1. Escaping local optimas Accept nonimproving neighbors – Tabu search and simulated annealing Iterating with different initial solutions – Multistart local search, greedy randomized adaptive search procedure (GRASP), iterative local search (ILS) Changing the neighborhood – Variable neighborhood search Changing the objective function or the input to the problem in a effort to solve the original problem more effectively. – Guided local search 1

2. Iterating with different initial solutions Multi-start local search – Several initial solutions – Obtain several local optimas using local search, tabu or SA ILS- Iterative local search – Improves the classical Multi-start local search – It does by perturbing the local optimas from the Multi-start local search and reconsidering them as initial solutions – Perturbation Perturbation is a large random move of the current solution Keep some part of the current solution and significantly change the other parts of the current solution. Complete change must be avoided because the perturbed solution loses the properties of the local optima (current solution) that was derived from the most recent search history. Such an extreme change is a random restart approach. The length of the perturbation can be fixed or varied during the search. – Acceptance Criteria: Deterministic simple conditions such as solution is better than previous best (tabu) or probabilistic (SA). 2

3. GRASP- greedy randomized adaptive search procedure A randomized greedy heuristic generates initial solutions – Adaptive because the greedy heuristic takes into account the precedent solutions Using the initial solution perform a local search Repeat the above steps several times (iterations) Greedy solution examples – TSP- selection of the nearest neighbor – Knapsack- Choosing objects that minimize w i /b i where w i is the weight and b i is the benefit – Minimum spanning tree- choose least costly edges 3

4. GRASP for capacitated minimum spanning tree Minimum cost spanning tree There is a root (central node) r Initially all nodes are connected to the root (called gates) in a star arrangement and every node is a sub-tree. Value G j is the cost of links to the root Objective- find the minimum cost spanning tree such that every sub-tree has a certain capacity c where c could be the max number of nodes in a sub-tree or is a range (min and max value) – If every node has a weight, then c could be the maximum weight or a range for the weight Exhaustive enumeration would be to find all possible spanning trees and select the best, which will take exponential time Application: networking of computers, communication between cell phone towers. Solution is to apply GRASP 4

5. GRASP for capacitated minimum spanning tree Start with different partial initial solutions and expand the tree Create a Restricted Candidate List RCL with edges that meet the capacity constraints. Randomly select from these edges to add to the sub-tree at each iteration – For multiple edges from node i to all j’s, (i,j) in RCL, find t i = G j -c ij and choose the edge with the largest t i – Add the edge only if the capacity constraint is met – If edge c ij is added then delete gate G j (link r-j) from the list of gates. – Continue until no further pair of nodes with positive t i or not exceeding capacity constraint can be added to the sub-tree 5

6. GRASP for capacitated minimum spanning tree Max Capacity = 5 6 r 4 3 2 1 2 3 1 4 1 2 2 14 5 3 3 3 1 RCL 1-2, 2-3, 1-3, 3-4 Capacity ≤ 5 2-4, 1-4 exceeds capacity Capacity 2-4 = 7 Capacity 1-4 = 6

7. 7 GRASP for capacitated minimum spanning tree 130x 2-11x 3-233 4 xx2 12341234 RCL 1-2, 2-3, 1-3, 3-4 Capacity ≤ 5 2-4, 1-4 exceeds capacity Capacity 2-4 = 7 Capacity 1-4 = 6 For multiple edges from node i to all j’s, (i,j) in RCL, find t i = G j -c ij Choose the edge with the largest positive t i (in red) Node i Node j t i = G j -c ij

8. Connect 1-2 remove r-2 link Connect 3-4 remove r-4 link Connect 3-2 remove r-2 link All nodes are connected. Choose the smallest gate among the leftover gates. Choose r-1 Total cost = r-1-2-3-4 = 6 In a star type connection total cost is (r-1)+(r-2)+(r-3)+(r-4) = 13 A regular spanning tree without capacity would be r-1-4-3-2 =5 8 GRASP for capacitated minimum spanning tree

9. Variable Neighborhood Search VNS exploits the second idea: change neighborhood structure. VNS uses different neighborhood structures during search A neighborhood is substituted by another neighborhood as soon as local search can not improve the current best solution. 9

10. Guided Local Search 10

11. Dispatching Rules See page 442 of text. The rules are useful to obtain initial solutions 11