Outline ► Introduction to ► BRKGA Metaheuristics Metaheuristics methods ► Hybrid Metaheuristics ► Local Search ► Matheuristics ► GRASP ► SimHeuristics ► Tabu Search ► Learnheuristics Helena Ramalhinho ► Iterated Local Search Metaheuristics 1 Metaheuristics 2 Solution Methods Solution Methods ► Branch-and-bound Local Search Methods ► Branch-and-cut Metaheuristics ► Column generation ► Cutting and price ► Dynamic programming Integer Programming ► Lagrangean relaxation Exact Methods Integer Programming ► Linear relaxation Exact Methods ► Surrogate relaxation ► Etc… Metaheuristics 3 Metaheuristics 4 Solution Methods Solution Methods ► Good solutions for ► Local Improvement Local Search Methods complex and large- Local Search Methods ► Tabu Search Metaheuristics scale problems Metaheuristics ► Iterated Local Search ► Short running times ► Simulated annealing ► Easily adapted • Mathematical proved ► Genetic Algorithms optimal solutions ► Evolutionary algorithms • Important information on the characteristics and Integer Programming ► Ant Colony Optimization properties of the Exact Methods ► Scatter Search problem. ► Memetic Algorithms ► Etc…. Metaheuristics 5 Metaheuristics 6 © Helena Ramalhinho-Lourenço, (2019) 1 Algorithms Algorithms ► Heuristics ► Metaheuristics § Any approximate method build up on the basis of the structural § The process of finding a good solution (eventually the optimum) properties or on the characteristics of the problem solution, with consists in applying at each step (guiding) a subordinate heuristic reduced complexity with respect to exact methods and providing, which has to be design for each particular problem. in general, good feasible quality solutions, without a forma * C. Ribeiro [1996] guarantee of solution quality. § Cost vs. Time § Have been designed to attack complex optimization problem where classical heuristics and optimization have failed (so far) to be effective and efficient. Metaheuristics 7 Metaheuristics 8 Metaheuristics Metaheuristics for real problems ► Four attributes of Heuristics and Metaheuristics: ► Metaheuristics ► Real Problems § Accuracy § Highly effective on hard § Complex problems * Close to optimal problems § Rapid changes in reality § Speed § Modularity § Need to quick * Small computational time § Easy implementation implementation § Simplicity § Short updates § Different aspects in * No parameters adjustment / easy to program § Robust different sectors § Flexibility § Able to give good § Need to quick answer * Easy to adapt to other real problems solutions in short time. and multiple scenarios Cordeau JF, Gendreau M, Laporte G, Potvin JY, Semet F (2002) A guide to vehicle routing heuristics. J Oper Res Soc 53:512–522. Metaheuristics 9 Metaheuristics 10 Local optimization algorithms Local optimization algorithms ► Given a solution, attempts to improve this solution by making ► Local search local modifications at each iteration. § 1. Get a initial solution x (current solution). Use a constructive ► Neighborhood heuristic. § N: A® 2A subset of feasible solutions § 2. Search the neighborhood. While there is an untested neighbor § Define a local modification, move; * the neighborhood of a solution x is the subset of feasible solutions of x: obtained from applying this move to x. * 2.1 Let x’ be an untested neighbor of x; § Local optima solution x: * 2.2 If c(x’)<c (x) set x=x’; (x’ is the new current solution) * c(x) £ c(y) for all y Î N(x) § 3. Return x (local optimal solution). ► Search Strategy § the name of the metaheuristic come usually from the type of search. Metaheuristics 11 Metaheuristics 12 © Helena Ramalhinho-Lourenço, (2019) 2 Local optimization algorithms Traveling Salesman Problem ► Design of a local optimization algorithm: ► Traveling Salesman Problem § Obtain an initial solution § Traveling Salesman Problem * Heuristic * E: set of edges, each has a cost c(e); * Random solution * A: any subset of edges forming a Hamilton cycle; * Constructive method * c(x): total cost of the edges in x. § Define the neighborhood 5 * Specific for each problem 8 § How to search the neighborhood 10 10 * Complete search 12 18 * First improvement 5 14 6 Metaheuristics 13 Metaheuristics 14 Vehicle Routing Nearest neighbor algorithm ► Traveling salesman problem – initial solution ► Nearest neighbor algorithm on TSP § Heuristic of the nearest neighbor 1. stand on an arbitrary vertex as current vertex. 2. find out the shortest edge connecting current vertex and an 0 1 12 unvisited vertex V. 3. set current vertex be V. 8 4. mark V as visited. 4 4 20 5. if all the vertices in domain are visited then terminate. Else go to 8 8 step 2. 18 5 10 6 Metaheuristics 15 Metaheuristics 16 2 3 Vehicle Routing Vehicle Routing ► Heuristic of the nearest neighbor ► Heuristic of the nearest neighbor 0 0 1 12 1 12 8 8 4 4 4 4 20 20 8 8 8 8 18 18 5 5 10 10 6 6 2 3 2 3 Metaheuristics 17 Metaheuristics 18 © Helena Ramalhinho-Lourenço, (2019) 3 Vehicle Routing Vehicle Routing ► Heuristic of the nearest neighbor ► Heuristic of the nearest neighbor 0 1 0 1 12 12 8 8 4 4 4 4 20 20 8 8 8 8 18 18 5 5 10 10 6 6 2 3 2 3 Metaheuristics 19 Metaheuristics 20 Vehicle Routing Vehicle Routing ► Heuristic of the nearest neighbor ► Optimal solution 0 1 0 1 12 12 8 8 4 4 4 4 20 20 8 8 8 8 18 18 5 5 10 10 6 6 2 3 2 3 Total distance = 43 Total distance = 35 Metaheuristics 21 Metaheuristics 22 Neighborhood Neighborhood ► Traveling Salesman Problem: ► Traveling Salesman Problem: § 2-Opt Move § 3-Opt Move Metaheuristics 23 Metaheuristics 24 © Helena Ramalhinho-Lourenço, (2019) 4 Local Search Example TSP – local search application ► Examples… ► Initial solution and current § Construction heuristics solution § 2-opt local search method ► Nearest neighbor heuristic ► http://www-e.uni-magdeburg.de/mertens/TSP/TSP.html The closest vertex to Berlin is Paris The closest vertex to Paris is Londres The closest vertex to Londres is Sevilla The closest vertex to Sevilla is Roma The closest vertex to Roma is Moscow The nearest to the initial vertex is Moscow SOLUTION: Nearest-Neighbor ------------------------- Route: Berlin, Paris, Londres, Sevilla, Roma, Moscow, Berlin Length: 65 Metaheuristics 25 Metaheuristics 26 Example TSP – local search application Example TSP – local search application ► 2-opt neighborhood ► 2-opt neighborhood § 9 neighbors § 9 neighbors § Neighbor 1 § Neighbor 2 § Cost: 69 § Cost: 69 Metaheuristics 27 Metaheuristics 28 Example TSP – local search application Example TSP – local search application ► 2-opt neighborhood ► 2-opt neighborhood § 9 neighbors § 9 neighbors § Neighbor 3 § Neighbor 4 § Cost: 78 § Cost: 67 Metaheuristics 29 Metaheuristics 30 © Helena Ramalhinho-Lourenço, (2019) 5 Example TSP – local search application Example TSP – local search application ► 2-opt neighborhood ► 2-opt neighborhood § 9 neighbors § 9 neighbors § Neighbor 5 § Neighbor 6 § Cost: 63 § Cost: 73 Metaheuristics 31 Metaheuristics 32 Example TSP – local search application Example TSP – local search application ► 2-opt neighborhood ► 2-opt neighborhood § 9 neighbors § 9 neighbors § Neighbor 7 § Neighbor 8 § Cost: 76 § Cost: 78 Metaheuristics 33 Metaheuristics 34 Example TSP – local search application Example TSP – local search application ► 2-opt neighborhood ► Best neighbor (5) § 9 neighbors ► Cost 63 § Neighbor 9 ► New current solution § Cost: 69 § Repeat until find a local optimal § Local optimal solution: The current solution is better than any solution in the neighborhood. Metaheuristics 35 Metaheuristics 36 © Helena Ramalhinho-Lourenço, (2019) 6 Facility Location Models Location example – Local search ► Important problem in logistics § Where to locate new facilities. * Retailers, warehouses, factories. § Very complex problems ► Warehouse location problem § to locate a set of warehouses in a distribution network ► Cost of locating a warehouse at a particular site: § fixed cost vs variable cost * cost of open facility vs. transportation cost Example of M. Resende AT&T Labs Research Metaheuristics 37 Metaheuristics 38 Location example – Local search Location example – Local search Example of M. Resende Example of M. Resende AT&T Labs Research AT&T Labs Research Metaheuristics 39 Metaheuristics 40 Location example – Local search Location example – Local search Example of M. Resende Example of M. Resende AT&T Labs Research AT&T Labs Research Metaheuristics 41 Metaheuristics 42 © Helena Ramalhinho-Lourenço, (2019) 7 Location example – Local search Location example – Local search Example of M. Resende Example of M. Resende AT&T Labs Research AT&T Labs Research Metaheuristics 43 Metaheuristics 44 Location example – Local search Location example – Local search Example of M. Resende Example of M. Resende AT&T Labs Research AT&T Labs Research Metaheuristics 45 Metaheuristics 46 Location example – Local search Vehicle Routing ► A set of customers at known geographical locations has to be supplied by a fleet of vehicles from a single depot. ► Each customer has a specific demand. ► Each route starts and finish at the depot. ► The objective is to find the set of routes whose total length or cost is minimal. § Or minimize the number of routes. Example of M. Resende AT&T Labs Research Metaheuristics 47 Metaheuristics 48 © Helena Ramalhinho-Lourenço, (2019) 8 Vehicle Routing The Job-Shop Scheduling ► Careful with the Local search ►A set of n jobs, J1, ... , Jn vehicle capacity! ► Moves: § 2-opt like in TSP. ►A set of m machines, M1, ... , Mm § Interchange customers in routes. ►Schedule the processing of each job by the machines. ►The objective