## A Column Generation Approach to Bus Driver Scheduling (1996)

Citations: | 7 - 4 self |

### BibTeX

@MISC{Fores96acolumn,

author = {Sarah Fores and Les Proll and Anthony Wren},

title = {A Column Generation Approach to Bus Driver Scheduling},

year = {1996}

}

### OpenURL

### Abstract

This paper outlines an alternative solution method which has been incorporated into a system which originated from IMPACS. Improved results on a selection of real bus driver problems are presented. THE DRIVER SCHEDULING PROBLEM

### Citations

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Citation Context ...s to implicitly consider all valid columns without the need to generate them beforehand. Ford and Fulkerson [52] describe the method where the problem is a shortest-path problem and Dantzig and Wolfe =-=[53]-=- describe the decomposition of a linear programming problem. Early applications of a column generation technique were reported by Gilmore and Gomory [54] which used column generation principles to sol... |

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Citation Context ... a shortest-path problem and Dantzig and Wolfe [53] describe the decomposition of a linear programming problem. Early applications of a column generation technique were reported by Gilmore and Gomory =-=[54]-=- which used column generation principles to solve CHAPTER 2. REVIEW OF BUS DRIVER SCHEDULING METHODS 25 the cutting-stock problem. Although column generation techniques have successfully been applied ... |

116 |
Solving airline crew scheduling problems by branchand-cut
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Citation Context ... et al. [50] describe an interactive system which solves a sequence of subproblems by mathematical or local search techniques and can produce optimal solutions for small problems. Hoffman and Padberg =-=[51]-=- have developed a system which uses a set partitioning method and can run independently of any company. This system also produced many integer LP solutions and for the remaining problems an integer so... |

71 | A column generation approach for graph coloring
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Citation Context ...ULING METHODS 25 the cutting-stock problem. Although column generation techniques have successfully been applied to many problems, e.g. the binary cutting stock problem [55], graph colouring problems =-=[56]-=-, and traffic assignment problems [57], sections 2.6.1, 2.6.2 and 2.6.3 refer only to column generation methods which have been implemented to solve transportation problems. 2.6.1 Bus Driver Schedulin... |

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Algorithms for Network Programming
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Citation Context ...hether a link is used or not. The previously defined costs can then be placed on each link with a zero cost to/from the newly defined source and sink nodes. By solving the network program using KHNET =-=[71]-=- on a Silicon Graphics (Iris Indigo Workstation with 33MHz R3000 MIPS processor) the optimal mealbreak chain is ascertained. However there remains the problem which occurred with the assignment proble... |

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Citation Context ...CS II ILP MODEL 89 Branching strategy Various different branching strategies were considered during the development of TRACS II. Initially a constraint branching strategy developed by Ryan and Foster =-=[34, 81]-=- was implemented but was superseded by a relief time branching strategy. The method considers the optimal LP solution containing fractions of duty variables covering pieces of work. For each relief ti... |

44 |
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Citation Context ...R 2. REVIEW OF BUS DRIVER SCHEDULING METHODS 25 the cutting-stock problem. Although column generation techniques have successfully been applied to many problems, e.g. the binary cutting stock problem =-=[55]-=-, graph colouring problems [56], and traffic assignment problems [57], sections 2.6.1, 2.6.2 and 2.6.3 refer only to column generation methods which have been implemented to solve transportation probl... |

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Citation Context ...prove the solution. Column generation methods attempt to create new columns as required so as to implicitly consider all valid columns without the need to generate them beforehand. Ford and Fulkerson =-=[52]-=- describe the method where the problem is a shortest-path problem and Dantzig and Wolfe [53] describe the decomposition of a linear programming problem. Early applications of a column generation techn... |

37 |
A column generation approach to the urban transit crew scheduling problem
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Citation Context ...ATION MODEL 97 where c k = cost of duty ksi = simplex multiplier for row i a ik = coefficient of duty k in constraint i. 6.4.2 The HASTUS Crew-Opt Method Method of Solution Papers have been published =-=[59, 60, 83, 84]-=- which describe a column generation approach to driver scheduling, the modelling of the subproblem, and the branching rule used to find the integer solution. Crew-opt first generates an initial subset... |

36 |
Very large-scale linear programming: A case study in combining interior point and simplex method
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Citation Context ... apart from the first entrant variable which is chosen to be the duty with the most negative pure reduced cost, the duties are then priced according to the steepest-edge strategy. Indeed Bixby et al. =-=[86]-=- observed that steepest-edge pricing produced much better results than using pure reduced costs, but also proposed an alternative pricing strategy lambda price which scales the reduced costs by dividi... |

33 | Airline crew scheduling: A new formulation and decomposition algorithm
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Citation Context ...ay of forming near-optimal integer solutions from fractional solutions is to apply an integer linear programming routine to the set covering problem consisting of the generated pairings. Vance et al. =-=[64]-=- present a column generation approach for solving the domestic daily flight problem which operates over a hub and spoke structure. The problem is decomposed into two stages of selecting duty periods a... |

27 |
A new approach for crew pairing problems by column generation with an application to air transportation
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Citation Context ... the problem lies in the large number of possible crew pairings from these periods. The problem of assigning crews to flight services at minimum cost has been investigated by Minoux and Lavoie et al. =-=[62, 63]-=-. The crew pairings are formulated as valid paths on a graph which is deduced from the flight service information. At each optimal subproblem the crew pairing with the minimum reduced cost amongst the... |

25 |
Exact Solution of Crew Scheduling Problems Using the Set Partitioning Model
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Citation Context ...atical programming based methods which tend to be solved more easily than similarly solved bus driver scheduling problems. CHAPTER 2. REVIEW OF BUS DRIVER SCHEDULING METHODS 24 Marsten and Shepardson =-=[49]-=- use a set partitioning method to solve the relaxed LP model over a set of generated duties and then perform a branch and bound search to form an integer schedule. Results report that in many cases th... |

25 |
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Citation Context ... non-basic variables, whereas Harris [79] suggested that it would be more beneficial to establish the steepest edge over all variables. Smith [15] incorporated a method developed by Goldfarb and Reid =-=[80]-=- into ZIP which calculates the edge gradients using recurrence relations. Due to the number of columns in a typical driver scheduling model, calculating the gradient of every non-basic variable for ea... |

20 | A Bus Crew Scheduling System Using a Set Covering Formulation - Smith, Wren - 1988 |

18 |
A genetic algorithm for public transport driver scheduling. Computers and operations research
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Citation Context ... result in different duty types appearing in the schedule estimate. CHAPTER 2. REVIEW OF BUS DRIVER SCHEDULING METHODS 21 2.3.2 Genetic Algorithms A feasibility study was carried out by Wren and Wren =-=[43]-=- in order to test whether or not genetic algorithms could be used to solve larger driver scheduling problems more robustly, more quickly and more cost-effectively than any of the methods detailed in t... |

17 |
A column generation approach to the multiple-depot vehicle scheduling problem
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Citation Context ...reement rules need to be considered. Since the model differs from that used to schedule drivers no algorithmic details regarding the column generation procedure will be given here. Riberio and Soumis =-=[67]-=- and Bianco et al. [68] use column generation techniques to solve the multiple depot vehicle scheduling problem. Bianco et al. [69] extend the multiple depot vehicle scheduling problem to freight tran... |

15 | Bus driver scheduling - an overview. In
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Citation Context ... the 1970s and a series of workshops [5, 6, 7, 8, 9, 10] have been organised since then to discuss the latest techniques and software evolving with rapid developments in technology. Wren and Rousseau =-=[11]-=- give an outline of the driver scheduling problem and computer approaches to solving it, with emphasis on the more recent developments. The limitations of computer technology meant that earlier system... |

13 |
Improving personnel scheduling at airline stations
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Citation Context ... the objective improvement between iterations is sufficiently small and the deadheads then provide the integer programming package with a good set of crew pairings from which to choose. Brusco et al. =-=[66]-=- consider the problem of scheduling ground personnel at airports. The relaxed model is solved over a subset of possible legal duties and the dual variables are used to select a further subset of candi... |

12 |
Subproblem modeling in a column generation approach to the urban crew scheduling problem
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Citation Context ...are required to adequately solve such a large problem. Columns are generated which have negative reduced costs relative to the Lagrangean multipliers by using a k-shortest paths enumeration. Crew-Opt =-=[59, 60]-=- is a new driver scheduling module implemented within the HASTUS system, which uses a column generation technique to solve a set covering formulation of the problem. A detailed description of the syst... |

11 |
Zip: A zero-one integer programming package for scheduling, Research Report CSS 85
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Citation Context ...driver scheduling problem, firstly using heuristics to reduce the problem size and then a three-stage approach to develop the driver duties. The set partitioning model is solved using the ZIP package =-=[34]-=- which is also used within TRACS II. The approach was developed to apply to the New Zealand bus driver scheduling rules which require scheduled mealbreaks, all work to be covered, and the possibility ... |

11 |
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Citation Context ...n the solution of set covering and set partitioning problems. There are core routines which contain the basic structure of the standard algorithms, and these call a set of Harwell Library subroutines =-=[78]-=- which have been developed to handle sparse linear programming bases. There is also a set of user routines which control the route through the program in order to reflect the characteristics of the dr... |

9 |
An optimal column-generation-withranking algorithm for very large scale set partitioning problems in traffic assignment
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Citation Context ...blem. Although column generation techniques have successfully been applied to many problems, e.g. the binary cutting stock problem [55], graph colouring problems [56], and traffic assignment problems =-=[57]-=-, sections 2.6.1, 2.6.2 and 2.6.3 refer only to column generation methods which have been implemented to solve transportation problems. 2.6.1 Bus Driver Scheduling Carraresi et al. [58] describe mathe... |

9 |
Pivot Selection Method of the Devex
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Citation Context ...oosing the entering variable with the most negative reduced cost. In terms of the convex polytope however this only considers the edges in the space of the current non-basic variables, whereas Harris =-=[79]-=- suggested that it would be more beneficial to establish the steepest edge over all variables. Smith [15] incorporated a method developed by Goldfarb and Reid [80] into ZIP which calculates the edge g... |

8 |
Improved Algorithms for Bus Crew Scheduling
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Citation Context ...six workshops on computer-aided scheduling of public transport [5, 6, 7, 8, 9, 10], this chapter will concentrate on those published more recently and any others which relate to current work. Willers =-=[14]-=- presents a fuller review of the material in the proceedings. Complete scheduling packages which are in most common use are : IMPACS [15, 16, 17, 18], HASTUS [19, 20, 21, 22, 23, 24], HOT II [25, 26, ... |

8 |
Experiences with a crew scheduling system based on set covering
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Citation Context ... recently and any others which relate to current work. Willers [14] presents a fuller review of the material in the proceedings. Complete scheduling packages which are in most common use are : IMPACS =-=[15, 16, 17, 18]-=-, HASTUS [19, 20, 21, 22, 23, 24], HOT II [25, 26, 27], UMA(Trapeze) and Teleride Sage. Other systems are in use such as TransTec EDP in Germany, and a system called BERTA [28] is being developed in B... |

8 | Producing train driver shifts by computer - Kwan, Kwan, et al. - 1996 |

7 |
HASTUS: An interactive system for buses and crew scheduling
- Rousseau, Blais
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(Show Context)
Citation Context ...which relate to current work. Willers [14] presents a fuller review of the material in the proceedings. Complete scheduling packages which are in most common use are : IMPACS [15, 16, 17, 18], HASTUS =-=[19, 20, 21, 22, 23, 24]-=-, HOT II [25, 26, 27], UMA(Trapeze) and Teleride Sage. Other systems are in use such as TransTec EDP in Germany, and a system called BERTA [28] is being developed in Berlin to provide automatic bus an... |

7 |
Computer-aided vehicle and duty scheduling using HOT programme system
- Daduna, Mojsilovic
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Citation Context ...lers [14] presents a fuller review of the material in the proceedings. Complete scheduling packages which are in most common use are : IMPACS [15, 16, 17, 18], HASTUS [19, 20, 21, 22, 23, 24], HOT II =-=[25, 26, 27]-=-, UMA(Trapeze) and Teleride Sage. Other systems are in use such as TransTec EDP in Germany, and a system called BERTA [28] is being developed in Berlin to provide automatic bus and driver schedules fo... |

6 |
A matching based heuristic for scheduling mass transit crews and vehicles
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Citation Context ...schedule the vehicles and drivers simultaneously to attain global optimality. However, for the majority of cases it is impracticable and the vehicle and driver schedules are compiled separately. Ball =-=[1, 2]-=- has proposed methods for scheduling the processes simultaneously, but in practice bus operators generally wish to inspect and amend the bus schedule in order to satisfy local conditions before alloca... |

6 |
editors. Computer-aided transit scheduling
- Daduna, Wren
- 1995
(Show Context)
Citation Context ...es which can be incorporated into a computerised system. 1.3.5 Computer Scheduling The earliest successful computer systems for driver scheduling were developed in the 1970s and a series of workshops =-=[5, 6, 7, 8, 9, 10]-=- have been organised since then to discuss the latest techniques and software evolving with rapid developments in technology. Wren and Rousseau [11] give an outline of the driver scheduling problem an... |

6 |
The development of Micro-BUSMAN: scheduling on micro-computers
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Citation Context ...st formulation to create a vehicle schedule. 2.2.4 The COMPACS System COMPACS (COMPuter Assisted Crew Scheduling) is an interactive driver scheduling system which is described by Wren and Chamberlain =-=[32]-=-. The system first estimates the number of drivers needed to cover the bus work and then aids the construction of the driver schedule by verifying and costing duties entered by the scheduler and by su... |

6 |
Deadhead selection for the long-haul crew pairing problem
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Citation Context ...n provided by the LP relaxation of the decomposed system gives a better bound than that from the traditional set partitioning problem over the pairings but is more difficult to solve. Barnhart et al. =-=[65]-=- tackle the problem of the long-haul flights which typically do not operate on a daily basis. The problem becomes one of reducing the amount of extended rest periods for crews by assigning them to fli... |

5 |
Computer scheduling of public transport
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(Show Context)
Citation Context ...es which can be incorporated into a computerised system. 1.3.5 Computer Scheduling The earliest successful computer systems for driver scheduling were developed in the 1970s and a series of workshops =-=[5, 6, 7, 8, 9, 10]-=- have been organised since then to discuss the latest techniques and software evolving with rapid developments in technology. Wren and Rousseau [11] give an outline of the driver scheduling problem an... |

5 |
Bus crew scheduling using mathematical programming
- Smith
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(Show Context)
Citation Context ... this only considers the edges in the space of the current non-basic variables, whereas Harris [79] suggested that it would be more beneficial to establish the steepest edge over all variables. Smith =-=[15]-=- incorporated a method developed by Goldfarb and Reid [80] into ZIP which calculates the edge gradients using recurrence relations. Due to the number of columns in a typical driver scheduling model, c... |

5 |
Hybrid genetic algorithms for the bus driver scheduling
- Kwan, Wren
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Citation Context ...ations into the use of genetic algorithms to solve the driver scheduling problem involved combining its strengths with the strengths of the intelligent rule based Estimator described in section 2.3.1 =-=[45]-=-. This hybrid technique uses the genetic algorithm approach to determine which set of relief opportunities will produce the best schedules. The `fitness' (quality) of any subset of relief opportunitie... |

5 |
Equivalent Weights for Lexicographic Multi-Objective Programs
- Sherali
- 1982
(Show Context)
Citation Context ...thod is used to determine which duties will appear in the final schedule. The customised branch and bound process is described in detail in section 5.4.3. 5.3 An Alternative Solution Strategy Sherali =-=[76]-=- noted that there are several disadvantages in using a sequential approach to find a solution which satisfies two objectives. Apart from the fact that two separate linear programs have to be solved wh... |

4 |
A glossary of terms in bus and crew scheduling
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(Show Context)
Citation Context ...different. It is, however, useful to have the ability to analyse quickly the impact that changes in a vehicle schedule may have on the number of drivers in a driver schedule. Please note that Hartley =-=[3]-=- provides a complete description of all terms used in bus and driver scheduling, including variations used in different countries. 1.2 Bus Scheduling The bus scheduling process takes a set of predeter... |

4 |
Computer Scheduling of Public Transport 2
- Rousseau
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(Show Context)
Citation Context ...es which can be incorporated into a computerised system. 1.3.5 Computer Scheduling The earliest successful computer systems for driver scheduling were developed in the 1970s and a series of workshops =-=[5, 6, 7, 8, 9, 10]-=- have been organised since then to discuss the latest techniques and software evolving with rapid developments in technology. Wren and Rousseau [11] give an outline of the driver scheduling problem an... |

4 |
editors. Computer-aided transit scheduling
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(Show Context)
Citation Context |

4 |
The use of digital computers in the economic scheduling for both man and machine in public transportation
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Citation Context ... it, with emphasis on the more recent developments. The limitations of computer technology meant that earlier systems found heuristic solutions rather than using known mathematical programming models =-=[12, 13]-=-. Most of these systems used heuristics to refine some initial schedule and are not easily portable to other bus companies. More recently mathematical programming techniques have been applied to solve... |

4 |
Enhancements to the HASTUS crew scheduling algorithm
- Rousseau, Lessard, et al.
- 1985
(Show Context)
Citation Context ...which relate to current work. Willers [14] presents a fuller review of the material in the proceedings. Complete scheduling packages which are in most common use are : IMPACS [15, 16, 17, 18], HASTUS =-=[19, 20, 21, 22, 23, 24]-=-, HOT II [25, 26, 27], UMA(Trapeze) and Teleride Sage. Other systems are in use such as TransTec EDP in Germany, and a system called BERTA [28] is being developed in Berlin to provide automatic bus an... |

4 |
The HASTUS System: New Algorithms and Modules for the 90s
- Hamer, S'eguin
- 1992
(Show Context)
Citation Context ...which relate to current work. Willers [14] presents a fuller review of the material in the proceedings. Complete scheduling packages which are in most common use are : IMPACS [15, 16, 17, 18], HASTUS =-=[19, 20, 21, 22, 23, 24]-=-, HOT II [25, 26, 27], UMA(Trapeze) and Teleride Sage. Other systems are in use such as TransTec EDP in Germany, and a system called BERTA [28] is being developed in Berlin to provide automatic bus an... |

4 |
Scheduling of rail driver duties
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Citation Context ...methods described in this thesis are based on a version of TRACS II, which is described in full in Chapter 4. 2.5 Overview of Other Driver Scheduling Systems 2.5.1 Train Driver Scheduling Wren et al. =-=[46]-=- observed that although many bus driver scheduling systems have been developed and reported over the years, there appeared to be no practical driver scheduling system for the rail industry. Some of th... |

4 |
A Set Partitioning Approach to the Multiple Depot Vehicle Scheduling Problem
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(Show Context)
Citation Context ...e considered. Since the model differs from that used to schedule drivers no algorithmic details regarding the column generation procedure will be given here. Riberio and Soumis [67] and Bianco et al. =-=[68]-=- use column generation techniques to solve the multiple depot vehicle scheduling problem. Bianco et al. [69] extend the multiple depot vehicle scheduling problem to freight transport. Reddington [70] ... |

4 |
A Dual Strategy for Solving the Linear Programming Relaxation of a Driver Scheduling System
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(Show Context)
Citation Context ...ved which may increase execution times over a simple model for which efficient solution codes exist. An alternative approach to solving the driver scheduling model has been explored by Willers et al. =-=[77]-=- and Willers [14]. The approach used is that of merging the two objectives using CHAPTER 5. SOLVING THE TRACS II ILP MODEL 80 equivalent weights to create a single objective, simultaneously reflecting... |

3 |
EXPRESS: Set Partitioning for Bus Crew Scheduling in Christchurch
- Falkner, Ryan
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(Show Context)
Citation Context ...his interactive system is best used to amend manually or add to a schedule already produced as COMPACS itself does not contain any optimisation techniques. 2.2.5 The EXPRESS system The EXPRESS system =-=[33]-=- is based on a set partitioning model of the driver scheduling problem, firstly using heuristics to reduce the problem size and then a three-stage approach to develop the driver duties. The set partit... |

3 |
RUCUS II: a review of system capabilities
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Citation Context ... the best duties from many possible valid duties. A matching algorithm is used to determine the optimal combination of two-part duties. 2.2.9 The RUCUS II-System RUCUS-II (RUn CUtting and Scheduling) =-=[36, 37]-=- is a newer version of the original RUCUS system [38] which was developed in the 1960s and is now believed to be obsolete. The original RUCUS system used the technique of making improvements to an ini... |

3 |
Driver Scheduling Using Intelligent Estimation Techniques with Heuristic Searches
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- 1992
(Show Context)
Citation Context ...r the bus work. This idea provoked further investigations into how the estimator may be made more sophisticated, providing a tighter lower bound, by incorporating knowledge from individual schedulers =-=[39, 40]-=-. There have been few previous attempts to analyse the bus scheduling problem in order to take advantage of the structure of the problem and match duties to its critical features. Work began in 1992 o... |

3 |
Greedy genetic algorithms, optimizing mutations and bus driver scheduling
- Clement, Wren
- 1995
(Show Context)
Citation Context ...t the option of mutation and with limited constraints on small test problems produced very good solutions quickly, encouraging further investigations. Further work was carried out by Clement and Wren =-=[44]-=- which depicted chromosomes as an unordered set of duties, each with a binary value dependent on whether or not the duty is present in the schedule. Different methods of crossover and mutation techniq... |