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A time–space network based exact optimization model for multi-depot bus scheduling
European Journal of Operational Research 175 (2006) 1616–1627 www.elsevier.com/locate/ejor
A time–space network based exact optimization model for multi-depot bus scheduling Natalia Kliewer *, Taı¨eb Mellouli, Leena Suhl Decision Support & Operations Research Lab, University of Paderborn, Warburger Street 100, 33098 Paderborn, Germany Available online 6 April 2005
Abstract The vehicle scheduling problem, arising in public transport bus companies, addresses the task of assigning buses to cover a given set of timetabled trips with consideration of practical requirements, such as multiple depots and vehicle types as well as depot capacities. An optimal schedule is characterized by minimal fleet size and minimal operational costs including costs for unloaded trips and waiting time. This paper discusses the multi-depot, multi-vehicle-type bus scheduling problem (MDVSP), involving multiple depots for vehicles and different vehicle types for timetabled trips. We use time–space-based instead of connection-based networks for MDVSP modeling. This leads to a crucial size reduction of the corresponding mathematical models compared to well-known connection-based network flow or set partitioning models. The proposed modeling approach enables us to solve real-world problem instances with thousands of scheduled trips by direct application of standard optimization software. To our knowledge, the largest problems that we solved to optimality could not be solved by any existing exact approach. The presented research results have been developed in co-operation with the provider of transportation planning software PTV AG. A software component to support planners in public transport was designed and implemented in context of this co-operation as well. 2005 Elsevier B.V. All rights reserved. Keywords: Transportation; Vehicle scheduling; Multi-depot bus scheduling; Time–space network; Decision support systems
1. Introduction 1.1. Motivation Been obliged to act market-oriented instead of the traditional monopolistic approach, public *
Corresponding author. E-mail address: [email protected] (N. Kliewer).
transport companies focus on efficient use of resources, especially vehicles and drivers. Operations research based decision support systems play hereby a crucial role. Service trips are provided as products with fixed departure and arrival times as well as start and end locations. The transport resources, operating these trips have to be allocated guaranteeing their availability at the right time on
0377-2217/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2005.02.030
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the right place, by minimizing total costs of resources. We consider the scheduling of vehicles under constraints and objectives arising in urban and suburban public transport. Thus, each timetabled trip can be served by a vehicle belonging to a given set of vehicle types. Each vehicle has to start and end its work day in one of the given depots (here: parking garage). After serving one timetabled (loaded) trip, each bus can serve one of the trips starting later from the station where the vehicle is standing, or it can change its location by moving unloaded to an another station (deadhead trip) in order to serve the next loaded trip starting there. The cost components include fixed costs for required vehicles as well as variable operational costs. The variable costs consist of distance-dependent travel costs and time-dependent costs for time spent outside the depot—the case where a driver is obliged to stay with the bus. All cost components depend on vehicle type. Since the fixed vehicle cost components are usually orders of magnitude higher than the operational costs, the optimal solution always involves the minimal number of vehicles. The combinatorial complexity of the multidepot bus scheduling problem (MDVSP1 in the following) is determined by numerous possibilities to assign vehicle type to each trip, to build sequences of trips for particular buses, and to assign buses to certain depots. To represent these sequences of trips, exact modeling approaches known in the literature consider explicitly all possible connections—pairs of trips that can be served successively. In context of this paper, we refer to all models that represent connections explicitly as ‘‘connection-based’’ models. Because connection-based MDVSP models inherently include a large number of arcs, such models of practical dimension with thousands of trips and multiple depots can often not be solved with standard optimization software packages. Known publications describe in detail applications of techniques such as column genera-
1 MDVSP means in the sense of this paper the MDMVTBSP— the multi-depot, multi-vehicle-type bus scheduling problem.
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tion and Lagrangean relaxation for the multi-commodity flow formulation (see Lo¨bel (1999)), or an auction algorithm for the quasi-assignment formulation (see Freling (1997)) of the MDVSP. In cases where suboptimal solutions are sufficient, other authors propose heuristic methods, such as schedule first—cluster second approaches (Daduna and Paixa˜o (1995)), or shortest pathbased heuristics (DellÕAmico et al. (1993)). Another approximate approach to cope with real-world instances is the artificial limitation of degrees of freedom for vehicle type and depot assigning. Furthermore, one may consider only a subset of possible deadhead trips (see, e.g., Lo¨bel (1999)). From our point of view, the explicit modeling of all possible connections seems to be inefficient for large scale planning problems, because the number of such pairs grows quadratically with the number of scheduled trips. 1.2. Intention The objective of this paper is to introduce a new modeling approach by applying a time–space network technique which avoids the explosive increase of the model size with a growing timetable. The new model should enable us to solve multi-depot vehicle scheduling problems of practical sizes and complexity, arizing in large public transport companies. Thus, we might be able to use standard optimization software (such as ILOG CPLEX or MOPS (see ILOG (2003) and Suhl (2003))), and profit from the recent significant improvements in optimization software. If we would use the specific algorithms developed for connection based models, it would not be possible to take advantage of the improvements in standard software. 1.3. Results We implemented the time–space network based modeling approach as a software component which has been integrated in commercial software packages to support planning processes in public transport. This software component generates mathematical models for given instances and
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solves them to optimality. We have carried out tests on real-life timetables of several public transport companies in Germany, such as Munich. Instances with more than 11,000 trips were provided by PTV AG. Concerning groups of vehicle types allowed for certain trips, this problem decomposes into four disjunct multi-depot problems. The largest of them has more than 7000 scheduled trips. 1.4. Outline The paper is organized as follows: Section 2 contains a definition of MDVSP. Section 3 describes MDVSP models known in literature, having substantial similarity in the representation of possible connections. In Section 4, a time–space network MDVSP model is introduced, and its advantages compared to existing models are discussed. Section 5 shows computational results on test instances and MIP-solver performance depending on problem complexity. Section 6 gives conclusions.
2. The multi-depot multi-vehicle type scheduling problem We define the vehicle scheduling problem (VSP), arising in public bus transportation, as the task of building an optimal set of rotations (vehicle schedule), such that each trip of a given timetable is covered by exactly one rotation. For each trip the timetable specifies a departure time and an arrival time with start and end stations respectively. Within a bus tour consisting of several (loaded) service trips chained with each other, the use of deadhead trips (unloaded trips between two end stations) often provides an improvement in order to serve all trips of a given timetable by a minimum number of buses. Thus a work day for a given bus is defined as a sequence of trips, deadheads, waiting times at stations (parking stops) and pull-out/pull-in trips from/to the assigned depot. Since deadhead trips mean an additional cost factor, minimization of this cost and minimization of waiting time cost are important optimization goals.
There are several variations of the bus scheduling problem involving different side constraints or numbers of depots and/or of bus types. The restrictions and optimization criteria may differ from one problem setting to another. The multi-depot vehicle scheduling problem involves several depots, so that a vehicle has to return in the evening to the same depot from which it started in the morning. Multi-vehicle-type VSP considers heterogeneous fleet of vehicles—for example normal bus, minibus, and kneel bus. For a given trip we define a group of vehicle types this trip can be served by. For example, a trip to a bus stop near a hospital can be served only by handicapped accessible vehicles such as kneel bus or normal bus with a ramp extension. In a feasible solution each rotation is assigned to exactly one depot and at least to one vehicle type. Furthermore, it is possible to state capacity restrictions for depots, which means that a depot has only a restricted number of (overnight) parking slots for buses. Other kinds of capacity constrains set a limit for the number of available vehicles of certain bus types and for the number of certain type vehicles in a given depot. It is well known that the single-depot homogeneous fleet vehicle scheduling problem is a polynomially solvable minimum cost flow problem, while the multiple depot and the heterogeneous fleet versions and their combinations imply that the optimization problem becomes NP-hard (see Bertossi et al. (1987)). The complexity of instances of the MDVSP depends on various factors, such as: • the number of timetabled trips, • the number of depots, or more precisely, the average number of depot-vehicle type combinations per timetabled trip, • the number of possible unloaded trips, which can vary depending on the completeness of the distance matrix for stop points. Fig. 1 shows a simple example for a timetable with three trips and a station topology with distances expressed in time units. Stations k3 and k4 are depots. Trips a and c can be served by buses
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d1(k3) Stop points and depots
Timetable Trip From a b c
k1 k2 k2
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To k2 k1 k1
Dep. time 1 3 5
Arr. time 2 4 6
d1 + +
D2 + + +
k1
1
1 time unit k2
d2(k4)
Fig. 1. An example for a timetable with three trips and distance matrix for 2 stop points and 2 depots.
from depots d1 and d2, whereas trip b can only be served from depot d2. In the following, we discuss state-of-the-art solution techniques for the MDVSP as well as present a new model for solving them with standard optimization software. Differences between connection-based and time–space-network-based models will be illustrated on the example from Fig. 1.
3. Existing connection-based MDVSP models Traditionally, mathematical optimization models for public mass transit are based on a network where trips and depots are represented by nodes (trip-nodes and depot-nodes, respectively), and possible connections between trips given by arcs (see, e.g., Daduna and Paixa˜o (1995)). The rotation of a vehicle is thus generated as a flow of one unit starting in the depot and returning back into it. The nodes connected by a flow represent the scheduled trips, carried out by the corresponding vehicle. The single-depot case of the VSP can be modeled as a minimum cost flow problem. Vehicles represent the flow units circulating in this network. Each arc is provided with the cost of its corresponding activity—stand time cost, deadhead or fixed vehicle cost. The objective is to minimize the total flow costs over all arcs. All flow units involved in the single-depot problem are of the same type, so that the incoming and outgoing flow of each trip-node has to be equal to one. The mathematical model consists of the minimum cost flow problem with flow conservation constraints for each network node. No integrality constraints are needed because of the unimodularity of the
coefficient matrix of the linear programming relaxation. For the multi-depot case, the network is multiplied, so that there is a network layer for each depot. Mathematical formulation of the model contains additional restrictions, called cover constraints, guaranteeing that each scheduled trip is carried out by exactly one vehicle belonging to one of the required vehicle types. Because of the explicit integrality constraints (all flow variables must be equal to 0 or 1), the optimization model is no more a minimum cost flow model and is now difficult to solve. Fig. 2 illustrates an application of the traditional modeling approach to the example from Fig. 1. Each trip to be scheduled is represented by a departure node and an arrival node, together with an arc joining them. Other arcs join arrival nodes with later departure nodes, representing the possibility of a single vehicle undertaking the two corresponding trips in succession. Further arcs represent bus movements to and from depot nodes. Each arc has an associated cost factor (deadhead cost for arcs joining arrival nodes to later departure nodes, vehicle cost for pull-out and/or pull-in arcs), and capacity (one for service trips, depot capacities for circular arcs). The network involves two network layers: one layer for each depot. A flow solution is feasible only if the flow value is equal to one on exactly one of all service trip arcs representing a certain service trip in different layers (see for example Fig. 2: unbroken arcs a in d1 and d2 layers). The optimal flow (feasible flow with minimum total costs) for this network now determines the optimal vehicle schedule for the original timetable. A circulation flow down to the right gives an example for feasible and fleet-minimal solution.
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Service trip arcs Trip compatibility arcs
a
Pull-in/pull-out arcs
depot d1 c
Circulation flow arc d1-Layer
a depot d2
b c d2-Layer Fig. 2. The example from Fig. 1 modeled with the connection-based approach.
We can clearly identify three different approaches among the existing modeling techniques: • path-oriented—leading to set partitioning formulation (see Ribeiro and Soumis (1994)), • arc-oriented—leading to multi-commodity flow formulation (compare e.g. Forbes et al. (1994)), and • combinations of these two approaches (see Carpaneto et al. (1989)). In all these modeling approaches, the possible trip connections are considered explicitly and the number of such connections, corresponding to the number of integer variables, grows quadratically in dependence of the number of loaded trips. Therefore, models with several thousand scheduled trips become too large to be solved directly by standard optimization tools in a reasonable time. Various techniques to reduce the number of possible connections have been proposed in the literature. Some approaches discard arcs with too long waiting times, others generate arcs applying the column generation idea to the network flow representation. 4. A new time–space network flow model for MDVSP Time–space network (TSN) models have been proposed for routing problems in airline schedul-
ing (see Hane et al. (1995) as an example), because they are advantageous in modeling possible connections between arriving and departing flights. In a time–space network, connections within a location are realized by using a time line that connects all possible landing and takeoff events within the location. Thus, there is no need to explicitly model connections for each feasible pair of events within a location. Time–space network models were not used for bus scheduling problems until now, because, compared to airline scheduling where deadheading is generally not allowed, bus scheduling permits unrestricted deadheading. Thus the advantages given by TSN remained negligible, because of too many deadhead arcs. However, we show in this section that a new modeling technique leads to a drastic reduction of arcs, simultaneously allowing all possible deadheads, so that we may exploit the advantages of TSN models for bus scheduling problems. 4.1. Modeling of connections within the time–space network The new TSN model is based on aggregation of possible connection arcs so that it is possible to carry several connections on one single arc simultaneously. Basically, we have to take into account three types of potential connections:
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• connections within one station, • connections over a depot with pull in/pull out trips to/from the depot, and • connections through a directly connecting trip (deadhead).
4.1.1. Connections within one station A connection within one station means the (standard) case where the successor trip starts at the same station where the predecessor trip ends. We build a time line for each station, as well as for depots. A time line contains all arrivals and departures at the given location at their corresponding points of time. The network contains a trip arc (Fig. 3: unbroken arcs) for each scheduled trip—from the start event on the time line of the
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start station to the end event on the time line of the end station. In contrast to connection-based models from the literature, the connections within one station must not be modeled explicitly by connection arcs, but can be implicitly represented by the flow through the timeline. The compatible trip pairs (a, b) and (a, c) from Fig. 2 can now be connected by waiting arcs (Fig. 3: dotted arcs) in the time line of station k2. 4.1.2. Connections over a depot with pull in/pull out trips to/from the depot In analogy to stations we build a time line for each depot, although there may not be scheduled trips starting or ending directly in a depot. To each scheduled trip i we introduce, if necessary, arcs for
Fig. 3. The example from Fig. 1 modeled with new time–space network approach.
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potential pull-out and pull-in trips from/to each depot directly before/after carrying out i (with associated deadhead costs) (Fig. 3: punctured arcs). Because it is more favorable for buses to stand at a depot than at other stations, we place a higher cost for waiting arcs outside the depots, therefore avoiding long waiting times outside the depots. 4.1.3. Connections through a directly connecting trip (deadhead) Basically it is possible to connect each pair of compatible trips through deadheading. However, as the number of possible deadhead trips can be extremely high, a direct modeling of all of them in large networks implies a problem size which cannot be handled by state-of-the-art solvers.
Thus, a crucial modeling technique is aggregation of possible matches—directly connecting trips between scheduled trips. Fig. 4 shows the equivalence to the connection-based approach relating to the feasible solutions set of VSP. The aggregation of possible trip connections is carried out in three steps, as explained below: First stage aggregation For each arriving scheduled trip i at station k we determine the first trip compatible with i at each other station l(k 5 l). We call this trip firstmatch (i, l). We introduce only those deadheading arcs into the model, which correspond to the first matches. Thus, the number of arcs is reduced significantly compared to the original situation. Nevertheless, all possible connections remain feasible. Each scheduled trip j compatible to a scheduled
1. Stage with all possible matches Space
Station k
Time
Matches Station l
2. Stage with waiting arcs in station l
Station k
Station l
First Matches
Station k Latest First Matches Station l
Fig. 4. Two-stage aggregation of possible connections.
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trip i can now be reached in the model network over first-match (i, start-station (j))—possibly via waiting arcs at start-station (j). Second stage aggregation At the second stage, the number of arcs is further reduced. We aggregate the set of first matches in a smaller set of latest first matches (see Fig. 4). The latest first matches can be determined in the following way: Let S be the set of incoming trips i at station k, having the same first match jS at station l. Let iL be the latest incoming trip in S. Then the latest first match for each element in S is the first match (iL, k). By removing all arcs corresponding to first matches but being no latest first matches, we reduce the network significantly, but do not loose any possible connections. Every scheduled trip j compatible to scheduled trip i can now be reached over the corresponding latest first match to the start station of j, and possibly via waiting arcs in one or both connection lines. Third stage aggregation There are still some redundancies in the network after the elimination all possible connections being no latest first matches. For deadhead trips, having enough time for temporarily return to depot, we evaluate costs arising for such a temporary return and compare them to costs of directly connecting with a deadhead trip. In case of positive cost savings by a depot-return we eliminate this specific latest first match in our network, guaranteeing the connection still by depotreturn. 4.1.4. Model reduction: Computational results The total number of matches, which determines the number of variables in connection-based network models for MDVSP, depends quadratically on the number of trips given by the timetable.
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However, the number of first matches is limited by the number of bus stations multiplied by the number of timetabled trips. Since the number of stations is always smaller than the number of trips, we notice a crucial model reduction compared to existing approaches. The last two aggregation stages yield an additional strengthening of the model size. Table 1 illustrates the impact of the three-stage aggregation process for timetables of some German public transport companies. We define here matches as possible connections between compatible trips at different stations. The table shows explicitly the number of matches, first matches, and latest first matches. Furthermore, the rightmost column denotes the number of latest first matches that are cheaper than traveling over depot, thus leaving out the direct connection trips that are dominated by trips over depot. The largest reduction arises for urban timetables with short distances between stations (see T2 to T4). Only 1% of the matches remain in the reduced set of latest first matches. Even for the regional timetable (T1), having long distances between stations, we achieve a reduction of 97%. 4.2. Construction of the network Because we have to assure that each bus returns back to its original depot at the end of a day, we need to distinguish between buses from different home depots. Furthermore, certain trips have to be taken over by buses of a group of given bus types, so that we need to distinguish between bus types as well. Therefore, we construct one network layer for each depot-vehicle-type-combination, called depot for simplicity, containing the following arcs:
Table 1 Scale of problem reduction by the new formulation Timetable
Scheduled trips
Stations
Matches
First matches
Latest first matches
T1 T2 T3 T4
682 2047 3054 7068
50 21 49 124
195,618 1,143,868 3,906,045 20,555,784
19,457 42,863 81,265 513,393
6292 17,791 52,898 246,580
T1: regional bus company; T2–T4: urban.
Lfm not-over-depot 6262 11,885 48,572 213,433
(3%) (1%) (1%) (1%)
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• a loaded trip arc for each timetabled trip which can be operated from this depot, • deadhead arcs, according to latest first matches, which are cheaper than temporarily return to the depot, and • pull-in/pull-out arcs to the depot after/before each loaded trip. For each station, including depots, we build a connection line representing all possible arrival and departure events at the given station—after pre-sorting of arcs according to arrival/departure time respectively. One node of this network connects a group of possible arrivals to the following group of possible departures (see also Fig. 3). In this way all stations, including depots, are represented as ordered sets of (connection) nodes, linked together by waiting arcs. On each layer, there is one circulation flow arc connecting the last node of the depot line with its first node. This arc is provided with fixed cost for corresponding vehicle type and represents vehicles parking over night in depot. There are different ways to express the capacity of a depot. Our approach incorporates the following specific modeling approaches: • We set flow upper bounds on circulation flow arcs for capacities of depot-vehicle type combinations. • The number of parking slots in certain depots generates additional restrictions in the mathematical model formulation constraining the sum of flow values on corresponding circulation flow arcs. • For a limited number of vehicles of certain types we also constrain the sum of corresponding flow values. • If it is required to use a fixed number of buses of certain type or of certain depot, we set a lower bound equal to the upper bound (variable fixing in mathematical model) on the corresponding arc. Waiting arcs together with arcs for unloaded trips are also provided with corresponding operational costs. Upper bounds on the loaded trip-arcs are equal to one.
In case of minimizing the number of buses before considering operational cost components as an objective, we should choose the cost factors properly: The costs on a circulation arc must be large enough in comparison to the operational costs (costs for deadheads) in order to induce the effect of minimizing the number of buses. Fig. 3 shows the complete time–space network for the MDVSP—example from Fig. 1. Down to the right a feasible and fleet-minimal solution is given by some arcs. Flow values on these arcs are equal to one; missing arcs from the above network have a null-flow in this solution. The resulting flow model contains one network layer for each depot (as defined above), where 0/1variables on trip arcs and integral flow variables on other arcs are defined. In addition to the flow balance equations, it is required by cover constraints that for each scheduled trip the sum over flow variables on the corresponding trip arcs of all network layers is equal to one. The optimal solution of the mathematical model describes flow values in this network with minimal total costs. 4.3. Decomposition of the aggregated flows It is an important characteristic of the time– space network formulation that due to the aggregation of possible connections, any feasible flow, including also an optimal flow, represents a bundle or a class of vehicle schedules. All of them have minimal total costs but different distributions of waiting times. With the help of a suitable flow decomposition procedure, we may extract a vehicle schedule with an optimal flow and desired characteristics. A large number of possible flow decomposition algorithms may be constructed to decompose a given optimal flow. For example, the first-in-first-out (FIFO) strategy connects the bus that arrived first with the trip that leaves first, whereas the last-in-first-out (LIFO) strategy connects the bus that arrived last with the trip that leaves first, both strategies having specific typical properties. Fig. 5 demonstrates the number of choices in the case that the flow of the given connection arc has the value of three units in the optimal solution.
N. Kliewer et al. / European Journal of Operational Research 175 (2006) 1616–1627
3 flow units
Fig. 5. Degree of freedom in flow decomposition.
There are 3! = 6 variations to connect arrivals to the departures.
5. Implementation and optimization runs Based on the description given in the previous section, our implementation of the time–space network based solution procedure to solve a given MDVSP involves the following steps: • Read the data and construct the layers of a time–space network. • Create a mathematical model for min cost flow with additional cover constraint for each unloaded trip. • Optimize the mixed-integer programming model with standard software, and compute an optimal flow representing a class of aggregated schedules. • Decompose this flow to rotations of one concrete optimal vehicle schedule, according to LIFO, FIFO or any intermediate strategy. In the context of a co-operation with the software developer PTV AG, we designed and implemented a software component which supports public transport planners in constructing schedules for buses. PTV AG provided us test instances of several public transport companies. The largest timetable (T4 in Table 1: City of Munich) contains: • • • •
11.018 scheduled trips, 33 vehicle types in 8 groups, 24 depots, and results in 4 disjunctive MDVS problems; largest problem has 7.068 scheduled trips and 124 end stations.
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Table 2 summarizes the sizes of the mathematical models, objective values and optimization times for 25 test instances based on three timetables. All solution times are shown in seconds of CPU time on a 2.1 GHz processor using ILOG CPLEX 8.0 as a standard optimization package. Except for the larges problem instances, we also received acceptable run times with the standard optimizer MOPS (Suhl (2000)). The number of depots means actually depotvehicle-type combinations, as mentioned above, and thus corresponds to the terminology in Lo¨bel (1999). The given number of depots may be different from the average depot group size—for such cases we additionally provide the average depot group size in parenthesis (last 10 instances). The entry ‘‘3 (2.5)’’ means, that for a timetable-trip on the average 2.5 different depot-assignments are possible, although there are three depots in the network. Note that the number of trip matches in the corresponding column gives a lower bound for the number of variables in the existing connection-based multi-commodity flow formulation. With the proposed modeling approach, we were able to solve quite large problems in a reasonable amount of time without special tuning of the used optimization software. Despite the fact that this problem is NP-hard, most of the solution time is spent solving the first LP relaxation. Comparing various LP solvers leads to the conclusion that the barrier optimizer is the best choice to solve large problems with many degrees of freedom for each loaded trip. This time–space network based model has a very good MIP behavior; the IP-gap is infinitesimal small or null, and almost all variables have integer values in the optimal (basis) solution of the LP-relaxation. The reason for this lies in the model properties: because of the aggregation of possible connections, the mathematical model tends to use one general integer variable instead of several binary variables. Thus, we practically shift the corresponding decisions to the postprocessing phase, where we construct an optimal vehicle schedule from the optimal network flow via flow decomposition.
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Table 2 Computational results for timestables T2, T3 and T4
Trips
7.068
3.054
2.047
Trip matches arcs Stop Depots* Trip points matches MCF
Model size TS-MCF
Obj./1000 MIP optimization time (in seconds)
First matches Latest first Rows · Cols · Nzs TS-MCF matches TS-MCF(d)
20,555,784 246,580 41,111,568 493,160 61,667,352 739,740 82,223,136 986,320 102,778,920 1,232,900
213,433 432,737 616,432 825,624 1,039,137
20,575 34,072 47,574 60,690 73,866
· · · · ·
248,021 · 503,110 501,903 · 1,017,942 720,181 · 1,461,566 963,175 · 1,954,622 1,210,542 · 2,456,424
Network Dual
1,014,213 12 1,014,110 136 1,014,050 7079 1,013,999 24,762 1,013,913 53,143
Best LP solver
Barrier Best time
124
1 2 3 4 5
12 207 12 150 1,313 136 4608 3435 3435 10,030 5863 5863 49,394 11,887 11,887
49
1 2 3 4 5 6 7 8 9 10
3,906,045 7,812,090 11,718,135 15,624,180 19,530,225 23,436,270 27,342,315 31,248,360 35,154,405 39,060,450
52,898 105,796 158,694 211,592 264,490 317,388 370,286 423,184 476,082 528,980
48,572 98,057 136,290 187,299 233,316 273,126 322,513 363,118 409,191 450,937
9597 · 64,231 · 131,516 16,060 · 129,295 · 264,698 22,611 · 183,195 · 375,552 29,076 · 249,775 · 511,766 35,267 · 310,703 · 636,686 41,370 · 365,185 · 748,694 47,766 · 429,933 · 881,244 54,096 · 485,834 · 996,100 60,416 · 547,138 · 1,121,762 66,747 · 604,151 · 1,238,842
508,223 508,103 508,056 508,049 508,002 507,939 507,922 507,870 507,829 507,822
2 24 246 805 1106 1115 1448 2039 2617 3264
1 20 136 482 646 858 1237 1334 2893 4423
22 158 420 988 1510 1510 1826 2568 4126 3136
2 20 36 482 646 858 1237 1334 2617 3136
21
3 (2.5) 6 (5) 9 (7.5) 12 (10) 15 (12.5) 18 (15) 21 (17.5) 24 (20) 27 (22.5) 30 (25)
1,143,868 2,115,896 3,397,601 4,754,636 6,153,772 7,473,783 9,005,549 10,468,848 12,014,270 13,558,845
17,791 34,955 52,864 70,882 90,390 108,458 128,764 148,143 168,591 188,927
11,885 23,118 40,813 56,101 72,007 85,873 101,448 119,162 138,008 157,526
8798 · 27,599 · 58,485 14,997 · 53,249 · 113,039 21,939 · 87,128 · 184,111 29,031 · 118,768 · 250,675 36,235 · 151,369 · 319,161 43,232 · 181,409 · 382,525 50,647 · 214,110 · 451,211 57,841 · 248,580 · 523,435 65,223 · 284,539 · 598,637 72,636 · 321,198 · 675,239
421,582 421,545 421,544 421,537 421,533 421,531 421,525 421,521 421,517 421,516
4 37 148 371 972 1566 3385 2798 3708 6069
4 40 106 271 669 1179 1652 3099 3551 4689
15 67 146 244 356 824 598 568 785 905
4 37 106 244 356 824 598 568 785 905
Dual = Network Network Barrier Barrier Barrier Dual Dual Dual Dual Dual Dual Dual Dual Network Barrier Dual Dual Dual Barrier Barrier Barrier Barrier Barrier Barrier Barrier
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Instance
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Several practical side constraints that could not be addressed in this paper have been successfully included into the software component, such as: outsourcing of the parts of timetables to private bus companies, returns to different depots, different types of fleet size limitations, and so on.
6. Conclusion We propose a new application of the time–space network flow model for the vehicle scheduling problem arising in public bus transport networks. The model size has been substantially reduced through aggregation of incoming and outgoing arcs within each station, without loss of generality. Thus, we were able to solve very large practical instances to optimality through direct application of standard optimization software. Many of these instances were not solvable with any existing approach according the current state-of-the-art. The main advantage of the time–space network model is the reduction of the number of variables in the exact optimization model. Inherently, there is no straightforward way to explicitly model costs of a given trip-to-trip connection within the aggregated model. In bus scheduling case with linear cost structures the individual connections of trips or flights are not necessary, and therefore the time–space network model is to be preferred to the traditional connection based models.
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References Bertossi, I., Carraresi, P., Gallo, G., 1987. On some matching problems arising in vehicle scheduling models. Networks 17, 271–281. Carpaneto, G., DellÕAmico, M., Fischetti, M., Toth, P., 1989. A branch and bound algorithm for the multiple depot vehicle scheduling problem. Networks 19, 531–548. Daduna, J.R., Paixa˜o, J.M.P., 1995. Vehicle scheduling for public mass transit—an overview. In: Daduna, J.R. et al. (Eds.), Computer-aided Transit Scheduling, LNEMS, vol. 430. Springer, Berlin, pp. 76–90. DellÕAmico, M., Fischetti, M., Toth, P., 1993. Heuristic algorithms for the multiple depot vehicle scheduling problems. Management Science 39, 115–125. Forbes, M.A., Holt, J.N., Watts, A.M., 1994. An exact algorithm for multi-depot bus scheduling. European Journal of Operational Research 72, 115–124. Freling, R., (1997). Models and Techniques for Integrating Vehicle and Crew Scheduling. Ph.D. thesis, Tinbergen Institute, Erasmus University Rotterdam. Hane, C., Barnhart, C., Johnson, E.L., Marsten, R.E., Nemhauser, G.L., Sigismondi, G., 1995. The fleet assignment problem: Solving a large integer program. Mathematical Programming 70 (2), 211–232. ILOG, 2003. Cplex v8.0 UserÕs Manual. ILOG, Gentilly, France, 2002. Lo¨bel, A., 1999. Solving large-scale multiple-depot vehicle scheduling problems. Computer-Aided Transit Scheduling 471, 193–220. Ribeiro, C., Soumis, F., 1994. A column generation approach to the multiple-depot vehicle scheduling problem. Operations Research 42 (1), 41–52. Suhl, U.H., 2000. MOPS—Mathematical OPtimization System. OR News 8, 11–16. Suhl, U.H., 2003. MOPS—Mathematical Optimization System. Available from:.
A time–space network based exact optimization model for multi-depot bus scheduling Natalia Kliewer *, Taı¨eb Mellouli, Leena Suhl Decision Support & Operations Research Lab, University of Paderborn, Warburger Street 100, 33098 Paderborn, Germany Available online 6 April 2005
Abstract The vehicle scheduling problem, arising in public transport bus companies, addresses the task of assigning buses to cover a given set of timetabled trips with consideration of practical requirements, such as multiple depots and vehicle types as well as depot capacities. An optimal schedule is characterized by minimal fleet size and minimal operational costs including costs for unloaded trips and waiting time. This paper discusses the multi-depot, multi-vehicle-type bus scheduling problem (MDVSP), involving multiple depots for vehicles and different vehicle types for timetabled trips. We use time–space-based instead of connection-based networks for MDVSP modeling. This leads to a crucial size reduction of the corresponding mathematical models compared to well-known connection-based network flow or set partitioning models. The proposed modeling approach enables us to solve real-world problem instances with thousands of scheduled trips by direct application of standard optimization software. To our knowledge, the largest problems that we solved to optimality could not be solved by any existing exact approach. The presented research results have been developed in co-operation with the provider of transportation planning software PTV AG. A software component to support planners in public transport was designed and implemented in context of this co-operation as well. 2005 Elsevier B.V. All rights reserved. Keywords: Transportation; Vehicle scheduling; Multi-depot bus scheduling; Time–space network; Decision support systems
1. Introduction 1.1. Motivation Been obliged to act market-oriented instead of the traditional monopolistic approach, public *
Corresponding author. E-mail address: [email protected] (N. Kliewer).
transport companies focus on efficient use of resources, especially vehicles and drivers. Operations research based decision support systems play hereby a crucial role. Service trips are provided as products with fixed departure and arrival times as well as start and end locations. The transport resources, operating these trips have to be allocated guaranteeing their availability at the right time on
0377-2217/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2005.02.030
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the right place, by minimizing total costs of resources. We consider the scheduling of vehicles under constraints and objectives arising in urban and suburban public transport. Thus, each timetabled trip can be served by a vehicle belonging to a given set of vehicle types. Each vehicle has to start and end its work day in one of the given depots (here: parking garage). After serving one timetabled (loaded) trip, each bus can serve one of the trips starting later from the station where the vehicle is standing, or it can change its location by moving unloaded to an another station (deadhead trip) in order to serve the next loaded trip starting there. The cost components include fixed costs for required vehicles as well as variable operational costs. The variable costs consist of distance-dependent travel costs and time-dependent costs for time spent outside the depot—the case where a driver is obliged to stay with the bus. All cost components depend on vehicle type. Since the fixed vehicle cost components are usually orders of magnitude higher than the operational costs, the optimal solution always involves the minimal number of vehicles. The combinatorial complexity of the multidepot bus scheduling problem (MDVSP1 in the following) is determined by numerous possibilities to assign vehicle type to each trip, to build sequences of trips for particular buses, and to assign buses to certain depots. To represent these sequences of trips, exact modeling approaches known in the literature consider explicitly all possible connections—pairs of trips that can be served successively. In context of this paper, we refer to all models that represent connections explicitly as ‘‘connection-based’’ models. Because connection-based MDVSP models inherently include a large number of arcs, such models of practical dimension with thousands of trips and multiple depots can often not be solved with standard optimization software packages. Known publications describe in detail applications of techniques such as column genera-
1 MDVSP means in the sense of this paper the MDMVTBSP— the multi-depot, multi-vehicle-type bus scheduling problem.
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tion and Lagrangean relaxation for the multi-commodity flow formulation (see Lo¨bel (1999)), or an auction algorithm for the quasi-assignment formulation (see Freling (1997)) of the MDVSP. In cases where suboptimal solutions are sufficient, other authors propose heuristic methods, such as schedule first—cluster second approaches (Daduna and Paixa˜o (1995)), or shortest pathbased heuristics (DellÕAmico et al. (1993)). Another approximate approach to cope with real-world instances is the artificial limitation of degrees of freedom for vehicle type and depot assigning. Furthermore, one may consider only a subset of possible deadhead trips (see, e.g., Lo¨bel (1999)). From our point of view, the explicit modeling of all possible connections seems to be inefficient for large scale planning problems, because the number of such pairs grows quadratically with the number of scheduled trips. 1.2. Intention The objective of this paper is to introduce a new modeling approach by applying a time–space network technique which avoids the explosive increase of the model size with a growing timetable. The new model should enable us to solve multi-depot vehicle scheduling problems of practical sizes and complexity, arizing in large public transport companies. Thus, we might be able to use standard optimization software (such as ILOG CPLEX or MOPS (see ILOG (2003) and Suhl (2003))), and profit from the recent significant improvements in optimization software. If we would use the specific algorithms developed for connection based models, it would not be possible to take advantage of the improvements in standard software. 1.3. Results We implemented the time–space network based modeling approach as a software component which has been integrated in commercial software packages to support planning processes in public transport. This software component generates mathematical models for given instances and
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solves them to optimality. We have carried out tests on real-life timetables of several public transport companies in Germany, such as Munich. Instances with more than 11,000 trips were provided by PTV AG. Concerning groups of vehicle types allowed for certain trips, this problem decomposes into four disjunct multi-depot problems. The largest of them has more than 7000 scheduled trips. 1.4. Outline The paper is organized as follows: Section 2 contains a definition of MDVSP. Section 3 describes MDVSP models known in literature, having substantial similarity in the representation of possible connections. In Section 4, a time–space network MDVSP model is introduced, and its advantages compared to existing models are discussed. Section 5 shows computational results on test instances and MIP-solver performance depending on problem complexity. Section 6 gives conclusions.
2. The multi-depot multi-vehicle type scheduling problem We define the vehicle scheduling problem (VSP), arising in public bus transportation, as the task of building an optimal set of rotations (vehicle schedule), such that each trip of a given timetable is covered by exactly one rotation. For each trip the timetable specifies a departure time and an arrival time with start and end stations respectively. Within a bus tour consisting of several (loaded) service trips chained with each other, the use of deadhead trips (unloaded trips between two end stations) often provides an improvement in order to serve all trips of a given timetable by a minimum number of buses. Thus a work day for a given bus is defined as a sequence of trips, deadheads, waiting times at stations (parking stops) and pull-out/pull-in trips from/to the assigned depot. Since deadhead trips mean an additional cost factor, minimization of this cost and minimization of waiting time cost are important optimization goals.
There are several variations of the bus scheduling problem involving different side constraints or numbers of depots and/or of bus types. The restrictions and optimization criteria may differ from one problem setting to another. The multi-depot vehicle scheduling problem involves several depots, so that a vehicle has to return in the evening to the same depot from which it started in the morning. Multi-vehicle-type VSP considers heterogeneous fleet of vehicles—for example normal bus, minibus, and kneel bus. For a given trip we define a group of vehicle types this trip can be served by. For example, a trip to a bus stop near a hospital can be served only by handicapped accessible vehicles such as kneel bus or normal bus with a ramp extension. In a feasible solution each rotation is assigned to exactly one depot and at least to one vehicle type. Furthermore, it is possible to state capacity restrictions for depots, which means that a depot has only a restricted number of (overnight) parking slots for buses. Other kinds of capacity constrains set a limit for the number of available vehicles of certain bus types and for the number of certain type vehicles in a given depot. It is well known that the single-depot homogeneous fleet vehicle scheduling problem is a polynomially solvable minimum cost flow problem, while the multiple depot and the heterogeneous fleet versions and their combinations imply that the optimization problem becomes NP-hard (see Bertossi et al. (1987)). The complexity of instances of the MDVSP depends on various factors, such as: • the number of timetabled trips, • the number of depots, or more precisely, the average number of depot-vehicle type combinations per timetabled trip, • the number of possible unloaded trips, which can vary depending on the completeness of the distance matrix for stop points. Fig. 1 shows a simple example for a timetable with three trips and a station topology with distances expressed in time units. Stations k3 and k4 are depots. Trips a and c can be served by buses
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d1(k3) Stop points and depots
Timetable Trip From a b c
k1 k2 k2
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To k2 k1 k1
Dep. time 1 3 5
Arr. time 2 4 6
d1 + +
D2 + + +
k1
1
1 time unit k2
d2(k4)
Fig. 1. An example for a timetable with three trips and distance matrix for 2 stop points and 2 depots.
from depots d1 and d2, whereas trip b can only be served from depot d2. In the following, we discuss state-of-the-art solution techniques for the MDVSP as well as present a new model for solving them with standard optimization software. Differences between connection-based and time–space-network-based models will be illustrated on the example from Fig. 1.
3. Existing connection-based MDVSP models Traditionally, mathematical optimization models for public mass transit are based on a network where trips and depots are represented by nodes (trip-nodes and depot-nodes, respectively), and possible connections between trips given by arcs (see, e.g., Daduna and Paixa˜o (1995)). The rotation of a vehicle is thus generated as a flow of one unit starting in the depot and returning back into it. The nodes connected by a flow represent the scheduled trips, carried out by the corresponding vehicle. The single-depot case of the VSP can be modeled as a minimum cost flow problem. Vehicles represent the flow units circulating in this network. Each arc is provided with the cost of its corresponding activity—stand time cost, deadhead or fixed vehicle cost. The objective is to minimize the total flow costs over all arcs. All flow units involved in the single-depot problem are of the same type, so that the incoming and outgoing flow of each trip-node has to be equal to one. The mathematical model consists of the minimum cost flow problem with flow conservation constraints for each network node. No integrality constraints are needed because of the unimodularity of the
coefficient matrix of the linear programming relaxation. For the multi-depot case, the network is multiplied, so that there is a network layer for each depot. Mathematical formulation of the model contains additional restrictions, called cover constraints, guaranteeing that each scheduled trip is carried out by exactly one vehicle belonging to one of the required vehicle types. Because of the explicit integrality constraints (all flow variables must be equal to 0 or 1), the optimization model is no more a minimum cost flow model and is now difficult to solve. Fig. 2 illustrates an application of the traditional modeling approach to the example from Fig. 1. Each trip to be scheduled is represented by a departure node and an arrival node, together with an arc joining them. Other arcs join arrival nodes with later departure nodes, representing the possibility of a single vehicle undertaking the two corresponding trips in succession. Further arcs represent bus movements to and from depot nodes. Each arc has an associated cost factor (deadhead cost for arcs joining arrival nodes to later departure nodes, vehicle cost for pull-out and/or pull-in arcs), and capacity (one for service trips, depot capacities for circular arcs). The network involves two network layers: one layer for each depot. A flow solution is feasible only if the flow value is equal to one on exactly one of all service trip arcs representing a certain service trip in different layers (see for example Fig. 2: unbroken arcs a in d1 and d2 layers). The optimal flow (feasible flow with minimum total costs) for this network now determines the optimal vehicle schedule for the original timetable. A circulation flow down to the right gives an example for feasible and fleet-minimal solution.
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Service trip arcs Trip compatibility arcs
a
Pull-in/pull-out arcs
depot d1 c
Circulation flow arc d1-Layer
a depot d2
b c d2-Layer Fig. 2. The example from Fig. 1 modeled with the connection-based approach.
We can clearly identify three different approaches among the existing modeling techniques: • path-oriented—leading to set partitioning formulation (see Ribeiro and Soumis (1994)), • arc-oriented—leading to multi-commodity flow formulation (compare e.g. Forbes et al. (1994)), and • combinations of these two approaches (see Carpaneto et al. (1989)). In all these modeling approaches, the possible trip connections are considered explicitly and the number of such connections, corresponding to the number of integer variables, grows quadratically in dependence of the number of loaded trips. Therefore, models with several thousand scheduled trips become too large to be solved directly by standard optimization tools in a reasonable time. Various techniques to reduce the number of possible connections have been proposed in the literature. Some approaches discard arcs with too long waiting times, others generate arcs applying the column generation idea to the network flow representation. 4. A new time–space network flow model for MDVSP Time–space network (TSN) models have been proposed for routing problems in airline schedul-
ing (see Hane et al. (1995) as an example), because they are advantageous in modeling possible connections between arriving and departing flights. In a time–space network, connections within a location are realized by using a time line that connects all possible landing and takeoff events within the location. Thus, there is no need to explicitly model connections for each feasible pair of events within a location. Time–space network models were not used for bus scheduling problems until now, because, compared to airline scheduling where deadheading is generally not allowed, bus scheduling permits unrestricted deadheading. Thus the advantages given by TSN remained negligible, because of too many deadhead arcs. However, we show in this section that a new modeling technique leads to a drastic reduction of arcs, simultaneously allowing all possible deadheads, so that we may exploit the advantages of TSN models for bus scheduling problems. 4.1. Modeling of connections within the time–space network The new TSN model is based on aggregation of possible connection arcs so that it is possible to carry several connections on one single arc simultaneously. Basically, we have to take into account three types of potential connections:
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• connections within one station, • connections over a depot with pull in/pull out trips to/from the depot, and • connections through a directly connecting trip (deadhead).
4.1.1. Connections within one station A connection within one station means the (standard) case where the successor trip starts at the same station where the predecessor trip ends. We build a time line for each station, as well as for depots. A time line contains all arrivals and departures at the given location at their corresponding points of time. The network contains a trip arc (Fig. 3: unbroken arcs) for each scheduled trip—from the start event on the time line of the
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start station to the end event on the time line of the end station. In contrast to connection-based models from the literature, the connections within one station must not be modeled explicitly by connection arcs, but can be implicitly represented by the flow through the timeline. The compatible trip pairs (a, b) and (a, c) from Fig. 2 can now be connected by waiting arcs (Fig. 3: dotted arcs) in the time line of station k2. 4.1.2. Connections over a depot with pull in/pull out trips to/from the depot In analogy to stations we build a time line for each depot, although there may not be scheduled trips starting or ending directly in a depot. To each scheduled trip i we introduce, if necessary, arcs for
Fig. 3. The example from Fig. 1 modeled with new time–space network approach.
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potential pull-out and pull-in trips from/to each depot directly before/after carrying out i (with associated deadhead costs) (Fig. 3: punctured arcs). Because it is more favorable for buses to stand at a depot than at other stations, we place a higher cost for waiting arcs outside the depots, therefore avoiding long waiting times outside the depots. 4.1.3. Connections through a directly connecting trip (deadhead) Basically it is possible to connect each pair of compatible trips through deadheading. However, as the number of possible deadhead trips can be extremely high, a direct modeling of all of them in large networks implies a problem size which cannot be handled by state-of-the-art solvers.
Thus, a crucial modeling technique is aggregation of possible matches—directly connecting trips between scheduled trips. Fig. 4 shows the equivalence to the connection-based approach relating to the feasible solutions set of VSP. The aggregation of possible trip connections is carried out in three steps, as explained below: First stage aggregation For each arriving scheduled trip i at station k we determine the first trip compatible with i at each other station l(k 5 l). We call this trip firstmatch (i, l). We introduce only those deadheading arcs into the model, which correspond to the first matches. Thus, the number of arcs is reduced significantly compared to the original situation. Nevertheless, all possible connections remain feasible. Each scheduled trip j compatible to a scheduled
1. Stage with all possible matches Space
Station k
Time
Matches Station l
2. Stage with waiting arcs in station l
Station k
Station l
First Matches
Station k Latest First Matches Station l
Fig. 4. Two-stage aggregation of possible connections.
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trip i can now be reached in the model network over first-match (i, start-station (j))—possibly via waiting arcs at start-station (j). Second stage aggregation At the second stage, the number of arcs is further reduced. We aggregate the set of first matches in a smaller set of latest first matches (see Fig. 4). The latest first matches can be determined in the following way: Let S be the set of incoming trips i at station k, having the same first match jS at station l. Let iL be the latest incoming trip in S. Then the latest first match for each element in S is the first match (iL, k). By removing all arcs corresponding to first matches but being no latest first matches, we reduce the network significantly, but do not loose any possible connections. Every scheduled trip j compatible to scheduled trip i can now be reached over the corresponding latest first match to the start station of j, and possibly via waiting arcs in one or both connection lines. Third stage aggregation There are still some redundancies in the network after the elimination all possible connections being no latest first matches. For deadhead trips, having enough time for temporarily return to depot, we evaluate costs arising for such a temporary return and compare them to costs of directly connecting with a deadhead trip. In case of positive cost savings by a depot-return we eliminate this specific latest first match in our network, guaranteeing the connection still by depotreturn. 4.1.4. Model reduction: Computational results The total number of matches, which determines the number of variables in connection-based network models for MDVSP, depends quadratically on the number of trips given by the timetable.
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However, the number of first matches is limited by the number of bus stations multiplied by the number of timetabled trips. Since the number of stations is always smaller than the number of trips, we notice a crucial model reduction compared to existing approaches. The last two aggregation stages yield an additional strengthening of the model size. Table 1 illustrates the impact of the three-stage aggregation process for timetables of some German public transport companies. We define here matches as possible connections between compatible trips at different stations. The table shows explicitly the number of matches, first matches, and latest first matches. Furthermore, the rightmost column denotes the number of latest first matches that are cheaper than traveling over depot, thus leaving out the direct connection trips that are dominated by trips over depot. The largest reduction arises for urban timetables with short distances between stations (see T2 to T4). Only 1% of the matches remain in the reduced set of latest first matches. Even for the regional timetable (T1), having long distances between stations, we achieve a reduction of 97%. 4.2. Construction of the network Because we have to assure that each bus returns back to its original depot at the end of a day, we need to distinguish between buses from different home depots. Furthermore, certain trips have to be taken over by buses of a group of given bus types, so that we need to distinguish between bus types as well. Therefore, we construct one network layer for each depot-vehicle-type-combination, called depot for simplicity, containing the following arcs:
Table 1 Scale of problem reduction by the new formulation Timetable
Scheduled trips
Stations
Matches
First matches
Latest first matches
T1 T2 T3 T4
682 2047 3054 7068
50 21 49 124
195,618 1,143,868 3,906,045 20,555,784
19,457 42,863 81,265 513,393
6292 17,791 52,898 246,580
T1: regional bus company; T2–T4: urban.
Lfm not-over-depot 6262 11,885 48,572 213,433
(3%) (1%) (1%) (1%)
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• a loaded trip arc for each timetabled trip which can be operated from this depot, • deadhead arcs, according to latest first matches, which are cheaper than temporarily return to the depot, and • pull-in/pull-out arcs to the depot after/before each loaded trip. For each station, including depots, we build a connection line representing all possible arrival and departure events at the given station—after pre-sorting of arcs according to arrival/departure time respectively. One node of this network connects a group of possible arrivals to the following group of possible departures (see also Fig. 3). In this way all stations, including depots, are represented as ordered sets of (connection) nodes, linked together by waiting arcs. On each layer, there is one circulation flow arc connecting the last node of the depot line with its first node. This arc is provided with fixed cost for corresponding vehicle type and represents vehicles parking over night in depot. There are different ways to express the capacity of a depot. Our approach incorporates the following specific modeling approaches: • We set flow upper bounds on circulation flow arcs for capacities of depot-vehicle type combinations. • The number of parking slots in certain depots generates additional restrictions in the mathematical model formulation constraining the sum of flow values on corresponding circulation flow arcs. • For a limited number of vehicles of certain types we also constrain the sum of corresponding flow values. • If it is required to use a fixed number of buses of certain type or of certain depot, we set a lower bound equal to the upper bound (variable fixing in mathematical model) on the corresponding arc. Waiting arcs together with arcs for unloaded trips are also provided with corresponding operational costs. Upper bounds on the loaded trip-arcs are equal to one.
In case of minimizing the number of buses before considering operational cost components as an objective, we should choose the cost factors properly: The costs on a circulation arc must be large enough in comparison to the operational costs (costs for deadheads) in order to induce the effect of minimizing the number of buses. Fig. 3 shows the complete time–space network for the MDVSP—example from Fig. 1. Down to the right a feasible and fleet-minimal solution is given by some arcs. Flow values on these arcs are equal to one; missing arcs from the above network have a null-flow in this solution. The resulting flow model contains one network layer for each depot (as defined above), where 0/1variables on trip arcs and integral flow variables on other arcs are defined. In addition to the flow balance equations, it is required by cover constraints that for each scheduled trip the sum over flow variables on the corresponding trip arcs of all network layers is equal to one. The optimal solution of the mathematical model describes flow values in this network with minimal total costs. 4.3. Decomposition of the aggregated flows It is an important characteristic of the time– space network formulation that due to the aggregation of possible connections, any feasible flow, including also an optimal flow, represents a bundle or a class of vehicle schedules. All of them have minimal total costs but different distributions of waiting times. With the help of a suitable flow decomposition procedure, we may extract a vehicle schedule with an optimal flow and desired characteristics. A large number of possible flow decomposition algorithms may be constructed to decompose a given optimal flow. For example, the first-in-first-out (FIFO) strategy connects the bus that arrived first with the trip that leaves first, whereas the last-in-first-out (LIFO) strategy connects the bus that arrived last with the trip that leaves first, both strategies having specific typical properties. Fig. 5 demonstrates the number of choices in the case that the flow of the given connection arc has the value of three units in the optimal solution.
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3 flow units
Fig. 5. Degree of freedom in flow decomposition.
There are 3! = 6 variations to connect arrivals to the departures.
5. Implementation and optimization runs Based on the description given in the previous section, our implementation of the time–space network based solution procedure to solve a given MDVSP involves the following steps: • Read the data and construct the layers of a time–space network. • Create a mathematical model for min cost flow with additional cover constraint for each unloaded trip. • Optimize the mixed-integer programming model with standard software, and compute an optimal flow representing a class of aggregated schedules. • Decompose this flow to rotations of one concrete optimal vehicle schedule, according to LIFO, FIFO or any intermediate strategy. In the context of a co-operation with the software developer PTV AG, we designed and implemented a software component which supports public transport planners in constructing schedules for buses. PTV AG provided us test instances of several public transport companies. The largest timetable (T4 in Table 1: City of Munich) contains: • • • •
11.018 scheduled trips, 33 vehicle types in 8 groups, 24 depots, and results in 4 disjunctive MDVS problems; largest problem has 7.068 scheduled trips and 124 end stations.
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Table 2 summarizes the sizes of the mathematical models, objective values and optimization times for 25 test instances based on three timetables. All solution times are shown in seconds of CPU time on a 2.1 GHz processor using ILOG CPLEX 8.0 as a standard optimization package. Except for the larges problem instances, we also received acceptable run times with the standard optimizer MOPS (Suhl (2000)). The number of depots means actually depotvehicle-type combinations, as mentioned above, and thus corresponds to the terminology in Lo¨bel (1999). The given number of depots may be different from the average depot group size—for such cases we additionally provide the average depot group size in parenthesis (last 10 instances). The entry ‘‘3 (2.5)’’ means, that for a timetable-trip on the average 2.5 different depot-assignments are possible, although there are three depots in the network. Note that the number of trip matches in the corresponding column gives a lower bound for the number of variables in the existing connection-based multi-commodity flow formulation. With the proposed modeling approach, we were able to solve quite large problems in a reasonable amount of time without special tuning of the used optimization software. Despite the fact that this problem is NP-hard, most of the solution time is spent solving the first LP relaxation. Comparing various LP solvers leads to the conclusion that the barrier optimizer is the best choice to solve large problems with many degrees of freedom for each loaded trip. This time–space network based model has a very good MIP behavior; the IP-gap is infinitesimal small or null, and almost all variables have integer values in the optimal (basis) solution of the LP-relaxation. The reason for this lies in the model properties: because of the aggregation of possible connections, the mathematical model tends to use one general integer variable instead of several binary variables. Thus, we practically shift the corresponding decisions to the postprocessing phase, where we construct an optimal vehicle schedule from the optimal network flow via flow decomposition.
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Table 2 Computational results for timestables T2, T3 and T4
Trips
7.068
3.054
2.047
Trip matches arcs Stop Depots* Trip points matches MCF
Model size TS-MCF
Obj./1000 MIP optimization time (in seconds)
First matches Latest first Rows · Cols · Nzs TS-MCF matches TS-MCF(d)
20,555,784 246,580 41,111,568 493,160 61,667,352 739,740 82,223,136 986,320 102,778,920 1,232,900
213,433 432,737 616,432 825,624 1,039,137
20,575 34,072 47,574 60,690 73,866
· · · · ·
248,021 · 503,110 501,903 · 1,017,942 720,181 · 1,461,566 963,175 · 1,954,622 1,210,542 · 2,456,424
Network Dual
1,014,213 12 1,014,110 136 1,014,050 7079 1,013,999 24,762 1,013,913 53,143
Best LP solver
Barrier Best time
124
1 2 3 4 5
12 207 12 150 1,313 136 4608 3435 3435 10,030 5863 5863 49,394 11,887 11,887
49
1 2 3 4 5 6 7 8 9 10
3,906,045 7,812,090 11,718,135 15,624,180 19,530,225 23,436,270 27,342,315 31,248,360 35,154,405 39,060,450
52,898 105,796 158,694 211,592 264,490 317,388 370,286 423,184 476,082 528,980
48,572 98,057 136,290 187,299 233,316 273,126 322,513 363,118 409,191 450,937
9597 · 64,231 · 131,516 16,060 · 129,295 · 264,698 22,611 · 183,195 · 375,552 29,076 · 249,775 · 511,766 35,267 · 310,703 · 636,686 41,370 · 365,185 · 748,694 47,766 · 429,933 · 881,244 54,096 · 485,834 · 996,100 60,416 · 547,138 · 1,121,762 66,747 · 604,151 · 1,238,842
508,223 508,103 508,056 508,049 508,002 507,939 507,922 507,870 507,829 507,822
2 24 246 805 1106 1115 1448 2039 2617 3264
1 20 136 482 646 858 1237 1334 2893 4423
22 158 420 988 1510 1510 1826 2568 4126 3136
2 20 36 482 646 858 1237 1334 2617 3136
21
3 (2.5) 6 (5) 9 (7.5) 12 (10) 15 (12.5) 18 (15) 21 (17.5) 24 (20) 27 (22.5) 30 (25)
1,143,868 2,115,896 3,397,601 4,754,636 6,153,772 7,473,783 9,005,549 10,468,848 12,014,270 13,558,845
17,791 34,955 52,864 70,882 90,390 108,458 128,764 148,143 168,591 188,927
11,885 23,118 40,813 56,101 72,007 85,873 101,448 119,162 138,008 157,526
8798 · 27,599 · 58,485 14,997 · 53,249 · 113,039 21,939 · 87,128 · 184,111 29,031 · 118,768 · 250,675 36,235 · 151,369 · 319,161 43,232 · 181,409 · 382,525 50,647 · 214,110 · 451,211 57,841 · 248,580 · 523,435 65,223 · 284,539 · 598,637 72,636 · 321,198 · 675,239
421,582 421,545 421,544 421,537 421,533 421,531 421,525 421,521 421,517 421,516
4 37 148 371 972 1566 3385 2798 3708 6069
4 40 106 271 669 1179 1652 3099 3551 4689
15 67 146 244 356 824 598 568 785 905
4 37 106 244 356 824 598 568 785 905
Dual = Network Network Barrier Barrier Barrier Dual Dual Dual Dual Dual Dual Dual Dual Network Barrier Dual Dual Dual Barrier Barrier Barrier Barrier Barrier Barrier Barrier
N. Kliewer et al. / European Journal of Operational Research 175 (2006) 1616–1627
Instance
N. Kliewer et al. / European Journal of Operational Research 175 (2006) 1616–1627
Several practical side constraints that could not be addressed in this paper have been successfully included into the software component, such as: outsourcing of the parts of timetables to private bus companies, returns to different depots, different types of fleet size limitations, and so on.
6. Conclusion We propose a new application of the time–space network flow model for the vehicle scheduling problem arising in public bus transport networks. The model size has been substantially reduced through aggregation of incoming and outgoing arcs within each station, without loss of generality. Thus, we were able to solve very large practical instances to optimality through direct application of standard optimization software. Many of these instances were not solvable with any existing approach according the current state-of-the-art. The main advantage of the time–space network model is the reduction of the number of variables in the exact optimization model. Inherently, there is no straightforward way to explicitly model costs of a given trip-to-trip connection within the aggregated model. In bus scheduling case with linear cost structures the individual connections of trips or flights are not necessary, and therefore the time–space network model is to be preferred to the traditional connection based models.
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