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A bus crew scheduling system using set covering formulation
Abstract
A bus crew scheduling system which uses mathematical programming is described. The system is based on a set covering formulation, and includes a number of heuristics to keep the problem to a manageable size. It has been in regular use by London Buses Ltd. since the beginning of 1985 and has been adopted by other bus companies. The crew scheduling problem is described, the solution process is presented and results are discussed briefly.
... The research and development of TRACS and its predecessors, notably IMPACS [10], can be tracked throughout this series of international conferences/workshops. Around the time of the last event, CASPT 2000 in Berlin, TRACS was adopted by First, which is the largest group of bus companies in the UK. ...
... • Set covering or set partitioning approaches which usually involve mathematical programming [10,11] or metaheuristics such as evolutionary algorithms. [12,13] The classifications are not distinctive because it is usual to find that some Mathematical Programming approaches may involve heuristic techniques to some extent; some metaheuristic approaches may involve Mathematical programming, etc. ...
... TRACS belongs to the set covering category. Its predecessor, the IMPACS system [10], was developed in the 70's and was implemented in Manchester and London for bus in the early 1980s, the BUSMAN system derived from it was used by about 40 bus companies since the 1980s. Since the early 1990s, extensive research, development and trials have been carried out for the rail industry [15,16,17] resulting in the TRACS system. ...
... Government and EU policies, labor regulations and other practical conditions further challenge transport companies to efficiently utilize their resources. As a consequence, over the years, there has been an increase in the development of decision sup-port tools based on mathematical programming approaches to aid transport companies in planning ( Desrochers & Soumis (1989) ; Lourenço, Paixão & Portugal (2001) ; Smith & Wren (1988) ; Wren et al. (2003) ). Typically, the transport planning process involves solving several planning subproblems as it is too complex to solve the entire planning problem in one integrated step. ...
... Due to potentially being a large number of possible duties, the formulation is intractable by exhaustive enumeration techniques. Some authors, e.g., Smith and Wren (1988) and Wren et al. (2003) , have considered reducing the number of duties being generated before solving the SCP. Smith & Wren (1988) heuristically generate a feasible subset of duties for the SCP. ...
... Some authors, e.g., Smith and Wren (1988) and Wren et al. (2003) , have considered reducing the number of duties being generated before solving the SCP. Smith & Wren (1988) heuristically generate a feasible subset of duties for the SCP. The SCP is solved by relaxing the integrality constraints and an integer solution is found using a branch-and-bound algorithm. ...
In the public bus transport industry, it is estimated that the cost of a driver schedule accounts for approximately 60% of a transport company's operational expenses. Hence, it is important for transport companies to minimize the overall cost of driver schedules. A duty is defined as the work of a driver for a day and the driver scheduling problem (DSP) is concerned with finding an optimal set of driver duties to cover a set of timetabled bus trips. Numerous labor regulations and other practical conditions enforce drivers to travel within the city network to designated bus stops to start/end duty, to take a break or to takeover a bus from another driver. This paper focuses on the driver scheduling problem with staff cars (DSPSC), where staff cars can be utilized by the drivers to fulfill their travel activities. However, staff cars should always be returned to the depot and can perform multiple round trips during the day. The problem is restricted by the number of cars available at the depot. We present a matheuristic for solving the DSPSC and the proposed method is tested on instances from Danish and Swedish companies. A comparison with a state-of-the-art mixed integer programming (MIP) solver indicates that the matheuristic provides better solutions, with comparable computation times, for 6 out of 10 large instances. For instances that have more than 6 staff cars and 1200 bus trips, the improvement is 13–15% on average.
... Esse problema tem sido muito estudado e a abordagem mais empregada é aquela que formula o PPT como um problema de recobrimento ou de particionamento (set covering ou set partitioning model) e utiliza a técnica de geração de colunas, branch-and-bound, branchand-price e a relaxação lagrangeana para encontrar uma solução inteira (Smith e Wren [11], Desrochers e Soumis [5], Desrochers et al. [6], Fores et al. [8], Barnhart et al. [2], Friberg e Haase [9]). Diferentemente dos modelos de programação linear, os algoritmos de fluxo em redes são capazes de resolver problemas de grandes dimensões produzindo soluções que já são inteiras. ...
... Os valores dos parâmetros β, γ, θ, δ, ε, ω e σ são baseados nos pesos da função de avaliação proposta por Silva et al. [11]. Foi utilizado γ = 0 quando o fator horas ociosas não estava sendo considerado na determinação do peso no arco. ...
... Para avaliar a qualidade dos resultados obtidos, foi utilizada uma versão adaptada da função proposta por Silva et al. [11], que visa medir a qualidade de cada resultado obtido. Esta função de avaliação está baseada em penalidades e é apresentada abaixo. ...
São apresentados modelos de fluxo em redes para a resolução de escalas de tripulações de ônibus urbano
... Regarding the crew-scheduling problem, Smith and Wren (1988) described a bus crewscheduling system based on a set covering formulation and used many heuristics to keep it to a manageable size [20]. Chen and Niu (2012) presented an approach for solving the bus crew-scheduling problem, which considers early, day, and late duty modes with time-shift and work-intensity constraints [21]. ...
... Step 6: Use Formula (20) to update ε, let o = o + 1. If the number of iterations is greater than Maxgen, stop and output the external archive set; otherwise, return to Step 4. ...
Replacing conventional buses with electric buses is in line with the concept of sustainable development. However, electric buses have the disadvantages of short driving range and high purchase price. Many cities must implement a semi-electrification strategy for bus routes. In this paper, a bi-level, multi-objective programming model is established for the collaborative scheduling problem of vehicles and drivers on a bus route served by the mixed bus fleet. The upper-layer model minimizes the operation cost and economic cost of carbon emission to optimize the vehicle and charging scheme; while the lower-layer model tries to optimize the crew-scheduling scheme with the objective of minimizing driver wages and maximizing the degree of bus-driver specificity, considering the impact of drivers’ labor restrictions. Then, the improved multi-objective particle swarm algorithm based on an ε-constraint processing mechanism is used to solve the problem. Finally, an actual bus route is taken as an example to verify the effectiveness of the model. The results show that the established model can reduce the impact of unbalanced vehicle scheduling in mixed fleets on crew scheduling, ensure the degree of driver–bus specificity to standardize operation, and save the operation cost and driver wage.
... O problema tem sido largamente estudado e as abordagens exatas mais exploradas utilizam o modelo de recobrimento ou de particionamento de conjuntos. Para resolver tais modelos são empregadas as técnicas de seleção de colunas (Smith e Wren, 1988;Silva et al., 2007), de geração de colunas (Desrochers e Soumis, 1989;Desrochers et al., 1992;Fores et al., 1999), assim como as técnicas de busca em árvore com estratégia do tipo branch-and-price (Barnhart et al., 1998) e branchand-cut (Friberg e Haase, 1999) associada à geração de colunas. A técnica de relaxação Lagrangeana também é utilizada para resolver o problema, como em Freling et al. (2001). ...
... Embora os pesos adotados neste trabalho não reflitam fidedignamente os custos operacionais da escala, seus valores foram escolhidos tendo em vista uma série de trabalhos clássicos que adotam estratégia semelhante. Smith e Wren (1988), que abordam o PPT por meio de um modelo de recobrimento, utilizam uma rotina WAGES para calcular o custo de cada jornada, dado em função do tempo total de trabalhado. Este tempo total pode incluir horas extras, que são sobretaxadas. ...
p> Resumo: Este artigo apresenta uma nova abordagem para a resolução do Problema de Programação de Tripulações no Sistema de Transporte Público (PPT). O modelo se baseia na metaheurística GRASP cuja busca local é realizada pelo método da Busca em Vizinhança de Grande Porte, conhecida na literatura como Very Large-Scale Neighborhood Search. O grande diferencial da aplicação desta técnica de busca para o PPT é que, além de incorporar os movimentos de realocação e troca de tarefas, realizados tradicionalmente, ela também permite considerar trocas do tipo 3-optimal, 4-optimal, até o limite de n-optimal, para uma solução com n tripulações. A implementação da heurística proposta foi testada com dados de problemas reais de uma empresa que opera em Belo Horizonte, e os resultados foram comparados com as soluções adotadas pela empresa. Desta forma foi possível observar que o modelo apresentado neste trabalho produziu soluções mais econômicas do que aquelas praticadas pela empresa.
Abstract: This paper presents a new approach to solve the Crew Scheduling Problem (CSP) for public mass transport system. The proposed model is based on the GRASP metaheuristic framework, where the local search is performed by the Very Large-Scale Neighborhood (VLSN) search technique. The great differential of this search technique applied to the CSP is that, in addition to task reassigning and swapping movements, adopted in previous work, it also allows considering 3-optimal, 4-optimal, up to n-optimal task movements, for a solution with n crews, yielding to improved solutions. The proposed heuristic was tested with data from real problems of a bus company operating in the city of Belo Horizonte, and the results compared to the manual solution adopted by the company. Thus it was observed that the model presented in this work have produced more economical solutions than those used by the company.
... Esse tema tem sido largamente estudado e seus resultados são geralmente utilizados nos países mais desenvolvidos. A abordagem mais explorada é aquela que formula o PPT como um problema de recobrimento ou de particionamento (set covering ou set partitioning model) e utiliza a técnica de geração de colunas para resolvê-lo (Smith e Wren 1988, Desrochers e Soumis 1989, Desrochers et al. 1992, Fores et al. 1999, Barnhart et al. 1998, Friberg e Haase 1999. A variedade de trabalhos deriva das diferentes maneiras de gerar as colunas e diferentes metodologias para resolver o problema, tais como: branch-and-bound, branchand-price e a relaxação lagrangeana. ...
... No presente trabalho, o PPT é formulado como um problema de particionamento. Para reduzir a dimensão do problema é implementada a metodologia baseada naquela proposta por Smith e Wren (1988), considerando a realidade brasileira e as regras operacionais seguidas por uma empresa que atua em Belo Horizonte. ...
... Esse tema tem sido largamente estudado e seus resultados são geralmente utilizados nos países mais desenvolvidos. A abordagem mais explorada é aquela que formula o PPT como um problema de recobrimento ou de particionamento (set covering ou set partitioning model) e utiliza a técnica de geração de colunas para resolvê-lo (Smith e Wren 1988, Desrochers e Soumis 1989, Desrochers et al. 1992, Fores et al. 1999, Barnhart et al. 1998, Friberg e Haase 1999, Fores 1996. A variedade de trabalhos deriva das diferentes maneiras de gerar as colunas e diferentes metodologias para resolver o problema. ...
... A primeira etapa é a geração das janelas de troca, as quais são geradas por um processo similar àquele descrito por Smith e Wren (1988), no qual são combinadas as tarefas a fim de se obterem intervalos de tempo nos quais deverá ocorrer a troca da tripulação dos veículos de uma dada linha. Esse intervalo é definido tendo em vista as seguintes características: Tempo máximo de trabalho da solução da empresa; Tempo médio de trabalho da solução da empresa. ...
span style="font-family: Times New Roman; font-size: xx-small;"> Este trabalho tem como objetivo implementar um método de otimização para o Problema da Programação de Tripulações (PPT), o qual visa determinar um conjunto de jornadas de trabalho para as tripulações, de tal forma que a programação dos veículos seja realizada com o menor custo possível. Como restrições, cada jornada deve atender à legislação e à convenção coletiva de trabalho do setor. Neste trabalho são apresentadas e comparadas quatro diferentes metodologias de geração de colunas para o PPT, definindo assim problemas de programação linear inteira com variáveis binárias. A primeira metodologia consiste em definir um intervalo de tempo durante o qual poderá ocorrer a troca de tripulações. Na segunda metodologia as jornadas possuem pelo menos um dado tempo mínimo de duração. Na terceira metodologia é implementada a heurística de Chvátal. A quarta metodologia apresenta a implementação de um método híbrido de geração de colunas para o PPT. Neste trabalho apresentam-se resultados comparativos obtidos com a aplicação das metodologias a problemas reais.
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... In the commercially viable BUSMAN and IMPACS systems, this task is handled by integer linear programming methods (Smith,1988;Smith & Wren, 1988;Ryan, 1980;Nemhauser & Wolsey,1988). The current research project has the specific aim of evaluating whether genetic algorithms (GAs) are able to produce competitive results in a similar time. ...
... GA are based on a natural analogy (population genetics (Shorrocks, 1978) and evolution (Darwin, 1950)) and are a form of random search. To explain GAs and to demonstrate that they are much more effective than more straightforward search methods we describe and evaluate several random search variations. ...
Genetic algorithms have been applied to bus driver scheduling and compared to other approaches such as simulated annealing. Bus driver scheduling is a more difficult domain than most genetic algorithm applications. Special purpose genetic algorithms have been developed that search constrained versions of the initial search space. Greedy algorithms are used for crossovers, though these had to be randomized to give good results. Special purpose optimizing mutation improves search in domains too large for traditional mutation to be useful. The greedy genetic algorithm produces schedules typically within a few duties of the optimum solution. Further theoretical analyses are expected to result in new methods that will improve results. The technology developed may also have applications in other areas of transport scheduling.
... Traditional public transit systems predominantly consist of fixed-route services, such as buses, and light and heavy rail, which operate on regular schedules. Over the years, researchers have extensively explored the design and operation of public transit systems, delving into areas like service line planning (1)(2)(3)(4)(5), timetabling (6)(7)(8)(9), crew scheduling (10)(11)(12)(13), and demand estimation and prediction (14,15). One of the critical challenges in the design of public transit systems is the distribution of finite service resources across time and space to meet the fluctuating and diverse travel demands. ...
Public transit plays an essential role in mitigating traffic congestion, reducing emissions, and enhancing travel accessibility and equity. One of the critical challenges in designing public transit systems is distributing finite service supplies temporally and spatially to accommodate time-varying and space-heterogeneous travel demands. Particularly, for regions with low or scattered ridership, there is a dilemma in designing traditional transit lines and corresponding service frequencies. Dense transit lines and high service frequency increase operation costs, while sparse transit lines and low service frequency result in poor accessibility and long passenger waiting time. In the coming era of Mobility-as-a-Service, the aforementioned challenge is expected to be addressed by on-demand services. In this study, we design an On-Demand Multimodel Transit System (ODMTS) for regions with low or scattered travel demands, in which some low-ridership bus lines are replaced with flexible on-demand ride-sharing shuttles. In the proposed ODMTS, riders within service regions can request shuttles to finish their trips or to connect to fixed-route services such as bus, metro, and light rail. Leveraging the integrated transportation system modeling platform, POLARIS, a simulation-based case study is conducted to assess the effectiveness of this system in Austin, Texas.
... Different variants of BDSPs have been studied from the early 60's [17]. Regarding exact methods, the BDSP is often modelled as a Set Partitioning Problem and column generation is often used [6,8,13,15]. However, due to the need to solve very large real-world problems in a reasonable time, several heuristic methods have been studied for BDSP: some examples are Greedy [12], Tabu Search [14], Simulated Annealing [5], GRASP [2], and Genetic Algorithm [7,11]. ...
The Bus Driver Scheduling Problem (BDSP) is a combinatorial op-timisation problem that consists of assigning bus drivers to vehicles with predetermined routes. The objective is to optimise the em-ployees' operating costs and work quality based on goals like the number of vehicle changes. This problem is highly constrained due to the complex rules specified by a collective agreement and law. Hence, solving real-life instances with exact methods in a reasonable time is very challenging. In this work, we investigate and compare metaheuristics based on Tabu Search and Iterated Local Search for solving this problem. We analyse the impact of different solution components, including neighbourhoods, acceptance criteria, tabu lists, and perturbation moves. Further, we provide a new set of large real-life-based instances that extends the existing benchmark. We compare our methods with the state-of-the-art approaches on the extended set of instances and show that our algorithms provide very good solutions for large instances.
... Since the 1950s, foreign scholars, especially European scholars, began to study the Crew Planning Problem (CPP) [1] and gained greater momentum with the advances in computational power in 1980s [2,3]. Early studies were mainly concentrated on airlines [4,5], and followed by urban public transportation [6][7][8]. So far, many achievements have been made in the field of crew planning problems. ...
This paper presents a novel mathematical formulation in crew scheduling, considering real challenges most railway companies face such as roundtrip policy for crew members joining from different crew depots and stricter working time standards under a sustainable development strategy. In China, the crew scheduling is manually compiled by railway companies respectively, and the plan quality varies from person to person. An improved genetic algorithm is proposed to solve this large-scale combinatorial optimization problem. It repairs the infeasible gene fragments to optimize the search scope of the solution space and enhance the efficiency of GA. To investigate the algorithm’s efficiency, a real case study was employed. Results show that the proposed model and algorithm lead to considerable improvement compared to the original planning: (i) Compared with the classical metaheuristic algorithms (GA, PSO, TS), the improved genetic algorithm can reduce the objective value by 4.47%; and (ii) the optimized crew scheduling plan reduces three crew units and increases the average utilization of crew unit working time by 6.20% compared with the original plan.
... , ξ in ] denotes a vector of binary parameters such that ξ ij = 1 if element j is able to cover target i and ξ ij = 0 otherwise. Set covering models find a variety of real-world applications, including scheduling [41], production planning [46], facility location [19], and vehicle routing [6], etc. Consider, for example, we wish to build medical facilities among n locations to cover I target residential regions. In this context, ξ i describes the connection between target i and all candidate locations and it may depend on their distances, e.g., ξ ij = 1 if target i is near location j. ...
We study a generalized distributionally robust chance-constrained set covering problem (DRC) with a Wasserstein ambiguity set, where both decisions and uncertainty are binary-valued. We establish the NP-hardness of DRC and recast it as a two-stage stochastic program, which facilitates decomposition algorithms. Furthermore, we derive two families of valid inequalities. The first family targets the hypograph of a "shifted" submodular function, which is associated with each scenario of the two-stage reformulation. We show that the valid inequalities give a complete description of the convex hull of the hypograph. The second family mixes inequalities across multiple scenarios and gains further strength via lifting. Our numerical experiments demonstrate the reliability of the DRC model and the effectiveness of our proposed reformulation and valid inequalities.
... Reformuler un problème sous forme d'un problème de Set Packing (SSP), de Set Covering (SCP) ou de Set Partitioning (SPP) est une technique classique et populaire, qui s'est avérée efficace pour résoudre beaucoup de problèmes combinatoires tel que les problèmes de planification des tournées de véhicules (Vehicle Routing and Scheduling Problem) [Agarwal et al., 1989, Bramel and Simchi-Levi, 1997, Kelly and Xu, 1999, Avella et al., 2004, Baldacci et al., 2008 et les problèmes de planification des horaires d'équipages (Crew Scheduling Problem) [Rubin, 1973, Smith and Wren, 1988, Hoffman and Padberg, 1993, Wedelin, 1995a, Mingozzi et al., 1999, Medard and Sawhney, 2007, Froger et al., 2015. Ces reformulations sont largement abordées dans la littérature. ...
Dans le cadre du développement durable et des innovations dans les systèmes agroalimentaires, les systèmes mixtes horticoles (vergers et maraîchage) visent à répondre aux enjeux actuels auxquels l’agriculture est confrontée, à savoir une diminution de la pollution des sols, une meilleure gestion des ressources (eau, énergies) et un enrichissement de la biodiversité, tout en continuant d’assurer des fonctions alimentaires.
Ils combinent des productions à la fois diversifiées et relativement intensifiées, leur permettant de s’insérer en périphérie urbaine. Ces systèmes agroforestiers reposent sur un ensemble complexe d’interactions modifiant l’utilisation de la lumière, de l’eau et des nutriments. La conception d’un tel système doit donc optimiser l’utilisation de ces ressources en maximisant les interactions positives (facilitations) et en minimisant celles négatives (compétitions). Nous définissons le problème de verger-maraîcher comme un
problème d’allocation des arbres et des cultures dans les dimensions spatio-temporelles. Nous proposons trois formulations mathématiques : modèle quadratique en variables binaires (BQP), modèle linéaire en variables mixtes (MILP) et modèle linéaire en variables binaires (01LP). Les limites des méthodes exactes pour résoudre ce problème sont présentées, montrant la nécessité d’appliquer des méthodes approchées, capables de résoudre des systèmes à grande échelle avec des solutions de bonne qualité en temps raisonnable. Pour cela, nous avons développé un solveur open source, baryonyx, qui est une version parallèle de l’heuristique de Wedelin (généralisée). Nous avons utilisé l’analyse de sensibilité pour identifier les paramètres les plus influents. Une fois trouvés, nous avons fixé les autres et utilisé un algorithme génétique pour régler les plus importants sur un ensemble d’instances d’entraînement. Le jeu de paramètres optimisé
peut alors être utilisé pour résoudre d’autres instances de plus grande taille du même type de problème. baryonyx avec son réglage automatique obtient des résultats améliorant l’état-de-l’art sur des problèmes de partitionnement. Les résultats sont plus mitigés sur le problème de verger-maraîchage, bien que capable de passer à l’échelle.
... Since the control of the rules and regulations can be quite complex, so the most popular approach for the vehicle/crew scheduling system is to solve the problem with a set partitioning model [5], and the relaxation is to use a set covering formulation [6], [7]. ...
The aim of the operative planning process is to offer a high quality of service for the passengers while, on the other hand, the costs for setting up and running the transit system should be small. Due to the dramatically increased processing capabilities today, manual work of planning and scheduling of metropolitan public transport operations should use advanced optimization methods. This paper briefly outlines the tasks of operative planning for transport management systems. The goal of this research is to create and implement the model in a real environment so that algorithms can be tested.
... A abordagem clássica para tratar esse problema formula-o como um problema de programação linear inteira de recobrimento ou particionamento (set covering ou set partitioning model). Para resolver o problema, normalmente é utiliza a estratégia de geração de colunas (Smith e Wren 1988, Desrochers e Soumis 1989, Fores et al. 1999, Barnhart et al. 1998). Entretanto, modelos exatos são limitados quando aplicados a problemas de grande porte. ...
This paper explores different methods associated with metaheuristics Variable Neighborhood Search – VNS and Iterated Local Search – ILS, associated with different local search methods to solve the Crew Scheduling Problem from Public Transportation System. Into implementation of metaheuristics a classical descent method and other one based in network algorithm, called Very Large-scale Neighborhood Search – VLNS, were used. Initially, the metaheuristics were implemented in their own forms. For the VNS, normally the classical Variable Neighborhood Descent – VND as local search method, and for ILS, the literature allows the researchers choose the refinement heuristic. Thereby,
firstly the First Improvement Method was used to reach the Classical forms of both metaheuristics and after, also the VLNS. The VLNS perform search into a larger space than the Classical, allowing tasks reallocation in a sequence of different crews. Consequently, local optimal solutions must be better than those obtained by classical search methods. Both versions of VNS and ILS have been applied to a set of problems from a company that operates in Belo Horizonte, producing better results than the solutions adopted by the company. The results also show that the solutions obtained by the VNS/ILS-VLNS are more economical and present characteristics more appropriate to the operation than those obtained by the Classical methods.
... Cogent reviews are presented by den Bergh et al. [1] and Brunner [2], and a comprehensive bibliography is provided by Ernst et al. [3]. Reported application areas include retail sales [4,5], manufacturing [6], transportation [7,8,9], health care delivery [10,11,12], and the telecommunications industry [13,14]. Solution approaches have incorporated diverse industrial engineering methods such as mathematical programming [15,16,17], simulation [18], dynamic programming [19], genetic algorithms [20], and other heuristic procedures [21,22]. ...
... IMPACS (Integer Mathematical Programming for Automatic Crew Scheduling) was developed for bus operation in the late 1970s. Parker and Smith presented the prototype and Wren and Smith [8] gave a full description of the system. It was installed in London Transport in 1984 and in Greater Manchester Buses in 1985. ...
We study the problem of vehicle scheduling in urban public transport systems taking into account the vehicle-type and size (VTSP). It is modeled as a multiobjective combinatorial optimization problem. A heuristic based on MOCell (Multi-Objective Cellular evolutionary algorithm) is proposed to solve the problem. A set of non-dominated solutions represents different assignments of vehicles to specific routes. The conflicting objectives of provider and users (passenger) are to minimize the total operating cost, and maximize the quality of service, taking into account restrictions of government agencies in context of smart cities to improve the Intelligent Transport Systems (ITS).
... Os métodos clássicos de busca em vizinhança se baseiam em realocação e trocas aos pares, de elementos entre os dois subconjuntos aos quais pertencem. Uma troca cíclica pode ser definida por uma sequência de elementos i 1 -i 2 Note que a vizinhança de troca cíclica contempla a troca aos pares e ainda explora uma infinidade de outras soluções não alcançáveis pela troca aos pares. Portanto, é de se esperar que soluções ótimas locais obtidas por meio de múltiplas trocas sejam, em média, superiores àquelas obtidas pela trocas aos pares. ...
... Ref. [25] trace back the origin of staff scheduling to [23]'s work, in which toll booth operations are analyzed and the relief times for toll collectors are determined. Since then, employee scheduling has been one of the most important topics for various organizations, such as airlines [12,17,26,28], railways [43], transit bus operators [42], retailers [49,37], call centers [9,14,22] and health-care organizations [32,48,33]. We refer the reader to den Bergh et al. [21] for a recent survey of the literature on employee scheduling. ...
We consider an employee scheduling problem arising in service industries with flexible employee availability and flexible demand. In the system to be planned, there is a given set of service requirements and a set of employees at any time. Each employee belongs to one of various skill levels, each service requirement specifies the requested employee skill level and the timing of the service delivery, and each requirement has a weight that indicates the importance of that requirement. Employees have individual flexible contracts with the organization, which are characterized by weekly/monthly contracted work hours, days the employee is available for work and availability of overtime. Furthermore, there are regulations on maximum work hours and minimum rest requirements of employees enforced by the government and the labor union. The problem that we investigate is to generate an assignment of employees to service requirements which (i) ensures that the maximum weighted number of service requirements is met, (ii) satisfies government and labor union regulations, (iii) honors individual employee contracts with minimum deviation from the contracted work hours, and (iv) ensures a fair balance between employee schedules in terms of work assignments on holidays. We model the problem as a mixed-integer programming problem and discuss a reformulation strategy, which allows us to solve practical problems in a reasonable amount of time. We also report our experience in a large health-care organization in Belgium.
... O PPT é um problema clássico, sobre o qual existem diversos trabalhos que exploram diferentes técnicas de programação linear inteira para resolvê-lo. Entre estas técnicas se destacam o Branch-and-Bound (Fores et al., 1999, Smith e Wren, 1988), o Branch-and-Price (Desrochers e Soumis, 1989, Barnhart et al., 1998) e o Branch-and-Cut (Friberg e Haase, 1999. Por outro lado, diversas metaheurísticas foram implementadas para resolver problemas de grande porte. ...
Neste artigo é aplicada a metaheurística Guided Local Search (GLS) para resolver o Problema de Programação de Tripulações de Ônibus Urbano (PPT). O PPT consiste em encontrar um conjunto de jornadas a serem designadas aos motoristas que realizarão a operação diária com o menor custo. A GLS tem como princípio penalizar características indesejáveis presentes na solução corrente, com o objetivo de escapar de soluções ótimas locais. Como heurística de busca local, foi utilizada a heurística Variable Neighborhood Descent, que explora diferentes estruturas de vizinhança para encontrar um mínimo local. De acordo com pesquisa realizada pelos autores, esta abordagem é inédita para resolver o PPT. A implementação proposta foi testada com dados de problemas reais de uma empresa que opera em uma região metropolitana de Belo Horizonte. Os resultados obtidos são similares àqueles presentes na literatura, havendo possibilidades de melhorias, visto que a GLS pode ser explorada em diferentes aspectos.
... A resolução do PPTé de grande importância, uma vez que nos gastos totais de uma empresa, a mão-de-obra operacional representa um dos maiores custos [2], motivo pelo qual esse tema tem sido largamente estudado. A abordagem mais exploradaé aquela que formula o PPT como um problema de recobrimento ou de particionamento e utiliza a técnica de geração de colunas para resolvê-lo [20,5,8,1,9]. A variedade de trabalhos deriva das diferentes maneiras de gerar as colunas e diferentes metodologias para resolver o problema, tais como: branch-and-bound, branch-and-price e a relaxação lagrangeana. ...
Resumo. Este trabalho aborda o Problema de Programação de Tripulações (PPT) no Sistema de Transporte Público. Tal problema consiste em atribuir um conjunto de tarefas aos tripulantes de uma dada empresa participante do sistema de forma que todas as viagens das linhas sob responsabilidade desta sejam executadas com o menor custo possível. A solução do PPTéPPTé um conjunto de jornadas diárias de trabalho de tripulantes. Neste trabalho, o PPT foi abordado utilizando as meta-heurísticas Simulated Annealing (SA), Método de Pesquisa em Vizinhança Variável e Busca Tabu (BT). Esses métodos exploram o espaço de soluções utilizando diferen-tes estruturas de vizinhança, as quais modificam as jornadas de trabalho através de operações realizadas com suas tarefas. Cada solução gerada pelos métodosmétodosé avaliada por uma função baseada em penalidades que visa atender a legislação tra-balhista, as regras operacionais da empresa, assim como melhorar o aproveitamento da mão-de-obra operacional. Os algoritmos foram testados com dados reais de uma empresa que opera na cidade de Belo Horizonte.
... The most widely used approach to deal with this problem models it as a set covering or a set partitioning problem. The strategy of column generation is largely used to solve the problem, as can be seen in the works of Smith and Wren (1988), Desrochers andSoumis (1989), Fores et al. (1999) and Barnhart et al. (1998). However, exact models are limited in practical applications, since they are unable to solve very large problems. ...
This paper explores different local search methods associated with the metaheuristic Iterated Local Search (ILS) to solve the Crew Scheduling Problem (CSP) of a Public Transportation System. The results from ILS were compared to those obtained in a previous work from the same authors that used the Variable Neighborhood Search (VNS). Initially, both metaheuristics were implemented using, as local search, the classical First Improvement Method, performing "guided" reallocation and exchange of crew tasks. The guided reallocation/exchange replaces random components from the classical method by searching the best position to insert the task. Further, the Very Large-scale Neighborhood Search (VLNS) technique was used as a local search procedure in the metaheuristics. This technique has substantially more neighbors than the 2-opt neighborhoods, since it performs a chain exchange of tasks from different crews. Both versions of metaheuristics were applied to a set of real data from a company operating in the city of Belo Horizonte, producing more economical schedules than those adopted by the company. The results are presented and discussed in this work.
... Due to the large number of variables this entails, these formulations are intractable by exhaustive enumeration techniques. Some approaches to solve these formulations are reducing the number of duties being considered (e.g., Smith and Wren, 1988), using Column Generation techniques (e.g., Desrochers and Soumis, 1989), and obtaining dual bounds through heuristics that are then fed into a B&B algorithm (e.g., Mingozzi et al. (1999)). ...
The efficiency of a transport system depends on several elements, such as available technology, governmental policies, the planning process, and control strategies. Indeed, the interaction between these elements is quite complex, leading to intractable decision making problems. The planning process and real-time control strategies have been widely studied in recent years, and there are several practical implementations with promising results. In this paper, we review the literature on Transit Network Planning problems and real-time control strategies suitable to bus transport systems. Our goal is to present a comprehensive review, emphasizing recent studies as well as works not addressed in previous reviews.
Özet: Günümüzün ekonomik şartları altında ekipman seçimi, şirketler için hayati öneme sahip bir karar verme problemidir. Uygun olmayan ekipman seçimi, bir üretim sisteminin üretkenliğini ve toplam performansını olumsuz yönde etkiler, maliyetleri arttırır ve firma prestijini zedeler. Ekipman seçimi, içinde birçok çelişen kriteri, olası alternatifi ve kavramı barındıran, yönetilmesi karmaşık ve zor bir süreçtir. Bu çalışmada, Ankara'da faaliyet gösteren bir üretim firmasının ekipman seçimi problemi ele alınmış ve bu problemin çözümü için yeni bir yaklaşım geliştirilmiştir. Bu yaklaşım, seçim sürecinin çok kriterli, bulanık ve çelişkiler içeren yapısının en etkin şekilde üstesinden gelebilmek ve matematiksel modellerin sağladığı güvenilirlik ve doğruluktan yararlanabilmek temelinde geliştirilmiştir. Çalışmada, değerlendirmede kullanılan dilsel ifadelerin etkisinin seçim sürecine doğru ve tam olarak yansıtılabilmesi ve matematiksel bir model yardımı ile belirlenen hedeflerinin ağırlıklarının dikkate alınması sağlanmıştır.
Anahtar Kelimeler :Ekipman seçimi, çok kriterli karar verme, F-PROMETHEE yöntemi, 0-1 hedef programlama yöntemi
Abstract: Equipment selection is a decision making problem which has a vital importance, according to the economical circumstances of today. An unsuiatble equipment selection effects productivity and the overall performance of the production system negatively, increases costs and damages the company?s prestige. Equipment selection is a complicated and difficult process to be managed, containing a lot of conflicting criteria, possible alternatives and conceptions. In this study, an equipment selection problem of a company operating in Ankara is handled and a new approach for solution of this problem is developed. This approach is constituted with the aim of successfully overcoming the multi criteria, fuzzy and having confilctions structure of the selection process and benefiting from reliability and accuracy that mathematical models provide. Reflecting the effects of the linguistic terms used in the evaluation to the selection process truely and properly and, developing an analytical solution method with the help of a mathematical model which stands for the weights of the determined goals are provided in the study.
Key Words : Equipment selection, multi criteria decision making, F- PROMETHEE method, 0 ? 1 Goal programming method
This article addresses a particular bus Crew Scheduling Problem (CSP) that arises in a public transport company in Spain. The classical CSP problem tries to create bus driver schedules covering all the bus schedules published by the company. The problem posed does not require satisfying any restriction due to the lack of drivers. This relaxed restriction emerged in the time of Covid-19 for different reasons. This is why the need arose in the company for a tool that allows optimizing the daily work of the drivers. The objective function of the problem requires servicing the greatest number of passengers possible, instead of minimizing the schedule cost. A model and strategies for solving the problem exactly are introduced: clustering, re-optimization, etc. Additionally, a re-optimization model is proposed using prior feasible solutions to speed up the resolution of the problem. The integration and practical use of the solutions obtained and their corresponding monitoring in the decision-making process of the company are described.
We study a generalized distributionally robust chance-constrained set covering problem (DRC) with a Wasserstein ambiguity set, where both decisions and uncertainty are binary-valued. We establish the NP-hardness of DRC and recast it as a two-stage stochastic program, which facilitates decomposition algorithms. Furthermore, we derive two families of valid inequalities. The first family targets the hypograph of a “shifted” submodular function, which is associated with each scenario of the two-stage reformulation. We show that the valid inequalities give a complete description of the convex hull of the hypograph. The second family mixes inequalities across multiple scenarios and gains further strength via lifting. Our numerical experiments demonstrate the out-of-sample performance of the DRC model and the effectiveness of our proposed reformulation and valid inequalities.
Public transportation is a shared transportation service and includes urban public transit and intercity public transportation. The main transportation modes in urban public transit are buses, trams, trolleybuses, trains, and the metro. Ferries also appear in some cities in the world as an urban transportation mode. Airlines, buses, intercity rail, and, in some countries, high-speed rail are the transportation modes that appear in intercity public transportation. Some of the biggest transit operators in the world operate a few thousand buses and serve a few hundred bus routes. They employ thousands of drivers. The world’s largest airline serves approximately 100 airports in 41 countries with a fleet of nearly 700 aircraft. This chapter considers planning and operations in public transportation systems and includes public transportation network design, service frequencies determination, vehicle and crew scheduling, disruption management, and revenue management problems. The chapter deals with urban public transit systems, interurban road transport systems, and air transportation.
Given a set of timetabled bus trips, transport companies are faced with the challenge of finding a feasible driver schedule that covers all trips and abides by various labor union regulations. The regulations are primarily concerned with providing sufficient breaks for the drivers during the day. Practical limitations in the city network enforce drivers to travel by cars between bus stops to have breaks. Transport companies have a limited number of cars, known as staff cars, which have to be returned to their respective depots at the end of the day. The simultaneous scheduling of drivers and staff cars for the drivers is known as the driver scheduling problem with staff cars (DSPSC). It is estimated that the DSPSC accounts for 60% of a bus company’s operational expense, and this paper proposes a column generation approach that attempts to minimize operational expense. The column-generation framework iterates between a master problem, a subproblem for generating driver variables and a subproblem for generating staff car variables. The subproblem related to the drivers is formulated as a resource constrained shortest path problem, which is solved by a dynamic programming approach. Several heuristic branching strategies are explored to find integer solutions. The proposed methodology is tested on eight real-life instances from seven Northern European bus companies. A comparison with a state-of-the-art mixed integer programming (MIP) solver and an adaptive large neighborhood search (ALNS) heuristic indicate that the column generation approach provides improved solutions for six instances and the average improvement is 1.45%.
This paper presents Bus Driver Allocation Problem (BDAP) which deals with the assignment of the drivers to the scheduled duties so that operations costs are minimized while its solution meets hard constraints resulting from the safety regulations of public transportation authorities as well as from the internal company’s agreements. Another concern in this study is the overall satisfaction of the drivers since it has important consequences on the quality and economic return of the company operations. This study proposes a non-dominated bi-criteria allocation approach of bus drivers to trips while minimizing the operations cost and maximizing the level of satisfaction of drivers. This paper proposes a new mathematical formulation to model the allocation problem. Its complexity has lead to solving the BDAP by using techniques such as: Genetic algorithms in order to assign the drivers with minimal operations cost and Fuzzy Logic and Rough Set techniques in order to fuzzyfied some parameters required for the satisfaction module followed by applying the Rough Set technique to evaluate the degree of satisfaction for the drivers. The application of the proposed approach to a medium size Transport Company Bus Driver Allocation Problem is evaluated.
The Crew Reserve Assignment Problem (CRAP) considers the assignment of the crew members to a set of reserve activities covering all the scheduled flights in order to ensure a continuous plan so that operations costs are minimized while its solution must meet hard constraints resulting from the safety regulations of Civil Aviation as well as from the airlines internal agreements. The problem considered in this study is of highest interest for airlines and may have important consequences on the service quality and on the economic return of the operations. A new mathematical formulation for the CRAP is proposed which takes into account the regulations and the internal agreements. While current solutions make use of Artificial Intelligence techniques run on main frame computers, a low cost approach is proposed to provide on-line efficient solutions to face perturbed operating conditions. The proposed solution method uses a dynamic programming approach for the duties scheduling problem and when applied to the case of a medium airline while providing efficient solutions, shows good potential acceptability by the operations staff. This optimization scheme can then be considered as the core of an on-line Decision Support System for crew reserve assignment operations management.
Abstract: The Crew Scheduling Problem is to determine the minimum number of crew members and specify their trips in such a way as to cover all trips of the operating fleet at the lowest possible cost and according the restrictions related to operation and labor standards. It is classified as a NP-Hard problem, so it is important to apply heuristic methods to solve large problems. This paper presents a heuristic method that uses a constructive method to obtain an initial solution and a search with different types of schedule works for reoptimization. The method was implemented in C ++ in problems available in the literature and the results are compared and proved to be quite satisfactory.
The authors discuss how Boston Public Schools (BPS) and a team from Massachusetts Institute of Technology addressed the BPS bus routing problem. They developed an algorithm that decomposes and separately solves subproblems of assigning students to bus stops, assigning stops to bus trips, and connecting trips into an itinerary for each bus.
This paper presents an adaptation of the adaptive large neighborhood search (ALNS) heuristic in order to solve the crew scheduling problem (CSP) of urban buses. The CSP consists in minimizing the total of crews that will drive a fleet in daily operation as well as the total overtime. The solution for the CSP is a set of duties performed by the crews throughout the day, and those duties must comply with labor laws, labor union agreements, and the company’s operational rules. The CSP is a NP-hard problem and it is usually solved by metaheuristics. Therefore, an ALNS-like heuristic was developed to solve the CSP. Its implementation was tested with real data from a bus company which operates in Belo Horizonte, MG-Brazil, and it provided solutions quite superior to those both adopted by the company and the ones generated by other methods in the literature.
This study deals with the bus & driver scheduling problems with mealtime windows for a single public transport bus route. Firstly, we develop three explicit Integer Linear Programming (ILP) models to formulate: the bus driver scheduling, bus & driver scheduling, and bus & driver scheduling with mealtime windows, respectively. These ILP models enable bus operators to solve their bus & driver scheduling problems by directly invoking an available optimization solver such as IBM ILOG CPLEX. However, our numerical experiments do not show remarkable improvements, as CPLEX takes more than 12, 17, and 35 h respectively to reduce the optimality gap to at most 0.5% for solving the above three models in an over 12-hour bus & driver scheduling problem. To improve the computational efficiency, we develop a valid inequality approach that can generate valid cuts incorporating with the CPLEX. An investigation into the valid inequality approach proves its high computational efficiency. In addition, we develop a self-adaptive search method for determining the upper and lower bounds of driver group and bus fleet sizes. Finally, we test our models and approaches on a real round-trip bus route. The results indicate that compared with the only use of CPLEX to solve the above three models, the valid inequality approach can address large-scale instances in a reasonable CPU time.
Significance
Spreading start times allows school districts to reduce transportation costs by reusing buses between schools. However, assigning each school a time involves both estimating the impact on transportation costs and reconciling additional competing objectives. These challenges force many school districts to make myopic decisions, leading to an expensive and inequitable status quo. For instance, most American teenagers start school before 8:00 AM, despite evidence of significant associated health issues. We propose an algorithm to jointly solve the school bus routing and bell time selection problems. Our application in Boston led to $5 million in yearly savings (maintaining service quality despite a 50-bus fleet reduction) and to the unanimous approval of the first school start time reform in 30 years.
Public transportation is a shared transportation service, and includes urban public transit and intercity public transportation. The main transportation modes in urban public transit are buses, trams, trolleybuses, trains, and metro. Ferries also appear in some cities in the world as an urban transportation mode. Airlines, buses, intercity rail, and, in some countries, high-speed rail are the transportation modes that appear in intercity public transportation. Some of the biggest transit operators in the world operate a few thousand buses and serve a few hundred bus routes. They employ thousands of drivers. The world's largest airline serves approximately 100 airports in 41 countries with a fleet of nearly 700 aircraft. This chapter considers planning and operations in public transportation systems and includes public transportation network design, service frequencies determination, vehicle and crew scheduling, disruption management, and revenue management problems. The chapter deals with urban public transit systems, interurban road transport systems, and air transportation.
Public transport driver scheduling is a complex problem because the number of possible schedules is usually astronomical and the rules regarding legality and quality of duties are often many, ill-defined, and conflicting. This paper describes some research into using Knowledge Based Systems (KBS) for overcoming these difficulties. The study has led to a new approach to driver scheduling. In this new approach, an intelligent duty estimator is used every time a duty is formed. Corrective actions are taken if the new estimate is different from before. The work on the new approach as currently evolved is outlined.
The 3-flip neighborhood local search (3FNLS) is an excellent heuristic algorithm for the set covering problem which has dominating performance on the most challenging crew scheduling instances from Italy railways. We introduce a method to further improve the effectiveness of 3FNLS by incorporating random flat move to its search process. Empirical studies show that this can obviously improve the solution qualities of 3FNLS on the benchmark instances. Moreover, it updates two best known solutions within reasonable time.
The VAMPIRES bus scheduling program has its origins in railway locomotive scheduling work over the period 1960-1965 (Wolfenden and Wren (7)), and was adapted to bus scheduling in 1970 (Wren (8)). Throughout the 1970s continual improvements were made, and the definitive version of the stand-alone program is given by Smith and Wren (4). VAMPIRES is based on a powerful heuristic which seeks out all opportunities of dead running and identifies changes which would have to be made to trip timings if a solution is to be achieved with a particular number of buses.
In Brazil, bus services, which have to meet extraordinary levels of demand and traffic congestion, are generally planned by a city transport authority, and franchised out to several private companies for operation. The nature of the scheduling problem is different from most of those described at previous workshops, because vehicle and driver schedules are developed simultaneously, but on a line by line basis. The present paper describes work carried out on representative Brazilian city schedules using enhancements of the Leeds scheduling systems reported at previous workshops. Although the precise nature of the problem varies among the cities studied, the approaches adopted in this research are similar. The Leeds scheduling software, adapted for the Brazilian situation, is used in a novel way to produce efficient combined bus and driver schedules, with buses being switched between lines to improve efficiency. Already savings have been indicated. The paper includes a detailed exposition of the processes used in one of the cities, serving to emphasize the practical problems which must be addressed in any implementation.
The goal of the present study was to develop and apply a genetic algorithm-based column generation heuristic to solve a passenger rail crew scheduling problem in North America. The crew scheduling problem minimized the total cost of payment to crew members on the basis of the number of hours on board, the number of hours held away from a crew base, the number of nights of lodging, and the number of onboard and away meals. Payment regulations also dictated an overtime payment and a guaranteed salary per week. Additional problem constraints included restrictions on the maximum number of continuous working hours, the maximum number of days worked per week, and the minimum number of hours of rest. The proposed heuristic produced solutions with improvements of total cost ranging from 3.0% to 27.9%.
The privatization of British Rail into twenty five train operating companies and three freight companies has highlighted the need for each company to have efficient operating schedules. Manpower costs are a significant element in any transport organization, and the ability to minimize these is seen as crucial to the effective running of these companies. In addition, the need to try out different operating strategies is gaining in importance as the search for cost cutting progresses.
Following a feasibility study conducted with British Rail in 1990–91 the paper describes more recent work in which a new driver scheduling system TRACS II has been developed to take account of the conditions relating specially to the rail situation.
Taking data from several train operating companies and talking to many others, both passenger and freight, a database of train work exhibiting a wide variety of different operating strategies has been established. This has enabled the development of new software tools and indicated new areas where further research could be useful.
The driver schedules produced for the train companies have shown savings in manpower requirements over those currently in operation. This situation has been achieved consistently, and savings improved as new techniques developed through the course of the work. The ease with which the drivers’ labor agreement could be changed has greatly impressed the staff in the companies with whom we have worked and further research has been independently promoted. The paper will show how these savings have been achieved and indicate the types of computer techniques employed.
The traditional manner to design public transportation system is to sequentially design the transit network and public bicycle network. A new public transportation system design problem that simultaneously considers both bus network design and public bicycle network design is proposed. The chemical reaction optimization (CRO) is designed to solve the problem. A shortcoming of CRO is that, when the two-molecule collisions take place, the molecules are randomly picked from the container. Hence, we improve CRO by employing different mating strategies. The computational results confirm the benefits of the mating strategies. Numerical experiments are conducted on the Sioux-Falls network. A comparison with the traditional sequential modeling
framework indicates that the proposed approach has a better performance and is more robust. The practical applicability of the approach is proved by employing a real size network.
A hybrid approach for bus driver scheduling is described, which incorporates genetic algorithms (GAs), a rule-based driver duties estimator, and an integer linear programming method called IMPACS. IMPACS produces near-optimal schedules. However, a scheduling run might take hours, and larger problems may have to be decomposed into sub-problems. Earlier research indicates that GAs for directly producing driver schedules are quick, but so far they tend to converge at suboptimal solutions. In the hybrid approach, GAs take on an indirect role. They do not form schedules, but instead would yield a population of vastly condensed versions of the problem. IMPACS is then applied using the elite population members to yield the optimal schedule almost trivially. The estimator plays an essential role in the evaluation of population members during the evolution process. The resulting scheduling system captures the complementary strengths of its components and has potential for solving large problems optimally and quickly.
The application of a bus driver scheduling system using linear programming with heuristics, to the problem of scheduling train drivers in the United Kingdom is described. The speed with which the existing proven system could be amended to model this new situation was of the utmost importance. It was required to produce a model which would allow the user to explore the consequences of implementing possible changes to an existing set of rules for constructing train driver schedules and to determine quickly whether the proposed changes would produce desirable schedules.
The prototype was developed on a set of data relating to one depot, and in the first instance the existing constraints were modelled. The data set was then extended to include more depots. Testing on another set of data which exhibited different features to the original, enabled the model to be more thoroughly checked, and allowed the user the opportunity to become familiar with using it Methods were developed which enabled new styles of constraints to be modelled, and the system had to maintain a degree of flexibility to cater for rules which only became necessary as work and testing progressed.
The problems which were encountered in modifying the existing system to embrace the practices of railway operation are outlined. Because of the time constraints placed upon the exercise both the model and the data input were the subject of some simplifying assumptions. However, the ease of use of the model by the user and the degree to which the model was successfully used, demonstrated the viability of the project.
CRU-SCHED—A computer based bus crew scheduling system using integer programming
- Mitra
Overview of the RUCUS package driver run cutting program—RUNS
- Wilhelm
ZIP—A zero-one integer programming package for scheduling
- Ryan