The results obtained showed that the proposed algorithm was capable of finding the optimal solution in most cases when considering a time limit of 500 seconds. The methodology is also applied to solve a real-life instance that arises in the transportation system in Colombia (2 depots and 719 services), resulting in a decrease of the required fleet size and a balanced allocation of services, thus reducing deadhead trips.
International Journal of Industrial Engineering Computations 10 (2019) 361–374 Contents lists available at GrowingScience International Journal of Industrial Engineering Computations homepage: www.GrowingScience.com/ijiec A hybrid algorithm for the multi-depot vehicle scheduling problem arising in public transportation César Augusto Marín Morenoa, Luis Miguel Escobar Falcóna*, Rubén Iván Bolosa, Anand Subramanianb, Antonio Hernando Escobar Zuluagac and Mauricio Granada Echeverric aUniversidad Tecnológica de Pereira – Integra S.A., Colombia Federal da Paraíba, Brazil cUniversidad Tecnológica de Pereira, Colombia CHRONICLE ABSTRACT bUniversidade Article history: Received December 18 2018 Received in Revised Format January 26 2019 Accepted February 2019 Available online February 2019 Keywords: Vehicle Scheduling Matheuristics Set Partitioning Tactical Planning Bus Rapid Transit In this article, a hybrid algorithm is proposed to solve the Vehicle Scheduling Problem with Multiple Depots The proposed methodology uses a genetic algorithm, initialized with three specialized constructive procedures The solution generated by this first approach is then refined by means of a Set Partitioning (SP) model, whose variables (columns) correspond to the current itineraries of the final population The SP approach possibly improves the incumbent solution which is then provided as an initial point to a well-known MDVSP model Both the SP and MDVSP models are solved with the help of a mixed integer programming (MIP) solver The algorithm is tested in benchmark instances consisting of 2, and depots, and a service load ranging from 100 to 500 The results obtained showed that the proposed algorithm was capable of finding the optimal solution in most cases when considering a time limit of 500 seconds The methodology is also applied to solve a real-life instance that arises in the transportation system in Colombia (2 depots and 719 services), resulting in a decrease of the required fleet size and a balanced allocation of services, thus reducing deadhead trips © 2019 by the authors; licensee Growing Science, Canada Introduction From a technical point of view, the Operational Planning of public transportation systems includes scheduling of work shifts for bus drivers, scheduling of preventive maintenance work, scheduling of the services, as well as personnel allocation for each shift Recently, this task has involved the control in real time of the operational system fleet with the incorporation of Communication and Information Technologies Each one of the aforementioned problems represents a challenge when they are faced by the companies All of these problems have already been studied widely in the specialized literature, and due to their computational complexity (they are classified as NP-hard), they are usually approached independently This work deals with the Multiple Depot Vehicle Scheduling Problem (MDVSP), in which a set of vehicles must perform a set of trips with a given frequency at specific moments during the day When examining the vehicle scheduling literature, we identified several works in the context of public transportation system It was verified that applying optimization approaches contributed to the development of an efficient transportation system, capable of meeting the mobility needs required in cities Ibarra-Rojas et al (2015) * Corresponding author Tel.: +573217938145 E-mail: luismescobarf@utp.edu.co (L M Escobar Falcón) 2019 Growing Science Ltd doi: 10.5267/j.ijiec.2019.2.002 362 The creation of itineraries for each vehicle is one of the most complex tasks in transportation planning and is known to be NP-hard Despite its difficulty, this has been a problem of great interest given that the cost represented by the vehicles, either by their acquisition or by use, is one of the highest in the budget of operation of public transportation systems (Ceder, 2007) The reality of public transportation companies highlights the importance of efficiently solving the MDVSP, thus motivating the study of new alternatives that fit the particular environment of each company dedicated to the operation of public transportation services However, regardless of the particularities of each scenario, the objective is likely to be framed in the total fulfillment of itineraries and reduction of costs related to system operation Each itinerary or route is a description of the trips that must be done in specific times and areas, satisfying a frequency according to the conditions of the service and the needs of the mass transportation service determined by tactical planning Therefore, the combination of route and departure times is denoted as service and a group of services in the same area is defined as timetable The public transportation system routes are defined by their strategic planning and usually not have substantial changes in the short and medium terms Each route must be served within a given frequency, at a defined average speed This must be defined in the strategic planning of the transportation system, since all these route requirements are defined from the design of the service network itself and are designed precisely to meet the requirements identified during the strategic planning In this work, we propose a matheuristic algorithm to solve an Operational Planning problem of a Public Transportation System that can be modeled as a MDVSP The algorithm was first tested by benchmark instances consisting of 2, and depots, and a service load ranging from 100 to 500 The results obtained show that the proposed algorithm was capable of finding the optimal solution in most cases when considering a time limit of 500 seconds We then applied the methodology to solve a real-life scenario of a Colombian public transportation system involving depots and 719 services, and the results obtained imply in a decrease of the required fleet size and a balanced allocation of services, thus reducing deadhead trips The remainder of the paper is organized as follows Section presents a literature on the MDVSP, Section formally describes the problem Section includes the proposed methodology Section contains the results of the computational experiments Section presents the concluding remarks Literature Review In this section we review the recent MDVSP literature, briefly describing the corresponding solution approaches Huisman et al (2004) proposed a dynamic model to solve the VSP The approach addressed by the authors consists of solving a set of optimization problems in a sequential manner, taking into account different scenarios in future travel times During the first phase, trips are assigned to the different depots (clustering), solving the static problem Next, in the second stage, a simple VSP is solved dynamically The proposed methodology was evaluated in a bus operator company in the Netherlands The data set consists of 1,104 trips and four depots Gintner et al (2005), considered the MDVSP with multiple types of vehicles The authors proposed a two-phase method that provides near-optimal solutions The mathematical formulation of the problem is based on a space-time network and a vehicle is allowed to start from one depot and return to a different one, aiming at minimizing deadhead and stop times In practical cases, the number of trips is over a thousand, which is the reason the authors combined the model of a space-time network with a heuristic approach, in order to solve large problems and add new practical considerations Hadjar et al (2006) proposed a Branch-and-Bound algorithm to solve the MDVSP, which combines Column Generation (CG), Fixed Variables and Cutting Planes The authors studied two mathematical formulations that are based on CG schemes to solve the Lagrange relaxation of the Linear Programming problem The algorithm was tested on randomly generated cases as well as benchmark instances In addition, they also applied their algorithm on a set of real world instances that were derived from data belonging to the Montreal Transport Society (MTS) The MTS operates a network that includes seven depots, 665 bus lines with 380 completion points and 17,037 trips C A Marín Moreno et al / International Journal of Industrial Engineering Computations 10 (2019) 363 In the study conducted by Wang and Shen (2007), a new version of the problem – denoted as VSP with Route and Fueling Time Constraints (VSPRFTC) – was presented in which they consider electric buses This implies taking into account two new constraints: the duration of the route and the time of vehicle recharging The authors propose a new mathematical formulation and use of the ant colony algorithm as a solution method Laurent and Hao (2009) proposed an iterated local search (ILS) algorithm to solve the MDVSP The authors developed a new neighborhood operator called block movement (Block Moves) The methodology uses a so-called auction algorithm to generate the initial vehicle scheduling It then integrates a Two-Step Perturbation mechanism as a diversification procedure The developed approach was tested on a set of 30 benchmark MDVSP instances Hassold and Ceder (2014) presented a methodology based on a minimum cost network flow model, where the authors dealt with the heterogeneous fleet VSP (MVT-VSP) to solve a real case in New Zealand and the results showed an improvement of 15%, in terms of the cost of the vehicle fleet By means of a heuristic framework that makes use of a space-time network, the work presented by Guedes and Borenstein (2015) addressed the heterogeneous fleet MDVSP using truncated column generation and reduction of the state space The results obtained were promising and constitute a feasible alternative to efficiently solve the problem Shui et al (2015) put forward a cloning algorithm and two heuristics of travel time readjustments The approach obtained small CPU times and the ability to solve large instances such as the operation of buses in Nanjing, China Recently, Wen et al (2016), solved the VSP involving electric buses denoted as Electric VSP (E-VSP), whose main constraint lies in the buses driving ranges associated with battery recharging, which can be fully or partially recharged The mathematical formulation proposed for the E-VSP involves, in the first step, minimizing the number of vehicles needed to carry out all scheduled trips (Timetable) and, secondly, minimizing the traveled distance, which is equivalent to minimizing deadhead trips The problem was formulated as a mixed integer programming problem and was solved using an Adaptive Large Neighborhood Search (ALNS) heuristic The solution methodology was capable of finding good quality solutions for large size problems and near-optimal solutions in small cases Schöbel (2017) presented an integrated approach in the context of planning process of public transportation system The author argues that, instead of optimizing each stage of transportation systems, it would be more beneficial to consider the entire process in an integrated fashion To this end, a model that jointly covers route planning, Timetables and VSP was proposed as well as an iterative heuristic algorithm The results obtained were promising and the author listed a series of challenges for future studies regarding the integrated point of view of optimizing public transportation system planning Problem description and formulation In the context of public transportation, the MDVSP can be represented through a set , , … , of trip timetables that must be performed in a time horizon τ Each trip is characterized by the time and place of origin, as well as the destination Let , , … , be the set of depots and the number of vehicles located at depot , is compatible, if trip can be executed immediately after trip by the same A pair of trips vehicle This is possible if the time of trip completion plus the time of empty travel (deadhead) from to (added with a sufficient safety margin), is less than the start time of trip The objective of MDVSP is thus to connect a subset of trips to be performed by a vehicle, starting and ending at the same depot in such a way that the sum of the total travel costs is minimized The formulation proposed by Mesquita and Paixão (1992) can be considered a suitable alternative in our case, mainly because it contains a polynomial number of variables and constraints Let , be a directed graph where a set of vertices 1,2, … , is partitioned into 1,2, … , , which corresponds to trips , , … , , and a set 1, 2, … , , which refers to depots , , … , Set 364 contains the arcs , representing the feasible trip pairs , and all the arcs leaving and arriving the depots , , 1, , for 1,2, … , and 1,2, … , A cost is associated to each arc, which estimates the fuel consumption necessary for the deadhead trip from to , and other penalties that the transportation company intends to apply The costs associated to the arcs from or to a depot are denoted as: 0, 1,2, … , and ∞, , 1,2, … , , , , In the particular case in which transportation companies seek to simultaneously minimize the size of the represents the cost of the deadhead trip from fleet and the costs of deadhead trips, then , , depot to the starting point of the trip (from the place of completion of trip to depot , plus half the cost incurred by the use of the vehicle belonging to depot The MDVSP aims at finding the set of least-cost circuits in such a way that: (i) Each vertex ∈ is covered exactly once by a circuit, (ii) Each circuit contains exactly one vertex of set and (iii) The number of circuits that covers vertex must never exceed 1,2, … , 3.1 Definition of decision variables The decision variables of the mathematical model proposed by Mesquita and Paixão (1992) are specified as follows: 1, if trip immediately proceeds trip 0, otherwise 1, if after trip the bus returns to depot 0, otherwise , , 1, if a bus starts a trip from depot 0, otherwise , 1, if trip is carried out by a vehicle from depot 0, otherwise 3.2 Mathematical formulation , , , , (1) , subject to 1,2, … , (2) 1,2, … , (3) 1,2, … , (4) , 0 , , , 1,2, … , ; 1,2, … , ; 1,2, … , ; 1,2, … , ∈ 0,1 , 1,2, … , 1,2, … , 1,2, … , , 1,2, … , ; , , ; , (5) (6) (7) (8) 1,2, … , (9) 365 C A Marín Moreno et al / International Journal of Industrial Engineering Computations 10 (2019) ∈ 0,1 , 1,2, … , ; 1,2, … , (10) (11) , The meaning of the constraints is described as follows: Constraints (2) ensure that, for each 1,2, … , ; when finishing trip , the bus must return to a depot or must start another trip ∈ Constraints (3) assign, for each 1,2, … , , a bus to a trip , either from the depot or from another trip Constraints (4) enforce that, for 1,2, … , the capacity of a depot must not be violated Constraints (5) impose, for each pair and , the assignment between trip and depot , as long as trip is the first trip of an itinerary done by a vehicle that starts its service from depot Constraints (6) ensure that, for each pair and , the assignment between trip and depot , as long as trip is the last trip of an itinerary done by a vehicle that returns to depot Constraints (7) establish that for each pair , ∈ , if a trip is connected directly with trip and trip is assigned to depot , then, trip is assigned to depot Constraints (8) guarantee that each trip ∈ is assigned exactly to one depot Constraints (9) and (10) define the domain of the decision variables Constraint (11) impose an upper bound on the maximum number of vehicles that is obtained by a simple constructive procedure as described in Section 4.2 Proposed methodology This section describes the proposed hybrid algorithm that combines (i) a GA procedure; (ii) a SP approach; and (iii) a MIP formulation The GA metaheuristic is based on the methodology presented by Chu and Beasley (1998), whose initial pool of itineraries is built using different constructive procedures The best combination of the itineraries generated while executing the GA is then obtained by the SP approach which in turn possibly returns an improved solution Such solution is then provided as a starting point to a MIP solver which aims at finding an even better solution using the formulation proposed by Mesquita and Paixão (1992) The algorithm will be referred to as GA+SP+MIP and its basic description can be seen in Fig Genetic Algorithm Heuristics Set Partitioning (Subramanian, Uchoa, & Ochi, 2013) Sbest MIP Solver (Mesquita & Paixão, 1992) S* Fig Methodology 366 4.1 Constructive Procedures The three constructive algorithms implemented to generate initial solutions are described as follows 4.1.1 Clustered Concurrent Scheduler (CCS) In this procedure, we apply the traditional Concurrent Scheduler method (Dell'Amico et al., 1993) adding a first clustering stage, that is, the algorithm starts by determining from which depot each of the trips should be attended To this end, each trip is assigned in a heuristic fashion taking into account the lowest value , , as shown in Fig Fig Clustering of trips for each depot The next step is to sort the trips assigned to each depot in ascending order with respect to the start time Each itinerary is then generated by only allowing feasible assignments When the first infeasibility occurs, the itinerary is completed and assigned to a vehicle of the corresponding depot The procedure is repeated until there are no trips to be assigned to each of the clusters (Figure 3) Fig Construction of itineraries for each depot’s fleet In case the least-cost arc that connects a given depot to service runs out of vehicles, the procedure attempts to reassign such service to the next depot also using the least-cost arc criterion, as shown in Fig The algorithm terminates when all trips have been assigned C A Marín Moreno et al / International Journal of Industrial Engineering Computations 10 (2019) 367 Fig Reassignment of trip 13 for fleet availability 4.1.2 Minimal Cost Attention Sequence (MCAS) The method is based on the construction of a general sequence with all trips , that is, only the nodes of set are taken into account The first trip in the sequence corresponds to the one with smaller start time The following trips in the sequence are assigned according to the feasible nearest neighbor criterion In addition, the terminal node must have at least degree (Jungnickel, 2007) When there are no more feasible assignments or the minimum degree requirement is not met, the procedure sets the subsequence as an itinerary The construction of the general sequence continues with the unassigned trips, taking as a junction trip the one which has the highest degree of feasible departure The previous procedure is repeated until all trips are assigned to the general sequence, as illustrated in Figure Fig Sequence generated connecting routes using the nearest neighbor criterion Finally, the subsequences become supernodes (Figure 6) and the Generalized Assignment Problem (GAP) is solved in order to relate it to the set of depots, and to have a complete MDVSP solution 368 Fig Supernodes corresponding to travel itineraries (GAP) Fig shows the GAP solution and the corresponding node decoding Fig Assignment of resolved supernodes 4.1.3 Division of Attention Sequence (DAS) This algorithm corresponds to an adaptation of the sequence division approach proposed by Prins (2004) and extended by Liuet al (2009) for vehicle routing problems The method starts with a sequence of all trips sorted in ascending order according to their start time (Fig 8) Fig Ascending ordered sequence (or any combination of criteria) and its corresponding subgraph 369 C A Marín Moreno et al / International Journal of Industrial Engineering Computations 10 (2019) From this sequence, we obtain a subgraph of the original problem and through its exhaustive exploration one can build a digraph or auxiliary graph with all possible feasible itineraries and sub-itineraries (Fig 9) Fig Subgraph (or auxiliary graph) after removing infeasibilities Because the problem involves multiple depots, each itinerary is repeated as many times as the number of depots The minimum path between the trip that starts earlier in the sequence and the last one in the digraph, as shown in Fig 10, yields a MDVSP solution In this paper, a minimum flow-based mathematical formulation is employed to determine the shortest path Fig 10 Digraph with all feasible routes and subroutes for each one of the depots We now compare the performance of the constructive procedures with the optimal solutions of the benchmark instances by Fischetti et al (1999) Each figure provides the results for the alternatives of (Fig 11), (Fig 12) and depots (Fig 13) Depots 40% 35% 30% 25% 20% 15% 10% Gap CCS Gap MCAS 2-500-01 2-500-02 2-500-03 2-500-04 2-500-05 2-500-06 2-500-07 2-500-08 2-500-09 2-500-10 2-400-01 2-400-02 2-400-03 2-400-04 2-400-05 2-400-06 2-400-07 2-400-08 2-400-09 2-400-10 2-300-01 2-300-02 2-300-03 2-300-04 2-300-05 2-300-06 2-300-07 2-300-08 2-300-09 2-300-10 2-200-01 2-200-02 2-200-03 2-200-04 2-200-05 2-200-06 2-200-07 2-200-08 2-200-09 2-200-10 0% 2-100-01 2-100-02 2-100-03 2-100-04 2-100-05 2-100-06 2-100-07 2-100-08 2-100-09 2-100-10 5% Gap DAS Fig 11 Gap constructive procedures vs optimal solution (2 Depots) 370 Gap CCS Gap MCAS 3-400-01 3-400-02 3-400-03 3-400-04 3-400-05 3-400-06 3-400-07 3-400-08 3-400-09 3-400-10 3-300-01 3-300-02 3-300-03 3-300-04 3-300-05 3-300-06 3-300-07 3-300-08 3-300-09 3-300-10 3-200-01 3-200-02 3-200-03 3-200-04 3-200-05 3-200-06 3-200-07 3-200-08 3-200-09 3-200-10 50% 45% 40% 35% 30% 25% 20% 15% 10% 5% 0% 3-100-01 3-100-02 3-100-03 3-100-04 3-100-05 3-100-06 3-100-07 3-100-08 3-100-09 3-100-10 Depots Gap DAS Fig 12 Gap constructive procedures vs optimal solution (3 Depots) Depots 90% 80% 70% 60% 50% 40% 30% 20% Gap CCS Gap MCAS 5-300-01 5-300-02 5-300-03 5-300-04 5-300-05 5-300-06 5-300-07 5-300-08 5-300-09 5-300-10 5-200-01 5-200-02 5-200-03 5-200-04 5-200-05 5-200-06 5-200-07 5-200-08 5-200-09 5-200-10 0% 5-100-01 5-100-02 5-100-03 5-100-04 5-100-05 5-100-06 5-100-07 5-100-08 5-100-09 5-100-10 10% Gap DAS Fig 13 Gap constructive procedures vs optimal solution (5 Depots) It is important to mention that during the execution of the tests, it was observed that the constructive procedures did not appear to dominate each other for the distinct alternatives In fact, they are complementary to obtain good quality results when considering the different cases of the existing benchmark instances 4.2 Genetic Algorithm (GA) After building several initial solutions with the constructive procedures, a population-based metaheuristic could be an interesting strategy to benefit from the generated schedules Once the initial population is created (Fig 14), one applies a procedure that seeks to quickly obtain an UB from the fleet required to in ascending order meet the services This procedure simply consists in sorting all the services according to their start time, and constructing itineraries consecutively with this sequence 371 C A Marín Moreno et al / International Journal of Industrial Engineering Computations 10 (2019) Clustered Concurrent Scheduler (CCS) Random Solutions Chu-Beasley GA Minimal Cost Attention Sequence (MCAS) Initial Population Division of Attention Sequence (DAS) Upper Bound (Fleet) Fig 14 Generating the initial population In our case, we chose to implement a Chu-Beasley-based Genetic Algorithm as a population-based metaheuristic An elitist acceptance criterion is used, and when it is possible to update the population, only the worst solution is modified The coding of the solutions that represent the population is composed of two vectors The first corresponds to the order in which the services are added to the itineraries, and the second indicates the vehicles to which the services of the first vector are assigned, as proposed in Park (2001) The criterion adopted for assigning trips to itineraries is improved using a trip insertion operator to each of the itineraries under construction Coding the solutions in vectors facilitates the application of recombinations In this work, the five forms of recombination presented in Leung et al (2001) are used 4.3 Set Partitioning (SP) After the GA terminates, a SP-based procedure is applied as an attempt to further improve the incumbent solution In this phase, a solution is built from the best itineraries of the final population obtained by the GA The strategy consists of partitioning each solution (each one is a set of itineraries) and performing the optimal reconstruction of a solution for the MDVSP In Subramanian et al (2013) a compact model of SP is proposed, oriented to vehicle routing problems with a homogeneous fleet In this work, such model is applied to the MDVSP as described next Let be the set of all possible itineraries assigned to ⊆ the set of itineraries that contains the trip or service Define as the the different depots and binary variable associated with the itinerary ∈ with associated cost Moreover, let ⊆ be the set of itineraries assigned to depot ∈ and let be the number of buses or available vehicles at depot The formulation can be written as follows: (12) ∈ subject to ∈ ∈ ∈ 0,1 , ∀ ∈ (13) ∀ ∈ (14) ∀ ∈ (15) 372 Objective function (12) minimizes the sum of the costs of choosing the best combination of itineraries Constraints (13) guarantee that service is served exactly by one itinerary Constraints (14) ensure that the selected itineraries not exceed the maximum number of vehicles available at each depot Enumerating all elements of set would result in a prohibitively large number of itineraries Therefore, we decided to limit the size of by only considering the set of solutions associated with the final population obtained by the GA procedure We will thus obtain a compact model that will contain about thousands of itineraries for the largest cases As a result, the model can be easily solved with the wellknown MIP solver CPLEX® Version 12.7 4.4 Initial primal bound and MIP-based solution The best solution found during the SP procedure is provided as an initial primal solution to the the MIP model proposed by Mesquita and Paixão (1992), which in turn is the last step of our methodology It was observed that providing an initial solution substantially helped improving the efficiency of the model However, as it still may be hard to determine the optimal solution, especially for large-size instances, a time limit was imposed to avoid prohibitively long runs Computational Results The proposed algorithm was implemented in C ++ ® running Ubuntu Linux 16.04.03 of 64-bits LTS operating system and CPLEX® 12.7 as a MIP solver The experiments were performed on an Intel® Core ™ i5-4570 CPU of 3.20 GHz × 4, and GB of RAM A time limit of 500 seconds was imposed to the MIP model presented by Mesquita and Paixão (1992) In order to evaluate the performance of the developed approach, we made use of the benchmark instances introduced by Fischetti et al (1999) These test cases are composed of 10 instances for each set The sets combine 100, 200, 300, 400 and 500 services with depots For and depots they handle a maximum of 400 and 300 services, respectively In Tables 1, and 3, the 10 instances of each benchmark set are grouped and the corresponding averages are presented for each set (OPT, optimal value of the objective function) Also, we compare the average number of vehicles (NV) as well as the CPU times in seconds (T (s)) and the Gaps found by the different stages of the proposed methodology Table Consolidated (average) of instances groups with depots Instances 2-100 2-200 2-300 2-400 2-500 Fischetti et al (1999) OPT NV TS 309258,4 27,9 0,6 606028,1 54,8 23,3 833318,7 75,5 84,16 1115927,9 101,4 297,94 1373243,4 125,1 1160,73 Gap GA 2,46% 3,12% 3,35% 3,11% 3,94% Gap SP 1,21% 1,13% 1,77% 1,98% 1,89% GA+SP+MIP Gap MIP 0,00% 0,00% 0,00% 0,00% 0,19% NV 27,9 54,8 75,5 101,4 125,1 T(s) 2,345 30,014 130,189 361,886 481,911 Gap SP 1,19% 1,36% 1,99% 2,13% GA+SP+MIP Gap MIP 0,00% 0,01% 0,02% 0,40% NV 28 53,6 76,5 99,9 T(s) 39,051 233,718 472,109 847,763 Gap SP 1,40% 1,76% 2,20% GA+SP+MIP Gap MIP 0,00% 0,08% 0,61% NV 29,5 54,3 76,8 T(s) 249,597 461,199 499,16 Table Aggregate results for the benchmark instances with depots Instances 3-100 3-200 3-300 3-400 Fischetti et al (1999) OPT NV TS 309642,3 28,2 2,8 584726 53,6 72,74 831986,7 76,5 339,84 1084336,6 99,9 3177,63 Gap GA 2,65% 3,49% 3,91% 3,54% Table Aggregate results for the benchmark instances with depots Instances 5-100 5-200 5-300 Fischetti et al (1999) OPT NV TS 320728,9 29,5 22,49 587420,1 54,3 341,43 827447 76,8 3130,49 Gap GA 2,95% 3,66% 4,28% 373 C A Marín Moreno et al / International Journal of Industrial Engineering Computations 10 (2019) In addition to achieving optimality in most of the test cases, the proposed algorithm also finds the optimal number of vehicles required in all instances, thus showing the efficiency of our approach for the benchmark instances considered We therefore believe that the algorithm can be extended to a real life instance in the context of mass public transportation 5.1 Application in a Real Case The results for the real world scenario obtained by both the constructive procedures and the GA+SP+MIP algorithm are shown in Table and Fig 15(a) and Fig 15(b) The real test case derived from operations that arise in a business day is composed of two depots and 719 services that must be served by the available fleet of 37 vehicles (available at https://www.researchgate.net/publication/327288601_Study_Case_MDVSP_Integra_SA_Pereira_Colombia) Table Comparative table constructives real case, business day Real Case NV Manual Scheduling 36 719 Trips – Depots CCS 66 MCAS 35 DAS 66 As illustrated in Table 4, by only applying the constructive procedures, we were capable of obtaning a reduction of one vehicle on the fleet necessary to meet all services on a business day In later stages, a decrease in the number of vehicles was not achieved, given that the demand is very high However, the GA+SP+MIP algorithm decreases deadhead trips, finding a uniform distribution on the allocation of services to buses In Fig 15 and Fig 16, where the vehicle scheduling is presented using 35 vehicles; the x-axis represents the time of the day, the y-axis contains each schedule, and the numbers in the lines correspond to the “ids” of the trips or services assigned to the vehicles Each service has a starting time and a completion time, which are the points between the ids The deadhead trips or idle times for the vehicles are highlighted in green, showing how the methodology achieves a more balanced solution (with cost 16208, Fig 16) when compared with the initial solution in Fig 15 (with cost 20240) BKS Instance: MDVSP_Integra; N.Vehicles: 35; Cost: 16208 700 701 422 423 401 402 40 41 359 360 361 682 591 576 577 135 558 559 560 683 295 11 589 590 279 280 281 663 446 608 571 282 572 573 404 665 468 506 443 80 533 469 81 507 269 270 271 272 273 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610 125 472 645 381 181 182 311 441 275 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 548 644 625 684 609 183 184 185 186 187 188 643 466 595 575 557 10 605 594 574 340 124 235 236 237 78 593 319 339 111 112 113 529 592 318 588 713 562 338 120 380 257 258 274 317 294 310 297 579 121 122 123 293 309 140 141 142 143 144 39 681 308 296 624 304 585 586 587 680 642 106 107 108 75 76 77 621 259 260 261 262 263 599 600 715 601 716 189 366 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 602 717 603 604 718 147 148 292 05:00 06:00 07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 Time of the day Vehicle ID Vehicle ID S_0: MDVSP_Integra; N.Vehicles: 35; Cost: 20240 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 121 122 123 359 360 338 50 51 295 444 445 426 12 214 665 319 663 183 625 626 643 666 79 150 80 81 532 508 279 264 401 402 11 446 10 363 605 606 208 466 681 467 554 492 41 381 700 48 448 449 450 84 15 16 572 573 13 17 425 324 668 386 614 688 367 368 557 669 411 344 536 125 472 473 412 559 592 410 61 161 162 163 164 461 674 26 95 481 69 169 439 488 501 502 484 485 98 99 396 483 397 712 145 566 503 569 570 567 568 489 580 581 400 582 224 225 226 227 419 420 421 292 603 604 73 74 289 75 290 259 622 312 34 699 178 179 176 76 77 177 313 196 197 198 199 200 201 202 203 204 205 206 207 273 180 105 679 231 551 399 490 175 100 101 102 398 315 262 263 288 256 257 258 621 291 149 602 698 71 620 623 601 697 33 173 174 38 584 505 287 32 70 440 303 553 585 586 587 696 286 463 546 170 525 97 482 285 660 260 261 583 358 695 718 106 107 304 552 278 487 357 230 504 37 717 234 715 337 114 115 116 117 118 438 711 395 714 486 284 103 528 356 462 283 138 545 94 335 694 656 460 617 561 350 334 637 65 247 248 249 250 251 349 355 524 636 414 619 354 333 716 46 171 172 618 96 459 521 522 345 346 347 220 221 222 480 408 409 654 64 595 45 140 141 142 143 616 413 594 44 661 36 465 228 229 336 437 560 593 693 464 42 710 690 418 43 301 104 678 31 28 352 615 677 30 35 300 600 311 27 133 134 558 591 574 417 253 310 692 635 689 29 659 232 599 252 458 223 309 147 148 146 565 691 653 456 658 527 598 136 709 457 634 192 193 515 332 137 351 479 308 302 233 72 379 564 597 642 299 550 657 563 676 655 544 24 436 25 434 272 455 216 217 218 219 348 535 500 392 370 613 687 66 393 394 328 478 633 612 415 371 166 454 19 474 475 85 126 154 155 156 701 18 127 511 702 391 327 632 686 513 341 510 495 390 477 191 611 406 512 499 92 130 131 132 129 514 326 476 190 323 306 184 185 186 187 120 380 78 703 325 631 365 427 83 498 91 369 87 630 68 641 378 254 255 526 139 596 144 442 298 549 377 562 435 93 23 366 86 404 238 239 469 556 494 215 54 682 468 555 493 110 305 684 62 548 67 277 441 640 297 376 523 331 578 245 246 452 453 416 673 330 433 271 497 375 577 135 542 388 705 189 646 447 40 235 236 237 387 704 608 607 39 506 496 296 547 374 276 713 639 675 353 167 672 432 270 372 520 575 364 667 470 403 389 168 329 576 541 275 579 638 165 22 540 537 538 609 645 517 244 407 53 281 267 90 539 373 63 519 518 89 516 269 385 342 509 280 266 571 443 47 362 533 293 210 211 212 507 265 647 671 430 431 88 543 652 708 651 157 158 471 82 590 670 243 451 384 531 57 429 194 195 60 707 650 240 241 242 589 119 383 628 530 56 428 322 629 307 588 529 109 683 268 55 343 321 627 644 649 159 160 213 320 21 59 706 685 610 282 664 318 58 648 52 111 112 113 491 20 128 534 151 152 153 382 317 405 188 424 209 294 181 182 14 340 124 423 49 274 361 339 422 680 662 314 316 624 108 05:00 06:00 07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 Time of the day Fig 15 Initial solution Fig 16 Algorithm (GA+SP+MIP) impact in the resulting assignment Therefore, it is possible to conclude that the minimum number of vehicles were used and that the deadhead trips were reduced significantly Concluding Remarks In this paper, we proposed a hybrid methodology that involves constructive algorithms, a metaheuristic, a Set Partitioning approach and a MIP-based procedure Such methodology is able to solve test cases of considerable size both in the benchmark instances and in a real case, within reasonable CPU times The 374 MDVSP approach, combined with techniques commonly used in vehicle routing problems, presents an adequate development and offers an acceptable scalability when the number of depots and trips increase Future work may include the development of parallel algorithms so as to mitigate the excessive CPU times that may arise in complex single-thread algorithms, such as the one described in this paper Acknowledgments The authors thank the support of Integra S.A., COLCIENCIAS, SENA, the Master Program in Electrical Engineering and the Doctoral Program in Engineering of the Universidad Tecnológica de Pereira References Ceder, A (2007) Public transit planning 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