Travelling Salesman Problem using Partical Swarm Optimization
- 1. Travelling Salesman Problem using Partical Swarm Optimization METAHEURISTICS ILGIN KAVAKLIOĞULLARI CSE - 273213005
- 2. What is The Travelling Salesman Problem? The travelling salesman problem (TSP) asks the following question: "Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the origin city?" It is an NP-hard problem in combinatorial optimization, important in operations research and theoretical computer science. TSP is a special case of the travelling purchaser problem and the vehicle routing problem. 2 ILGIN KAVAKLIOĞULLARI
- 3. What is The Travelling Salesman Problem? In the theory of computational complexity, the decision version of the TSP (where, given a length L, the task is to decide whether the graph has any tour shorter than L) belongs to the class of NP-complete problems. Thus, it is possible that the worst- case running time for any algorithm for the TSP increases superpolynomially (but no more than exponentially) with the number of cities. The problem was first formulated 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 algorithms are known, so that some instances with tens of thousands of cities can be solved completely and even problems with millions of cities can be approximated within a small fraction of 1%. 3 ILGIN KAVAKLIOĞULLARI
- 4. What is The Travelling Salesman Problem? 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. The TSP also appears in astronomy, as astronomers observing many sources will want to minimize the time spent moving the telescope between the sources. In many applications, additional constraints such as limited resources or time windows may be imposed. 4 ILGIN KAVAKLIOĞULLARI
- 5. What is The Travelling Salesman Problem? Solution of a travelling salesman problem: the black line shows the shortest possible loop that connects every red dot 5 ILGIN KAVAKLIOĞULLARI
- 6. History of TSP • 1920: Karl Menger introduced the concept to colleagues in Vienna • 1930: Intensive discussion in math community in Princeton University • 1940: Merrill Meeks Flood publicized TSP to mass • 1948: Flood presented TSP to RAND Corp. RAND is a non-profit organization that focuses in intellectual research and development within the US • 1950: Linear Programming was becoming a vital force in computing solutions to combinatorial optimization problems. The US Airforce needed the method to optimize solutions of their combinatorial transportation problem • 1960’s: The TSP could not be solved in polynomial time using Linear Programming techniques 6 ILGIN KAVAKLIOĞULLARI
- 7. History of TSP • 1970’s and 1980’s : Richard M. Karp showed in 1972 that the Hamiltonian cycle problem was NP-complete, which implies the NP-hardness of TSP. This supplied a mathematical explanation for the apparent computational difficulty of finding optimal tours. Great progress was made in the late 1970s and 1980, when Grötschel, Padberg, Rinaldi and others managed to exactly solve instances with up to 2392 cities, using cutting planes and branch-and-bound. • 1990’s : Applegate, Bixby, Chvátal, and Cook developed the program Concorde that has been used in many recent record solutions. Gerhard Reinelt published the TSPLIB in 1991, a collection of benchmark instances of varying difficulty, which has been used by many research groups for comparing results. In 2006, Cook and others computed an optimal tour through an 85,900-city instance given by a microchip layout problem, currently the largest solved TSPLIB instance. For many other instances with millions of cities, solutions can be found that are guaranteed to be within 2-3% of an optimal tour. • The problem is sometimes, especially in newer publications, referred to as Travelling Salesperson Problem. 7 ILGIN KAVAKLIOĞULLARI
- 8. Description Polynomial Time Nondeterministic- Polynomial Time NP-Complete NP-Hard Find the shortest possible route that visits each city exactly once and returns to the origin city TRAVELLING SALESMAN PROBLEM 8 ILGIN KAVAKLIOĞULLARI
- 9. P (Polynomial Time) P is the set of all decision problems which can be solved in polynomial time by a deterministic Turing machine. Since it can be solved in polynomial time, it can also be verified in polynomial time. 9 ILGIN KAVAKLIOĞULLARI
- 10. NP (Non-Deterministic Polynomial) NP is the set of all decision problems (question with yes-or-no answer) for which the 'yes'-answers can be verified in polynomial time (O(nk) where n is the problem size, and k is a constant) by a deterministic Turing machine. Polynomial time is sometimes used as the definition of fast or quickly • P is a subset of NP 10 ILGIN KAVAKLIOĞULLARI
- 11. NP-Complete A problem x that is in NP is also in NP-Complete if and only if every other problem in NP can be quickly transformed into x. In other words: • x is in NP, and • Every problem in NP is reducible to x So what makes NP-Complete so interesting is that if any one of the NP-Complete problems was to be solved quickly then all NP problems can be solved quickly. 11 ILGIN KAVAKLIOĞULLARI
- 12. NP-Hard NP-Hard are problems that are at least as hard as the hardest problems in NP. Note that NP-Complete problems are also NP-hard. However not all NP- hard problems are NP (or even a decision problem), despite having 'NP' as a prefix. That is the NP in NP-hard does not mean ‘non-deterministic polynomial time’. 12 ILGIN KAVAKLIOĞULLARI
- 13. P, NP, NP-Complete, and NP-Hard Correlation 13 ILGIN KAVAKLIOĞULLARI • TSP is NP-HARD. • U$ 1m • IF P = NP - It is solved. • Millenium Prize Problem and credits from scientists around the World. • TSP has never been solved.
- 14. Solving the TSP using PSO Particle Swarm Optimization (PSO) has a good potential for problem solving. The susceptibilities and charms of this nature based algorithm convinced researchers to use the PSO to solve NP-Hard problems such as TSP and Job-Scheduling. Here, we investigate some of these proposed approaches for solving the TSP. One of the attractive works for solving the TSP was cited in (Yuan et al.., 2007). They propose a novel hybrid algorithm which invokes the sufficiency of both PSO and COA (Chaotic Optimization Algorithm). In fact, they exert the COA to restrain the particles from getting stock on local optima’s in rudimentary iterations. In other word, they claim that the COA could considerably useful to keep particle’s global searching ability. 14 ILGIN KAVAKLIOĞULLARI
- 15. Solving the TSP using PSO One of the other exciting algorithms based on PSO for solving TSP is introduced in (Pang et al., 2004). In this paper they propose an algorithm based on PSO which uses the fuzzy matrices for velocity and position vectors. In addition, they use the fuzzy multiplication and addition operators for velocity and position updating formulas. The mentioned PSO algorithm in previous sections modified to an algorithm which works based on fuzzy means such as fuzzification and defuzzification. In each iteration, the position of each generated solution has been defuzzified to determine the cost of the individual. This cost will be used for updating the local best position. 15 ILGIN KAVAKLIOĞULLARI
- 16. Solving the TSP using PSO Some Equations for solving • Update velocity and position 16 ILGIN KAVAKLIOĞULLARI
- 17. Solving the TSP using PSO (a) Create a ‘population’ of agents (called particles) uniformly distributed over X (feasible region) and Evaluate each particle’s position according to the objective function, (b) Update particles’ velocities according to equation (1), (c) Move particles to their new positions according to equation (2), (d) If a particle’s current position is better than its previous best position, update it. 17 ILGIN KAVAKLIOĞULLARI
- 18. TSP Owerview-1 Find the shortest possible route that visits each city exactly once and returns to the origin city => Hamiltonian cycle Posed such computational complexity that any programmable efforts to solve such problems would grow super-polynomially with the problem size Can be used in : • Transportation: school bus routes, service calls, delivering meals • Manufacturing: an industrial robot that drills holes in printed circuit boards • VLSI (microchip) layout • Communication: planning new telecommunication networks 18 ILGIN KAVAKLIOĞULLARI
- 19. TSP Owerview-2 One way to solve TSP is to use exhaustive search to find all possible combinations of the next city to visit • However, the method is costly, since the number of possible tours of a map with n cities is (n − 1)! / 2 • 25 cities will require: 310,224,200,866,619,719,680,000 19 ILGIN KAVAKLIOĞULLARI
- 20. TSP Owerview-3 Vehicle Routing - Meet customers demands within given time windows using lorries of limited capacity . It is much more difficult than TSP. 8am-10am 2pm-3pm 3am-5am7am-8am10am-1pm 4pm-7pm Depot 6am-9am 6pm-7pm 20 ILGIN KAVAKLIOĞULLARI
- 21. TSP Owerview-4 Until this very day, an efficient solution to the general case TSP, or even to any of its NP-hard variations, has not been found. However, there are approximation solutions to solve the TSP: • Polynomial Time Approximation Scheme (PTAS) • Christofides Algorithm • Double MST Algorithm • Arora’s Algorithm • Mitchell’s Algorithm 21 ILGIN KAVAKLIOĞULLARI