δglobal using minψ∈{1,...,}cψ 27: end if 28: if minψ∈{1,...,}eψ<eglobal and g= 1then 29: recalculate δglobal using minψ∈{1,...,}eψ 30: end if 31: for ψ←1to do 32: evaluate Equation (44)andstore the value in δψ 33: end for 34: if minψ∈{1,...,}δψ<δ global then 35: store the solution of (argminψ∈{1,...,}δψ) individual 36: ←0 37: else 38: ←+1 39: end if 40: return the updated δglobal and its corresponding solution details, total cost and emission values, the and the updated cglobal,eglobal and . 41: end if 123 686 Annals of Operations Research (2025) 351:667–726 Fig. 3 Network of the West German canal system. Data: Fachserie Binnenschiffahrt. (StatistischeBundesamt, 2019) Layout: OpenStreetMap 5 Real-life case study In this section, we consider the real case study of the West German canal system. The case study is used to evaluate heuristic performance in Sect.6.3 and to derive the managerial insights presented in Sect.7. We examine historical goods and inland waterway transportation flows across the canal system for the research project Preview. We also evaluate the stake-holding industries and their expected cost impact on transport due to disruptions, such as queues at locks or unforeseen infrastructure failures. The expected cost impact on the entire supply chain is more severe because, among others, these costs include the management effort for modal shift to trucks, time-sensitive market premiums, and subsequent costs if supply chain operations are delayed. Since the infrastructure of the West German canal system is old, critical points from the logistics perspective must be identified, and the need for a multimodal split will be analyzed. Wehrle et al. (2020) provided more insights into the resiliency issues of the West German canal system and noted the importance of inland waterway transportation in Germany. Their model helps to detect critical infrastructure to schedule the maintenance of the infrastructure of the system. Figure3provides a detailed view of the different canals. There are four canals that belong totheWest German canalsystem.Duisburg wasthe starting pointforthe base scenario. Inland waterway transportation must include canals. Consequently, if one port is not available, the distance from one port to the other is often greater than is the relative failure of roads because there are more alternatives for trucks in the event of any failure. We used 16 ports and 14 locks. The water level is 1.9ms in the baseline scenario. All locks and ports were available, and the lock time was 40min. Because of the high demand for industries in that region, we compared inland waterway transportation and a package 123 Annals of Operations Research (2025) 351:667–726 687 Table 2 Ports and locks in the case study Index Port Lock 0 Emsland Meiderich 1 Münster Oberhausen 2 Dortmund Gelsenkirchen 3 Rhein-Lippe Wanne-Eickel 4 Marl Herne-Ost 5 Lünen Henrichenburg 6 Bergkamen Datteln 7 Hamm Ahsen 8 Schmehausen Flaesheim 9 Bottrop Dorsten 10 Essen Hünxe 11 Coelln-Neuessen Friedrichsfeld 12 Ruhr Öl Hamm 13 Gelsenkirchen Münster 14 Wanne-Eickel – 15 Duisburg (depot) – of ten trucks in our case study as a reasonable relaxation. Effects such as platooning have not yet been considered, even though they have been evaluated in practice to reduce fuel costs and emissions. Therefore, four ships with different capacities and input parameters were included. Trucks and inland waterway transportation use diesel engines. We focused on the chemistry industry. The international standard for measuring greenhouse gas emissions, ISO14083, is the new equivalent to the German DIN EN 16,258 standard for reporting and for quantifying greenhouse gas emissions from transportation operations (ISO, 2019). The demand is based on historical data of the last ten years of freight spending in canal systems and ports published in the “Fachserie Binnenschifffahrt” (StatistischeBundesamt, 2019). The distances are calculated with the Here-API and the website of Institut für Energieund Umweltforschung Heidelberg gGmbH (2023) for inland waterways because the HereAPI does not include canal navigation. In Managerial Insights 1–3, we consider situations where the decision-maker decides that the two objective functions are equally important to the project. Consequently, we set the importance weights (a1and a2) to 0.5. Let us note that a decision-maker may utilize a multicriteria decision-making tool, such as the analytic hierarchy process, to assess the importance of each objective. Table 2lists the ports and locks considered in this case study, where Duisburg is the actual depot for the model. Tables 9,10, 11,12,13 and 14 in Appendix Eprovide the data used in this study. The real-life case study provides answers to the following questions: •What situations make the use of multimodal transportation more attractive than unimodal transportation? •What effects do variable lock times have on the total cost? •What are the advantages of multimodal transportation in the event of infrastructure failures? •Whatisthe trade-offbetween costsand emissions when usingmultimodal transportation? 123 688 Annals of Operations Research (2025) 351:667–726 6 Numerical experiments In this section, we study the problem complexity to understand the impact of the number of ports, the demand and the number of vehicles on the complexity of the exact solution. We also analyze the impact of heuristic parameters (maximum number of iterations G,maximum number of individuals and the number of iterations without improvement in the solution ) on the solution quality and time. Finally, we present the heuristic performance against the exact approach. All experiments were conducted by using a computer equipped with an Intel(R) Core(TM) i7-9750H CPU @ 2.6 GHz and 16 GB of RAM running the Windows 10 Home 64-bit operating system. Let us note that since the heuristic uses a randomized process to generate its initial population and in applying operators, we solved each instance ten times and computed the average to determine the performance and the robustness. We used CPLEX 20.1.0 to obtain the exact solution; we set a time limit of three hours as the stopping criterion for the solver. 6.1 Impact of system parameters on the exact solution complexity This biobjective multimodal transportation problem is NP-hard since it can be reduced to a TSP problem, which is also proven to be NP-hard. We conduct a complexity study to understand the contributions of parameters to the problem complexity. We vary the number of ports |L|={8,13,18,23}. Let us note that for each problem size, the first three ports are the virtual depot, the depot and its duplicate. We also increase the demand by a factor FD, FD={1,1.3,1.5,1.7}. In addition to understand the impact of vehicles, we increase the number of vehicles FKm,FKm={1,1.5,2,2.5}. Let us note that we consider the instance with |L|=5, FD=1, and FKm=1 as a baseline case. In each experiment, we record the exact solving time and the solver’s gap, and then, we combine the two to form a complexity score (complexity score = 0.5×Computation time Time limit +0.5×MIP gap). Let us note that in these experiments, we average the complexity score of the three objective functions (cost only, emission only and the biobjective). A score between 0 and 0.5 means that an exact optimal solution (MIP gap = 0) is found within the time limit. A complexity score higher than 0.5 means that the time limit (computation time = time limit) is reached and that the increased part represents the MIP gap, and a score greater than 1 means that the MIP gap when the time limit is reached is greater than 100%. Figure4shows the complexity score under the experimental setup from the baseline case. Figure4a shows the complexity score as a function of the demand factor and the number of ports. It is clear that increasing both the demand and the number of ports increase the complexity. However, it is difficult to say that increasing demand results in only greater complexity for a given number of ports (e.g., at |L|=20, see FD=1.3andFD=1.5). This is due to the high variability in the MIP Gap of the different instances. However, increasing the numberof portsonly increases thecomplexity(see, for instance,the differentproblem sizes at FD=1). Figure4b illustrates the effect of the number of vehicles (represented by the vehicle factor, FKm) and the number of ports on the problem complexity from the baseline case. A greater number of vehicles increases the complexity score; although in some instances, the increased number of vehicles slightly reduces the complexity (see |L|= 18, from FKm=2to FKm=2.5). Instances with more than 13 ports and high demand (FD≥1.5) and more than 23 ports and a high number of vehicles (FKm≥2) experience high complexity and cannot be solved by using the exact approach (Fig.4, for which the score is greater than 0.5). Three-way analysis of variance (ANOVA) was used to understand the impact of the three parameters. 123 Annals of Operations Research (2025) 351:667–726 689 Fig. 4 The average complexity score using the multimodal model The test results indicate that the number of ports only and the number of vehicles only have significant impacts on the complexity score (pvalues = 0 and 0.002, respectively), while the demand factor does not have a statistically significant impact on the complexity score (p value = 0.1771). In addition, the combination of both the number of ports and the number of vehicles and the combination of demand and number of vehicles and the combination of all three factors were found to have a statistically significant impact on the complexity score (p values = 0.007, 0 and 0, respectively). 123 690 Annals of Operations Research (2025) 351:667–726 Table 3 Characteristics of the 21 instances Instance Size |L|FD|Km|Instance Size |L|FD|Km| 1 Small 8 1 11 12 Medium 23 1.546 281.313 13 281 40 381.514 14 281.350 4 13 1 17 15 28 1.560 5131.3 21 16 Large 33 1 46 6131.523 17 331.361 7 Medium 18 1 37 18 33 1.569 8181.345 19 381 55 9181.555 20 381.367 10 23 1 32 21 38 1.580 11 23 1.341 6.2 Impact of heuristic parameters We generated 21 instances with different characteristics to analyze the impact of heuristic parameters on the solution quality and time of the multimodal model. Table 3shows the characteristics of these instances. In the following experiments, we varied the number of iterations from 1000 to 10,000 with a step of 1000. First, we fixed the number of individuals to 54 (i.e., 3 multiplies of ), and we varied the maximum number of iterations with improvement as follows: =100, 500, 1000, 1500, and 2000. Figure5shows the average impact of all instances (ten runs of each instance and the three objective functions; cost, emission and the biobjective formulation). Figure5a shows the average relative difference in the objective value with respect to the best heuristic objective value found (i.e., the smallest among all iterations and values) under different values as a function of the number of iterations. Figure5b shows the average computation time savings (with respect to the largest computation time) as a function of the number of iterations. In terms of quality, setting to 500 provides, on average, the best solution quality at (and beyond) 5000 iterations. However, in terms of time savings, choosing =2000 or 1500 results in the fastest algorithm setting, with 83% at 5000 iterations compared to 96% at 1000 iterations. It is worth noting that setting =500 at 5000 iterations is 3.5% slower than that of =2000. Given that the difference is small and that the solution quality difference is more favorable, we chose to perform our experiments by using =500. Let us note that the higher the value of is, the less there is a need for the algorithm to generate a new population, thus making it faster. However, this may lead to fewer opportunities to explore new potential individuals (through generating completely new populations) but a greater possibility of exploring modifications to the current individuals. Next, we explore the impact of the number of individuals in the population as follows:  = 54, 108 and 162 (3, 6 and 9 multiplies of the number of operators ). In these experiments, we set to 500, as it was found to perform well in terms of the solution quality. Figure6 shows the average relative difference of the objective value with respect to the best solution found and the average time savings with respect to the slowest case. Figure6a shows that the difference between the best solution found when using = 54 and when using = 108 is 0.02% at 5000 iterations, while the algorithm becomes slower as the time savings decrease 123 Annals of Operations Research (2025) 351:667–726 697 Fig. 7 Objective function values for inland waterways, trucks and multimodal transportation under various demand levels Observation 2: Although the increase in the lock time increases the total cost, the multimodal model reduces this impact by 23% compared to the inland waterway transportation model. We analyzed the effect of longer lock times due to traffic jams on the canal system. We varied the lock times from 40 to 120min in 20-minute steps. An increase in the number of lock times requires more inland waterway transportation when demand is too high; in practice, this approach increases the likelihood of long queues at the lock, further increasing the number of lockage times. We also illustrate how the multimodal model minimizes costs. The results are shown in Fig.8. Different scenarios for longer lock times do not influence the singletruck model because they influence only the optimal solutions for models involving inland waterway transportation. Emissions are also not affected. However, costs are affected. By comparing the results while minimizing costs, we can see that the costs of inland waterway transportation increase faster than those of multimodal transportation systems (Fig.8). The total costs for inland waterway transportation increase by e165.7 per minute of increase in the lock time (linear fit, with a coefficient of determination of 0.999) compared with e126.2 per minute when using multimodal transportation (linear, with a coefficient of determination of 0.992). This is an approximately 23.8% reduction in the cost per minute of lock time. The multimodal model absorbs the impact of the increase in lock time due to the greater flexibility in routing decisions provided by the availability of trucks as an option. The findings demonstrate the cost-saving potential of the different modes. Figure8bshows the relative difference in the cost with respect to the baseline case of 40min of lock time. The figure shows that the cost increases, on average, by 5.05% for inland waterways and by 3.71% for multimodal transportation for each 20-minute increase in lock time. We conclude that multimodal transportation provides a statistically significant decrease in cost compared to inland waterway transportation (normal with pvalue = 0.67, one-tailed paired t test with pvalue = 0.03). 123 698 Annals of Operations Research (2025) 351:667–726 Fig. 8 Effect of varying lock times on total costs Fig. 9 Effects of lock failure scenarios on total costs while minimizing only costs Observation 3: Infrastructure failures impose higher costs of nearly 28% per day; in this case, multimodal transportation is more efficient than a single transportation mode. We investigated the effect of the failure of each lock in the system on performance. This analysis allows decision-makers to know the effects of their respective port locations if one lock fails. Only some scenarios allow inland waterway transportation to fulfill the total demand (see Table 11 in the Appendix). Others must be served by inland waterway transportation and trucks. Figure9shows the total cost of inland waterway transportation and multimodal transportation under the various lock failure scenarios. Let us note that scenario 0 represents 123 Annals of Operations Research (2025) 351:667–726 699 Fig. 10 Effects of lock failures for the biobjective model the baseline case with no lock failures. Lock failure under scenarios 4 and 7–12 results in infeasible service due to accessibility issues (see Table 11). The total cost under multimodal transportation is significantly lower than that under inland waterway transportation (normal with pvalue = 0.2, one-tailed paired t test with pvalue = 0.045). As Fig.9shows, minimizing the total costs leads to a maximum increase of 27.59% of the costs when using inland waterway transportation (see the increase in the total cost of scenario 1). This increase is with respect to the baseline scenario (scenario 0 - no lock failure). The locks in scenarios 1, 2, and 3 are the most critical due to the high increase in the cost and require a higher priority to modernize the infrastructure or maintenance operations. Scenario 5 was of interest. We see that taking the alternative northern route via RheinLippe is more beneficial even if the locks fail. The reason for this is the high demand for Marl and the number of locks that have to be passed to reach Marl. To achieve cost efficiency, decision-makers should select a route via Rhein-Lippe for accessing the port, considering the high demand for chemical products in this region. The multimodal model allows savings of up to 78% compared with a single model. Infrastructure dependency demonstrates the benefits of having an additional mode. Figure9shows that inland waterway transportation results in a 10.72% increase in the total cost on average in Scenarios 1, 2, 3, 5 and 6 compared to an average increase of 5.72% when using multimodal transportation. Figure10 shows the effect of minimizing both costs and emissions (the biobjective formulation). We can again observe the greatest difference for scenarios 1 to 3. The total cost and emissions when using the multimodal model are lower than those of inland waterway transportation. The mean total cost difference is not significantly different between the two models (normal with pvalue = 0.54, two-tailed paired t test with p value = 0.87). The mean total emission was found to be significantly lower in the multimodal model (not normal with pvalue = 0.006, one-tailed sign test with pvalue = 0.03). The trade-off between emissions and costs is clear. For some scenarios, higher costs are compensated for by the use of fewer emissions. These results show the variations in both objectives, which are further discussed in the next section. The increase in emissions under the multimodal model in Fig.10 is 123 700 Annals of Operations Research (2025) 351:667–726 with respect to the baseline scenario (no lock failure). However, the increase in emissions under these scenarios is still lower than that under inland waterway transportation. Decisionmakers should assign different weights to both objectives depending on the importance and prioritization of the decision. Observation 4: Multimodal transportation provides a faster emission reduction rate when the total cost increases by 1%. Here, we calculated the cost of reducing emissions using inland waterway ships, trucks, and multimodal transportation. The analysis is performed by varying a1and a2in Equation (44), whichresultsindifferentParetooptimalsolutions.Weemphasizethatthechosenscalarization method was selected due to the availability of decision-maker preferences, its simplicity, and its suitability for the heuristic approach. Although this method might not yield every Pareto optimal point, it aligns well with our analysis, given that the decision-maker’s preferences are predefined. Moreover, since the decision-maker predetermines the importance of each objective function, this approach adequately provides the necessary insights for informed decision-making. The relative difference between any point on the Pareto optimal set and its subsequent point (next point found on the front) is then calculated as (ci−ci+1 ci,ei−ei+1 ei), where iis the current point, i+1 is the next (subsequent) point on the front, and ciand eiare the cost and emission values of point i, respectively. All the resulting relative differences are averaged. Figure11 shows the total cost and the total emission when varying the importance weights from 0 to 1 with an increment of 0.1 when using the three models (inland waterways, trucks and multimodal). All solutions were checked, and only nondominated unique solutions were retained. Figure11 illustrates the trade-off between the total cost and total emissions for each model. A 1% reduction in emissions increased the total cost by 1.32%, 2.93%, and 0.90%, respectively, when using inland waterway transportation, trucks, and multimodal transportation. This highlights that using multimodal transportation helps reduce emissions while incurring lower costs when compared to other options. The findings also highlight that if decision-makers give less preference to the environment, a reduction in cost is associated with a greater rate of increase in emissions (compared with the ideal case of minimizing emissions) when using multimodal transportation. That is, reducing costs by 1% results in increases in emissions of 0.76%, 0.34%, and 1.12% for inland waterway ships, trucks, and multiple modes, respectively. It is also true that increasing costs by 1% reduces emissions by 1.12% when using multimodal transportation. These percentages reflect the rate of change or how quickly costs or emissions increase and decrease. We utilized the Wilcoxon rank sum test (the data were not normally distributed, pvalue <0.05) to compare the Pareto points of each transportation mode and to assess whether these differences were statistically significant. We found that the differences in cost and emissions between inland waterways and multimodal transportation are not statistically significant (p value>0.08). This means that although multimodal waterways provide slightly better tradeoffs than do inland waterways, these differences are not statistically significant and that both modes have comparable efficiencies in terms of balancing cost and emissions. In contrast, the differences between multimodal transportation and trucks are statistically significant, indicating that multimodal transportation or inland waterway transportation offers a more cost-effective solution for reducing emissions than trucks. To assess the heuristic performance in generating the biobjective solutions, we used the hypervolume indicator to quantify the volume of the objective function space covered by the Pareto points with respect to a reference point. This measure indicates how close the 123 Annals of Operations Research (2025) 351:667–726 701 Fig. 11 Total cost versus total emissions for each transportation mode. The circles ’o’ and ’x’ represent the solutions obtained by using the exact and heuristic approaches, respectively Table 8 Hypervolume results of the exact and heuristic Pareto optimal sets Hypervolume RD (%) Transportation mode Exact Heuristic Inland waterways 767,895,800 763,730,000 0.54 Truck 25,730,000 25,730,000 0.00 Multimodal 10,322,413,400 10,256,591,500 0.64 heuristic Pareto set is to the exact approach Pareto set. We used the “ecr” package in R Studio to calculate the nondominated hypervolume of the Pareto optimal set by using the exact and heuristic approaches (Table 8). The relative difference in the hypervolume between the two sets was very small (less than 1%), meaning that the two sets had very similar results. 8 Conclusion In this study, we gained insights into how multimodal transportation can be used in the West German region, more precisely, what impact the choice of transportation mode has on total costs and emissions. Furthermore, the viability of this choice was evaluated under different waterway infrastructure failure scenarios, which affected the multimodal balance. While using a cost function proposed by the German Federal Institution and an emission function based on ISO14083, the results are reliable regarding historic routing problems in the company and cost estimations of transportation by the German Federal Institution. To create different scenarios, we utilized actual data from the West German canal system and its 16 ports and 14 locks. Managerial insights, such as the effect of infrastructure failures on the logistics flow in the region, allow decision-makers to know the monetary and emissionrelated costs for their companies. Moreover, the effect of varying demand on the network 123 702 Annals of Operations Research (2025) 351:667–726 was analyzed, and decision-makers were able to understand how increased demand volumes impact the cost and emissions of different transportation modes. We also identified additional costs incurred when the number of lock times increased and demonstrated the advantages of using the two modes. We also describe the effects of using only one objective optimization compared to that of the biobjective function and its effect on the objective values. The proposed heuristic approach for our model leads to an average time savings of 82% and acceptable accuracy with a relative difference of less than 5% when compared to that of the best exact solution found. This formulation allowed us to generate good results within an acceptable time. This study was based on the literature on transportation and sustainable routing problems. To the best of our knowledge, there is no algorithm that selects transshipment nodes in the models, as we do here. The biobjective approach allows decision-makers to consider the future pricing of emissions. Single biobjective and multimodal biobjective transportation modelscanbeappliedtoanywaterwaysystemwithmultimodalhubssubjecttocomputational boundaries,suchastheinlandwaterwaysystemoftheNetherlands.Businessdecision-makers benefit from obtaining transparency about the expected cost increase in their supply relations regarding the current state of infrastructure availability. Public decision-makers benefit from the possibility of prioritizing the operation of infrastructure regarding the monetary and the emission costs of infrastructure failures in the system. The model determines the quantitative evaluation of modal shift and of rerouting in response to infrastructure failure as a risk mitigation strategy that can be integrated into a broader risk management or supply chain resilience perspective by decision-makers. The analysis in this study allows a better understanding of the impact of varying demands on transportation mode choice, and the effects of varying lock times and infrastructure failure can be interpreted by decision-makers with this model. The proposed formulation allows the decision-maker to optimize the transshipment port, delivery routes, and quantities, as well as the number of vehicles used in each mode, based on a reasonable setting of parameters. Moreover, other data on stake-holding industries in the West German region, such as the coal, arc, and stone industries, can be used to obtain insights into the optimal transportation mode choice. This analysis allows decision-makers to gauge the locks and bridges that can have the most substantial impact on costs and emissions in the event of failure. System cost levels are minimized under the assumption that shippers cooperate and consolidate their shipments for optimal cost-efficient utilization. This behavior, namely, tramp shipping, is evident in short-sea shipping and inland waterway transport. However, the real costs of the transport system and the adverse effects of infrastructure failure are greater in practice, as the system contains more underutilized point-to-point transport. The algorithm is a single-product, multivehicle model. In the future, considering multiple products of similar industries can be advantageous for decision-makers. However, considering a bundle of 10 trucks can be criticized because it leads to higher costs if the capacity is not fully used. The formulation complexity is significantly affected by the problem size because the runtime exponentially increases with the addition of new ports and vehicles. As transshipment emissions (e.g., those stemming from prolonged storage times) are difficult to account for, this aspect was not considered in the case study. This aspect can be explored in future studies. An advantage of the model is that it allows the addition of new transportation modes without changing constraints; therefore, adding rail can benefit decision-makers because it is a sustainable substitution, especially for trucks. Furthermore, speed optimization can be added to reduce emissions and costs. It is of interest to implement the algorithm in other case studies, such as the coal, ore, and stone industries. Moreover, it is of interest to evaluate the algorithm with another entrance port, that is, the depot, in the model to analyze its influence on the network. We also performed a sensitivity analysis of 123 Annals of Operations Research (2025) 351:667–726 703 the vehicles and their effect on the optimal solution and runtime. Another scenario may be the development of new technologies in the truck market with greener propulsion technologies. Let us note that decision-makers can use the results of this study for the allocation of production sites at a strategic level because of the vulnerability of the existing infrastructure. Appendix A Problem complexity The problem complexity is characterized in Lemma 1: Lemma 1 The multimodal transportation problem with transshipment port allocation defined in Equations (18)–(41)is an NP-hard problem. Proof We prove that this problem is an NP-hard by reducing it to a well-known NP-hard problem (Arora & Barak, 2009). Let m=1andm∈MKm=1, that is, one vehicle with unlimited capacity is available in the service. Thus, Constraints (26)and(31) can be eliminated. Having one vehicle requires that one service be possible and that a secondary service (transshipment) is impossible, resulting in the removal of decision variables Tk,m iand UQk,m iand their associated constraints. Since all demand must be satisfied (Constraint (30)), Constraint (21) becomes redundant as all ports are forced to be visited by the same vehicle. Thus, decision variables yk,m i,Qk,m iand fk,m i,jbecome unnecessary, and their associated constraints can be removed. This reduces this problem to the well-known traveling salesman problem, which is NP-hard; see (Garey & Johnson, 1979; Jungnickel, 1999), among others. Since the simplified case is NP-hard, the general case presented in this work is also NP-hard, which completes the proof.  Appendix B Reducing multimodal transportation to a single-mode model The multimodal model can be reduced to a single model for inland waterway transportation or trucks. In the single model, we no longer have a virtual depot but rather an actual ordinary depot of the network. Input data, such as the distances for inland waterway transportation, trucks, and demand, must be adapted. The decision variables UQk,m iand Tk,m iwere not needed. Moreover, some constraints remain unchanged, others need to be adapted, and others are not needed. The following equations are subject to the same objective function and are unchanged: (20), (21), (22), (25), (26), (29)and(30). The constraints (23), (27), (28), and (31) are changed to (B1), (B2), (B3), and (B4), respectively. S×yk,m i≤Qk,m i≤Di×yk,m i∀i∈L\{i0},∀k∈Km∀m∈M,(B1)  j∈L fk i,j= j∈L fk,m j,i−Qk,m i,∀i∈L,∀k∈Km,∀m∈M,(B2)  j∈L fk i0,j= j∈L Qk,m j,∀k∈Km,∀m∈M,(B3)  i∈L Qk,m i≤Ck,m∀k∈Km∀m∈M.(B4) Constraint (B1) ensures that if we do not select the port, then the quantity equals zero. If we select the port to be visited, then the quantity is nonzero, less than or equal to the demand. 123 704 Annals of Operations Research (2025) 351:667–726 Fig. 12 Main algorithm and subroutines connections Constraint (B2) defines the flow from ito jas the flow that enters ifrom all possible jvalues minus the quantities. Constraint (B3) defines the first flow, which is the total loaded quantity, and Constraint (B4) ensures that the total quantities loaded in each vehicle do not exceed its capacity. Appendix C Heuristic subroutines In this section, we detail the various subroutines used in the proposed heuristic. The connections between different subroutines are illustrated in Fig.12. C.1 Population generation Subroutine The Population Generation subroutine (shown in Algorithm 3) generates randomly feasible initial individuals. The population generation subroutine creates two chromosomes for each individual (ψ), called the “route chromosome” and the “quantity chromosome.” The route chromosome carries information about the vehicle type and the sequence of the port visit route, and the quantity chromosome carries information about the quantity delivered to each port and the transshipment quantities to be transported by using another transportation model. First, vehicle information is generated by randomly deciding the number of vehicles (NV ψ) and the type of each vehicle (m∈M). The route is then generated by randomly choosing the numberand sequence ofports visited (NPveh ψandRouteveh ψ)by each vehicle.The visitedports are created randomly such that a vehicle cannot visit the same port more than once, satisfying Constraint (20). Likewise, the sequence in the route chromosome implies the satisfaction of Constraint (20). Finally, each vehicle is loaded with a random quantity for each visited port, andtheinformationisstoredonthequantitychromosome.Oncethevehicleinformation,route sequence, and quantity delivered are generated for each individual, the algorithm compares the total loaded quantity of each vehicle with its capacity. If a violation is detected, it is called the Fix Vehicle Capacity subroutine, which reduces the total loaded quantity and ensures the fulfillment of the model constraint (31). Subsequently, the total quantity carried by all vehicles is compared with the total demand (Constraint (30)). If a violation is detected, then the Fix Total Quantity subroutine is called. Finally, it initializes the transshipment quantity (stored in the quantity chromosome) and checks whether the total quantity delivered to each port fulfills the demand of the port. The Fix Individual Quantity subroutine is used to fix any violations in Constraint (30). Algorithm 3 Population generation subroutine 1: Input: Context-related, vehicle-related, heuristic-related parameters,  2: Output: P: an initial feasible population of individuals Generate initial individuals 3: for ψ←1to do 4: decide randomly the number of vehicles to be used (NV ψ); NV ψ≤m∈M|Km| 5: for veh ←1to NV ψdo 123 Annals of Operations Research (2025) 351:667–726 705 6: decide randomly the vehicle type (m∈M) 7: decide randomly the number of ports (NPveh ψ) to be included in the vehicle route (Routeveh ψ) 8: select randomly NPveh ψports and add them to Routeveh ψ 9: for j←1to NPveh ψdo 10: load vehicle veh with a random quanitity (Qveh ψ,Routeveh ψ,j ) for port Routeveh ψ,j, calculated as 11: Qveh ψ,Routeveh ψ,j =rand() ×DRouteveh ψ,j where rand() represents a randomly generated number between 0 and 1. 12: end for 13: calculate the total qunatity (NPveh ψ j=1Qveh ψ,Routeveh ψ,j ) carried by vehicle veh and compare it with the vehicle capacity (Cveh) 14: if NPveh ψ j=1Qveh ψ,Routeveh ψ,j >Cveh then 15: Fix Vehicle Capacity subroutine 16: end if 17: end for 18: calculate the total quantity NV ψ veh=1NPveh ψ j=1Qveh ψ,Routeveh ψ,jand compare it with the total demand (|L| j=1Dj) 19: if NV ψ veh=1NPveh ψ j=1Qveh ψ,Routeveh ψ,j = |L| j=1Djthen 20: Fix Total Quantity subroutine 21: end if 22: initialize unloaded quantities UQveh ψ 23: for j←1to NPveh ψdo 24: if NV ψ veh=1Qveh ψ,Routeveh ψ,j −NV ψ veh=1UQveh ψ,Routeveh ψ,j −DRouteveh ψ,j = 0then 25: Fix Individual Quantity subroutine 26: end if 27: end for 28: end for C.2 Fix vehicle capacity subroutine The Fix Vehicle Capacity subroutine (Algorithm 4) is called during feasible population initialization when a vehicle within an individual (potential solution) has a quantity greater than its capacity. The subroutine calculates the excess quantity carried (Q+) as the difference between the total quantity loaded and the vehicle capacity. The algorithm then randomly selects a port from the visited port list of the vehicle (Routeveh ψ,rnd) such that the quantity transported to that port is nonzero. The selected port quantity (Qveh ψ,Routeveh ψ,rnd ) is reduced by Q+.IfQ+is larger than Qveh ψ,Routeveh ψ,rnd ,Qveh ψ,Routeveh ψ,rnd is set equal to zero to avoid a negative quantity. The algorithm terminates when the vehicle capacity constraint is not violated. Algorithm 4 Fix vehicle capacity subroutine 1: Input: NPveh ψ,Qveh ψ,Routeveh ψ,j ,Cveh 2: Output: Qveh ψ,Routeveh ψ,j 123 706 Annals of Operations Research (2025) 351:667–726 3: while NPveh ψ j=1Qveh ψ,Routeveh ψ,j >Cveh do calculate excess Q 4: Q+←NPveh ψ j=1Qveh ψ,Routeveh ψ,j −Cveh 5: select randomly (rnd =randi([1NPveh ψ])) a port (Routeveh ψ,rnd) such that Qveh ψ,Routeveh ψ,rnd >0 where randi([1NPveh ψ])returns a random integer between 1 and NPveh ψthat represents a port within the route 6: Qveh ψ,Routeveh ψ,rnd ←max{0,Qveh ψ,Routeveh ψ,rnd −Q+} 7: end while 8: return Qveh ψ,Routeveh ψ,j . C.3 Fix total quantity subroutine After the Population Generation subroutine fixes the vehicle capacity issues by calling the Fix Vehicle Capacity subroutine, the total demand is compared with the total loaded quantity for allvehicles. Oncea violation isdetected, the Fix Total Quantity subroutine(Algorithm 5) is called. This algorithm is based on two cases. The first case is the case of an excess quantity (Q+) that exceeds the total demand. The algorithm selects the vehicle with the lowest load and then selects a visited port on the route of the selected vehicle. Subsequently, it reduces the loaded quantity to that port to either zero or by Q+. The selection of the vehicle with the lowest load helps reduce the number of vehicles used. Let us note that if a vehicle carries no quantity, it is excluded from the search. The second case represents the situation in which the total quantity loaded is less than the total demand (called the shortage case), in which case the missing quantity (Q−) is calculated as the difference between the total demand and the total quantity loaded on the vehicle. The remaining capacity of each vehicle was calculated, and vehicles with no remaining capacity were excluded. This decision led to two possible scenarios. In the first scenario, where all used vehicles have no remaining capacity, a new vehicle is added of a randomly available type, and random ports are assigned. The total load of this new vehicle can be calculated as the maximum between the vehicle’s capacity and Q−, which is then distributed randomly among the assigned ports. In the second scenario, where some vehicles are not fully utilized, select the vehicle with the lowest utilization and transfer to a random port a minimum quantity between the vehicle’s remaining capacity and Q−. The algorithm terminates when the total demand is satisfied and returns the updated individual chromosomes to the Population Generation subroutine. Algorithm 5 Fix total quantity subroutine 1: Input: Routeψ,Qψ,NP ψ,NV ψ,C,D,L 2: Output: Routeψ,Qψ,NP ψ,NV ψ 3: while NV ψ veh=1NPveh ψ j=1Qveh ψ,Routeveh ψ,j = |L| j=1Djdo 4: while NV ψ veh=1NPveh ψ j=1Qveh ψ,Routeveh ψ,j >|L| j=1Djdo 5: Q+←NV ψ veh=1NPveh ψ j=1Qveh ψ,Routeveh ψ,j −|L| j=1Dj 6: Exclude a vehicle veh if NPveh ψ j=1Qveh ψ,Routeveh ψ,j =0 123 Annals of Operations Research (2025) 351:667–726 713 Fig. 17 Illustration of Operator 11 1 and 3 are chosen randomly, and then Port 3 on Vehicle 1 and Port 4 on Vehicle 3 are chosen randomly. Exchanging the quantities does not violate the capacity constraint; thus, it is permitted. Consequently, Vehicle 1 delivers 150 to Port 4 (previously, it was assigned to deliver 50), and Vehicle 3 delivers 100 to Port 3. Let us note that Port 4 is removed from Vehicle 3 and that Port 3 is removed from Vehicle 1 then added to Vehicle 3 since it was not served before. Operator 12 is similar to Operator 11; however, instead of doing one change in each generation (iteration), it tries to exchange more than one port between the two vehicles, where the number of exchanges is decided randomly based on the smallest number of ports visited by the vehicles. Operator 13 considers exchanging unique ports between two vehicles (Fig.18). Operator 13 randomly identifies two vehicles (Vehicles 1 and 3 in Fig.18). Then, it checks the route of each vehicle and identifies unique ports on each vehicle (i.e., ports not visited by the other vehicle). In Fig.18, the unique ports are 2 and 3 for Vehicle 1 and 8 for Vehicle 3. It then selects one port served by the first vehicle and tries to exchange it with one or more unique ports served by the second vehicle based on the remaining capacity. In Fig.18,Port 3 served by the first vehicle (Vehicle 1) is chosen as a unique port, and since the second vehicle (Vehicle 3) has only one unique port (port 8), the operator tries to exchange 3 and 8. First, the remaining capacity of Vehicle 1 and the quantity delivered to port are calculated as (C−Q−Q3), which represents the allowable exchange limit of Vehicle 1 after removing Port 3. Then, for the second vehicle (Vehicle 3), the algorithm searches for an equivalent exchange (i.e., a unique port with a quantity close to the allowable exchange limit of Vehicle 1). If no port exists with a matching quantity, the algorithm searches for multiple unique ports not exceeding the allowable exchange limit of Vehicle 1. Then, it swaps the quantities between the two vehicles and adds the new ports to the vehicles’ routes if capacity limit allows this exchange. In Fig.18, the second vehicle (Vehicle 3) has only one unique port (Port 8). Thus, it checks the remaining capacity as C−Q−Q8and compares it with the quantity of Port 3. The exchange of Ports 3 and 8 is permitted since the capacity is not violated. Operator 14 has a similar functionality to Operator 13, except that it tries to swap 123 714 Annals of Operations Research (2025) 351:667–726 Fig. 18 Illustration of Operator 13 ports between Vehicle 1 and multiple other vehicles, where the number of vehicles involved is chosen randomly based on the number of vehicles involved in the service. D.6 Quantity transfer operators Operators 15–18 are unidirectional quantity movement operators. Operator 15, as shown in Fig.19, identifies vehicles that start from the depot and are not fully utilized (i.e., have remaining capacities) and selects two random vehicles. It identifies the vehicle with the highest utilization (smallest remaining capacity) to receive quantities, called the receiving vehicle, and the vehicle with the lowest utilization, named the sending vehicle. In Fig.19,the receiving and sending vehicles are assumed to be Vehicles 1 and 3, respectively. It calculates the remaining capacity of the receiving vehicle and then randomly selects a subset of the ports served by the sending vehicle. Figure19 shows an example of a subset of one port (Port 7). It then transfers quantities from each chosen port on the sending vehicle (Port 7 in this example) as long as the transferable quantities are within the vehicle’s remaining capacity. The transfer process stops when either the remaining capacity reaches zero or a random 123 Annals of Operations Research (2025) 351:667–726 715 Fig. 19 Illustration of Operator 15 number of ports is reached. Let us note that if the receiving vehicle does not originally visit a certain port, it is then added to the route randomly. We note also that this operator does not necessarily result in a fully utilized receiving vehicle (e.g., if the total quantity transferred due to the number of selected ports is less than the remaining capacity). The route sequence chromosome is updated accordingly. Operator 16 follows the same concept as Operator 15 but ensures that the receiving vehicle is fully utilized by allowing more than one vehicle to be a sending vehicle. In the case where the receiving vehicle receives all quantities transported by a sending vehicle, the sending vehicle is removed from the service. Operator 17, shown in Fig.20, represents a simplified version of Operator 11, where the quantity movement occurs in one direction. That is, one port served by the sending vehicle is chosen randomly (Port 3 of Vehicle 1), and the possibility of moving its quantity to the receiving vehicle (Vehicle 3 in Fig.20) is checked. If the remaining capacity is nonnegative, then the port is moved to the receiving vehicle. Operator 18 is the multiple port transfer version of Operator 17. 123 716 Annals of Operations Research (2025) 351:667–726 Fig. 20 Illustration of Operator 17 123 Annals of Operations Research (2025) 351:667–726 717 Appendix E Case study data Table 9 Input data for the different transportation modes Notation Inland waterway transportation Truck Unit αk,m58.88 −103.45 0.1506 [e/h], [e/km] Ck,m900 −2000 26 [t] βk,m37.52 −44.95 19.33 [e/h] nk,m2−2.43 −− dk,m i,jsee Table 13 see Table 14 [km] qi,jsee Table 10 −− h1 −[h] τ0.67 −[h] η230 −[t/h] μk,m20.36 −24.25 0.3208 [e/km] p0.44 −[e/t] k,m3,25 2 [e/t] γk,m2,30 3,25 [e/t] sk,m10 65 [km/h] k,m−0.08 [e/h] k,m−13.54 [e/h] Tk,m−0.155 [e/km] κk,m i,j0,15 see Table 12 − z3.6 rk,m0,45 fk,m i,j/Ck,m− ρ23,2 23,2 [g/MJ] φ0,03−1,4 0,03 −1,4 [g/km] ξ0,0957 0,277 [g/MJ] ωk,m152 −181 −[kW] χk,m−−[g] Ek,m−10,9 [MJ/km] 123 718 Annals of Operations Research (2025) 351:667–726 Table 10 Matrix of the number of locks between port iand jfor the single inland waterway transportation model Ports0123456789101112131415 0 0127411224443336 1 1016300113332225 2 2107411224443336 3 7670366772223330 4 43430334466 6 5 5 5 8 5 1016300113332225 6 1016300113332225 7 2127411004443336 8 2127411004443336 9 4342633440001112 10 43426334400 0 1 1 1 2 11 43426334400 0 1 1 1 2 12 32335223311 1 0 0 0 3 13 32335223311 1 0 0 0 3 14 32335223311 1 0 0 0 3 15 65608556622 2 3 3 3 0 123 Annals of Operations Research (2025) 351:667–726 719 Table 11 Infrastructure failure scenarios Scenario Infrastructure failure Accessibility Scenario Infrastructure failure Accessibility 1 Meiderich Yes 7 Hamm No Oberhausen Yes 8 Münster No 2 Gelsenkirchen Yes 9 Dortmund-Ems-Kanal No 3 Wanne Eickel Yes Herne Ost 4 Henrichenburg No 10 Datteln-Hamm-Kanal No 5 Datteln Yes 11 Rhein-Herne-Kanal No Ahsen Flaesheim 6 Dorsten Yes 12 Wesel-Datteln Kanal No Hünxe Friedrichsfeld 123 720 Annals of Operations Research (2025) 351:667–726 Table 12 Empty percentage for trucks from port ito jPorts01234567 0 – 0.38 0.30 0.30 0.38 0.30 0.38 0.30 1 0.38 – 0.38 0.30 0.38 0.38 0.44 0.38 2 0.30 0.38 – 0.38 0.44 0.44 0.44 0.38 3 0.30 0.30 0.38 – 0.38 0.38 0.38 0.30 4 0.38 0.38 0.44 0.38 – 0.44 0.38 0.38 5 0.30 0.38 0.44 0.38 0.44 – 0.44 0.44 6 0.38 0.44 0.44 0.38 0.38 0.44 – 0.44 7 0.30 0.38 0.38 0.30 0.38 0.44 0.44 – 8 0.30 0.38 0.38 0.30 0.38 0.44 0.44 0.44 9 0.30 0.38 0.44 0.44 0.44 0.44 0.38 0.38 10 0.30 0.38 0.44 0.44 0.44 0.44 0.38 0.38 11 0.30 0.38 0.44 0.44 0.44 0.44 0.38 0.38 12 0.30 0.38 0.44 0.38 0.44 0.44 0.38 0.38 13 0.30 0.38 0.44 0.44 0.44 0.44 0.38 0.38 14 0.30 0.38 0.44 0.38 0.44 0.44 0.38 0.38 15 0.30 0.38 0.38 0.44 0.44 0.38 0.38 0.30 Ports8 9 101112131415 0 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 1 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.30 2 0.38 0.44 0.44 0.44 0.44 0.44 0.44 0.38 3 0.30 0.44 0.44 0.44 0.38 0.44 0.38 0.44 4 0.38 0.44 0.44 0.44 0.44 0.44 0.44 0.44 5 0.44 0.44 0.44 0.44 0.44 0.44 0.44 0.38 6 0.44 0.38 0.38 0.38 0.44 0.38 0.44 0.38 7 0.44 0.38 0.38 0.38 0.38 0.38 0.38 0.30 8 – 0.38 0.38 0.38 0.38 0.38 0.38 0.30 9 0.38 – 0.44 0.44 0.44 0.44 0.44 0.44 10 0.38 0.44 – 0.44 0.44 0.44 0.44 0.44 11 0.38 0.44 0.44 – 0.44 0.44 0.44 0.44 12 0.38 0.44 0.44 0.44 – 0.44 0.44 0.44 13 0.38 0.44 0.44 0.44 0.44 – 0.44 0.44 14 0.38 0.44 0.44 0.44 0.44 0.44 – 0.44 15 0.30 0.44 0.44 0.44 0.44 0.44 0.44 – 123 Annals of Operations Research (2025) 351:667–726 721 Table 13 Distance matrix for inland waterway transportation from port ito j Ports01234567 0040.15 104.74 144.39 107.29 98.33 110.35 131.92 140.15 0 64.58 104.24 67.14 58.18 70.20 91.77 2 104.74 64.58 0 78.54 41.43 28.87 40.89 62.46 3 144.39 104.24 78.54 0 37.10 72.13 84.15 105.72 4 107.29 67.14 41.43 37.10 0 35.03 47.05 68.62 598.33 58.18 28.87 72.13 35.03 0 12.02 33.59 6 110.35 70.20 40.89 84.15 47.05 12.02 0 21.57 7 131.92 91.77 62.46 105.72 68.62 33.59 21.57 0 8 134.41 94.25 64.94 108.21 71.10 36.07 24.06 11.46 9 120.70 80.55 44.47 54.17 57.40 44.84 56.85 78.43 10 120.70 80.55 44.47 54.17 57.40 44.84 56.85 78.43 11 120.70 80.55 44.47 54.17 57.40 44.84 56.85 78.43 12 108.80 68.65 32.57 66.39 45.50 32.94 44.95 66.53 13 112.50 72.35 36.27 62.37 49.20 36.64 48.66 70.23 14 105.86 65.70 29.62 69.02 42.55 29.99 42.01 63.58 15 135.70 95.55 59.47 40.17 72.40 59.83 71.85 93.42 Ports8 9 101112131415 0 134.41 120.70 120.70 120.70 108.80 112.50 105.86 135.70 194.25 80.55 80.55 80.55 68.65 72.35 65.70 95.55 264.94 44.47 44.47 44.47 32.57 36.27 29.62 59.47 3 108.21 54.17 54.17 54.17 66.39 62.37 69.02 40.17 471.10 57.40 57.40 57.40 45.50 49.20 42.55 72.40 536.07 44.84 44.84 44.84 32.94 36.64 29.99 59.83 624.06 56.85 56.85 56.85 44.95 48.66 42.01 71.85 711.46 78.43 78.43 78.43 66.53 70.23 63.58 93.42 8080.91 80.91 80.91 69.01 72.71 66.06 95.91 980.91 0 0 0 12.22 8.20 14.85 15.00 10 80.91 0 0 0 12.22 8.20 14.85 15.00 11 80.91 0 0 0 12.22 8.20 14.85 15.00 12 69.01 12.22 12.22 12.22 0 4.02 2.94 27.21 13 72.71 8.20 8.20 8.20 4.02 0 6.65 23.19 14 66.06 14.85 14.85 14.85 2.94 6.65 0 29.84 15 95.91 15.00 15.00 15.00 27.21 23.19 29.84 0 123 722 Annals of Operations Research (2025) 351:667–726 Table 14 Distance matrix for trucks from port ito j Ports01234567 00.00 57.28 115.24 136.53 96.09 102.96 83.63 113.78 157.23 0.00 75.09 108.84 55.75 62.80 43.48 73.62 2 121.79 80.36 0.00 76.92 46.99 10.62 45.78 62.97 3 135.66 110.02 77.41 0.00 55.82 77.96 97.69 114.87 497.42 56.31 46.76 54.69 0.00 47.30 67.04 84.22 5 104.85 63.42 10.52 76.78 46.85 0.00 13.44 46.03 687.29 45.86 23.46 91.43 61.50 13.44 0.00 36.83 7 113.34 71.90 63.90 114.22 84.29 45.21 37.33 0.00 8 114.34 72.90 64.90 115.22 85.29 46.21 38.33 3.05 9 134.29 80.17 40.91 44.16 25.97 48.66 68.40 85.58 10 135.93 81.82 40.79 44.04 27.62 47.12 66.86 84.04 11 134.28 80.17 40.13 43.37 25.97 48.66 68.39 85.57 12 115.08 73.98 28.62 51.46 24.38 36.26 55.99 73.18 13 138.86 81.00 33.66 46.18 23.24 39.99 59.72 76.91 14 114.99 73.89 27.04 51.04 30.51 36.17 55.90 73.08 15 145.67 99.65 59.20 29.93 45.45 67.58 87.32 104.50 Ports8 9 101112131415 0 114.77 135.14 136.78 135.14 115.63 121.06 113.89 149.81 174.62 79.56 81.20 79.56 75.29 80.72 73.54 108.45 263.96 41.33 41.21 40.54 28.57 34.00 26.83 58.38 3 115.86 42.55 43.71 43.04 50.92 48.23 51.65 43.06 485.21 25.40 27.05 25.40 30.94 23.06 29.19 48.49 547.02 48.35 46.80 46.13 34.17 39.60 32.42 70.58 637.83 63.00 61.45 60.79 48.82 54.25 47.08 85.23 73.05 85.79 84.24 85.79 71.61 77.04 69.87 108.03 80.00 86.79 85.24 86.79 72.61 78.04 70.87 109.02 986.57 0.00 2.43 0.79 14.37 11.68 17.08 30.29 10 85.03 2.43 0.00 1.65 14.25 11.56 16.96 30.17 11 86.57 0.79 1.65 0.00 13.58 10.89 16.29 29.50 12 74.17 15.81 15.69 15.02 0.00 5.47 5.90 37.59 13 77.90 10.53 10.41 9.75 5.47 0.00 9.83 32.31 14 74.08 17.46 17.34 16.68 4.71 10.14 0.00 39.24 15 105.49 25.57 25.45 24.78 32.65 29.96 35.36 0.00 Funding The project (FKZ: 13N14700) is partially funded by the German Federal Ministry of Education and Research (BMBF). Availability of data and materials Available upon request. Code Availability Available upon request. 123