2Classical Vehicle Routing n customers must be served from a single depot utilizing vehicle with capacity Q for delivering goodsEach customer requires a quantity qi ≤ Q of goodsCustomer orders cannot be split
3Additional Features Depots Multiple locations Vehicles Multiple vehicle types and capacitiesRelease, maximum and down timesCustomersTime windows (soft or hard)Accessibility restrictionsPriorityPickup and deliveryRoutesMaximum timeLink costsObjective FunctionsMinimize total traveled distanceMinimize total traveled timeMinimize number of vehiclesMaximize quality of serviceMultiple objective functions
4How Can It Be Solved ??? Heuristics that Grow Fragments Nearest neighborDouble-ended nearest neighborMultiple fragment heuristicHeuristics that Grow ToursNearest additionFarthest additionRandom additionHeuristics Based on TreesMinimum spanning treeChristofides heuristicFast recursive partitioningNearest neighbor Randomly select a starting node2. Add to the last node the closest node until no more nodes are available3. Connect the last node with the first node4. O(n2) running timeDouble-ended nearest neighbor Conceptually the same as nearest neighbor heuristic2. The fragment is allowed to grow from both endsMultiple Fragment Heuristic - This heuristic considers the edges of the graph in increasing order of length and it adds any edge that will not make it impossible to complete a tour (i.e., it avoids cycles and vertices of degree three)Minimum Spanning Tree Construct a minimum spanning tree2. Traverse the tree to build a tour by eliminating edges from vertices with degree three and adding edges to vertices with degree oneClarke and Wright Heuristic – 1. Start with an initial solution where each customer is servicedindividually from the depotAND MANY MORE
5Ant Colony Optimization (ACO) OUR CHOICE OF ALGORITHEMAnt Colony Optimization(ACO)
6ACO Concepts Ants (blind) navigate from nest to food source Shortest path is discovered via pheromone trailseach ant moves at randompheromone is deposited on pathants detect lead ant’s path, inclined to followmore pheromone on path increases probability of path being followedAnt Colony Optimisation has been inspired by the foraging behaviour of real ants. Antsrandomly explore the surroundings of the anthill; when they find food, they return to the nestdepositing a pheromone trail, a trace of a chemical substance that can be smelled by otherants. Ants can follow various paths to the food source and back, but it has been observedthat, thanks to the reinforcement of the pheromone trail by successive passages, only theshortest path remains in use, since ants prefer to follow stronger pheromone concentrations.Pheromone reinforcement is autocatalytic, since the shortest the path, the least time will betaken to travel back and forth, and therefore, while ants on longer paths are still in transit, theants on the shortest path can restart the route again, reinforcing the pheromone trail on theshortest path. Over time, the majority of the ants will travel on that path, while a minoritywill still choose alternative paths. The behaviour of this minority is important, since it allowsto explore the environment to find even better solutions, which initially were not considered.The choice of the path is therefore probabilistic and, while it is strongly influenced by thepheromone intensity, it still allows for random deviations from the current best solution.
8ACO System Virtual “trail” accumulated on path segments Starting node selected at randomPath selected at randombased on amount of “trail” present on possible paths from starting nodehigher probability for paths with more “trail”Ant reaches next node, selects next pathContinues until reaches starting nodeFinished “tour” is a solution
9ACO System, cont. A completed tour is analyzed for optimality “Trail” amount adjusted to favor better solutionsbetter solutions receive more trailworse solutions receive less trailhigher probability of ant selecting path that is part of a better-performing tourNew cycle is performedRepeated until most ants select the same tour on every cycle (convergence to solution)
11The AlgorithmAt the beginning of the search process, a constant amount of pheromone is assigned to all arcs. When located at a node i an ant k uses the pheromone trail to compute the probability of choosing j as the next node:α - is a weight function based on arc cost etc..β – is a weight function base on arc lengthiWhen all ants have comleted a tour each ant compute the quantity per unit of length , the pheromone value changes as follows:By using this rule, the probability increases that forthcoming ants will use this arc.Ant Colony Algorithms are typically use to solve minimum cost problemsWe may usually have N nodes and A undirected arcsThere are two working modes for the ants: either forwards or backwards.Pheromones are only deposited in backward mode.The ants evaluate the cost of the paths they have traversed.The shorter paths will receive a greater deposit of pheromones. An evaporation rule will be tied with the pheromones, which will reduce the chance for poor quality solutions.at the end of each iteration, global update uses only the best solution, computed so far, to update the pheromone trail. The only edges that are modified are those edges i–j belonging to the best solution.