An Exact Solution Approach for the Bus Line Planning Problem with Integrated Passenger Routing

Evert Vermeir; Wouter Engelen; Johan Philips; Pieter Vansteenwegen

doi:10.1155/2021/6684795

An Exact Solution Approach for the Bus Line Planning Problem with Integrated Passenger Routing

Vermeir, Evert;Engelen, Wouter;Philips, Johan;Vansteenwegen, Pieter 2021-10-04 00:00:00 Hindawi Journal of Advanced Transportation Volume 2021, Article ID 6684795, 18 pages https://doi.org/10.1155/2021/6684795 Research Article An Exact Solution Approach for the Bus Line Planning Problem with Integrated Passenger Routing 1 2 3 1 Evert Vermeir , Wouter Engelen, Johan Philips , and Pieter Vansteenwegen KU Leuven, Leuven Mobility Research Centre-CIB, Celestijnenlaan 300, 3001 Leuven, Belgium UC Leuven-Limburg, Research & Expertise Digital Solutions, Geldenaaksebaan 335, 3001 Leuven, Belgium KU Leuven, Division RAM-Flanders Make, Celestijnenlaan 300, 3001 Leuven, Belgium Correspondence should be addressed to Evert Vermeir; evert.vermeir@kuleuven.be Received 16 November 2020; Revised 19 July 2021; Accepted 3 September 2021; Published 4 October 2021 Academic Editor: Gonçalo Correia Copyright © 2021 Evert Vermeir et al. *is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. *e bus line planning problem or transit network design problem with integrated passenger routing is a challenging combinatorial problem. Although well-known benchmark instances for this problem have been available for decades, the state of the art lacks optimal solutions for these instances. *e branch and bound algorithm, presented in this paper, introduces three novel concepts to determine these optimal solutions: (1) a new line pool generation method based on dominance, (2) the introduction of essential links, i.e., links which can be determined beforehand and must be present in the optimal solution, and (3) a new network representation based on adding only extra edges. Next to presenting the newly obtained optimal solutions, each of the abovementioned concepts is examined in isolation in the experiments, and it is shown that they contribute signiﬁcantly to the success of the algorithm. *ere are two main conﬂicting goals present in nearly every 1. Introduction line planning problem. At the one hand, a public transit *e design of a public transport network is a multistep company wants to spend as little money as possible, and at process [1, 2]. It starts with the infrastructure network de- the other hand, the passengers want to get to their desti- sign. Decisions on the physical network, such as bus stops or nation as fast as possible. One of the common approaches is bus only lanes, are made in this step. *e next step is the line to optimize the quality of the service for the passengers planning or the transit network design. In this step, the (described by the average travel time and/or the number of public transport operator decides where its vehicles will transfers) and to limit the costs by putting constraints on the drive and which stops will be served in which order. *en, number of lines that can be used and the length of the lines the timetabling step produces a ﬁxed time schedule for each (e.g., [4–10]). In order to evaluate the average travel time, the bus. *e setting of frequencies is sometimes included in the transit assignment, i.e., how the passengers will use the diﬀerent bus lines to travel through the network, needs to be line planning step and sometimes in the timetabling step [3]. *e next step is to plan the rolling stock and the crews. *e integrated in the decision process. If frequencies are not ﬁnal component of the planning process is the dispatching considered, an “all-or-nothing” shortest path assignment is strategy. While all these subproblems inﬂuence each other, typically used [4, 8, 9, 11–15]. Mandl’s Swiss network [14], is they are mostly solved sequentially in practice [1]. *is paper the only widely used benchmark network for the uncapa- focuses on the second step, namely, the line planning citated line planning problem [4, 7–10, 12, 13, 15–21]. problem, without considering frequencies. *is is also called Mandl’s network with 15 nodes and 21 links is illustrated in the uncapacitated line planning problem. Figure 1. *e length of each link is indicated with the *e uncapacitated line planning problem is a diﬃcult number next to the link. Benchmark instances are available combinatorial problem with an enormous search space. with two diﬀerent maximum line lengths and with diﬀerent 2 Journal of Advanced Transportation tackled in this paper. Section 4 discusses the solution ap- 8 proach. It shows the branch and bound algorithm and ex- plains all the details about the essential links and the Direct Link Network representation. Section 5 contains the results of the experiments on Mandl’s Swiss Network [14]. *e paper ends in Section 6 with the conclusions. 6 2. Literature Review *is section contains a short survey of the literature on the line planning problem, with and without frequency setting. Both (meta) heuristics and exact approaches will be covered. A more extensive overview can be found in the following papers and books: [1, 3, 22, 24–27]. It should be noted that, depending on the community, this planning problem is called the transit network design problem or the line planning problem. In this paper, the term line planning will be used. 2.1. Heuristics and Metaheuristics. Early research on dif- ferent variants of the line planning problem focused on Figure 1: Mandl’s Swiss network [14]. heuristics. One of the earliest heuristics to solve the line planning was by Patz [28]. In this work, lines are iteratively numbers of lines. Despite it being a small network with only removed from a large set of lines based on a penalty ﬁfteen nodes, there are no known optimal solutions for structure. Mandl [29] combined both passenger demand and shortest paths to create an initial solution. *en, this solution instances with more than three bus lines on this network [22]. is iteratively improved aiming to optimize the total travel *e main contribution of this paper is that, for the ﬁrst time. *ese early heuristics were unable to solve large in- time, all these benchmark instances for the uncapacitated stances, but they provide the basis for the development of the line planning problem with integrated passenger routing are more modern metaheuristics [11]. solved to optimality. *e amount of lines necessary to oﬀer *e ﬁrst metaheuristics for solving the line planning each passenger a direct connection along his/her shortest problem were Genetic Algorithms (GA). Chakroborty and path between origin and destination is also determined for Wivedi [30] and Pattnaik et al. [31] were some of the ﬁrst to the ﬁrst time. Furthermore, a number of novel concepts are solve the line planning problem with a GA. Because they introduced which are required to obtain these optimal so- were one of the ﬁrst ones to use metaheuristics during the lutions, but which are also interesting for future research on optimization process, the performance of their algorithms was a lot better than the previously known methods. Since this complex planning problem. First, a new method is developed to construct a line pool then, many reﬁnements have happened to these GA’s. Zhao of all feasible lines that can be present in an end solution. and Zeng [32] added a local search component to the GA, Secondly, the concept of “essential links” is presented. *ese resulting in a memetic algorithm. Nayeem et al. [8] intro- are parts of lines, determined during the branch and bound duced elitism in their GA as well as a guided local search. process, which must be part of the optimal solution. *irdly, Islam et al. [7] used a stochastic beam search to tackle the the “Direct Link Network” representation (DLN) is intro- line planning problem. One of their other contributions is duced. *is is a novel network representation allowing a the development of a new heuristic to get a very strong faster evaluation of solutions, compared to the well-known starting solution. *eir method combines edge lengths with change-and-go network representation [23]. Finally, a new demand served into a single cost, while previous methods branch and bound method limits the amount of required just tried to serve a high demand along (quasi) shortest evaluations before obtaining the optimal solution. *e paths. Chai and Liang’s [33] work is an example of a recent method branches over all possible lines containing a certain paper using a GA, and they developed a modiﬁed version of Origin-Destination (OD) pair. *e experimental results il- the well-known NSGA-II algorithm to solve the line plan- lustrate the importance of each of these concepts. We also ning problem. Fan and Mumford [12] used both hill make clear that research on the line planning problem climbing and simulated annealing. Also swarm intelligence should shift to larger and more realistic instances. techniques are being used in line planning such as Blum and *e paper is organized as follows. Section 2 contains an Mathew [5] who use Ant Colony Optimization or Nikolic ´ overview of the line planning problem literature, focusing on and Teodorovic [9] who use Bee Colony Optimization. (meta) heuristics and exact approaches for the line planning Vermeir et al. [34] use an iterated local search combined problem with integrated passenger routing. Section 3 de- with a local evaluation method which only looks at a cut of scribes in detail which variant of the line planning problem is the network to quickly evaluate candidate solutions. Journal of Advanced Transportation 3 maximizing the amount of direct travelers [46]. Here, the Mauttone and Urquhart [35] were the ﬁrst ones to model the line planning problem as a multiobjective problem. *ey integer program is solved with a branch and bound approach which greedily chooses lines with maximal current direct used a greedy randomized adaptive search procedure to generate a Pareto front with minimal values for operator and travelers. Bussieck et al. [47] and Bussieck [42] expanded this passenger cost. Recently, Duran-Micco et al. [6] developed a work by taking the vehicle capacities into consideration memetic algorithm to add emissions as an extra objective. through introducing constraints on the edge frequencies. *ey showed that emissions can be greatly reduced while *e repeated passenger calculations can also be avoided only having a minimal impact on the travel time and op- through a so-called “system split,” where links are catego- erator costs. Research focusing only on minimizing the rized by speed. Passengers are then assumed to switch to the fast levels as soon as possible and leave it as late as possible. operator cost is not discussed here, but the interested reader is referred to the above mentioned survey papers or e.g., [17]. With these assumptions, the passenger ﬂows can be esti- mated beforehand and do not have to be recalculated during the optimization process. *is method was promoted by 2.2. Exact Approaches. A number of publications on exact Bouma and Oltrogge [48], where it is used in a branch-and- solution approaches start by deﬁning a mathematical model bound approach applied to the Dutch railway network. and then try to solve the problem or a relaxation of the Later, the objective function evolved to the minimization of problem. As illustrated by the paper below, most of these line the travel time of the passengers [23, 37–40]. planning models originate from train line planning. Since More recent research integrates passenger routing in trains typically have lower frequencies than urban bus the line planning problem. In order to still model the line networks, the selected frequencies have a much larger im- planning problem as MIPs, assumptions about passenger pact on the transfer times than for a high frequency network. routing are made, such as passengers choosing minimal Capacity limitations at stops (stations) are more important in vehicle travel-time routes (and thus not considering for rail systems and are typically checked through con- transfers) or routes with a minimal number of transfers straints based on frequencies [27]. Hence, most commonly, [40, 49, 50] or such as passenger routes that can be the setting of frequencies is included in the line planning assigned by the public transport operator [23, 37, 40]. To process [36]. While frequencies should have a direct relation avoid these limitations, a bilevel model (like in most to transfer times, including frequency-based transfer times metaheuristics) can be used. In this bilevel model, net- lead to non-linear optimization models [36]. *erefore, even work decisions are made at an upper level and the pas- when including frequency setting, typically transfer times senger routing decisions are made at the lower level are considered through a transfer penalty [23, 37–40]. *ese [36, 51, 52]. An early implementation of this is formu- line planning problems are typically formulated as a (mixed- lated by Constantin and Florian [52]. Goerigk and ) integer program. Schmidt [36] use this bilevel model to completely inte- *ere exist many ways to calculate passenger routing in grate passenger routing and solve, to optimality, in- line planning. Distributing all the passengers along their stances with up to ten bus nodes and a line pool of 30 lines shortest path is the most commonly used strategy in line randomly generated beforehand. For larger networks, planning. While this is a simple method, it is already they use a genetic algorithm. However, none of these computationally expensive to execute during an optimiza- methods have solved the instances on Mandl’s Swiss tion [40]. In cost-oriented models the operational cost is Network [14] (ﬁfteen nodes) with integrated passenger minimized subject to certain constraints on the service level. routing to optimality. *ese constraints are typically very easy to check and avoid the expensive calculations. Since costs are easy to calculate, it 2.3. Benchmark Instances. Despite being too small to be is possible to use exact approaches even on real-world data. Claessens et al. [41] use a branch and bound procedure to representative of a city bus network, Mandl’s Swiss Network is the most commonly used benchmark instance in bus line obtain results for the Dutch railway system. Bussieck [42] used this work as a basis to make a cut and branch algorithm planning [22]. Nevertheless, fair comparisons in line plan- ning research remains an issue since diﬀerent variants of the which was used to obtain results for both the German and Dutch railway system. Goossens et al. [43] built further on problem are considered [4]. *ere are a lot of real-life in- this work with a branch and cut approach which allowed to stances used in literature, but they each have very speciﬁc loosen some constraints regarding the lines. Bussieck et al. objectives and constraints or the data is not publicly [44] created a fast procedure that obtains good solutions in a available. *ese issues are also mentioned in Ceder [1], small computation time. Canca et al. [45] managed to in- Ibarra-Rojas et al. [25], Farahani et al. [24], Schobel [27], tegrate several other planning steps in the optimization Kepaptsoglou and Karlaftis [26], and Guihaire and Hao [3]. Fortunately, Mumford [15] recently made four larger process. When looking at passenger-oriented models, explicit datasets publicly available. *ese networks are based on actual bus route networks from Chinese and British cities passenger routing is very important and cannot be easily avoided. In these models, an evolution of the objective and are being used more and more [4, 6, 8, 18–20, 53]. One of the datasets was already used earlier by Fan and Mumford function can be observed, which directly inﬂuences the speed at which the service level can be calculated. *e earliest [12], and another one was already used by Nikolic ´ and work circumvents the expensive passenger routing by Teodorovic ´ [9] with diﬀerent parameters. 4 Journal of Advanced Transportation be high enough to serve all passengers. *e integration of the 3. Problem Description passenger routing and the fact that no limited set of lines is *e available infrastructure network is depicted by a directed considered are two aspects signiﬁcantly complicating the graph G � V, E , which contains vertices V � v , v , . . . , v { } 􏼈 􏼉 uncapacitated line planning problem considered in this 1 2 n representing the bus stops and edges E � {e , e , . . ., e } which ij kl yz paper. are the connections available between these bus stops. *e cost or travel time on an edgee is indicated byt . *e demand of the ij e 4. Methodology passengers traveling through the public transport network is represented by an Origin Destination (OD) matrix D. *e *is section starts with a detailed discussion of a number of number of passengers per hour that want to travel from bus stop essential concepts and components implemented in our i to bus stop j is then depicted by d . In the line planning ij algorithm to optimally solve the uncapacitated line planning problem, the goal is to select the best possible set L of bus lines. problem with integrated passenger routing. First, it is In the uncapacitated line planning problem with integrated explained how the line pool is generated and how “essential passenger routing, the objective function is to minimize the total links” can be determined. *en, a new transit network travel time (TTT) of all passengers. *is means the sum of all representation is constructed, and it is shown how it can be travel times of all passengers. *is travel time also includes used to eﬃciently calculate the passenger routing. Finally, transfer times. *is time spent waiting on the next bus (and the the actual branch and bound algorithm is explained in detail. discomfort of transferring) is modelled with a transfer penalty In Section 4.6, a summary of the method is given together TP, which penalizes each transfer. It should be noted that a with the pseudocode and a ﬂowchart, and two alternative timetable or frequencies are not available at this stage, and approaches are brieﬂy introduced. therefore, a more accurate modelling is not feasible and waiting times are not considered. In order to evaluate and minimize the TTT, the routing each passenger will take has to be known. *is 4.1. Line Pool Generation. To ensure that the entire search routing results in a setπ � {e ,e , . . .,e } of edges used and the ij ik kl xj space is explored, a pool of all possible lines is generated. To amount of necessary transfers τ . *e total travel time of all ij construct this pool, an important property of the problem is passengers TTT is then represented by exploited. *e total travel time of all passengers will never increase when a line is extended by adding an extra stop at ⎝ ⎝ ⎠⎠ one of the ends. *erefore, all lines that are a subline of ⎛ ⎛ ⎞⎞ TTT(L) � 􏽘 􏽘 d 􏽘 t + TP∗τ . (1) ij e ij another feasible line are dominated by that line. When a i∈V j∈V e∈π ij dominated line would be part of the optimal solution, it can To completely integrate the passenger routing in the be substituted for its dominator. *is means that only optimization, the line planning problem is formulated as a considering nondominated lines will be suﬃcient to ﬁnd the bilevel problem as mentioned in the literature review. *e optimal solution. *is is illustrated in Figure 2, where line (a) design of the lines is the upper problem, and the routing of is dominated by line (b). All passengers that use line (a) in the passengers is the lower problem. For the lower problem, their shortest path will still be able to use the same shortest the “transit assignment,” a shortest path allocation is used. It path if we transform line (a) to line (b). *e situation can is assumed that each passenger will travel along its shortest only improve for the passengers. For example, passengers path, considering both the in-vehicle travel time and the that want to travel from node 5 to node 3 now have a direct transfer penalties. Although more complex and accurate connection along the shortest path. Because the uncapaci- transit assignment methods are available [54–57], most line tated line problem does not consider frequencies or costs, planning problems are still solved making this assumption line (b) is always at least as good as line (a). *is means that [4, 7–13, 16, 29, 53]. When operator costs are used as an we can guarantee to obtain an optimal solution without objective and/or frequencies are considered, frequency- having to consider line (a). *is is the power of domination, based assignment models are regularly used (e.g., [58–61]). and it signiﬁcantly limits the size of the line pool. For ex- *e objective and constraints used in this paper are the ample, on Mandl’s network with inﬁnite line lengths, this most commonly used in literature for the uncapacitated line results in 581 lines that could be present in the ﬁnal solution, planning problem. *is is required to allow a fair com- while without the dominance rules, there would be 8180 parison with the state-of-the-art algorithms. First of all, as candidate lines. Note that this dominance rule only holds for mentioned above, a passenger-oriented objective function is the uncapacitated line planning problem. If frequencies are used. *erefore, the operator cost is limited by imposing included or when demand elasticity is considered the constraints. *ere is a maximum number of nodes that can method will have to be adjusted. be present in each line, and there is a maximum number of A pool of nondominated lines is now constructed re- lines that can be selected in the line plan. *e shape of the cursively. *e algorithm starts by selecting any node in the bus lines is only limited by not allowing any stop to be visited network and connecting one of its adjacent nodes to con- twice. *is excludes all loops in a given line. *is also means struct a bus line. *is bus line is extended by adding new that, in this paper, the set of feasible lines is not limited or adjacent nodes until it is no longer possible without violating ﬁxed beforehand, as is the case in some other papers (e.g., any constraints (maximum line length or visiting a stop [36, 37, 51, 62]). When a line is selected, it is assumed to be twice). A line that cannot be extended is a possible candidate served in both directions and the bus capacity is assumed to for the optimal solution and is added to the line pool. After Journal of Advanced Transportation 5 23 23 1 1 4 5 45 (a) (b) Figure 2: Example of domination. Line (a) is dominated by line (b). undoing the last extension, a diﬀerent adjacent node can be 4.3. Direct Link Network Representation. To take transfers into account when looking for the passenger routes, the chosen for extending the line. When there are no adjacent nodes left, the previous extension is undone, and so on. *is available infrastructure network needs to be extended to obtain a proper representation. Typically, this is done by process continues until only the starting node remains. If this is repeated for all nodes in the network, a pool of all adding a dummy node for every stop on every bus line. *is possible lines is generated. Since all lines are considered type of extended network is also called the Change and Go bidirectionally, symmetric lines can be eliminated. Network (CNG) [23, 36] or the Train Service Network (TSN) [63]. A disadvantage of this method is that the addition of many extra nodes signiﬁcantly impacts the time required to 4.2. Essential Links. An essential link is a link for which it calculate the passenger routes. can be determined beforehand that it has to be present in the *is paper does not use the TSN or CNG. Rather than optimal solution. All candidate solutions that do not contain adding extra nodes, extra links are added to the network. For all essential links do not have to be evaluated. *is results in any two nodes that are connected by a single bus line, a direct fewer candidate solutions that have to undergo a time- link is added to the network. *e total travel time of the bus consuming evaluation. It will be shown in the experimental between these two nodes is then used as travel time for this results (Section 5) that this makes the algorithm signiﬁcantly link. If multiple bus lines connect the same nodes, only the faster. link with the shortest travel time is kept in the ﬁnal network, Essential links are determined by removing a link from which we call the “Direct Link Network (DLN).” the infrastructure network and then solving the all pair Figure 3 illustrates both the CNG and the DLN on a shortest path problem. *is assumes every passenger will small toy network. Figure 3(a) depicts a small toy network have a direct connection along its shortest path, but the with ﬁve stops and two bus lines: a full black line and a removed link cannot be used. If the total travel time of all dotted orange line. Figure 3(b) is the DLN representation of passengers obtained this way is worse than the best-known the same toy network. *ree extra links have been added to solution, then the removed link is an essential link. In other the network. *e links in the DLN are color coded to make it words, a lower bound is calculated for the network where a clear from which line each link originates. Node one has a certain link is removed. If this lower bound is higher than an direct connection to every other node in the network; hence, upper bound that is already available at the start of the it has a link to every other node in the network. But node one algorithm (the best-known solution), then the removed link can reach node two with a direct connection through each of is essential. Without this link, the optimal solution can never the two lines. Because the connection through the orange be obtained. After repeating this process for all links in the line is shorter, this is the only link that is kept in the rep- network, a list of all essential links is constructed. resentation. Node three and node ﬁve are not connected *is paper uses the optimal solution with one less line as the through a direct connection. *us, passengers traveling best-known solution to determine the essential links. For ex- between these nodes need to use a transfer. In the DLN, this ample, when starting to solve the problem for six lines, the is represented by the absence of a link connecting the two optimal solution for ﬁve lines is used as the best-known solution nodes. If a shortest path is calculated on the DLN, then each to determine the essential links for six lines. Results known from link beyond the ﬁrst that is part of the shortest path also literature or (meta) heuristics could also be used as the best- represents a transfer (and comes with a penalty). Figure 3(c) known solution. If a problem currently has no known solution, is the CNG representation of the toy network. Seven nodes any solution method can be used to get a ﬁrst upper bound. and seven links have been added to the network. Obviously, the better the quality of this upper bound, the more *ere are two ways to consider transfers in this DLN. *e essential links can be identiﬁed. Note that the lower bound can simplest way is to add the transfer penalty to the length of be obtained when not including a link is constant and can be every link in the network. Any method to solve the all pair precalculated. *is means that the list of essential links could be shortest path problem can then be used to calculate the updated on the go every time a new best solution is found. passenger routes. Since every link now contains a transfer However, in our algorithm, the list of essential links is con- penalty, a single transfer penalty has to be subtracted from structed during a precalculation phase. *e presence of each of each shortest path calculated this way. Because of the nature the essential links is also precalculated for each line in the of the DLN, every additional link used beyond the ﬁrst generated line pool. *is makes checking these added con- implies an actual transfer (for which the penalty is indeed straints during optimization very fast. included). However, we decided to incorporate the transfer 6 Journal of Advanced Transportation 23 23 2.1 3.1 1 1 1.1 2 3 45 4 5 1 2.2 1.2 4.2 5.2 4 5 (a) (b) (c) Figure 3: (a) Toy network with 2 bus lines; (b) Direct Link Network representation. (c) Change and Go representation. penalties by slightly modifying Floyd–Warshall’s algorithm, the detour (compared to the shortest possible path) between which is very simple for the DLN representation proposed in A and B is less than a single transfer penalty are considered this paper. Direct connections in the network are repre- as branches in the branch and bound tree. *ese lines are sented by a single link. *erefore, every combination of links then sorted according to their travel time between A and B. implies a transfer. *en, a transfer penalty can simply be Now, two diﬀerent scenarios are possible. One of these lines added to the main operator of the Floyd–Warshall algo- is chosen as part of the solution, each leading to a diﬀerent rithm, which is illustrated in branch (OD1 L1 is part of the solution or OD1 L2 is part of the solution), or none of these lines is chosen, leading to one ∀k, i, j: dist(i, j) � min(dist[i][j], dist[i][k] + dist[k][j] + TP). additional branch (≥TP in Figure 4). (2) When a line is chosen as a part of the solution, this implies that, in the end solution, all passengers traveling Also, in Dijkstra’s algorithm, the transfer penalties can from A to B will travel along this line. *is is due to the fact be incorporated directly. Actually, Dijkstra’s algorithm is that shorter alternatives for traveling from A to B have been most commonly used in line planning research. *is paper, considered in previous branches. For instance, if OD1 L2 is however, uses Floyd–Warshall’s algorithm since it per- selected as part of the solution, OD1 L1 was not selected and formed better in the initial testing. Note that the method the shortest path to travel from A to B in the solution will be used in this paper creates a much more dense network than along OD1 L2. *is path is then called the “optimal path” the traditional CNG network. Floyd–Warshall tends to between A and B in this branch. *is has an important perform better on dense networks, while Dijkstra tends to do implication. If the optimal path is longer than the “shortest better on sparse networks [64, 65]. possible path” between A and B, then the lower bound for It should be noted as well that this DLN representation this branch can be adjusted with the diﬀerence between the can also be used in line planning research using meta- “optimal path” and the “shortest possible path,” multiplied heuristics. It can be especially useful when a high number of with all the demand between A and B. Obviously, if this lines are considered. *is network can also be adjusted easily makes the lower bound of this branch worse than the current to work for line planning with frequencies. Whenever a upper bound available, this branch can be pruned. *ere is single OD pair is connected by multiple bus lines, this would also an eﬀect for all OD pairs lying on the part of the chosen also result in multiple links, one for each bus line available line between A and B. Since they contribute to the “optimal between these nodes, instead of only keeping the shortest path” between A and B, their own “optimal path” cannot one as explained above. improve the connection between A and B. *is is also taken into account in the lower bound. 4.4. Branch and Bound. At the start of the algorithm, the If none of these lines (OD1 L1, OD1 L2, etc.) is chosen, the additional branch is followed and the “optimal path” lower bound corresponds to the ideal situation. *is would mean that every passenger is able to travel from its origin to between A and B in that branch will be at least a single its destination along the shortest possible path in the in- transfer penalty longer than the “shortest possible path.” In frastructure network without any transfers. *en, as illus- Figure 4, this is represented by the branches called “≥TP.” In trated in Figure 4, the branching process starts by selecting the best-case scenario in this branch, traveling from A to B is the ﬁrst OD pair (OD1) from the sorted list of OD pairs. *e possible along the shortest possible path with a single way these OD pairs are sorted is explained in Section 4.5. In transfer. *erefore, in this branch, the lower bound can be the explanation below, we assume that OD1 has node A as adjusted by adding a single transfer penalty multiplied with origin and node B as destination. From the pool of lines, all the total demand betweenA andB. *e new lower bound has to be compared to the current upper bound to decide lines that contain both node A and node B are selected (OD1 L1, OD1 L2, etc.). In this selection, only those lines for which whether to continue along this branch or not. Journal of Advanced Transportation 7 Sorted OD List Start Lines OD1 OD1 OD1 L1 OD2 OD1 L1 OD1 L2 … ≥TP OD1 L2 OD3 OD1 L3 OD2 L2 ≥TP OD2 L1 … Figure 4: *e branching process. In the next branching step, the next OD pair from the transfer penalty compared to the shortest path (plus one sorted list is chosen (OD2) and the process described above extra branch for where the detour is assumed to be at least a is repeated for this OD pair. Again, all lines from the line transfer penalty, and thus, no line is selected). *e complex sort calculates the total amount of branches and divides this pool with an optimal path that deviates less than a single transfer penalty from the minimum possible are selected to by the square root of the demand of the chosen OD pair to branch upon. *e square root is used to add more weight to branch upon, together with one branch were none of these lines is selected. *ere is one diﬀerence however. All can- the number of branches compared to the size of the demand. didate lines with an optimal path for OD1 that is shorter In this way, the algorithm tries to limit the number of top than the chosen one should not be considered again and thus branches, while also maximizing the impact of not selecting cannot be branched upon. Since the lines were sorted by a line. Finally, the second alternative sorting method is a length (or travel time), all previously branched upon lines random sort. *is is used to prove whether the sorting are excluded from being further explored. In Figure 4, this actually has an eﬀect on the performance of the algorithm. means that if one of the possible branches of OD2 is a line already explored in an upper branch, it is immediately 4.6. Summary of the Exact Algorithm. Figure 5 shows an pruned. If, for example, OD1 L2 is the current branch being overview of the entire algorithm in a ﬂowchart, and Algo- explored, then the lines OD1 L1 and OD1 L2 will not be rithm 1 presents the pseudocode. *e algorithm starts with considered as lines for OD2. But OD1 L3 could be a valid some precalculations. *e most important precalculation is candidate since it has not been explored before. the generation of the full set of bus lines that could end up in *is branching process continues either until the new an optimal solution (line 2 in Algorithm 1). All origin- lower bound is worse than the current upper bound, after destination (OD) pairs are sorted by the number of nodes which the branch gets pruned, or until the required number along the shortest path and their demand (line 3 in Algo- of lines is chosen. After suﬃcient lines are chosen, the rithm 1). All the branches for each OD pair are also de- presence of all nodes and essential links is checked. If they termined in this step, so it is checked for each OD pair which are all present, the solution is evaluated and the upper bound lines contain that OD pair and how long the detour is is adjusted if a new best solution is found. compared to the shortest path (line 4 in Algorithm 1). *en, the essential links are calculated (line 5 in Algorithm 1). 4.5. Sorting of OD Pairs. *e order in which OD pairs are Finally, the initial lower bound corresponds to the (probably selected to be branched upon greatly impacts the calculation unfeasible) solution where every passenger travels along the time of the algorithm. *ere are two elements considered shortest possible path in the infrastructure network without when selecting the next OD pair. One is to have as little lines any transfers. Based on all these precalculations, the branch as possible with a detour smaller than the transfer penalty. and bound algorithm can commence (line 6 in Algorithm 1). *e other is to have a large demand for an OD pair. *e ﬁrst *e branch and bound algorithm branches on all pos- limits the number of branches that need to be constructed, sible lines from the line pool that connects a chosen OD pair with a detour smaller than a transfer penalty (line 16 in and the latter increases the lower bound faster, allowing to prune more frequently. In this paper, experimental results Algorithm 1). One additional branch is considered where none of these lines are allowed (line 24 in Algorithm 1). for three diﬀerent sorting methods for OD pairs are dis- cussed. *e main sorting method combines both elements Here, the lower bound is increased with the transfer penalty mentioned earlier in this paragraph by ﬁrst sorting the OD multiplied with the demand of the OD pair (line 25 in pairs by decreasing the number of nodes in between and Algorithm 1). *e OD pair to branch on next is always then breaking any ties by putting the highest demand ﬁrst. selected based on the precalculations. *e branch and bound *is is the main sorting method used in the experiments. It algorithm keeps branching deeper and deeper until the set makes sure that the top levels have as little branches as amount of bus lines is selected (line 7 in Algorithm 1) or possible. until the lower bound of a branch exceeds the current best A ﬁrst alternative sorting method, which will be called solution (line 20 in Algorithm 1). Whenever the set amount of lines is selected, the presence of all nodes and essential “complex sort,” combines the number of branches and the demand in a single variable. *e number of branches is equal links is checked (line 8 in Algorithm 1). If this is the case, the Direct Link Network is constructed and the passenger routes to the amount of lines that contain the OD pair under consideration and that do not make a detour of at least one are determined using the adapted Floyd–Warshall 8 Journal of Advanced Transportation Generate dominant line pool Sort OD Pairs Pre-Calculations List of lines per OD pair Determine Essential Links Branch on OD-pair Select Branch Yes Branch and Bound Max #lines? Evaluate Solution No Update Bounds Update Bounds Close Branch Figure 5: Flowchart of the algorithm. algorithm. *e solution is then evaluated and compared to the results found by heuristics and are, hence, not included in the results. To use the insights gained in this paper for the current upper and lower bound. At the end, all relevant parts of the search space will have been explored and the larger networks, extra adjustments have to be made. *is will optimal solution is determined. be discussed further in Section 6. 5. Results 4.7. Alternative Approaches. *e most obvious approach to ﬁnd an optimal solution is a simple brute force solution. By *is section contains the results of the experiments on selecting all possible combinations of n lines out of the Mandl’s Swiss Network [14]. First Mandl’s Swiss Network is feasible line pool, all feasible solutions can be evaluated. *e introduced. *e actual experiments start with determining, computation time of this approach increases drastically for for the ﬁrst time, the optimal solutions for all available each extra line that can be part of the solution. *is approach instances on the network with the exact algorithm discussed becomes intractable in even very small instances, such as above. *ese solutions are also compared to the solutions Mandl’s network with ﬁve lines. found by state-of-the-art (meta) heuristics. *en, the im- Preliminary experiments on the Mumford0 instance portance of the newly introduced concepts is analyzed: the with 30 nodes and 90 links show that it takes up too much sorting method used, the essential links, the Direct Link memory to calculate all possible lines beforehand. To address Network, and ﬁnally using the entire branch and bound this, the line pool can be generated on the go. After selecting method instead of brute forcing a solution. an OD pair to branch on, the pool of lines with a detour of *e software algorithms were implemented in C++17 less than one transfer penalty can be calculated. Initial ex- and compiled with g++ (GCC) 9.3.0, and Docker was used to periments on larger networks and with a higher maximum setup a standalone image to run the experiments. *e number of lines took too much time to even come close to Docker containers were executed on a dedicated virtual Journal of Advanced Transportation 9 (1) Precalculations: (2) Generate pool of dominant lines (Section 4.1); (3) Sort OD Pairs (Section 4.5); (4) Construct list of lines per OD pair (Section 4.5); (5) Determine Essential Links (Section 4.2); (6) Recursive Branch and Bound (Section 4.4): (7) If (size of Current_Line_Plan equals maximum) (8) If (All essential links and nodes are present in Current_Line_Plan) (9) If (TTT<UpperBound) (10) Optimal_Line_Plan⟵Current_Line_Plan; (11) UpperBound⟵ TTT; (12) end (13) end (14) Else (15) Select OD pair (16) For (All lines serving the OD pair with a detour less than TP) (17) Next_Line_Plan⟵Current_Line_Plan; (18) Add line to the Next_Line_Plan; (19) Next_LowerBound⟵LowerBound + detour ∗ Demand of OD pair; (20) IF (Next_LowerBound<UpperBound) (21) Go one step deeper in the branch and bound with Next_Line_Plan (22) end (23) end (24) Go one step deeper in branch and bound without selecting a line (deviation> TP) (25) LowerBound⟵LowerBound + TP ∗ Demand of OD pair; (26) end (27) Return Optimal_Line_Plan ALGORITHM 1: Pseudocode of the algorithm. machine, running CoreOS, to minimize context switching 5.2. Exact Solution Algorithm. *e algorithm used here and external interference. *e virtual machine ran on works entirely as described in Section 4.6. Table 1 shows the Intel(R) Xeon(R) CPU E5-2640 v4@2.40 GHz hardware and optimal solutions with respect to Total Travel Time for all was granted 4 dedicated CPU cores (8 Hyper*reads) and instances (Number of Lines) with at most eight nodes per 16 GB dedicated RAM. *e C++ implementation exploits line. *e third column shows the Total Travel Time (TTT) the multicore setup by parallelizing the execution on several and the fourth column the Average Travel Time (ATT) in worker threads. All the data of the experiments and the minutes, which is the TTT per trip. *e ﬁfth column shows the CPU time of the algorithm, in seconds. *is CPU time is instances is available at https://www.mech.kuleuven.be/en/ cib/lp/mainpage#section-12. the time the algorithm spent in the branch and bound procedure. *e entire precalculations require close to 40 seconds (of which nearly all time is spent preparing the 5.1.Mandl’sSwissNetwork. Mandl’s Swiss Network (Figure 1) branches), but does not have to be repeated for every in- is a small network with 15 nodes and 21 links originally used in stance. *e ﬁnal column shows the number of solutions that Mandl [14]. In total, there are 15570 trips in the network and the were evaluated, in millions. Table 2 shows the actual lines demand is symmetric. It is one of the only publicly available that make up the optimal line plan for each instance. As a datasets for the line planning problem and, therefore, the most visual example, Figure 6 represents the optimal solution for used benchmark instance. *e most commonly used param- the instance with four lines and at most eight stops per line. eters for this network are used in this work. *e most common Note that every line passes by node ten. *is is expected since limiter of line length is by limiting the number of nodes per line this node alone is responsible for more or less one quarter of to eight. In this paper, both a maximum of eight nodes per line all the demand in the network. and an unlimited number of nodes per line are used. *ese *e CPU time keeps increasing until nine lines. Every “unlimited” lines will then be limited by the fact that nodes can extra line increases the amount of possible solutions dras- only be included once in a single line. For ﬁnding the optimal tically. *erefore, it is expected that more lines result in solutions, the results are shown for all instances with a relevant longer CPU times and more solutions checked. But, at ten amount of lines. When the newly introduced concepts are lines, the search time decreases again. *ere are several analyzed, only a subset of instances will be used to somewhat eﬀects at play here. More lines make it easier for the al- limit the required calculation time. *e transfer penalty is set to gorithm to get good upper bounds quickly because more ﬁve minutes, and this value is used in most publications lines result in better solutions. *ese upper bounds will also [4, 7, 8, 10]. be closer to the absolute lower bound. Both of these things 10 Journal of Advanced Transportation Table 1: Optimal solutions for Mandl’s Swiss network with at most 8 nodes per line. Number of lines Max nodes per line TTT (min) ATT (min) CPU time (s) Solutions checked (M) 3 169450 10.88 13 <1 4 163210 10.48 297 15 5 160450 10.31 3317 466 6 158500 10.18 18856 3785 7 157260 10.10 39753 6737 8 156750 10.07 161902 25704 9 156300 10.04 200872 24934 10 155940 10.02 29296 4032 11 155850 10.01 17054 1369 12 155820 10.01 16320 1044 13 155800 10.01 24883 1634 14 155790 10.01 1019 177 Table 2: Optimal solutions for 3 to 14 lines with at most 8 nodes per line. 3 lines 4 lines 5 lines Line 1: 1, 2, 3, 6, 8, 10, 11, and 13 Line 1: 1, 2, 3, 6, 8, 10, 11, and 12 Line 1: 1, 2, 3, 6, 8, 10, 14, and 13 Line 2: 5, 4, 6, 15, 7, 10, 14, and 13 Line 2: 1, 2, 5, 4, 6, 8, 10, and 11 Line 2: 1, 2, 3, 6, 15, 7, 10, and 11 Line 3: 2, 4, 12, 11, 10, 8, 15, and 9 Line 3: 10, 7, 15, 6, 3, 2, 4, and 12 Line 3: 1, 2, 5, 4, 6, 8, 10, and 11 Line 4: 9, 15, 8, 10, 14, 13, 11, and 12 Line 4: 2, 4, 12, 11, 10, 7, 15, and 9 Line 5: 7, 15, 8, 6, 4, 12, 11, and 13 6 lines 7 lines 8 lines Line 1: 1, 2, 3, 6, 8, 10, 11, and 13 Line 1: 1, 2, 3, 6, 8, 10, 11, and 13 Line 1: 1, 2, 3, 6, 8, 10, 11, and 13 Line 2: 1, 2, 3, 6, 15, 7, 10, and 14 Line 2: 1, 2, 3, 6, 15, 7, 10, and 14 Line 2: 1, 2, 3, 6, 15, 7, 10, and 14 Line 3: 1, 2, 5, 4, 6, 8, 10, and 14 Line 3: 1, 2, 5, 4, 6, 8, 10, and 14 Line 3: 1, 2, 5, 4, 6, 8, 10, and 14 Line 4: 3, 2, 5, 4, 6, 8, 15, and 7 Line 4: 3, 2, 5, 4, 6, 8, 15, and 7 Line 4: 3, 2, 5, 4, 6, 15, 7, and 10 Line 5: 5, 4, 12, 11, 10, 7, 15, and 9 Line 5: 5, 4, 12, 11, 10, 7, 15, and 9 Line 5: 5, 4, 12, 11, 10, 7, 15, and 9 Line 6: 1, 2, 4, 12, 11, 13, 14, and 10 Line 6: 9, 15, 6, 3, 2, 4, 12, and 11 Line 6: 9, 15, 6, 3, 2, 4, 12, and 11 Line 7: 1, 2, 4, 12, 11, 13, 14, and 10 Line 7: 1, 2, 4, 12, 11, 13, 14, and 10 Line 8: 12, 4, 6, 8, 15, 7, 10, and 13 9 lines 10 lines 11 lines Line 1: 1, 2, 3, 6, 8, 10, 11, and 13 Line 1: 1, 2, 3, 6, 8, 10, 11, and 13 Line 1: 1, 2, 3, 6, 8, 10, 11, and 13 Line 2: 1, 2, 3, 6, 15, 7, 10, and 13 Line 2: 1, 2, 3, 6, 15, 7, 10, and 8 Line 2: 1, 2, 3, 6, 15, 7, 10, and 8 Line 3: 1, 2, 3, 6, 15, and 9 Line 3: 1, 2, 3, 6, 15, and 9 Line 3: 1, 2, 3, 6, 15, and 9 Line 4: 12, 4, 2, 3, 6, 8, 10, and 14 Line 4: 12, 4, 2, 3, 6, 8, 10, and 14 Line 4: 12, 4, 2, 3, 6, 8, 10, and 14 Line 5: 1, 2, 5, 4, 6, 8, 10, and 11 Line 5: 5, 4, 6, 8, 10, 11, 13, and 14 Line 5: 5, 4, 6, 8, 10, 11, 13, and 14 Line 6: 3, 2, 5, 4, 6, 8, 15, and 7 Line 6: 1, 2, 5, 4, 6, 15, 7, and 10 Line 6: 1, 2, 5, 4, 6, 15, 7, and 10 Line 7: 5, 4, 12, 11, 10, 7, 15, and 9 Line 7: 5, 4, 12, 11, 10, 7, 15, and 9 Line 7: 5, 4, 12, 11, 10, 7, 15, and 9 Line 8: 9, 15, 8, 6, 4, 12, 11, and 10 Line 8: 3, 2, 5, 4, 6, 8, 15, and 9 Line 8: 3, 2, 5, 4, 6, 15, and 9 Line 9: 1, 2, 4, 12, 11, 13, 14, and 10 Line 9: 12, 4, 6, 8, 15, 7, 10, and 14 Line 9: 9, 15, 8, 6, 4, 12, 11, and 10 Line 10: 1, 2, 4, 12, 11, 13, 10, and 7 Line 10: 12, 4, 6, 8, 15, 7, 10, and 14 Line 11: 1, 2, 4, 12, 11, 13, 10, and 7 12 lines 13 lines 14 lines Line 1: 1, 2, 3, 6, 8, 10, 11, and 13 Line 1: 1, 2, 3, 6, 8, 10, 11, and 13 Line 1: 1, 2, 3, 6, 8, 10, 11, and 13 Line 2: 1, 2, 3, 6, 15, 7, 10, and 8 Line 2: 1, 2, 3, 6, 15, 7, 10, and 8 Line 2: 1, 2, 3, 6, 15, 7, 10, and 8 Line 3: 1, 2, 3, 6, 15, and 9 Line 3: 1, 2, 3, 6, 15, and 9 Line 3: 1, 2, 3, 6, 15, and 9 Line 4: 1, 2, 3, 6, 8, 10, 14, and 13 Line 4: 1, 2, 3, 6, 8, 10, 14, and 13 Line 4: 1, 2, 3, 6, 8, 10, 14, and 13 Line 5: 2, 5, 4, 6, 8, 10, 11, and 13 Line 5: 1, 2, 5, 4, 6, 8, 10, and 11 Line 5: 1, 2, 5, 4, 6, 8, 10, and 11 Line 6: 1, 2, 5, 4, 6, 15, 7, and 10 Line 6: 1, 2, 5, 4, 6, 15, 7, and 10 Line 6: 1, 2, 5, 4, 6, 15, 7, and 10 Line 7: 5, 4, 12, 11, 10, 7, 15, and 9 Line 7: 5, 4, 12, 11, 10, 7, 15, and 9 Line 7: 5, 4, 12, 11, 10, 7, 15, and 9 Line 8: 3, 2, 5, 4, 6, 15, and 9 Line 8: 3, 2, 5, 4, 6, 15, and 9 Line 8: 3, 2, 5, 4, 6, 15, and 9 Line 9: 9, 15, 8, 6, 4, 12, 11, and 10 Line 9: 9, 15, 8, 6, 4, 12, 11, and 10 Line 9: 7, 10, 11, 12, 4, 6, 15, and 9 Line 10: 12, 4, 6, 15, 7, 10, 14, and 13 Line 10: 2, 5, 4, 6, 8, 10, 14, and 13 Line 10: 2, 5, 4, 6, 8, 10, 14, and 13 Line 11: 1, 2, 4, 12, 11, 13, 14, and 10 Line 11: 1, 2, 4, 12, 11, 13, 14, and 10 Line 11: 1, 2, 4, 12, 11, 13, 10, and 7 Line 12: 7, 15, 8, 6, 3, 2, 4, and 12 Line 12: 7, 15, 8, 6, 3, 2, 4, and 12 Line 12: 9, 15, 8, 6, 3, 2, 4, and 12 Line 13: 12, 4, 6, 15, 7, 10, 14, and 13 Line 13: 7, 15, 6, 4, 12, 11, 13, and 14 Line 14: 12, 4, 6, 8, 15, 7, 10, and 14 Journal of Advanced Transportation 11 Tables 5 and 6 contain the comparison with the results from literature for a limit of eight and fourteen nodes per line, respectively. In Table 5, the results from our exact algorithm are compared with the results available from [4, 7, 14, 30]. To the best of our knowledge, these papers have the best ATT for this same problem in literature. In Table 6, the results obtained by [8, 10] are chosen since they solve 4 instances with a limit of fourteen nodes per line. *ese 15 papers report results for instances with four, six, seven, and eight lines. Obviously, the CPU times for the metaheuristics in Table 6 are much lower than for our exact algorithm. Unfortunately, no CPU times are reported for the two approaches reported in Table 6. Only for four lines with a maximum of eight nodes per line does one of the algorithms from literature ﬁnd the optimal solution (without knowing it is optimal), but in general, they ﬁnd a solution close to the optimal. However, the question can be raised whether it is useful to keep competing for ﬁnding better metaheuristic solutions on this small network. On this small network, many complex and computationally expensive methods (such as our exact al- gorithm) can be used to ﬁnd very good solutions. However, Figure 6: Optimal solution on Mandl’s Swiss network with 4 lines this does not mean the proposed method would work on a with at most 8 nodes each. realistic network. Hence, researchers should be careful when basing their conclusions only on results found for Mandl’s result in faster pruning in the branch and bound algorithm. Swiss Network. When lines get longer or when there are simply more lines, lower and upper bounds will be updated more often. *is results in even more pruning and is an important con- 5.3. Sorting. In this and following sections, the importance of diﬀerent parts of the algorithm is evaluated. In Section 4.5, tributor to keeping the CPU times under control for the higher amount of lines. Another added beneﬁt of having three diﬀerent sorting methods are introduced. All previous experiments were executed with the main sorting method. In more lines is that the likelihood of not selecting any of the lines at a branch decreases, which results in having to travel this section, it is analyzed what the impact is of sorting. *e main sorting method is compared with the complex sort and less deep to get to a solution. For example, when the absolute the random sort. *e sorting itself takes much less than a lower bound is achievable, this means that every passenger second (sorting 120 OD pairs based on a criteria). Based on can travel along its shortest path without transfers. *ere- the results from the previous section, only instances with fore, no matter which OD pair gets chosen to branch upon, four, eight, and twelve lines are considered in order to there will always be a line selected from the pool. For the somewhat limit the required total CPU time. With this instances with eight nodes per line, this happens at fourteen lines. *is is thus the minimum amount of lines necessary to subset, both a very small number of lines as well as a large number of lines are being tested. Eight lines is also chosen reach the lower bound with at most eight nodes per line. Table 3 shows the optimal solutions for the instances because it took a very long time to ﬁnd the optimal solution in the earlier experiments. *e random sort is executed three without any restriction on the number of nodes per line, times, and the best results are reported here. *e CPU time which results in a maximum of fourteen nodes per line in per instance (and per execution) is limited to 24 hours. *e this network. *e analysis is the same as for the previous results are presented in Table 7. results. Because of the longer lines, the quality of the A ﬁrst obvious conclusion is that the main sort used in solution is obviously better than with shorter lines. With the algorithm performs much better than random sort. thirteen lines, every passenger has a direct connection along its shortest possible path. *e actual details of the *ese results make it clear that sorting your OD pairs is very important in our branch and bound algorithm. *e complex optimal solutions can be found in Table 4. To the best of our knowledge, only the result for three lines were sort performs very good when considering a small amount of lines. With four lines, it is signiﬁcantly faster than the main previously known and published in Fiss and Ritt [66]. *eir Mixed Integer Programming implementation re- variant for both line lengths. For eight lines with at most eight nodes, it is a lot faster than the main sort. While with quired 78992 seconds (on their system) to ﬁnd the op- unlimited line lengths, the results are almost equal. For timal solution with three lines, and they calculated it twelve lines, the complex sort can never ﬁnish within a day. would take them 77 days to ﬁnd the optimal solution for Hence, it needs a lot more time than the main sort for high four lines. In comparison, our algorithm only needs 15 line numbers. *e main sort is chosen in our algorithm seconds (for three lines) and 168 seconds (for four lines) because it can ﬁnd the optimal solution for all instances. to ﬁnd the optimal solution. 12 Journal of Advanced Transportation Table 3: Optimal solutions for Mandl’s Swiss network with no limitations to route length. Number of lines Max nodes per line TTT (min) ATT (min) CPU time (s) Solutions checked (M) 3 163430 10.50 15 2 4 159990 10.28 168 44 5 158220 10.16 1641 433 6 157040 10.09 5357 1429 7 156550 10.05 33898 8002 8 14 156090 10.03 20359 5013 9 155930 10.01 14376 1829 10 155840 10.01 8002 816 11 155820 10.01 17590 1025 12 155800 10.01 46067 4036 13 155790 10.01 849 206 Table 4: Optimal solutions for 3 to 13 lines with at most 14 nodes per line. 3 lines 4 lines 5 lines Line 0: 1, 2, 3, 6, 8, 10, 14, 13, 11, 12, 4, and 5 Line 0: 1, 2, 3, 6, 8, 10, 11, 13, and 14 Line 0: 1, 2, 3, 6, 8, 10, 11, 13, and 14 Line 1: 1, 2, 5, 4, 6, 8, 10, 7, 15, and 9 Line 1: 1, 2, 4, 6, 8, 15, 7, 10, 11, and 12 Line 1: 1, 2, 4, 6, 8, 15, 7, 10, 14, 13, 11, and 12 Line 2: 5, 4, 2, 3, 6, 15, 7, 10, 11, and 12 Line 2: 13, 11, 12, 4, 5, 2, 3, 6, 15, 7, 10, and 14 Line 2: 13, 11, 12, 4, 5, 2, 3, 6, 15, 7, 10, and 14 Line 3: 1, 2, 5, 4, 6, 8, 10, 7, 15, and 9 Line 3: 1, 2, 5, 4, 6, 8, 10, 7, 15, and 9 Line 4: 7, 10, 11, 12, 4, 2, 3, 6, 15, and 9 6 lines 7 lines 8 lines Line 0: 1, 2, 3, 6, 8, 10, 11, 13, and 14 Line 0: 1, 2, 3, 6, 8, 10, 11, 13, and 14 Line 0: 1, 2, 3, 6, 8, 10, 11, 13, and 14 Line 1: 1, 2, 3, 6, 15, 7, 10, 14, 13, 11, 12, 4, and Line 1: 1, 2, 3, 6, 15, 7, 10, 14, 13, 11, 12, 4, and Line 1: 1, 2, 3, 6, 15, 7, 10, 14, 13, 11, 12, 4, and 5 5 5 Line 2: 1, 2, 5, 4, 6, 8, 10, 11, 13, and 14 Line 2: 1, 2, 5, 4, 6, 8, 10, 11, 13, and 14 Line 2: 1, 2, 3, 6, 15, and 9 Line 3: 3, 2, 5, 4, 6, 8, 15, 7, 10, 14, 13, 11, and Line 3: 3, 2, 5, 4, 6, 8, 15, 7, 10, 14, 13, 11, and Line 3: 1, 2, 5, 4, 6, 8, 10, 11, 13, and 14 12 12 Line 4: 1, 2, 4, 12, 11, 10, 7, 15, and 9 Line 4: 1, 2, 4, 12, 11, 10, 7, 15, and 9 Line 4: 3, 2, 5, 4, 6, 15, 7, 10, 14, 13, 11, and 12 Line 5: 8, 10, 14, 13, 11, 12, 4, 2, 3, 6, 15, and 9 Line 5: 8, 10, 14, 13, 11, 12, 4, 2, 3, 6, 15, and 9 Line 5: 1, 2, 4, 12, 11, 10, 7, 15, and 9 Line 6: 7, 10, 11, 12, 4, 6, 8, 15, and 9 Line 6: 7, 10, 11, 12, 4, 6, 8, 15, and 9 Line 7: 6, 3, 2, 4, 12, 11, 13, 14, 10, 8, 15, and 7 9 lines 10 lines 11 lines Line 0: 1, 2, 3, 6, 8, 10, 11, 13, and 14 Line 0: 1, 2, 3, 6, 8, 10, 11, 13, and 14 Line 0: 1, 2, 3, 6, 8, 10, 11, 13, and 14 Line 1: 1, 2, 3, 6, 15, 7, 10, 11, 12, 4, and 5 Line 1: 1, 2, 3, 6, 15, 7, 10, 11, 12, 4, and 5 Line 1: 1, 2, 3, 6, 15, 7, 10, and 8 Line 2: 1, 2, 3, 6, 15, and 9 Line 2: 1, 2, 3, 6, 15, and 9 Line 2: 1, 2, 3, 6, 15, and 9 Line 3: 11, 12, 4, 2, 3, 6, 8, 10, 14, and 13 Line 3: 11, 12, 4, 2, 3, 6, 8, 10, 14, and 13 Line 3: 1, 2, 3, 6, 8, 10, 14, 13, 11, 12, 4, and 5 Line 4: 1, 2, 5, 4, 6, 8, 10, 11, 13, and 14 Line 4: 1, 2, 5, 4, 6, 8, 10, 11, 13, and 14 Line 4: 1, 2, 5, 4, 6, 8, 10, 11, 13, and 14 Line 5: 1, 2, 5, 4, 6, 15, 7, 10, 14, 13, 11, and 12 Line 5: 1, 2, 5, 4, 6, 15, 7, 10, 14, 13, 11, and 12 Line 5: 1, 2, 5, 4, 6, 15, 7, 10, and 8 Line 6: 1, 2, 4, 12, 11, 10, 7, 15, and 9 Line 6: 1, 2, 4, 12, 11, 10, 7, 15, and 9 Line 6: 1, 2, 4, 12, 11, 10, 7, 15, and 9 Line 7: 3, 2, 5, 4, 6, 8, 15, and 9 Line 7: 3, 2, 5, 4, 6, 15, and 9 Line 7: 3, 2, 5, 4, 6, 15, and 9 Line 8: 7, 15, 8, 6, 4, 12, 11, 10, 13, and 14 Line 8: 7, 10, 11, 12, 4, 6, 8, 15, and 9 Line 8: 7, 10, 11, 12, 4, 6, 8, 15, and 9 Line 9: 7, 15, 8, 6, 4, 12, 11, 10, 13, and 14 Line 9: 11, 12, 4, 6, 15, 7, 10, 14, and 13 Line 10: 3, 2, 4, 12, 11, 10, 7, 15, 8, and 6 12 lines 13 lines Line 0: 1, 2, 3, 6, 8, 10, 11, 13, and 14 Line 0: 1, 2, 3, 6, 8, 10, 11, 13, and 14 Line 1: 1, 2, 3, 6, 15, 7, 10, and 8 Line 1: 1, 2, 3, 6, 15, 7, 10, and 8 Line 2: 1, 2, 3, 6, 15, and 9 Line 2: 1, 2, 3, 6, 15, and 9 Line 3: 1, 2, 3, 6, 8, 10, 14, 13, 11, 12, 4, and 5 Line 3: 1, 2, 3, 6, 8, 10, 14, 13, 11, 12, 4, and 5 Line 4: 1, 2, 5, 4, 6, 8, 10, 11, and 12 Line 4: 1, 2, 5, 4, 6, 8, 10, 11, and 12 Line 5: 1, 2, 5, 4, 6, 15, 7, 10, and 8 Line 5: 1, 2, 5, 4, 6, 15, 7, 10, and 8 Line 6: 1, 2, 4, 12, 11, 10, 7, 15, and 9 Line 6: 1, 2, 4, 12, 11, 10, 7, 15, and 9 Line 7: 1, 2, 5, 4, 6, 15, and 9 Line 7: 1, 2, 5, 4, 6, 15, and 9 Line 8: 7, 10, 11, 12, 4, 6, 8, 15, and 9 Line 8: 7, 10, 11, 12, 4, 6, 15, and 9 Line 9: 3, 2, 5, 4, 6, 8, 10, 14, 13, 11, and 12 Line 9: 3, 2, 5, 4, 6, 8, 10, 14, 13, 11, and 12 Line 10: 3, 2, 4, 12, 11, 10, 7, 15, 8, and 6 Line 10: 3, 2, 4, 12, 11, 10, 7, 15, 8, and 6 Line 11: 11, 12, 4, 6, 15, 7, 10, 14, and 13 Line 11: 7, 15, 6, 4, 12, 11, 10, and 8 Line 12: 7, 10, 14, 13, 11, 12, 4, 6, 8, 15, and 9 Journal of Advanced Transportation 13 Table 5: Comparison of the exact algorithm with literature for at most 8 nodes per line. Number of lines Max nodes per line Paper ATT (min) CPU time (s) Exact 10.48 297 Ahmed 10.48 450 4 Islam 10.51 7 Mandl 12.90 NA Chakroborty 11.90 NA Exact 10.18 18856 Ahmed 10.18 450 Islam 10.18 7 8 Chakroborty 10.48 NA Exact 10.10 39753 Ahmed 10.10 450 Islam 10.12 8 Chakroborty 10.42 NA Exact 10.07 161902 Ahmed 10.08 450 Islam 10.07 8 Chakroborty 10.36 NA Table 6: Comparison of the exact algorithm with literature for unlimited line length. Number of lines Max nodes per line Paper ATT (min) CPU time (s) Exact 10.28 168 4 Nayeem 10.33 NA Wu 10.35 NA Exact 10.09 5357 6 Nayeem 10.10 NA Wu 10.10 NA Exact 10.05 33898 7 Nayeem 10.07 NA Wu 10.07 NA Exact 10.03 20359 8 Nayeem — NA Wu 10.04 NA It can be explained why the complex sort is a lot faster impact on the lower bound. Hence, it is logical that the for instances with fewer lines, but it is unable to ﬁnd an complex sort performs better for smaller number of lines. optimal solution for larger number of lines. *e complex sort selects OD pairs with a low number of branches but a high demand. *is results in slightly more branches than 5.4.EssentialLinks. In this experiment, the algorithm is also executed without using the concept of essential links. Table 8 with the main sort, but the average quality of each branch shows the results for instances with four, eight, and twelve is higher. *e high demand means that when a detour lines. It is immediately clear that essential links result in a lot from the shortest path is selected, the lower bounds will less solutions that need to be evaluated. It is expected that increase more. *is results in more pruning. However, as the amount of lines rises, the number of branches be- this impact would be larger for fewer lines and fewer nodes per line. When there are fewer lines (and/or nodes) in a line comes more important for the total CPU time. Re- member, from Section 5.2, that it is more likely a line will plan, then there are fewer edges being served in that line plan. *erefore, it is more likely that certain essential links be selected at each branching point when more lines can be selected in the solution. So, for the early lines, more will not be present in solutions that are evaluated. For up to eight nodes per line, the experiments conﬁrm this trend. For branches will just result in exponentially more branches in the lower depths. At the same time, more and more the unlimited line length, this trend is less pronounced. For eight lines, there is only a minimal impact, while for twelve (almost all) of the high OD streams will be covered by the lines, the eﬀect is much larger. In all cases, the essential links actual shortest path. *is means there is almost no beneﬁt to selecting these lines ﬁrst since there will be no extra do provide a clear positive eﬀect on the CPU time, and for 14 Journal of Advanced Transportation Table 7: Impact of diﬀerent sorting methods on the exact algorithm. Number of lines Max nodes per line Sorting method CPU time (s) Solutions checked (M) Main 297 15 4 Complex 37 5 Random 1666 57 Main 161902 25704 8 8 Complex 6812 983 Random >86400 >9518 Main 16320 1044 12 Complex >86400 >524 Random >86400 >11685 Main 168 44 4 Complex 75 25 Random 3822 701 Main 20359 5013 8 14 Complex 21992 4164 Random >86400 >18 Main 46067 4036 12 Complex >86400 >10418 Random >86400 >4867 Table 8: *e impact of essential links on the exact algorithm. low line numbers, the CPU time is reduced by 35–45%. We could not measure a signiﬁcant diﬀerence between the Number of Max nodes Essential CPU Solutions precalculation times with or without essential links. lines per line links time (s) checked (M) Yes 297 15 No 545 188 5.5. Direct Link Network. In this section, the algorithm is run Yes 161902 25704 8 8 with the Change and Go (CNG) network representation instead No 225915 50765 of the proposed Direct Link Network (DLN) representation. Yes 16320 1044 Both algorithms still use the Floyd–Warshall algorithm to No 19067 2238 evaluate a solution. Table 9 shows the results for four, eight, and Yes 168 44 twelve lines. All tests ran for up to 24 hours. *e number of No 256 106 Yes 20359 5013 solutions checked is of course identical. Both methods use the 8 14 No 22975 6180 exact same sorting, so the same solutions have to be evaluated. Yes 46067 4036 *e impact of the DLN is signiﬁcant and appears to increase No 60121 7206 when the number of lines or nodes per line increases. *is behavior is expected, even for four lines with eight nodes per line, the CPU time increases with a factor four, and for ﬁfteen 5.6.BruteForce. We also developed a brute force algorithm to nodes per line, almost with a factor twenty. *e CNG creates an calculate the optimal solution. *e brute force algorithm simply extra node for each stop on each bus line, so more bus lines tries all combinations of nondominated lines and keeps the best result in more nodes in the CNG, while the DLN adds new links combination. *e algorithm started with three lines with eight and only keeps the best links. *erefore, the impact of more and nodes per line and kept increasing its number of lines until it larger lines is more limited. Note that, in literature, Dijkstra’s needs more than 24 h to reach the optimal solution. *e results algorithm is used more often than Floyd–Warshall, but we are shown in Table 10. For only three lines, the CPU time is choose to compare with our Floyd–Warshall algorithm to keep similar to that of our branch and bound algorithm. But, for four the comparison as fair as possible. Nevertheless, some pre- lines, it already needs more than ten times as much time. Since liminary tests comparing CNG with DLN using Dijkstra’s al- the brute force algorithms require exponentially more time gorithm gave the same type of results as shown here. Obviously, when adding an extra line, the algorithm was terminated after it this does not mean that DLN will always perform better than ran for 24 hours for ﬁve lines. *us, ﬁnding the optimal solution CNG, but these results deﬁnitely warrant further research into for ﬁve lines or more with this brute force algorithm is not using the DLN in line planning. It can be concluded that the possible in a reasonable amount of time. We conclude that the DLN representation is a major contributor to the speed of our branch and bound is eﬀective in limiting the amounts of so- algorithm. lutions that have to be evaluated. Journal of Advanced Transportation 15 Table 9: Impact of the Direct Link Network instead of the Change and Go network representation on the exact algorithm. Number of lines Max nodes per line Network model CPU time (s) Solutions checked (M) DLN 297 15 CNG 1377 15 DLN 161902 25704 8 8 CNG >86400 >125 DLN 16320 1044 CNG >86400 >69 DLN 168 44 CNG 3277 44 DLN 20359 5013 8 14 CNG >86400 >165 DLN 46067 4036 CNG >86400 >29 Table 10: Comparison of the branch and bound with a brute force algorithm. Number of lines Max nodes per line Method CPU time (s) Solutions checked (M) Branch and bound 13 <1 Brute force 27 16 Branch and bound 297 15 4 8 Brute force 3213 1957 Branch and bound 3317 466 Brute force >86400 >50605 (5.3). *ree diﬀerent sorting principles are implemented and 6. Conclusion tested, and the best way of sorting is implemented in our In this paper, a novel branch and bound algorithm is de- algorithm. By choosing OD pairs with minimal branches but veloped to ﬁnd optimal solutions for the uncapacitated line with high demand ﬁrst, the CPU time is signiﬁcantly planning problem with integrated passenger routing. *e decreased. objective is to minimize the total travel time of all pas- *e newly-developed Direct Link Network representa- sengers, and the available resources are constrained by the tion also reduces the required CPU time. For our algorithm, number of lines that can be operated and the maximum the gains were signiﬁcant compared to the traditionally used length of those lines. *e algorithm is applied to Mandl’s Change and Go Network (5.5). *ese results deﬁnitely Swiss network [14]. *is is by far the most used benchmark warrant to consider the use of the DLN in metaheuristics and instance in line planning research. However, until now, no other networks as well as looking into adjusting the DLN to optimal solutions have been determined for instances with deal with frequencies. Finally, in order to decide to actually more than three lines on this network. Our algorithm ob- evaluate a solution or not, the concept of essential links is tains optimal solutions for all available instances on this used. A link is essential if it has to be present in the optimal network, i.e., diﬀerent number of lines and two diﬀerent line solution. Essential links can be determined beforehand (4.2). *is had a positive eﬀect on the CPU time and was even lengths: for inﬁnite line length and for at most eight nodes per line (5.2). Furthermore, the minimum number of lines more pronounced for smaller line numbers and line lengths necessary to reach the lower bound, where every passenger (5.4). travels along his/her shortest path without transfers, is Finally, by calculating all optimal solutions for the determined. uncapacitated line planning problem, we hope to show that *e success of the algorithm is due to a number of new it is no longer useful to play the “up-the-wall” game [67] of concepts and the way the algorithm is constructed. *is is trying to beat the best algorithms on this small benchmark illustrated by the experimental results. By deﬁning all ex- network. Actually, it can be concluded that several state-of- tendable lines as dominated, the size of the pool of feasible the-art solution approaches for the uncapacitated line lines can already be greatly diminished (4.1). *e branch and planning problem obtain near-optimal solutions for almost bound algorithm itself chooses an OD pair to branch on. all these instances. *erefore, aiming to further improve Every feasible line connecting this OD pair directly with a these solutions for these instances is not very useful any- detour of less than one transfer penalty is considered as a more, but the focus should shift to making the approaches branch as well as not selecting a line. When a line is selected, much faster and, more importantly, to develop new concepts this line is assumed to provide the optimal routing for the for solving larger and more realistic instances. OD pair that was branched upon. *is creates many bounds According to us, our algorithm can be adapted to search for the OD pair and all possible pairs on the route. *e order for optimal solutions for other variants of the line planning in which the OD pairs are chosen is also very important problem, for example, by including frequencies as well. With 16 Journal of Advanced Transportation Swarm and Evolutionary Computation, vol. 46, pp. 154–170, frequencies, the line pool generation method has to be adjusted because the dominance criterion is no longer true. [8] M. A. Nayeem, M. K. Rahman, and M. S. Rahman, “Transit *us, there will be a larger line pool. *e passenger route network design by genetic algorithm with elitism,” Trans- choice will no longer be a shortest path problem, probably portation Research Part C: Emerging Technologies, vol. 46, resulting in having more links in the DLN. Both of these pp. 30–45, 2014. things will increase the required CPU time. Another pos- [9] M. Nikolic ´ and D. Teodorovic, ´ “Transit network design by bee sibility is to apply the method to larger networks. Although colony optimization,” Expert Systems with Applications, several problems occurred during preliminary testing with vol. 40, no. 15, pp. 5945–5955, 2013. an exact algorithm, it should be possible to integrate at least [10] R. Wu and S. Wang, “Discrete wolf pack search algorithm parts of this algorithm in an eﬃcient (meta) heuristic. If based transit network design,” in Proceedings of the 2016 7th instances are considered with a limited pool of lines gen- IEEE International Conference on Software Engineering and erated beforehand, our algorithm should be able to address Service Science, pp. 509–512, Beijing, China, 2016. larger instances as well. Furthermore, we are convinced that [11] M. H. Baaj and H. S. Mahmassani, “An AI-based approach for both the essential links and the DLN representation can be transit route system planning and design,” Journal of Ad- useful for developing exact and (meta) heuristic algorithms vanced Transportation, vol. 25, no. 2, pp. 187–209, 1991. for this or other line planning problems. For instance, the [12] L. Fan and C. L. Mumford, “A metaheuristic approach to the urban transit routing problem,” Journal of Heuristics, vol. 16, DLN representation could be tested in a metaheuristic no. 3, pp. 353–372, 2010. setting. Or the essential links could also be updated “dy- [13] P. N. Kechagiopoulos and G. N. Beligiannis, “Solving the namically,” i.e., every time the lower bound is increased. urban transit routing problem using a particle swarm opti- mization based algorithm,” Applied Soft Computing, vol. 21, Data Availability pp. 654–676, 2014. [14] C. E. Mandl, Applied Network Optimization, Academic Press, All the data of the experiments and the instances is available London, UK, 1979. at the following location: https://www.mech.kuleuven.be/ [15] C. Mumford, “Data and results 2013,” 2013, http://users.cs.cf. en/cib/lp/mainpage#section-12. ac.uk/C.L.Mumford/Research%20Topics/UTRP/ CEC2013Supp/. Conflicts of Interest [16] I. M. Cooper, M. P. John, R. Lewis, C. L. Mumford, and A. Olden, “Optimising large scale public transport network *e authors declare that there are no conﬂicts of interest design problems using mixed-mode parallel multi-objective regarding the publication of this paper. evolutionary algorithms,” in Proceedings of the 2014 IEEE Congress on Evolutionary Computation (CEC), pp. 2841–2848, Acknowledgments IEEE, Beijing, China, 2014. [17] X. Feng, X. Zhu, X. Qian, Y. Jie, F. Ma, and X. Niu, “A new *is research was funded by the Research Foundation transit network design study in consideration of transfer time Flanders, Project G.0853.16N. composition,” Transportation Research Part D: Transport and Environment, vol. 66, pp. 85–94, 2019. References [18] M. P. John, C. L. Mumford, and R. Lewis, “An improved multi-objective algorithm for the urban transit routing [1] A. Ceder, Public Transit Planning and Operation: Modeling, problem,” in Evolutionary Computation in Combinatorial Practice and Behavior, CRC Press, Boca Raton, FL, USA, 2nd Optimisation, C. Blum and G. Ochoa, Eds., Springer, Berlin, edition, 2016. Germany, pp. 49–60, 2014. [2] R. M. Lusby, J. Larsen, and S. Bull, “A survey on robustness in [19] F. Kılıç and M. Gok, ¨ “A demand based route generation railway planning,” European Journal of Operational Research, algorithm for public transit network design,” Computers & vol. 266, no. 1, pp. 1–15, 2018. Operations Research, vol. 51, pp. 21–29, 2014. [3] V. Guihaire and J.-K. Hao, “Transit network design and [20] J. Yang and Y. Jiang, “Application of modiﬁed NSGA-II to the scheduling: a global review,” Transportation Research Part A: transit network design problem,” Journal of Advanced Policy and Practice, vol. 42, no. 10, pp. 1251–1273, 2008. Transportation, vol. 2020, Article ID 3753601, 24 pages, 2020. [4] L. Ahmed, C. Mumford, and A. Kheiri, “Solving urban transit [21] B. Yao, P. Hu, X. Lu, J. Gao, and M. Zhang, “Transit network route design problem using selection hyper-heuristics,” Eu- design based on travel time reliability,” Transportation Re- ropean Journal of Operational Research, vol. 274, no. 2, search Part C: Emerging Technologies, vol. 43, pp. 233–248, pp. 545–559, 2019. [5] J. J. Blum and T. V. Mathew, “Intelligent agent optimization of [22] C. Iliopoulou, K. Kepaptsoglou, and E. Vlahogianni, “Meta- urban bus transit system design,” Journal of Computing in heuristics for the transit route network design problem: a Civil Engineering, vol. 25, no. 5, pp. 357–369, 2011. review and comparative analysis,” Public Transport, vol. 11, [6] J. Duran-Micco, E. Vermeir, and P. Vansteenwegen, “Con- no. 3, pp. 487–521, 2019. sidering emissions in the transit network design and fre- [23] A. Schobel ¨ and S. Scholl, “Line planning with minimal quency setting problem with a heterogeneous ﬂeet,” European traveling time,” in Proceedings of the 5th Workshop on Al- Journal of Operational Research, vol. 282, no. 2, pp. 580–592, gorithmic Methods and Models for Optimization of Railways, [7] K. A. Islam, I. M. Moosa, J. Mobin, M. A. Nayeem, and Palma de Mallorca, Spain, 2006. [24] R. Z. Farahani, E. Miandoabchi, W. Y. Szeto, and H. Rashidi, M. S. Rahman, “A heuristic aided stochastic beam search algorithm for solving the transit network design problem,” “A review of urban transportation network design problems,” Journal of Advanced Transportation 17 European Journal of Operational Research, vol. 229, no. 2, planning problem with maximum proﬁt,” Transportation pp. 281–302, 2013. Research Part E: Logistics and Transportation Review, vol. 127, [25] O. J. Ibarra-Rojas, F. Delgado, R. Giesen, and J. C. Muñoz, pp. 1–30, 2019. “Planning, operation, and control of bus transport systems: a [46] H. Dienst, Linienplanung im spurgefuhrten personenverkehr literature review,” Transportation Research Part B: Method- mithilfe eines heuristischen verfahrens, Ph.D. thesis, Techni- ological, vol. 77, pp. 38–75, 2015. sche Universitat ¨ Braunschweig, Braunschweig, Germany, [26] K. Kepaptsoglou and M. Karlaftis, “Transit route network 1978. design problem: review,” Journal of Transportation Engi- [47] M. R. Bussieck, U. Zimmerman, and P. Kreuzer, “Optimal lines for railway systems,” European Journal of Operational neering, vol. 135, no. 8, pp. 491–505, 2009. [27] A. Schobel, ¨ “Line planning in public transportation: models Research, vol. 96, pp. 54–63, 1996. and methods,” OR Spectrum, vol. 34, no. 3, pp. 491–510, 2012. [48] A. Bouma and C. Oltrogge, “Linienplanung und simulation [28] A. Patz, “Die richtige auswahl von verkehrslinien bei grossen fur ¨ oﬀentliche ¨ verkehrswege in praxis und theorie,” Eisen- strassenbahnnetzen,” Verkehrstechnik, vol. 51, 1925. bahntechnische Runschau, vol. 43, pp. 369–378, 1994. [29] C. E. Mandl, “Evaluation and optimization of urban public [49] J. F. Guan, H. Yang, and S. C. Wirasinghe, “Simultaneous transportation networks,” European Journal of Operational optimization of transit line conﬁguration and passenger line Research, vol. 5, no. 6, pp. 396–404, 1980. assignment,” Transportation Research Part B: Methodological, [30] P. Chakroborty and T. Wivedi, “Optimal route network vol. 40, no. 10, pp. 885–902, 2006. design for transit systems using genetic algorithms,” Engi- [50] M. Schmidt and A. Schobel, ¨ “*e complexity of integrating neering Optimization, vol. 34, no. 1, pp. 83–100, 2002. passenger routing decisions in public transportation models,” [31] S. B. Pattnaik, S. Mohan, and V. M. Tom, “Urban bus transit Networks, vol. 65, no. 3, pp. 228–243, 2015. route network design using genetic algorithm,” Journal of [51] E. Cipriani, S. Gori, and M. Petrelli, “Transit network design: a Transportation Engineering, vol. 124, no. 4, pp. 368–375, 1998. procedure and an application to a large urban area,” Trans- [32] F. Zhao and X. Zeng, “Simulated annealing-genetic algorithm portation Research Part C: Emerging Technologies, vol. 20, for transit network optimization,” Journal of Computing in no. 1, pp. 3–14, 2012. Civil Engineering, vol. 20, no. 1, pp. 57–68, 2006. [52] I. Constantin and M. Florian, “Optimizing frequencies in a [33] S. Chai and Q. Liang, “An improved NSGA-II algorithm for transit network: a nonlinear bi-level programming approach,” transit network design and frequency setting problem,” International Transactions in Operational Research, vol. 2, Journal of Advanced Transportation, vol. 2020, Article ID no. 2, pp. 149–164, 1995. [53] C. L. Mumford, “New heuristic and evolutionary operators for 2895320, 20 pages, 2020. [34] E. Vermeir, J. Duran ´ Micco, and P. Vansteenwegen, “*e grid the multi-objective urban transit routing problem,” in Pro- based approach, a fast local evaluation technique for line ceedings of the 2013 IEEE Congress on Evolutionary Compu- planning,” 4OR, 2021. tation, pp. 939–946, IEEE, Cancun, Mexico, 2013. [35] A. Mauttone and M. E. Urquhart, “A multi-objective meta- [54] B. Beltran, S. Carrese, E. Cipriani, and M. Petrelli, “Transit heuristic approach for the transit network design problem,” network design with allocation of green vehicles: a genetic Public Transport, vol. 1, no. 4, pp. 253–273, 2009. algorithm approach,” Transportation Research Part C: [36] M. Goerigk and M. Schmidt, “Line planning with user-op- Emerging Technologies, vol. 17, no. 5, pp. 475–483, 2009. timal route choice,” European Journal of Operational Re- [55] W. Fan and R. B. Machemehl, “Optimal transit route network design problem with variable transit demand: genetic algo- search, vol. 259, no. 2, pp. 424–436, 2017. [37] R. Borndorfer, ¨ M. Grotschel, ¨ and M. E. Pfetsch, “A column- rithm approach,” Journal of Transportation Engineering, generation approach to line planning in public transport,” vol. 132, no. 1, pp. 40–51, 2006. Transportation Science, vol. 41, no. 1, pp. 123–132, 2007. [56] H. Shimamoto, N. Murayama, A. Fujiwara, and J. Zhang, [38] R. Borndorfer ¨ and M. Karbstein, “A direct connection ap- “Evaluation of an existing bus network using a transit network proach to integrated line planning and passenger routing,” optimisation model: a case study of the Hiroshima city bus OpenAccess Series in Informatics, vol. 25, pp. 47–57, 2012. network,” Transportation, vol. 37, no. 5, pp. 801–823, 2010. [39] K. Nachtigall and K. Jerosch, “Simultaneous network line [57] H. Shimamoto, J.-D. Schmocker, ¨ and F. Kurauchi, “Opti- planning and traﬃc assignment,” OpenAccess Series in In- misation of a bus network conﬁguration and frequency considering the common lines problem,” Journal of Trans- formatics, vol. 9, 2008. [40] M. E. Schmidt, “Integrating routing decisions in public portation Technologies, vol. 2, no. 3, pp. 220–229, 2012. [58] R. O. Arbex and C. B. da Cunha, “Eﬃcient transit network transportation problems,” Springer Optimization and Its Applications, Springer, New York, NY, USA, 2014. design and frequencies setting multi-objective optimization [41] M. T. Claessens, N. M. Van Dijk, and P. J. Zwaneveld, “Cost by alternating objective genetic algorithm,” Transportation optimal allocation of rail passenger lines,”EuropeanJournalof Research Part B: Methodological, vol. 81, pp. 355–376, 2015. Operational Research, vol. 110, no. 3, pp. 474–489, 1998. [59] A. T. Buba and L. S. Lee, “A diﬀerential evolution for si- [42] M. R. Bussieck, Optimal Lines in Public Transport, Technische multaneous transit network design and frequency setting Universitat ¨ Braunschweig, Braunschweig, Germany, 1998. problem,” Expert Systems with Applications, vol. 106, pp. 277–289, 2018. [43] J.-W. Goossens, S. van Hoesel, and L. Kroon, “A branch-and- cut approach for solving railway line-planning problems,” [60] M. Nikolic ´ and D. Teodorovic, ´ “A simultaneous transit net- Transportation Science, vol. 38, no. 3, pp. 379–393, 2004. work design and frequency setting: computing with bees,” [44] M. R. Bussieck, T. Lindner, and M. E. Lubbecke, ¨ “A fast Expert Systems with Applications, vol. 41, no. 16, pp. 7200– algorithm for near cost optimal line plans,” Mathematical 7209, 2014. Methods of Operations Research, vol. 59, no. 2, pp. 205–220, [61] M. Owais, M. K. Osman, and G. Moussa, “Multi-objective 2004. transit route network design as set covering problem,” IEEE [45] D. Canca, A. De-Los-Santos, G. Laporte, and J. A. Mesa, Transactions on Intelligent Transportation Systems, vol. 17, “Integrated railway rapid transit network design and line no. 3, pp. 670–679, 2016. 18 Journal of Advanced Transportation [62] A. Ceder and N. H. M. Wilson, “Bus network design,” Transportation Research Part B: Methodological, vol. 20, no. 4, pp. 331–344, 1986. [63] H. Fu, L. Nie, L. Meng, B. R. Sperry, and Z. He, “A hierarchical line planning approach for a large-scale high speed rail network: the China case,” Transportation Research Part A: Policy and Practice, vol. 75, pp. 61–83, 2015. [64] D. B. Johnson, “Eﬃcient algorithms for shortest paths in sparse networks,” Journal of the ACM, vol. 24, no. 1, pp. 1–13, [65] R. R. Williams, “Faster all-pairs shortest paths via circuit complexity,” SIAM Journal on Computing, vol. 47, no. 5, pp. 1965–1985, 2018. [66] B. C. Fiss and M. Ritt, “Exact and metaheuristic algorithms for the urban transit routing problem,” in Proceedings of the Congreso Latino-Iberoamericano de Investigacion Operativa, Simposio Brasileiro de Pesquisa Operacional, pp. 3717–3728, Rio de Janeiro, Brazil, 2012. [67] K. Sorensen, ¨ “Metaheuristics-the metaphor exposed,” Inter- national Transactions in Operational Research, vol. 22, no. 1, pp. 3–18, 2015. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Advanced Transportation Hindawi Publishing Corporation http://www.deepdyve.com/lp/hindawi-publishing-corporation/an-exact-solution-approach-for-the-bus-line-planning-problem-with-yVvKkRzm6a

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References (71)

C. Mandl (1980)
Evaluation and optimization of urban public transportation networks
European Journal of Operational Research, 5
M. Baaj, H. Mahmassani (1991)
AN AI-BASED APPROACH FOR TRANSIT ROUTE SYSTEM PLANNING AND DESIGN
Journal of Advanced Transportation, 25
Ian Cooper, Matthew John, Rhyd Lewis, C. Mumford, A. Olden (2014)
Optimising large scale public transport network design problems using mixed-mode parallel multi-objective evolutionary algorithms
2014 IEEE Congress on Evolutionary Computation (CEC)
K. Nachtigall, K. Jerosch (2008)
Simultaneous Network Line Planning and Traffic Assignment
Fang Zhao, Xiaogang Zeng (2006)
Simulated Annealing-Genetic Algorithm for Transit Network Optimization
Journal of Computing in Civil Engineering, 20
Hiroshi Shimamoto, Jan-Dirk Schmöcker, F. Kurauchi (2012)
Optimisation of a Bus Network Configuration and Frequency Considering the Common Lines Problem
Journal of Transportation Technologies, 2
I. Constantin, M. Florian (1995)
Optimizing frequencies in a transit network: a nonlinear bi-level programming approach
International Transactions in Operational Research, 2
(1978)
Linienplanung im spurgeführten personenverkehr mithilfe eines heuristischen verfahrens
Ryan Williams (2013)
Faster all-pairs shortest paths via circuit complexity
Proceedings of the forty-sixth annual ACM symposium on Theory of computing
B. Fiss (2012)
Exact and metaheuristic algorithms for the urban transit routing problem
A. Ceder (2015)
Public Transit Planning and Operation: Modeling, Practice and Behavior
Xuesong Feng, Xiaojing Zhu, Xuepeng Qian, Yuanpeng Jie, Fei Ma, X. Niu (2019)
A new transit network design study in consideration of transfer time composition
Transportation Research Part D: Transport and Environment
P. Kechagiopoulos, G. Beligiannis (2014)
Solving the Urban Transit Routing Problem using a particle swarm optimization based algorithm
Appl. Soft Comput., 21
C. Iliopoulou, K. Kepaptsoglou, E. Vlahogianni (2019)
Metaheuristics for the transit route network design problem: a review and comparative analysis
Public Transport, 11
O. Ibarra-Rojas, Felipe Delgado, Ricardo Giesen, J. Muñoz (2015)
Planning, operation, and control of bus transport systems: A literature review
Transportation Research Part B-methodological, 77
M. Sokolova, A. Caballero (2012)
Data and Results
Donald Johnson (1977)
Efficient Algorithms for Shortest Paths in Sparse Networks
Journal of the ACM (JACM), 24
Kazi Islam, Ibraheem Moosa, Jaiaid Mobin, Muhammad Nayeem, Mohammad Rahman (2019)
A heuristic aided Stochastic Beam Search algorithm for solving the transit network design problem
Swarm Evol. Comput., 46
R. Lusby, J. Larsen, Simon Bull (2018)
A survey on robustness in railway planning
Eur. J. Oper. Res., 266
R. R. Williams (2018)
Faster all-pairs shortest paths via circuit complexity,
SIAM Journal on Computing, 47
K. Sörensen (2015)
Metaheuristics - the metaphor exposed
Int. Trans. Oper. Res., 22
Jie Yang, Yangsheng Jiang (2020)
Application of Modified NSGA-II to the Transit Network Design Problem
Journal of Advanced Transportation
A. Buba, L. Lee (2018)
A differential evolution for simultaneous transit network design and frequency setting problem
Expert Syst. Appl., 106
K. Kepaptsoglou, M. Karlaftis (2009)
Transit Route Network Design Problem: Review
Journal of Transportation Engineering-asce, 135
Ruirui Wu, Sheng-sheng Wang (2016)
Discrete wolf pack search algorithm based transit network design
2016 7th IEEE International Conference on Software Engineering and Service Science (ICSESS)
R. Arbex, C. Cunha (2015)
Efficient transit network design and frequencies setting multi-objective optimization by alternating objective genetic algorithm
Transportation Research Part B-methodological, 81
S. Pattnaik, S. Mohan, V. Tom (1998)
Urban Bus Transit Route Network Design Using Genetic Algorithm
Journal of Transportation Engineering-asce, 124
M. Goerigk, M. Schmidt (2017)
Line planning with user-optimal route choice
Eur. J. Oper. Res., 259
Milos Nikolic, D. Teodorovic (2014)
A simultaneous transit network design and frequency setting: Computing with bees
Expert Syst. Appl., 41
J. Guan, Hai Yang, S. Wirasinghe (2006)
Simultaneous Optimization of Transit Line Configuration and Passenger Line Assignment
Transport Logistics
Evert Vermeir, Javier Durán-Micco, P. Vansteenwegen (2021)
The grid based approach, a fast local evaluation technique for line planning
4OR, 20
M. Bussieck, T. Lindner, M. Lübbecke (2004)
A fast algorithm for near cost optimal line plans
Mathematical Methods of Operations Research, 59
Lang Fan, C. Mumford (2010)
A metaheuristic approach to the urban transit routing problem
Journal of Heuristics, 16
J. Blum, Tom Mathew (2011)
Intelligent Agent Optimization of Urban Bus Transit System Design
J. Comput. Civ. Eng., 25
A. Bouma, C. Oltrogge (1994)
Linienplanung und Simulation für öffentliche Verkehrswege in Praxis und Theorie
, 43
Jan-Willem Goossens, S. Hoesel, L. Kroon (2004)
A Branch-and-Cut Approach for Solving Railway Line-Planning Problems
Transp. Sci., 38
M. Owais, M. Osman, Ghada Moussa (2016)
Multi-Objective Transit Route Network Design as Set Covering Problem
IEEE Transactions on Intelligent Transportation Systems, 17
Hiroshi Shimamoto, Naoki Murayama, A. Fujiwara, Junyi Zhang (2010)
Evaluation of an existing bus network using a transit network optimisation model: a case study of the Hiroshima City Bus network
Transportation, 37
B. Beltrán, S. Carrese, E. Cipriani, M. Petrelli (2009)
Transit network design with allocation of green vehicles: A genetic algorithm approach
Transportation Research Part C-emerging Technologies, 17
Javier Durán-Micco, Evert Vermeir, P. Vansteenwegen (2020)
Considering emissions in the transit network design and frequency setting problem with a heterogeneous fleet
Eur. J. Oper. Res., 282
Matthew John, C. Mumford, Rhyd Lewis (2014)
An Improved Multi-objective Algorithm for the Urban Transit Routing Problem
A. Schöbel (2012)
Line planning in public transportation: models and methods
OR Spectrum, 34
M. Bussieck, P. Kreuzer, U. Zimmermann (1997)
Optimal lines for railway systems
European Journal of Operational Research, 96
C. Mandl (1980)
Applied Network Optimization
C. Mumford
Data and results 2013,
W. Fan, R. Machemehl (2006)
Optimal Transit Route Network Design Problem with Variable Transit Demand: Genetic Algorithm Approach
Journal of Transportation Engineering-asce, 132
R. Borndörfer, M. Grötschel, M. Pfetsch (2007)
A Column-Generation Approach to Line Planning in Public Transport
Transp. Sci., 41
C. Mumford (2013)
New heuristic and evolutionary operators for the multi-objective urban transit routing problem
2013 IEEE Congress on Evolutionary Computation
R. Borndörfer, Marika Karbstein (2012)
A Direct Connection Approach to Integrated Line Planning and Passenger Routing
M. Schmidt, A. Schöbel (2015)
The complexity of integrating passenger routing decisions in public transportation models
Networks, 65
M. Claessens, N. Dijk, P. Zwaneveld (1998)
Cost optimal allocation of rail passenger lines
Eur. J. Oper. Res., 110
Milos Nikolic, D. Teodorovic (2013)
Transit network design by Bee Colony Optimization
Expert Syst. Appl., 40
Leena Ahmed, C. Mumford, A. Kheiri (2019)
Solving urban transit route design problem using selection hyper-heuristics
Eur. J. Oper. Res., 274
(1998)
Optimal Lines in Public Transport
R. Farahani, E. Miandoabchi, W. Szeto, Hannaneh Rashidi-Bajgan (2013)
A review of urban transportation network design problems
Eur. J. Oper. Res., 229
Antonio Mauttone, María Urquhart (2009)
A multi-objective metaheuristic approach for the Transit Network Design Problem
Public Transport, 1
P. Chakroborty, Tathagat Wivedi (2002)
Optimal Route Network Design for Transit Systems Using Genetic Algorithms
Engineering Optimization, 34
(1925)
Die richtige auswahl von verkehrslinien bei grossen strassenbahnnetzen
A. Ceder, N. Wilson (1986)
Bus network design
Transportation Research Part B-methodological, 20
(1998)
European Journal of Operational Research
E. Cipriani, S. Gori, M. Petrelli (2012)
Transit network design: A procedure and an application to a large urban area
Transportation Research Part C-emerging Technologies, 20
Shushan Chai, Q. Liang (2020)
An Improved NSGA-II Algorithm for Transit Network Design and Frequency Setting Problem
Journal of Advanced Transportation
M. Schmidt (2014)
Integrating Routing Decisions in Public Transportation Problems
, 89
D. Canca, Alicia De-Los-Santos, G. Laporte, J. Mesa (2019)
Integrated Railway Rapid Transit Network Design and Line Planning problem with maximum profit
Transportation Research Part E: Logistics and Transportation Review
Muhammad Nayeem, Md. Rahman, M. Rahman (2014)
Transit network design by genetic algorithm with elitism
Transportation Research Part C-emerging Technologies, 46
Baozhen Yao, P. Hu, Xiao Lu, Junjie Gao, Mingheng Zhang (2014)
Transit network design based on travel time reliability
Transportation Research Part C-emerging Technologies, 43
Huiling Fu, L. Nie, L. Meng, B. Sperry, Zhenhuan He (2015)
A hierarchical line planning approach for a large-scale high speed rail network: The China case
Transportation Research Part A-policy and Practice, 75
(2019)
planning problem with maximum proﬁt
Fatih Kılıç, Mustafa Gök (2014)
A demand based route generation algorithm for public transit network design
Comput. Oper. Res., 51
A. Schöbel, Susanne Scholl (2005)
Line Planning with Minimal Traveling Time
Valérie Guihaire, Jin-Kao Hao (2008)
Transit network design and scheduling: A global review
Transportation Research Part A-policy and Practice, 42

Publisher: Hindawi Publishing Corporation
Copyright: Copyright © 2021 Evert Vermeir et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ISSN: 0197-6729
eISSN: 2042-3195
DOI: 10.1155/2021/6684795
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Abstract

Hindawi Journal of Advanced Transportation Volume 2021, Article ID 6684795, 18 pages https://doi.org/10.1155/2021/6684795 Research Article An Exact Solution Approach for the Bus Line Planning Problem with Integrated Passenger Routing 1 2 3 1 Evert Vermeir , Wouter Engelen, Johan Philips , and Pieter Vansteenwegen KU Leuven, Leuven Mobility Research Centre-CIB, Celestijnenlaan 300, 3001 Leuven, Belgium UC Leuven-Limburg, Research & Expertise Digital Solutions, Geldenaaksebaan 335, 3001 Leuven, Belgium KU Leuven, Division RAM-Flanders Make, Celestijnenlaan 300, 3001 Leuven, Belgium Correspondence should be addressed to Evert Vermeir; evert.vermeir@kuleuven.be Received 16 November 2020; Revised 19 July 2021; Accepted 3 September 2021; Published 4 October 2021 Academic Editor: Gonçalo Correia Copyright © 2021 Evert Vermeir et al. *is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. *e bus line planning problem or transit network design problem with integrated passenger routing is a challenging combinatorial problem. Although well-known benchmark instances for this problem have been available for decades, the state of the art lacks optimal solutions for these instances. *e branch and bound algorithm, presented in this paper, introduces three novel concepts to determine these optimal solutions: (1) a new line pool generation method based on dominance, (2) the introduction of essential links, i.e., links which can be determined beforehand and must be present in the optimal solution, and (3) a new network representation based on adding only extra edges. Next to presenting the newly obtained optimal solutions, each of the abovementioned concepts is examined in isolation in the experiments, and it is shown that they contribute signiﬁcantly to the success of the algorithm. *ere are two main conﬂicting goals present in nearly every 1. Introduction line planning problem. At the one hand, a public transit *e design of a public transport network is a multistep company wants to spend as little money as possible, and at process [1, 2]. It starts with the infrastructure network de- the other hand, the passengers want to get to their desti- sign. Decisions on the physical network, such as bus stops or nation as fast as possible. One of the common approaches is bus only lanes, are made in this step. *e next step is the line to optimize the quality of the service for the passengers planning or the transit network design. In this step, the (described by the average travel time and/or the number of public transport operator decides where its vehicles will transfers) and to limit the costs by putting constraints on the drive and which stops will be served in which order. *en, number of lines that can be used and the length of the lines the timetabling step produces a ﬁxed time schedule for each (e.g., [4–10]). In order to evaluate the average travel time, the bus. *e setting of frequencies is sometimes included in the transit assignment, i.e., how the passengers will use the diﬀerent bus lines to travel through the network, needs to be line planning step and sometimes in the timetabling step [3]. *e next step is to plan the rolling stock and the crews. *e integrated in the decision process. If frequencies are not ﬁnal component of the planning process is the dispatching considered, an “all-or-nothing” shortest path assignment is strategy. While all these subproblems inﬂuence each other, typically used [4, 8, 9, 11–15]. Mandl’s Swiss network [14], is they are mostly solved sequentially in practice [1]. *is paper the only widely used benchmark network for the uncapa- focuses on the second step, namely, the line planning citated line planning problem [4, 7–10, 12, 13, 15–21]. problem, without considering frequencies. *is is also called Mandl’s network with 15 nodes and 21 links is illustrated in the uncapacitated line planning problem. Figure 1. *e length of each link is indicated with the *e uncapacitated line planning problem is a diﬃcult number next to the link. Benchmark instances are available combinatorial problem with an enormous search space. with two diﬀerent maximum line lengths and with diﬀerent 2 Journal of Advanced Transportation tackled in this paper. Section 4 discusses the solution ap- 8 proach. It shows the branch and bound algorithm and ex- plains all the details about the essential links and the Direct Link Network representation. Section 5 contains the results of the experiments on Mandl’s Swiss Network [14]. *e paper ends in Section 6 with the conclusions. 6 2. Literature Review *is section contains a short survey of the literature on the line planning problem, with and without frequency setting. Both (meta) heuristics and exact approaches will be covered. A more extensive overview can be found in the following papers and books: [1, 3, 22, 24–27]. It should be noted that, depending on the community, this planning problem is called the transit network design problem or the line planning problem. In this paper, the term line planning will be used. 2.1. Heuristics and Metaheuristics. Early research on dif- ferent variants of the line planning problem focused on Figure 1: Mandl’s Swiss network [14]. heuristics. One of the earliest heuristics to solve the line planning was by Patz [28]. In this work, lines are iteratively numbers of lines. Despite it being a small network with only removed from a large set of lines based on a penalty ﬁfteen nodes, there are no known optimal solutions for structure. Mandl [29] combined both passenger demand and shortest paths to create an initial solution. *en, this solution instances with more than three bus lines on this network [22]. is iteratively improved aiming to optimize the total travel *e main contribution of this paper is that, for the ﬁrst time. *ese early heuristics were unable to solve large in- time, all these benchmark instances for the uncapacitated stances, but they provide the basis for the development of the line planning problem with integrated passenger routing are more modern metaheuristics [11]. solved to optimality. *e amount of lines necessary to oﬀer *e ﬁrst metaheuristics for solving the line planning each passenger a direct connection along his/her shortest problem were Genetic Algorithms (GA). Chakroborty and path between origin and destination is also determined for Wivedi [30] and Pattnaik et al. [31] were some of the ﬁrst to the ﬁrst time. Furthermore, a number of novel concepts are solve the line planning problem with a GA. Because they introduced which are required to obtain these optimal so- were one of the ﬁrst ones to use metaheuristics during the lutions, but which are also interesting for future research on optimization process, the performance of their algorithms was a lot better than the previously known methods. Since this complex planning problem. First, a new method is developed to construct a line pool then, many reﬁnements have happened to these GA’s. Zhao of all feasible lines that can be present in an end solution. and Zeng [32] added a local search component to the GA, Secondly, the concept of “essential links” is presented. *ese resulting in a memetic algorithm. Nayeem et al. [8] intro- are parts of lines, determined during the branch and bound duced elitism in their GA as well as a guided local search. process, which must be part of the optimal solution. *irdly, Islam et al. [7] used a stochastic beam search to tackle the the “Direct Link Network” representation (DLN) is intro- line planning problem. One of their other contributions is duced. *is is a novel network representation allowing a the development of a new heuristic to get a very strong faster evaluation of solutions, compared to the well-known starting solution. *eir method combines edge lengths with change-and-go network representation [23]. Finally, a new demand served into a single cost, while previous methods branch and bound method limits the amount of required just tried to serve a high demand along (quasi) shortest evaluations before obtaining the optimal solution. *e paths. Chai and Liang’s [33] work is an example of a recent method branches over all possible lines containing a certain paper using a GA, and they developed a modiﬁed version of Origin-Destination (OD) pair. *e experimental results il- the well-known NSGA-II algorithm to solve the line plan- lustrate the importance of each of these concepts. We also ning problem. Fan and Mumford [12] used both hill make clear that research on the line planning problem climbing and simulated annealing. Also swarm intelligence should shift to larger and more realistic instances. techniques are being used in line planning such as Blum and *e paper is organized as follows. Section 2 contains an Mathew [5] who use Ant Colony Optimization or Nikolic ´ overview of the line planning problem literature, focusing on and Teodorovic [9] who use Bee Colony Optimization. (meta) heuristics and exact approaches for the line planning Vermeir et al. [34] use an iterated local search combined problem with integrated passenger routing. Section 3 de- with a local evaluation method which only looks at a cut of scribes in detail which variant of the line planning problem is the network to quickly evaluate candidate solutions. Journal of Advanced Transportation 3 maximizing the amount of direct travelers [46]. Here, the Mauttone and Urquhart [35] were the ﬁrst ones to model the line planning problem as a multiobjective problem. *ey integer program is solved with a branch and bound approach which greedily chooses lines with maximal current direct used a greedy randomized adaptive search procedure to generate a Pareto front with minimal values for operator and travelers. Bussieck et al. [47] and Bussieck [42] expanded this passenger cost. Recently, Duran-Micco et al. [6] developed a work by taking the vehicle capacities into consideration memetic algorithm to add emissions as an extra objective. through introducing constraints on the edge frequencies. *ey showed that emissions can be greatly reduced while *e repeated passenger calculations can also be avoided only having a minimal impact on the travel time and op- through a so-called “system split,” where links are catego- erator costs. Research focusing only on minimizing the rized by speed. Passengers are then assumed to switch to the fast levels as soon as possible and leave it as late as possible. operator cost is not discussed here, but the interested reader is referred to the above mentioned survey papers or e.g., [17]. With these assumptions, the passenger ﬂows can be esti- mated beforehand and do not have to be recalculated during the optimization process. *is method was promoted by 2.2. Exact Approaches. A number of publications on exact Bouma and Oltrogge [48], where it is used in a branch-and- solution approaches start by deﬁning a mathematical model bound approach applied to the Dutch railway network. and then try to solve the problem or a relaxation of the Later, the objective function evolved to the minimization of problem. As illustrated by the paper below, most of these line the travel time of the passengers [23, 37–40]. planning models originate from train line planning. Since More recent research integrates passenger routing in trains typically have lower frequencies than urban bus the line planning problem. In order to still model the line networks, the selected frequencies have a much larger im- planning problem as MIPs, assumptions about passenger pact on the transfer times than for a high frequency network. routing are made, such as passengers choosing minimal Capacity limitations at stops (stations) are more important in vehicle travel-time routes (and thus not considering for rail systems and are typically checked through con- transfers) or routes with a minimal number of transfers straints based on frequencies [27]. Hence, most commonly, [40, 49, 50] or such as passenger routes that can be the setting of frequencies is included in the line planning assigned by the public transport operator [23, 37, 40]. To process [36]. While frequencies should have a direct relation avoid these limitations, a bilevel model (like in most to transfer times, including frequency-based transfer times metaheuristics) can be used. In this bilevel model, net- lead to non-linear optimization models [36]. *erefore, even work decisions are made at an upper level and the pas- when including frequency setting, typically transfer times senger routing decisions are made at the lower level are considered through a transfer penalty [23, 37–40]. *ese [36, 51, 52]. An early implementation of this is formu- line planning problems are typically formulated as a (mixed- lated by Constantin and Florian [52]. Goerigk and ) integer program. Schmidt [36] use this bilevel model to completely inte- *ere exist many ways to calculate passenger routing in grate passenger routing and solve, to optimality, in- line planning. Distributing all the passengers along their stances with up to ten bus nodes and a line pool of 30 lines shortest path is the most commonly used strategy in line randomly generated beforehand. For larger networks, planning. While this is a simple method, it is already they use a genetic algorithm. However, none of these computationally expensive to execute during an optimiza- methods have solved the instances on Mandl’s Swiss tion [40]. In cost-oriented models the operational cost is Network [14] (ﬁfteen nodes) with integrated passenger minimized subject to certain constraints on the service level. routing to optimality. *ese constraints are typically very easy to check and avoid the expensive calculations. Since costs are easy to calculate, it 2.3. Benchmark Instances. Despite being too small to be is possible to use exact approaches even on real-world data. Claessens et al. [41] use a branch and bound procedure to representative of a city bus network, Mandl’s Swiss Network is the most commonly used benchmark instance in bus line obtain results for the Dutch railway system. Bussieck [42] used this work as a basis to make a cut and branch algorithm planning [22]. Nevertheless, fair comparisons in line plan- ning research remains an issue since diﬀerent variants of the which was used to obtain results for both the German and Dutch railway system. Goossens et al. [43] built further on problem are considered [4]. *ere are a lot of real-life in- this work with a branch and cut approach which allowed to stances used in literature, but they each have very speciﬁc loosen some constraints regarding the lines. Bussieck et al. objectives and constraints or the data is not publicly [44] created a fast procedure that obtains good solutions in a available. *ese issues are also mentioned in Ceder [1], small computation time. Canca et al. [45] managed to in- Ibarra-Rojas et al. [25], Farahani et al. [24], Schobel [27], tegrate several other planning steps in the optimization Kepaptsoglou and Karlaftis [26], and Guihaire and Hao [3]. Fortunately, Mumford [15] recently made four larger process. When looking at passenger-oriented models, explicit datasets publicly available. *ese networks are based on actual bus route networks from Chinese and British cities passenger routing is very important and cannot be easily avoided. In these models, an evolution of the objective and are being used more and more [4, 6, 8, 18–20, 53]. One of the datasets was already used earlier by Fan and Mumford function can be observed, which directly inﬂuences the speed at which the service level can be calculated. *e earliest [12], and another one was already used by Nikolic ´ and work circumvents the expensive passenger routing by Teodorovic ´ [9] with diﬀerent parameters. 4 Journal of Advanced Transportation be high enough to serve all passengers. *e integration of the 3. Problem Description passenger routing and the fact that no limited set of lines is *e available infrastructure network is depicted by a directed considered are two aspects signiﬁcantly complicating the graph G � V, E , which contains vertices V � v , v , . . . , v { } 􏼈 􏼉 uncapacitated line planning problem considered in this 1 2 n representing the bus stops and edges E � {e , e , . . ., e } which ij kl yz paper. are the connections available between these bus stops. *e cost or travel time on an edgee is indicated byt . *e demand of the ij e 4. Methodology passengers traveling through the public transport network is represented by an Origin Destination (OD) matrix D. *e *is section starts with a detailed discussion of a number of number of passengers per hour that want to travel from bus stop essential concepts and components implemented in our i to bus stop j is then depicted by d . In the line planning ij algorithm to optimally solve the uncapacitated line planning problem, the goal is to select the best possible set L of bus lines. problem with integrated passenger routing. First, it is In the uncapacitated line planning problem with integrated explained how the line pool is generated and how “essential passenger routing, the objective function is to minimize the total links” can be determined. *en, a new transit network travel time (TTT) of all passengers. *is means the sum of all representation is constructed, and it is shown how it can be travel times of all passengers. *is travel time also includes used to eﬃciently calculate the passenger routing. Finally, transfer times. *is time spent waiting on the next bus (and the the actual branch and bound algorithm is explained in detail. discomfort of transferring) is modelled with a transfer penalty In Section 4.6, a summary of the method is given together TP, which penalizes each transfer. It should be noted that a with the pseudocode and a ﬂowchart, and two alternative timetable or frequencies are not available at this stage, and approaches are brieﬂy introduced. therefore, a more accurate modelling is not feasible and waiting times are not considered. In order to evaluate and minimize the TTT, the routing each passenger will take has to be known. *is 4.1. Line Pool Generation. To ensure that the entire search routing results in a setπ � {e ,e , . . .,e } of edges used and the ij ik kl xj space is explored, a pool of all possible lines is generated. To amount of necessary transfers τ . *e total travel time of all ij construct this pool, an important property of the problem is passengers TTT is then represented by exploited. *e total travel time of all passengers will never increase when a line is extended by adding an extra stop at ⎝ ⎝ ⎠⎠ one of the ends. *erefore, all lines that are a subline of ⎛ ⎛ ⎞⎞ TTT(L) � 􏽘 􏽘 d 􏽘 t + TP∗τ . (1) ij e ij another feasible line are dominated by that line. When a i∈V j∈V e∈π ij dominated line would be part of the optimal solution, it can To completely integrate the passenger routing in the be substituted for its dominator. *is means that only optimization, the line planning problem is formulated as a considering nondominated lines will be suﬃcient to ﬁnd the bilevel problem as mentioned in the literature review. *e optimal solution. *is is illustrated in Figure 2, where line (a) design of the lines is the upper problem, and the routing of is dominated by line (b). All passengers that use line (a) in the passengers is the lower problem. For the lower problem, their shortest path will still be able to use the same shortest the “transit assignment,” a shortest path allocation is used. It path if we transform line (a) to line (b). *e situation can is assumed that each passenger will travel along its shortest only improve for the passengers. For example, passengers path, considering both the in-vehicle travel time and the that want to travel from node 5 to node 3 now have a direct transfer penalties. Although more complex and accurate connection along the shortest path. Because the uncapaci- transit assignment methods are available [54–57], most line tated line problem does not consider frequencies or costs, planning problems are still solved making this assumption line (b) is always at least as good as line (a). *is means that [4, 7–13, 16, 29, 53]. When operator costs are used as an we can guarantee to obtain an optimal solution without objective and/or frequencies are considered, frequency- having to consider line (a). *is is the power of domination, based assignment models are regularly used (e.g., [58–61]). and it signiﬁcantly limits the size of the line pool. For ex- *e objective and constraints used in this paper are the ample, on Mandl’s network with inﬁnite line lengths, this most commonly used in literature for the uncapacitated line results in 581 lines that could be present in the ﬁnal solution, planning problem. *is is required to allow a fair com- while without the dominance rules, there would be 8180 parison with the state-of-the-art algorithms. First of all, as candidate lines. Note that this dominance rule only holds for mentioned above, a passenger-oriented objective function is the uncapacitated line planning problem. If frequencies are used. *erefore, the operator cost is limited by imposing included or when demand elasticity is considered the constraints. *ere is a maximum number of nodes that can method will have to be adjusted. be present in each line, and there is a maximum number of A pool of nondominated lines is now constructed re- lines that can be selected in the line plan. *e shape of the cursively. *e algorithm starts by selecting any node in the bus lines is only limited by not allowing any stop to be visited network and connecting one of its adjacent nodes to con- twice. *is excludes all loops in a given line. *is also means struct a bus line. *is bus line is extended by adding new that, in this paper, the set of feasible lines is not limited or adjacent nodes until it is no longer possible without violating ﬁxed beforehand, as is the case in some other papers (e.g., any constraints (maximum line length or visiting a stop [36, 37, 51, 62]). When a line is selected, it is assumed to be twice). A line that cannot be extended is a possible candidate served in both directions and the bus capacity is assumed to for the optimal solution and is added to the line pool. After Journal of Advanced Transportation 5 23 23 1 1 4 5 45 (a) (b) Figure 2: Example of domination. Line (a) is dominated by line (b). undoing the last extension, a diﬀerent adjacent node can be 4.3. Direct Link Network Representation. To take transfers into account when looking for the passenger routes, the chosen for extending the line. When there are no adjacent nodes left, the previous extension is undone, and so on. *is available infrastructure network needs to be extended to obtain a proper representation. Typically, this is done by process continues until only the starting node remains. If this is repeated for all nodes in the network, a pool of all adding a dummy node for every stop on every bus line. *is possible lines is generated. Since all lines are considered type of extended network is also called the Change and Go bidirectionally, symmetric lines can be eliminated. Network (CNG) [23, 36] or the Train Service Network (TSN) [63]. A disadvantage of this method is that the addition of many extra nodes signiﬁcantly impacts the time required to 4.2. Essential Links. An essential link is a link for which it calculate the passenger routes. can be determined beforehand that it has to be present in the *is paper does not use the TSN or CNG. Rather than optimal solution. All candidate solutions that do not contain adding extra nodes, extra links are added to the network. For all essential links do not have to be evaluated. *is results in any two nodes that are connected by a single bus line, a direct fewer candidate solutions that have to undergo a time- link is added to the network. *e total travel time of the bus consuming evaluation. It will be shown in the experimental between these two nodes is then used as travel time for this results (Section 5) that this makes the algorithm signiﬁcantly link. If multiple bus lines connect the same nodes, only the faster. link with the shortest travel time is kept in the ﬁnal network, Essential links are determined by removing a link from which we call the “Direct Link Network (DLN).” the infrastructure network and then solving the all pair Figure 3 illustrates both the CNG and the DLN on a shortest path problem. *is assumes every passenger will small toy network. Figure 3(a) depicts a small toy network have a direct connection along its shortest path, but the with ﬁve stops and two bus lines: a full black line and a removed link cannot be used. If the total travel time of all dotted orange line. Figure 3(b) is the DLN representation of passengers obtained this way is worse than the best-known the same toy network. *ree extra links have been added to solution, then the removed link is an essential link. In other the network. *e links in the DLN are color coded to make it words, a lower bound is calculated for the network where a clear from which line each link originates. Node one has a certain link is removed. If this lower bound is higher than an direct connection to every other node in the network; hence, upper bound that is already available at the start of the it has a link to every other node in the network. But node one algorithm (the best-known solution), then the removed link can reach node two with a direct connection through each of is essential. Without this link, the optimal solution can never the two lines. Because the connection through the orange be obtained. After repeating this process for all links in the line is shorter, this is the only link that is kept in the rep- network, a list of all essential links is constructed. resentation. Node three and node ﬁve are not connected *is paper uses the optimal solution with one less line as the through a direct connection. *us, passengers traveling best-known solution to determine the essential links. For ex- between these nodes need to use a transfer. In the DLN, this ample, when starting to solve the problem for six lines, the is represented by the absence of a link connecting the two optimal solution for ﬁve lines is used as the best-known solution nodes. If a shortest path is calculated on the DLN, then each to determine the essential links for six lines. Results known from link beyond the ﬁrst that is part of the shortest path also literature or (meta) heuristics could also be used as the best- represents a transfer (and comes with a penalty). Figure 3(c) known solution. If a problem currently has no known solution, is the CNG representation of the toy network. Seven nodes any solution method can be used to get a ﬁrst upper bound. and seven links have been added to the network. Obviously, the better the quality of this upper bound, the more *ere are two ways to consider transfers in this DLN. *e essential links can be identiﬁed. Note that the lower bound can simplest way is to add the transfer penalty to the length of be obtained when not including a link is constant and can be every link in the network. Any method to solve the all pair precalculated. *is means that the list of essential links could be shortest path problem can then be used to calculate the updated on the go every time a new best solution is found. passenger routes. Since every link now contains a transfer However, in our algorithm, the list of essential links is con- penalty, a single transfer penalty has to be subtracted from structed during a precalculation phase. *e presence of each of each shortest path calculated this way. Because of the nature the essential links is also precalculated for each line in the of the DLN, every additional link used beyond the ﬁrst generated line pool. *is makes checking these added con- implies an actual transfer (for which the penalty is indeed straints during optimization very fast. included). However, we decided to incorporate the transfer 6 Journal of Advanced Transportation 23 23 2.1 3.1 1 1 1.1 2 3 45 4 5 1 2.2 1.2 4.2 5.2 4 5 (a) (b) (c) Figure 3: (a) Toy network with 2 bus lines; (b) Direct Link Network representation. (c) Change and Go representation. penalties by slightly modifying Floyd–Warshall’s algorithm, the detour (compared to the shortest possible path) between which is very simple for the DLN representation proposed in A and B is less than a single transfer penalty are considered this paper. Direct connections in the network are repre- as branches in the branch and bound tree. *ese lines are sented by a single link. *erefore, every combination of links then sorted according to their travel time between A and B. implies a transfer. *en, a transfer penalty can simply be Now, two diﬀerent scenarios are possible. One of these lines added to the main operator of the Floyd–Warshall algo- is chosen as part of the solution, each leading to a diﬀerent rithm, which is illustrated in branch (OD1 L1 is part of the solution or OD1 L2 is part of the solution), or none of these lines is chosen, leading to one ∀k, i, j: dist(i, j) � min(dist[i][j], dist[i][k] + dist[k][j] + TP). additional branch (≥TP in Figure 4). (2) When a line is chosen as a part of the solution, this implies that, in the end solution, all passengers traveling Also, in Dijkstra’s algorithm, the transfer penalties can from A to B will travel along this line. *is is due to the fact be incorporated directly. Actually, Dijkstra’s algorithm is that shorter alternatives for traveling from A to B have been most commonly used in line planning research. *is paper, considered in previous branches. For instance, if OD1 L2 is however, uses Floyd–Warshall’s algorithm since it per- selected as part of the solution, OD1 L1 was not selected and formed better in the initial testing. Note that the method the shortest path to travel from A to B in the solution will be used in this paper creates a much more dense network than along OD1 L2. *is path is then called the “optimal path” the traditional CNG network. Floyd–Warshall tends to between A and B in this branch. *is has an important perform better on dense networks, while Dijkstra tends to do implication. If the optimal path is longer than the “shortest better on sparse networks [64, 65]. possible path” between A and B, then the lower bound for It should be noted as well that this DLN representation this branch can be adjusted with the diﬀerence between the can also be used in line planning research using meta- “optimal path” and the “shortest possible path,” multiplied heuristics. It can be especially useful when a high number of with all the demand between A and B. Obviously, if this lines are considered. *is network can also be adjusted easily makes the lower bound of this branch worse than the current to work for line planning with frequencies. Whenever a upper bound available, this branch can be pruned. *ere is single OD pair is connected by multiple bus lines, this would also an eﬀect for all OD pairs lying on the part of the chosen also result in multiple links, one for each bus line available line between A and B. Since they contribute to the “optimal between these nodes, instead of only keeping the shortest path” between A and B, their own “optimal path” cannot one as explained above. improve the connection between A and B. *is is also taken into account in the lower bound. 4.4. Branch and Bound. At the start of the algorithm, the If none of these lines (OD1 L1, OD1 L2, etc.) is chosen, the additional branch is followed and the “optimal path” lower bound corresponds to the ideal situation. *is would mean that every passenger is able to travel from its origin to between A and B in that branch will be at least a single its destination along the shortest possible path in the in- transfer penalty longer than the “shortest possible path.” In frastructure network without any transfers. *en, as illus- Figure 4, this is represented by the branches called “≥TP.” In trated in Figure 4, the branching process starts by selecting the best-case scenario in this branch, traveling from A to B is the ﬁrst OD pair (OD1) from the sorted list of OD pairs. *e possible along the shortest possible path with a single way these OD pairs are sorted is explained in Section 4.5. In transfer. *erefore, in this branch, the lower bound can be the explanation below, we assume that OD1 has node A as adjusted by adding a single transfer penalty multiplied with origin and node B as destination. From the pool of lines, all the total demand betweenA andB. *e new lower bound has to be compared to the current upper bound to decide lines that contain both node A and node B are selected (OD1 L1, OD1 L2, etc.). In this selection, only those lines for which whether to continue along this branch or not. Journal of Advanced Transportation 7 Sorted OD List Start Lines OD1 OD1 OD1 L1 OD2 OD1 L1 OD1 L2 … ≥TP OD1 L2 OD3 OD1 L3 OD2 L2 ≥TP OD2 L1 … Figure 4: *e branching process. In the next branching step, the next OD pair from the transfer penalty compared to the shortest path (plus one sorted list is chosen (OD2) and the process described above extra branch for where the detour is assumed to be at least a is repeated for this OD pair. Again, all lines from the line transfer penalty, and thus, no line is selected). *e complex sort calculates the total amount of branches and divides this pool with an optimal path that deviates less than a single transfer penalty from the minimum possible are selected to by the square root of the demand of the chosen OD pair to branch upon. *e square root is used to add more weight to branch upon, together with one branch were none of these lines is selected. *ere is one diﬀerence however. All can- the number of branches compared to the size of the demand. didate lines with an optimal path for OD1 that is shorter In this way, the algorithm tries to limit the number of top than the chosen one should not be considered again and thus branches, while also maximizing the impact of not selecting cannot be branched upon. Since the lines were sorted by a line. Finally, the second alternative sorting method is a length (or travel time), all previously branched upon lines random sort. *is is used to prove whether the sorting are excluded from being further explored. In Figure 4, this actually has an eﬀect on the performance of the algorithm. means that if one of the possible branches of OD2 is a line already explored in an upper branch, it is immediately 4.6. Summary of the Exact Algorithm. Figure 5 shows an pruned. If, for example, OD1 L2 is the current branch being overview of the entire algorithm in a ﬂowchart, and Algo- explored, then the lines OD1 L1 and OD1 L2 will not be rithm 1 presents the pseudocode. *e algorithm starts with considered as lines for OD2. But OD1 L3 could be a valid some precalculations. *e most important precalculation is candidate since it has not been explored before. the generation of the full set of bus lines that could end up in *is branching process continues either until the new an optimal solution (line 2 in Algorithm 1). All origin- lower bound is worse than the current upper bound, after destination (OD) pairs are sorted by the number of nodes which the branch gets pruned, or until the required number along the shortest path and their demand (line 3 in Algo- of lines is chosen. After suﬃcient lines are chosen, the rithm 1). All the branches for each OD pair are also de- presence of all nodes and essential links is checked. If they termined in this step, so it is checked for each OD pair which are all present, the solution is evaluated and the upper bound lines contain that OD pair and how long the detour is is adjusted if a new best solution is found. compared to the shortest path (line 4 in Algorithm 1). *en, the essential links are calculated (line 5 in Algorithm 1). 4.5. Sorting of OD Pairs. *e order in which OD pairs are Finally, the initial lower bound corresponds to the (probably selected to be branched upon greatly impacts the calculation unfeasible) solution where every passenger travels along the time of the algorithm. *ere are two elements considered shortest possible path in the infrastructure network without when selecting the next OD pair. One is to have as little lines any transfers. Based on all these precalculations, the branch as possible with a detour smaller than the transfer penalty. and bound algorithm can commence (line 6 in Algorithm 1). *e other is to have a large demand for an OD pair. *e ﬁrst *e branch and bound algorithm branches on all pos- limits the number of branches that need to be constructed, sible lines from the line pool that connects a chosen OD pair with a detour smaller than a transfer penalty (line 16 in and the latter increases the lower bound faster, allowing to prune more frequently. In this paper, experimental results Algorithm 1). One additional branch is considered where none of these lines are allowed (line 24 in Algorithm 1). for three diﬀerent sorting methods for OD pairs are dis- cussed. *e main sorting method combines both elements Here, the lower bound is increased with the transfer penalty mentioned earlier in this paragraph by ﬁrst sorting the OD multiplied with the demand of the OD pair (line 25 in pairs by decreasing the number of nodes in between and Algorithm 1). *e OD pair to branch on next is always then breaking any ties by putting the highest demand ﬁrst. selected based on the precalculations. *e branch and bound *is is the main sorting method used in the experiments. It algorithm keeps branching deeper and deeper until the set makes sure that the top levels have as little branches as amount of bus lines is selected (line 7 in Algorithm 1) or possible. until the lower bound of a branch exceeds the current best A ﬁrst alternative sorting method, which will be called solution (line 20 in Algorithm 1). Whenever the set amount of lines is selected, the presence of all nodes and essential “complex sort,” combines the number of branches and the demand in a single variable. *e number of branches is equal links is checked (line 8 in Algorithm 1). If this is the case, the Direct Link Network is constructed and the passenger routes to the amount of lines that contain the OD pair under consideration and that do not make a detour of at least one are determined using the adapted Floyd–Warshall 8 Journal of Advanced Transportation Generate dominant line pool Sort OD Pairs Pre-Calculations List of lines per OD pair Determine Essential Links Branch on OD-pair Select Branch Yes Branch and Bound Max #lines? Evaluate Solution No Update Bounds Update Bounds Close Branch Figure 5: Flowchart of the algorithm. algorithm. *e solution is then evaluated and compared to the results found by heuristics and are, hence, not included in the results. To use the insights gained in this paper for the current upper and lower bound. At the end, all relevant parts of the search space will have been explored and the larger networks, extra adjustments have to be made. *is will optimal solution is determined. be discussed further in Section 6. 5. Results 4.7. Alternative Approaches. *e most obvious approach to ﬁnd an optimal solution is a simple brute force solution. By *is section contains the results of the experiments on selecting all possible combinations of n lines out of the Mandl’s Swiss Network [14]. First Mandl’s Swiss Network is feasible line pool, all feasible solutions can be evaluated. *e introduced. *e actual experiments start with determining, computation time of this approach increases drastically for for the ﬁrst time, the optimal solutions for all available each extra line that can be part of the solution. *is approach instances on the network with the exact algorithm discussed becomes intractable in even very small instances, such as above. *ese solutions are also compared to the solutions Mandl’s network with ﬁve lines. found by state-of-the-art (meta) heuristics. *en, the im- Preliminary experiments on the Mumford0 instance portance of the newly introduced concepts is analyzed: the with 30 nodes and 90 links show that it takes up too much sorting method used, the essential links, the Direct Link memory to calculate all possible lines beforehand. To address Network, and ﬁnally using the entire branch and bound this, the line pool can be generated on the go. After selecting method instead of brute forcing a solution. an OD pair to branch on, the pool of lines with a detour of *e software algorithms were implemented in C++17 less than one transfer penalty can be calculated. Initial ex- and compiled with g++ (GCC) 9.3.0, and Docker was used to periments on larger networks and with a higher maximum setup a standalone image to run the experiments. *e number of lines took too much time to even come close to Docker containers were executed on a dedicated virtual Journal of Advanced Transportation 9 (1) Precalculations: (2) Generate pool of dominant lines (Section 4.1); (3) Sort OD Pairs (Section 4.5); (4) Construct list of lines per OD pair (Section 4.5); (5) Determine Essential Links (Section 4.2); (6) Recursive Branch and Bound (Section 4.4): (7) If (size of Current_Line_Plan equals maximum) (8) If (All essential links and nodes are present in Current_Line_Plan) (9) If (TTT<UpperBound) (10) Optimal_Line_Plan⟵Current_Line_Plan; (11) UpperBound⟵ TTT; (12) end (13) end (14) Else (15) Select OD pair (16) For (All lines serving the OD pair with a detour less than TP) (17) Next_Line_Plan⟵Current_Line_Plan; (18) Add line to the Next_Line_Plan; (19) Next_LowerBound⟵LowerBound + detour ∗ Demand of OD pair; (20) IF (Next_LowerBound<UpperBound) (21) Go one step deeper in the branch and bound with Next_Line_Plan (22) end (23) end (24) Go one step deeper in branch and bound without selecting a line (deviation> TP) (25) LowerBound⟵LowerBound + TP ∗ Demand of OD pair; (26) end (27) Return Optimal_Line_Plan ALGORITHM 1: Pseudocode of the algorithm. machine, running CoreOS, to minimize context switching 5.2. Exact Solution Algorithm. *e algorithm used here and external interference. *e virtual machine ran on works entirely as described in Section 4.6. Table 1 shows the Intel(R) Xeon(R) CPU E5-2640 v4@2.40 GHz hardware and optimal solutions with respect to Total Travel Time for all was granted 4 dedicated CPU cores (8 Hyper*reads) and instances (Number of Lines) with at most eight nodes per 16 GB dedicated RAM. *e C++ implementation exploits line. *e third column shows the Total Travel Time (TTT) the multicore setup by parallelizing the execution on several and the fourth column the Average Travel Time (ATT) in worker threads. All the data of the experiments and the minutes, which is the TTT per trip. *e ﬁfth column shows the CPU time of the algorithm, in seconds. *is CPU time is instances is available at https://www.mech.kuleuven.be/en/ cib/lp/mainpage#section-12. the time the algorithm spent in the branch and bound procedure. *e entire precalculations require close to 40 seconds (of which nearly all time is spent preparing the 5.1.Mandl’sSwissNetwork. Mandl’s Swiss Network (Figure 1) branches), but does not have to be repeated for every in- is a small network with 15 nodes and 21 links originally used in stance. *e ﬁnal column shows the number of solutions that Mandl [14]. In total, there are 15570 trips in the network and the were evaluated, in millions. Table 2 shows the actual lines demand is symmetric. It is one of the only publicly available that make up the optimal line plan for each instance. As a datasets for the line planning problem and, therefore, the most visual example, Figure 6 represents the optimal solution for used benchmark instance. *e most commonly used param- the instance with four lines and at most eight stops per line. eters for this network are used in this work. *e most common Note that every line passes by node ten. *is is expected since limiter of line length is by limiting the number of nodes per line this node alone is responsible for more or less one quarter of to eight. In this paper, both a maximum of eight nodes per line all the demand in the network. and an unlimited number of nodes per line are used. *ese *e CPU time keeps increasing until nine lines. Every “unlimited” lines will then be limited by the fact that nodes can extra line increases the amount of possible solutions dras- only be included once in a single line. For ﬁnding the optimal tically. *erefore, it is expected that more lines result in solutions, the results are shown for all instances with a relevant longer CPU times and more solutions checked. But, at ten amount of lines. When the newly introduced concepts are lines, the search time decreases again. *ere are several analyzed, only a subset of instances will be used to somewhat eﬀects at play here. More lines make it easier for the al- limit the required calculation time. *e transfer penalty is set to gorithm to get good upper bounds quickly because more ﬁve minutes, and this value is used in most publications lines result in better solutions. *ese upper bounds will also [4, 7, 8, 10]. be closer to the absolute lower bound. Both of these things 10 Journal of Advanced Transportation Table 1: Optimal solutions for Mandl’s Swiss network with at most 8 nodes per line. Number of lines Max nodes per line TTT (min) ATT (min) CPU time (s) Solutions checked (M) 3 169450 10.88 13 <1 4 163210 10.48 297 15 5 160450 10.31 3317 466 6 158500 10.18 18856 3785 7 157260 10.10 39753 6737 8 156750 10.07 161902 25704 9 156300 10.04 200872 24934 10 155940 10.02 29296 4032 11 155850 10.01 17054 1369 12 155820 10.01 16320 1044 13 155800 10.01 24883 1634 14 155790 10.01 1019 177 Table 2: Optimal solutions for 3 to 14 lines with at most 8 nodes per line. 3 lines 4 lines 5 lines Line 1: 1, 2, 3, 6, 8, 10, 11, and 13 Line 1: 1, 2, 3, 6, 8, 10, 11, and 12 Line 1: 1, 2, 3, 6, 8, 10, 14, and 13 Line 2: 5, 4, 6, 15, 7, 10, 14, and 13 Line 2: 1, 2, 5, 4, 6, 8, 10, and 11 Line 2: 1, 2, 3, 6, 15, 7, 10, and 11 Line 3: 2, 4, 12, 11, 10, 8, 15, and 9 Line 3: 10, 7, 15, 6, 3, 2, 4, and 12 Line 3: 1, 2, 5, 4, 6, 8, 10, and 11 Line 4: 9, 15, 8, 10, 14, 13, 11, and 12 Line 4: 2, 4, 12, 11, 10, 7, 15, and 9 Line 5: 7, 15, 8, 6, 4, 12, 11, and 13 6 lines 7 lines 8 lines Line 1: 1, 2, 3, 6, 8, 10, 11, and 13 Line 1: 1, 2, 3, 6, 8, 10, 11, and 13 Line 1: 1, 2, 3, 6, 8, 10, 11, and 13 Line 2: 1, 2, 3, 6, 15, 7, 10, and 14 Line 2: 1, 2, 3, 6, 15, 7, 10, and 14 Line 2: 1, 2, 3, 6, 15, 7, 10, and 14 Line 3: 1, 2, 5, 4, 6, 8, 10, and 14 Line 3: 1, 2, 5, 4, 6, 8, 10, and 14 Line 3: 1, 2, 5, 4, 6, 8, 10, and 14 Line 4: 3, 2, 5, 4, 6, 8, 15, and 7 Line 4: 3, 2, 5, 4, 6, 8, 15, and 7 Line 4: 3, 2, 5, 4, 6, 15, 7, and 10 Line 5: 5, 4, 12, 11, 10, 7, 15, and 9 Line 5: 5, 4, 12, 11, 10, 7, 15, and 9 Line 5: 5, 4, 12, 11, 10, 7, 15, and 9 Line 6: 1, 2, 4, 12, 11, 13, 14, and 10 Line 6: 9, 15, 6, 3, 2, 4, 12, and 11 Line 6: 9, 15, 6, 3, 2, 4, 12, and 11 Line 7: 1, 2, 4, 12, 11, 13, 14, and 10 Line 7: 1, 2, 4, 12, 11, 13, 14, and 10 Line 8: 12, 4, 6, 8, 15, 7, 10, and 13 9 lines 10 lines 11 lines Line 1: 1, 2, 3, 6, 8, 10, 11, and 13 Line 1: 1, 2, 3, 6, 8, 10, 11, and 13 Line 1: 1, 2, 3, 6, 8, 10, 11, and 13 Line 2: 1, 2, 3, 6, 15, 7, 10, and 13 Line 2: 1, 2, 3, 6, 15, 7, 10, and 8 Line 2: 1, 2, 3, 6, 15, 7, 10, and 8 Line 3: 1, 2, 3, 6, 15, and 9 Line 3: 1, 2, 3, 6, 15, and 9 Line 3: 1, 2, 3, 6, 15, and 9 Line 4: 12, 4, 2, 3, 6, 8, 10, and 14 Line 4: 12, 4, 2, 3, 6, 8, 10, and 14 Line 4: 12, 4, 2, 3, 6, 8, 10, and 14 Line 5: 1, 2, 5, 4, 6, 8, 10, and 11 Line 5: 5, 4, 6, 8, 10, 11, 13, and 14 Line 5: 5, 4, 6, 8, 10, 11, 13, and 14 Line 6: 3, 2, 5, 4, 6, 8, 15, and 7 Line 6: 1, 2, 5, 4, 6, 15, 7, and 10 Line 6: 1, 2, 5, 4, 6, 15, 7, and 10 Line 7: 5, 4, 12, 11, 10, 7, 15, and 9 Line 7: 5, 4, 12, 11, 10, 7, 15, and 9 Line 7: 5, 4, 12, 11, 10, 7, 15, and 9 Line 8: 9, 15, 8, 6, 4, 12, 11, and 10 Line 8: 3, 2, 5, 4, 6, 8, 15, and 9 Line 8: 3, 2, 5, 4, 6, 15, and 9 Line 9: 1, 2, 4, 12, 11, 13, 14, and 10 Line 9: 12, 4, 6, 8, 15, 7, 10, and 14 Line 9: 9, 15, 8, 6, 4, 12, 11, and 10 Line 10: 1, 2, 4, 12, 11, 13, 10, and 7 Line 10: 12, 4, 6, 8, 15, 7, 10, and 14 Line 11: 1, 2, 4, 12, 11, 13, 10, and 7 12 lines 13 lines 14 lines Line 1: 1, 2, 3, 6, 8, 10, 11, and 13 Line 1: 1, 2, 3, 6, 8, 10, 11, and 13 Line 1: 1, 2, 3, 6, 8, 10, 11, and 13 Line 2: 1, 2, 3, 6, 15, 7, 10, and 8 Line 2: 1, 2, 3, 6, 15, 7, 10, and 8 Line 2: 1, 2, 3, 6, 15, 7, 10, and 8 Line 3: 1, 2, 3, 6, 15, and 9 Line 3: 1, 2, 3, 6, 15, and 9 Line 3: 1, 2, 3, 6, 15, and 9 Line 4: 1, 2, 3, 6, 8, 10, 14, and 13 Line 4: 1, 2, 3, 6, 8, 10, 14, and 13 Line 4: 1, 2, 3, 6, 8, 10, 14, and 13 Line 5: 2, 5, 4, 6, 8, 10, 11, and 13 Line 5: 1, 2, 5, 4, 6, 8, 10, and 11 Line 5: 1, 2, 5, 4, 6, 8, 10, and 11 Line 6: 1, 2, 5, 4, 6, 15, 7, and 10 Line 6: 1, 2, 5, 4, 6, 15, 7, and 10 Line 6: 1, 2, 5, 4, 6, 15, 7, and 10 Line 7: 5, 4, 12, 11, 10, 7, 15, and 9 Line 7: 5, 4, 12, 11, 10, 7, 15, and 9 Line 7: 5, 4, 12, 11, 10, 7, 15, and 9 Line 8: 3, 2, 5, 4, 6, 15, and 9 Line 8: 3, 2, 5, 4, 6, 15, and 9 Line 8: 3, 2, 5, 4, 6, 15, and 9 Line 9: 9, 15, 8, 6, 4, 12, 11, and 10 Line 9: 9, 15, 8, 6, 4, 12, 11, and 10 Line 9: 7, 10, 11, 12, 4, 6, 15, and 9 Line 10: 12, 4, 6, 15, 7, 10, 14, and 13 Line 10: 2, 5, 4, 6, 8, 10, 14, and 13 Line 10: 2, 5, 4, 6, 8, 10, 14, and 13 Line 11: 1, 2, 4, 12, 11, 13, 14, and 10 Line 11: 1, 2, 4, 12, 11, 13, 14, and 10 Line 11: 1, 2, 4, 12, 11, 13, 10, and 7 Line 12: 7, 15, 8, 6, 3, 2, 4, and 12 Line 12: 7, 15, 8, 6, 3, 2, 4, and 12 Line 12: 9, 15, 8, 6, 3, 2, 4, and 12 Line 13: 12, 4, 6, 15, 7, 10, 14, and 13 Line 13: 7, 15, 6, 4, 12, 11, 13, and 14 Line 14: 12, 4, 6, 8, 15, 7, 10, and 14 Journal of Advanced Transportation 11 Tables 5 and 6 contain the comparison with the results from literature for a limit of eight and fourteen nodes per line, respectively. In Table 5, the results from our exact algorithm are compared with the results available from [4, 7, 14, 30]. To the best of our knowledge, these papers have the best ATT for this same problem in literature. In Table 6, the results obtained by [8, 10] are chosen since they solve 4 instances with a limit of fourteen nodes per line. *ese 15 papers report results for instances with four, six, seven, and eight lines. Obviously, the CPU times for the metaheuristics in Table 6 are much lower than for our exact algorithm. Unfortunately, no CPU times are reported for the two approaches reported in Table 6. Only for four lines with a maximum of eight nodes per line does one of the algorithms from literature ﬁnd the optimal solution (without knowing it is optimal), but in general, they ﬁnd a solution close to the optimal. However, the question can be raised whether it is useful to keep competing for ﬁnding better metaheuristic solutions on this small network. On this small network, many complex and computationally expensive methods (such as our exact al- gorithm) can be used to ﬁnd very good solutions. However, Figure 6: Optimal solution on Mandl’s Swiss network with 4 lines this does not mean the proposed method would work on a with at most 8 nodes each. realistic network. Hence, researchers should be careful when basing their conclusions only on results found for Mandl’s result in faster pruning in the branch and bound algorithm. Swiss Network. When lines get longer or when there are simply more lines, lower and upper bounds will be updated more often. *is results in even more pruning and is an important con- 5.3. Sorting. In this and following sections, the importance of diﬀerent parts of the algorithm is evaluated. In Section 4.5, tributor to keeping the CPU times under control for the higher amount of lines. Another added beneﬁt of having three diﬀerent sorting methods are introduced. All previous experiments were executed with the main sorting method. In more lines is that the likelihood of not selecting any of the lines at a branch decreases, which results in having to travel this section, it is analyzed what the impact is of sorting. *e main sorting method is compared with the complex sort and less deep to get to a solution. For example, when the absolute the random sort. *e sorting itself takes much less than a lower bound is achievable, this means that every passenger second (sorting 120 OD pairs based on a criteria). Based on can travel along its shortest path without transfers. *ere- the results from the previous section, only instances with fore, no matter which OD pair gets chosen to branch upon, four, eight, and twelve lines are considered in order to there will always be a line selected from the pool. For the somewhat limit the required total CPU time. With this instances with eight nodes per line, this happens at fourteen lines. *is is thus the minimum amount of lines necessary to subset, both a very small number of lines as well as a large number of lines are being tested. Eight lines is also chosen reach the lower bound with at most eight nodes per line. Table 3 shows the optimal solutions for the instances because it took a very long time to ﬁnd the optimal solution in the earlier experiments. *e random sort is executed three without any restriction on the number of nodes per line, times, and the best results are reported here. *e CPU time which results in a maximum of fourteen nodes per line in per instance (and per execution) is limited to 24 hours. *e this network. *e analysis is the same as for the previous results are presented in Table 7. results. Because of the longer lines, the quality of the A ﬁrst obvious conclusion is that the main sort used in solution is obviously better than with shorter lines. With the algorithm performs much better than random sort. thirteen lines, every passenger has a direct connection along its shortest possible path. *e actual details of the *ese results make it clear that sorting your OD pairs is very important in our branch and bound algorithm. *e complex optimal solutions can be found in Table 4. To the best of our knowledge, only the result for three lines were sort performs very good when considering a small amount of lines. With four lines, it is signiﬁcantly faster than the main previously known and published in Fiss and Ritt [66]. *eir Mixed Integer Programming implementation re- variant for both line lengths. For eight lines with at most eight nodes, it is a lot faster than the main sort. While with quired 78992 seconds (on their system) to ﬁnd the op- unlimited line lengths, the results are almost equal. For timal solution with three lines, and they calculated it twelve lines, the complex sort can never ﬁnish within a day. would take them 77 days to ﬁnd the optimal solution for Hence, it needs a lot more time than the main sort for high four lines. In comparison, our algorithm only needs 15 line numbers. *e main sort is chosen in our algorithm seconds (for three lines) and 168 seconds (for four lines) because it can ﬁnd the optimal solution for all instances. to ﬁnd the optimal solution. 12 Journal of Advanced Transportation Table 3: Optimal solutions for Mandl’s Swiss network with no limitations to route length. Number of lines Max nodes per line TTT (min) ATT (min) CPU time (s) Solutions checked (M) 3 163430 10.50 15 2 4 159990 10.28 168 44 5 158220 10.16 1641 433 6 157040 10.09 5357 1429 7 156550 10.05 33898 8002 8 14 156090 10.03 20359 5013 9 155930 10.01 14376 1829 10 155840 10.01 8002 816 11 155820 10.01 17590 1025 12 155800 10.01 46067 4036 13 155790 10.01 849 206 Table 4: Optimal solutions for 3 to 13 lines with at most 14 nodes per line. 3 lines 4 lines 5 lines Line 0: 1, 2, 3, 6, 8, 10, 14, 13, 11, 12, 4, and 5 Line 0: 1, 2, 3, 6, 8, 10, 11, 13, and 14 Line 0: 1, 2, 3, 6, 8, 10, 11, 13, and 14 Line 1: 1, 2, 5, 4, 6, 8, 10, 7, 15, and 9 Line 1: 1, 2, 4, 6, 8, 15, 7, 10, 11, and 12 Line 1: 1, 2, 4, 6, 8, 15, 7, 10, 14, 13, 11, and 12 Line 2: 5, 4, 2, 3, 6, 15, 7, 10, 11, and 12 Line 2: 13, 11, 12, 4, 5, 2, 3, 6, 15, 7, 10, and 14 Line 2: 13, 11, 12, 4, 5, 2, 3, 6, 15, 7, 10, and 14 Line 3: 1, 2, 5, 4, 6, 8, 10, 7, 15, and 9 Line 3: 1, 2, 5, 4, 6, 8, 10, 7, 15, and 9 Line 4: 7, 10, 11, 12, 4, 2, 3, 6, 15, and 9 6 lines 7 lines 8 lines Line 0: 1, 2, 3, 6, 8, 10, 11, 13, and 14 Line 0: 1, 2, 3, 6, 8, 10, 11, 13, and 14 Line 0: 1, 2, 3, 6, 8, 10, 11, 13, and 14 Line 1: 1, 2, 3, 6, 15, 7, 10, 14, 13, 11, 12, 4, and Line 1: 1, 2, 3, 6, 15, 7, 10, 14, 13, 11, 12, 4, and Line 1: 1, 2, 3, 6, 15, 7, 10, 14, 13, 11, 12, 4, and 5 5 5 Line 2: 1, 2, 5, 4, 6, 8, 10, 11, 13, and 14 Line 2: 1, 2, 5, 4, 6, 8, 10, 11, 13, and 14 Line 2: 1, 2, 3, 6, 15, and 9 Line 3: 3, 2, 5, 4, 6, 8, 15, 7, 10, 14, 13, 11, and Line 3: 3, 2, 5, 4, 6, 8, 15, 7, 10, 14, 13, 11, and Line 3: 1, 2, 5, 4, 6, 8, 10, 11, 13, and 14 12 12 Line 4: 1, 2, 4, 12, 11, 10, 7, 15, and 9 Line 4: 1, 2, 4, 12, 11, 10, 7, 15, and 9 Line 4: 3, 2, 5, 4, 6, 15, 7, 10, 14, 13, 11, and 12 Line 5: 8, 10, 14, 13, 11, 12, 4, 2, 3, 6, 15, and 9 Line 5: 8, 10, 14, 13, 11, 12, 4, 2, 3, 6, 15, and 9 Line 5: 1, 2, 4, 12, 11, 10, 7, 15, and 9 Line 6: 7, 10, 11, 12, 4, 6, 8, 15, and 9 Line 6: 7, 10, 11, 12, 4, 6, 8, 15, and 9 Line 7: 6, 3, 2, 4, 12, 11, 13, 14, 10, 8, 15, and 7 9 lines 10 lines 11 lines Line 0: 1, 2, 3, 6, 8, 10, 11, 13, and 14 Line 0: 1, 2, 3, 6, 8, 10, 11, 13, and 14 Line 0: 1, 2, 3, 6, 8, 10, 11, 13, and 14 Line 1: 1, 2, 3, 6, 15, 7, 10, 11, 12, 4, and 5 Line 1: 1, 2, 3, 6, 15, 7, 10, 11, 12, 4, and 5 Line 1: 1, 2, 3, 6, 15, 7, 10, and 8 Line 2: 1, 2, 3, 6, 15, and 9 Line 2: 1, 2, 3, 6, 15, and 9 Line 2: 1, 2, 3, 6, 15, and 9 Line 3: 11, 12, 4, 2, 3, 6, 8, 10, 14, and 13 Line 3: 11, 12, 4, 2, 3, 6, 8, 10, 14, and 13 Line 3: 1, 2, 3, 6, 8, 10, 14, 13, 11, 12, 4, and 5 Line 4: 1, 2, 5, 4, 6, 8, 10, 11, 13, and 14 Line 4: 1, 2, 5, 4, 6, 8, 10, 11, 13, and 14 Line 4: 1, 2, 5, 4, 6, 8, 10, 11, 13, and 14 Line 5: 1, 2, 5, 4, 6, 15, 7, 10, 14, 13, 11, and 12 Line 5: 1, 2, 5, 4, 6, 15, 7, 10, 14, 13, 11, and 12 Line 5: 1, 2, 5, 4, 6, 15, 7, 10, and 8 Line 6: 1, 2, 4, 12, 11, 10, 7, 15, and 9 Line 6: 1, 2, 4, 12, 11, 10, 7, 15, and 9 Line 6: 1, 2, 4, 12, 11, 10, 7, 15, and 9 Line 7: 3, 2, 5, 4, 6, 8, 15, and 9 Line 7: 3, 2, 5, 4, 6, 15, and 9 Line 7: 3, 2, 5, 4, 6, 15, and 9 Line 8: 7, 15, 8, 6, 4, 12, 11, 10, 13, and 14 Line 8: 7, 10, 11, 12, 4, 6, 8, 15, and 9 Line 8: 7, 10, 11, 12, 4, 6, 8, 15, and 9 Line 9: 7, 15, 8, 6, 4, 12, 11, 10, 13, and 14 Line 9: 11, 12, 4, 6, 15, 7, 10, 14, and 13 Line 10: 3, 2, 4, 12, 11, 10, 7, 15, 8, and 6 12 lines 13 lines Line 0: 1, 2, 3, 6, 8, 10, 11, 13, and 14 Line 0: 1, 2, 3, 6, 8, 10, 11, 13, and 14 Line 1: 1, 2, 3, 6, 15, 7, 10, and 8 Line 1: 1, 2, 3, 6, 15, 7, 10, and 8 Line 2: 1, 2, 3, 6, 15, and 9 Line 2: 1, 2, 3, 6, 15, and 9 Line 3: 1, 2, 3, 6, 8, 10, 14, 13, 11, 12, 4, and 5 Line 3: 1, 2, 3, 6, 8, 10, 14, 13, 11, 12, 4, and 5 Line 4: 1, 2, 5, 4, 6, 8, 10, 11, and 12 Line 4: 1, 2, 5, 4, 6, 8, 10, 11, and 12 Line 5: 1, 2, 5, 4, 6, 15, 7, 10, and 8 Line 5: 1, 2, 5, 4, 6, 15, 7, 10, and 8 Line 6: 1, 2, 4, 12, 11, 10, 7, 15, and 9 Line 6: 1, 2, 4, 12, 11, 10, 7, 15, and 9 Line 7: 1, 2, 5, 4, 6, 15, and 9 Line 7: 1, 2, 5, 4, 6, 15, and 9 Line 8: 7, 10, 11, 12, 4, 6, 8, 15, and 9 Line 8: 7, 10, 11, 12, 4, 6, 15, and 9 Line 9: 3, 2, 5, 4, 6, 8, 10, 14, 13, 11, and 12 Line 9: 3, 2, 5, 4, 6, 8, 10, 14, 13, 11, and 12 Line 10: 3, 2, 4, 12, 11, 10, 7, 15, 8, and 6 Line 10: 3, 2, 4, 12, 11, 10, 7, 15, 8, and 6 Line 11: 11, 12, 4, 6, 15, 7, 10, 14, and 13 Line 11: 7, 15, 6, 4, 12, 11, 10, and 8 Line 12: 7, 10, 14, 13, 11, 12, 4, 6, 8, 15, and 9 Journal of Advanced Transportation 13 Table 5: Comparison of the exact algorithm with literature for at most 8 nodes per line. Number of lines Max nodes per line Paper ATT (min) CPU time (s) Exact 10.48 297 Ahmed 10.48 450 4 Islam 10.51 7 Mandl 12.90 NA Chakroborty 11.90 NA Exact 10.18 18856 Ahmed 10.18 450 Islam 10.18 7 8 Chakroborty 10.48 NA Exact 10.10 39753 Ahmed 10.10 450 Islam 10.12 8 Chakroborty 10.42 NA Exact 10.07 161902 Ahmed 10.08 450 Islam 10.07 8 Chakroborty 10.36 NA Table 6: Comparison of the exact algorithm with literature for unlimited line length. Number of lines Max nodes per line Paper ATT (min) CPU time (s) Exact 10.28 168 4 Nayeem 10.33 NA Wu 10.35 NA Exact 10.09 5357 6 Nayeem 10.10 NA Wu 10.10 NA Exact 10.05 33898 7 Nayeem 10.07 NA Wu 10.07 NA Exact 10.03 20359 8 Nayeem — NA Wu 10.04 NA It can be explained why the complex sort is a lot faster impact on the lower bound. Hence, it is logical that the for instances with fewer lines, but it is unable to ﬁnd an complex sort performs better for smaller number of lines. optimal solution for larger number of lines. *e complex sort selects OD pairs with a low number of branches but a high demand. *is results in slightly more branches than 5.4.EssentialLinks. In this experiment, the algorithm is also executed without using the concept of essential links. Table 8 with the main sort, but the average quality of each branch shows the results for instances with four, eight, and twelve is higher. *e high demand means that when a detour lines. It is immediately clear that essential links result in a lot from the shortest path is selected, the lower bounds will less solutions that need to be evaluated. It is expected that increase more. *is results in more pruning. However, as the amount of lines rises, the number of branches be- this impact would be larger for fewer lines and fewer nodes per line. When there are fewer lines (and/or nodes) in a line comes more important for the total CPU time. Re- member, from Section 5.2, that it is more likely a line will plan, then there are fewer edges being served in that line plan. *erefore, it is more likely that certain essential links be selected at each branching point when more lines can be selected in the solution. So, for the early lines, more will not be present in solutions that are evaluated. For up to eight nodes per line, the experiments conﬁrm this trend. For branches will just result in exponentially more branches in the lower depths. At the same time, more and more the unlimited line length, this trend is less pronounced. For eight lines, there is only a minimal impact, while for twelve (almost all) of the high OD streams will be covered by the lines, the eﬀect is much larger. In all cases, the essential links actual shortest path. *is means there is almost no beneﬁt to selecting these lines ﬁrst since there will be no extra do provide a clear positive eﬀect on the CPU time, and for 14 Journal of Advanced Transportation Table 7: Impact of diﬀerent sorting methods on the exact algorithm. Number of lines Max nodes per line Sorting method CPU time (s) Solutions checked (M) Main 297 15 4 Complex 37 5 Random 1666 57 Main 161902 25704 8 8 Complex 6812 983 Random >86400 >9518 Main 16320 1044 12 Complex >86400 >524 Random >86400 >11685 Main 168 44 4 Complex 75 25 Random 3822 701 Main 20359 5013 8 14 Complex 21992 4164 Random >86400 >18 Main 46067 4036 12 Complex >86400 >10418 Random >86400 >4867 Table 8: *e impact of essential links on the exact algorithm. low line numbers, the CPU time is reduced by 35–45%. We could not measure a signiﬁcant diﬀerence between the Number of Max nodes Essential CPU Solutions precalculation times with or without essential links. lines per line links time (s) checked (M) Yes 297 15 No 545 188 5.5. Direct Link Network. In this section, the algorithm is run Yes 161902 25704 8 8 with the Change and Go (CNG) network representation instead No 225915 50765 of the proposed Direct Link Network (DLN) representation. Yes 16320 1044 Both algorithms still use the Floyd–Warshall algorithm to No 19067 2238 evaluate a solution. Table 9 shows the results for four, eight, and Yes 168 44 twelve lines. All tests ran for up to 24 hours. *e number of No 256 106 Yes 20359 5013 solutions checked is of course identical. Both methods use the 8 14 No 22975 6180 exact same sorting, so the same solutions have to be evaluated. Yes 46067 4036 *e impact of the DLN is signiﬁcant and appears to increase No 60121 7206 when the number of lines or nodes per line increases. *is behavior is expected, even for four lines with eight nodes per line, the CPU time increases with a factor four, and for ﬁfteen 5.6.BruteForce. We also developed a brute force algorithm to nodes per line, almost with a factor twenty. *e CNG creates an calculate the optimal solution. *e brute force algorithm simply extra node for each stop on each bus line, so more bus lines tries all combinations of nondominated lines and keeps the best result in more nodes in the CNG, while the DLN adds new links combination. *e algorithm started with three lines with eight and only keeps the best links. *erefore, the impact of more and nodes per line and kept increasing its number of lines until it larger lines is more limited. Note that, in literature, Dijkstra’s needs more than 24 h to reach the optimal solution. *e results algorithm is used more often than Floyd–Warshall, but we are shown in Table 10. For only three lines, the CPU time is choose to compare with our Floyd–Warshall algorithm to keep similar to that of our branch and bound algorithm. But, for four the comparison as fair as possible. Nevertheless, some pre- lines, it already needs more than ten times as much time. Since liminary tests comparing CNG with DLN using Dijkstra’s al- the brute force algorithms require exponentially more time gorithm gave the same type of results as shown here. Obviously, when adding an extra line, the algorithm was terminated after it this does not mean that DLN will always perform better than ran for 24 hours for ﬁve lines. *us, ﬁnding the optimal solution CNG, but these results deﬁnitely warrant further research into for ﬁve lines or more with this brute force algorithm is not using the DLN in line planning. It can be concluded that the possible in a reasonable amount of time. We conclude that the DLN representation is a major contributor to the speed of our branch and bound is eﬀective in limiting the amounts of so- algorithm. lutions that have to be evaluated. Journal of Advanced Transportation 15 Table 9: Impact of the Direct Link Network instead of the Change and Go network representation on the exact algorithm. Number of lines Max nodes per line Network model CPU time (s) Solutions checked (M) DLN 297 15 CNG 1377 15 DLN 161902 25704 8 8 CNG >86400 >125 DLN 16320 1044 CNG >86400 >69 DLN 168 44 CNG 3277 44 DLN 20359 5013 8 14 CNG >86400 >165 DLN 46067 4036 CNG >86400 >29 Table 10: Comparison of the branch and bound with a brute force algorithm. Number of lines Max nodes per line Method CPU time (s) Solutions checked (M) Branch and bound 13 <1 Brute force 27 16 Branch and bound 297 15 4 8 Brute force 3213 1957 Branch and bound 3317 466 Brute force >86400 >50605 (5.3). *ree diﬀerent sorting principles are implemented and 6. Conclusion tested, and the best way of sorting is implemented in our In this paper, a novel branch and bound algorithm is de- algorithm. By choosing OD pairs with minimal branches but veloped to ﬁnd optimal solutions for the uncapacitated line with high demand ﬁrst, the CPU time is signiﬁcantly planning problem with integrated passenger routing. *e decreased. objective is to minimize the total travel time of all pas- *e newly-developed Direct Link Network representa- sengers, and the available resources are constrained by the tion also reduces the required CPU time. For our algorithm, number of lines that can be operated and the maximum the gains were signiﬁcant compared to the traditionally used length of those lines. *e algorithm is applied to Mandl’s Change and Go Network (5.5). *ese results deﬁnitely Swiss network [14]. *is is by far the most used benchmark warrant to consider the use of the DLN in metaheuristics and instance in line planning research. However, until now, no other networks as well as looking into adjusting the DLN to optimal solutions have been determined for instances with deal with frequencies. Finally, in order to decide to actually more than three lines on this network. Our algorithm ob- evaluate a solution or not, the concept of essential links is tains optimal solutions for all available instances on this used. A link is essential if it has to be present in the optimal network, i.e., diﬀerent number of lines and two diﬀerent line solution. Essential links can be determined beforehand (4.2). *is had a positive eﬀect on the CPU time and was even lengths: for inﬁnite line length and for at most eight nodes per line (5.2). Furthermore, the minimum number of lines more pronounced for smaller line numbers and line lengths necessary to reach the lower bound, where every passenger (5.4). travels along his/her shortest path without transfers, is Finally, by calculating all optimal solutions for the determined. uncapacitated line planning problem, we hope to show that *e success of the algorithm is due to a number of new it is no longer useful to play the “up-the-wall” game [67] of concepts and the way the algorithm is constructed. *is is trying to beat the best algorithms on this small benchmark illustrated by the experimental results. By deﬁning all ex- network. Actually, it can be concluded that several state-of- tendable lines as dominated, the size of the pool of feasible the-art solution approaches for the uncapacitated line lines can already be greatly diminished (4.1). *e branch and planning problem obtain near-optimal solutions for almost bound algorithm itself chooses an OD pair to branch on. all these instances. *erefore, aiming to further improve Every feasible line connecting this OD pair directly with a these solutions for these instances is not very useful any- detour of less than one transfer penalty is considered as a more, but the focus should shift to making the approaches branch as well as not selecting a line. When a line is selected, much faster and, more importantly, to develop new concepts this line is assumed to provide the optimal routing for the for solving larger and more realistic instances. OD pair that was branched upon. *is creates many bounds According to us, our algorithm can be adapted to search for the OD pair and all possible pairs on the route. *e order for optimal solutions for other variants of the line planning in which the OD pairs are chosen is also very important problem, for example, by including frequencies as well. With 16 Journal of Advanced Transportation Swarm and Evolutionary Computation, vol. 46, pp. 154–170, frequencies, the line pool generation method has to be adjusted because the dominance criterion is no longer true. [8] M. A. Nayeem, M. K. Rahman, and M. S. Rahman, “Transit *us, there will be a larger line pool. *e passenger route network design by genetic algorithm with elitism,” Trans- choice will no longer be a shortest path problem, probably portation Research Part C: Emerging Technologies, vol. 46, resulting in having more links in the DLN. Both of these pp. 30–45, 2014. things will increase the required CPU time. Another pos- [9] M. Nikolic ´ and D. Teodorovic, ´ “Transit network design by bee sibility is to apply the method to larger networks. Although colony optimization,” Expert Systems with Applications, several problems occurred during preliminary testing with vol. 40, no. 15, pp. 5945–5955, 2013. an exact algorithm, it should be possible to integrate at least [10] R. Wu and S. Wang, “Discrete wolf pack search algorithm parts of this algorithm in an eﬃcient (meta) heuristic. If based transit network design,” in Proceedings of the 2016 7th instances are considered with a limited pool of lines gen- IEEE International Conference on Software Engineering and erated beforehand, our algorithm should be able to address Service Science, pp. 509–512, Beijing, China, 2016. larger instances as well. Furthermore, we are convinced that [11] M. H. Baaj and H. S. Mahmassani, “An AI-based approach for both the essential links and the DLN representation can be transit route system planning and design,” Journal of Ad- useful for developing exact and (meta) heuristic algorithms vanced Transportation, vol. 25, no. 2, pp. 187–209, 1991. for this or other line planning problems. For instance, the [12] L. Fan and C. L. Mumford, “A metaheuristic approach to the urban transit routing problem,” Journal of Heuristics, vol. 16, DLN representation could be tested in a metaheuristic no. 3, pp. 353–372, 2010. setting. Or the essential links could also be updated “dy- [13] P. N. Kechagiopoulos and G. N. Beligiannis, “Solving the namically,” i.e., every time the lower bound is increased. urban transit routing problem using a particle swarm opti- mization based algorithm,” Applied Soft Computing, vol. 21, Data Availability pp. 654–676, 2014. [14] C. E. Mandl, Applied Network Optimization, Academic Press, All the data of the experiments and the instances is available London, UK, 1979. at the following location: https://www.mech.kuleuven.be/ [15] C. Mumford, “Data and results 2013,” 2013, http://users.cs.cf. en/cib/lp/mainpage#section-12. ac.uk/C.L.Mumford/Research%20Topics/UTRP/ CEC2013Supp/. Conflicts of Interest [16] I. M. Cooper, M. P. John, R. Lewis, C. L. Mumford, and A. Olden, “Optimising large scale public transport network *e authors declare that there are no conﬂicts of interest design problems using mixed-mode parallel multi-objective regarding the publication of this paper. evolutionary algorithms,” in Proceedings of the 2014 IEEE Congress on Evolutionary Computation (CEC), pp. 2841–2848, Acknowledgments IEEE, Beijing, China, 2014. [17] X. Feng, X. Zhu, X. Qian, Y. Jie, F. Ma, and X. Niu, “A new *is research was funded by the Research Foundation transit network design study in consideration of transfer time Flanders, Project G.0853.16N. composition,” Transportation Research Part D: Transport and Environment, vol. 66, pp. 85–94, 2019. References [18] M. P. John, C. L. Mumford, and R. Lewis, “An improved multi-objective algorithm for the urban transit routing [1] A. Ceder, Public Transit Planning and Operation: Modeling, problem,” in Evolutionary Computation in Combinatorial Practice and Behavior, CRC Press, Boca Raton, FL, USA, 2nd Optimisation, C. Blum and G. Ochoa, Eds., Springer, Berlin, edition, 2016. Germany, pp. 49–60, 2014. [2] R. M. Lusby, J. Larsen, and S. Bull, “A survey on robustness in [19] F. Kılıç and M. Gok, ¨ “A demand based route generation railway planning,” European Journal of Operational Research, algorithm for public transit network design,” Computers & vol. 266, no. 1, pp. 1–15, 2018. Operations Research, vol. 51, pp. 21–29, 2014. [3] V. Guihaire and J.-K. Hao, “Transit network design and [20] J. Yang and Y. Jiang, “Application of modiﬁed NSGA-II to the scheduling: a global review,” Transportation Research Part A: transit network design problem,” Journal of Advanced Policy and Practice, vol. 42, no. 10, pp. 1251–1273, 2008. Transportation, vol. 2020, Article ID 3753601, 24 pages, 2020. [4] L. Ahmed, C. Mumford, and A. Kheiri, “Solving urban transit [21] B. Yao, P. Hu, X. Lu, J. Gao, and M. Zhang, “Transit network route design problem using selection hyper-heuristics,” Eu- design based on travel time reliability,” Transportation Re- ropean Journal of Operational Research, vol. 274, no. 2, search Part C: Emerging Technologies, vol. 43, pp. 233–248, pp. 545–559, 2019. [5] J. J. Blum and T. V. Mathew, “Intelligent agent optimization of [22] C. Iliopoulou, K. Kepaptsoglou, and E. Vlahogianni, “Meta- urban bus transit system design,” Journal of Computing in heuristics for the transit route network design problem: a Civil Engineering, vol. 25, no. 5, pp. 357–369, 2011. review and comparative analysis,” Public Transport, vol. 11, [6] J. Duran-Micco, E. Vermeir, and P. Vansteenwegen, “Con- no. 3, pp. 487–521, 2019. sidering emissions in the transit network design and fre- [23] A. Schobel ¨ and S. Scholl, “Line planning with minimal quency setting problem with a heterogeneous ﬂeet,” European traveling time,” in Proceedings of the 5th Workshop on Al- Journal of Operational Research, vol. 282, no. 2, pp. 580–592, gorithmic Methods and Models for Optimization of Railways, [7] K. A. Islam, I. M. Moosa, J. Mobin, M. A. Nayeem, and Palma de Mallorca, Spain, 2006. [24] R. Z. Farahani, E. Miandoabchi, W. Y. Szeto, and H. Rashidi, M. S. Rahman, “A heuristic aided stochastic beam search algorithm for solving the transit network design problem,” “A review of urban transportation network design problems,” Journal of Advanced Transportation 17 European Journal of Operational Research, vol. 229, no. 2, planning problem with maximum proﬁt,” Transportation pp. 281–302, 2013. Research Part E: Logistics and Transportation Review, vol. 127, [25] O. J. Ibarra-Rojas, F. Delgado, R. Giesen, and J. C. Muñoz, pp. 1–30, 2019. “Planning, operation, and control of bus transport systems: a [46] H. Dienst, Linienplanung im spurgefuhrten personenverkehr literature review,” Transportation Research Part B: Method- mithilfe eines heuristischen verfahrens, Ph.D. thesis, Techni- ological, vol. 77, pp. 38–75, 2015. sche Universitat ¨ Braunschweig, Braunschweig, Germany, [26] K. Kepaptsoglou and M. Karlaftis, “Transit route network 1978. design problem: review,” Journal of Transportation Engi- [47] M. R. Bussieck, U. Zimmerman, and P. Kreuzer, “Optimal lines for railway systems,” European Journal of Operational neering, vol. 135, no. 8, pp. 491–505, 2009. [27] A. Schobel, ¨ “Line planning in public transportation: models Research, vol. 96, pp. 54–63, 1996. and methods,” OR Spectrum, vol. 34, no. 3, pp. 491–510, 2012. [48] A. Bouma and C. Oltrogge, “Linienplanung und simulation [28] A. Patz, “Die richtige auswahl von verkehrslinien bei grossen fur ¨ oﬀentliche ¨ verkehrswege in praxis und theorie,” Eisen- strassenbahnnetzen,” Verkehrstechnik, vol. 51, 1925. bahntechnische Runschau, vol. 43, pp. 369–378, 1994. [29] C. E. Mandl, “Evaluation and optimization of urban public [49] J. F. Guan, H. Yang, and S. C. Wirasinghe, “Simultaneous transportation networks,” European Journal of Operational optimization of transit line conﬁguration and passenger line Research, vol. 5, no. 6, pp. 396–404, 1980. assignment,” Transportation Research Part B: Methodological, [30] P. Chakroborty and T. Wivedi, “Optimal route network vol. 40, no. 10, pp. 885–902, 2006. design for transit systems using genetic algorithms,” Engi- [50] M. Schmidt and A. Schobel, ¨ “*e complexity of integrating neering Optimization, vol. 34, no. 1, pp. 83–100, 2002. passenger routing decisions in public transportation models,” [31] S. B. Pattnaik, S. Mohan, and V. M. Tom, “Urban bus transit Networks, vol. 65, no. 3, pp. 228–243, 2015. route network design using genetic algorithm,” Journal of [51] E. Cipriani, S. Gori, and M. Petrelli, “Transit network design: a Transportation Engineering, vol. 124, no. 4, pp. 368–375, 1998. procedure and an application to a large urban area,” Trans- [32] F. Zhao and X. Zeng, “Simulated annealing-genetic algorithm portation Research Part C: Emerging Technologies, vol. 20, for transit network optimization,” Journal of Computing in no. 1, pp. 3–14, 2012. Civil Engineering, vol. 20, no. 1, pp. 57–68, 2006. [52] I. Constantin and M. Florian, “Optimizing frequencies in a [33] S. Chai and Q. Liang, “An improved NSGA-II algorithm for transit network: a nonlinear bi-level programming approach,” transit network design and frequency setting problem,” International Transactions in Operational Research, vol. 2, Journal of Advanced Transportation, vol. 2020, Article ID no. 2, pp. 149–164, 1995. [53] C. L. Mumford, “New heuristic and evolutionary operators for 2895320, 20 pages, 2020. [34] E. Vermeir, J. Duran ´ Micco, and P. Vansteenwegen, “*e grid the multi-objective urban transit routing problem,” in Pro- based approach, a fast local evaluation technique for line ceedings of the 2013 IEEE Congress on Evolutionary Compu- planning,” 4OR, 2021. tation, pp. 939–946, IEEE, Cancun, Mexico, 2013. [35] A. Mauttone and M. E. Urquhart, “A multi-objective meta- [54] B. Beltran, S. Carrese, E. Cipriani, and M. Petrelli, “Transit heuristic approach for the transit network design problem,” network design with allocation of green vehicles: a genetic Public Transport, vol. 1, no. 4, pp. 253–273, 2009. algorithm approach,” Transportation Research Part C: [36] M. Goerigk and M. Schmidt, “Line planning with user-op- Emerging Technologies, vol. 17, no. 5, pp. 475–483, 2009. timal route choice,” European Journal of Operational Re- [55] W. Fan and R. B. Machemehl, “Optimal transit route network design problem with variable transit demand: genetic algo- search, vol. 259, no. 2, pp. 424–436, 2017. [37] R. Borndorfer, ¨ M. Grotschel, ¨ and M. E. Pfetsch, “A column- rithm approach,” Journal of Transportation Engineering, generation approach to line planning in public transport,” vol. 132, no. 1, pp. 40–51, 2006. Transportation Science, vol. 41, no. 1, pp. 123–132, 2007. [56] H. Shimamoto, N. Murayama, A. Fujiwara, and J. Zhang, [38] R. Borndorfer ¨ and M. Karbstein, “A direct connection ap- “Evaluation of an existing bus network using a transit network proach to integrated line planning and passenger routing,” optimisation model: a case study of the Hiroshima city bus OpenAccess Series in Informatics, vol. 25, pp. 47–57, 2012. network,” Transportation, vol. 37, no. 5, pp. 801–823, 2010. [39] K. Nachtigall and K. Jerosch, “Simultaneous network line [57] H. Shimamoto, J.-D. Schmocker, ¨ and F. Kurauchi, “Opti- planning and traﬃc assignment,” OpenAccess Series in In- misation of a bus network conﬁguration and frequency considering the common lines problem,” Journal of Trans- formatics, vol. 9, 2008. [40] M. E. Schmidt, “Integrating routing decisions in public portation Technologies, vol. 2, no. 3, pp. 220–229, 2012. [58] R. O. Arbex and C. B. da Cunha, “Eﬃcient transit network transportation problems,” Springer Optimization and Its Applications, Springer, New York, NY, USA, 2014. design and frequencies setting multi-objective optimization [41] M. T. Claessens, N. M. Van Dijk, and P. J. Zwaneveld, “Cost by alternating objective genetic algorithm,” Transportation optimal allocation of rail passenger lines,”EuropeanJournalof Research Part B: Methodological, vol. 81, pp. 355–376, 2015. Operational Research, vol. 110, no. 3, pp. 474–489, 1998. [59] A. T. Buba and L. S. Lee, “A diﬀerential evolution for si- [42] M. R. Bussieck, Optimal Lines in Public Transport, Technische multaneous transit network design and frequency setting Universitat ¨ Braunschweig, Braunschweig, Germany, 1998. problem,” Expert Systems with Applications, vol. 106, pp. 277–289, 2018. [43] J.-W. Goossens, S. van Hoesel, and L. Kroon, “A branch-and- cut approach for solving railway line-planning problems,” [60] M. Nikolic ´ and D. Teodorovic, ´ “A simultaneous transit net- Transportation Science, vol. 38, no. 3, pp. 379–393, 2004. work design and frequency setting: computing with bees,” [44] M. R. Bussieck, T. Lindner, and M. E. Lubbecke, ¨ “A fast Expert Systems with Applications, vol. 41, no. 16, pp. 7200– algorithm for near cost optimal line plans,” Mathematical 7209, 2014. Methods of Operations Research, vol. 59, no. 2, pp. 205–220, [61] M. Owais, M. K. Osman, and G. Moussa, “Multi-objective 2004. transit route network design as set covering problem,” IEEE [45] D. Canca, A. De-Los-Santos, G. Laporte, and J. A. Mesa, Transactions on Intelligent Transportation Systems, vol. 17, “Integrated railway rapid transit network design and line no. 3, pp. 670–679, 2016. 18 Journal of Advanced Transportation [62] A. Ceder and N. H. M. Wilson, “Bus network design,” Transportation Research Part B: Methodological, vol. 20, no. 4, pp. 331–344, 1986. [63] H. Fu, L. Nie, L. Meng, B. R. Sperry, and Z. He, “A hierarchical line planning approach for a large-scale high speed rail network: the China case,” Transportation Research Part A: Policy and Practice, vol. 75, pp. 61–83, 2015. [64] D. B. Johnson, “Eﬃcient algorithms for shortest paths in sparse networks,” Journal of the ACM, vol. 24, no. 1, pp. 1–13, [65] R. R. Williams, “Faster all-pairs shortest paths via circuit complexity,” SIAM Journal on Computing, vol. 47, no. 5, pp. 1965–1985, 2018. [66] B. C. Fiss and M. Ritt, “Exact and metaheuristic algorithms for the urban transit routing problem,” in Proceedings of the Congreso Latino-Iberoamericano de Investigacion Operativa, Simposio Brasileiro de Pesquisa Operacional, pp. 3717–3728, Rio de Janeiro, Brazil, 2012. [67] K. Sorensen, ¨ “Metaheuristics-the metaphor exposed,” Inter- national Transactions in Operational Research, vol. 22, no. 1, pp. 3–18, 2015.

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An Exact Solution Approach for the Bus Line Planning Problem with Integrated Passenger Routing

An Exact Solution Approach for the Bus Line Planning Problem with Integrated Passenger Routing

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An Exact Solution Approach for the Bus Line Planning Problem with Integrated Passenger Routing

An Exact Solution Approach for the Bus Line Planning Problem with Integrated Passenger Routing

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