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TSP: an introduction

TSP: an introduction. Vittorio Maniezzo University of Bologna. Travelling Salesman Problem (TSP). n locations given (home of salesperson and n -1 customers). [ d ij ] distance matrix given, contains distances (Km, time, … ) between each pair of locations.

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TSP: an introduction

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  1. TSP: an introduction Vittorio Maniezzo University of Bologna Vittorio Maniezzo - University of Bologna - Transportation Logistics

  2. Travelling Salesman Problem (TSP) • n locations given (home of salesperson and n-1 customers). • [dij] distance matrix given, contains distances (Km, time, … ) between each pair of locations. • The TSP asks for the shortest tour starting home, visiting each customer once and returning home • Symmetric / asymmetric versions. Vittorio Maniezzo - University of Bologna - Transportation Logistics

  3. Introduction • Which is the shortest tour? • Difficult to say … Vittorio Maniezzo - University of Bologna - Transportation Logistics

  4. TSP history • The general form of the TSP appears to have been first studied by mathematicians starting in the 1930s by Karl Menger in Vienna. • Mathematical problems related to the traveling salesman problem were treated in the 1800s by the Irish mathematician Sir William Rowan Hamilton. • The picture shows Hamilton's Icosian Game that requires players to complete tours through the 20 points using only the specified connections. (http://www.math.princeton.edu/tsp/index.html) Vittorio Maniezzo - University of Bologna - Transportation Logistics

  5. Past practice … Vittorio Maniezzo - University of Bologna - Transportation Logistics

  6. Current practice • Actual tours of persons or vehicles (deliveries to customers) • Given:Graph of streets, where some nodes represent customers. • Find: • shortest tour among customers. • Precondition: • Solve shortest path problems  Distance matrix Vittorio Maniezzo - University of Bologna - Transportation Logistics

  7. Application: PCB • Minimization of time for drilling holes in PCBs (printed circuit boards) Vittorio Maniezzo - University of Bologna - Transportation Logistics

  8. Application: setup costs • Minimization of total setup costs (or setup times) when setup costs are sequence dependent • All product types must be produced and the setup must return to initial configuration Vittorio Maniezzo - University of Bologna - Transportation Logistics

  9. Application: paint shop • Minimization of total setup costs (or setup times) when setup costs are sequence dependent • All colors must be processed Vittorio Maniezzo - University of Bologna - Transportation Logistics

  10. Solution Methods • n nodes (n-1)! possible tours • Some possible solution method: • Complete enumeration • IP: Branch and Bound (and Cut, and Price, … ) • Dynamic programming • Approximation algorithms • Heuristics • Metaheuristics n 5 10 15 20 n! 120 3,628.800 1,31*10^12 2,43*10^18 Vittorio Maniezzo - University of Bologna - Transportation Logistics

  11. i \ j • 1 • 2 • 3 • 4 • 5 • 1 •  • 18 • 17 • 20 • 21 • 2 • 18 •  • 21 • 16 • 18 • 3 • 16 • 98 •  • 19 • 20 • 4 • 17 • 16 • 20 •  • 17 • 5 • 18 • 18 • 19 • 16 •  Standard Example • Example with 5 customers (1 home, 4 customers) • Zimmermann, W.,O.R., Oldenbourg, 1990: • e.g. cost of 1-2-3-4-5: 18+21+19+17+18 = 93 • First: simplify by reducing costs Vittorio Maniezzo - University of Bologna - Transportation Logistics

  12. i \ j • 1 • 2 • 3 • 4 • 5 • 1 •  • 1 • 0 • 3 • 3 • 2 • 2 •  • 5 • 0 • 1 • 3 • 0 • 82 •  • 3 • 3 • 4 • 1 • 0 • 4 •  • 0 • 5 • 2 • 2 • 3 • 0 •  Lower bound • Subtract minimum from each row & column • Reduction constant = 17+16+16+16+16 + 1 = 82 • all tours have costs which are 82 greater, • e.g. cost of 1-2-3-4-5:1+5+3+0+2 = 11[= 93 - 82] • Optimal tour does not change (see [Toth, Vigo, 02]). Vittorio Maniezzo - University of Bologna - Transportation Logistics

  13. Complete Enumeration • 5 customers  4! = 234 = 24 tours Vittorio Maniezzo - University of Bologna - Transportation Logistics

  14. A wider scope • A combinatorial optimizationproblem is defined over a set C= {c1, … ,cn}of basic components. • A solution of the problem is a subset S C; • F 2C is thesubset of feasible solutions, (a solution S is feasible iff SF). • z: 2C is the cost function, • the objective isto find a minimum cost feasible solutionS°, i.e., to find S° F such that z(S°) z(S), S  F. • Failing this,the algorithm anyway returns the best feasible solution found, S* F. Vittorio Maniezzo - University of Bologna - Transportation Logistics

  15. TSP as a CO example • TSP is defined over a weighted undirected complete graph G=(V,E,D), where V is the set of vertices, E is the set of edges and D is the set of edge weights. • The component set corresponds to E (C=E), • F corresponds to the set of Hamiltonian cycles in G • z(S) is the sum of the weights associated to the edges belonging to the solution S. Vittorio Maniezzo - University of Bologna - Transportation Logistics

  16. LP-Formulation • LP formulation is a binary LP (IP) which is almost identical to the assignment problem: • plus conditions to avoid short cycles Vittorio Maniezzo - University of Bologna - Transportation Logistics

  17. 1 2 3 4 5 Example for short cycles • Optimal solution of LP without conditions to avoid short cycles = solution of assignment problem • short cycles 1-3-1 and 2-5-4-2 •  conditions to avoid short cycles are needed Vittorio Maniezzo - University of Bologna - Transportation Logistics

  18. Conditions to avoid short cycles • Several formulations in the literature • e.g. Dantzig-Fulkerson-Johnson • exponentially many! Vittorio Maniezzo - University of Bologna - Transportation Logistics

  19. Example: avoid short cycles • Above example with n = 5 cities: • must consider subsets Q: • {1, 2}; {1, 3}; {1, 4}; {1, 5}; {2, 3}; {2, 4}; {2, 4}; {3, 4}; {3, 5}; {4, 5} • constraint for subset Q = {1, 2} and V-Q = {3, 4, 5} x13 + x14 + x15 + x23 + x24 + x25  1 • n = 6  subsets (constraints) • exponentially many! Vittorio Maniezzo - University of Bologna - Transportation Logistics

  20. Heuristics for the TSP Vittorio Maniezzo University of Bologna Vittorio Maniezzo - University of Bologna - Transportation Logistics

  21. Computational issues • The size of the instances met in realworld applications rules out the possibility of solving them to optimality inan acceptable time. • Nevertheless, these instances must be solved. • Thus the needto look forsuboptimal solutions, provided that they are of “acceptablequality” and that they can be found in “acceptable time”. Vittorio Maniezzo - University of Bologna - Transportation Logistics

  22. Heuristic algorithms • How to cope with NP-completeness: • small instances; • polynomial specialcases; • approximationalgorithms guarantee to find asolution within a known gap from optimality; • probabilistic algorithms guarantee that for instancesbig enough, the probability of getting a bad solution is very small; • heuristic algorithms: no guarantee, but historically, on the average, these algorithms have the best quality/time trade off for the problem of interest. Vittorio Maniezzo - University of Bologna - Transportation Logistics

  23. heuristics: focus on solution structure • Simplest heuristics exploit structural properties of feasiblesolutions in order to quickly come to a good one. • Theybelong to one of two main classes: constructive heuristics or localsearch heuristics. Vittorio Maniezzo - University of Bologna - Transportation Logistics

  24. Constructive heuristics • 1. Sort the components in C by increasing costs. • 2. Set S*= and i=1. • 3. Repeat • If (S*  ci is a partial feasible solution) thenS* =S*ci. • i=i+1. • Until S* F. Vittorio Maniezzo - University of Bologna - Transportation Logistics

  25. Constructive heuristics • A constructive approach can yield optimal solutions for certainproblems, eg. the MST. • In other cases itcould be unable to construct a feasible solution. • TSP: order all edges byincreasing edge cost, take the least cost one and add to it increasing costedges, provided they do not close subtours, until a Hamiltonian circuit iscompleted. • More involved constructive strategies give rise to well-known TSPheuristics, like the Farthest Insertion, the Nearest Neighbor, the Best Insertion or the Sweepalgorithms. Vittorio Maniezzo - University of Bologna - Transportation Logistics

  26. Nearest Neighbor Heuristic • 0. Start from any city; e.g. the first one • 1. Find nearest neighbor (to the last city) not already visited • 2. repeat 1 until all cities are visited. • Then connect last city with starting point • Ties can be broken arbitrarily. • Starting from different starting points gives different solutions! Some will be bad some will be good. • GREEDYthe last connections tend to be long and improvement heuristics should be used afterwards Vittorio Maniezzo - University of Bologna - Transportation Logistics

  27. Best Insertion Heuristic • 0. Select 2 cities A, B and start with short cycle A-B-A • 1. Insert next city (not yet inserted) in the best position in the short cycle • 2. Repeat 1. until all cities are visited • Using different starting cycles and different choices of the next city different solutions are obtained • Often when symmetric: starting cycle = 2 cities with maximum distancenext city = maximizes min distance to inserted cities Vittorio Maniezzo - University of Bologna - Transportation Logistics

  28. An approximation algorithm • In some cases, it is possible to construct solutions which are guaranteed to be “not too bad”. • Approximation algorithms provide a guarantee on worst possible value for the ratio (zh – z*)/z*. • TSP hypothesis: it is always cheaper to go straight from a node u to another node v instead of traversing an intermediate node w (triangular inequality). Vittorio Maniezzo - University of Bologna - Transportation Logistics

  29. Approx-TSP • An approx. Algorithm for the TSP with triangular inequality is: • Approx-TSP-Tour(G,w) • 1 select a root vertex rV • 2 construct a MST T for G rooted in r • 3 let L be the list of vertices visited in a preorder tree walk of T • 4 return the hamiltonian cycle H which visits all vertices in the order L Vittorio Maniezzo - University of Bologna - Transportation Logistics

  30. Approx-TSP: example a d e b f g c h Vittorio Maniezzo - University of Bologna - Transportation Logistics

  31. Approx-TSP: example a d e b f g c h Vittorio Maniezzo - University of Bologna - Transportation Logistics

  32. Approx-TSP: example a d e b f g c h Vittorio Maniezzo - University of Bologna - Transportation Logistics

  33. Approx-TSP: example a d e b f g c h Vittorio Maniezzo - University of Bologna - Transportation Logistics

  34. Approx-TSP: performance • Theorem • Approx-TSP-Tour is an approximation algorithm with a ratio bound of 2 for the traveling salesman problem with triangle inequality. • Proof • Because the optimal tour solution must have one more edge than the MST of T, so c(MST)  c(T*). • c(Tour) = 2c(MST) • Because of the assumption of triangle inequality, c(H)  c(Tour). • Therefore, c(H)  2c(T*). • Since the input is a complete graph, so the implementation of Prim’s algorithm runs in O(V2). Therefore the total time complexity of • Approx-TSP-Tour also is O(V2). Vittorio Maniezzo - University of Bologna - Transportation Logistics

  35. Local Search: neighborhoods • The neighborhood of a solution S, N(S), is a subset of 2C defined by a neighborhood functionN: 2C 22c. • Often only feasible solutions considered, thus neighborhood functions N: F  2F. • The specific function used has a deep impact on the algorithm performance and its choice is left to the algorithm designer. Vittorio Maniezzo - University of Bologna - Transportation Logistics

  36. Local search • 1. Generate an initial feasible solution S. • 2. Find S'N(S), such that z(S')=min z(S^), S^ N(S). • 3. If z(S') < z(S) then S=S' • go to step 2. • 4. S* = S. • The update of a solution in step 3 is called a move from S to S'. • It could be made to the first improving solution found. Vittorio Maniezzo - University of Bologna - Transportation Logistics

  37. Local search • There are problems forwhich a local search approach guarantees to find an optimal solution (ex. the simplexalgorithm). • ForTSP, two LS are 2-opt and 3-opt, which take a solution (a list of nvertices) and exhaustively swap the elements of all pairs or triplets ofvertices in it. • More sophisticated neighborhoods give rise to moreeffective heuristics, among which Lin and Kernighan[LK73]. Vittorio Maniezzo - University of Bologna - Transportation Logistics

  38. 2 – opt (arcs): try allinversions of some subsequence if improvement  start again from beginning 3 – opt (arcs): try allshifts of some subsequence to different positions if improvement  start again from beginning Local search heuristics 3-opt (nodes): swap every triplet of nodes in the permutation representing the solution. if improvement  start again from beginning 2-opt (nodes): swap every pair of nodes in the permutation representing the solution. if improvement  start again from beginning Vittorio Maniezzo - University of Bologna - Transportation Logistics

  39. Metaheuristcs for the TSP(some of them) Vittorio Maniezzo University of Bologna, Italy Vittorio Maniezzo - University of Bologna - Transportation Logistics

  40. Metaheuristics: focus on heuristic guidance • Simple heuristics can perform very well, but can get trapped in poor quality solutions (local optima). • Newapproaches have been presented starting from the mid '70ies. • They are algorithms which manage the trade-offbetween search diversification, when search is goingon in bad regions of the search space, and intensification, aimed atfinding the best solutions within the region being analyzed. • These algorithms have been named metaheuristics. Vittorio Maniezzo - University of Bologna - Transportation Logistics

  41. Metaheuristics • Metaheuristics include: • Simulated Annealing • Tabu Search • GRASP • Genetic Algorithms • Variable Neighborhood Search • ACO • Particle Swarm Optimization (PSO) • … Vittorio Maniezzo - University of Bologna - Transportation Logistics

  42. Simulated Annealing • Simulated Annealing (SA) [AK89] modifies local search in order to accept, in probability, worsening moves. • 1. Generate an initial feasible solution S, • initialize S* = S and temperature parameter T. • 2. Generate S’N(S). • 3. If z(S') < z(S) then S=S', if(z(S*) > z(S)) S* = S • else accept to set S=S' with probability p = e-(z(S')-z(S))/kT. • 4. If (annealing condition) decrease T. • 5. If not(end condition) go to step 2. Vittorio Maniezzo - University of Bologna - Transportation Logistics

  43. SA example: ulysses 16 • Simulated annealing trace Vittorio Maniezzo - University of Bologna - Transportation Logistics

  44. Tabu search • Tabu Search (TS) [GL97] escapes from local minima by moving onto the best solution of theneighborhood at each iteration, eventhough it is worse than the current one. • A memory structure called tabu list, TL, forbids to return to already explored solutions. • 1. Generate an initial feasible solution S, • set S* = S and initialize TL=. • 2. Find S' N(S), such that • z(S')=min {z(S^), S^ N(S), S^ TL}. • 3. S=S', TL=TL {S}, if (z(S*) > z(S)) set S* = S. • 4. If not(end condition) go to step 2. Vittorio Maniezzo - University of Bologna - Transportation Logistics

  45. TS example: ulysses 16 • Tabu Search trace Vittorio Maniezzo - University of Bologna - Transportation Logistics

  46. GRASP • GRASP (Greedy Randomized Adaptive Search Procedure) [FR95] restarts search from another promising region of the searchspace as soon as a local optimum is reached. • GRASP consists in amultistart approachwith a suggestion on how toconstruct the initial solutions. • 1. Build a solution S (=S* in the first iteration) by a constructive greedy randomized procedure based on candidate lists. • 2. Apply a local search procedure starting from S and producing S'. • 3. If z(S')<z(S*) then S*=S'. • 4. If not(end condition) go to step 2. Vittorio Maniezzo - University of Bologna - Transportation Logistics

  47. Genetic algorithms • Genetic Algorithms (GA) [G89] do not rely on construction or localsearch but on the parallel update of a set R of solutions. • This is achievedby recombining subsets of 2elements in R, the parent sets, toobtain new solutions. • To be effective on CO problems they are often hybridized with SA, TSor local search. Vittorio Maniezzo - University of Bologna - Transportation Logistics

  48. Genetic algorithms • 1. Initialize a set R of solutions. • 2. Construct a set Rc of solutions by recombining pairs of randomly chosen elements in R. • 3. Construct a set Rmof solutions by randomly modifying elements in Rc. • 4. Construct the new set R by extracting elements from Rm, following a montecarlo strategy with repetitions. • 5. If not(end condition) go to step 2. Vittorio Maniezzo - University of Bologna - Transportation Logistics

  49. Components of a GA • A problem to solve, and ... • Encoding technique (gene, chromosome) • Initialization procedure (creation) • Evaluation function (fitness) • Selection of parents (reproduction) • Genetic operators (mutation, recombination) • Parameter settings (practice and art) Vittorio Maniezzo - University of Bologna - Transportation Logistics

  50. GA jargon • Procedure GA • { initialize population; • evaluate population; • while not(End_condition) • { selectparents for reproduction; • perform recombination and mutation; • evaluate population; • } • } Vittorio Maniezzo - University of Bologna - Transportation Logistics

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