Multiobjective capacitated arc routing problem

Transcription

1 Multiobjective capacitated arc routing problem Philippe Lacomme 1, Christian Prins 2, Marc Sevaux 3 1 University Blaise-Pascal, Clermont-Ferrand, France 2 University of Technology of Troyes, France 3 University of Valenciennes, France Lacomme, Prins, Sevaux EMO - April 8-11,

2 Outline Outline The Capacitated Arc Routing Problem A simple example A bi-objective NSGA-II implementation Computational experiments Conclusion Lacomme, Prins, Sevaux EMO - April 8-11,

3 The Capacitated Arc Routing Problem Data: The Capacitated Arc Routing Problem undirected network G n nodes including a depot with vehicles of capacity W m edges including a set of t required edges or tasks each edge has a demand and a traversal cost Goal: process all tasks with a min-cost set of trips Applications: urban waste collection, winter gritting etc. NP-hard: Solved in practice with constructive heuristics, and metaheuristics Tabu Search (Eglese,1994,1996; Hertz et al. 2000), GLS (Beullens et al. 2001) or HGA (Lacomme et al. 2001). Lacomme, Prins, Sevaux EMO - April 8-11,

4 The Capacitated Arc Routing Problem A small example: gdb nodes, 22 edges, capacity = 5, unit demands Lacomme, Prins, Sevaux EMO - April 8-11,

5 The Capacitated Arc Routing Problem Optimal solution: gdb1 2 3 T1: T2: Lacomme, Prins, Sevaux EMO - April 8-11,

6 The Capacitated Arc Routing Problem Optimal solution: gdb1 (continued) 2 3 T3: T4: Lacomme, Prins, Sevaux EMO - April 8-11,

7 The Capacitated Arc Routing Problem Optimal solution: gdb1 (end) 2 3 T5: Optimal solution: 5 trips; total cost= 316 Lacomme, Prins, Sevaux EMO - April 8-11,

8 The Capacitated Arc Routing Problem The Biobjective Capacitated Arc Routing Problem Data: undirected network G n nodes including a depot with vehicles of capacity W m edges including a set of t required edges or tasks each edge has a demand and a traversal cost Goals: Minimize f 1 the total cost of the trips Minimize f 2 the cost of the longest trip Application: urban waste collection Once all streets are collected, the crews are employed for selective sorting at the depot.. Lacomme, Prins, Sevaux EMO - April 8-11,

9 A bi-objective NSGA-II implementation A bi-objective NSGA-II implementation The NSGA-II template has been improved by adding A few good solutions in the initial population A special encoding of CARP solutions (chromosomes) An optimal chromosome evaluation procedure Hybridization with bi-objective local search procedures Lacomme, Prins, Sevaux EMO - April 8-11,

10 A bi-objective NSGA-II implementation Initial population The initial population is randomly generated, then three good individuals are added using the following classical heuristics for the single objective CARP: Path-Scanning Augment-Merge Ulusoy s heuristic Lacomme, Prins, Sevaux EMO - April 8-11,

11 A bi-objective NSGA-II implementation Solution encoding and crossover G coded as a symmetric digraph with 2m arcs, 2 per edge A chromosome is an ordered list S of t tasks Implicit shortest paths between consecutive tasks No trip delimiter: giant tour or priority order for 1 vehicle Classical crossover OX for sequencing problems P1 P Offspring Lacomme, Prins, Sevaux EMO - April 8-11,

12 A bi-objective NSGA-II implementation Optimal chromosome evaluation Consider a chromosome (a, b, c, d, e) viewed as a giant tour: c(5) b(3) 5 5 d(1) a(4) e(6) 20 depot 16 Shortest path in a graph modelling all feasible trips (W =9): bcd(80) bc(56) de(50) a(37) b(27) c(40) d(32) e(33) ab(51) cd(64) Lacomme, Prins, Sevaux EMO - April 8-11,

13 A bi-objective NSGA-II implementation Bi-objective local search procedure (LS) LS is applied to offspring with a fixed probability. Scanned moves: inverse the traversal direction of a edge move one task after one other move two adjacent tasks after one other swap two tasks perform two-opt moves. LS performs the 1 st improving move detected, until no such moves are found. 3 versions: LS1: accept moves improving f 1 (total cost of trips) LS2: accept moves improving f 2 (max trip cost) LS3: accept if new solution dominates current one. Lacomme, Prins, Sevaux EMO - April 8-11,

14 Computational evaluations Computational evaluations Tested on the 23 Golden, De Armon and Baker s instances. PopSize =60. Test Max Initial Local Search Protocol # it. heuristics LS %LS MO1 100 Yes No MO2 100 Yes LS1 10 MO3 100 Yes LS2 10 MO4 100 Yes LS3 10 MO5 300 No No MO6 100 Yes LS3 20 Lacomme, Prins, Sevaux EMO - April 8-11,

15 Computational evaluations Measuring the deviation from a reference front f2 d1 reference front front to compare d2 d3 d4 extrapolated front f1 Lacomme, Prins, Sevaux EMO - April 8-11,

16 Computational evaluations Results of the tests protocols (Celeron, 650 MHz, Delphi 6) Test Efficient Measure CPU Protocol solutions Std. Per sol. time (s) MO MO MO MO MO MO Lacomme, Prins, Sevaux EMO - April 8-11,

17 Computational evaluations Comparing to the lower bounds Leftmost solution Rightmost solution Test f 1 f 2 f 1 f 2 Protocol dev. # opt. dev. dev. dev. # opt. MO MO MO MO MO MO Lacomme, Prins, Sevaux EMO - April 8-11,

18 Computational evaluations Lacomme, Prins, Sevaux EMO - April 8-11,

19 Conclusion and future work Conclusion and future work Study of a bi-objective CARP with real applications (waste collection) Efficient MOGA hybridized with bi-objective local search Able to retrieve most optimal solutions of the single objective case (f 1 or f 2 ) Need some improvement Use ε-dominance Use gender in the Local Search procedure Compute for the other classical instances Lacomme, Prins, Sevaux EMO - April 8-11,