1 Introduction

Multimodality within transportation has significantly expanded its scope to encompass the use and combination of multiple transit modes for a vast range of applications [1]. By reducing reliance on private vehicles and promoting a more integrated approach to transportation, multimodality promotes accessible commuting choices along with a reduction of traffic congestion and air pollution. As cities continue to develop, the design and optimisation of multimodal infrastructure becomes crucial for improving urban mobility.

Previous research on multimodality comprises of several literature reviews, particularly in terms of freight, shipping, and supply chains. SteadieSeifi et al. [2] for example provided an overview of the traditional strategic-, tactical-, and operational planning approaches of multimodal transportation, focusing on a collection of studies on containerized freight. Dua and Sinha, [3] reviewed the research trends of multimodal transport within the scope of supply chains. Agamez-Arias and Moyano-Fuentes [4] carried out a systematic literature review on intermodal transport in freight distribution. Kumar and Anbanandam [5] evaluated the state of strategic multimodal freight network design for sustainable supply chains using a classification and coding approach. More recently, Rentschler et al. [6] conducted a systematic literature review on the promotion of sustainability through synchromodal transport, a form of multimodality that builds on the collaboration between shippers and logistic service providers. Similarly, Archetti et al. [7] surveyed studies on the optimisation of multimodal freight, where an increase in synchromodality and drone delivery optimisation was noted. Pencheva et al. [8] reviewed the challenges multimodal freight transport faces, as well as the interaction between modes.

Past literature reviews are also often found to focus on a specific problem and scenario. Verga et al. [9] reviewed the approaches used in multimodal shortest path problems, where the additional modes increase the complexity. Various approaches were identified and categorised as either classical, metaheuristic, or fuzzy, dealing with uncertainty stemming from real life conditions. Tirachini and Hensher [10] examined literature on the pricing policies of multimodal transport, with a special focus on public transit. Park and Goldberg [11] examined multimodal spatial accessibility, alongside temporal changes, to investigate the methodological progress of place-based accessibility measures. Beresford et al. [12] presented a thorough review of a multimodal cost/time-distance model, offering insights into the relations between modes, methods and more, trying to assist organisations by using the model to review door-to-door supply chain costs. It shed light on the shortcomings of the oversimplification of multimodality.

Considering the vast literature that multimodal mobility may cover, it remains an enduringly pivotal topic of interest. In 2021, Matei et al. [13] suggested that the normalization of multimodal systems supports the progression of the global economy, as well as sustainability, mitigating global warming. Efforts focused on optimising these systems were reviewed and metaheuristic approaches like the genetic algorithm, ant colony optimisation, and simulated annealing were examined. In consideration of real-time decision making, the problem may become dynamic. This motivated Mutlu et al. [14] to review multimodal freight studies at the operational level, systematically classifying different planning and solution methods. Metaheuristics were again highlighted as an appropriate solution method, due to their suitability of processing different data types. Elbert et al. [15] reviewed the problem of multimodal service network design and flow planning at a tactical level. The problem characteristics, model formulation and solution approaches were rigorously reviewed, classified and split into either dynamic or non-dynamic problems, with and without given network structure. Metaheuristics were used in most cases for solving the problems, with alternatives, like exact methods, used less frequently. The authors noted that no specific metaheuristic stood out as overwhelmingly dominant in terms of popularity.

Generally, studies outline metaheuristics as a leading solution method in multimodal transportation problems, due to their effective handling of complex problems like the Transport Network Design Problem [16]. Due to their low computational costs and adaptability to diverse problem structures, metaheuristics possess vast applicability. This is of particular value to the vast spectrum of optimisation problems that arise through the modelling of multimodal transport problems. Common multimodal urban transportation problems include the Shortest Path Problem (SPP), the Transport Network Design Problem (TNDP), the Transit Assignment Problem (TAP), the Orienteering Problem (OP), the Hub Location Problem (HLP), the Vehicle Scheduling- (VSP) and the Vehicle Routing Problem (VRP). These problem categories are reviewed in the remainder of our study.

Classically, the SPP is solved to find the shortest route between pairs of Origin–Destination (OD) vertices to minimise passenger travel times and/or operating costs [17]. When applied on a multimodal urban transport network, private-, public- and pedestrian modes are typically modelled using graph theory, where each respective transport mode may be represented using different methods, including subgraph- or multilayer approaches [9]. The heuristic solution approach of Dijkstra’s algorithm is typically used to solve SPP problems, and novel approaches solving the SPP are typically benchmarked against it. Within the domain of urban mobility, there exist multiple considerations. For instance, travellers may require alternative paths when the shortest route is crowded, resulting in the K-shortest Path Problem. For fixed modes like the metro, where each vertex may only be visited exactly once, a Hamiltonian SPP can be considered [17].

As SPP can be used for path generation, it is frequently integrated within the formulation of the TNDP. Depending on the underlying infrastructure and input data for designing the considered transportation network, the TNDP often varies in definition. Broadly, it typically defines a set of transit lines, stops and passenger demand, and can also be extended towards considering operational frequencies [17]. Inherently complex just by considering a network with homogeneous vehicles due to the many criteria the TNDP may target, this optimisation problem is often approached using multi-objective optimization techniques. Furthermore, passenger distribution is typically modelled in the TNDP as part of a bi-level sub-problem by solving an assignment problem. For multimodal networks, the TNDP requires a multi-commodity formulation, further increasing its complexity. Multimodality is typically captured in the TNDP by defining a set of different transit modes of a multigraph network, where again modes may be denoted using subgraphs.

Similarly, other classical transport optimisation problems like the Orienteering Problem or Hub Location Problem are adapted to meet the needs of transporting passengers using different modes. The OP is often extended to plan touristic itineraries, while the HLP may accommodate the growing demand of multimodal transfers that occur with the change of transportation. Additionally, mode specific optimisation models like the Vehicle Scheduling- and Routing Problem are used to improve the service performance of multimodal transport. The diversity of multimodal transport problem variants, combined with the modelling requirements needed to support the needs of multimodal networks and urban passenger mobility, result in the growing complexity of transportation networks, necessitating sophisticated solution approaches to effectively support and enhance the services.

Passenger transport, specifically within urban settings, diverges from freight transport, particularly due to the different requirements and considerations intrinsic to the problem characteristics and modelling. This was highlighted by Udomwannakhet et al. [18], who reviewed the optimisation of multimodal transportation according to the two categories of freight and passenger transport. More recently, Xu et al. [19] reviewed studies on the emergency evacuation of passengers from multimodal transportation hubs. Similarly, in 2023, Alessandretti et al. [20] comprehensively assessed studies on multilayer transport networks and multimodal urban mobility to introduce a mathematical framework. In reviewing modelling approaches on urban mobility and public transport system dynamics, multimodal quantifiers and infrastructures were surveyed. Alongside methodological advancements, a rise in datasets and novel computational tools, especially open source, were additionally reviewed.

Parallel to the development of multimodal infrastructure is the rapid implementation of Internet of Things (IoT), resulting in the extensive collection of diverse datatypes. Harris et al. [21] extensively reviewed 33 EU framework programs focusing on ICT developments within multimodal European freight transportation systems, analysing their slow adoption. The potential of emerging technologies such as cloud computing and IoT was explored, along with existing barriers and the transformation of the sector.

In addition to IoT, the continuously increasing availability of data regarding real-time conditions is particularly relevant to transportation design. The accessibility of this information supports the capability of dynamic optimisation, a necessity for real-world modelling and control. Nonetheless, the inherent complexities associated with multimodality, combined with numerous user preferences (for e.g. comfort and routes) and conflicting modelling criteria (for e.g. sustainability and timing), present an intricate optimisation challenge. Furthermore, as the range of transportation modes expands, so does the complexity of the problem. This expands to established Non-deterministic Polynomial time (NP) problems, often modelled within the field of multimodality. Metaheuristics thus naturally emerge as a central methodology for the design and operations of multimodal urban mobility.

Motivated by the above, this literature review systematically examines the current landscape of research regarding the metaheuristic optimisation of multimodal transportation systems, with a special emphasis on the urban transportation of passengers. To the best of our knowledge, no such systematic analysis exists, hence this paper sets out to determine the following:

Q1: Unveil emerging research interests in multimodal optimisation, along with the classification of studies and observational trends.

Q2: Assess the leading metaheuristics employed by the aforementioned scope, along with how they relate/differ to one another.

Q3: Outline prospective fields in need of further research, along with opportunities and challenges.

To achieve this, the remainder of this study is structured as follows. In Sect. 2, the research methodology behind the bibliographic search is presented. The gathered literature and reported findings are then split into three sub-categories of Sect. 3, based on their taxonomical classification. Papers employing multiple methods, either studying multiple metaheuristics in parallel or in terms of hybridisations, are relegated to their own sub-section. Throughout, the formulations, problem scope, complexity, static/dynamic nature, and employed frameworks are comprehensively examined, with the studies categorised accordingly. Metaheuristic performances are analysed, along with their performance indicators. This paper endeavours to provide a comprehensive review of the current state of research, offering insights into potential gaps and recommendations, along with guidance and considerations for future investigation, all of which is concluded in the final section.

2 Research methodology

The bibliographic search was conducted in concordance with the scientific strategies employed by Cook et al. [22] for systematic reviews, used to reduce bias for the assembly, critical appraisal, and synthesis of studies. To achieve this, five systematic steps adapted from the recommendations of Denyer and Tranfield [23] were referenced as follows:

2.1 Defining the scope of study

The metaheuristic optimisation of multimodal transportation systems, with a special emphasis on the urban transportation of passengers, was identified as this paper’s scope of study in the previous section. Following the formulation of the three directions of research (Q1–Q3), keywords and inclusion criteria are successfully outlined below.

2.2 Study retrieval

To narrow and locate the most significant studies related to this systematic review, a reproducible search of past literature was implemented through the Scopus database using the following logical search combinations: keyword 1 (“multimodal” OR “multi-modal”) AND keyword 2 (“transport*” OR “transit”) AND lastly keyword 3, which comprised of every metaheuristic listed below in (i)–(xvii), each individually searched with the other two keywords:

i. "Genetic Algorithms", ii. "Particle Swarm Optimisation", iii. "Differential Evolution", iv. "Ant Colony Optimisation", v. "Simulated Annealing", vi. "Tabu Search", vii. "Variable Neighbourhood Search", viii. "Iterated Local Search", ix. "Scatter Search", x. "Greedy Randomised Adaptive Search Procedure", xi. "Path Relinking", xii. "Harmony Search", xiii. "Memetic Algorithm", xiv. "Artificial Bee Colony", xv. "Bat Algorithm", xvi. "Cuckoo Search", xvii. "Firefly Algorithm".

The list of metaheuristic keywords was selected after a comprehensive initial search, combining the first two keywords 1 AND 2 (together representing the scope of multimodal transportation) AND the keyword “metaheuristic”. The selection allows for a more specialised search, due to the methodologies primarily being referred to by their algorithmic name, rather than as a metaheuristic specifically, or at times being labelled as “heuristics” due to their study-specific applications.

Thus, variations of keywords 1, 2 and 3 were all exhaustively combined as research queries, resulting in works that contained all three keywords in either their title, abstract, or keywords list. Due to the large volume of available literature, the most popular metaheuristic keywords (i)–(iv) were narrowed down, such that works containing all three additional keywords “container”, “aircraft” and “freight” were filtered out using the AND NOT operator. In total, 451 works were retrieved on the 16th of January 2025.

2.3 Paper selection and evaluation

Duplicates and retracted studies were removed, resulting in 362 unique papers. After a thorough review of the collected papers’ abstracts, studies centred on non-urban related transportation systems, mathematical research, emergency relief, supply chains, or freight/cargo operations, were excluded. This was to maintain the coherency of the scope of multimodality to urban-based passenger transportation systems. Non-English written papers were similarly not included. As a result, 69 studies were found to be relevant to the topic of this paper.

2.4 Analysis and synthesis

From the 69 papers, 31 employed Genetic Algorithms (GA), 2 Differential Evolution (DE), 11 Ant Colony Optimisation (ACO), 6 Particle Swarm Optimisation (PSO), 1 Artificial Bee Colony (ABC), 3 Simulated Annealing (SA), 2 Tabu Search (TS), 2 Variable Neighbourhood Search (VNS) and 1 Greedy Randomised Adaptive Search Procedure (GRASP). The remaining 10 consisted of the use of multiple metaheuristics, 3 of which used hybridised methods. In answering the research questions by rigorously evaluating and synthesising the underlying themes among the body of works, a classification system was used to categorise the analysed research by outlining the following of each paper: problem scope, investigated transport modes, corresponding objective(s), Multi-Objective Optimisation Problem (MOOP) (including bi-level structures), mathematical formulation, metaheuristic used, whether the problem was framed within a Static (S) or Dynamic (D) context, and, lastly, the size of the multimodal testbed (Small (S), Medium (M) or Large (L)).

Studies whose mathematical formulations were not explicitly stated or did not possess the relevant information to infer the formulations were left blank within the categorisation of “Formulation”. For clarity, in the case of bi-level problems, the upper-level mathematical programming is presented as the formulation.

By construction, metaheuristics employ stochastic search processes [17], which is why the classification of the studies’ metaheuristic approaches into deterministic and stochastic categories is not taking place throughout this paper’s review. Instead, a Dynamic/Static categorisation is used to present whether the optimisation problems were solved using frameworks which were respectively online/offline and/or in real-time or not.

Urban transportation modelling often varies in scale with respect to the specific context of the metropolitan planning taking place, depending on factors such as the region and infrastructure being optimized for, as well as the specificity/generalizability of the optimisation problem being solved. Smaller networks may focus on localized transportation issues or numerical benchmark networks for validation, while larger planning problems may consist of complex transit networks involving hundreds to thousands of links and nodes (examples include [24,25,26]). In developing the criteria of our systematic review and analysis, the categorisation of the reviewed problems’ sizes and scale were explicitly pre-determined. As many of the authors of the reviewed literature have expressed their own scaling of their optimized networks by describing them as either large, medium or small, alongside the details regarding the number of links and nodes of their network(s), the scaling used in this review paper considers this, as well as the relative difference in sizes of the collected literature to one another. In papers where the numbers were not provided, “N/A” is written in place for the classification. This review adopts a vertex/node-based scale to determine the optimisation problem’s size, where the number of vertices/nodes is used to determine the categorisation. The following node-based scaling of 0 < “Small” ≤ 30, 30 < “Medium” ≤ 80, 80 < “Large” was applied throughout the review and was found to mostly coincide with what the papers classified their own networks as, supporting the use of its categorisation.

2.5 Reporting results to present a research agenda

Insights and key thematic aspects drawn throughout the systematic review are then lastly discussed to present a research agenda, determining the three directions of research outlined in the introduction, concluding with recommendations for future research.

3 Results

The distribution of multimodal transportation problems that have been optimised using metaheuristic methodologies has been assorted into a pie chart that has split the studies according to the problem being solved, as presented in Fig. 1 below.

Fig. 1
figure 1

The assortment of literature according to their optimised multimodal transportation problem

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Variants of the TNDP arise as the most prominent multimodal transportation problem being solved, followed by the SPP. The third largest group includes a collection of different and relatively lesser researched multimodal transportation problems that do not fit the problem formulation of the other more popularly solved transportation problems. Other optimisation problems, that can be grouped as a collection of studies, include the OP, VRP, HLP, and VSP.

Below, Fig. 2 denotes that throughout the reviewed publication history, Evolutionary Algorithms (specifically GA) appear to broadly dominate the field. It is worth noting that Swarm Intelligence methods (ACO, PSO and ABC) make up a substantial portion of the reviewed literature. Together, the two groupings present population-based metaheuristics as the most applied solution methods for multimodal passenger transportation problems within urban settings. Collectively, single-solution methods (SA, TS, VNS and GRASP) are often found to be considered alongside population-based approaches for either comparative purposes, or to be hybridised into a Memetic Algorithm. A vast range of methods are observed to be developed in the past decade.

Fig. 2
figure 2

Publication of literature according to their metaheuristic taxonomy

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Conversely, accounting for the explosive advancements in data collection and processing, facilitated by novel technologies such as machine learning, this study strives to analyse and assess the existing progress of metaheuristic methods used to optimise urban multimodal passenger transportation. Thus, our study narrows down implications for what its current state of research entails and how it fits within the rapid developments of technological tools, providing direction for future research.

To determine this, the collected works are sorted according to their employed solution methodology, such that all metaheuristics are organised into chapters classified by their taxonomical categories. This results in Evolutionary Algorithms and Swarm Intelligence methods being categorised within Sect. 3.1 of Population-based approaches. All remaining metaheuristics fall within Sect. 3.2 of Single-solution approaches. Studies that have investigated multiple metaheuristics either individually or for hybridization purposes, are discussed in Sect. 3.3.

3.1 Analysis of population-based approaches

This section examines research works which employ a subset of optimisation algorithms utilizing a population of candidate solutions to explore the search space. The metaheuristics are grouped according to their shared solution approach characteristics, resulting in the sub-sections of “Evolutionary Algorithms” (EAs) and “Swarm Intelligence” (SI) as follows.

3.1.1 Evolutionary algorithms

3.1.1.1 An overview

Mimicking the process of natural evolution, EAs draw inspiration from the principle of natural selection, genetic inheritance, and mutation to solve complex optimisation problems. Typically, a population of candidate solutions is evolved and maintained throughout the algorithm, following the key steps of: Initialisation, selection, followed by the genetic operators of mutation and crossover, both evolving the population of solutions iteratively until a termination criterion is met. As a result, the evolutionary operators produce the new generation of diverse solutions [17].

Like other metaheuristics, EAs are typically used as solution search strategies, capable of being tailored according to the model of the optimisation problem. Using sensitivity analysis, the algorithm’s encoding, parameters and termination criteria can be fine-tuned according to the specific needs of the problem at hand. While metaheuristics can broadly be defined by their basic routines and operators, they typically do not abide to a standardised approach [17]. The GA is an adaptive and robust optimisation method, performing effective searches on poorly defined spaces using an ordered pool of chromosomes to represent regions within the search space. The genetic operators generate new solutions from existing ones, efficiently navigating information of the problem’s search space. This guided search is thus not particularly susceptible to local optima, nor greatly influenced by non-continuous functions [27]. The processing of the solution variables, using randomised operators and stochastic search, allows for the intelligent exploration of search spaces, regardless of whether they are discrete or continuous. This is evident in its significant use among combinatorial optimisation problems, making up most of the literature presented hereinafter.

Starting with an initial random population of individuals, these candidate solutions from the solution search space are characterised by a set of the optimisation problem’s decision variables, making up the genes of the chromosome. Depending on the mathematical program being solved for, the encoding scheme of the chromosome and fitness function may vary. For binary and integer decision variables, bit-string representations can be used, whilst for continuous formulations, floating-point vector representation would be used instead. The encoded representation affects the genetic operators that follow, possibly requiring modifications as they are typically based on bit-string formulations. After this encoding, the GA framework proceeds with initialisation. Iteratively, the algorithm evolves its population of solutions to determine the fittest individuals to be chosen for reproduction of the next generation. This is achieved using a fitness function, representing the optimisation problem’s objective, which evaluates the performance of the candidates, guiding the reproduction’s selection process to favour the better performing solutions. Crossover and mutation operators generate the offspring population of the next generation of solutions, resulting in a diverse exploration of the solution space. This search and generation process is iteratively repeated until a termination criterion is satisfied, evolving the population of individuals. Independently, the population can explore the search space simultaneously, enabling parallelism. GAs can additionally be hybridised with other metaheuristics, popularly embedding local search routines or customising genetic operators [17].

Unlike the bitstrings characteristic to a GA, Differential Evolution employs vectors of real-valued variables to represent the population of candidate solutions. Moreover, in place of the genetics-inspired transformations, vector operations are used to navigate the search [28]. Generally, DE is considered as better suited for continuous constrained optimisation [17]. Similar to GAs, DE uses hyperparameter values to tune its algorithm according to the optimisation problem. Examples of this include population size, crossover probability and the mutation constant (also respectively referred to as the recombination constant and differential weight within the context of DE). As an evolutionary algorithm, DE follows the core steps of initialising a population of solutions, after which for each population member, also known as an agent, a set of distinct and random agents are chosen to compute a donor vector and randomly select a position within the search space. The fitness function evaluates a resulting trial vector and compares it to the population member, replacing it should the trial vector yield a better result. This iteratively repeats until a termination criterion is triggered [17].

3.1.1.2 Classification of studies

To begin, studies utilizing EAs as part of their solution methods for the famously combinatorial Shortest Path Problem (SPP) for multimodal routes are reviewed.

The optimisation of multimodal routes based on the hybridisation of Dijkstra and GA was presented by Zidi and Hammadi [29] for prospective deployment in real-time decision-making systems, to support travellers in the continuity of their trip. A distributed approach of formulating multi-objective criteria including travel costs, time and comfort into a single objective was employed, designed for frameworks to interactively provide users information to reduce travel uncertainty in the navigation of dynamic networks. To resolve such NP-Hard problems in large networks, the GA was adapted into a method of Control Genetic Operators with Management of Final Population. Abbaspour and Samadzadegan [30] meanwhile developed a dual-module framework to optimise time-dependent multimodal shortest paths within large and complex urban areas. This approach integrated user contextual information, such as age, into its multimodal shortest path module. The system utilized parameters like Origin–Destination (OD) data, start time and transport mode to compute the optimal paths using a GA. The EA also considered the necessary spatial information from a geodatabase, generated from mode pathways, including bus and subway, as well as a Stations and Service Line Timetable. Evaluated on 250 randomly selected OD pairs of a Tehran dataset, the algorithm’s capability of finding optimum paths according to applied constraints was demonstrated, with high performance supported by a high average optimisation success rate. Furthermore, multimodal routes were found to not necessarily be the shortest, however they were found to result in the most reduced costs, as opposed to paths of single-mode alternatives.

Medssia and Ghedira [31] presented a distributed guided GA to enhance the quality of multimodal transportation services both generally, as well as in cases of disturbances. Public transport services including metro, bus, tram and regional trains were considered and the multi-objective criteria of time, cost and comfort were aggregated into a sum with penalty coefficients. Tested on disturbances of the Lille Road network of 78 stations and 80 arcs, and compared to both Dijkstra and an Ant Colony Optimisation algorithm for performance, the proposed GA was found to reach the best solution in three out of four scenarios with the best execution time. Along the same lines, Bouazzi et al. [32] extended their prior research on Superimposed Graphs (SGs), using an agglomeration of them to represent Guangzhou’s multimodal transportation system and thus optimise the network. This resulted with a Hamiltonian circuit (HC), a circuit that visits each node on the graph exactly once, known to be used in multiple fields including multimodal transport. Framed as a Constraint Satisfaction Optimisation Problem (CSOP), the Integer Program (IP) was solved using different methods to test the efficacy of their newly presented solution framework. Simple- and Superimposed Graphs were implemented using the exact method of Cut and Bound, both evaluated in terms of CPU time. The HC problem was also used to examine the use of a hybridised GA (HGA), where Dijkstra was developed to obtain optimal solutions for the expanded capacity of available paths of the SGs, and then embedded within the metaheuristic. HGA was observed to outperform the standard GA in terms of solution quality, minimising the distance of the circuit for a shorter period.

With the intention of creating a travel plan to encourage passengers to prioritise public transport, Peng et al., [33] set out to minimise the travel time of routes by considering the factor of real-world uncertainty and shared multimodal lines. The unpredictable nature of stochasticity was accounted for by optimising the SPP under a Monte Carlo simulation, deterministically transforming the network. The simulation was combined with both single- and multi-population-based GAs, used to minimise the objective. The two simulated optimisation models performed well, however the multiple independent parallel sub-populations GA proved to obtain better results than the single one within a shorter computation time. To reduce traffic congestion, a visual route recommendation system was devised by Zou et al. [34], constructing multi-layer complex networks based on processed taxi and bike-sharing trajectory data from Xiamen City. An initial network was created to combine the modes using feature calculations and clustering, after which the SPP was solved using a GA, modified with an A* search algorithm for speed. Converging after 10 iterations, multimodal routes were swiftly generated as a result. The hybrid routes were proven to effectively reduce travel time cost within congested areas.

To support the analysis, Table 1 categorises the discussed studies according to the classification system outlined in section “Research Methodology”.

Table 1 Categorisation of studies that use EAs to optimise the SPP
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Continuing from the review of the SPP, the often similarly combinatorial Network Design Problem (NDP) and its transport related subvariants are also found to popularly employ the GA. Cipriani et al. [35] recontextualised the Transport NDP (TNDP) in terms of urban multimodality by considering demand as a variable to estimate the effect of externalities and the service level on modal split. Choice behaviour was deterministically established through a User Equilibrium (UE) assignment model, and modality was represented as a binary logit model. The network was optimised using a Heuristic Route Generation Algorithm (HRGA) which generated a set of feasible routes for the GA’s input. In 2009, Beltran et al. [24] addressed the problem of assigning low-emission vehicles to public transport services, while considering their limited availability. The resulting TNDP accounted for modal shifts and demand variations. The objective was to minimise a weighted sum of operator, user, and external costs in a multimodal, demand-elastic context of car and transit modes. A solving procedure was devised, involving a HRGA to produce a combined set of green and non-green routes. A numerical experiment on the 49-node neighbourhood network in Rome compared the results to the existing urban infrastructure. It was revealed that increasing the weight of externalities guided the search towards solutions of greater green line usage, and increased demand along their routes, irrespective of overall transit benefits. In another work, Zhang and Huang [36] employed a Geographic Information System-based platform “TransitNet” to optimise bus route networks in large cities, focusing on the setbacks caused by the expansion of urban areas and the associated complexities introduced by the added construction of rail transit. To do so, the methodology consisted of three key stages: first, existing bus route performance was quantitatively evaluated, and the identified key issues were then used to extract Existing Good Bus Routes (EGBRs). A multi-criteria evaluation was also conducted, where a kth shortest path method based on Dijkstra’s algorithm generated the candidate route set, which was then optimised by a GA. The EGBRs were included in the resulting optimal set. Tested on Wuhan’s central area, comparing existing- and optimised networks, results revealed that the identified problems were improved, and coverage was increased, validating the importance of the used approach for transit planners and policy makers.

Characterised by a complex road topology and multimodal public transport systems, bus NDPs were explored by Cipriani et al. [25]. A single-objective sum of operator costs, user costs and a penalty related to unsatisfied demand was considered using a HRGA and a parallel GA, determining a suboptimal set of routes with associated frequencies. Tested on the large-scale real-size urban network of Rome, consisting of 1300 nodes, including rapid rail-, bus- and tramway systems, the implemented methodology ran 12 h of 250 iterations and was proven to be robust and effective. The bus network’s waiting time was reduced by 30%, lines reduced by 50% for efficiency, and a 20% decrease in operating costs, still providing the same coverage. The combinatorial optimisation of Exclusive Bus Lanes (EBLs) and variable Bus Frequencies (BFs) was solved by Yao et al. [37]. The objective of minimising the sum of road user travel- and transit operating costs was represented in a bi-level formulation. The upper-level Integer Program (IP) optimally decided the set up for the EBLs and BFs, while the lower-level presented a multimodal transportation network user equilibrium model, using a logit function for modal split. A path-based traffic assignment algorithm and the method of successive averages were used to solve the lower level. Typically non-convex, the bi-level problem was solved using a GA due to its proven efficiency in TNDP. As the number of feasible schemes grows exponentially along with increases of links and lines, enumeration methods were considered infeasible, even for small networks. Results were illustrated on a small network of 13 nodes, showing that the optimisation scheme of EBLs and BFs performed well with the formulated model, particularly under high traffic demand.

To model a multimodal TNDP, an NP-hard bi-level multi-objective optimisation problem was defined by Brands et al. [38]. This model’s aim was to optimise accessibility, urban space use, operating deficits, climate impact and modal split. Transport was categorised as either cars or public transit, facilitated using a mode chain. Modal split was captured using a nested logit model. To solve the Integer Linear Program (ILP), a Pareto set was estimated using an Epsilon Non-Dominated Sorting GA (ε-NSGAII), which was then compared to a standard Non-Dominated Sorting GA (NSGA II). Evaluated using data from the Amsterdam Metropolitan Area, it was determined that whilst ε-NSGAII was comparable to its counterpart in high performance, it was outperformed in terms of early-stage algorithmic execution. Investigating the ramifications of demand uncertainty on the optimisation of multi-objective passenger transportation networks to incorporate robustness into their design, Brands et al. [39] continued their work by analysing the performance of various decision variables under three different demand scenarios. The upper-level of the Discrete Optimisation Problem (DOP) aimed to minimise multiple system objectives concerned with sustainability, including accessibility, use of urban space by parking, operating deficits and climate impact. Presented as a stochastic multimodal UE assignment model, the equilibrium in the lower level sets the network state and load, serving as a constraint for the upper-level, allowing it to interact with the behaviour of the travellers in the network determined by the traffic assignment problem. As established earlier, notable for outperforming its NSGAII predecessor, ε-NSGAII was used to approximate the Pareto set of the variables on different demand forecasts of 2020, 2030. It was found that whilst solutions were susceptible to losing Pareto optimality when assessed under different transportation demand, the loss in performance was limited.

To improve the integration of mass transit systems, Sachan and Mathew [40] proposed a methodology where Dijkstra-generated potential routes were optimised using the GA, due to the large search space and non-convexity of the integer program. Their approach aimed to balance feeder and non-feeder demand, with a case study of rerouting buses alongside the introduction of a new metro system in the Mumbai Metropolitan Region. While a trend towards a global minimum was observed, convergence proved elusive due to the imbalanced dominance of conflicting user and operator costs within the single objective function. Population size was constrained due to the computational limitations of processing the vast candidate route set, resulting in 1000 generations over 52 h of CPU time. Later in 2022, Cai et al. [41] presented a method for optimising transport integration to encourage sustainable and cost-effective use of pre-existing multimodal infrastructure. An Integrated Multimodal NDP (IMNDP) was formulated as a bi-level Mixed Integer Non-Linear Problem (MINLP). Combining mode and route choice, multimodality was captured using a nested logit model, allowing the lower level to be defined as a unified Generalised Modal Split/Traffic Assignment Problem (GMS/TAP). The upper-level problem set out to promote modal-shift, and when optimised, facilitates a balanced, well-connected multimodal network. For the hubs to be collaboratively optimised, they were categorised as either location- or route based. To solve the IMNDP, a Hybrid Genetic Search with advanced Diversity Control (HGSADC) was developed and tested, whilst an efficient hybrid successive average algorithm targeted the GMS/TAP using a solver. HGSADC’s biased fitness function and diversification mechanism successfully expanded its exploration capacity, outperforming the standard GA which in comparison was found to be trapped in local solutions. Diversification, however, increased the average CPU time and decreased the convergence speed/iterations. The metaheuristic occupied 23% of the optimisation’s computation time, significantly less than the solver. Supporting the spread and integration of novel demand-responsive transit systems within pre-existing multimodal networks was similarly of particular interest to Cheng et al. [42] particularly with the increased availability of data provided by online platforms and smartphones. The study focused on the design of Customised Bus (CB) routes implemented in China, where most were typically developed under the assumption that CBs were the sole travel choice, potentially limiting attractiveness due to mode competition between CBs and the metro. To mitigate this and recontextualise the design, the problem was formulated as a leader–follower game. The leader was a CB route planner, maximising bus use according to travel constraints, whilst the followers i.e. commuters, would choose modes and routes according to the state of the network, maximising utility. This relationship was presented as a bi-level Mixed Integer Program (MIP). A case study in Shanghai validated the model, showing that the proposed route network attracted around 5,000 users during the morning peak period.

Moghaddam et al. [43] studied the Multimodal Network Design Problem (MMNDP). Bike, car, bus links and routes were optimised to minimise traffic-generated air pollution, with multiple objectives including demand coverage, travel time and exposure variables. Exclusive bus and bike lane implementations were also examined. The Method of Successive Average was used for the lower-level TAP, optimising routes and mode choice according to traffic and passenger flow. This was then used by the NSGA-II to optimise the upper-level of the bi-level model, determining the network’s route distribution. Mandle’s network, as well as other small networks of 9 nodes and 12 links were used to determine the multi-objective trade-offs. The model was deemed effective, with a slight increase in travel time found to occur by considering air pollution, however the exposure to it was significantly reduced by 47%, with exclusive lanes similarly reducing pollution exposure by 60%. When considering network demand, which was increased by 47%, naturally travel time and exposure increased by 28% and 58% respectively. For further analysis, Table 2 categorises the discussed studies according to the classification system adopted throughout this paper.

Table 2 Categorisation of studies that use EAs to optimise the NDP
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In addition to the SPP and TNDP, multimodality has been largely optimised by EAs for HLPs, as follows. Rahimi et al. [44] proposed a multi-objective MINLP to deal with the multimodal HLP under uncertainty considering congestion in hubs. The model was supported by means of a M/M/c/K queueing system to analyse the wating time of flow units at each hub, as the capability of hub facilities is limited. Fuzzy parameters were considered due to environmental impacts, and to further cope with the uncertainty, a hybrid approach was created. A Differential Evolution algorithm was developed to cope with the computational complexity of the problem, as well as large-sized instances, to obtain near-optimal Pareto solutions. The methodology was tested on a real case study of passenger transport in Iran, investigated to better validate the model’s performance and solution approach. Using computational experiments, DE’s efficacy was proven to be superior for problems of all sizes, ranging from small to large, in comparison to a SA algorithm. A t-test proved that DE outperformed specifically in terms of solution quality, as no statistical difference in CPU time was found.

Seeking to assist the local government in transit hub location planning for the Suzhou Industrial Park, Yuan and Yu [45], developed a cluster-based optimisation framework for multimodal transportation networks. The Hub Location Problem was modelled to optimise Traffic Analysis Zones, defined in terms of geographical and administrative features to increase real-world applicability, which included the simultaneous allocation of clusters, hubs and links. The UE principle was used to establish a multimodal traffic flow distribution of a network given the candidate inter-/intra-hub and links locations and settings. The overall model resulted in an NLP with discrete variables, thus necessitating a metaheuristic approach. A GA, embedded with Augmented Lagrangian Dual algorithm to solve the equilibrium subproblem, was used to solve the HLP to meta-optimality. Converging after 10 h, the obtained results of various design capacities and budget levels were compared to those of alternative strategies and further investigated through sensitivity analyses.

Colovic et al. [46] recently introduced a Micromobility Maximal Coverage Parking Location Model (M-MCPL) in light of the need for regulating micromobility. Specifically, shared electric kick scooters (e-kscooters) were targeted by the model, intended to maximise the coverage of shared e-kscooters’ optimal parking locations. The large-scale case study of Rome, with 1364 census zones, 993 Points of Interests (POIs), 477 and 14 bus stops and railway/metro stations respectively, was targeted by the multi-objective problem. The M-MCPL was solved using the NSGA-II elitist variant of the GA. CPU time of the metaheuristic optimisation ranged from 1.38 to 214.23 s for different instances and cases, with a multi-objective coverage of 39% being achieved for the case of all objectives sharing the same prioritisation.

Noting the growing divide between Transit Oriented Developments (TOD) and Affordable Housing Communities (AHC), Qian et. al [47] examined the potential of optimally allocating Bikeshare Stations (BSS) to alleviate the spatial gap. Addressing this, a multimodal Agent-Based Modelling (ABM) and simulation framework was designed where, given the NP-hard nature of the location optimisation problem, the GA was used within the framework to optimise BSS locations, maximising accessibility to AHC and transit services. Based on the Sacramento network, data comprising of 149 AHC locations, 93 POIs, transit services, and demographic information was used to realistically run the trip generation, distribution and mode split simulation. 500 candidate locations using spatial K-means clustering based on geographic data was used for the optimisation, employing the Manhattan distance criteria. Results underscored the advantages of the strategic positioning of BSS, successfully improving AHC and transit accessibility by reducing travel time and walking distances, as well as by increasing the number of transit-accessible destinations. Bikeshare was additionally noted to not only complement existing transit networks, but also to serve as a substitute for long trips involving multiple mode transfers and prolonged waiting times.

A bi-level bikeshare station location design problem was developed by Song et al. [48] to consider electric bikes alongside conventional bike-sharing systems. A bike-sharing road network was designed over a modelling horizon, representing the daily operational time of bike-sharing systems. The upper-level binary NLP which chose potential locations by maximising social welfare was solved using the GA. Meanwhile, the lower-level multi-period multimodal network equilibrium problem was solved using the rolling-horizon method. The lower-level convex program was decomposed into horizon-based subproblems and solved using Gauss–Seidel decomposition with a revised simplex method, alongside column generation. The Anaheim network of 416 nodes was used to tune the GA’s parameters for accuracy and efficiency, as well as to showcase the framework in a real-world scenario. Alongside other small numerical examples, the Nguyen-Dupuis network containing 13 nodes was used to compare the GA methodology against the ABC metaheuristic and enumeration methods. Results found an increase in social welfare, with GA outperforming the ABC and enumeration in computational efficiency, particularly for increasing network sizes, matching the globally optimal results of the enumeration method. Similarly, a high-quality solution within a reasonable time frame was produced by the GA for the large network.

Xanthopoulos et al. [49] presented their location and capacity model to optimise multimodal mobility hubs, maximising the total utility of passengers employing traditional and/or shared modes. The framework’s allocation impact was investigated in terms of modal split, service level, as well as environmental factors, while ensuring economic feasibility. The sum of utilities for trips of all modes was maximised by optimising the number of shared vehicles and docks/spaces, number of relocated vehicles, and the ratio of satisfied demand. This objective was used by the GA module to determine the fitness of the algorithm’s candidate solution, determining the optimal hub distribution. The algorithm was tested on different hub-construction costs scenarios in Amsterdam. Findings demonstrated that increasing hub-numbers with lower capacities of shared vehicles was more advantageous than having less hubs with higher capacity. This was attributed to travel time savings as a result of increased coverage of the area by the hub network. Table 3 categorises the HLP studies accordingly below.

Table 3 Categorisation of studies that use EAs to optimise HLPs
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EAs have similarly been applied for various other applications, including routing and tourist planning problems, and more. For instance, the GA was repurposed by Yu and Lu [50] for multimodal route planning subject to multiple criteria, including travel time, fare and transfer numbers. Due to the incompatibility of a standard GA’s evolutionary computation for multimodal optimisation, the original operators were adjusted, presenting inter-modal variants hypercrossover and hypermutation, facilitating the algorithm’s capability to solve multimodal problems. Unions of bus, subway, taxi, and pedestrian subgraphs were used to logically define the modes, with transfers being additionally defined. The multi-objective environment was represented using a p-dimensional vector of criteria, assigning each its own corresponding fitness function. Multi-objective ranking was applied, and a non-dominating set of solutions was provided and evaluated. Solutions were found to be capable of adapting to different situations, conforming to real-world experiences of the network. Operationalising the conceptual framework of a Microsimulation Learning-based Approach to Transit Assignment, Wahba and Shalaby [51] modelled travellers’ behaviour pre-trip and enroute using a construct based on Markovian Decision processes and Reinforcement Learning principles. Simulated using the peak morning data from the multimodal network of Toronto, the dynamics of route and passenger interaction were captured, representing the network’s evolving effects. The learning-based transit assignment problem, solved using a mesoscopic model, was integrated with a parallel GA engine known as GenoTrans. The cluster computing used for parameter calibration of the model sped up the GA’s convergence, accelerating its learning curve. The model’s predictability was found to be very promising, with agents successfully adjusting decisions in response to experienced congestion, supplied information and control measures. Thus, a coherent behavioural framework where aggregated travel patterns were extracted for individual choice was presented. This experiential learning-based approach serves as a different perspective of travel behaviour modelling, compared to UE analysis.

Meng and Liu [52] investigated the impact of cordon-based congestion pricing on the modal split of a bimodal transportation network with auto and rail modes. Essentially, a special type of the second-best pricing problem, the toll-charge scheme’s impact on mode-distribution was estimated by solving a combined modal split and traffic assignment problem. For the modes, a binary logit model was used, and a probit-based stochastic user equilibrium (SUE) principal was employed for the traffic assignment. Asymmetric link travel times and a continuously distributed value of time were assumed to better reflect practical conditions. The probit-based SUE problem with elastic demand was formulated using a fixed-point model, solved using a convergent cost average method incorporating a two-stage Monte Carlo simulation-based stochastic network loading method. The fixed-point model was formulated as a constraint and the problem was converted to a Mathematical Non-Linear Programming (NLP) with Equilibrium Constraints (MPEC), solved for 3449.7 s using a GA. Numerically validated on a 15-node network built on a portion of car and rail network in Singapore, the problem was simplified such that only one OD pair was defined to cover the network. In addressing transit congestion and departure delays from airports to multimodal transit stations, Du et al. [53] presented their distribution optimisation model. Designed to minimise disparities in passenger departure times arising from multimodality, it operates under a departure chain capacity, which includes transfer routes, platforms/waiting rooms, and subway-, bus-, taxi transit modes. Differences in passenger choices were captured in terms of rigid and elastic departures. Tested on the Shanghai Hongqiao International Airport, the problem was solved through a GA. The case study concluded that rigid passengers allowed managers to partially balance passenger distribution and improve the average departure time. When passenger volume approached its peak, the optimised distribution significantly improved the departure time.

A two-stage method tailored for supporting multimodal transport network scheduling was proposed by Pop et al. [54], adapted to scenarios specific to touristic-service requirements. Approached in formulation similarly to that of orienteering problems, the tourist trip planning timetable problem captured multimodality in terms of minibus and train transitions according to their schedules. A Delay Time Petri Nets model was developed to first describe the routes of the network, associated with activities of a considered area. This was then followed using a GA to determine the convenient time intervals appropriate for transitions at critical points in the transport network, reducing delays to avoid unwanted scenarios. The efficiency of the methodology was verified using scenarios rooted in reality, where touristic routes of a small transport infrastructure were considered. Results eliminated the critical points in time that would lead to unwanted behaviour, proving the utility of the method where resources were found to be shared or in conflict.

Azizi et al. [55] devised a GA-based strategy to support Tour Planning (TP) for tourists in public transportation networks. Initial parameters including starting days and times, duration, city priority rates, a list of activities and their duration, and transportation modes (bus, train, airplane, and taxi), underwent rigorous testing. Addressing the personal or activity-centric nature of TP, constraints were based on activity duration and periods of enduring activities. Two nested modules were embedded into the model, such that the main module determined the city visitation order using a GA, while the subordinate module used an adapted Dijkstra algorithm to calculate the resulting multimodal shortest paths. Evaluated using real-world data of the transportation networks of 15 major cities in Iran, 100 tour planners were employed to validate the performance. The number of transition nodes were found to increase in the presence of specific networks. To reduce this, the weight of said network’s edges was suggested to be increased, such that the resulting route was not necessarily the earliest in terms of arrival time, but the best in terms of a tourist’s choice. Haqqani et al. [56] addressed the limitations of popular journey planning systems by integrating passenger preferences into the optimisation of a personalised multi-criteria journey planning problem. Modelled for large and complex multimodal urban networks, a k-dimensional cost vector for a sequence of stations approach was used to tackle the NP-complete problem, factoring for the multi-objective criteria of travel time, expenses, emissions, and personal energy consumption. Initialised by a user query, the proposed framework used a Personalised Multi-Criteria Genetic Algorithm (PMGA), i.e., a modified NSGA-II based solver where only a certain region defined by the user-based preference vector of the Pareto front was relevant to the optimisation. The crowding distance indicator of NSGA-II was replaced with a user-specific journey utility indicator, such that the Pareto-optimal journey resulted in an increased utility. Multimodality (including public transit, bicycle and pedestrian modes) was mathematically depicted using subsets. Evaluated on 100 user queries using real General Transit Feed Specification (GTFS) data from Melbourne’s large transit network, PMGA excelled over the unmodified NSGA-II in diversity and convergence.

Mutlu et al. [26] considered the trade-off between private- and public transport user preferences to minimise the costs of the two modes, formulating a bi-level optimisation model of bus line frequencies. Due to the inherent complexity, Differential Evolution was proposed as the solution method for the upper-level of the Transit Frequency Setting Problem (TFSP), determining optimal headways. Elastic demand was considered in the formulation of the lower-level model of mode choice. Mode distribution was calculated using a logit-type mode choice function, and car and bus flow interactions were accounted for. The proposed model was calibrated and tested on Mandl’s benchmark network of 15 nodes, running on average for 280 min, evaluating performance and applicability. Multimodal assignment results suggested that bus travel time was significantly affected by car flows, supporting the potential benefit of exclusive bus lanes to enhance public transit competitiveness. Due to the computational complexity of the bi-level formulation, the less expensive static assignment model was used, prioritizing processing time over the realism provided by dynamic assignment models.

Addressing China’s urban transportation network planning, Tian et. al [57] modelled a bi-objective bus-metro transfer scheduling optimisation framework to minimise overall passenger transfer time, as well as bus schedule changes. The multiple objectives integrated passenger demand and transit systems’ operations. Bus departure time and departure interval decision variables were used to develop the model, for which an augmented generalised Chebyshev algorithm, an iterative method, was tested. The NSGA-II metaheuristic was used for comparisons. Tested using Shenzhen’s metro and public transport integrated circuit card data, specifically of the Shenzhen 66 bus road, the network comprised of five bus stops and four different metro stations. The improved Chebyshev method outperformed the multi-objective GA in both computational efficiency (the prolonged computation time attributed to the NSGA-II’s algorithmic complexity) as well as convergence and solution quality, yielding a 68.6% reduction in transfer time.

The integration of sustainability and personalisation within Tourist Trip Planning was investigated by Zeinab Aliahmadi et al. [58] where tourists would visit different scenic spots with varying schedules, as well as diverse resting places. Economic, social and environmental aspects of sustainability were considered in the tri-objective IP model which minimised total travel costs (rental, service, transport), maximised total trip utility to capture preferences (optimising used amenity, hotel, and resting-spots star ratings), and lastly minimised CO2 emissions. The intricacies of an urban transportation system were modelled as well, including traffic lights, weather conditions, and individual features of different modes. The uncertainty associated to the costs and travel time of the model was captured using a credibility-based fuzzy approach. Due to the problem’s complexity, a Self-Adaptive NSGA-II (SA-NSGA-II) was developed to tackle large scale instances of the problem within a reasonable computation time. The methodology’s performance was evaluated against the augmented ε-constraint technique, alongside different variants of the NSGA-II. The framework was applied on a case study of Montreal, where decisions for different walking scenarios, including hotel and mode-choices were compared. To conclude the classification of EA-based studies, Table 4 categorises the above as follows.

Table 4 Categorisation of studies that use EAs to optimise various transportation problems
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3.1.1.3 Further reflections

Overall, the reviewed studies above investigating the optimisation of multimodal urban transit using EAs largely consisted of the use of genetic algorithms for the Transport Network Design Problem. The Shortest Path and Hub Location Problems were also equally targeted as the second most studied optimisation problems of this section, following the TNDP. While EAs have been more recently applied towards HLPs, most of its reviewed work has been set within a static context. In contrast, SPPs were equally considered in terms of both static and dynamic scenarios, with recent years focusing more on the static variant. Alongside the TNDP and SPP, a vast range of optimisation problems were reflected on, including parameter calibration for the Transit Assignment Problem (TAP), touristic activities’ planning, route planning and more. With a total of 31 studies, the GA is the most widely used metaheuristic among the reviewed approaches. Whilst DE is capable of being extended to mixed/discrete problems, the consensus that it is more appropriate for continuous optimisation [17] distinguishes it from the GA, resulting in less applications of it. In contrast, the GA is more widely acknowledged and applied within the realm of discrete optimisation.

Multimodal transportation systems were broadly optimised in terms of the minimisation of overall system costs, travel time, routes, and mode distribution. Within the latest research developments, environmentally friendly initiatives mark a shift within optimisation objectives, as can be seen with the increasing consideration for bike-modes [43, 47,48,49]. Due to its sustainability and cost-effective appeal, public transit is similarly significant within this context. Conversely, with the rise of Mobility as a Service, the necessity for seamless multimodal transfer synchronisations, particularly within public transit, has been recently raised [47, 57]. The appeal of connecting modes appears to persist to current times, marked with the increase of optimising multimodal Hub Location Problems.

Naturally, with the diversity of problems, there exists an extensive variety of conditions, often times in conflict with one another, that are being optimised throughout the literature, such as sustainability, expenses or user preferences. Hence, problems are increasingly being formulated as multi-objective, within which a bi-level structure is frequently implemented. This is particularly relevant for the extensively used UE/Logit based modelling, both commonly incorporated to capture passenger behaviour and modal split. The two are widely embraced in combination, particularly for the lower level of a bi-level MOOP where a Transit Assignment Problem is typically used to optimise mode share specifically, feeding back to the higher level of the formulation to determine factors like transfers. Algorithmic modifications appear to be increasingly prevalent within recent years, taking advantage of the GA’s flexibility, both in defining parameters and subroutines, particularly in terms of hybridization. In the context of this section, the hybridisations exclude the combination of multiple metaheuristics, which will be specifically explored in Sect. 3.3. The GA is capable of effectively accounting for Pareto optimality through its NSGA-II variant. The deployment of solution frameworks, oftentimes modelled and tested within a static scope using deterministic or historical data, occasionally aided by simulation tools, has evolved to increasingly become a point of interest within a dynamic context, particularly within real world/time environments. Conversely, the GA has been observed to be used for frameworks explicitly designed for dynamic purposes, either by solving complex problems in a very short time, or because of dynamically updated data input. Frequently, data sourced from an online database would be initialised through a user request. Adding modes, alongside improving computational efficiency and processing time are further topics of consistent interest. Continued research into multi-objective modelling and the development of the GA is additionally suggested as a key point necessary of further research. The concern for computational power is particularly noteworthy, especially considering recent technological advances proliferated by machine learning research and graphic design, particularly in terms of GPU usage and parallel computing. In-depth reflections and how this relates to the current landscape of state-of-the-art advancements will be further discussed and concluded at the end of this literature review in Sect. 4.

3.1.2 Swarm intelligence

3.1.2.1 An overview

Swarm Intelligence (SI) represents the collective learning of the self-organising agents of a population (swarm). Within metaheuristics, they are typically based on the decentralised natural systems of the social behaviour of species in search of food. The central characteristics of SI algorithms include the cooperation of simple and non-sophisticated agents through indirect means of communication, facilitating the robustly efficient navigation of a global search. The general principles of SI, supporting the algorithm’s adaptability, include the principle of proximity (the simple computation of the basic units of the swarm related to the surrounded environment), quality (the ability to respond to factors like food and safety), diverse response (varied distribution of resources, not narrowly concentrated in certain regions), and lastly the principle of stability and adaptability (swarms adapt to environmental fluctuations without rapid changes, costing energy) [59,60,61].

As detailed in Sect. 2, ACO and PSO were found to be the most used approaches among SI-based metaheuristics for optimising multimodal urban mobility problems, followed by the Artificial Bee Colony algorithm. ACO, inspired by the foraging of ants, is a probabilistic method particularly effective for the optimisation of combinatorial problems [62]. A population of simulated artificial ants builds a solution by randomly traversing an iteratively built path of a network. A trail of pheromones is excreted along the travelled path, concentration determined by the success of the path in terms of the amount of food identified (reinforcement). Strong pheromone levels encourage other ants to follow the same route. The more a path is visited, the stronger the concentration of pheromones. With time, the pheromone trail evaporates, resulting in the favouring of shortest paths [61]. The algorithm’s process of pheromone initialisation, solution construction based on the trail, and the proceeding pheromone update based on the evaluated quality of the solution using the evaporation and reinforcement operators, all iterate until the termination criteria are met. The indirect sharing of information based on the experience of other agents guides the colony towards promising areas of the search space. Formally, the metaheuristic’s pheromone values are associated with candidate solutions which are explored and iterated through using the path an artificial ant builds, traversing a fully connected construction graph, consisting of a set of vertices and edges. Depending on the problem, solutions may be specifically represented using either the vertex or edge set, with pheromone values being deposited and updated on the associated solution representation [62].

Based on the social behaviour of organisms like flocks of birds or schools of fish, PSO uses the decentralised control of local movements to coordinate behaviour. Initialised on a population of randomly generated candidate solutions, namely particles, they are designed to traverse the search space according to the mimicked physical quantities exhibited by the agent, such as the velocity and position of a bird within the flock. The algorithm defines a neighbourhood, using a global- and local best method, for each solution, denoting the social influence between the particles. The particle is composed of a mix of vectors and fitness values, whereby each particle of the swarm is iteratively updated in parallel in terms of its velocity, position, and resulting best local solution, updating the overall best global solution of the swarm. At each iteration, every particle changes its position according to its own experience and that of neighbouring particles. The population members navigate the search space according to the guidance of the best-known position of both the particles (personal), as well as the swarm (global). The swarm of particles update between iterations, each candidate moves in the direction of its previous personal and related global best position. With the goal of finding the swarm’s best-performing overall position, a particle is defined by its position and velocity vector. At each iteration, the main search step updates the velocity and position vectors using parameters such as the real-valued inertia weight (balancing global and local exploration) alongside uniformly distributed random variables and acceleration- and cognitive coefficients. Collaborative efforts between the particles are also captured using a cooperation component within the formulation, promoting a search for overall optimal solutions [17]. Unlike ACO, where ants indirectly communicate through pheromones to update their values, PSO particles update the current solution directly. Thus, PSO has notably few hyperparameters, making it a relatively straightforward algorithm to deploy, and a popular metaheuristic approach [61]. Similar to DE, PSO is found to often be considered for constrained continuous optimisation [17].

Inspired by the foraging of bee colonies, ABC executes its search for optimal solutions using artificial bees to search for food sources. Different bee agents are designated with different roles, for e.g. scout bees are randomly mapped out throughout the search space, visiting different solution areas to evaluate the quality. Bees with the highest fitness value are chosen and their visited sites are selected for a neighbourhood search. Onlooker bees are then assigned to search the neighbourhood for the best sites, again evaluating based on fitness values associated to the solution quality. The bee with the highest value is selected to generate the next bee population, whilst remaining scout bees continue to conduct randomised searches, resulting with two sets of groups within the colony at the end of the iterative search [61].

3.1.2.2 Classification of studies

To analyse and classify research employing SI as part of their solution framework, papers optimising the SPP for the multimodal urban transportation of passengers are first evaluated.

Building on their past work on transfer graphs, used to abstract and simplify a multimodal transit network of its properties Hedi et al. [63] outlined a dynamic user-request method to find the shortest path. The modes’ transfer points were reformulated into a simplified “Relevant Graph”, for ease of processing. Despite being expandable to solve for MOOP, a single objective example was used to create and compute a stored database. For each computation, Dijkstra minimised the travel time. The pre-calculations proved effective for the exact method as it optimised large networks within appropriate CPU time. Found, however, to consume significant memory space, mainly due to the pre-processing of data structures, ACO was suggested as an alternative for computing the database. The construction graph of the metaheuristic was replaced with the problem’s graph itself and directly optimised. An efficient gain in memory was observed due to the lack of pre-calculation of data. As a result, the two algorithms were proposed to be hybridised. Investigating this in their next paper, Hedi et al. [64] combined Dijkstra and ACO to make use of their respective strengths, compensating for their opposing drawbacks. Tested on multiple instances of various large-scale networks, Dijkstra was confirmed to outperform ACO in terms of CPU time, however as the problem size increased, ACO significantly outperformed in terms of efficient memory use. Thus, ACO was used for the pre-calculation step, resulting in a further simplified variation of the aforementioned relevant graph, namely an “abstract graph”, for Dijkstra to solve. The hybrid algorithm proved promising, resulting in an ideal trade-off between computation time and memory, solving the multimodal SPP in comparable speed to that of only Dijkstra, with the efficiency of ACO. Furthermore, in adjusting the ACOs hyper-parameters like ant number, changes in computation time and path generation were observed. Thus, a real-time user-itinerary request methodology was devised.

To support travellers in their journeys as increasing amounts of information needs to be considered along with growing computational demand, state-of-the-art methodologies were considered by Katona et al. [65]. Specifically, the use of parallel computation of ACO. Capable of dealing with multiple criteria, ACO is a resource-heavy algorithm. Thus, its parallelisation was proposed, noted to be suitable for potentially large networks. As multimodality increases the complexity of the SPP, calculation time becomes central. A hypothesis was presented to be tested on the multimodal GTFS database of Hungarian and Austrian networks, as follows. If ACO’s iteration number is decreased so that parallel computing can supplement for it, the total processing time decreases, and as a result, resource utilization increases. Each ant corresponded to a processor’s thread. Different parameters and thread counts were investigated during testing to validate the applicability of multi-threaded systems, used by cloud computing, to increase the probability of finding valid routes. Improved effectiveness was confirmed to be achieved with the increased threads used, verifying the hypothesis of enhanced timing and resource utilization. Future developments suggested a detailed approach for large networks on multi-core systems with more available threads. Additionally, autonomous vehicles were mentioned to be considered in the problem’s design, alongside variables like travel time and other factors.

The presented studies are categorised and classified in Table 5 below.

Table 5 Categorisation of studies that use SI to optimise the SPP
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This section continues with the review of SI in their use of solving multimodal problems designed as TNDPs.

The architecture of a multimodal urban transit system encompassing busses, BRT and metro was defined using a hyper-graph by An et al. [66]. Designed to help solve a bi-level programming model to determine the optimal service of a multimodal transit network, the upper level included performance coefficients flexible for integrating several performance indicators in the case of multi-objectivity, and the lower level was set as a transit assignment problem. The hyper graph captured the system’s strategy cost, i.e. overall passenger cost, and was used as the comprehensive performance indicator. An incremental assignment approach was used for solving the TAP, representing the strategy cost in terms of a utility function and a logit model to evaluate transit travel. PSO, with modifications suitable for discrete problems, was used to generate new BRT lines to improve overall system performance based on an initial set of networks. A test bed of 68 junctions and 108 candidate platforms was used to validate performance and search efficiency. Improved service quality for long-distance travellers was observed, highlighting inadequacies in performance metrics. Crucial to strategic choice behaviour, the requirement for a deep field investigation regarding waiting time before being incorporated into assignment models was raised, especially due to its relevance for multimodal problems. In contrast to a traditional GA, PSO outperformed in proficiency and convergence, however an increased susceptibility to local optima emerged with increasing iterations, highlighting the efficiency-performance trade-off as a field for further study. Despite the efficiency, incremental assignment (used as a faster alternative to UE) accelerated the solution algorithm for cases where PSO took too long to converge, proving to be useful.

Vitins and Axhausen [67] then studied the long-term transport network planning for larger regions in terms of the interactions and interference between infrastructure projects, formulating a discrete NDP to maximise overall benefits under budget constraints. Each infrastructure project and associated bundles were first assessed through a cost–benefit analysis. Exact complete enumeration techniques of all possible project combinations were estimated to require 114 days’ worth of computation. Alternatives like the Knapsack problem and its extensions were rejected due to not satisfying the NDP preconditions because of the necessary relaxations violating the constraints. Thus, due to its suitability for discrete problems and robust simplicity, ACO was used for optimisation. The equilibrium assignment problem was solved for public and private transportation modes. Additional factors like environmental impact and safety were proposed to be included within the model formulation. Due to the lack of data availability, however, they were excluded. For simplicity, the evening rush hour of the static network of Bern with 724 nodes was tested. Results showed sensitivity towards the overall modelling, exemplified by the different modal split approaches both derived from multinomial logit models. Project bundles were susceptible to changes due to factors such as destination choice and trip generation. Whilst ACO was successful in solving the large problem scope, the ants of the algorithm were suggested to significantly benefit from preceding experiences, highlighting the importance of parameter calibration or its automation. The resulting computation times and the need for improvements to avoid local optima also implied the necessity for a faster and more efficient methodology. Issues of fixed budget constraints were raised and the need for generalisable algorithmic and modelling adaptations to include for changes in travel demand and population distribution were discussed.

In discussing the need for the diversification of modes within a transit network, Mohaymany and Gholami [68] note that by designing a network in accordance to the performance measures and characteristics of different feeder modes, system costs can be reduced, with an increase in coverage and demand. In extension of its unimodal variant, ACO was used to optimise a Multimodal Feeder NDP (MFNDP’s) routes, frequency and mode assignment, minimising a single-objective combination of user, operator and social costs. Results between a bus feeder- and a multimodal bus and van feeder network after 600 iterations were compared, where convergence and stability in results supported the metaheuristic’s reliability. It was concluded that multimodal networks are capable of significantly reducing user costs due to convenience, increasing demand, and returning increased profits for the operator. It was mentioned that at the time, the problem was based on assumed unit costs and a lack of deterministic data to compare single mode to multiple modes. Seeking to further support the design of feeder lines in a multimodal transit network, minibuses were considered by Gholami and Mohaymany [69] to reduce the system’s total user, operator and social costs, as well as help spread the area of operation. ACO was used to construct the network for which the frequency of modes of each route was calculated, whereby the mode choice was determined by optimising the route’s costs. Once every route had been optimised, the total network cost was used to update the pheromone trails for the next iteration, whereby ACO creates a new network. This process ran until the termination criterion of 700 iterations was reached. Performance measures between conventional and mini-busses were compared to highlight their respective impacts and it was determined that through costs and benefits, such as reduced pollution and depreciation, as well as serving lower demand, total network costs can be lowered when mini-busses are appropriately allocated under the right conditions.

A hub-and-spoke framework was introduced by Huang et al. [70] for the reconfiguration of bus services to better integrate newly constructed rail systems serving as the backbone of a multimodal network, central to Mobility as a Service. A three-stage methodology was developed to address the TNDP, where a cluster-based algorithm was first used to locate hubs from rail stations. The bus sub-network was hierarchically designed such that main bus lines were developed using a heuristic line generation approach, and feeder lines were solved through a Traveling Salesman Problem. The resulting set of this stage, filtered through using practical feasibility constraints, served as input for the final step of a bi-level model proposed to determine the frequencies of each mode. Solved using the Artificial Bee Colony algorithm, ultimately both operator and user travel costs were decreased. Passenger mode-choice behaviour was estimated by a multinomial logit model on the upper level, which defined the statically modelled transit assignment problem depicting passenger line choice behaviour. Numerical examples in terms of both small and large networks were used to demonstrate the framework’s effectiveness. For the large scenario, the public transport system of Nanjing was considered. Including urban area and smart-card data, with a daily average passenger demand of 1.5 million, the size of the network was noted to cause overwhelming computational burden. To reduce complexity, the area was partitioned into 9 zones and each separately clustered for hub locations. Overall, results confirmed the practicality of the proposed method for real-world implementation and demonstrated its potential to enhance public transit systems by strategically selecting hub locations and creating hub-and-spoke transit systems, efficiently reducing travel time, making better use of rapid transit modes.

To assist authorities in the systematic planning of urban networks, Barahimi et al. [71] enhanced a dual-mode medium-sized network’s reliability under the consideration of demand uncertainty. This was countered through simulation, where Monte Carlo was used for time and flow, and travel demand was set to follow a lognormal distribution. Favoured due to its homogeneous performance and low computational cost in terms of memory and speed, as well as established in its use for optimising bi-level formulations, PSO was chosen to solve the non-convex MOOP. Evaluated simultaneously, capacity and travel time reliability were successfully improved, and strategic recommendations were made. Solving the NP-hard high-level problem proved to be a computationally expensive task, processed on a CPU for over 382 h in a network of 41 nodes and 108 links.

Solved for the past decade under deterministic conditions like peak-hour, Ghaffari et al. [72] set out to re-contextualise the Transit Priority NDP (TPNDP) in terms of uncertain demand. The objective of this study was to find the optimum set of exclusive bus lanes at a network level, minimising the risk-measure of expected social cost on travel time excess of auto/transit users, while considering the uncertainty of travel demand. This was modelled as a bi-level NLP, where the upper-level optimised the transit priority scheme, propagating the lower-level calculations which captured mode and route choice through the use of traffic UE, and transit assignment problems, as well as a multinomial logit model and utility function. For any combination of the upper-level transit prioritisation, the lower-level model was solved for all demand scenarios. Due to the complexity and concern for computation time of the NP-complete NLP, ACO was applied using MATLAB and the PTV-VISUM scenario manager tool. Run on random scenarios of the four-step demand calibrated travel distribution of the real-world transport network of Zanjan in Iran, demand uncertainty was shown to have a significant impact on solutions, supporting the importance of considering risk in problem formulation. To summarise this section of the review, the papers classified in Table 6 accordingly.

Table 6 Categorisation of studies that use SI to optimise the NDP. Acronyms include: Multi-objectivity: MOOP | Static vs Dynamic: S / D | Network Size: Large (L), Medium (M), Small (S)
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Similar to EAs, SI is also found to be used for a vast range of applications, including routing, calibration, HLPs, and more.

ACO was first used by Zidi et al. [73] to solve a spatial reconfiguration problem of a multimodal transportation network, specifically to find the minimal travel time and distance for a maximum capacity of passengers in real-time, within a disrupted public transportation system. ACO was improved for search space exploration through use of a “second search”, avoiding local optima by skipping over the first solution found. Pheromone trails of chosen arcs were also reinforced, prioritizing optimal paths. The changes were implemented to the algorithm and proved to efficiently compute a single-objective multicriteria search for network reconfiguration against disturbances. The hyper-parameter of 6 ants was used for the path optimisation of single-mode reconfiguration whereas for a dual-mode network the number was increased to 30, increasing execution time. It was concluded that the algorithm proved to be suitable for reconfiguration modelling, and taking the metaheuristic’s hyper-parameter characteristics into account, it was mentioned that premature convergence can be ensured for simple graphs, suitable for real-time reconfiguration. Zidi et al.’s [74] interest in applying their algorithm for the previously described real-time application of network reconfiguration (close to the Capacitated Arc Routing Problem (CARP)) was further investigated through premature convergence. The objective function was modified to include a third criterion, namely comfort, and ACO’s execution time was additionally analysed. The metaheuristic was found to converge to a feasible solution for their unimodal scenario after around 30 iterations within seconds of CPU execution time, highlighted as suitable for real-time applications. For their multimodal scenario, ACO and GA were applied to four examples of network disturbances. The two were generally comparable in performance, with GA outperforming in one instance and ACO noted for its more diverse results. It was concluded that ACOs capability for early convergence and fast optimisation proved suitable for the optimisation of real-time network configuration.

In Faroqi and Mesgari’s [75] routing problem of the public transport network of Tehran, a multi-objective ACO algorithm was initialised with two different sets of parameters. Every objective function was assigned its own colony and corresponding pheromone matrix for the multi-colony (MCACO) approach, whereas for multi-pheromone (MPACO), every objective had a pheromone matrix of its own, of which all were within the same single colony. As iterations typically influence running time compared to the number of ants, the study was capped at a maximum of 100 iterations, where MCACO ran on average 42.4 s. The two were deployed and found optimal paths, defined as a sequence of different modes, with MCACO performing better in terms of run time, solution variation and convergence.

Yu et al. [76] addressed Tourist Trip Design Problems by using PSO to tackle the multimodal Team Orienteering Problem with Time Windows (MM-TOPTW). Their objective was to maximise collected scores for customizing tourist routes. The NP-hard MIP was solved with CPLEX, however performance deteriorated with the increase of model complexity. To overcome this, Global Local Neighbourhood Particle Swarm Optimisation (GLNPSO) was employed because of its swift convergence due to its multiple social learning terms and ease of implementation in handling continuous and discrete problems. GLNPSO was adapted into 2L-GLNPSO, implementing a two-level solution approach. The first level constructed paths, while the second determined transportation modes. Local search improvement was embedded as a stopping criterion through the metaheuristic’s hyper-parameter settings. The proposed algorithm’s efficacy was demonstrated on various networks, showcasing its ability to provide optimal solutions for small- and medium-scale problems within a reasonable time frame. For larger networks, 2L-GLNPSO produced high-quality solutions, outperforming its variant without local search. The algorithm demonstrated competitive performance on standard TOPTW benchmarks, delivering solutions close to the best-known methods. Its computational efficiency in CPU-time, tested by means of the Super Pi benchmark, proved to be competitive when compared to other metaheuristics.

Tavassoli et al. [77] introduced a transit assignment calibration and validation methodology using data from South-East Queensland’s large-scale multimodal transit network. The framework combined smart-card data from the Automatic Fare Collection (AFC) with network, geographic, and geometric data to create a high-volume, high-quality dataset of passenger boardings and alightings across various modes. Their approach involved revising the network, employing an optimal strategy for transit assignment, utilizing the logit model to distribute flow, and assuming constant service frequency and passenger demand while not considering congestion. This allowed for the development of a frequency-based assignment. The calibration process utilized PSO to find the optimal transit assignment parameters. The objective function aimed to minimise an error term, which was defined as the difference between the percentage of root mean square error (measuring model performance at the segment level for each mode) and the mean absolute percent error (encompassing deviations from observed data, comparing total mode volumes). The framework emphasized achieving mode levels that closely matched observations. The final stage involved model validation by testing the calibrated model with AFC data from a different day, determining the applicability of calibration parameters to different conditions. Results revealed that modelled bus modes exhibited greater variability and uncertainty due to greater variety in passenger route options, while rail and ferry choices are more limited, yielding better predictions. Typically modelled in terms of deterministic parameters and single-mode transport, Kaveh et al., [78] expanded on the Hub Location Problem by designing a multimodal incomplete hub network, to better reflect the dynamic nature of real-world planning and environmental uncertainty. Each node was designed to count the availability of busses and trains, however the model is easily extendable to include more than two distinct modes of travel. Adapted from an originally static model to include elastic demand, a hybrid two-stage method was used to optimise the network’s total profits and travel time by converting a Fuzzy Bi-objective Mixed Integer Linear Programming model into a mono-objective, auxiliary parametric Linear Program (LP). Tested on Qom Monorail Project data as a small-scale instance, it was determined that elasticity provides more flexibility and balance between multiple goals and can be tuned to benefit under-served communities. Concerned with the timing of solving larger-scaled problems, Multi-Objective Particle Swarm Optimisation (MOPSO) was tested against Non-dominated Sorting Genetic Algorithm (NSGA-II) to solve the model using a Turkish Data Set. Four comparison metrics were employed to examine the metaheuristics and MOPSO provided a generally better Pareto solution in terms of quality, solution-space spread and closeness with an ideal solution. NSGA-II outperformed in terms of having a more uniform Pareto frontier.

The PSO algorithm was integrated within the computationally efficient Discrete-Event Simulation (DES) framework by Khattak and Hussain [79] to optimise commuters’ circulation space in a multimodal transport interchange. Based on random variates of phase-type distribution, DES-PSO accommodates fluctuations in commuter arrival flow, considering speed- and flow-density relations for state-dependency. Thus, an alternative to stochastic analytical models, the simulation-optimisation method outclassed such existing designs and codes which typically assume a uniform distribution, overlooking dynamic factors like congestion or randomness. Supported using parallel computing, DES-PSO significantly improved the commute’s width value compared to established designs.

Lastly, in 2022, He et al. [80] set out to solve a Multi-Criteria Journey Planning (MCJP) problem for passengers traversing multimodal public transport networks. Spatiotemporal constraints like waiting time or transfer feasibility between modes were considered, and a transfer graph was used to characterise the transfer opportunities among all bus/metro lines. To model the passenger’s multi-objective personal preferences, a weighted sum vector was used as the cost function and solved using a Machine Learning based Max–Min Ant System. The deep learning-pheromone prediction model was designed to estimate evolving states of the search space during the optimisation process. The experiment was limited to six travel criteria, namely, journey distance, journey time, wating time, travel fare, walking distance and number of transfers. Tested on the multimodal public transit network of Singapore and three datasets of passenger demand with varying distributions, results found the deep learning method to achieve near optimality, performing significantly faster than exact methods and its original ACO based counterpart.

A classification of the aforementioned studies is presented in Table 7.

Table 7 Categorisation of studies that use SI to optimise multimodal transportation problems
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3.1.2.3 Further reflections

To conclude, the reviewed studies of this section utilize SI to investigate the optimisation of multimodal urban mobility. Similarly to the previous literature employing EAs as their solution approach, the researched problem scope overwhelmingly consists of TNDPs. Less in comparison, the SPP is the second most studied multimodal optimisation problem. Overall, the papers once again reflect a rich variety of multimodal problem settings, comparable to those studied in Sect. 3.1.1.2, again including parameter calibration for the TAP, an HLP, touristic team-orienteering problem and more. In summary, 18 studies employed SI-based methodologies, 11 of which used ACO specifically, the second most applied metaheuristic based on our literature review.

Generally, the multimodal systems were solved for the minimisation of multi-criteria system costs, travel time, mode distribution, routes, and social costs. Implicit to multimodal problem modelling, the arising complexities being accounted for by applying multi-objective formulations are comparably fewer in proportion to that of the GA. In contrast, however, within the multi-objective papers, bi-level formulation appears to be more frequently adopted here. It is worth noting that despite being among the most used metaheuristics in this paper, only the relatively more recent ACO-based studies have used multi-objective formulations. In contrast, PSO literature overwhelmingly applied multi-objectivity for their optimisation problems, making up half of their studies. Furthermore, the bi-level structure that appears well suited to user-based mode distribution by formulating a UE-Logit based TAP in the lower level only appears in the earliest PSO study, the sole ABC paper, and a more recent ACO study [72]. Notably, any framework that was designed within a dynamic setting was solved using ACO. All PSO and ABC studies were framed as static. In validating the confirmed success of mode diversification through the use of SI-based approaches for their work, performance metrics such as CPU processing time, methodological efficiency, solution quality, and the performance-efficiency trade-off, all appear to be a central theme of investigation. Hyperparameter variability and their resulting effect on the optimisation process, alongside the design of different data types and representations, driven by recent advances in data collection were additionally raised as topics of interests, alongside the further study of Pareto friendly methodologies. To better account for real-world conditions, further mode additions and integration of uncertainty have also been discussed, alongside concern for the SIs being susceptible to local minima.

3.2 Analysis of single-solution approaches

3.2.1 Overview and classification of studies

While population-based methods iterate through their search to improve a population of potential solutions, single-solution-based metaheuristics set out to locally improve one sole solution. Referred to as walks through the neighbourhood, the search of the solution space of the problem is navigated from the sole current solution to another. For the sake of clarity, this chapter sets out to review each of the single-solution metaheuristics in a self-contained manner chronologically.

Classically established for its significant impact on the field of heuristic searches due to its simplicity and efficiency in solving combinatorial optimisation problems, Simulated Annealing (SA) has since been extended to execute continuous optimisation. Based on the annealing process of metallurgy, SA is a probabilistic metaheuristic where its objective function is representative of the energy state of the system, with decision variables analogous to molecular positions. The algorithm’s objective is to delay convergence for the escaping of local optima (a metastable state), ultimately reaching a global optimum corresponding to the metaphorical ground state of the metal’s physical system. Interestingly, SA does not retain and use additional information gathered throughout its search beyond the iterated solution, essentially rendering it as a memoryless algorithm [61]. Each iteration of the algorithm includes an update in the parameter value of temperature and the selection of a random neighbour solution of the current state, where depending on the quality of its performance, the decision of solution replacement may be based on stochastic distributions (like Boltzmann) [17]. The three studies that employ the SA solution method are reviewed as follows.

García and Marín [81] introduced their, at the time, newly presented NDP that incorporates the use of generalized costs as design variables. Originally subjected to budget constraints, the bi-criterion model was reformulated as single-objective, a compromise of social- and improvement costs, to then outline a bi-level model. The upper-level optimised total travel costs for the tactical planning of capacity and fare of combined modes park-N-ride trips. The continuous NDP was further developed to counter non-linear cost functions as well as by projecting candidates into a feasible set. The lower-level model represented a user-equilibrium traffic assignment problem, where demand was modelled as a nested logit distribution. In line with prior research at the time, SA was deemed as viable for this NDP, and a series of tests (three of which were based on existing road networks, and the last generated by the authors) were run, outlining computational issues of convergence and execution time. Emphasis on the computational burden of solving the equilibrium model was placed, where SA making direct use of path information generated at its first stage, simplifying each iteration of equilibrium optimisation was noted. This effect was further highlighted with the reformulated model outperforming the original, requiring less TAP calculations in comparison. SA was deemed appropriate for small to moderate sized networks and suggested to be hybridised with other algorithms for future works.

Ghaderi and Pahlavani [82] proposed an integrated approach to optimise a multimodal multi-criteria route planning problem for a real-world, static, urban network. Fare, travel time, comfort and path length were defined as criteria significant to a user’s trip. The weight of each was calculated using a fuzzy analytical hierarchical process weighting method, accounting for user uncertainty, after which, based on these factors, the SA was implemented to optimise routes. The model was tested using MATLAB, running the method on the small central transportation network of Tehran City. Five different urban modes were represented using sub-graphs, connected with walking edges. Results were found to be highly efficient and fast. Testing the model on a dynamic variant of the network with real-time information updates was suggested for further study, along with the use of a multi-objective variant of SA. The Optimal Traffic Calming Implementation in a Multimodal Transportation Problem (OTCIMPT), essentially network design from the pedestrian’s perspective, was proposed by Rashidi et al. [83] to study the effect of side- and crosswalks (S&Cs) installation as traffic calming facilities of transportation networks. The mathematical programming model was developed as a bi-level MINLP, using multimodal transport cost functions to minimise pedestrian safety hazards on the upper level, as well as enforce optimal flow on the lower level to be at user equilibrium. For the problem to optimally locate S&Cs in the urban network, the model was implemented in YALMIP to be solved using the BARON solver. Due to the computational complexity of the formulation, a customised Greedy Heuristic (GH) and SA were used on three reconstructed networks. GH and SA similarly reduced total costs for all three samples, however for the larger network specifically, SA outperformed in both solution quality as well as processing time. Results indicated that S&C installation at optimal location successfully improves pedestrian safety and reduces overall transportation costs. Conducted under deterministic conditions, future work was suggested to consider the variable of uncertainty. This concludes the review for SA.

In response to the necessary randomisation of SA to escape local optima, memory-based algorithms were researched and designed, such as the Tabu Search (TS). Known for its use of memory to store search-related information throughout the process, the metaheuristic has been long established as a widespread single-solution approach. TS employs local search methods alongside short-term memory structures of the neighbourhoods it iterates through, effective for complex combinatorial problems [61]. TS operates on a set of mappings, based on a given trial solution of the search space of the problem. This set of moves that maps feasible solutions, uniquely defines the solution’s neighbourhood. The memory structures forbid certain moves which may return the algorithm to a recently visited solution, also known as cycling. Constraining the search by classifying forbidden, i.e. tabu moves and freeing it up using its short-term memory function that provides strategic forgetting, are two key elements of the algorithm. Additional memory structures, like intermediate and long term, can respectively also be used to intensify the search towards a specific area or diversify it towards a new region when stuck in local optima [17]. The metaheuristic is mostly applied for multimodal Vehicle Routing Problems (VRP), as described below.

With the demand for Home Health Care (HHC) on the rise due to social demographic developments, Rest and Hirsch [84] addressed the need for increased organizational effort and anticipatory risk management of HHC within an urban setting. First, a vulnerability analysis was conducted, highlighting critical success factors, as well as blackouts, epidemics, and heat waves as possible threats. Real-world data of Vienna’s HHC services was provided by the Austrian Red Cross (ARC) for which a Tabu Search was implemented to support the daily scheduling, as well as disaster management, of a time-dependent multimodal urban region. Typically limited to rural regions where nurses primarily use cars, a sensitivity analysis determined that due to the short distances between clients, cycling posed as the most efficient transport mode in urban environments. While the presented decision support system could compute a schedule from scratch, it was found to be unsuitable for the requirement of repeated adjustments throughout the 24-h period of disaster events. A new methodology, operational within a real-time environment and capable of overcoming the mentioned limitation was further proposed by Rest and Hirsch [85]. Scheduling was solved using three different formulations of TS using travel-time matrices as input, created through an efficient exact solution that processed the timetables of different public transportation on a minute-by-minute basis. Designed to solve real-world instances within reasonable running time, especially in consideration of the additional computational strain time-dependency invokes, a TS that adapts to neighbourhood searches dynamically (TSDYN) was formulated. Tested on real data, the ARC was found that for small instances of 30 jobs, exact solutions were not possible due to the minute-based travel times serving as a restriction. For larger instances of 202 jobs, two scenarios were outlined, where the first was based on a predefined roster and the second included uncertainties, assuming flexible working hours not limited by a given roster. For the first scenario, the standard TS which searches the whole neighbourhood outperformed both TSDYN, as well as a TS that used restricted neighbourhood of fixed size. For the second, the opposite was found. As opposed to the first test restricting search space due to the availability of rosters, the second with flexible times allowed the dynamic method to adjust the neighbourhood size as necessary, successfully overcoming the trade-off of solution quality typically associated with improved computation time. Scheduling without the roster was optimised by 51.08%, concluding that time-dependent formulation saved significant time, creating more reliable schedules. Future work was suggested to integrate additional modes such as pick-up services to better reflect user requirements.

We now turn our attention to the Variable Neighbourhood Search (VNS). VNS successively explores different pre-defined neighbourhoods’ worth of solutions, either systematically or at random. Facilitating diversification to escape as well as arrive at local optima, VNS exploits the fact that a globally optimal solution is a local optimum for a given neighbourhood. The deterministic variant of VNS is known as Variable Neighbourhood Descent (VND) [61]. Works employing VNS for their urban multimodal optimisation are reviewed as follows.

Divsalar et al. [86] first developed a multi-objective green variant of the orienteering problem (OP) for multimodal travel, determining the maximal trip score with minimal costs, along with simultaneous reduced emissions. The tour is presented in terms of connected Points of Interest (POI), with different transportation facilities available for tourists to use. The choice of POIs and modes was determined by using exact approaches for two different types of tourists, first using a lexicographic method for those who prioritize total trip score, then costs, and emissions last. The other group was said to compromise between objectives, for which an ε-constraint method was used. Additionally, a Multi-Objective Variable Neighbourhood Search (MOVNS) was designed and evaluated in performance compared to the exact approaches. 30 new instances of different sizes were generated from existing OP benchmarks. MOVNS proved to competitively provide high quality solutions in terms of multiple metrics including hypervolume and Pareto uniformity. Computationally, it was found that with increases in problem size, the ε-constraint exact approach was rendered impractical for real-world applications, requiring over 4 h. In contrast, MOVNS remained efficient. MOVNS was examined further against a basic VNS, as well as a redesign of it, implementing an Iterated Local Search (ILS) structure. ILS was found to outperform in terms of Mean Ideal Distance (a measure of mean deviation of Pareto solutions from ideal solutions) and CPU; however, this was attributed to the decrease in quality of its solution, consolidating MOVNS strong performance. Lastly, the metaheuristic was verified for its applicability to real-world tourist planning on a small-scaled study of Tehran, successfully providing a set of non-dominated solutions catering to diverse preferences.

A bi-objective INLP model was then formulated by Wang et al. [87] to make a trade-off between passenger waiting time and operating costs, jointly optimising train schedules and a flexible routing plan of inter-city and multiple inner-city modes. To bridge and improve the scheduling coordination between the metro and other transportation modes during normal- or emergency operations, a multimodal passenger arrival flow prediction method was based on Monte Carlo simulation, and the network queuing system was embedded into the model to supply real-time passenger volume at the metro hub station’s platform. The ε-constraint method was used alongside a tailored VNS to generate approximate Pareto Optimal solutions, including the number of effective trains, route pattern, train dwell time and demand-sensitive headway. The use of simulation for passenger demand was compared to the estimations based on historical data and it was found that this method is effective in supporting multimodal scheduling coordination during abrupt changes of arrival flow. Two sets of numerical experiments, a small-scale case, and real-world hub-metro network instance of the Beijing South Railway Station were examined. Results showed that the proposed VNS algorithm found high-quality solutions within acceptable computing time, compared with the performance of the GUROBI solver and GA. GUROBI could not provide a feasible solution through a 24-h window due to the explosion of decision variables in the large-scale instance.

Lastly, a study on the Greedy Randomised Adaptive Search Procedure is reviewed. GRASP is an iterative greedy metaheuristic designed to solve combinatorial optimisation problems, such that each iteration contains two steps: construction (building a feasible solution using a randomised greedy algorithm), proceeded by a local search [61]. Due to the limited development of personalized group tourism itinerary tools, Ruiz-Meza et al. [88] addressed the Tourist Trip Design Problem (TTDP) to better capture the complexities of real-world planning by extending the Team Orienteering Problem (TOP) with Time Windows into a multi-constraint, multimodal MIP. The single-objective formulation constructed routes in terms of the cost and time limits associated with each individual, mode choice and heterogeneous preferences within the group. The paper’s proposed GRASP methodology delivered competitive and occasionally optimal results for test instances of 20 and 50 nodes, underscoring its computational efficiency, particularly for when exact solutions were unable to solve instances of larger complexity, with the processor running out of memory. A statistical analysis highlighted the effectiveness of the Insert-Replace-Swap sequence operator for the neighbourhood search in enhancing solution quality. For future research directions, it was suggested to include parameters under uncertainty to better emulate reality, as well as alternative metaheuristics and hybridizations, and the consideration of environmental concerns under Sustainable Development Goals. The studies of this section are categorised and comparatively summarised in Table 8 as follows.

Table 8 Categorisation of studies that use single-solution approaches to optimise the transportation problems
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3.2.2 Further reflections

Covering a plethora of single-solution methods, simulated annealing was found to be the most widely applied metaheuristic for this section, closely followed by Tabu Search and VNS. Whilst touristic planning and the VRP emerged as the more researched problems, overall, the scope of this section’s multimodality varied significantly, with no single optimisation problem being studied considerably more than the others. Most SA-based studies benefited from the established bi-level structure for multimodal MOOPs, assigning modes according to UE in the lower level through a traffic assignment problem. VNS similarly appeared to be well suited to multi-objectivity. Additionally, the remaining single-objective study solved using SA mentioned the use of multi-objectivity for future research, further supporting it as a principal field of interest.

Over the course of this section, hybridisation, real-world uncertainty, dynamic optimisation within a real-time environment, and the addition of modes were again highlighted as core topics in need of further investigation. Collectively, computational performance is observed to persist as a key performance indicator of metaheuristics throughout this literature review, especially within the context of the solution optimality-efficiency trade-off. Throughout this section, SA has consistently been deemed as appropriate for small to moderately sized networks. This appears to be in-line with all single-solution metaheuristics reviewed in this paper, with only TS being favourably considered for larger multimodal urban networks.

3.3 Analysis of multiple approaches

Research on multiple metaheuristics, either for individual method comparisons or hybridization, is frequently recommended throughout the literature for further exploration. In considering the integration of multiple metaheuristic algorithms, the Memetic Algorithm (MA) stands out. MAs consist of the embedding of local search processes (single-solution metaheuristics) within evolutionary algorithms, for the further refinement of individual solutions. Accounting for one another’s strengths to compensate for their respective weaknesses, single-solution- and population-based algorithms are often integrated with one another, such that the single-solution searches optimise locally, while the population-based methods attempt global optimisation. Such approaches are typically considered state of the art, having become established with their increased applications [61]. A collection of both individual comparisons of methods, as well as hybrids, designed to solve multimodal human mobility optimisation problems, are chronologically reviewed respectively in the following sub-sections.

3.3.1 Individual use and comparison of multiple metaheuristics

To solve the NP-hard multimodal K-shortest viable path problem, Niksirat et al. [89] employed hybridizations of ACO and SA algorithms, respectively. A Bi-directional search through Ant Colony Systems (BACS), making use of two separate colonies of ants for a divide and conquer approach was formulated, alongside a greedy algorithm Simulated Annealing method. The multi-objective problem was described as priority-based, with viability constraints extended to restrict travel time, mode, line changes, and walk duration. The hybrid methods were analysed using numerical simulations of different scales, tested for computation time and costs. BACS parameters were set to consume the least CPU time and when compared to performance to the standard ACO algorithm, it was found to outperform in both speed and objective function performance. Furthermore, the SA method was considered more suitable for smaller networks, whereas BACS was better suited for medium and large problems. The metaheuristics were further evaluated on Tehran’s public transport, the two optimising under 0.2 s, in contrast to CPLEX’s 30 s execution time. Thus, CPLEX was deemed unsuitable for large-scale networks, with the two hybrid methods poised as suitable for the study’s proposed real-time multimodal information system of urban networks, available to advise travellers on their journey. The greedy SA provided better solutions, explained to be due to Tehran’s public transport being sparse, however in aggregating costs with computation time, BACS was found to mostly outperform.

Rendl et al. [90] aimed to construct a generalisable framework for the Homecare Scheduling Problem (HSP), adaptable to other homecare scenarios with similar constraints. Multimodality was innovatively introduced for the HSP by having each nurse state their preferred mode of either public transport or car. The objective was to find the optimal roster for nurses that optimises employer costs, customer-, and nurse satisfaction. A weighted sum of all influencing factors was used along with soft constraint violations. To solve, a two-step framework was developed such that an initial solution was first generated. A Constraint Programming (CP) approach with clustering decomposition was devised for this and compared to a random construction heuristic. The solutions were then improved through multiple methodologies, for which VND, VNS, a memetic algorithm, scatter search, and simulated annealing hyper heuristic were all considered. VNS, which embeds VND in its local search phase for its general search scheme uses diversification to escape local optima, which VNS is susceptible to. Tested on data from a Viennese public health care company, for which exact methods were considered too expensive to use due to the scale, the metaheuristics were limited to 75 min of computation time when evaluated. The MA produced the best results with the fastest convergence, attributed to the nature of randomization of its EA, diminishing the effects of the initial solutions provided by the CP. VNS produced similarly good results, however required more convergence time. VND, VNS and SS all benefited in performance from the hybrid setup of the CP. The impact of different metaheuristics used was concluded and the potential for the solution framework to effectively tackle the large-scale real world multimodal HSP was validated, finding promising solutions.

Due to the significant impact of initialization settings on metaheuristics, Katona et al. [91] studied the influence of the settings on ACO and GA on solving a multimodal SPP. A parallelisation framework was additionally introduced. Run on a simulation of a multimodal GTFS database using a multi-threaded server, ACO was configured to avoid loops and dead ends in its search, as well as to incorporate an elite ant to promote faster convergence. On the other hand, the standard GA formulation was used. Ants were employed in parallel to explore the network and provide feedback. Increasing the number of ants was found to enhance convergence and potentially reduce iterations, making ACO suitable for parallel operations on unknown network structures. GA’s iteration number, population size, parent number, and mutation rate were optimised for faster execution. Parallelisation using the island model with different parameter settings for each population improved conversion rates and population diversity. This research supports the applicability of metaheuristics for distributed computing, like Cloud systems.

GA, SA and Water Cycle Algorithm (WCA) were hybridised by Almasi et al. [92] to build an integrated multimodal public transit model of different feeder services. Using a benchmark as well as real-case medium-sized network of rail stations and bus routes in Petaling Jaya, Malaysia, total costs were intended to be minimised by strategically assigning coordinated urban mass transit stops with different feeder modes according to an optimal demand ratio. User and operator objectives were framed as either several single-objective functions or multi-objective, using fuzzy-membership to evaluate the best trade-off. For single-optimisation, the WCA was incorporated into the separate optimisation methods of GA and SA specifically to proportion demand, whereas for the multiple, separate non-dominated sorting GA and -WCA approaches were used. Results showed that multimodal networks produced improved optimal statistical conditions compared to those of the single mode, with GA outperforming both in costs as well as convergence time. Similarly, whilst the two non-dominated algorithms obtained similar demand proportion rates, NSGA bore better total costs. It was proposed that social costs would be considered for future work.

Afrasyabi et al. [93] developed a Crossover-Based Multi-Objective Discrete Particle Swarm Optimisation (CBMODPSO) algorithm to solve a multi-objective VRP, optimising distance, traffic, comfort, and safety for passengers. Graph theory was used to model the Tehran transportation network, consisting of 408 BRT, subway, and taxi stations, with walking used to transfer between modes. Evaluated in terms of routes, convergence, reproducibility and computation time, the CBMODPSO’s performance was compared to multiple multi-objective variants of the ABC, ACO, Biogeography-Based Optimisation (BBO), Gray Wolf Optimisation (GWO), and NSGA-II. The CBMODPSO had the fastest convergence rate and outperformed the other solution methods in terms of reproducibility, providing various multimodal routes for passengers.

Ghiasvand Ghiasi et al. [94] studied the Home Health Care Routing and Scheduling Problem (HHCRSP). Formulated as a MILP, minimising the sum of the travel distance and overtime costs, the HHCRSP was represented as a directed graph based on a set of vertices representing patients, starting, and destination depots. Developed with a daily planning horizon at the beginning of each day, each nurse was allocated a service time, along with a time window. A mode was allocated to each nurse based on their determined routes. Public transport modes had lower travel costs but higher travel time, compared to private modes. Solutions were represented using multiple string sequences, capturing the depots, patient-visitation order and assigned vehicles. Three metaheuristic methodologies were employed, namely Invasive Weed Optimisation (IWO), Grasshopper Optimisation Algorithm (GOA), and Simulated Annealing. The Taguchi method was used to stochastically set the metaheuristics’ parameters and operators. 32 numerical experiments were used to analyse the NP-hard problem and compare algorithmic performances. IWO was found to perform best for medium and large cases, with no significant difference observed between algorithms for the small-sized instances. The HHCRSP was suggested to be extended to a multi-objective variant related to sustainability for future study, as well as to consider uncertainty.

Incorporating metro, bus, car-sharing and pedestrian modes into their network, Yan et. al [95] presented their multimodal transport framework to promote seamless connectivity. Designed as scalable, it was intended to accommodate future extensions to emerging transport modes. A real-time path planning algorithm, Q_EDQ, was presented, improving the reinforcement learning algorithm of Q_learning by using dynamic exploration, combining double-Q learning and simulated annealing to avoid slow and unstable convergence. The network represented stations as nodes, connecting different travel modes or routes. Various attributes including timetable, cost, distance and transfer information were captured at the edges. Multiple objectives of transfer, economic and travel time costs were minimised as total costs with equal weights. Using the deep learning framework of PyTorch, and CUDA for GPU-accelerated computation, multiple experiments were run using real bus and metro data from Xi’an, Shaanxi Province, comparing the Q_EDQ algorithm to multiple methodologies, including the GA, an improved ACO of previous literature, hybridised GA with VNS, and the Q_learning algorithm. The GA-VNS method outperformed the GA, attributed to its more efficient search and avoidance of local optima. The Q_EDQ outperformed the GA in computation time (0.05 s and 5 s respectively), as well as in terms of solution quality. Similarly, the Q_EDQ achieved greater minimisations than all the other methodologies, performing well in both simple and complex path planning instances, validating its scalability.

3.3.2 Hybridizations

Concerned with the inherent nature of stochasticity and time-dependence of public transportation, Dib et al. [96] posited that multimodal route planning is integral in supporting a seamlessly advanced traveller information system. Railway, bus and pedestrian networks were labelled as either public or private and a SPP was optimised using a memetic algorithm. The MA was constructed as a scalable hybrid GA and Variable Neighbourhood Search metaheuristic and was compared in performance to both a pure GA, as well as Dijkstra’s exact shortest path algorithm, modified for multimodality. A large urban network of 275,606 nodes was constructed using transfer links, connecting the multimodal sub-networks of Ile-de France. The approximate approaches outperformed in terms of CPU computation, with the GA optimising the tri-modal scenario in 0.0058 s, and the MA in 0.008 s, both ideal for dynamic decision making. In contrast, whilst Dijkstra guarantees an optimal result, it ran for 4.125 s. The hybrid was found to be much closer to optimality than the pure GA, possibly due to the MA’s nature of enhanced initial solutions and smarter search, at the cost of slightly slower computation time of the standard GA. The noted trade-off between computation and optimisation capabilities was in line with how augmentation of a problem and/or method’s scope comes at the expense of further computational cost, accounting for the size of networks and added variables like multimodality. An online journey planner for Ile-de-France’s public transportation system was then further developed [97] using a hybrid algorithm that combined the population-based GA with the local-search algorithm of VNS to address multi-objective SPPs. Initially, the VNS showed limitations in the multi-objective environment, which were mitigated by using weighted average ranking to convert the problem into a single-objective one. VNS was integrated into the GA as a mutation operator to improve the spread of solutions explored, avoiding local minima. This diversification was also used to enhance initial solutions, boosting overall performance. The study successfully modelled individual transit modes as sub-networks within a multimodal urban system, all integrated into a larger network when required within less than 12 s, validating the data structure’s suitability for real-time computation. Motivated by passenger expectations extending beyond fast travel, efficient routing systems must consider multiple criteria such as travel time, cost, transfers, and walking. In a test of 10,000 routing queries, the hybrid method outperformed individual metaheuristics and Dijkstra’s exact approach. While Dijkstra worked well for single-criterion instances, it slowed down exponentially to over 3 min as criteria multiplied, making it unsuitable for real-time optimisation. Metaheuristics responded within 170 ms, with the pure GA and VNS methods showing the best run times due to their rapid convergence, costing them in terms of solution quality. The hybrid GA-VNS offered a well-balanced compromise between solution quality and computation, suitable for solving dynamic real-life itinerary problems. Future studies were suggested to consider incorporating urban modes beyond public transit, such as car-sharing and bicycles. This expansion was expected to increase the search space, and to adapt to this, parallelising genetic operators was recommended to improve the method’s global computation time and solution search.

In light of human mobility being increasingly organized within a multimodal context, Dib et al. [98] continued their work by positing that an efficient multi-objective routing system is a necessity in meeting the demands of both passengers and system complexity. To do so, an MA was formulated, where a GA was combined with a Hill Climbing (HC) local search procedure to solve the multi-criteria SPP in stochastic networks, factoring for uncertainty. Each transportation mode was separately modelled as a directed graph, then combined. Routes were optimised under the criteria of stochastic travel time, travel cost, transfers, and walking time. Assessed upon the historical data of the real-life itinerary of the large urban network of Ile-de France, compared to classical deterministic algorithms and pure GA/HC methods, MA achieved solutions of better quality. This was in line with how MA benefits from the GAs exploration capacity, and the local search’s procedure within genetic operations, resulting in an improved search of regions of interest. As MA incorporates the framework of the two methods, its run time is comparably slower, however, at 19 s, it is still considered efficient enough to be integrated within a real-world journey planning system. Future work mentioned improvement of the MA’s efficiency.

3.3.3 Classification of studies and further reflections

Below, Table 9 categorises the discussed studies into the classifications adopted throughout this review.

Table 9 Categorisation of studies that use multiple metaheuristics
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This section covers the testing of multiple metaheuristics either individually or combined for hybridization purposes. The SPP emerges overall as the central problem type studied. For the individual comparisons, both SPPs directly compared ACO performance to another method within a dynamic setting. For the multi-objective study [89] a bi-directional ACO search, along with a Greedy SA method were designed and evaluated on a multimodal network. Comparable in speed, SA provided better solutions, attributed to the network’s sparse design, in line with the SA’s favouring of smaller scaled systems as established in the previous section of single-solution methods. BACS was concluded to generally outperform in terms of aggregated performance-time-costs, once again prioritizing a balanced trade-off between the two metaheuristic performance indicators. ACO was then studied alongside the GA on a parallelised framework [91], where ACO was found to be ideal for parallel operations on unknown network structures. The GA’s parallelisation was observed to improve the quality of the search. Ultimately, metaheuristics were deemed as suitable for distributed computing. The MA was comparatively found to produce the most optimised results the fastest, attributed to its EA. This is in line with the findings of [93], who’s outperforming PSO algorithm embedded evolutionary search operators (mutation and crossover) into its multi-objective search process. Similarly, [92] found the GA to outperform its SA and WCA counterparts on both counts of speed and solution quality, regardless of the single- or multi-objective nature of its use. VNS was comparable in solution quality, however required more convergence time [90], so when hybridised [95] with the GA, as per the findings of [97], the algorithm was deemed to perform a more efficient search than the GA. Conversely, the hybrid Simulated Annealing and Reinforcement Learning model [95] performed the best in both computation time and solution quality, intentionally designed to be scalable to large multimodal networks, as well as to be deployed in real-time settings. This was highlighted as a necessity within multimodality and the need for seamless transfers, mirroring the motivation of [47, 57] as discussed in Sect. 3.1.1.3. In support of [95]‘s framework’s development, GPU-accelerated computation was employed.

Specifically in terms of hybridisation, different variants of the MA were primarily assessed on a multimodal SPP. The hybrid metaheuristic was generally found to directly benefit from its evolutionary algorithmic features, swiftly optimising large networks to deliver strong results. Consisting of the co-integration of two metaheuristics, while they significantly outperformed Dijkstra’s exact approach, hybrid methods performed comparably slower to their pure metaheuristic counterparts.

4 Research agenda and outlook

The systematic synthesis of the presented literature validates the efficacy and suitability of metaheuristics for the optimisation of urban human mobility by means of multimodal transportation. This review provides a classification of the studies, along with further reflections, threading together several continuous trends and a coherent picture of what both the current and future landscape of research entails. The overarching research agenda is thus presented below, concluding with an outlook for future investigations.

Comprising of a vast range of optimisation problems including various settings of the TNDP, SPP, HLP and more, multiple graphical representations (transfer, abstract-, hyper graphs, etc.…) have been used to capture the complexities intrinsic to multimodality, with modes often individually represented as sub-graphs. These complexities are evident in the mathematical formulations, often characterised as non-convex, NP-hard and/or combinatorial. Alongside the diverse range of problems, various facets of multimodal transport (mode split/assignment, transfers, routes, timetabling, hubs, user experience and utility) are considered. This results in the optimisation of a plethora of criteria, such as user preferences, travel time, emissions, etc., broadly considering various social and/or operational costs, classically in conflict with one another. Naturally, the situation-specific trade-offs between diverse criteria becomes a necessity, particularly for real-world modelling, resulting in the need for Pareto-optimality. Multi-objective formulation is thus increasingly applied, within which bi-level programming is frequently adopted. Typically representing mode split/choice behaviour by means of User Equilibrium and/or logit-based modelling in the lower level as a Transit/Traffic Assignment Problem, the program is then optimised for the central objective(s) of the upper level, where the resulting mode distribution of the lower level cyclically feeds into the upper’s decision making. Thus, bi-level formulation is primed as highly suited to support the heavy demands of multimodal network design problems.

The multi-objectivity of multimodal transport problems further introduces variables into the search space, rapidly growing in complexity with the addition of objective(s). In adherence to the standards of green initiatives, multimodality captures the concept of sustainability in its integration of existing infrastructure with the introduction of novel systems. Sustainability and user-oriented transportation of passengers present emerging state-of-the-art means of mobility (for e.g. EVs, scooter-sharing, CCAM), contributing to the addition of modes within the optimisation problems. Prevalent as a point of continued interest throughout the analysed literature, the increase of modes, alongside multi-objective modelling, depict the considerations for the future study of problems associated with urban multimodal transport optimisation, illustrated to only continue to rise in complexity.

Spanning over 31 studies, the Genetic Algorithm is the predominantly applied metaheuristic of the review, attributed to its ease of use and flexible encoding, adaptable to diverse modelling needs through algorithmic modifications. On par with the established frequently discrete nature of the multimodal studies, the bitstring representation of the genetic encoding is highly suited to the optimisation of multimodal mobility, capturing variables like mode type or route composition. This is corroborated by its comprehensive scope of problem applications. Consistently considered for hybridization with, and in comparison to, other approaches throughout the reviewed studies, the Genetic Algorithm stands as a benchmark in terms of both solution quality and performance markers among metaheuristics. Generally possessing the best efficacy in the optimisation of multimodal passenger mobility, GA is further shown to be considered as the methodological standard for Pareto-optimisation, where its NSGAII variant is found to be the most used multi-objective metaheuristic of this paper.

With the rapid shifts of technological developments, driven by advanced processors, sensors, and data collection, optimisation problems like the SPP are increasingly being considered within a real-time context. In terms of the static/dynamic problem settings of the reviewed literature, GA largely optimised multimodal urban transportation systems within a static context, outweighing dynamic papers by four times. The second most applied metaheuristic approach, namely ACO, notably has a more balanced share of static/dynamic papers. This is depicted in Fig. 3.

Fig. 3
figure 3

Categorisation of studies according to their metaheuristic and static/dynamic design

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In contrast to alternate approaches often requiring additional pre-processing of data structures, consuming significant memory and thus impacting computation, the graphical structure employed by the ACO algorithm is compatible with the aforementioned representations of multimodal networks [63]. Found to additionally perform well in the optimisation of unknown network structures, especially through the support of parallel computation [91], the metaheuristic appears to be characteristically well suited to both multimodality, as well as real-time problems. ACO’s reviewed field of study, however, is found to be comparatively sparse in terms of multi-objective optimisation (let alone bi-level programming), seeing an uptake in interest only within recent years. In contrast, half of PSO-based studies have promisingly been researched within multi-objective contexts, yet simultaneously limited to statically designed problems. Thus, future investigations of ACO/PSO optimisation stand to benefit from these considerations to compensate for where the metaheuristics may lack in comparison to the comprehensive field of the GA and its applications.

Similar to population-based methods, single-solution approaches were found to be considered for a diverse range of applications. Their limitations lie however in their scalability, as they mostly favoured small-sized test instances specifically. The memory-retaining capabilities of Tabu Search proved useful to its application, as demonstrated by its considerations for larger experiments with dynamic qualities. Overall, the qualitative performances of local-search approaches are particularly worth considering for hybridization purposes. With the multitude of methods and their variable associated strengths and short-comings, their ease of algorithmic embedding and modifications can account for one another, naturally resulting in hybridizations. In line with the growing research of hybrid frameworks, regardless of whether they are purely metaheuristic or not, the advantageous compensation of the mixed methodologies additionally presents a suitable optimisation approach to support multimodal modelling and minimise the performance quality-efficiency trade-off.

Overall, the applicability and success of metaheuristics with respect to the optimisation of urban multimodal human mobility has been thoroughly validated throughout the reviewed papers, conventionally delivering high-calibre optimisation of NP-hard problems. They are predominantly presented as efficient alternatives to exact approaches for the exponentially growing search space that multimodal problems capture in terms of various considered modes and criteria. Despite this, their continued improvement in terms of performance criteria (CPU time, optimality gap, hypervolume (a reliable measure of convergence and diversity for multi-objective optimisation [86]) remains a central topic for further research.

Given the explosive complexity that prospective multimodal transportation problems are bound to be defined by, especially in view of the heightened interest in terms of the addition of modes and criteria, alongside the considerations of real-world/time characteristics, computational burden remains as a fundamental area of concern. For instance, to create the Pareto front relevant to multi-objectivity, the calculations required by the ε-constraint method increases computation time considerably.

To support the development and deployment of metaheuristic-based frameworks for multimodal systems, parallel processing is proposed. While distributed and/or parallel computing has been studied and mentioned as a tool in the reviewed literature, GPU technology is sparsely considered. Throughout the review, the reporting of CPU specifications is traditionally included as they physically perform the computation. Their serial processors are typically equipped with a core count of between 2–64, whilst the parallel processors of GPUs are found to consist of within the thousands [99]. Researched within the past decade, the conversion of CPU-based programs to that of GPU, along with the barrier of data-representation, have made implementation traditionally difficult. Conversely, the proliferation of machine learning research has similarly evolved and extensively fine-tuned the field of GPU usage, made expandable to an increasing range of applications.

This is of particular interest, as in contrast to methodologies like deep- or reinforcement learning [95], metaheuristics are disadvantaged by their noted lack of scalability [100], where performance is found to fall in face of increasingly complex problems. To possibly overcome the aforementioned obstacles of high dimensionality, posed by multimodal transportation problems, GPU-operated parallelised metaheuristics are suggested for future research.

Accelerated optimisation through the support of a single GPU is recently considered as highly efficient, significantly reducing computation time [101]. With the appropriate data structures to match the programming platform, a parallelised PSO algorithm [102] was found to maintain solution quality at a remarkably enhanced speed when optimised on a single GPU for large, high-dimensional problems. With the increase in dimensions of the objective function, the speedup ratio was found to efficiently increase, mitigating the concern of scalability for metaheuristics. This is particularly significant as the implementation of multiple GPUs was suggested for future study, further supporting the advantageous potential scalability of metaheuristic optimisation to account for large-scale problems of high complexity, capable of supporting future developments of multimodal frameworks.

This need for maintaining efficiency along with scalability is highly relevant in the realm of real-time deployment, where speed is a necessity. Such considerations are important for developing prospective solution pipelines, where for example both supervised machine learning and metaheuristic approaches could be integrated. Data processing through the means of methodologies, such as neural networks, for time series forecasting may be considered to account for the inherent uncertainty and elasticity of real-world and/or dynamic optimisation. Furthermore, supported by GPU equipped hardware to train the neural models, the processing units offer an attractive computing power to price ratio, as opposed to the typically general-purpose CPU counterparts [101]. Thus, the integration of metaheuristics with such technology would be a natural shift.

Drawing upon the reflections and findings discussed throughout, the holistic research agenda is presented. To conclude, the below suggestions aim to guide future research in addressing the evolving challenges of optimising multimodal urban transport.

  • Parallel Processing and GPU Utilisation: Investigate the use of parallel processing and GPU technology to accelerate the metaheuristic optimisation, particularly for large-scale/high-dimensional problems. The population-based framework of the global search metaheuristics primes them as attractive methodologies to parallelise.

  • Scalability and Efficiency: Support the scalability and efficiency of metaheuristic optimisation to account for the increasing complexities intrinsic to multimodal transport development.

  • Alternative Metaheuristics: Further explore different metaheuristic optimisation approaches such as SI. Future research of ACO and PSO optimisation could benefit from addressing their respective sparse exploration in multi-objectivity and dynamic problems.

  • Apply Genetic Algorithms for benchmarking: Due to its consistent efficacy, GA/NSGA II can serve as a benchmark metaheuristic for the research of novel problems and/or the comparison to different methodologies. Its continued algorithmic development is also recommended, particularly to account for the computational complexities of the higher dimensions that multimodal optimisation problems can result in. Its use in dynamic optimisation is relatively less established than its static counterpart and stands to benefit from additional research.

  • Hybridisation: Continue the research and development of hybridising different metaheuristics and methodologies to minimise performance quality-efficiency trade off. Due to its evolutionary algorithmic features, the memetic algorithm is recommended to serve as a benchmark hybrid metaheuristic.

  • Explore diverse problem settings: Address various multimodal mobility challenges by further investigating different optimisation problems under unaccounted for settings, including real-time/world considerations, additional modes, and criteria.

  • Multi-objectivity: Investigate Pareto friendly optimisation methods, along with bi-level programming, to capture and address the higher dimensionality imposed by prospective diverse problem settings.