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Optimal Tours & Routes: Mathematics of Traveling Salesman Problems

Explore the mathematics behind Traveling Salesman Problems (TSP) and learn how to find optimal tours and routes for various real-life applications like touring cities, exploring moons, planning Mars missions, routing school buses, delivering packages, and fabricating circuit boards.

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Optimal Tours & Routes: Mathematics of Traveling Salesman Problems

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  1. 6 The Mathematics of Touring 6.1 Hamilton Paths and Hamilton Circuits 6.2 Complete Graphs? 6.3 Traveling Salesman Problems 6.4 Simple Strategies for Solving TSPs 6.5 The Brute-Force and Nearest-Neighbor Algorithms 6.6 Approximate Algorithms 6.7 The Repetitive Nearest-Neighbor Algorithm 6.8 The Cheapest-Link Algorithm

  2. Traveling Salesman Problems The “traveling salesman” is a convenientmetaphor for many different real-life applications. The next few examples illustrate a few of the many possible settings for a “traveling salesman” problem,starting, of course, with the traveling salesman’s“traveling salesman” problem.

  3. Example 6.4 A Tour of Five Cities Meet Willy, the traveling salesman. Willy has customers in five cities, which forthe sake of brevity we will call A, B, C, D, and E. Willy needs to schedule a salestrip that will start and end at A (that’s Willy’s hometown) and goes to each of theother four cities once. We will call the trip “Willy’s sales tour.” Other than startingand ending at A, there are no restrictions as to the sequence in which Willy’s salestour visits the other four cities.

  4. Example 6.4 A Tour of Five Cities The graph shows the cost of a one-way airline ticket between eachpair of cities. Like most people, Willy hates to waste money. Thus, among the many possibilitiesfor his sales tour, Willy wants to find the optimal (cheapest) one. How? We willreturn to this question soon.

  5. Example 6.5 Touring the Outer Moons It is the year 2020. An expedition to explore the outer planetary moons in oursolar system is about to be launched from planet Earth. The expedition is scheduled to visit Callisto, Ganymede, Io, Mimas, and Titan (the first three are moonsof Jupiter; the last two, of Saturn), collect rock samples at each, and then return toEarth with the loot.The next slide shows the mission time (in years) between any two moons. An important goal of the mission planners is to complete the mission in the leastamount of time.

  6. Example 6.5 Touring the Outer Moons What is the optimal (shortest) tour of the outer moons?

  7. Example 6.6 Roving the Red Planet The figure shows seven locations on Mars where NASA scientists believe there isa good chance of finding evidence of life.

  8. Example 6.6 Roving the Red Planet Imagine that you are in charge of planning a sample-return mission. First, youmust land an unmanned rover in the Ares Vallis (A). Then you must direct the rover to travel to each site and collect and analyze soil samples. Finally, you mustinstruct the rover to return to the Ares Vallis landing site, where a return rocketwill bring the best samples back to Earth. A Mars tour like this will take severalyears and cost several billion dollars, so good planning is critical.

  9. Example 6.6 Roving the Red Planet Here are the estimated distances (in miles) that a rover would haveto travel to get from one Martian site to another. What is the optimal (shortest)tour for the Mars rover?

  10. Traveling Salesman Examples In each case the problem is to find a tour of the sites (i.e., a trip that starts and ends at a designated siteand visits each of the other sites once) and has the property of being optimal (i.e.,has the least total cost). Any problem that shares these common elements (atraveler, a set of sites, a costfunction for travel between pairs of sites, a need totour all the sites, and a desire to minimize the total cost of the tour) is known as atraveling salesman problem, or TSP.

  11. Routing School Buses A school bus (the traveler) picks up children in themorning and drops them off at the end of the day at designated stops (thesites). On a typical school bus route there may be 20 to 30 such stops. Withschool buses, total time on the bus is always the most important variable (students have to get to school on time), and there is a known time of travel (thecost) between any two bus stops. Since children must be picked up at everybus stop, a tour of all the sites (starting and ending at the school) is required.Since the bus repeats its route every day during the school year, finding anoptimal tour is crucial.

  12. Delivering Packages Package delivery companies such as UPS and FedExdeal with TSPs on a daily basis. Each truck is a traveler that must deliverpackages to a specific list of delivery destinations (the sites). The travel timebetween any two delivery sites (the cost) is known or can be estimated. Eachday the truck must deliver to all the sites on its list (that’s why sometimesyou see a UPS truck delivering at 8 P.M.), so a tour is an implied part of therequirements. Since one can assume that the driver would rather be homethan out delivering packages, an optimal tour is a highly desirable goal.

  13. Fabricating Circuit Boards In the process of fabricating integrated-circuitboards, tens of thousands of tiny holes (the sites) must be drilled in eachboard. This is done by using a stationary laser beam and moving the board(the traveler). To do this efficiently, the order in which the holes are drilledshould be such that the entire drilling sequence (the tour) is completed in theleast amount of time (optimal cost). This makes for a very high tech TSP.

  14. Fabricating Circuit Boards On a typical Saturday morning, an average Joeor Jane (the traveler) sets out to run a bunch of errands around town, visitingvarious sites (grocery store, hair salon, bakery, post office). When gas wascheap,time used to be the key costvariable, but with the cost of gas thesedays, people are more likely to be looking for the tour that minimizes thetotal distance traveled.

  15. Modeling a TSP Every TSP can be modeled bya weighted graph, that is, a graph such that there is a number associated with eachedge (called the weightof the edge). The beauty of this approach is that themodel always has the same structure: The vertices of the graph are the sites of theTSP, and there is an edge between X and Y if there is a direct link for the travelerto travel from site X to site Y. Moreover, the weight of the edge XY is the cost oftravel between X and Y.

  16. Modeling a TSP In this setting a tour is a Hamilton circuit of the graph,and an optimal tour is the Hamilton circuit of least total weight. In all the applications and examples we will be considering in this chapter, wewill make the assumption that there is an edge connecting every pair of sites,which implies that the underlying graph model is always a complete weightedgraph. The following is a summary of the preceding observations.

  17. GRAPH MODEL OF A TRAVELING SALESMAN PROBLEM ■Sites g vertices of the graph. ■Costs g weights of the edges. ■Tour g Hamilton circuit. ■Optimal tour g Hamilton circuit of least total cost

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