An Evolutionary Algorithm with Heuristic Longest Cycle Crossover for Solving the Capacitated Vehicle Routing Problem Depar artment ment of C Computer er Science ence and Information mation Engineer neering, ng, Natio ional nal T aiwan an Normal mal Univer versit ity, , T aiwan an Personal contact Contributed by thammarsat@gmail.com * , tcchiang@ieee.org Thammarsat Visutarrom * , and Tsung-Che Chiang 2019 IEEE CONGRESS ON EVOLUTIONARY COMPUTATION, WELLINGTON, NEW ZEALAND, 10-13 JUNE 2019
Contents 1 I ntroduction CVRP: The problem introduction Brief literature review 2 T he Research Motivation The crossover operator’s performance Strategies for improvement (overvie view) 3 E volutionary Algorithm The EA’s mechanism 4 E xperiments and Results Parameter setting Crossover-only EA Complete EA 5 C onclusion Conclusion The Research Motivation Evolutionary Algorithm Contents Experiments and Results Introduction
Introduction (CVRP: Capacitated Vehicle Routing Problem) CVRP: The Problem Introduction 10 Kg. 7 X 3 X 3 1 15 Kg. 7 Kg. 8 9 15 Kg. 4 Minimize mize travel vel distan tance ce 45 Kg. 10 CVRP’s constraints Depot 3 Customer 45 Kg. Customer visited only 24 Kg. • 6 11 one time 12 Kg. 29 Kg. Vehicle 2 No over load • 30 Kg. 5 No extra vehicle • 22 Kg. Conclusion Contents The Research Motivation Evolutionary Algorithm Introduction Experiments and Results
Introduction (CVRP: Capacitated Vehicle Routing Problem) CVRP: The Problem Introduction 10 Kg. 7 1 15 Kg. 7 Kg. 8 9 15 Kg. 4 45 Kg. 10 Depot 3 45 Kg. 24 Kg. 6 11 12 Kg. 29 Kg. 0 1 9 10 6 6 0 2 5 11 0 3 4 8 8 7 7 0 2 30 Kg. Start point End point 5 22 Kg. Conclusion Contents The Research Motivation Evolutionary Algorithm Introduction Experiments and Results
Introduction Brief Literature Review T abu Search Algorithm Simulated Annealing Algorithm CVRP Evolu olutiona tionary y Algor orith ithm m (EA) A) Ant Colony Optimization Artificial Bee Colony Algorithm P roblem A lgorithms Conclusion Contents The Research Motivation Evolutionary Algorithm Introduction Experiments and Results
Introduction Brief Literature Review One - point crossover (1PX) • Two - point crossover (2PX) • Crossover operator Partially mapped crossover (PMX) • T abu Search Algorithm Heuristic crossover (HX) • Cycle crossover (CX) • Simulated Annealing Algorithm CVRP Evolu olutiona tionary y Algor orith ithm m (EA) A) Ant Colony Optimization Swap Mutation • Artificial Bee Colony Algorithm Insertion Mutation • Mutation operator Inversion Mutation • Scramble Mutation • P roblem A lgorithms Conclusion Contents The Research Motivation Evolutionary Algorithm Introduction Experiments and Results
Introduction Brief Literature Review One - point crossover (1PX) • Two - point crossover (2PX) • Crossover operator Partially mapped crossover (PMX) • T abu Search Algorithm Heuristic crossover (HX) • * Cycle cle crossov ssover er (CX) X) Simulated Annealing Algorithm CVRP Evolu olutiona tionary y Algor orith ithm m (EA) A) Ant Colony Optimization Swap Mutation • Artificial Bee Colony Algorithm Insertion Mutation • Mutation operator Inversion Mutation • Scramble Mutation • P roblem A lgorithms O perators Conclusion Contents The Research Motivation Evolutionary Algorithm Introduction Experiments and Results
The Research Motivation The Crossover O perator’s Performance Average age of the best fitnes ess s value ue in e each ch iterati ation n over 30 runs Initialization 1700 * Only ly Crossover ver 1600 Operat rator or Reproduction 1500 No Fitness value 1400 selection Generation < Max 1300 Yes 1200 Final Population 1100 1000 1 10 20 30 40 50 60 70 80 90 100 Generation (1- point crossover) (2- point crossover) (Cycle crossover) (Heuristic crossover) (Partially mapped crossover ) Conclusion Contents The Research Motivation Evolutionary Algorithm Introduction Experiments and Results
The Research Motivation Strategies for Improvement Longest cycle selection Create large difference between parents and offspring. CX CX HLCX Heuristi stic c Longest est Cycle le Cycle le Crossover over Nearest neighbor heuristic Crossover over Keep short travel distance during the big change. Conclusion Contents The Research Motivation Evolutionary Algorithm Introduction Experiments and Results
Evolutionary Algorithm (EA) The E A’s Mechanism Initialization Encoding → Decoding Reproduction ournament Selection → Crosso sover ver → Local Refinement Duplicate Removal → T No selection Selection→ Mutation → 2 -opt Generation < Max Yes Final Population Conclusion Contents Introduction The Research Motivation Evolutionary Algorithm Experiments and Results
Evolutionary Algorithm (EA) (Encoding) The E A’s Mechanism Random the 1 st group to put inside customer sequence, then • move to the next closet group and NP/2 will created following Initiali ializat zation ion clockwise and the other counterclockwise. Encoding 2 Decoding 1 The order inside each can be arrange randomly. • Reproduct ction ion 5 3 4 clockw kwise ise Duplicate Removal G8 G8 G1 G1 7 6 6 3 8 10 11 9 7 4 1 2 5 G7 G7 G2 G2 T ournament Selection G3 G3 G6 G6 Crossover 8 G2 G3 G4 G5 G6 G7 G8 G1 G5 G5 G4 G4 9 Local Refinement 11 11 counterclock lockwise wise Selec ection ion 10 10 4 7 9 11 10 8 3 6 2 5 1 selection G: Grou oup Numb mber er Mutation G7 G6 G5 G4 G3 G2 G1 G8 2-opt Conclusion Contents Introduction The Research Motivation Evolutionary Algorithm Experiments and Results
Evolutionary Algorithm (EA) (Decoding: Vehicle assignment) The E A’s Mechanism Initiali ializat zation ion 1 4 7 9 11 10 8 3 6 5 2 X 3 (Maximum capacity : 100 Kg) Encoding Decoding 15 45 8 18 12 75 25 24 48 12 6 Reproduct ction ion Duplicate Starts with the 1 st vehicle, check through customer sequence and select the customer which do not Removal make the total demand violate the maximum capacity till the vehicle cannot serve any customer else. T ournament Selection 1 4 7 9 11 - - - - - - {1,4,7,9,11} ∈ Crossover 15+45+8+18+12 = 98 Local Refinement - - - - - 10 8 - - - - {10,8} ∈ Selec ection ion 75 + 25= 98 selection Mutation - - - - - - - 3 6 5 2 {3,6,5,2} ∈ 2-opt 24+48+12+6 = 90 Conclusion Contents Introduction The Research Motivation Evolutionary Algorithm Experiments and Results
Evolutionary Algorithm (EA) (Decoding: Greedy insertion heuristic) The E A’s Mechanism 0 0 Initiali ializat zation ion {3,6,5,2} Encoding Decoding Reproduct ction ion 0 3 0 {-,6,5,2} Duplicate Removal 48Km 70Km 6 6 T ournament Selection 0 3 0 0 3 6 0 {-,-,5,2} Crossover 97Km 82Km 75Km Local 5 5 5 Refinement 0 3 6 0 0 3 6 5 0 {-,-,-,2} Selec ection ion selection 93Km 97Km 80Km 110Km 2 2 2 2 Mutation 0 3 6 2 5 0 0 3 6 5 0 2-opt {-,-,-,2} Conclusion Contents Introduction The Research Motivation Evolutionary Algorithm Experiments and Results
Evolutionary Algorithm (EA) (Decoding) The E A’s Mechanism Initiali ializat zation ion 0 1 4 7 9 11 0 Encoding Decoding 0 10 8 0 Reproduct ction ion Duplicate 0 3 6 5 2 0 Removal T ournament Selection 0 1 4 7 9 11 0 10 8 0 3 6 5 2 0 Crossover Local Refinement Selec ection ion T otal Distance Calculation selection Fitness ness Value ue Mutation 2-opt Distance Matrix Conclusion Contents Introduction The Research Motivation Evolutionary Algorithm Experiments and Results
Evolutionary Algorithm (EA) The E A’s Mechanism • Duplicate Removal Initiali ializat zation ion Solution 1 205 Remove solution 2 Encoding Solution 2 205 New Random Solution Decoding . . Solution 3 218 . . . . Reproduc oduction tion Duplicate Removal Solution 100 530 T ournament • 4-T ournament Selection Selection Crossover Solution A Solution B Local Refinement B < A Solution B B < C Selec ection ion Solution B selection Winner C < D Mutation Solution C 2-opt Solution C Solution D Conclusion Contents Introduction The Research Motivation Evolutionary Algorithm Experiments and Results
Evolutionary Algorithm (EA) (Original Cycle Crossover) The E A’s Mechanism 1. Start with the first unassigned customer in Parent 1 and drop down to the same position in Parent 2. 1 st 1 6 4 2 3 7 10 9 8 5 Parent 1 2 4 8 10 7 3 9 5 6 1 Parent 2 2. Then, look for customer 4 in Parent 1 and drop down to the same position in Parent 2. 1 st 2 nd 1 6 4 2 3 7 10 9 8 5 Parent 1 2 4 8 10 7 3 9 5 6 1 Parent 2 Conclusion Contents Introduction The Research Motivation Evolutionary Algorithm Experiments and Results
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