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Metaheuristics for The Bus-Driver Scheduling Problem
Abstract
We present new metaheuristics for solving real crew scheduling problems in a public transportation bus company. Since the crews of these companies are drivers, we will designate the problem by the bus-driver scheduling problem. Crew scheduling problems are well known and several mathematical programming based techniques have been proposed to solve them, in particular using the set-covering formulation. However, in practice, there exists the need for improvement in terms of computational efficiency and capacity of solving large-scale instances. Moreover, the real bus-driver scheduling problems that we consider can present variant aspects of the set covering, as for example a different objective function, implying that alternative solutions methods have to be developed. We propose metaheuristics based on the following approaches: GRASP (greedy randomized adaptive search procedure), tabu search and genetic algorithms. These metaheuristics also present some innovation features based on and genetic algorithms. These metaheuristics also present some innovation features based on the structure of the crew scheduling problem, that guide the search efficiently and able them to find good solutions. Some of these new features can also be applied in the development of heuristics to other combinatorial optimization problems. A summary of computational results with real-data problems is presented.
... Because civilian airline crew costs often exceed $1.3 billion every year, even very small incremental savings can save the airlines a significant amount of money each year (Hoffman and Padberg, 1993:658). The success of tabu search on similar combinatorial problems motivates its use in this research (Barnes, et al., 1995;Dowsland, 1998;Lourenco, et al., 1998;Shen and Kwan, 2000). ...
... Although tabu search has not been used to solve the airline CSP, it has been used to schedule other types of crews (Dowsland, 1998;Lourenco, et al., 1998;Shen and Kwan, 2000). Lourenco, et al. (1998) and Shen and Kwan (2000) describe tabu search approaches to the bus driver CSP. ...
... Although tabu search has not been used to solve the airline CSP, it has been used to schedule other types of crews (Dowsland, 1998;Lourenco, et al., 1998;Shen and Kwan, 2000). Lourenco, et al. (1998) and Shen and Kwan (2000) describe tabu search approaches to the bus driver CSP. Lourenco, et al. assume that the bus driver CSP is small enough to generate all feasible columns of the SPP a priori and their tabu search assumes such an approach while Shen and Kwan develop a methodology starting from an initial feasible solution, similar to Baker, et al. (1979Baker, et al. ( , 1981. ...
This research develops the first metaheuristic approach to the complete air crew scheduling problem. It develops the first dynamic, integrated, set-partitioning based vocabulary scheme for metaheuristic search. Since no benchmark flight schedules exist for the tanker crew scheduling problem, this research defines and develops a Java(Trademark) based flight schedule generator. The robustness of the tabu search algorithms is judged by testing them using designed experiments. An integer program is developed to calculate lower bounds for the tanker crew scheduling problem objectives and to measure the overall quality of solutions produced by the developed algorithms.
... A review of the literature reveals several works regarding crew scheduling, a problem than can be modelled as a Set Covering Problem (SCP) [5] or a Set Partition Problem (SPP) [2] . In some cases the crew scheduling is applied to the road passenger ...
... These limitations of our approach could be overcomed in a future with the combination of liner programming techniques when computing the estimation function. We are also studying other alternative mechanisms to accelerate the problems solution , as for example, symmetries [8] or learning techniques [5]. ...
In this paper we explain an estimation function for the crew scheduling problem in the coach transportation domain. Thanks to this function we obtain an estimation that is very close to the real cost of the solution and we can efficiently prune the search space with a branch and bound approach. Consequently we find the optimal solution in less time than with other estimations such as the Russian doll method.
... While, some [14,15] refer to the total working time to spread over for each duty in the limitation of maximum and minimum value. Some models [16][17][18] consider the duty cost as a whole including real cost, meal cost and so on, while PLOS sometimes [19,20] penalty cost function is added to measure duty working hours to limit hours in labor agreement rules. Moreover, unfairness function considering different duty working hours and overtime penalty is referred in some papers [21]. ...
This work proposes a model considering fairness in the problem of crew scheduling for bus drivers (CSP-BD) using a hybrid ant-colony optimization (HACO) algorithm to solve it. The main contributions of this work are the following: (a) a valid approach for cases with a special cost structure and constraints considering the fairness of working time and idle time; (b) an improved algorithm incorporating Gamma heuristic function and selecting rules. The relationships of each cost are examined with ten bus lines collected from the Beijing Public Transport Holdings (Group) Co., Ltd., one of the largest bus transit companies in the world. It shows that unfair cost is indirectly related to common cost, fixed cost and extra cost and also the unfair cost approaches to common and fixed cost when its coefficient is twice of common cost coefficient. Furthermore, the longest time for the tested bus line with 1108 pieces, 74 blocks is less than 30 minutes. The results indicate that the HACO-based algorithm can be a feasible and efficient optimization technique for CSP-BD, especially with large scale problems.
... A Figura 1, utilizada em Lourenço, Paixão & Portugal (2001), situa a programação de tripulações dentro do processo de planejamento de transporte. ...
... The construction of timetables in STCP is aided by GIST system: a Decision Support System to assist the planning department of public and private transportation companies or transit authorities in the operations management [12]. The system includes several modules and interfaces to aid the planner in activities as the production of timetables, the scheduling of vehicles, the generation of daily duties for drivers and the construction of rosters of individual drivers for a certain period [14,15]. ...
Reliability of public transport systems is one of the major concerns of both passengers and public transport companies. The recent evolution of some technologies like GPS made possible the storing of de-tailed information about past travels. The analysis of that information can provide important descriptions of the system performance, allowing the transit planners to take decisions in order to improve it. The need of tools especially designed for these kind of analysis motivated the de-velopment of the application described in this paper. The main goal of this application is to provide statistical information about travel times and offering decision support to transit planners. This application was developed for the bus transportation company STCP, situated in Porto.
... Na literatura destacam-se os trabalhos de Clement e Wren (1995), e Kwan et al. (2001) que utilizam Algoritmos Genéticos, enquanto Shen e utilizam Busca Tabu. Lourenço et al., (2001) utilizam as metaheurísticas Busca Tabu e Algoritmo Genético para selecionar soluções não dominadas, segundo o conceito multiobjetivo. As colunas de tais soluções, geradas ao longo do processo, são armazenadas para constituir o domínio sobre o qual é aplicado um método exato para o problema de particionamento, quando se tratar de instâncias de pequeno porte. ...
RESUMO Otimizar os custos de alocação da mão de obra, ou seja dos motoristas e cobradores, em sistemas reais de transporte público é um problema de tal complexidade que não é possível resolvê-lo por meio de métodos exatos. Além da magnitude do problema, que em caso reais o número de variáveis é da ordem milhões, existe a dificuldade de definir e calibrar uma função objetivo que reflita com precisão as características inerentes às situações práticas. Este trabalho apresenta uma metaheurística para minimizar os custos de alocação com a mão de obra e discute os resultados obtidos com diversas funções objetivo adotados para a realidade operacional de duas empresas dentre aquelas que atuam em um sistema de transporte público. A análise dos resultados mostra diferentes possibilidades de redução de custos em função de mudanças na forma de operação da mão de obra. Mais ainda, o trabalho deixa evidente a existência de objetivos conflitantes na abordagem do problema de alocação da tripulações no sistema. ABSTRACT The task of solving real crew scheduling problems is so complex that becomes impossible to solve it with the aid of exact methods. Besides the size of the problem, whose number of variables can achieve millions, it is not easy to define an objective function that can reflect precisely all characteristics inherent to real applications. This paper presents a metaheuristic to solve the crew scheduling problem and also discuss the results obtained through a set of different objective functions developed to represent the operational reality of two companies among those that operates on a public transportation system. The analysis of the obtained results shows different possibilities of cost reductions associated to changes in the crew operational way. Moreover, in this paper become evident how conflicting are the objectives contained in the mass transit crew scheduling problem. 1. INTRODUÇÃO O planejamento de um Sistema Municipal de Transporte Público é bastante complexo e conseqüentemente é dividido em alguns subproblemas. Os dados de saída referentes à resolução de um subproblema são os dados de entrada do subproblema seguinte. As principais etapas deste processo são: o planejamento das rotas, a definição da tabela de horários de cada rota, a programação dos veículos, a programação das tripulações e finalmente o rodízio das tripulações ao longo do mês.
... In fact, we can find several works in the literature regarding crew scheduling problems, which are known to be particular cases of the set covering problem (Hoffman & Padberg 2000). Crew scheduling, however, has often been studied in regular services (Loureno, Paixao, & Portugal 2001;Pezzella & Faggioli 1997;Cantillon & Pesendorfer 2006). Indeed, it is important to distinguish between scheduling and two other related problems: vehicle scheduling and the rostering problem (dealing with the rotating shifts of the crews). ...
In this paper we investigate the interactions between planning and scheduling in a specific scenario, road pas-senger transportation. The problem consists of finding the best assignment of available drivers (resources) to a set of requested services (actions) given a cost function and subject to the constraints provided by administra-tions. We have considered two approaches. In the first approach we deal with planning and scheduling as a sin-gle solution step. In the second approach, we consider planning and scheduling in separate steps. The results obtained from the experimentation show, in this particu-lar case, that keeping planning and scheduling separate, achieves a better performance.
... Our interest in studying a new heuristic for a problem for which many efficient heuristics have been proposed is motivated by several observations. GRASP has been applied with success to a number of scheduling problems [4,5,12131417,18,26, 29, 35, 36, 41]. A natural question is whether it can find good solutions to the job shop scheduling problem. ...
In the job shop scheduling problem (JSP), a finite set of jobs is processed on a finite set of machines. Each job is characterized by a fixed order of operations, each of which is to be processed on a specific machine for a specified duration. Each machine can process at most one job at a time and once a job initiates processing on a given machine it must complete pro-cessing uninterrupted. A schedule is an assignment of operations to time slots on the machines. The objective of the JSP is to find a schedule that mini-mizes the maximum completion time, or makespan, of the jobs. In this paper, we describe a greedy randomized adaptive search procedure (GRASP) for the JSP. A GRASP is a metaheuristic for combinatorial optimization. Although GRASP is a general procedure, its basic concepts are customized for the prob-lem being solved. We describe in detail our implementation of GRASP for job shop scheduling. Further, we incorporate to the conventional GRASP two new concepts: an intensification strategy and POP (Proximate Optimality Princi-ple) in the construction phase. These two concepts were first proposed by Fleurent & Glover (1999) in the context of the quadratic assignment problem. Computational experience on a large set of standard test problems indicates that GRASP is a competitive algorithm for finding approximate solutions of the job shop scheduling problem.
... Variants of this strategy have more recently been introduced by Aggarwal, Orlin and Tai (1997) as a proposal for modifying traditional genetic algorithms, and have also been applied to weighted clique problems by Balas and Niehaus (1998). A particularly interesting application occurs in the work of Lourenço, Paixao and Portugal (1998), who use such concepts to create " perfect children " for crew scheduling problems. In general, vocabulary building relies on destructive as well as constructive processes to generate desirable partial solutions, as in the early proposals for exploiting strongly determined and consistent variables – which essentially " break apart " good solutions to extract good component assignments, and then subject these assignments to heuristics to rebuild them into complete solutions. ...
The evolutionary approach called scatter search, and its generalized form called path relinking, originated from strategies for creating composite decision rules and surrogate constraints. Recent studies demonstrate the practical advantages of these approaches for solving a diverse array of optimization problems from both classical and real world settings. Scatter search and path relinking contrast with other evolutionary procedures, such as genetic algorithms, by providing unifying principles for joining solutions based on generalized path constructions (in both Euclidean and neighborhood spaces) and by utilizing strategic designs where other approaches resort to randomization. Additional advantages are provided by intensification and diversification mechanisms that exploit adaptive memory, drawing on foundations that link scatter search and path relinking to tabu search. The goal of this paper is to clarify the connection between these developments in evolutionary methods, and to highlight key ideas and research issues that offer promise of yielding future advances.
... A journey is then defined as the set of services that can be assigned to a single driver (journey duties driver). Journey generation is known as the crew scheduling problem, which is complemented by the rostering problem in which journeys are assigned to drivers (Ramalhinho Lourenc -o et al., 2001). In our experimental study, tackling the problem as a whole (service approach) or in two steps (journey approach) depends on the modeling capacities of the techniques. ...
Analyzing the state of the art in a given field in order to tackle a new problem is always a mandatory task. Literature provides surveys based on summaries of previous studies, which are often based on theoretical descriptions of the methods. An engineer, however, requires some evidence from experimental evaluations in order to make the appropriate decision when selecting a technique for a problem. This is what we have done in this paper: experimentally analyzed a set of representative state-of-the-art techniques in the problem we are dealing with, namely, the road passenger transportation problem. This is an optimization problem in which drivers should be assigned to transport services, fulfilling some constraints and minimizing some function cost. The experimental results have provided us with good knowledge of the properties of several methods, such as modeling expressiveness, anytime behavior, computational time, memory requirements, parameters, and free downloadable tools. Based on our experience, we are able to choose a technique to solve our problem. We hope that this analysis is also helpful for other engineers facing a similar problem.
The Labor Scheduling Problem consists of planning the shifts for the employees, and minimizing costs associated to the workforce, which is a NP-hard problem.This paper presents a mixed integer linear programming modelapplied to a real case, which minimizes labor costs, satisfies the requirements of demand and establishes adequate working conditions for employees by incorporating constraints that ensure well-being to generate an optimal assignment of physiotherapist shifts in the intensive and intermediate care areas in a Clinic.For this, different scenarios were defined by varying both the number of physiotherapists and the structure of the model, the result that the appropriate number of physiotherapists in the clinic schedule is 32, since it satisfies the requirements of demand, employment law and the needs of the company and employees
We suggest a decision support system for everyday operations of a public transit company. It will help to realize the timetabled trips and to give the best decisions throughout the process. Top objectives of this system are operating smoothly, minimizing the cost and, maximizing the employees' satisfaction. Each timetabled trip needs a vehicle and a driver. Operations are usually planned in monthly basis. As a result, each driver has a roster. It is very important to balance the work among the drivers. This corresponds to a very big sized multi-objective decision making problem. This problem is mathematically modeled and solved using Multi-Objective Tabu Search. As a result, we propose a nondominant set of solutions, where the end user will pick a solution.
Toplu taşıma firmalarının planlanmış seferlerini gerçekleştirebilmelerine ve bu sırada verilen kararlarını en iyi şekilde verebilmelerine yardımcı olacak bir karar destek sistemi önerilmiştir. Bu sistemin amaçları, günlük operasyonları doğru bir şekilde yürütmek, verilen doğru kararlarla maliyetleri azaltmak ve çalışanların memnuniyetini en yüksek seviyede tutmaktır. Seferleri yürütürken, her sefere bir araç ve bir şoför ataması gerekir. Bu planlar genelde aylık olarak yapılır. Bu planın sonucunda her şoförün kendine ait bir planı vardır. Seferleri dengeli olarak şoförlere dağıtmak çok önemlidir. Bu da çok büyük boyutlu ve çok amaçlı bir karar verme problemine denk gelir. Bu problem matematiksel olarak modellenip, meta-sezgisel bir çözüm yöntemi olan çok amaçlı Tabu araması yaklaşımı kullanılarak çözülmüş ve kullanıcıya arasında seçebileceği bir çözüm kümesi önerilmiştir.
This paper suggests O(m) polynomial time heuristic algorithm to obtain the solution for the driver scheduling problem, DSP, that has been classified as NP-complete problem. The proposed algorithm gets the initial assignment of n minimum number of drivers from given m schedules. Nextly, this algorithm gets the minimum total time (TC) using 5 rules of swap and insert for decrease of over times (OT) and idle times (IT). Although this algorithm is a heuristic polynomial time algorithm with O(m) time complexity rules to be find a optimal (or approximate) solution, this algorithm is equal to metaheuristic methods for the 5 experimental data. To conclude, this paper shows the DSP is not NP-complete problem but Polynomial time (P)-problem with polynomial time rules.
GRASP is a multi-start metaheuristic for combinatorial optimization problems, in whicheach iteration consists basically of two phases: Construction and local search. The construction phase builds a feasible solution, whose neighborhood is investigated until a local minimum isfound during the local search phase. The best overall solution is kept as the result. An intensification strategy based on path-relinking is frequently used to improve solution quality andtoreduce computation times by exploring elite solutions previously found along the search. This chapter describes the basic components of GRASP, successful implementation strategies, and effective hybridizations with path-relinking and other metaheuristics. We also list some tricks to be used in the quest for good implementations. The bibliography is enriched by an account of relevant applications and by links to surveys, software, and additional sources of material.
Abstract—The evolutionary approach called Scatter Search, and its generalized form called Path Relinking, have proved unusually effective for solving a diverse array of optimization,problems from both classical and real world settings. Scatter Search and Path Relinking differ from other evolutionary procedures, such as genetic algorithms, by providing unifying principles for joining solutions based on generalized ,path constructions (in both Euclidean and neighborhood,spaces) and by utilizing strategic designs where other approaches resort to randomization. Scatter Search and Path Relinking are also intimately related to the Tabu Search metaheuristic, and derive additional advantages by making ,use of adaptive ,memory,and associated memory-exploiting mechanisms ,that are capable of being adapted to particular contexts. We,describe the features of Scatter Search and Path Relinking that set them apart from other evolutionary approaches, and that offer opportunities for creating increasingly more versatile and effective methods in the future. ,,,,,,, ,,,,,,,
We present new multiobjective metaheuristics for solving real-world crew scheduling problems in public bus transport companies. Since the crews of these companies are drivers, we will designate the problem as bus-driver scheduling. Crew scheduling problems are well known, and several mathematical programming-based techniques have been proposed to solve them, in particular, using the single-objective set-covering formulation. However, in practice, there exists the need to consider multiple objectives, some of them in conflict with each other; for example, the cost and service quality, implying also that alternative solution methods have to be developed. We propose multiobjective metaheuristics based on the tabu search and genetic algorithms. These metaheuristics also present some innovation features related with the structure of the crew scheduling problem that guide the search efficiently and enable them to find good solutions. Some of these new features can also be applied to the development of heuristics to other combinatorial optimization problems. A summary of computational results with real-data problems is presented. The methods have been successfully incorporated in the GIST Planning Transportation Systems and are actually used by several companies.
The public transportation is gaining importance every year basically due the population growth, environmental policies and, route and street congestion. Too able an e¢cient management of all the resources related to public transportation, several techniques from di¤erent areas are being applied and several projects in Transportation Planning Systems, in di¤erent countries, are being developed. In this work, we present the GIST Planning Transportation Systems, a Portuguese project involving two universities and six public transportation companies. We describe in detail one of the most relevant modules of this project, the crew-scheduling module. The crew-scheduling module is based on the application of meta-heuristics, in particular GRASP, tabu search and genetic algorithm to solve the bus-driver-scheduling problem. The metaheuristics have been successfully incorporated in the GIST Planning Transportation Systems and are actually used by several companies.
So, the main purpose of this work is to propose an efficient method for determine optimal shifts in transportation, for any
local public transportation companies. The objective of it is to minimize the cost of the shifts programmed in the temporary
horizon considered (day, week, month,..), with the restriction of having enough number of drivers in all periods of times
to satisfy the demand required. This problem has been solved using two Grasp Metaheuristic and a Bio-inspired Algorithm.
In the job shop scheduling problem (JSP), a finite set of jobs is processed on a finite set of machines. Each job is characterized by a fixed order of operations, each of which is to be processed on a specific machine for a specified duration. Each machine can process at most one job at a time and once a job initiates processing on a given machine it must complete processing uninterrupted. A schedule is an assignment of operations to time slots on the machines. The objective of the JSP is to find a schedule that minimizes the maximum completion time, or makespan, of the jobs. In this paper, we describe a greedy randomized adaptive search procedure (GRASP) for the JSP. A GRASP is a metaheuristic for combinatorial optimization. Although GRASP is a general procedure, its basic concepts are customized for the problem being solved. We describe in detail our implementation of GRASP for job shop scheduling. Further, we incorporate to the conventional GRASP two new concepts: an ...
. GRASP (greedy randomized adaptive search procedure) is a metaheuristic for combinatorial optimization. GRASP usually is implemented as a multistart procedure, where each iteration is made up of a construction phase, where a randomized greedy solution is constructed, and a local search phase which starts at the constructed solution and applies iterative improvement until a locally optimal solution is found. This chapter gives an overview of GRASP. Besides describing the basic building blocks of a GRASP, the chapter covers enhancements to the basic procedure, including reactive GRASP, hybrid GRASP, and intensification strategies. 1. Introduction Consider a combinatorial optimization problem, where one is given a discrete set X of solutions and an objective function f(x) : x # X # to be minimized and seeks a solution x # # X such that f(x # ) # f(x), for all x # X . Problems of this type are sometimes easy to solve, i.e. they can be solved in polynomial time, but mor...
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.
Scope and Purpose-A summary is provided of some of the recent (and a few not-so-recent) developments that otTer promise for enhancing our ability to solve combinatorial optimization problems. These developments may be usefully viewed as a synthesis of the perspectives of operations research and artificial intelligence. Although compatible with the use of algorithmic subroutines, the frameworks examined are primarily heuristic, based on the supposition that etTective solution of complex combinatorial structures in some cases may require a level of flexibility beyond that attainable by methods with formally demonstrable convergence properties. Abstract-Integer programming has benefited from many innovations in models and methods. Some of the promising directions for elaborating these innovations in the future may be viewed from a framework that links the perspectives of artificial intelligence and operations research. To demonstrate this, four key areas are examined: (1) controlled randomization, (2) learning strategies, (3) induced decomposition and (4) tabu search. Each of these is shown to have characteristics that appear usefully relevant to developments on the horizon.
Today, a variety of heuristic approaches are available to the operations research practitioner. One methodology that has a strong intuitive appeal, a prominent empirical track record, and is trivial to efficiently implement on parallel processors is GRASP (Greedy Randomized Adaptive Search Procedures). GRASP is an iterative randomized sampling technique in which each iteration provides a solution to the problem at hand. The incumbent solution over all GRASP iterations is kept as the final result. There are two phases within each GRASP iteration: the first intelligently constructs an initial solution via an adaptive randomized greedy function; the second applies a local search procedure to the constructed solution in hope of finding an improvement. In this paper, we define the various components comprising a GRASP and demonstrate, step by step, how to develop such heuristics for combinatorial optimization problems. Intuitive justifications for the observed empirical behavior of the methodology are discussed. The paper concludes with a brief literature review of GRASP implementations and mentions two industrial applications.
A large class of computational problems involve the determination of properties of graphs, digraphs, integers, arrays of integers, finite families of finite sets, boolean formulas and elements of other countable domains. Through simple encodings from such domains into the set of words over a finite alphabet these problems can be converted into language recognition problems, and we can inquire into their computational complexity. It is reasonable to consider such a problem satisfactorily solved when an algorithm for its solution is found which terminates within a number of steps bounded by a polynomial in the length of the input. We show that a large number of classic unsolved problems of covering, matching, packing, routing, assignment and sequencing are equivalent, in the sense that either each of them possesses a polynomial-bounded algorithm or none of them does.
Short abstract, isn't it? P.A.C.S. numbers 05.20, 02.50, 87.10 1 Introduction Large Numbers "...the optimal tour displayed (see Figure 6) is the possible unique tour having one arc fixed from among 10 655 tours that are possible among 318 points and have one arc fixed. Assuming that one could possibly enumerate 10 9 tours per second on a computer it would thus take roughly 10 639 years of computing to establish the optimality of this tour by exhaustive enumeration." This quote shows the real difficulty of a combinatorial optimization problem. The huge number of configurations is the primary difficulty when dealing with one of these problems. The quote belongs to M.W Padberg and M. Grotschel, Chap. 9., "Polyhedral computations", from the book The Traveling Salesman Problem: A Guided tour of Combinatorial Optimization [124]. It is interesting to compare the number of configurations of real-world problems in combinatorial optimization with those large numbers arising in Cosmol...
The urban mass transit companies are forced to look for new paths to reduce the conflict between the demands for retention or even an expansion of available services and decreasing subsidies. A possible way of escape from this situation consists in establishing computer-aided planning systems on operational level (cf. Magnanti [10], Wren [13], Bodin et al. [1], Rousseau [12]).
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.
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.
The integer linear programming (ILP) based optimization approaches to driver scheduling have had most success. However there is scope for a Genetic Algorithm (GA) approach, which is described in this paper, to make improvements in terms of computational efficiency, robustness, and capability to tackle large data sets. The question “What makes a good fit amongst potential shifts in forming a schedule?” is pursued to identify combinatorial traits associated with the data set. Such combinatorial traits are embedded into the genetic structure, so that they would play some role in the evolutionary process. They could be effective in narrowing down the solution space and they could assist in evaluating the fitness of individuals in the population.
The first stage of research uses as a starting point the continuous solution resulting from relaxing the integer requirement of an ILP model for driver scheduling. The continuous solution consists of a relatively small set of shifts, which usually contains a high proportion of the shifts in the integer solution obtained. The aim is to derive a GA to evolve from the non-integer solution to yield some elite schedules for further exploitation of combinatorial traits. This first stage is already very effective, yielding in some test cases schedules as good as those found by ILP.
The second stage of research is still ongoing. The aim is to extract from the fittest individuals in the population various forms of combinatorial traits. The genetic structures are then dynamically transformed to make use of the traits in future generations.
This proceedings volume consists of selected papers presented at the Eighth International Conference on Computer-Aided Scheduling 0/Public Transport (CASPT 2000), which was held at the conference center of the Konrad rd Adenauer-Foundation in Berlin, Germany, from June 2pt to 23 , 2000. The CASPT 2000 is the continuation of aseries of international workshops and conferences presenting recent research and progress in computer-aided scheduling in public transport.Previous workshops and conferences were held in • Chicago (1975), • Leeds (1980), • Montreal (1983 and 1990), • Hamburg (1987), • Lisbon (1993) and • Cambridge, Mass. (1997).1 With CASPT 2000, our series of workshops and conferences celebrated th its 25 anniversary. Starting with a Workshop on Automated Techniques [or Scheduling 0/ Vehicle Operators [or Urban Public Transportation Services in 1975 the scope and purpose has broadened since and still continues to do so. The previous workshops and conferences were focused on public mass transit, and while this remained the primary focus ofthe 2000 conference, it included also computer-aided scheduling methods being developed and applied in re lated means of passenger transport systems. Commonalities regarding op erations research techniques such as, e.g., column generation techniques and 1 While there were no formal proceedings for the first workshop but only a p- printed copy of all papers issued to participants on arrival, the subsequent ones are weil documented as folIows: Wren, A. (Ed.) (1981). Computer Scheduling 0/ Public Transport. North Holland, Amsterdam.
We built a bid-line generator to help Federal Express perform what-if analyses of work rules during contract negotiations with ALPA, the bargaining unit for the company's pilots. The objectives were to minimize both the number of bid lines produced (a measure of required manning) and the amount of flying not assigned to bid lines (flying requirements that must be accommodated during subsequent phases of scheduling). The tool was useful in negotiations since it was automated and easily modified in-house to provide quick responses to bid line what-ifs. Using a two-step process, the tool produces a complete set of legal, flyable lines for an airplane fleet and identifies the remaining open (unscheduled) flying. First it uses simulated annealing to find as many good bid lines as possible; then it uses a greedy heuristic to complete as many more lines as possible.
This chapter reviews the operations research models in two major sectors of transportation: urban transportation and aviation. Airports and the air traffic control (ATC) system are the two principal types of infrastructure for the air transportation system. Most existing operations research models are concerned with issues related either to the runway and taxiway systems of airports or to passenger terminals or to ATC. The chapter also discusses the operations research techniques used to solve planning and operational problems in public transport authorities. It emphasizes operations research techniques that have been applied or have potential applications. It is demonstrated that in the transit area, operations research techniques are alive. They are being used to assist in the planning and operation of many transit networks around the world. New problems are arising from the implementation of new technology in transit. Better information will be available from the in-vehicle microcomputer and better information will be required by the public in general.
The consideration of sequence-dependent setup times is one of the most difficult aspects of production scheduling problems. This paper reports on the development of a heuristic procedure to address a realistic production and inventory control problem in the presence of sequence-dependent setup times. The problem considers known monthly demands, variable production rates, holding costs, minimum and maximum inventory levels per product, and regular and overtime capacity limits. The problem is formulated as a Mixed-Integer Program (MIP), where subtour elimination constraints are needed to enforce the generation of job sequences in each month. By relaxing the subtour elimination constraints, the MIP formulation can be used to find a lower bound on the optimal solution. CPLEX 3.0 is used to calculate lower bounds for relatively small instances of this production problem, which are then used to assess the merit of a proposed heuristic. The heuristic is based on a simple short-term memory tabu search method that coordinates linear programming and traveling salesperson solvers in the search for optimal or near-optimal production plans.
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 problem of constructing daily shifts for public transport (generally bus) drivers is explained, and some of the currently available computer based solution methods are introduced. The need for improved methods is set out, and one of the more widely applied current methods is outlined in sufficient detail to show where new approaches may profitably be introduced. A feasibility study is described in which a simple genetic algorithm has been developed in order to examine the suitability of such an approach. This has required the development of a new crossover operator. Such an algorithm could ultimately replace part of the presented existing method, making it more efficient in terms of both of quality of result and of time taken to produce a good schedule. The simple algorithm has been shown to produce comparable results to the existing method on a test problem. The results encourage further investigation, but some complexities which can exist in real problems require further study. The results of the present experiments are presented, and the further complexities are discussed in the context of the genetic approach.
This paper reports on a lower bound technique based on state space relaxation for a dynamic program associated with a particular class of covering problems related with crew scheduling. Both dynamic programming (DP) and state space relaxation (SSR) techniques may be applied to any type of set covering problem (SCP), but, in particular, the SSR described in this paper revealed itself as an interesting approach for the bus driver scheduling problem. SSR provides a lower bound on the optimal value for the SCP and some reduction tests are derived in order to reduce the number of columns and rows for the instances. Also, feasible solutions may be build upon the SSR solutions yielding an upper bound on the optimum. Computational experience with real life test problems shows that the technique described in this paper is worthwhile trying when dealing with such applications. In fact, for most of the cases, we were able to improve on the quality of the feasible solutions obtained through the using of the greedy heuristics described in the literature for the set covering problem.
Computer solution of timetabling, scheduling and rostering problems has been addressed in the literature since the 1950's. Early mathematical formulations proved impossible to solve given the limited computer power of the era. However, heuristics, often very specialised, were used for certain problems from a very early date, although the term heuristic was not generally recognised until later; a few guaranteed optimality, some consistently produced good solutions, but most became unwieldy when adjusted to deal with practical situations. In some cases, weaknesses in the heuristics were overcome by appeal to manual intervention. Mathematical approaches to some problems returned to favour, successfully, around 1980. Some of the subsequent developments of these are very powerful in practical situations, but they are no panacea, and metaheuristics are the flavour of the nineties.
This paper explores the relationships between the problem types, and traces the above developments as applied principally in the areas of Vehicle Routeing and Scheduling, Driver Scheduling, Job Shop Scheduling and Personnel Rostering. Parallels are drawn with Class and Examination Timetabling, but these subjects themselves are not examined, as they are covered extensively elsewhere in this volume.
The set covering problem (SCP) was one of the first problems shown to be NP-complete. Heuristics are commonly used on mainframe computers in order to efficiently solve large-scale SCPs. In this paper, we use a new heuristic and several existing heuristics written in FORTRAN to solve 31 large (up to 2000 variables) SCPs on an IBM PC/AT. The new heuristic, SCAMP (set covering algorithm for the microprocessor), performed the best, with solution values deviating only an average 1.8% from the optimum.
Thesis (doctoral)--Erasmus Universiteit Rotterdam, 1997.
In this paper, the set covering problem (SCP) is considered. Several algorithms have been suggested in the literature for solving it. We propose a new algorithm for solving the SCP which is based on the genetic technique. This algorithm has been implemented and tested on various standard and randomly generated test problems. Preliminary results are encouraging, and are better than the existing heuristics for the problem.
This book sets out to explain what genetic algorithms are and how they can be used to solve real-world problems. The first objective is tackled by the editor, Lawrence Davis. The remainder of the book is turned over to a series of short review articles by a collection of authors, each explaining how genetic algorithms have been applied to problems in their own specific area of interest. The first part of the book introduces the fundamental genetic algorithm (GA), explains how it has traditionally been designed and implemented and shows how the basic technique may be applied to a very simple numerical optimisation problem. The basic technique is then altered and refined in a number of ways, with the effects of each change being measured by comparison against the performance of the original. In this way, the reader is provided with an uncluttered introduction to the technique and learns to appreciate why certain variants of GA have become more popular than others in the scientific community. Davis stresses that the choice of a suitable representation for the problem in hand is a key step in applying the GA, as is the selection of suitable techniques for generating new solutions from old. He is refreshingly open in admitting that much of the business of adapting the GA to specific problems owes more to art than to science. It is nice to see the terminology associated with this subject explained, with the author stressing that much of the field is still an active area of research. Few assumptions are made about the reader's mathematical background. The second part of the book contains thirteen cameo descriptions of how genetic algorithmic techniques have been, or are being, applied to a diverse range of problems. Thus, one group of authors explains how the technique has been used for modelling arms races between neighbouring countries (a non- linear, dynamical system), while another group describes its use in deciding design trade-offs for military aircraft. My own favourite is a rather charming account of how the GA was applied to a series of scheduling problems. Having attempted something of this sort with Simulated Annealing, I found it refreshing to see the authors highlighting some of the problems that they had encountered, rather than sweeping them under the carpet as is so often done in the scientific literature. The editor points out that there are standard GA tools available for either play or serious development work. Two of these (GENESIS and OOGA) are described in a short, third part of the book. As is so often the case nowadays, it is possible to obtain a diskette containing both systems by sending your Visa card details (or $60) to an address in the USA.
In this paper we present a genetic algorithm-based heuristic for non-unicost set covering problems. We propose several modifications to the basic genetic procedures including a new fitness-based crossover operator (fusion), a variable mutation rate and a heuristic feasibility operator tailored specifically for the set covering problem. The performance of our algorithm was evaluated on a large set of randomly generated problems. Computational results showed that the genetic algorithm-based heuristic is capable of producing high-quality solutions. Keywords: genetic algorithms; set covering; optimisation. 1 Introduction The set covering problem (SCP) is the problem of covering the rows of a m-row, n- column, zero-one matrix (a ij ) by a subset of the columns at minimal cost. Defining x j = 1 if column j (with cost c j ? 0) is in the solution and x j = 0 otherwise, the SCP is Minimise n X j=1 c j x j (1) Subject to n X j=1 a ij x j 1, i = 1; : : : ; m (2) x j 2 f0; 1g, j = 1; ...
In this paper we present a genetic algorithm-based heuristic for solving the set partitioning problem. The set partitioning problem is an important combinatorial optimisation problem used by many airlines as a mathematical model for flight crew scheduling. We develop a steady-state genetic algorithm in conjunction with a specialised heuristic feasibility operator for solving the set partitioning problem. Some basic genetic algorithm components, such as fitness definition, parent selection and population replacement are modified. The performance of our algorithm is evaluated on a large set of real-world set partitioning problems provided by the airline industry. Computational results show that the genetic algorithm-based heuristic is capable of producing highquality solutions. In addition a number of the ideas presented (separate fitness, unfitness scores and subgroup population replacement) are applicable to any genetic algorithm for constrained problems. Keywords: combinator...
Genetic algorithms for the crew scheduling problem: a real experiment with relaxation models
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Producing train driver schedules under operating strategies
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