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A Genetic Algorithm for the Vehicle Routing Problem
Software Engineering (CS530)
Visvesvaraya Technological University
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University of Tennessee, KnoxvilleUniversity of Tennessee, Knoxville TRACE: Tennessee Research and CreativeTRACE: Tennessee Research and Creative ExchangeExchange
Masters Theses Graduate School
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A Genetic Algorithm for the Vehicle Routing ProblemA Genetic Algorithm for the Vehicle Routing Problem
Vickie Dawn Wester University of Tennessee, Knoxville
Follow this and additional works at: trace.tennessee/utk_gradthes Part of the Management Sciences and Quantitative Methods Commons
Recommended CitationRecommended Citation Wester, Vickie Dawn, "A Genetic Algorithm for the Vehicle Routing Problem. " Master's Thesis, University of Tennessee, 1993. trace.tennessee/utk_gradthes/
This Thesis is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research andCreative Exchange. It has been accepted for inclusion in Masters Theses by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact trace@utk.
To the Graduate Council: I am submitting herewith a thesis written by Vickie Dawn Wester entitled "A Genetic Algorithm for the Vehicle Routing Problem." I have examined the 昀椀nal electronic copy of this thesis for form and content and recommend that it be accepted in partial ful昀椀llment of the requirements for the degree of Master of Science, with a major in Management Science. Charles E. Noon, Major Professor We have read this thesis and recommend its acceptance: Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on 昀椀le with o昀케cial student records.)
STATEMENT OF PERMISSION TO USE
In presenting this thesis in partial fulfillment of the requirements for a Master's degree at The University of Tennessee, Knoxville, I agree that the Library shall make it available to borrowers under rules of the Library. Brief quotations from this thesis are allowable without special permission, provided that accurate acknowledgment of the source is made. Permission for extensive quotation from or reproduction of this thesis may be granted by my major professor, or in his absence, by the Head of Interlibrary Services when, in the opinion of either, the proposed use of the material is for scholarly purposes. Any copying or use of the material in this thesis for financial gain shall not be allowed without my written permission.
Signature Date
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####### The Genetic Algorithm
####### Problem The Genetic Algorithm Related to the Vehicle Routing
A Thesis Presented for the Master of Science Degree The University of Tennessee, Knoxville
Vickie Dawn Wester December 1993
TABLE OF CONTENTS
CHAPTER
####### The Vehicle Routing Problem.........................
- INTRODUCTION PAGE
- ROUTING PROBLEM........................... I. THE GENETIC ALGORITHM AND THE VEHICLE
- The Vehicle Routing Problem.........................
- The Genetic Algorithm
- Problem The Genetic Algorithm Related to the Vehicle Routing
- Data Format
- Encoding of Solutions
- Evaluating the Solutions
- Reproduction
- II. EXPERIMENTATION RES ULTS
- Penalization for Infeasibility
- Sampling Procedures
- Reproduction
- Parameters .....................
- Infeasibility
- Duplicate Solutions
- Reverse Paths
- Feasibility vs昀攀asibility
- Gene Selection Parameter
- Summary......................................
- III. ANALYSIS - Parameter Settings................................ - Initial Population................................. - Search Space.................................... - Elitism........................................ - Encoding of Solutions............................. - Crossover Methods............................... - Convergence............................. - - The Final Program - Results........................................ - Proposed Improvements............................ - Conclusion...
- REFERENCES......................................
- APPENDICES
- APPENDIX 1. GENETIC ALGORITHM
- APPENDIX 2. REPRESENTATION OF TOURS
- VITA
Introduction
The Vehicle Routing Problem (VRP) is a combinatorial optimization problem in which a number of customers, requiring either pick-ups or deliveries, must be serviced by a set of vehicles. The objective is to route the vehicles in such a manner that each customer is visited by exactly one vehicle and the total distance traveled is minimized. The vehicles may be constrained by a load capacity or a maximum time spent on the route. Vehicle routing problems are complex to solve, particularly to optimality, causing many "algorithm designers" to settle for an approximation of the optimal solution [Haimovich, et al 1988]. The focus of this thesis is to experiment with the genetic algorithm (GA) as a means of solving the VRP. According to Goldberg, the goal of genetic algorithms is to be efficient and robust over different environments in order to eliminate costly redesigns in the programs. Genetic algorithms are described as "computationally simple yet powerful" and are not limited by restrictive assumptions such as, "continuity, existence of derivatives, unimodality, and other matters" [Goldberg 1988]. However, according to Davis,"..., in general, the robustness of a genetic algorithm and its performance on a particular problem are inversely related" [Davis 1991]. Genetic algorithms are robust in that they can be used to solve several different problem types without changing the algorithm. A slight change in the problem could make a nonrobust algorithm inoperative [Davis 1987].
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Genetic algorithms first appeared in theory in the early 1970's, but John Holland is said to have founded the field of genetic algorithms in 1975. In more recent years, work on genetic algorithms has been focused on application [Davis 1991]. Holland's original idea in developing the algorithm was to create a program which would adapt to its environment [Goldberg 1988]. The genetic algorithm was first used in industry to optimize the design of a communications network [Davis 1987]. There are several areas in which GA performance has been studied. The following is a partial list of areas for which genetic algorithms have been studied:
- Davis (1985) - Job shop scheduling
- Glover (1987) - Keyboard configuration systems
- Goldberg (1983) - Optimizing gas pipeline systems
- Grefenstette (1985) - Traveling salesman problem
- Nygard and Kadaba (1990) - Multi-vehicle routing problem [Nygard 1992]. Atidel Ben Hadj-Alouane (1992) at the University of Michigan successfully used a genetic algorit栀洀 to solve multiple choice integer programs with nonlinear relaxation. The algorithm successfully solved 100% dense problems and had computation times superior to IBM's Optimization Subroutine Library (OSL). Hadj-Alouane noted three advantages of the genetic algorithm when compared to OSL after running the genetic algorithm on three facility location problems:
- The optimal solution was found for all three, yet less time was taken than with OSL.
- There was a small variation in solutions for different random seeds.
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CHAPTER 1 THE GENETIC ALGORITHM AND THE VEHICLE ROUTING PROBLEM
Chapter 1 is an introduction to the genetic algorithm and the type of vehicle routing problem addressed in this thesis. The first section describes the VRP and some of the common heuristic methods currently being used to solve problems of this type. There are several important aspects of the GA which must be considered during development. A few of these aspects are encoding of solutions, evaluation function, parameter values, selection methods, crossover methods, and mutation methods. The second section explains the terminology of the GA and describes various ways of representing some of these important aspects. The last section of this chapter specifically describes the aspects of the GA which were used to solve the VRP.
The Veh椀挀le Routin� Problem The vehicle routing problem addressed in this thesis is one consisting of a single depot, n customers, and m vehicles. For each customer, the vehicle must pick up a certain amount of weight, Wj, where j is the customer number. For problems with time considerations there is a constant stop time, s, at each customer. The objective (1) is to minimize the total distance traveled by all vehicles where dij is the distance from customer i to customer j. The binary variable Ym ij will equal 1 if vehicle m
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goes from customer i to customer j and O otherwise. A formulation for the VRP is given below: (1 ) mm LmLij dij Ymij (2) S. LmLi,i<>j Ymij + LmLk Ymjk = 2 (3) Lj Ym0j = 1 for all m (4) Li,i<>j Ymij � 1 for a11 m, j (5) Li Ymi0 = 1 for all m (6) Li Ymij - Lk Ymjk = 0 for all m, j
for all j
(7) LiLj Ymij � ISi -1 for all subsets S, for all m (8) Li,j Ymij Wj � w for all m (9) Li,j (dij + s) Ymij � t for all m [Noon, et al 1 991 ]. Constraint (2) ensures that exactly one vehicle visits and leaves each customer. Each vehicle is forced to leave the depot by constraint (3). Constraint (4) ensures that a vehicle does not visit a particular customer more than once and constraint (5) ensures that the vehicle retu爀渀s to the depot fNoon, et al 1991]. Flow conservation for each vehicle tour is enforced by constraint (6). Subtours are eliminated by constraint (7). The capacity constraint (8) ensures that the total amount of weight picked up by the vehicle does not exceed a weight limit of w. Constraint (9) ensures that the total time on the route for each vehicle can not exceed atime limit oft, where the time on a route is calculated by the distance on the route plus the sum of all the stop times on the route. "The vehicle routing problem is a hard combinatorial problem and to this day, only relatively small VRP instances can be solved to optimality." [Gendreau, et al 1991 ] There are four groups in which heuristic methods for solving the VR P can be divided. They are
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to be visited are relaxed. The idea is to have a dispatcher who assigns each vehicle an initial customer to visit and then assigns reward values to every other customer. The objective when preassigning these customers is to maximize the minimum distance between any two of them. Each vehicle driver then decides which customers to visit and the corresponding sequence of visits. The objective of the dispatcher is to assign the rewards so that each customer will be visited by exactly one vehicle. The major di昀昀erence between this approach and that of Fisher and Jaikumar is that the dispatcher in the Fisher method decides which customers each driver visits, and the driver is only responsible 昀漀r sequencing the route. In the Lagrangean Relaxation approach, the driver has the additional responsibility of deciding which customers to visit [Noon, et al 1991]. The third heuristic method is Incomplete Optimization. This approach uses an enumerative algorithm to find a good solution by means of an incomplete search tree [Gendreau, et al 1991]. Finally, the fourth heuristic method used is Improvement Methods which is the category in which tabu search falls. Among the tabu search methods that exist are one developed by Pureza and Franca (1991) in which cities are swapped between two routes and one developed by Semet and Taillard (1991) in which a city is moved from one route to an alte爀渀ate route [Gendreau, et al 1991]. The algorithm developed by Gendreau, Hertz, and Laporte inserts a node into a tour from another tour using a generalized insertion procedure (GENI). A tour improvement procedure which was also developed by Gendreau, Hertz, and Laporte, is used to improve each route. Once a customer is taken out of a particular vehicle's tour, it cannot be put back into that tour for a certain number of iterations.
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One di昀昀erence between this method and the other tabu search methods is that it allows in昀攀asible solutions whereas the others do not. An advantage to using this method, called TABROUTE, is that the risk of converging to a local optima is reduced in two ways. The 昀椀rst is by allowing infeasible solutions through the use of a penalty function. The second is by using GENI to perform the insertion of the customer into a di昀昀erent route [Gendreau, et al 199 1]. Another improvement method which has successfully been used was developed by Ibrahim Osman at the University of Canterbury [Osman 1993] and solves the vehicle routing problem using simulated annealing and tabu search. This method finds a route by first using a heuristic followed. by an improvement method in which a portion of one route is exchanged with a portion of a second route. An insertion/deletion procedure is used to recalculate the objective value, and the 2-opt arc exchange heuristic of Lin [Osman 1993] is used to correct any paths that are crossed. There are two selection strategies used for selecting alte爀渀ative solutions: best improvement and first improvement. The tabu search consists of a forbidding strategy, a freeing strategy, a short-term strategy, and a stopping criterion. The forbidding strategy keeps a list of the moves which are forbidden. The freeing strategy removes the moves from the tabu list after a certain number of iterations. The short-term strategy uses an aspiration criterion to overrule the tabu list and includes two possible selection strategies: Best Admissible (BA) and First Best Admissible (FBA). BA selects the move resulting in the greatest improvement or the least nonimprovement. FBA selects the first move resulting in an improvement in the objective value if one exists; otherwise, the best
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This chromosome represents a solution in which the sequence of customer visits is 5,1,4,3,6,2. Each position of this chromosome is cal led a gene, and the val ue of each gene is called an allele. For example, 5 is the allele of the first gene [Nygard 1992]. At each generation there are a number of methods which can be used to produce a new population of solutions. Although all genetic algorithms use some form of reproduction, crossover, and mutation, there are many di昀昀erent ways of carrying out these operations. The next section describes some of the alte爀渀ate methods. First, there are a number of di昀昀erent ways to select the chromosomes to be added to the mating pool. The challenge is to select the parents in such a way that the good parents reproduce enough to survive,. but not so much as to cause the population to prematurely converge [DeJong 1985]. There is still disagreement among researchers on the best method of parent selection. Four of the most common methods are listed below.
- Random selection of the chromosomes.
- Roulette sampling in which the probability of selecting a particular chromosome increases with its fitness.
- Rank based sampling which uses the roulette wheel to select two chromosomes, of which the one with the best fitness is added to the mating pool.
- Tou爀渀ament sampling in which solutions are sequentially chosen with the one having the higher fitness being added to the mating pool [Nygard 1992]. Next, there are several different methods of crossing over the two parent chromosomes. There is the one-point crossover in which a point on
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the chromosome is randomly selected, and the two chromosomes exchange the genes following this point. 吀栀e disadvantage of using the one-point crossover is that if good genetic material is at both ends of the chromosome, these two good traits will be separated during the crossover. The two-point crossover solves the one-point crossover problem by enabling two genes on opposite ends of the chromosome to remain on the same chromosome after the crossover. This is accomplished since two points are randomly selected on the chromosome, and the genes between these two points are exchanged between the two chromosomes. However, this still may present a problem if, for example, all of the good traits are on one of the chromosomes [Davis 1991]. The best crossover method seems to be the uniform crossover in which a random number (between 1 and 100) is generated for each gene. If this number is less than a certain user defined number (which is defined at the beginning of the GA as the gene selection parameter) , the child will receive this gene from the first parent. If the number is greater than the gene selection parameter, the child receives the gene from the second parent. Unlike the one-point and two-point crossovers, this crossover method has the ability to combine good traits irrespective of where they are located on the chromosome [Davis 1991]. Also, there are differences in the methods of producing mutations. One method is to simply mutate a single gene at a certain rate (i. 1 out of every 1000 genes). However, since mutation is the main means of producing variation in the population, mutating a single gene does not seem to be very efficient. Another, more efficient method is to mutate the entire chromosome for a low percentage of the chromosomes in the population
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A Genetic Algorithm for the Vehicle Routing Problem
Course: Software Engineering (CS530)
University: Visvesvaraya Technological University
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