Examensarbete Computation of Mileage Limits for Traveling Salesmen by Means of Optimization Techniques Johan Torstensson LiTH - MAT - EX - - 2008 / 08 - - SE Computation of Mileage Limits for Traveling Salesmen by Means of Optimization Techniques Division of Optimization, Department of Mathematics Linköpings Universitet Johan Torstensson LiTH - MAT - EX - - 2008 / 08 - - SE Examensarbete: 30 hp Level: D Examiner: Kaj Holmberg, Division of Optimization, Department of Mathematics Linköpings Universitet Supervisor: Kaj Holmberg, Professor, Linköpings Universitet Co-supervisor: Maria Mehlin, Scantech Sverige Linköping: June 2008 Abstract Many companies have traveling salesmen that market and sell their products. This results in much traveling by car due to the daily customer visits. This causes costs for the company, in form of travel expenses compensation, and en- vironmental effects, in form of carbon dioxide pollution. As many companies are certified according to environmental management systems, such as ISO 14001, the environmental work becomes more and more important as the environmen- tal consciousness increases every day for companies, authorities and public. The main task of this thesis is to compute reasonable limits on the mileage of the salesmen; these limits are based on specific conditions for each salesman’s district. The objective is to implement a heuristic algorithm that optimizes the customer tours for an arbitrary chosen month, which will represent a “standard” month. The output of the algorithm, the computed distances, will constitute a mileage limit for the salesman. The algorithm consists of a constructive heuristic that builds an initial solution, which is modified if infeasible. This solution is then improved by a local search algorithm preceding a genetic algorithm, which task is to improve the tours separately. This method for computing mileage limits for traveling salesmen generates good solutions in form of realistic tours. The mileage limits could be improved if the input data were more accurate and adjusted to each district, but the suggested method does what it is supposed to do. Keywords: Period Traveling Salesman Problem, Periodic, Travelling Salesman Problem, PTSP, TSP, Heuristic Algorithm, mileage limit, business trips, travel expenses compensation. i ii Abstract Acknowledgements I would like to thank my supervisor Professor Kaj Holmberg for his assistance and guidance. I would also like to thank my co-supervisor at Scantech Sverige, Maria Mehlin, for her help, engagement and encouragement throughout the whole period. I also would like to thank the rest of the staff at Scantech Sverige for their help with understanding the salesmen’s work situation and for compilation of data material. I would also like to thank my opponent Emelie Raba for her comments and criticism of the report. Finally, I would like to thank family and friends for their support. iii iv Acknowledgements Contents Abstract i Acknowledgements iii 1 Introduction 1 1.1 Background . .1 1.2 Purpose . .1 1.3 Delimitations . .2 1.4 Outline of the Thesis . .2 2 Background 3 2.1 Scantech - The Company . .3 2.1.1 Products and Business Idea . .3 2.2 The Environmental Work . .4 2.2.1 ISO Certification . .4 2.2.2 Reasons for Certification . .5 2.2.3 Scantech’s Environmental Work . .5 2.3 Business Trips . .6 2.3.1 The Salesmen’s Working Situation . .7 2.4 Problem Data . .7 2.4.1 Adjustment of Data . .8 2.4.2 Computation of Distances . 10 3 Theoretical Background 11 3.1 TSP . 11 3.2 PTSP . 12 3.2.1 Formulation in Terms of Graph Theory . 12 3.3 Heuristic Algorithms . 13 3.3.1 Genetic Algorithms . 13 3.4 Literature on the PTSP . 14 3.4.1 A Study in the PTSP Literature . 14 3.4.2 Mathematical Formulation of the PTSP . 15 v vi CONTENTS 4 Mathematical Formulation 17 4.1 Description of the Problem . 17 4.2 Parameters . 17 4.3 Sets . 18 4.4 Variables . 18 4.5 Mathematical Model . 18 4.5.1 Explanation of the Model . 19 4.6 Commentary on the Mathematical Model . 19 5 Solution Method 21 5.1 The Algorithm . 21 5.2 Description of the Algorithm . 21 5.2.1 Initial Tour . 23 5.2.2 Tour Construction . 23 5.2.3 Extra Improvement Procedure . 23 5.2.4 Empty Tours . 24 5.2.5 Improvement Procedure . 25 5.2.6 Tour Improvement . 25 5.2.7 Insertion Rule . 25 5.2.7.1 Check Feasibility . 26 5.2.8 Removal Rule . 27 5.3 Example of Algorithm Progress . 28 5.4 Alternative Solution Techniques . 32 6 Results 33 6.1 Runs and Parameters . 33 6.2 Computational Results . 33 6.2.1 An Illustration of the Tours . 35 6.3 Customization to Budgeted Sales . 35 7 Analysis 39 7.1 Computation of a Lower Boundary . 39 7.2 Analysis of Computational Results . 39 7.3 Analysis of the Algorithm . 40 7.3.1 Algorithm Fluctuation . 40 7.3.2 Running Times . 40 7.3.3 Essential Procedures . 41 8 Demarcation of New District Boundaries 43 8.1 Reason for Proposal of New Boundaries . 43 8.2 The Proposal . 43 CONTENTS vii 9 Conclusions and Further Research 47 9.1 Conclusions . 47 9.2 Further Research . 48 A More Illustrations of Tours 49 B Detailed Maps 53 Bibliography 59 viii CONTENTS List of Figures 2.1 Map of Scandinavia, with the sales offices marked . .4 2.2 Pollution of carbon dioxide (CO2) due to business trips . .6 2.3 Average monthly mileages for the districts . .7 2.4 Map of the current district boundaries . .9 3.1 Solutions to a TSP and a PTSP (the square represents the home- town) . 12 5.1 An overview of the algorithm progress . 22 5.2 Insertion Rule: Inserting customer s into a non-empty tour . 26 5.3 Insertion Rule: Inserting customer s into an empty tour . 26 5.4 An axis that illustrates the punishment intervals . 27 5.5 Removal Rule: Removing customer r ................ 28 5.6 Initial Tour after insertion of customer 2. 29 5.7 Tour Construction after insertion of customer 6. 29 5.8 Tour Construction after insertion of customer 5. 29 5.9 Tour Construction after insertion of all six customers. 30 5.10 The tours after the extension of one day and two customers. 30 5.11 Empty Tour after moving customer 8 to the empty tour on day 3. 31 5.12 Improvement Procedure after moving customer 7 to the tour on day3. ................................. 31 6.1 Mileage limits (green striped) and mileage means (black) . 34 6.2 The tours of district 15, Skaraborg . 37 7.1 The result of all 30 runs for district 2 . 40 8.1 Map of the proposed district boundaries . 45 A.1 The tours of district 7, Västmanland/Närke . 50 A.2 The tours of district 14, Bohuslän . 51 A.3 The tours of district 24, East Skåne . 52 ix x LIST OF FIGURES B.1 Map of the districts in southern Sweden . 54 B.2 Map of the districts in southern-central Sweden . 55 B.3 Map of the districts in northern-central Sweden . 56 B.4 Map of the districts in northern Sweden . 57 List of Tables 2.1 Information concerning the districts (a star (*) implies that no mileage limit will be computed for that district) . .8 6.1 Mileage limits . 34 6.2 Results for the tours on district 15, Skaraborg . 35 6.3 Budgeted mileage limits . 36 xi xii LIST OF TABLES Chapter 1 Introduction This chapter gives a description of the problem background and the purpose and delimitations of the thesis. 1.1 Background Almost every company has salesmen that market and sell their products. A traveling salesman visits customers every day and does spend much time in the car. The more traveling the more increases the cost for the company, in the form of travel expenses compensation to the salesmen, and the effects on the environment. To keep down the traveled distance is therefore essential both in the perspective of cost and environmental considerations. On account of that, it is important to get an idea of whether a traveling sales- man’s mileage1 is reasonable, with respect to the customers’ geographical lo- cations in relation to the salesman’s hometown. Taking an arbitrary “normal” month for a specific salesman and optimizing on which day and in which order the customers should be visited, that distance will constitute a mileage limit. The problem will be a periodic variant of the well-known Traveling Salesman Problem. This thesis concentrates on finding a minimal travel distance for every salesman, with the help of a heuristic algorithm. 1.2 Purpose The main purpose of this thesis is to solve the periodic Traveling Salesman Problem with a heuristic algorithm that consists of a constructive heuristic, a local search algorithm and a metaheuristic. A subproblem in this thesis is also to look at the boundaries of the districts and to demarcate these in order to reduce the environmental effetcs. 1mileage refers to the traveled distance in consequence of business trips 1 2 1.4. Outline of the Thesis 1.3 Delimitations Because the number of customers is huge for every district, it would be very time-consuming to compute the exact road distance between all of them. There- fore the distances are approximated with the air distances, with an addition to compensate the error. An estimation of how often the customers are visited are also done, where the basis is the order statistics. 1.4 Outline of the Thesis In Chapter 2 the background of the company and their business trips is pre- sented. Chapter 3 gives a theoretical background to the periodic variant of the Traveling Salesman Problem and the heuristics that are used.