Use of Genetic Algorithms in Discrete Optimalization Problems

Transcription

1 Use of Genetic Algorithms in Discrete Optimalization Problems Alena Rybičková supervisor: Ing. Denisa Mocková PhD Faculty of Transportation Sciences

2 Main goals: design of genetic algorithm for vehicle routing problem design of genetic algorithm for location problem implementation of suggested algorithms aplication of the program to real data comparison of the results with other methods interpretation and assessment of the results and suitability of chosen algorithms

3 Discrete optimization problems optization discrete - set of feasible solutions consists of discrete points or areas continuous general optimization in space of real parameters complexity of the problems - classes P and NP problems in transportation travelling salesman number of possible solutions n!, vehicle routing problem, location problem

4 Location problems solve how to best locate given number of facilities criterion of minimalization (maximalization) of objective function most common criterion is distance costs - fixed, variable types of location problems continuous, network, discrete deterministic, stochastic one, multiple criteria problems

5 Vehicle routing problem determine route of vehicles, which are to ensure customers supply regarding given limiting conditions vehicle capacity, route lenght (km, time) routes start in depot, visit subset of customers and returt to depot variants: more than one depot time windows delivery and pick-up heterogenous fleet stochastic problem more comodities

6 Solution methods exact methods evaluation of all possible solutions branch and bound, branch and cut classic heuristics methods which do not ensure finding of optimal solution deterministic problem specific heuristics Clarke-Wright common methods for more problems greedy heuristics, local search

7 Solution methods metaheuristics general approach for wide range of problems stochastic element enables to avoid getting stuck in local optimum inspiration in nature processes simulated annealing ant colonies tabu search neural networks evolution algorithms

8 Genetic algorithms terminology borrowed from nature genetics population, individual, gen, genome, chromosome fitness function individual evaluation representation binary, natural numbers, real numbers, chains operators: selection selects parents crossover - creates offsprings mutation - small changes in indiviudals crossover and mutation parameters

9 Real problem assignment spare parts supply to garages of Peugeot and Citröen 2 depots (Praha, Brno), 97 garages optimization of distribution routes limiting conditions - length of one route in km, number of depots on one route (limited capacity of the vehicle) criterion - minimization of route lenght in km optimization of depots location + creating new distribution routes criterion - minimization of sum of distances to each garage

10 Application of GA to VRP representation, fitness function calculation crossover and mutation have to ensure that only feasible solution are created (without duplicities) crossover - section in one parent is chosen randomly and is inserted at random point in second parent + points of inserted section are deleted crossover 2 - insert at the nearest point to the beginning of inserted section elitism implementation in Matlab - CellArrays

11 Application of GA to VRP - results Praha Brno total Clarke-Wright [km] 2492,1 1577,6 4069,7 GA [km] 2462,2 1462,5 3924,7 savings GA compared to CW 1,20% 7,30% 3,56%

12 Applicaton of GA to location problem division of ČR after 2 km both horizontaly and vertiacaly of possible depot location for each location distance in 97 depots GoogleMaps natural numbers representation fitness function - simple sum of distances from each garage to the nearest depot comparison of results with exact solution in approximately one third GA found optimal solution x less of tested solutions

13 Aplication of GA to location problem - results depo 1 depo 2 total Clarke-Wright [km] 2492,1 1577,6 4069,7 GA1 original depots [km] 2462,2 1462,5 3924,7 saving to CW 1,20% 7,30% 3,56% GA2 new depots [km] 2506, ,3 saving to CW -0,57% 20,26% 7,50% saving to GA1-1,79% 13,98% 4,08%

14 Results Clarke-Wright Genetic algorithm