Mathematics
The Travelling Salesman Problem
The Travelling Salesman Problem is a classic optimization problem in which the goal is to find the shortest possible route that visits a set of given locations exactly once and then returns to the starting point. It has applications in logistics, computer science, and operations research, and is known for its difficulty in finding an optimal solution.
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12 Key excerpts on "The Travelling Salesman Problem"
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Orange Apple(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter- 12 Travelling Salesman Problem The Travelling Salesman Problem ( TSP ) is an NP-hard problem in combinatorial optimization studied in operations research and theoretical computer science. Given a list of cities and their pairwise distances, the task is to find a shortest possible tour that visits each city exactly once. The problem was first formulated as a mathematical problem in 1930 and is one of the most intensively studied problems in optimization. It is used as a benchmark for many optimization methods. Even though the problem is computationally difficult, a large number of heuristics and exact methods are known, so that some instances with tens of thousands of cities can be solved. The TSP has several applications even in its purest formulation, such as planning, logistics, and the manufacture of microchips. Slightly modified, it appears as a sub-problem in many areas, such as DNA sequencing. In these applications, the concept city represents, for example, customers, soldering points, or DNA fragments, and the concept distance represents travelling times or cost, or a similarity measure between DNA fragments. In many applications, additional constraints such as limited resources or time windows make the problem considerably harder. In the theory of computational complexity, the decision version of the TSP (where, given a length L, the task is to decide whether any tour is shorter than L) belongs to the class of NP-complete problems. Thus, it is likely that the worst case running time for any algorithm for the TSP increases exponentially with the number of cities. History The origins of The Travelling Salesman Problem are unclear. A handbook for travelling salesmen from 1832 mentions the problem and includes example tours through Germany and Switzerland, but contains no mathematical treatment.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- White Word Publications(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 9 Travelling Salesman Problem The Travelling Salesman Problem ( TSP ) is an NP-hard problem in combinatorial optimization studied in operations research and theoretical computer science. Given a list of cities and their pairwise distances, the task is to find a shortest possible tour that visits each city exactly once. The problem was first formulated as a mathematical problem in 1930 and is one of the most intensively studied problems in optimization. It is used as a benchmark for many optimization methods. Even though the problem is computationally difficult, a large number of heuristics and exact methods are known, so that some instances with tens of thousands of cities can be solved. The TSP has several applications even in its purest formulation, such as planning, logistics, and the manufacture of microchips. Slightly modified, it appears as a sub-problem in many areas, such as DNA sequencing. In these applications, the concept city represents, for example, customers, soldering points, or DNA fragments, and the concept distance represents travelling times or cost, or a similarity measure between DNA fragments. In many applications, additional constraints such as limited resources or time windows make the problem considerably harder. In the theory of computational complexity, the decision version of the TSP (where, given a length L, the task is to decide whether any tour is shorter than L) belongs to the class of NP-complete problems. Thus, it is likely that the worst case running time for any algorithm for the TSP increases exponentially with the number of cities. History The origins of The Travelling Salesman Problem are unclear. A handbook for travelling salesmen from 1832 mentions the problem and includes example tours through Germany and Switzerland, but contains no mathematical treatment.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- College Publishing House(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 9 Travelling Salesman Problem The Travelling Salesman Problem ( TSP ) is an NP-hard problem in combinatorial optima-zation studied in operations research and theoretical computer science. Given a list of cities and their pairwise distances, the task is to find a shortest possible tour that visits each city exactly once. The problem was first formulated as a mathematical problem in 1930 and is one of the most intensively studied problems in optimization. It is used as a benchmark for many optimization methods. Even though the problem is computationally difficult, a large number of heuristics and exact methods are known, so that some instances with tens of thousands of cities can be solved. The TSP has several applications even in its purest formulation, such as planning, logistics, and the manufacture of microchips. Slightly modified, it appears as a sub-problem in many areas, such as DNA sequencing. In these applications, the concept city represents, for example, customers, soldering points, or DNA fragments, and the concept distance represents travelling times or cost, or a similarity measure between DNA frag-ments. In many applications, additional constraints such as limited resources or time windows make the problem considerably harder. In the theory of computational complexity, the decision version of the TSP (where, given a length L, the task is to decide whether any tour is shorter than L) belongs to the class of NP-complete problems. Thus, it is likely that the worst case running time for any algo-rithm for the TSP increases exponentially with the number of cities. History The origins of The Travelling Salesman Problem are unclear. A handbook for travelling salesmen from 1832 mentions the problem and includes example tours through Germany and Switzerland, but contains no mathematical treatment.
eBook - PDFOperations Planning
Mixed Integer Optimization Models
- Joseph Geunes(Author)
- 2014(Publication Date)
- CRC Press(Publisher)
9 Vehicle Routing and Traveling Salesman Problems 9.1 Introduction The traveling salesman problem (TSP) provides a classic example of a diffi-cult combinatorial optimization problem that can be easily explained to the layperson. A person located at a home base must visit a set of n cities and then return home, while traveling the least total distance (or incurring the lowest cost) possible. The data required in solving the problem includes the number of cities and the set of all pairwise inter-city distances. In the sym-metric version of the TSP, the inter-city distance from city i to city j is the same as that from city j to city i ; when considering straight-line Euclidean distances or two-way road travel distances, symmetric distances are not un-likely to apply. The general version of the TSP does not, however, impose such a symmetry requirement. Minimizing the total distance traveled serves as one possible goal; alternatively, one may wish to minimize the total cost incurred in visiting the set of cities, in which case asymmetric distances may be reasonable. For example, if air travel is involved, then it is not unlikely for the cost to travel from city i to city j to differ from the cost from j to i . Although the idea of a person visiting a set of cities and returning home aids in conceptualizing the problem, the TSP finds application in numerous seemingly unrelated contexts. For example, consider the problem of scheduling a set of n jobs on a single machine when changeover times between jobs depend on the job sequence (for example, the time to prepare the machine for some job k after processing job i differs from the time to prepare for job j ). We may think of each job as corresponding to a city and think of the changeover time when job i is performed immediately before job j as the “distance” from i to j .
eBook - PDFGeneralized Network Design Problems
Modeling and Optimization
- Petrica C. Pop(Author)
- 2012(Publication Date)
- De Gruyter(Publisher)
Chapter 3 The Generalized Traveling Salesman Problem (GTSP) The traveling salesman problem (TSP) certainly is one of the classical and most inten- sively studied and analyzed representatives of combinatorial optimization problems, with a lot of solution methodologies and several applications in, e.g. logistics, trans- portation, planning, microchips manufacturing, DNA sequencing, etc. Some mathematical problems related to the TSP were treated in the 1800s by W.R. Hamilton and the TSP already appeared in the literature in the early 19th century. It was published in Vienna, in the 1930s, by the mathematician and economist Karl Menger [120] and later was promoted by H. Whitney and M. Flood in the circles of Princeton, being named the traveling salesman problem. Comparing the TSP to other combinatorial optimization problems, the main dif- ference is that very powerful problem-specific methods, such as the Lin–Kernighan algorithm [113] and effective branch and bound methods are available, that are able to achieve a global optimal solution in very high problem dimensions. Several extensions of the TSP have been considered in the literature: the time- dependent TSP, the bottleneck TSP, TSP with profits, TSP with mixed deliveries and collections, TSP with backhauls, one commodity pickup and delivery TSP, the multi- ple TSP, etc. In this chapter, we are concerned with an extension of the traveling salesman prob- lem (TSP) called the the generalized traveling salesman problem (GTSP). Given a complete undirected graph whose nodes are partitioned into a number of subsets (clusters) and with nonnegative costs associated to the edges, the GTSP consists of finding a minimum-cost Hamiltonian tour including exactly one node from each clus- ter. Therefore, the TSP is a special case of the GTSP, where each cluster consists of exactly one node. A variant of the GTSP is the problem of finding a minimum cost Hamiltonian tour including at least one vertex from each cluster.
eBook - PDF- Federico Greco(Author)
- 2008(Publication Date)
- IntechOpen(Publisher)
1 Population-Based Optimization Algorithms for Solving The Travelling Salesman Problem Mohammad Reza Bonyadi, Mostafa Rahimi Azghadi and Hamed Shah-Hosseini Department of Electrical and Computer Engineering, Shahid Beheshti University, Tehran, Iran 1. Introduction The Travelling Salesman Problem or the TSP is a representative of a large class of problems known as combinatorial optimization problems. In the ordinary form of the TSP, a map of cities is given to the salesman and he has to visit all the cities only once to complete a tour such that the length of the tour is the shortest among all possible tours for this map. The data consist of weights assigned to the edges of a finite complete graph, and the objective is to find a Hamiltonian cycle, a cycle passing through all the vertices, of the graph while having the minimum total weight. In the TSP context, Hamiltonian cycles are commonly called tours. For example, given the map shown in figure l, the lowest cost route would be the one written (A, B, C, E, D, A), with the cost 31. Fig. 1. The tour with A=>B =>C =>E =>D => A is the optimal tour. In general, the TSP includes two different kinds, the Symmetric TSP and the Asymmetric TSP. In the symmetric form known as STSP there is only one way between two adjacent cities, i.e., the distance between cities A and B is equal to the distance between cities B and A (Fig. 1). But in the ATSP (Asymmetric TSP) there is not such symmetry and it is possible to have two different costs or distances between two cities. Hence, the number of tours in the ATSP and STSP on n vertices (cities) is (n-1)! and (n-1)!/2, respectively. Please note that the graphs which represent these TSPs are complete graphs. In this chapter we mostly consider the STSP. It is known that the TSP is an NP-hard problem (Garey & Johnson, 1979) and is often used for testing the optimization algorithms. Finding Hamiltonian cycles or traveling
eBook - PDFExplorations in Computing
An Introduction to Computer Science
- John S. Conery(Author)
- 2010(Publication Date)
- CRC Press(Publisher)
Before looking at the solution (which is shown in Figure 12.13 at the end of the chapter) can you try to figure out on your own what the shortest tour is? Hint: the best tour does not include the line segment drawn with a dashed line in this picture. It’s not likely you will ever be called upon to devise a tour that includes 25 destinations. But the same basic idea, of planning the optimal route that connects several different points, is an important part of several real-life situations, and mathematicians and computer scien-tists have devoted considerable effort to developing algorithms to solve this problem. Some familiar examples are a courier service that wants to plan the most efficient route for their delivery trucks at the start of each day, or a school district that needs to design the shortest route for a set of school buses that pick up and drop off students. A problem that has the same essential structure is figuring out how to connect circuits on the surface of a computer chip. In this case, the “destinations” are electronic components, and the goal is to figure out how to best place metal pathways on the chip to connect the components. In computer science this problem is known as the Traveling Salesman Problem , or TSP for short. The input to an algorithm that solves the TSP is a map with a set of cities, where the distance between each pair of cities is defined in a table. The goal is to create the shortest tour that includes all the cities. A tour can start at any city, but it must visit every other city exactly once and then return to the starting point. In applications of the TSP there are several ways to define the places to visit and the cost of moving from one place to another.
eBook - ePub- Lawrence V. Snyder, Zuo-Jun Max Shen(Authors)
- 2019(Publication Date)
- Wiley(Publisher)
Chapter 10 The Traveling Salesman Problem10.1 Supply Chain Transportation
Transportationis arguably the largest component of total supply chain costs. In 2017, United States businesses spent $966 billion moving freight along roads, rails, waterways, air routes, and pipelines (Council of Supply Chain Management Professionals, 2018a ). Even small improvements in transportation efficiency can have a huge financial impact. A transportation system has many aspects to optimize, from mode selection to driver staffing to contract negotiations, and mathematical models have been used for all of these issues and more. In this chapter and the next, we cover one important aspect of the transportation‐related decisions a firm must make, namely, routing vehicles among the locations they must visit.We discuss the famous traveling salesman problem (TSP) in this chapter. The TSP is important not only because of its practical utility but also because, over the past several decades, the study of the TSP has spurred many fundamental developments in the theory of optimization itself. Then, in Chapter 11, we discuss the vehicle routing problem (VRP), which generalizes the TSP by adding constraints that necessitate the use of multiple routes.10.2 Introduction to the TSP
10.2.1 Overview
The TSP involves finding the shortest route through n nodes that begins and ends at the same city and visits every node. The TSP is perhaps the best‐known combinatorial optimization problem and has been intensely studied by researchers in supply chain management, operations research, computer science, and other fields. Moreover, it serves as the foundation for a great many routing problems, and instances of these problems are solved thousands, if not millions, of times per day by companies and public agencies to plan package and mail deliveries, optimize robot movements, direct naval vessels, fabricate semiconductor chips, and more. (See Applegate et al. (2007
eBook - PDFExplorations in Computing
An Introduction to Computer Science and Python Programming
- John S. Conery(Author)
- 2014(Publication Date)
- Chapman and Hall/CRC(Publisher)
Before looking at the solution (which is shown in Figure 12.18 at the end of the chapter) can you try to figure out on your own what the shortest tour is? Hint: The best tour does not include the line segment drawn with a dashed line in this picture. It’s not likely you will ever be called upon to devise a tour that includes 25 destinations. But the same basic idea, of planning the optimal route that connects several different points, is an important part of several real-life situations, and mathematicians and computer scien-tists have devoted considerable effort to developing algorithms to solve this problem. Some familiar examples are a courier service that wants to plan the most efficient route for their delivery trucks at the start of each day, or a school district that needs to design the shortest route for a set of school buses that pick up and drop off students. A problem that has the same essential structure is figuring out how to connect circuits on the surface of a computer chip. In this case, the “destinations” are electronic components, and the goal is to figure out how to best place metal pathways on the chip to connect the components. In computer science this problem is known as the Traveling Salesman Problem , or TSP for short. The input to an algorithm that solves the TSP is a map with a set of cities, where the distance between each pair of cities is defined in a table. The goal is to create the shortest tour that includes all the cities. A tour can start at any city, but it must visit every other city exactly once and then return to the starting point. In applications of the TSP there are several ways to define the places to visit and the cost of moving from one place to another.
eBook - ePubLinear Integer Programming
Theory, Applications, Recent Developments
- Elias Munapo, Santosh Kumar(Authors)
- 2021(Publication Date)
- De Gruyter(Publisher)
Chapter 8 The Travelling Salesman Problem: Sub-tour elimination approaches and algorithmsAbstract
This chapter presents a few new approaches to The Travelling Salesman Problem by generating sub-tour elimination cuts and adding these to a binary LP formulation of the TSP. The binary LP is converted as a convex quadratic problem which is solved efficiently by interior point algorithms. The other approach is to deal with the TSP network and convert that into transshipment sub-problems. These separate transshipment problems are then combined to come up with a master formulation for the TSP problem. The proposed formulation has the advantage that in most cases the relaxed LP gives an optimal integer solution.Keywords: Traveling salesman problem, Sub-tour elimination cuts, Formulation of the TSP as a binary LP, Convex quadratic problem, Interior point algorithm,Linear integer model,Totally unimodular and transshipment,8.1 Introduction
A travelling salesman problem (TSP) can be defined as a problem of visiting n centers in such a way that:- each center is visited only once,
- after visiting all centers, finally return to the original center (i.e. the starting center) and
- the total distance covered is the shortest of all the possible routes available.
The TSP model has so many applications and it is for this reason, it has remained as an active research area. Some of the areas of practical applications are vehicle routing, crystallography, circuit board drilling, order collection in warehouses etc.The traveling salesman problem (TSP) was believed to be a difficult problem in combinatorial optimization until recently, see Munapo (2020) . Researchers were unaware of any consistent and efficient general-purpose algorithm for this NP hard problem. The TSP has been extensively discussed by Finke et al. (1983) , Gavish and Graves (1978) , Lenstra and Rinnoonkan (1975 ), Miller (1960 ), Schrijver (1998) , Vajda (1961) , and Wong (1980) . Several variants of the TSP have originated from various real-life problems, see Applegate et al. (2007) , and Gutine and Punnen (2006 ). Large scale TSP has been studied by Bland and Shallcross (1989) . Special class of TSP have been studied by Claus (1984) , Fox et al. (1980) , Grostschel et al. (1991 ), Van Dal (1992) and Plante et al. (1987) . Munapo et al. (2016) developed a minimum spanning tree with node index ≤2. Kumar et al. obtained a TSP using the special minimum spanning tree, see Kumar et al. (2016 , 2017 , 2020
eBook - PDF- Bellman(Author)
- 1970(Publication Date)
- Academic Press(Publisher)
216 The ‘‘ Travelling Salesman” and Other Scheduling Problems this figure by a set of points and a set of associated distances (see Fig. 2 ) . Examining the map, we can write down the array of distances between these points (see Table 1). We suspect from our previous experience that this reduction to a set of points with an associated matrix will greatly facilitate our analytic formulation. Exercise 1. Fill in the missing distances. 4. Enumeration Let us now consider the problem of starting at 1 (home), visiting each of the numbered sites once and only once, and returning to 1, in such a way as to minimize the total distance travelled. As we have commented before, in many situations this not is at all the same as minimizing the total time required for the trip. The first approach to be considered is, of course, that of enumeration of all possible paths. Some simple calculations which we carry out below show that this is not a feasible approach as soon as we have even a moder- ate number of errands to run. As in the routing problem, the totality of possibilities increases i n an alarming fashion with the number of places to visit. We are, in effect, competing with N ! , and this is a very uneven contest, as we already know. Let us go over some of the figures. In the problem just described, there are I1 different places to visit, exclusive of home. Starting from home, we can go to any of 11 places first, any of 10 places next, any of 9 placesafter that, and so on. It follows that there are 11! = 11 X 10 X 9 X * * * X 2 = 39,916,800 ( 1 ) different paths. The proliferation of possible paths is quite remarkable. Numbers such as IOO!, IOOO!, associated in this fashion with relatively moderately sized maps are unimaginably large. We can agree then that as far as we can see into the future, regardless of the kind of computer availa- ble, combinatorial problems of any significance cannot be resolved by brute force.
eBook - PDF- Emile Aarts, Jan Karel Lenstra, Emile Aarts, Jan Karel Lenstra, Emile Aarts, Jan Lenstra(Authors)
- 2018(Publication Date)
- Princeton University Press(Publisher)
Given this, its expected length would be roughly 0.5 (or perhaps 0.25, if one is picking the best of two possible shortcuts). The number of shortcuts that need to be made is typically 230 8 The traveling salesman problem: a case study proportional to N, so this suggests the percentage excess for Christofides would itself grow linearly with N. 3 2-OPT, 3-OPT, AND THEIR VARIANTS In this section, we consider local improvement algorithms for the TSP based on simple tour modifications, exchange heuristics in the terminology of Chapter 1. Such an algorithm is specified in terms of a class of operations (exchanges or moves) that can be used to convert one tour into another. Given a feasible tour, the algorithm then repeatedly performs operations from the given class, so long as each reduces the length of the current tour, until a tour is reached for which no operation yields an improvement, a locally optimal tour. Alternatively, we can view this as a neighborhood search process, where each tour has an asso- ciated neighborhood of adjacent tours, i.e., those that can be reached in a single move, and one continually moves to a better neighbor until no better neighbors exist. Among simple local search algorithms, the most famous are 2-Opt and 3-Opt. The 2-Opt algorithm was first proposed by Croes [1958], although the basic move had already been suggested by Flood [1956]. This move deletes two edges, thus breaking the tour into two paths, then reconnects those paths in the other possible way (Figure 8.1). Note that this picture is a schematic; if distances were as shown in the figure, the particular 2-change depicted here would be counter- productive and so would not be performed. In 3-Opt [Bock, 1958b; Lin, 1965] the exchange replaces up to three edges of the current tour (Figure 8.2). In Section 3.1 we shall describe what is known theoretically about these algorithms in the worst and average cases. Section 3.2 then presents experimental results from Johnson et al.
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