Related work

A Multimodal Transportation System (MTS) combines two or more modes of transport that can be used to reach the final destination. When planning an itinerary, a traveler may wish to reach their destination through multiple intermediate distant cities by multimodal transport within a given time frame. Planning itineraries becomes complex in multimodal transportation because of the plethora of possibilities for transport services, time schedules, and routes. The traveling Itinerary Problem (TIP) is a variant of the traveling salesman problem [1–3]. Many traveling salesman problem variants and generalizations have been studied extensively [4, 5]. Particularly relevant to this research is the shortest path problem [6–10] and the time-dependent shortest path problem with time windows also studied in [11–15].

Several approaches based on graph theory have been developed in the research community. Hedi et al. [16] proposed a transfer graph defined as a set of sub-graphs connected by a transfer point, which is a node that connects two distant components of the transfer graph, Ayed et al. [17] proposed a hybrid approach of transfer and relevant graph to solve the time dependent multimodal transportation network. In [18], the authors developed a hyper graph abstraction of a time-dependent multimodal transportation network that consists of two levels of abstraction, i.e., regions and transport modes. In [19], the authors modeled the network using a hierarchical structure. Dynamic segmentation and linear reference techniques are applied, and a three-layer transport system consisting of a physical layer (street network), logical layer (roadway networks), and application layer is constructed (bus, metro, BRT, etc), time-dependent graph [20, 21] and time expanded network [22]. Besides several algorithms, the Dijkstra algorithm [23] is still the most frequently used approach to compute the shortest path [17, 18, 24, 25]. For arcs having interval weights, [26] proposed a generalized Dijkstra algorithm to solve the shortest path problem. Enayattabar et al. [27] provided a solution to the shortest path problem using the Dijkstra algorithm, in which the arc is represented by interval-valued Pythagorean fuzzy numbers. At the same time, other algorithms like Tabu Search [28], Ant Colony algorithm [18], Artificial bee colony (ABC) algorithm [29], Branch-and-Cut [30], A* and ALT [31, 32], Fuzzy weighted Ant Colony optimization approach [33], hybrid of adaptive large neighborhood search with Tabu search [34], a hybrid of Genetic algorithm and Variable Neighborhood search (VNS) [24] and other heuristics and meta-heuristics [24, 30] have also been investigated.

Ref. [2] proposed a traveling itinerary problem to find the cheapest way how to visit multiple cities within a single mode of transportation. The author decomposed the problem into macroscopic and microscopic tours. An implicit enumeration algorithm was used to find the optimal combination of tours between the origin and destination city. A dynamic form of route planning was proposed by [31] to find the optimal path from a source to a destination based on the shortest path problem algorithm over a time-dependent multimodal graph. A hybrid of the Genetic algorithm and the Variable Neighborhood Search approach was proposed by [24] for route planning in a multimodal network. The hybrid approach gives better performance in terms of the quality of the final solution in comparison with the Genetic algorithm. Ref. [35] addressed a Profitable Tour Problem based on mixed-integer programming to elaborate travel itineraries to visit tourist attractions by considering the visitor profile and solved it using the exact branch and cut algorithm and heuristic Tabu search. Sori et al. [36] proposed fuzzy inference, that, considering cost, time, and risk factors, finds the optimal solution and the desirable path between origin and destinations. Ref. [37] studied the complexity of shortest paths in time-dependent graphs in which the costs of links vary as a function of time. Ref. [1] studied the time-dependent asymmetric traveling salesman problem with time windows with a constant travel time along with the links. Ref. [32] proposed a time-dependent shortest path problem with time windows intending to minimize the total arrival time at a destination in which costs along the links are also constant. Ref. [15] studied the K− shortest path problem in a time-scheduled network with constraints on links and proposed constant traverse costs along the links for different departure times. However, the real-world transportation network is more complex. The costs to traverse a link varies with different time schedules and modes of transportation.

New Mobility Services (NMS) include Mobility as a Service (MaaS) that provides users the option of choosing multimodal mode alternative [38, 39]. Integration of mobility services, enables offering a combination of different modes of transport for multiple cities with stay time to visit with a single interface. MaaS is still under development, but it could change the way we conceive transport and travel. The traveling itinerary problem in multimodal transportation networks is of significant and fast-growing importance. The aim is to find the route between origin and destination while utilizing multiple transport modes [19]. Different modifications of the time-dependent shortest path problems have been studied in the literature. Still, none of them addressed the problem for a given sequence of cities except [3], where the authors have studied the pre-determined travel plan in a single mode of transportation (flights). The authors decomposed the network into sub-networks based on time steps over the finite time horizon and searched for an optimal travel plan in a deterministic environment.

This paper studies the traveling itinerary problem for a fixed sequence of cities in multimodal transportation networks with time windows and other time constraints. The proposed approach is based on a directed scheduled multimodal transportation network (SMTN), constructed by integrating different modes of public transport in the sequence of cities. Next, we apply a network pruning algorithm to eliminate irrelevant transport connections. We developed an exact algorithm to find a low-cost journey plan in the SMTN under time constraints. Finally, to demonstrate the effectiveness of the proposed algorithm, we compare the results with solutions obtained by the general-purpose 0-1 integer programming solver. To the best of our knowledge, no study has previously addressed this problem in the literature.

Contribution

This paper is motivated by the traveling itinerary problem found in the domain of public transport. In particular, we address combined inter-regional and cross-border journey planning. A sequence of cities is considered with multiple days to stay in each city within a planning time horizon and other time constraints while utilizing multimodal transportation. An optimal journey plan is searched for in a deterministic environment. Nowadays, several solutions exist to solve conventional route planning in transportation systems, such as Google, Bing, and Baidu Maps. Most such routing tools support more than one transport mode. Still, they do not support journey planning for multiple intermediate cities with multiple days to stay at each city in a multimodal transportation network. Developing an efficient and economical journey plan is still an annoying experience for a traveler. Although journey planners operated by transport service providers can provide some predefined itineraries, they are not tailored to a specific traveler. These journey planners are also limited to national or regional-level journey planning and do not support cross-border and inter-regional trips. Moreover, most journey planners only consider a single day’s trip, while in real cases, most travelers wish to schedule an n− days itinerary. This deficiency can also be viewed against huge cross-border multimodal travel demand within the European Union (EU), leading to about one hundred million cross-border trips every year by EU residents and several hundred million trips by international tourists [40]. In this paper, we design a more general itinerary planning service to address the above limitations.

The contributions of this paper are as follows: (i) First, we model our traveling itinerary problem as a Scheduled Multimodal Transportation Network (SMTN) through a fixed sequence of cities. Taking advantage of the particular structure of this model, we proposed an exact approach that allows us to solve problems of realistic size to optimality. (ii) We propose an approach that satisfies the unprecedented demand for intelligent and sustainable tourism-driven advances in big data. Our work can form the foundation for extensive models and support systems for decision support that will emerge as functional websites or mobile applications to plan regional or cross-border traveling itineraries in multimodal transportation. (iii) We bridge the gap between urgent travel demand and supply at the inter-regional and cross-border levels. Our solution approach shows computational effectiveness when applied to inter-regional and cross-border itinerary planning situations.

Network pruning algorithm

The irrelevant transport connections that do not obey some of the constraints are eliminated from the graph representing the scheduled multimodal transportation network by the Network pruning algorithm. The Network pruning algorithm is divided into two parts. First, the irrelevant transport connections are removed based on the start of trip and end of trip . Second, transport connections are eliminated based on the earliest acceptable time of day of a departure and the latest time of day of an arrival . Below, we show the pseudo-code of the Network pruning algorithm.

Algorithm 1: Network pruning algorithm.

Input: G, , , ,

Output: G

1 for i = 1 to n − 1 do

2  for k = 1 to e_i do

    // Part I: eliminate transport connections outside the trip time window

3   if d_(i,k) ≤ then

4    remove (i, i + 1)_k from G

5   end

6   if a_(i,k) ≥ then

7    remove (i, i + 1)_k from G

8    end

    // Part II: eliminate transport connections with too early departure or too late arrival

9   if , are given then

10    if t[d_(i,k)] ≤ then

11     remove (i, i + 1)_k from G

12    end

13    if t[a_(i,k)] ≥ then

14     remove (i, i + 1)_k from G

15    end

16   end

17  end

18 end

Exact algorithm

The problem (1–10) can be interpreted as a problem to find the cheapest path in the SMTN that originates at city 1 and terminates at city n and obeys all constraints (2—10). To find such a path, we propose the Exact algorithm 1. The costs are additive, and hence to find the optimal path, the dynamic programming [41] can be applied. With each arc (i, i + 1)_k in the SMTN, we associate the value ϕ_(i,j), that represents the costs associated with the optimal path that is initiated at the origin city and terminates at city i + 1 and utilizes arc j exiting city i. To keep track of optimal paths, we define variables P_(i,j) that contains the index of the arc exiting city i − 1, that precedes arc j exiting city i in the optimal path that is initiated at the origin city and terminates at city i + 1.

First, all values ϕ_(i,k) are initialized. The computation starts from the city 1. Therefore, for the city 1, values ϕ_(1,k) are set to c_(1,k) for all k = 1, …, e₁ where constraint (7) is satisfied and for all other cities and all others arcs to value ∞. Furthermore, cities are recursively processed in the order of visits, and for each city i = 2, …, n−1 and for each arc exiting the city i, the value of ϕ_(i,j) and the corresponding value P_(i,j) are established. Knowing the values ϕ_(i−1,k) for all k = 1, …, e_i−1, the value ϕ_(i,j) is set to the minimum sum ϕ_{(i−1, k)} + c_i,j over all feasible k. The value of k which minimizes the sum is stored in P_(i,j). This way the costs of the optimal path utilizing the connection e_i,j from the city 1 to the city i are calculated. After calculating the values of ϕ_(n−1,k) for k = 1, …, e_n−1, costs associated with the optimal path that is connecting cities 1 and n are determined as . The optimal path can be found by backtracking utilizing values recorded in P_(i,j).

Algorithm 2: Exact algorithm.

Input: (G, s_i, δ)

Output: P

//Initialization

1 for i = 1 to n − 1 do

2  for k = 1 to e_i do

3   if (i == 1) and () then

4    ϕ_(i,k) ← c_(i,k)

5   end

6   else

7    ϕ_(i,k) ← ∞

8   end

9  end

10 end

11 for i = 2 to n − 1 do

12  for j = 1 to e_i do

13   for k = 1 to e_i−1 do

14    new_costs ← ϕ_(i−1,k) + c_(i,j)

15    if new_costs < ϕ_(i,j) then

16     if d_(i,j) ≥ a_(i−1,k) + s_i then

17      if a_(i,j) ≤ a_(i−1,k) + s_i + δ then

18       ϕ_(i,j) ← new_cost

19       P_(i,j) ← k

20      end

21     end

22    end

23   end

24  end

25 end

The optimality of the solution is implied by the Bellman’s optimality mechanism of the dynamic programming [41], and it can be simply proven by induction. According to this principle, an optimal solution to a problem includes an optimal solution to any related sub-problem. The proposed algorithm breaks down the problem into smaller sub-problems of finding an optimal itinerary for a shorter sub-sequences of cities. The algorithm stores the solution of sub-problems in the ϕ_(i,k). At each iteration, the algorithm examines all the possible connections between last city in the already optimized sub-problem and the next city. It is straightforward to see that values ϕ_(1,k) are set in the optimal way. Assuming that values ϕ_(i−1,k) are optimal, setting the values ϕ_(i,j) to the minimum value of ϕ_(i−1,k) + c_i,j over all feasible k is optimal, as the city i − 1 cannot be bypassed and there is no cheaper way how to reach city i + 1 while using arc j existing city i. The feasibility of the solution results from the data pruning and also from the used conditions. Application of the Network pruning algorithm 1 ensures that constraints (2), (3), (8) and (9) are obeyed. The constraints (4) and (10) are satisfied by definition as P_(i,j) maintains information about one arc index only. Constraints (5) are satisfied thanks to the condition on line 16 and similarly, constraints (6) are ensured by the condition on line 17. Finally, constraints (7) are enforced by the condition on the line 3. Thus, solutions encoded by variables ϕ_(i,j) and P_(i,j) obey all constraints (2)- (10) and are therefore feasible.

It is possible that the exact algorithm will not find a solution. For example, if all connections between a city pair are infeasible, the algorithm will return an incomplete itinerary up to the first city in the pair.

Data

First, we collected the data for five European cities, London, Paris, Berlin, Warsaw, and Amsterdam, from publicly available sources. Flight data were obtained from the travel agency Trip [42], and train and bus data were collected from Omio journey planner website [43]. Second, we assembled a data set concerning five Chinese cities, Beijing, Chongqing, Changchun, Qingdao, and Shanghai, where the data for both flights and high-speed trains were taken from [2].

The raw data, available at https://github.com/ralikk/Transportation, contains links between each pair of cities, including the transport details, origin, destination, departure time, arrival time, transfers (stopover), and cost. Currently, exchanges or stopovers are considered a part of input data. Thus, input data already include indirect connections that pass through other cities where a traveler must change the mean of transport. A generalization of the proposed approach that incorporates exchanges into the search for travel connections is left for future works. Tables 1 and 2 show the number of connections for each mode of transportation, for both data sets. The European cities data set contains connections from 2019-04-01 until 2019-04-13, while the Chinese data set contains connections from 2015-11-10 until 2015-11-20.

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Table 2. Numbers of transport connections by transport mode in the Chinese cities dataset.

https://doi.org/10.1371/journal.pone.0287604.t002

Illustrative example 1

Let us assume a traveler living in London (origin and destination city) with a plan to visit the sequence of cities: London → Paris → Berlin → Warsaw → Amsterdam → London. The traveler prefers to take the economy class and has a preference to stay in each intermediate city, namely, 2 days in Paris, 3 days in Berlin, 2 days in Warsaw, and 2 days in Amsterdam. The traveler has a preference for the earliest departure 08:00, and the latest arrival time 22:30 for every city. The traveler wants to plan the entire itinerary in the multimodal transportation network at once, instead of searching for each connection and travel mode separately, to achieve the minimum travel costs, under time constraints given by the time horizon, the stay time at each city, the earliest departure, and the latest arrival time.

First, the SMTN was constructed, including all available transport connections. Then, by using the Network pruning algorithm, all irrelevant transport connections were eliminated. The Exact algorithm is used to find the cheapest path for the sequence of cities under time constraints. The traveling itinerary obtained by the Exact algorithm is shown in Table 3 with total costs USD 308.

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Table 3. Traveling itinerary developed for Illustrative example 1.

https://doi.org/10.1371/journal.pone.0287604.t003

Taking into account the average number of daily flights, buses, and train connections, there are 102 for London→ Paris, 151 for Paris→ Berlin, 88 for Berlin→ Warsaw, 30 for Warsaw→ Amsterdam and 85 connections for Amsterdam→ London. Thus, in total, there are about 3.4 × 10⁹ potential itinerary planning strategies (including the infeasible). It would take an unmanageable time if a traveler had to find the optimal itinerary manually. In comparison, the Exact algorithm took 14.10 milliseconds to construct the SMTN and to find a traveling itinerary.

We developed single-mode and multimodal itineraries to demonstrate the effectiveness of traveling itineraries in the scheduled multimodal transportation network regarding travel costs and time. Fig 2 shows that the single-mode itineraries for buses and trains are not complete. This is because some pairs of cities in the sequence of travel do not have the desired transport connections, or there are no feasible connections available under time constraints. For example, in Fig 2 the train itinerary is incomplete and stops at Paris city because train connections are infeasible as traverse time exceeds the allowed waiting and traverse time (δ = 12hr). Thus, the results show that multimodal itineraries are more likely to be feasible and cost-effective than a single mode.

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Fig 2. Illustrative example 1: Comparison of costs and traverse times for single and multiple modes of transport.

https://doi.org/10.1371/journal.pone.0287604.g002

Illustrative example 2

As Illustrative example 2, we consider the trip developed in [2] that is considering a sequence of Chinese cities and a single mode of transport (high-speed trains) with a total cost of 6369 RMB (Chinese Yuan). The sequence of cities with the stay time to visit are Beijing (origin city), Chongqing (1 day), Wuhan (1 day), Shenzhen (2 days), Shanghai (2 days), and Changchun (destination city). The time horizon for travel starts on 2015/11/10 at 08:00 and terminates on 2015/11/20 at 08:00, with the earliest departure time at 7:00 and the latest arrival time at 23:00, for each visited city. The Exact algorithm is used to find the cheapest path for the given sequence. The traveling itineraries developed by the approach developed in [2] and the Exact algorithm are compared in Table 4. The itinerary developed by the Exact algorithm is associated with costs of 3906.5 RMB (that is an optimal solution). The Exact algorithm took 5.10 milliseconds to construct SMTN and to find a traveling itinerary. Considering the multimodal approach proposed in this paper, for the same situation but a single-mode transport as presented in [2], the costs were reduced by 38.5%, as shown in Table 4.

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Table 4. Traveling itinerary developed for Illustrative example 2.

https://doi.org/10.1371/journal.pone.0287604.t004

Analogously to Fig 2, in Fig 3 we compare single mode (high-speed train) with multimodal (high-speed train and flights) itineraries obtained for Illustration example 2. The results again imply that the multimodal itineraries are complete and cost-effective as compared to single modes, highlighting the superiority of the proposed approach over the approach presented in [2].

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Fig 3. Illustrative example 2: Comparison of costs and traverse times developed by [2] and this paper.

https://doi.org/10.1371/journal.pone.0287604.g003

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Table 5. Comparison of results obtained by CPLEX, and Exact algorithm (run times are reported in milliseconds).

https://doi.org/10.1371/journal.pone.0287604.t005

Computational analysis

Computational experiments are performed for a number of different travel sequences of cities with different time horizons and preferred earliest departure and arrival times. For each case, we used six different sequences of the same length of cities and calculated the run time (sequence details are available at https://github.com/ralikk/Transportation). We performed the experiments with both data sets (Chinese and European) with different cities in the sequence and sequence lengths to illustrate the computational efficiency of the proposed algorithms. The SMTN Graph Development refers to the run time taken by building a scheduled multimodal transportation network that also include the data management, while columns 3 and 4(b) refer to the run time taken by algorithms 1, and 2, respectively. Table 5 illustrates that the computation run time for the European cities (as we have three modes of transportation) varies from 2.13 milliseconds to 9.27 milliseconds for the Exact algorithm, while for the Chinese cities (as we have two modes of transportation only) it varies from 0.75 milliseconds to 6.17 milliseconds. The run time goes up with increasing the number of cities in the sequence and modes of transportation.

The comparison of the performance of the Exact algorithm with the CPLEX is shown in Table 5. The solution obtained by the Exact algorithm is optimal, and hence we do not report the optimality gap. As expected, the CPLEX features a more extensive run-time in comparison with the Exact algorithm. The CPLEX run-time varies from 46 milliseconds to 515 milliseconds for the European cities data-set and from 658 milliseconds to 2132 milliseconds for the Chinese cities data set. The Exact algorithm run time varies from 3.20 milliseconds to 11.73 milliseconds for the European cities data-set and from 2.68 milliseconds to 8.18 milliseconds for the Chinese cities data set. Hence, the Exact algorithm provided an optimal solution with a much smaller run-time compared to the optimal solution provided by CPLEX.

The size of travel plans in computational experiments is limited to six cities. This size is greater than the majority of real-world trips. We can derive some expectations about the performance of the Exact algorithm on larger networks based on the structure of algorithms 1 and 2 and results shown in Table 5. The number of computations in the Network pruning algorithm grows linearly with the length of the sequence of cities and with the number of connections between cities. The number of computations in the Exact algorithm grows linearly with the length of the sequence of cities and quadratically with the number of connections between cities. Table 5 confirms that computational times grow mildly with the problem size. Hence, extending the number of cities should not lead to a considerable increase in the computational time, and the algorithm will remain practically applicable.

Limitations and future outlook

As the number of intermediate cities increases, the number of strategies for connection selection increases exponentially so that the computation time for itinerary optimization rises rapidly. Given the complexity caused by the increase in the number of intermediate cities, we may implement the proposed algorithms with a Map-Reduce computing framework and perform them on some distributed Cloud computing platforms, which could definitely report the optimal travel itineraries with significantly reduced computational time. In this study, exchange or stopover points are considered as data and can be included as a point of transfer in the multimodal transportation network. We do not take hotel expenses into consideration, which are an important portion of the travel plan costs. In our future research, we will consider the costs of staying (hotel reservation costs) at intermediate cities, develop alternative traveling itineraries for a set of cities instead of a fixed sequence, including time-scheduled intra-cities public transport, and design more sophisticated rules to identify irrelevant data. The travel distance and geographical conditions between different cities are different and optional travel modes may also be different. In this paper, the algorithm does not provide optional travel modes passengers can choose in different links. In the future, we will develop more intelligent and robust algorithms, where the passengers will have the option to choose travel mode in different links and also will have the option to choose the service type (business class, economic class, high-speed train, etc). We have only studied the deterministic approach. In the future, we will extend this work to ensure the reliability of planning inter-city travel in advance before departure.