Selective Survey: Most Efficient Models and Solvers for Integrative Multimodal Transport

1 Introduction

The current context of World Globalization has raised many difficult problems regarding the transportation of goods. The products are hauled over large distances of land and water, and more often have to travel by more than one means of transport: by ships, planes, trucks (see Van Schijndel and Dinwoodie, 2000); all these lead to the Multimodal Transportation Systems (MMTS).

In contrast with classical, single mean transportation, multi-modal transportation has multiple constraints, for example, Litman (2017) names different optimization processes as parcel loading, and transfer between transports. In a context of today’s global warming and increased pollution, it is a necessity to also globally reduce gas emission. The environmental goal correlated with the economical performance could be reached through several ways including an optimized transport planning and using appropriate resources.

In the literature review on Multimodal freight transportation planning conducted by SteadieSeifi et al. (2014), several strategic planning issues within multi-modal freight transportation and tactical planning problems are shown. Complex operational planning for real-time requirements of multimodal operators, carriers and shippers, not previously addressed at strategic and tactical levels, are described. The main models with related solvers and proposed future research are included. A detailed review, with an analysis of the optimization-based decision-making models for the problem of Disaster Recovery Planning of Transportation Networks (DRPTN), is provided by Zamanifar and Hartmann (2020). The authors described the phases of optimization-based decision-making models and investigate their methodologies. Nevertheless, the authors identified some challenges and opportunities, discussed research improvement and made suggestions for possible future research.

A recent systematic review about dynamic pricing techniques for Intelligent Transportation System (ITS) in smart cities was published by Saharan et al. (2020). The authors included existing ITS techniques with pertinent overviews and discussions about problems related to electric vehicles (EVs) used for reducing the peak loads and congestion, at the same time increasing the mobility.

The current work overviews the multimodal transport. Section 2 presents prerequisites related to the multimodal transport and the context around it. Section 3 follows presenting the characteristics and the challenges related to the transport. Further on, methods of planning (Section 4) and optimization (Section 5) in terms of time (Section 5.1), cost (Section 5.2) and network topology (Section 5.3) are presented. Existing unimodal transport models and solvers with possible future extension to multimodal features are included in Section 6. Section 7 draws the important conclusions about multimodal transport.

2 What is Multimodal Transportation?

The multimodal transport is defined by the UN Convention on International Multimodal Transport of Goods as follows.

Definition 1 (See Peplowska-Dabrowska and Nawrot, 2019 ).

The multimodal transport is the transport of goods from one place to another, usually located in a different country, by at least two means of transportation.

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Fig. 1

The difference between a multimodal point-to-point network and a multimodal integrated network, see Rodrigue (2020).

Optimization. A classical transportation of goods implies direct links and one mode of transportation (Fig. 1 left), a shortest path route, from a sender to a receiver of goods, see Zografos and Androutsopoulos (2008); the multimodal transport (Fig. 1 right), implies complex links and more than one mode of transportation. Optimization of classical transportation routes is a fairly easy and intensely studied topic. There are already existing state-of-the-art algorithms, like Dijkstra’s Algorithm (see Jianya, 1999), or Clarke-Wright technique as in Golden et al. (1977). These approaches use a single mean of transport, a single warehouse and one or more clients (or receivers).

Real-life scenarios. In the complex reality, goods can be transported in any direction, for example, inside a country there are couriers delivering from any side of the country to another side; using just trucks to perform a complete task would be impossible, as the problem has O ( n 2 ) complexity for n cities to reach. Optimizing this case means designing a system with a central warehouse or hub, where all the goods are unloaded, sorted according to the destination and finally loaded on the respective trucks and dispatched towards the destination. This optimization alone reduces the required number of trucks (the same truck makes a round trip from each city to the central hub) (Zhang et al., 2013).

But what about the larger countries, or about international transport? The multimodal transportation has the advantage of moving a huge amount of goods, in the hundreds of thousands of tons at once, via large ships, over very large distances (Fig. 1).

This section focuses on the challenges of the multimodal transportation, both for passengers (Section 3.1) and freight (Section 3.2). The majority of received goods are moved by many transportation modes, e.g. ships, airplanes, trucks. In the first stages of the transport, the sorting hubs collect all the goods from different senders, establish their destination and assign them a way of dispatch. Routes may be calculated at this step to assess the most economical ones, both in cost and time. Goods with the same route are grouped and loaded on the same shipment mode.

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Fig. 2

Overview: diverse scales of multimodal transport, from continental to local (Rodrigue, 2020).

Figure 2 shows the direct consequence: when reaching the end of a route, for each transportation mode there must be a sorting/dispatching hub. These operations of unloading, sorting, grouping and dispatching are repeated at every hub and with a highly time consuming action. As a consequence, the intermediate hubs have to be very organized so as to limit the time spent at that point, and also their number has to be kept low enough. As a disadvantage, the very large number of shipping hubs will dramatically increase the transport cost, due to the number of transport units used.

Based on these features, several challenges arise: How can we make shipping from A to B cheaper, quicker, and with the least environmental impact? How can we calculate the optimum number of hubs with maximum benefits? What is the optimal way of transporting goods between hubs while avoiding their weaknesses?

> Real-life scenarios.

– HAZMAT: transport Security Vulnerability Assessment (SVA) by Reniers and Dullaert (2013). The hazardous materials HAZMAT transport SVA assesses the relative security risk levels of the different modes of hazardous freight transport models, e.g. road, inland waterways, pipelines or railway. The policymakers could use this tool to assess the user-friendly security in multi-modal transport. The HAZMAT model follows:

The intermodal risk is determined on the minimum security risk path, considering only the risks of the individual segments of a transport route and include also the number of intermodal transshipment.

The risk with the transshipment is defined as: R t = R n t ( 1 + x ( n t s ) ) where R t is the security with the transshipment, R n t is the security risk without transshipment, x is the weight factor for importance of transshipment risks, compared with the transportation risks and n t s is the number of transshipment. The model was implemented with CPLEX studio and OPL; it was successfully tested on two multimodal networks with highways and railways.

– New Delhi, Indian busy urban area MMTS by Kumar et al. (2013). In 2021 the Delhi population will be around 23 million, therefore public transit should be integrated. In Kumar et al. (2013) MMTS focuses on reducing congestion on roads and improving transfers and interchanges between modes. Delhi’s public transport will grow from a 60% of the total number of vehicular trips to at least at 80% in 2021; 15 million trips per day by 2021 in the Integrated Rail-cum-Bus Transit, plus 9 million by other modes are estimated. The Delhi public transport model is illustrated, evaluated and its performance is discussed in Kumar et al. (2013).

The performance of the MMTS is quantified using the following measures.

In particular, for the New Delhi case study: TTR = 1.3 shows a competitive public transport; LS, the mean OVTT / IVTT > 1 , thus people spend more time out-of-vehicle than in-vehicle; IR ∈ [ 0.2 , 0.5 ] value shows that inter-connectivity between transportation modes should be improved; PWI = 0.825 for the metro is recommended as the mean passenger waiting time is similar to metro’s frequency; RI = 0.7681 , indicates that passenger satisfaction should be improved.

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Fig. 3

ARKTRANS: the Norwegian framework for MMTS (Natvig et al., 2006): a) MMTS components; b) Functionality required related to transhipment and storage; c) Reference model and upper level functional decomposition of the Transport Network Management; d) Intertwined processes, a theoretical example.

– ARKTRANS. The Norwegian MMTS framework architecture by Natvig et al. (2006) and Natvig and Vennesland (2010). The framework offers an overview of all the major systems running in Norway that will hopefully further contribute to new and improved solutions. The transport, whether sea, air or railway, have similar needs and challenges with respect to communication, information, management, planning and costs.

Figure 3 illustrates: a) specific multimodal components, b) an example of functionality between components, c) an example of the Transport Network Management, d) and how a transport process includes functionality from sub-domains and how processes are intertwined; thus transport services work together and efficiently exchange information. Furthermore, the main MMTS specifications of the ARKTRANS frameworks could serve as a guide for other similar frameworks.

The detailed technical aspects conclude this list. A beneficent interaction within all MMTS process leads to an efficient multimodal transport framework. Overall conclusions specify that MMTS is especially suited for long distances; a major MMTS feature is the total travel time; the access, egress and transfer times could be reduced if there is an integrated MMTS, e.g. park and bicycles facilities, and card access on transit systems.

Other Frameworks. Frameworks for multimodal transport security and various policy applications are described in detail in the book by Szyliowicz et al. (2016). Other related frameworks and security challenges, for both passengers and freight and security and policy applications around the world, are analysed in the book by Wiseman and Giat (2016).

3.1 Passenger Multimodal Transportation

> Theoretical approaches.

– Designs for chains and networks by Bockstael-Blok (2003). As the author describes it: “The objective of this research is: Develop a design approach for improving inter-organizational multimodal passenger transport systems from a chain perspective”. The article raises some interesting aspects, like balancing the positive and negative impacts of mobility, a holistic approach for modes of transportation not necessarily reducing the number of kilometres for passengers, but improving the number of vehicle-kilometres, resulting in a more efficient usage of the resources (infrastructure, fuel).

– A highly conceptual approach by Chiabaut (2015). It is applied to a very idealized network. The authors aim to combine different transport modes by extending the concept of Macroscopic Fundamental Diagram (MFD) and therefore, the efficiency of the global transportation system can be assessed. This approach can be applied to a wide range of cases. Although it is an idealized analysis, it provides knowledge about how to compute the overall performance of a multimodal transportation network and methods to compare different traffic management strategies.

> Real-life scenarios.

– A multiobjective linear programming model for passenger pre-trip planning in Greece by Aifadopoulou et al. (2007). As a case study, trips in Greece using public transport were studied through an integrated web based information gateway. The introduced algorithm (with polynomial complexity) computes the compatibility of various modes based on user preferences, respective intermodal stations, and identifies the feasible paths. It was structured to check and certificate optimality; validation on how constrains impact the computational complexity linear model was made; and it focuses on a decomposition strategy. Hub selection is significant for compatibility and viability of MMTS; it leads to identify parameters in order to increase compatibility of MMTS services and fees.

– A detailed analysis of the Rhein-Ruhr area by Schöharting et al. (2003). The authors identified the Rhein-Ruhr area as a network of corridors (or mega-corridor). Good practices are featured and analysed with the aim of putting the Rhein-Ruhr area on the “map” of good examples to follow.

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Fig. 4

Passenger Multimodal Transportation: example of a “waiting time model” (Bouzir et al., 2014).

– A “waiting time model”: case study Tunisian Great Sahel by Bouzir et al. (2014). The model based on multiple variables was developed in order to optimize waiting time in stations. A case study was based on a survey in the Tunisian Great Sahel. Multiple Correspondence Analysis (MCA) and the General Linear Model were technically used. The new model from Fig. 4 depends on the following features: the travel cost, purpose and frequencies while using MMTS, and is based on the age of respondents.

The main results of the case study follow. –

The MMTS combination including bus & tram needs a longer waiting time than other transportation modes;

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Young people wait longer for transport services; they use public transport more often than workers; tourists wait more than daily passengers;

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MMTS trips, including waiting time of taxis, are shorter when two transport services are included;

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The semi-collective transportation seems beneficent as it reduces waiting time; the semi-public transport with just one transportation mode, e.g. taxi, cancels reduced waiting time;

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The travel cost has a major influence in the overall waiting time.

The waiting time within public transportation is a direct consequence of the quality of the transportation service.

– TRANSFER model-multimodal network in large cities by Carlier et al. (2005). The model was introduced for analysis of the multimodal network in large cities, as well as route generation. Building Park and Ride (P&R) keeps automobiles outside city centre. More P&R locations are planned for car drivers as they could park there and transfer to public transport to further arrive in the city centre. The main advantage is that it makes public and/or alternative transport more appealing to passengers.

As any other model, it could be successful if MMTS became more attractive than unimodal transport, e.g. car-only trip. The access and egress are also quantified. Here, MMTS are represented as supernetworks where unimodal networks are interconnected by transfer links, the possibility of transfer and related time and costs. TRANSFER components include the following:

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A multimodal route-set generation module based on network features and passenger preferences;

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An assignment module to distribute transport flows among routes;

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A path-size route-choice algorithm to avoid overlap among the routes in a route set.

The superbuilder tool was developed, combining some unimodal networks & transfering data in order to generate a multimodal supernetwork with features of unimodal networks and most relevant transfer possibilities.

–Transfer points with specific features by Sun et al. (2015). The authors analyse age-related transfer speed, the effect of the time of day, the effect of a single person in relation to others, crowding and the use of smart cards. The authors detailed the following: passenger behaviour related to transfer between MTTS modes; correctness of data in order to make a feasible model for passenger transport when complex real-world configurations are provided; efficient using of the Smart card data within MMTS. An overall conclusion includes the fact that passengers are faster in the morning whether or not it is crowded; children and seniors transfer more slowly than adults but children outperform adults in relation to overpasses; further models will have to support pedestrian behaviour and convenient facility design.

4 Multimodal Transport Planning

Multiple facets of planning multimodal transport exist, making it more difficult. For example, in a large city, somebody might suddenly decide to engage in a long distance travel. This implies an ad-hoc computation of the route and means of transport to be used, according to the individual personal preferences, e.g. not using metro system due to motion-sickness. Planning such a transport means using any available means at the specific time; the factors to consider could include: time, cost, weather, waiting time in hubs, etc. What does planning a diverse transport system for a large city imply? The designer must compute the available resources, the requirements and even the schedules/working hours of different companies.

In the planning phase, the designer could suggest the transport means (buses, trams, etc.) in order to obtain an economic and eco-friendly system. An Introduction in Multimodal Transportation Planning book was published by Litman (2017) in which he summarizes the basic principles for multimodal transportation planning for people. He studies transport options for pedestrians, like sidewalk design, bicycles, ride-sharing and public transit systems. He also has very good explanations for multimodal transport planning process, impact to be considered that is often overlooked and different traffic models, like the Four-Step Traffic Model. The first stage of planning a multimodal transport system is to understand its complexities. A complete and accurate model has to be created and analysed.

> Planning MMTS with uncertainties and limitations.

– Fuzzy cross-efficiency Data Envelopment Analysis by Dotoli et al. (2016). The planning of efficient multimodal transports using a fuzzy cross-efficiency Data Envelopment Analysis technique is presented by Dotoli et al. (2016). Other approaches like uncertainty conditions with complex traits and high discriminative power are described. They prove the effectiveness of their approach while studying the optimal transport planning and computing the boundaries of the multi-modal transport. In Sumalee et al. (2011) the multi-modal transport network with demand uncertainties and adverse weather conditions includes formulation of the fixed point problem; other future and existing related developments include works by Ticala (2017) and Xu and Gao (2009).

– Metaheuristics for real-time decisions by Mutlu et al. (2017). The planning part of multimodal transport with various limitations is reviewed by Mutlu et al. (2017). They discuss problems like real-time decisions in the context of short-term planning, restructuring and re-configuring logistic strategies, and collaborative planning (Fig. 5). Appropriate solution methods and intuitive meta-heuristic approaches to rapidly act upon changes are suggested.

> Passenger & freight flows Planning for Multimodal transport.

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Fig. 5

Freight Transport Network, Multimodal Freight Transportation (MTP) Collaborations, as in Mutlu et al. (2017) (upper); Network levels and transport modes (left); using Multimodal Routing Algorithm to find the optimum path: the number of paths, transfers and total time (right), see Bielli et al. (2006)

– Multi-Agents Systems for MMTS planning by Greulich et al. (2013). As the name suggests, the implementation uses intelligent agents representing various stakeholders and considers the effects of passenger behaviour.

– Genetic Local Search tested in the Java Island, Indonesia by Yamada et al. (2007). The research revealed that a procedure based on Genetic Local Search outperforms in order to find the best combination of alternatives.

> Multi-modal systems.

– A multimodal travel system by Bielli et al. (2006). The authors focused on the network object modelling (Fig.5). This enables to use the model for computing a shortest path while also integrating multimodal options. They also implement and test a solution for the problem of long-run planning in such systems.

– Syncromodal Transport Planning by Mes and Iacob (2016). It is a multimodal planning where the best possible combination of transport modes is selected for each package and is discussed in depth. The syncromodal algorithm is implemented in a 4PL service provider in the Netherlands and managed to obtain a 10.1% cost reduction and a 14.2% reduction in C O 2 .

– A multimodal transport path sequence: AND/OR graphs facilitate planning by Wang et al. (2020b). It proposes a triple-phase generate route method for a feasible multimodal transport path sequence, based on AND/OR graphs. Energy consumption evaluates the multimodal transport energy efficiency. A biobjective optimization model for both energy consumption and route risk is solved with an ant-based technique. The research is limited by the graph complexity; the simulation shows valid and promising results.

> Traffic Flow Risk Analysis and Predictions.

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Fig. 6

European Carlink framework for distributed time-dependant transport networks; a validation within a route from Arlon to Luxembourg and a related mobile application representations (Bock, 2010).

– Fuzziness approach for risk analysis by Stanković et al. (2020). A fuzzy Measurement Alternatives and Ranking according to the Compromise Solution, fuzzy MARCOS for Road Traffic Risk Analysis was proposed. The method defines reference points, determined relationships between alternatives & fuzzy ideal/anti-ideal values and defined utility degree of alternatives in relation to the fuzzy ideal and fuzzy anti-ideal solutions. A case study on a road network of 7.4 km was made. The method supported multi-criteria decision-making, within uncertain environments and its results, in terms of risk, could be further used for improving road safety. Other similar efficient method used to cope with multi-criteria optimization is described in Dzemyda and Petkus (2001). As a plus, parallel processes, see e.g. Dzemyda (1996), are effective to optimize objective functions.

– A Best-worst method & triangular fuzzy sets by Moslem et al. (2020). It is used for ranking and prioritizing critical driver uncertain behaviour criteria for road safety. The case study uses data from Budapest city: on how drivers perceived road safety issues.

– Intelligent transportation system-Bird Swarm Optimizer by Zhang and Lin (2020). It includes an Improved Bird Swarm Optimizer used to predict traffic flows; the prediction results are evaluated and an accurate prediction is obtained; the model has positive significance to prevent urban traffic congestion.

A distributed approach for time-dependant transport networks integrated in the multimodal transport service of the European Carlink platform and validated in real scenarios, was proposed by Galvez-Fernandez et al. (2009) (Fig. 6). A real-life validation is included for a specific route from a Belgian city Arlon to Luxembourg.

In the related mobile application implementation within MTS of the Carlink Platform, the requests are sent to the MTS and the users get the shortest path between two selected locations. A framework for selecting an optimal multi-modal route was designed by Kengpol et al. (2014) based on a multimodal transport cost-model, CO₂ emissions and even the integrated quantitative risk assessment. This complex optimization targets to minimize transportation costs, transportation time, risk and CO₂ emission all at once. Multi-node, Multi-mode, Multi-path Integrated Optimization Problems using Hybrid heuristics in the work of Kai et al. (2010) are studied. They propose an integrated Particle Swarm Optimization (PSO)-Ant Colony Optimization (ACO) double-layer optimization algorithm.

Hierarchical network structures of transport networks and how the main mechanisms lead to these network structures are the main interests of van Nes (2002). Optimizing Containerized Transport across multiple choice Multimodal Networks using Dynamic Programming was proposed and successfully tested on a real problem by Hao and Yue (2016). Route Optimisation Problem using Genetic Algorithms (GA) was proposed by Jing et al. (2012); the same technique was used to solve a Multi-Objective Transport System by Khan et al. (2019) and could be further extended for the multimodal transport. GA was also used by Kozan and Preston (1999) to optimize the time for container handling/transfer, and, respectively, the time at the port, by speeding up the handling of operations.

5.1 Time Optimization of Multimodal Transport

– Running time & Rescheduling; solver: Ant Colony Optimization. Zidi and Maouche (2006) propose an Ant-Colony Optimization (ACO) approach for the rescheduling of multimodal transport networks. The ant-colony approach is best in this case (rescheduling) as it is able to work from a given state and only adapt the solution to new conditions. Rescheduling is a must, as the system is subject to disturbances (traffic jams, collisions, strikes) which cannot be accounted for at the beginning of the transport, but are very likely to introduce delays or other discrepancies. Furthermore, Zidi et al. (2006), plans the public transportation system, by using the ant-colony optimization when the theoretical schedule cannot be followed; this approach overcomes the inherent overloading with information of the operators when some problematic situations occur.

– Transport time between nodes; solver: Genetic Algorithms & K-shortest path. Zeng et al. (2009) take into consideration the transport time between nodes, time needed for mode change and possible delays. They also present a model that aims to minimize transport and transfer costs, built on a GA based K-shortest-paths.

– Real-time system. We cannot discuss time optimizations without including (Bock, 2010) article about “real-time control of freight forwarder transportation networks”. The architecture is depicted in Fig. 7. His approach integrates multimodal transportation and multiple transshipments. The real-time system is continually optimized in order to adapt it to the current status of the live data.

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Fig. 7

Information flow in a real-time control system of transportation networks (Bock, 2010).

6 Future Possible Extensions from Unimodal to Multimodal Transport

As transportation quickly expands worldwide, some existing unimodal transport problems and their solvers could be furthermore extended while including specific requirements to solve multimodal transport problems. Some of these problems are further described briefly.

A. Supply Chain Networks. One of the two-stage supply chain network is considered here to optimize the cost from a manufacturer, to a given number of customers while using a set of distribution centres (Fig. 8).

Models & Solvers: Supply chain for further Multimodal extension.

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Multi-Objective Goal Programming by Roy et al. (2017). The mathematical model of Two-Stage Multi-Objective Transportation Problem (MOTP) with the use of a utility function for selecting the goals of the objective functions and numeric tests are included; real-world uncertainty with the use of grey parameters (reduced to numbers) are also involved. As for the metrics within objective functions, usually Euclidean distances are used, but today, for urban related ITS, for example, it could be a plus to use city-block distances as in Redondo et al. (2012), where an evolutionary multimodal optimization technique with suitable parameters obtains better results than existing techniques.

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Genetic Algorithm. Pop et al. (2016) proposes a heuristic-genetic approach with a hybrid based GA for the capacitated fixed-charge problem. Their algorithm was tested on benchmark instances and found to obtain competitive results with other state-of-the-art algorithms.

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Other heuristics. Chen et al. (2017) studies an Uncertain Bicriteria Solid Transportation problem; Moreno et al. (2016) employs a heuristic approach for the multiperiod location-transportation problem. Several versions of supply-chain problem including efficient reverse distribution system, secure and green features alongside related solvers are presented by Pop et al. (2014, 2016, 2019). A parallel fast solver where the search domain of solutions is efficiently reduced at each iteration was proposed by Cosma et al. (2020) for the two-stage transportation problem with fixed charges. It was identified as a very competitive approach when compared to existing ones using the literature dataset.

B. (Generalized) Vehicle Routing Problem. For a given set of vehicles and clients, the (G)VRP problem is to determine the optimal set of routes, see Toth and Vigo (2002). This is one of most studied combinatorial set of problems. Lee et al. (2019) consider an integrative three-echelon supply chain: Vehicle Routing and Truck Scheduling Problem with a Cross-Docking System; this promising logistics strategy distributes products by eliminating storage and order-picking while using warehouse: directly from inbound to outbound vehicles; a cost optimization EEA-based method was proposed which outperforms existing solvers.

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Fig. 8

Examples of feasible solutions for supply chain in order (Pop et al., 2014), GTSP (Helsgaun, 2015) and GVRP (Pop et al., 2011).

Due to its effectiveness many variations of the VRP were built on the basic VRP with extra features, e.g. the Generalized VRP (GVRP), Fig. 8. VRP with time windows and VRP pick-up and delivery problems, e.g. solved by Vaira and Kurasova (2014) with GA insertion operators, can be further extended to related complex problem.

A version of GVRP includes designing optimal delivery or collection routes, subject to capacity restrictions, from a given depot to a number of locations organized in clusters, with exactly one node visited from each cluster. See Ghiani and Improta (2000), Pop et al. (2013) for more details.

Models & Solvers: (G)VRP for further Multimodal extension. –

Capacitated VRP implies that the vehicles have fixed capacities and the locations have fixed demands in time, see Toth and Vigo (2002);

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VRP & Multiple Depots involves more depots from which each customer can be served as in Crevier et al. (2007);

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Heterogeneous Fixed Fleet VRP uses a heterogeneous (different types) fleet of vehicles as in Taillard (1999);

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Multi-Commodity VRP deals with more commodities per vehicle, which has a set of compartments in which only one commodity can be loaded, the same as in Repoussis et al. (2006);

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Tabu Search and hybridization. Various heuristics have been developed for solving the VRP including: an algorithm based on Tabu Search, adaptive memory and column generation described by Taillard (1999). Tarantilis et al. (2004) implemented a threshold accepting procedure where a worse solution is accepted only if it is within a given threshold. A multi-start adaptive memory procedure combined with Path Relinking and a modified Tabu Search was developed by Li et al. (2010).

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Iterated Local Search based & Set Partitioning. (Subramanian et al., 2012) described a hybrid algorithm composed by an Iterated Local Search based heuristic and Set Partitioning formulation.

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Bio-inspired algorithms. Matei et al. (2015) propose an improved immigration memetic algorithm which combines the power of genetic algorithms with the advantages of local search. The article describes the advantages of the immigrational approach on the overall quality of the algorithm (result quality and run-time speed). Diverse versions of the Genetic Algorithms for solving the current problem are presented by Matei and Pop (2010) and Petrovan et al. (2019). Ant colony methods were used to solve Generalized VRP by Pop et al. (2009a, 2009b) and Pintea et al. (2011).

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Heuristics. Leuveano et al. (2019) proposed a heuristic to find optimum inventory replenishment decision when solving transportation & quality problems into a Just-in-Time (JIT) environment. An vendor-buyer lot-sizing model was proposed; parameter study was included and both capacitated and incapacitated cases were studied. Some advantages of the proposals follow: it obtains feasible solution for inventory replenishment decisions; improves transport payload, reduces defectiveness of products and improves quality-related costs.

C. (Generalized) Travelling Salesman Problem. Since 1988, this is one of the most studied problems, and the problem of Applegate et al. (2006) from which the (G)VRP evolved. It is considered a particular case of GVRP when the capacity of the vehicles is infinite, and no intermediary return to the depot is required (Fig. 8). Some GTSP versions use a node from each cluster in a route solution, e.g. a city-node from a county-cluster. (G)TSP libraries (see TSP library, 2020; TSP data instances, 2020; GTSP Library, 2020) are continuously updated mainly based on Geographical Information Systems (GIS) as in Crisan et al. (2020a), Crisan et al. (2016), Crisan et al. (2017) and TSP world-wide countries instances (2020). Integer solvers, e.g. NEOS-Concorde integer TSP solver (2020), are feasible for TSP, but solving large-scale real-life problems requires updated strategies.

Models & Solvers (G)TSP for further Multimodal extension.

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Intelligent Transport System – (G)TSP related. In the context of multimodal transportation, the (G)TSP family of problems has many applications; for example, it could be extended to a related Intelligent Transport System (ITS) as in Crisan et al. (2020b) and Pintea et al. (2018). Recently, Internet of Things (IoT) was used by Luo et al. (2019) to connect platforms for ITS.

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Other – (G)TSP related models. Recent optimization models allow the instances of realistic freight rail transport to be solved; a stage-wise approach for solving the scheduling and routing problems separately nowadays is prevalent.

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Heuristics. As for VRP, the solvers of (G)TSP are using mainly heuristics: Tabu Search of Pedro et al. (2013) uses local search and accepts worsening moves, but introduces restrictions to discourage previously visited solutions; Dynamic Programmingby Bellman (1962); Approximation Algorithms by Malik et al. (2007);

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Bio-inspired algorithms used to solve (G)TSP include: Simulated Annealing by Wang et al. (2015); Genetic Algorithms by Lin et al. (2016) and Pop et al. (2017) are one of the most straightforward ways to tackle TSP. Ant Colony Optimization (ACO) used by Cheng and Mao (2007), Mavrovouniotis and Yang (2011), Dorigo and Gambardella (1997), Pintea (2015) and Pintea et al. (2007) uses pheromone trails to optimize routes; a successful interactive Machine Learning (iML) uses ACO to solve TSP with the human-in-the-loop approach as in Holzinger et al. (2019); some other related natural computing solvers include: Particle Swarm Optimization by Wang et al. (2003), Onwubolu and Clerc (2004) and Clerc (2004); Discrete Cuckoo Search Algorithm by Ouaarab et al. (2014) is inspired by the breeding behaviour of cuckoos using agents’ attraction.

An abstract Formalization of Multimodal Transportation, as a concept, is presented by Ayed et al. (2008). ITS could be expanded by using multi-objective facility location problem models and solvers including heuristics, e.g. Lancinskas et al. (2016).

The graph theory is applied within an algorithm in order to optimize routes and route guidance. The authors try to insert their approach into the Carlink project in order to assess its performance. Cosma et al. (2018) propose an efficient Hybrid Iterated local Search heuristic procedure to obtain high-quality solutions in a reasonable running-time.

7 Conclusions

The current selective survey presents a review of real-world problems, applications and optimization in the integrative multimodal transportation. Transportation is a key element of today’s society and a very important engine for economic growth. Some areas (like food, medical supplies) raise transportation to strategic importance, and thus indispensable.

The multimodal transport comes with diverse challenges, e.g. related to security, resource saving and reduction of emissions. In the context of today’s accelerated global warming, it is more important than ever to do everything to reduce pollution as much as possible. An example is the green multi-modal transport organization approach presented by Wang et al. (2020a), where the China-Europe railway network is validated, reducing the transportation time, increasing energy conservation and lowering carbon emissions by 40%, when compared with the classical unimodal water transport.

Uncertainties will coexist with the multimodal transportation problem, and as recent (Sharma et al., 2020) research shows, while using road, rail and air transportation, soft sets, for example, could be used to model these uncertainties related to the transportation attributes (cost, distance and duration of transport). Multi-criteria shortest path optimization, including time, travel cost and route length, for the NP-complete bus routing problem as in Widuch (2013) could be further extended for complex ITS problems.

Multimodal transportation research nowadays is a challenge that is being continued, on both theory (e.g. solving complex vehicle routing problem) and applicability, in order to obtain feasible models and solvers for various transportation means on complex conditions, with general and specific attributes.