A dual ascent heuristic for obtaining a lower bound of the generalized set partitioning problem with convexity constraints

Abstract In this paper we propose a dual ascent heuristic for solving the linear relaxation of the generalized set partitioning problem with convexity constraints, which often models the master problem of a column generation approach. The generalized set partitioning problem contains at the same time set covering, set packing and set partitioning constraints. The proposed dual ascent heuristic is based on a reformulation and it uses Lagrangian relaxation and subgradient method. It is inspired by the dual ascent procedure already proposed in literature, but it is able to deal with right hand side greater than one, together with under and over coverage. To prove its validity, it has been applied to the minimum sum coloring problem, the multi-activity tour scheduling problem, and some newly generated instances. The reported computational results show the effectiveness of the proposed method.

[1]  S. Ilker Birbil,et al.  Combination of Metaheuristic and Exact Algorithms for Solving Set Covering-Type Optimization Problems , 2010, INFORMS J. Comput..

[2]  David M. Ryan,et al.  The train driver recovery problem - A set partitioning based model and solution method , 2010, Comput. Oper. Res..

[3]  G. Mitra,et al.  An extension of set partitioning with application to scheduling problems , 1985 .

[4]  Egon Balas,et al.  A Dynamic Subgradient-Based Branch-and-Bound Procedure for Set Covering , 1992, Oper. Res..

[5]  Fabio Furini,et al.  ILP Models and Column Generation for the Minimum Sum Coloring Problem , 2018, Electron. Notes Discret. Math..

[6]  Vittorio Maniezzo,et al.  A set covering based matheuristic for a real-world city logistics problem , 2015, Int. Trans. Oper. Res..

[7]  Ruslan Sadykov,et al.  Automation and Combination of Linear-Programming Based Stabilization Techniques in Column Generation , 2018, INFORMS J. Comput..

[8]  M. Fisher,et al.  Optimal solution of set covering/partitioning problems using dual heuristics , 1990 .

[9]  William J. Cook,et al.  Maximum-weight stable sets and safe lower bounds for graph coloring , 2012, Mathematical Programming Computation.

[10]  Fabrizio Rossi,et al.  A set packing model for the ground holding problem in congested networks , 2001, Eur. J. Oper. Res..

[11]  Mikael Rönnqvist.  A method for the cutting stock problem with different qualities , 1995 .

[12]  Antonio Sassano,et al.  A Lagrangian-based heuristic for large-scale set covering problems , 1998, Math. Program..

[13]  Dennis Huisman,et al.  Combining Column Generation and Lagrangian Relaxation , 2005 .

[14]  Marco A. Boschetti,et al.  A dual ascent procedure for the set partitioning problem , 2008, Discret. Optim..

[15]  M. Balinski,et al.  On an Integer Program for a Delivery Problem , 1964 .

[16]  Jesper Larsen,et al.  Optimisation-Based Solution Methods for Set Partitioning Models , 2011 .

[17]  Andrew Lim,et al.  Pricing strategies for capacitated ring-star problems based on dynamic programming algorithms , 2017, Eur. J. Oper. Res..

[18]  Kenneth R. Baker,et al.  Workforce Allocation in Cyclical Scheduling Problems: A Survey , 1976 .

[19]  Hesham K. Alfares,et al.  Survey, Categorization, and Comparison of Recent Tour Scheduling Literature , 2004, Ann. Oper. Res..

[20]  Valentina Cacchiani,et al.  A set-covering based heuristic algorithm for the periodic vehicle routing problem , 2014, Discret. Appl. Math..

[21]  V. Maniezzo,et al.  An Exact Algorithm for the Resource-Constrained Project Scheduling Problem Based on a New Mathematical Formulation , 1998 .

[22]  Roberto Baldacci,et al.  An Exact Algorithm for the Period Routing Problem , 2011, Oper. Res..

[23]  R. R. Vemuganti.  Applications of Set Covering, Set Packing and Set Partitioning Models: A Survey , 1998 .

[24]  E. Balas,et al.  Set Partitioning: A survey , 1976 .

[25]  François Vanderbeck,et al.  Implementing Mixed Integer Column Generation , 2005 .

[26]  Francisco Barahona,et al.  On some difficult linear programs coming from set partitioning , 2002, Discret. Appl. Math..

[27]  Roberto Roberti,et al.  An exact solution framework for a broad class of vehicle routing problems , 2010, Comput. Manag. Sci..

[28]  Bjørn Nygreen,et al.  Column generation approaches to ship scheduling with flexible cargo sizes , 2010, Eur. J. Oper. Res..

[29]  J. Desrosiers,et al.  A Primer in Column Generation , 2005 .

[30]  Jorge J. Moré,et al.  Benchmarking optimization software with performance profiles , 2001, Math. Program..

[31]  Jin-Kao Hao,et al.  Algorithms for the minimum sum coloring problem: a review , 2015, Artificial Intelligence Review.

[32]  Matthias Ehrgott,et al.  Railway track allocation: models and methods , 2011, OR Spectr..

[33]  Andreas T. Ernst,et al.  Staff scheduling and rostering: A review of applications, methods and models , 2004, Eur. J. Oper. Res..

[34]  Roberto Baldacci,et al.  An Exact Algorithm for the Pickup and Delivery Problem with Time Windows , 2011, Oper. Res..

[35]  Roberto Baldacci,et al.  The Vehicle Routing Problem with Transhipment Facilities , 2017, Transp. Sci..

[36]  Nicos Christofides,et al.  An exact algorithm for the vehicle routing problem based on the set partitioning formulation with additional cuts , 2008, Math. Program..

[37]  Paolo Toth,et al.  An exact approach for the Vertex Coloring Problem , 2011, Discret. Optim..

[38]  Jacques Desrosiers,et al.  Crew Pairing at Air France , 1993 .

[39]  Ruslan Sadykov,et al.  Column generation based approaches for a tour scheduling problem with a multi-skill heterogeneous workforce , 2016, Eur. J. Oper. Res..