A Column Generation based Heuristic for Train Driver Rescheduling - PowerPoint PPT Presentation

Outline Introduction Model Algorithm Computational results Future research A Column Generation based Heuristic for Train Driver Rescheduling Daniel Potthoff potthoff@few.eur.nl joint work with D. Huisman and G. Desaulniers June 18, 2008 D. Potthoff A Column Generation based Heuristic forTrain Driver Rescheduling

Outline Introduction Model Algorithm Computational results Future research 1 Introduction 2 Model 3 Algorithm 4 Computational results 5 Future research D. Potthoff A Column Generation based Heuristic forTrain Driver Rescheduling

Outline Introduction Model Algorithm Computational results Future research Introduction Reasons for unexpected disruptions Infrastructure malfunctions Rails, switches, catenary, bridges Numbers from 2007 Computer problems in control centers Disruptions # Rolling stock breakdowns Small 933 Accidents with other traffic Medium 1011 Large 834 Weather conditions Crew no shows ... D. Potthoff A Column Generation based Heuristic forTrain Driver Rescheduling

Outline Introduction Model Algorithm Computational results Future research Introduction Reasons for unexpected disruptions Infrastructure malfunctions Rails, switches, catenary, bridges Numbers from 2007 Computer problems in control centers Disruptions # Rolling stock breakdowns Small 933 Accidents with other traffic Medium 1011 Large 834 Weather conditions Crew no shows ... D. Potthoff A Column Generation based Heuristic forTrain Driver Rescheduling

Outline Introduction Model Algorithm Computational results Future research Problem description Given a blocked route and an estimated duration The timetable has been modified according to emergency scenarios The rolling stock has been rescheduled Crew rescheduling Cover as much tasks as possible such that: each original duty gets a feasible extension the modifications to the crew schedule and the usage of taxis is as minimal as possible D. Potthoff A Column Generation based Heuristic forTrain Driver Rescheduling

Outline Introduction Model Algorithm Computational results Future research Feasible extensions - example Gn 107 D. Potthoff A Column Generation based Heuristic forTrain Driver Rescheduling

Outline Introduction Model Algorithm Computational results Future research Feasible extensions - example Gn 107 D. Potthoff A Column Generation based Heuristic forTrain Driver Rescheduling

Outline Introduction Model Algorithm Computational results Future research Feasible extensions - example Gn 107 D. Potthoff A Column Generation based Heuristic forTrain Driver Rescheduling

Outline Introduction Model Algorithm Computational results Future research Notation N : Set of tasks, where for every i ∈ N f i : Cost for canceling task i ∆: Set of original duties K δ : Set of all feasible extensions for original duty δ ∈ ∆ c δ k : Cost of extension k for original duty δ � 1 , if extension k is selected for original duty δ x δ k = 0 , otherwise � 1 , if task i is canceled y i = 0 , otherwise D. Potthoff A Column Generation based Heuristic forTrain Driver Rescheduling

Outline Introduction Model Algorithm Computational results Future research Mathematical model � � � c δ k x δ min + f i y i (1) k δ ∈ ∆ k ∈ K δ i ∈ N � � a δ ik x δ s.t. k + y i ≥ 1 ∀ i ∈ N (2) δ ∈ ∆ k ∈ K δ � x δ = 1 ∀ δ ∈ ∆ (3) k k ∈ K δ x δ { 0 , 1 } ∀ δ ∈ ∆ , ∀ k ∈ K δ ∈ (4) k y i ∈ { 0 , 1 } ∀ i ∈ N (5) D. Potthoff A Column Generation based Heuristic forTrain Driver Rescheduling

Outline Introduction Model Algorithm Computational results Future research Mathematical model Observations Only a few original duties need to be rescheduled in order to obtain a good solution The number of feasible extension for each original duty might be huge Solution approach Consider a core problem containing only a subset of the original duties and tasks Use a column generation based heuristic to solve the core problem D. Potthoff A Column Generation based Heuristic forTrain Driver Rescheduling

Outline Introduction Model Algorithm Computational results Future research Overview over the algorithm D. Potthoff A Column Generation based Heuristic forTrain Driver Rescheduling

Outline Introduction Model Algorithm Computational results Future research Overview over the algorithm D. Potthoff A Column Generation based Heuristic forTrain Driver Rescheduling

Outline Introduction Model Algorithm Computational results Future research Selection of the initial core problem Define N p as the subset of tasks, where at least on of the following conditions holds: 1 The task is canceled or modified (rerouted) 2 The task is performed on the obstructed route and the departure time of task i lies in the interval [ t 0 , t 1 + p ] 3 The task is part of the same train as one of the tasks selected in 1 and 2 The subset of original duties is now defined as ∆ := { δ ∈ ∆ : δ covers at least one task in N p } . The core problem is given by ∆ and N where N the set of tasks covered by a original duty in ∆. D. Potthoff A Column Generation based Heuristic forTrain Driver Rescheduling

Outline Introduction Model Algorithm Computational results Future research Selection of the initial core problem Define N p as the subset of tasks, where at least on of the following conditions holds: 1 The task is canceled or modified (rerouted) 2 The task is performed on the obstructed route and the departure time of task i lies in the interval [ t 0 , t 1 + p ] 3 The task is part of the same train as one of the tasks selected in 1 and 2 The subset of original duties is now defined as ∆ := { δ ∈ ∆ : δ covers at least one task in N p } . The core problem is given by ∆ and N where N the set of tasks covered by a original duty in ∆. D. Potthoff A Column Generation based Heuristic forTrain Driver Rescheduling

Outline Introduction Model Algorithm Computational results Future research Selection of the initial core problem Define N p as the subset of tasks, where at least on of the following conditions holds: 1 The task is canceled or modified (rerouted) 2 The task is performed on the obstructed route and the departure time of task i lies in the interval [ t 0 , t 1 + p ] 3 The task is part of the same train as one of the tasks selected in 1 and 2 The subset of original duties is now defined as ∆ := { δ ∈ ∆ : δ covers at least one task in N p } . The core problem is given by ∆ and N where N the set of tasks covered by a original duty in ∆. D. Potthoff A Column Generation based Heuristic forTrain Driver Rescheduling

Outline Introduction Model Algorithm Computational results Future research A column generation heuristic to solve a core problem D. Potthoff A Column Generation based Heuristic forTrain Driver Rescheduling

Outline Introduction Model Algorithm Computational results Future research A column generation heuristic to solve a core problem The RMP of the core problem in the n th column generation iteration reads. � � c δ k x δ � min + (1) f i y i k k ∈ K δ δ ∈ ∆ i ∈ N n � � a δ ik x δ s.t. k + y i ≥ 1 ∀ i ∈ N (2) k ∈ K δ δ ∈ ∆ n � x δ = 1 ∀ δ ∈ ∆ (3) k k ∈ K δ n x δ { 0 , 1 } ∀ δ ∈ ∆ , ∀ k ∈ K δ ∈ (4) k n y i ∈ { 0 , 1 } ∀ i ∈ N (5) D. Potthoff A Column Generation based Heuristic forTrain Driver Rescheduling

Outline Introduction Model Algorithm Computational results Future research Lagrangian relaxation If we relax the set covering constraints, the Lagrangian subproblem becomes � � c δ k x δ � � � � a δ ik x δ φ ( λ ) = min k + f i y i + λ i (1 − k − y i ) k ∈ K δ k ∈ K δ δ ∈ ∆ i ∈ N i ∈ N δ ∈ ∆ � � � ( c δ � λ i a δ ik ) x δ � φ ( λ ) = min λ i + k − k + ( f i − λ i ) y i k ∈ K δ i ∈ N δ ∈ ∆ i ∈ N i ∈ N n � x δ s.t. = 1 ∀ δ ∈ ∆ k k ∈ K δ n x δ { 0 , 1 } ∀ δ ∈ ∆ , ∀ k ∈ K δ ∈ k n ∈ { 0 , 1 } ∀ i ∈ N y i D. Potthoff A Column Generation based Heuristic forTrain Driver Rescheduling

Outline Introduction Model Algorithm Computational results Future research Lagrangian relaxation If we relax the set covering constraints, the Lagrangian subproblem becomes � � c δ k x δ � � � � a δ ik x δ φ ( λ ) = min k + f i y i + λ i (1 − k − y i ) k ∈ K δ k ∈ K δ δ ∈ ∆ i ∈ N i ∈ N δ ∈ ∆ � � � ( c δ � λ i a δ ik ) x δ � φ ( λ ) = min λ i + k − k + ( f i − λ i ) y i k ∈ K δ i ∈ N δ ∈ ∆ i ∈ N i ∈ N n � x δ s.t. = 1 ∀ δ ∈ ∆ k k ∈ K δ n x δ { 0 , 1 } ∀ δ ∈ ∆ , ∀ k ∈ K δ ∈ k n ∈ { 0 , 1 } ∀ i ∈ N y i D. Potthoff A Column Generation based Heuristic forTrain Driver Rescheduling

Outline Introduction Model Algorithm Computational results Future research Building blocks of the CG heuristic Master problem Apply Lagrangian relaxation and subgradient optimization Pricing problems Solve a resource constraint shortest path problem for each original duty on a dedicated acyclic graph Feasible solutions Use the Lagrangian multipliers in a greedy procedure to construct feasible solutions D. Potthoff A Column Generation based Heuristic forTrain Driver Rescheduling