ปัญหาการเดินทางของพนักงานขาย Traveling Salesman Problem (TSP)

1. ปัญหาการเดินทางของพนักงานขายปัญหาการเดินทางของพนักงานขาย Traveling Salesman Problem (TSP)

2. A photograph of Hamilton's Icosian Game that requires players to complete tours through the 20 points using only the specified connections. History of the TSP • Mathematical problems related to TSP were treated in the 1800s by the Irish mathematician Sir William Rowan Hamilton and by the British mathematician Thomas Penyngton Kirkman. • In the 1930s, the general form of the TSP first studied by Karl Menger in Vienna and Harvard. http://www.tsp.gatech.edu//

3. TSP is closely related to the Hamiltonian-cycle problem. Problem Statement • In TSP, a salesman must visit n cities. • The salesman wishes to make a tour or Hamiltonian cycle. • He must visit each city exactly once and finish at the city he starts from. • There is a cost c(i, j) or to travel from city i to city j. • For the symmetric TSP, c(i, j) = c(j, i). • For the asymmetric TSP, c(i, j) ≠c(j, i). • Euclidean TSP (triangle inequality)

4. The salesman wishes to make the tourwith minimum total cost. • For transportation problems, a vehicle has unlimited capacity. • TSP is NP-complete. • n!/(2n) possible tours (symmetric). • The problem can be modeled as a complete graph with n vertices. • A graph consists of vertices (nodes) and arcs (edges). 4 2 3 1 3 2 1 1 4 5

6. Mathematical Formulation as Integer Programming Subject to:

7. LINGO Program for TSP

8. Solution Approaches 1. Heuristic Approaches • Good enough solution. • Less computational time. • Usually followed by local search methods (r-opt exchange heuristic). • Nearest Neighbor Heuristic, Insertion Heuristic, etc. 2. Meta-heuristics • More efficient than heuristics. •Genetic Algorithm, GRASP, Ant Colony, etc. 3. Exact Algorithms • Optimal solution • Require more computational time. • Branch and Bound, Branch and Cut, etc.

9. Depot Customer 2 7 1 5 4 10 9 6 8 3 Traveling Salesman Problem-TSP 0 Assumptions 1. Undirected and symmetric arcs. 2. All possible arcs between nodes exist. 3. Each node is visited exactly once. 4. For transportation problems, a vehicle has unlimited capacity.

10. Select node j that is nearest node i and not in a subtour. Add node j to the tour (connected to node i) and set i = j. Yes Is there aNode not in a tour? No End TSP Heuristic Approaches Basic Nearest Neighbor Heuristic Randomly select an initial node i as a partial tour. j i 2 1 j i 3 0 8 4 5 7 6

11. Compare the total distance (cost) of all tourswith a different starting point. Choose the tour with smallest total distance as the solution. TSP Modified Nearest Neighbor Heuristic For every node i as the starting point of the tour, construct the tour using the basic nearest neighbor heuristic. 50 52 i 2 1 53 55 3 54 0 8 53 4 55 5 7 51 6 56

12. Randomly select a node not in the tour. Insert that node in the edge that incurs smallest increasing distance (cost). Yes Is there aNode not in a tour? No End TSP Arbitrary Insertion Heuristic Select a starting tour with k nodes (k ≥ 1). 2 1 3 0 8 4 5 7 6

13. Nearest Insertion Heuristic 1. Start with a subgraph consisting of city i only. 2. Find city k such that is minimal and form the subtour (i, k). 3. Find city k not in the subtour and city l in the current subtour such that , where j denotes a city not in the current subtour and i denotes a city in the current subtour. 4. Find the edge { i, j } in the subtour which minimizes . Insert k between i and j . 5.Go to step 3 unless we have a Hamiltonian cycle.

14. Farthest Insertion Heuristic 1. Start with a subgraph consisting of city i only. 2. Find city k such that is maximal and form the subtour (i, k). 3. Find city k not in the subtour and city l in the current subtour such that , where j denotes a city not in the current subtour and i denotes a city in the current subtour. 4. Find the edge { i, j } in the subtour which minimizes . Insert k between i and j . 5.Go to step 3 unless we have a Hamiltonian cycle.

15. Convex Hull Insertion Heuristic 1. Form the convex hull of the set of cities. The hull gives an initial subtour. 2. For each city k not yet contained in the subtour, decide between which two cities i and j on the subtour to insert city k such that is minimal. 3. From all (i, k, j) found in step 2, determine the (i*, k*, j*) such that is minimal. 4. Insert city k* in subtour between cities i* and j*. 5. Repeat step 2 through 4 until a Hamiltonian cycle is obtained.

16. Minimum Spanning Tree Based Algorithm 1. Construct an MST of the graph corresponding to an instance of TSP. 2. Starting at an arbitrary vertex, perform a depth-first traversal of the MST and add each vertex as visited to a list. 3. Iterate through the list of vertices, marking each vertex encountered as visited. When a previously visited vertex is encountered, remove this vertex from the list unless it is the starting vertex.

17. MST Algorithm 1. Begin with any node i and join node i to node j in the network that is closest to node i . Now arc (i , j ) is the spanning tree. 2. Choose a node in the network (not in the spanning tree) that is closet to a node in the spanning tree. 3. Connect the node chosen in step 2 to the spanning tree (to its closet node in the spanning tree). 4. Repeat steps 1 – 3 until a minimum spanning tree is found.

19. Neighborhood Search 2- Opt 2 2 3 3 4 4 1 1 5 5 Depot Depot 3- Opt 2 2 3 3 4 4 1 1 5 5 Depot Depot

21. Problem • A beer distributor has received orders from seven customers for delivery the next day. The number of cases required by each customer and travel times between each pair of customer are as follows Customer 1 2 3 4 5 6 7 Cases 46 55 33 30 24 75 30 Assume a delivery truck has unlimited capacity. Construct a vehicle route using any heuristics.

22. Greedy Randomized Adaptive Search Procedure(GRASP) • GRASP is an iterative process. • In each iteration, a solution is obtained. • Consist of two phases: a construction phase. a local search phase. • The best overall solution is kept as a result. • The process is terminated when some termination criterion is met.

23. Procedure grasp( ) 1. InputInstance ( ) ; 2. for GRASP stopping criterion not satisfied 3. ConstructGreedyRandomizedSolution (Solution) ; 4. LocalSearch (Solution) ; • UpdateSolution (Solution, BestSolutionFound) ; 6. rof ; 7. return (BestSolutionFound) end grasp ;

24. Construction Phase • A feasible solution is iteratively constructed, one element at a time. • In each construction iteration, a candidate list of elements is created by ordering the elements with respect to a greedy function. • One of the best candidates in the list is randomly chosen. • The benefits associated with each element are updated at each construction iteration. • The list of the best candidates is called the restricted candidate list (RCL). • The solutions obtained in the construction phase are not guarantee to be locally optimal.

25. ProcedureConstructGreedyRandomizedSolution (Solution) 1. Solution = { }; 2. for Solution construction not done 3. MakeRCL (RCL) ; 4. s = SelectElementAtRandom (RCL) ; • Solution = Solution {s} ; 6. AdaptGreedyFunction (s) ; 7. rof ; endConstructGreedyRandomizedSolution ;

26. Local Search Phase • Each constructed solution is improved by applying a local search. • In the local search algorithm, the current solution is successively replaced by a better solution in the neighborhood of the current solution. • The local search is terminated when no improved solution is found in the neighborhood.

27. Procedurelocal(P, N(P), s) 1. for s not locally optimal 2. Find a better solution t N(s) ; 3. Let s = t ; 4. rof ; 5. return ( s as local optimal for P ) end local ;