Ta-Chun Lien

Reference

Objective Introduce principle of Ant System(AS) Inspiration for your research Algorithm term project

Agenda Inspiration Biological Background Adapt to Computer System Ant System Travel Salesman Problem Model Definition Ant Cycle Algorithm Parameter Setting Basic Parameter setting from Model definition Synergistic Effects

Agenda(cont’d) Properties Initialization Elitist Strategy Problem Size Experiment Travel Salesman Problem Asymmetric Travel Salesman Problem Conclusions and Extensions Contribution of this Paper Extension

Inspiration Biological Background Adapt to Computer System

Biological background One of the problems studied by ethologists was to understand how almost blind animals like ants could manage to establish shortest route paths from their colony to feeding sources and back The Answer is Pheromone Trails

What is Pheromone(from wiki) A pheromone (from Greek φέρω phero "to bear" + ‘ορμόνη “hormone") is a chemical that triggers a natural behavioral response in another member of the same species. There are alarm pheromones, food trail pheromones, sex pheromones, and many others that affect behavior or physiology. Their use among insects has been particularly well documented, although many vertebrates and plants also communicate using pheromones.

Trail Pheromones(from wiki) Trail pheromones are common in social insects. For example, ants mark their paths with these pheromones, which are non-volatile hydrocarbons. Certain ants lay down an initial trail of pheromones as they return to the nest with food. This trail attracts other ants and serves as a guide. As long as the food source remains, the pheromone trail will be continually renewed. The pheromone must be continually renewed because it evaporates quickly. When the supply begins to dwindle, the trailmaking ceases. In at least one species of ant, trails that no longer lead to food are also marked with a repellent pheromone.

Biological background(cont’d)

Adapt to computer system Artificial Ants will have some Memory They will not be completely Blind They will live an environment where time is discrete

Adapt to computer system(cont’d)

Ant System TSP problem (AS is designed to solve TSP Problem originally) Model Definition Ant Cycle Algorithm

Travel Salesman Problem Given a graph G=(V,E), where: V:set of Cities E:set of edges between Cities and :Euclidean distance between i and j The problem is finding a minimal length closed tour that visits each town once

Travel Salesman Problem(cont’d) TSP NPC, you can reduce HC to NPC For this problem, each simple agent have the following characteristics: (1)it choose the town to go with a function of the town distance and amount of trail (Pheromone Trails) (2)used a tabu list to make a legal tours (3)lays a substance called trail on each edge(i,j) visited, when it completes a tour (4)Each agent at time t chooses the next town, where it will be at time t+1

Model Definition iteration and cycle tabu list Trail intensity update formula Transition probability formula(decide where to go)

Iteration and Cycle (t)(i=1,…,n) be the number of ants in town i at time t m=, total number of ants an iteration of the AS algorithm the m moves carried out by the m ants in the interval (t,t+1), and every n iteration of the algorithm called a cycle(completed a tour)

Tabu list associate with each ant a data structure called the tabu list, that saves the town already visited and forbids the ant to visit them again before a cycle completed. when a cycle is completed, the tabu list is used to compute the ant’s travel distance

Tabu Search(from wiki) Tabu search is a mathematical optimization method, belonging to the class of local search techniques. Tabu search enhances the performance of a local search method by using memory structures: once a potential solution has been determined, it is marked as "taboo" ("tabu" being a different spelling of the same word) so that the algorithm does not visit that possibility repeatedly. Tabu search is attributed to Fred Glover.

Trail intensity update formula :intensity of trail on edge (i,j) at time t Trail intensity update formula: ρ: is a coefficient such that(1 - ρ)represents the evaporation of trail between time t and t+n(ρ<1) is the quantity per unit of length of trail substance laid on edge (i,j) by the k-th ant between t and t+n

Trail intensity update formula(cont’d) definition: Q is a constant(user define) and is the tour length of the k-th ant.

Model Definition(5) visibility : transition probability from i to town j for the k-th ant: ={V- }

Ant Cycle Algorithm(1) 1.Initialization: set t:=0 {t is time counter} set NC:=0 {NC is the cycles counter} for every edge(i,j) set an initial value (t)=c for trail intensity and =0 place the m ants on the n nodes 2.set s:=1 {s is the tabu list index} For k:=1 to m do place the starting town of the k-th ant in

Ant Cycle Algorithm(2) 3.repeat until tabu list is full {this step will be repeated n-1 times} set s:=s+1 for k:=1 to m do choose the town j to move to, with probability given by Transition probability formula {at time t the k-th ant is on town i= } Move the k-th ant to the town j insert town j in

Ant Cycle Algorithm(3) 4.for k:=1 to m do move the k-th ant from to compute the length of k-th ant’s tabu list update the shortest tour found for every edge(i,j) for k:=1 to m do

Ant Cycle Algorithm(4) 5.For every edge (i,j) compute according to equation set t:=t+n set NC:=NC+1 For every edge (i,j) set 6.if(NC< ) and (not stagnation behavior) then empty all tabu list goto step2 else print shortest tour and STOP

A Simplified Flowchart initial starting travel estimate set trail terminal? Set counter and termination condition Reset tabu list travel by probability equation evaluate tour length by tabu list Estimate trail intensity by intensity formula

Time Complexity StepComplexity Step1 Step2 Step3 Step4 Step5 Step6 Time Complexity:

Take a break Okunohosomichi, Tohoku,Japan

What’s the difference between AS and SGA in TSP AS is more intuitive for this problem SGA can solve this problem well, but you should overcome following difficulties: (1)how to encode this problem? or DCABE? (2)how to learn linkage? (3)how to do crossover and how to do crossover without disrupt BB?

Parameter Setting Basic Parameter setting from Model definition Synergistic Effects

Basic Parameter setting from Model definition(1) α: the relative importance of the trail,α≥0 β: the relative importance of the visibility,β≥0 ρ: trail persistence,0≤ρ≤1 Q:a constant related to the quantity of trail laid by ants

Basic Parameter setting from Model definition(2) It will lead to premature convergence, if ρ is too high

Basic Parameter setting from Model definition(3)

Synergistic Effects

Synergistic Effects(cont’d)

Properties Initialization Elitist Strategy Problem Size

Initialization Is there any difference between the case in which all ants are in the same city and the case in which thy are uniformly distributed? the answer is distributing ants resulted in better performance Whether an initial random distribution of the ants over the cities performed better than a uniform one? the answer is no significant difference between the two choices

Elitist Strategy Modified algorithm in which at every cycle the trail laid on the edges belonging to the best-so-far tour is reinforced more than in the standard version Added to the trail of each arc of the best tour a quantity, where e is the number of elitist ants and is the length of the best found tour.

Elitist Strategy(cont’d)

Problem Size

Problem Size(cont’d) size cycles

Experiment Travel Salesman Problem Asymmetric Travel Salesman Problem

Travel Salesman Problem Comparison with TSP-tailored heuristics Comparison with general-purpose heuristics

Heuristic algorithm(from wiki) In computer science, a heuristic algorithm or simply a heuristic is an algorithm that ignores whether the solution to the problem can be proven to be correct, but which usually produces a good solution or solves a simpler problem that contains or intersects with the solution of the more complex problem. Heuristics are typically used when there is no known way to find an optimal solution, or when it is desirable to give up finding the optimal solution for an improvement in run time.

Comparison with TSP-tailored heuristics Compare AS with the “Travel” package Run on Oliver30 problem

Comparison with general-purpose heuristics Compare AS with the Simulated Annealing(SA) and Tabu Search(TS) Run on Oliver30 problem for 1 hour on a IBM- compatible PC with Intel processor

Asymmetric Travel Salesman Problem The asymmetric traveling salesman problem is a TSP in which the distance between two nodes is not symmetric(i.e., ) Application of the AS to the ATSP is straightforward, no modifications are necessary Run test problem the RY48P problem for 4000 cycles Result: the best found tour was 14899, that is 3.3% longer than the optimal one average number of cycles to find it was 1517

Conclusions Employing positive feedback as a search and optimization tool Showing how synergy can arise and be useful in distribution systems Showing how to apply the AS to different combinatorial optimization problem

Extension M.Dorigo, Optimization, Learning and Natural Algorithms, Ph.D Thesis, Dip. Elettronica e Informazione, Politecnico di Milano, Italy, 1992 M.Dorigo T.Sttzle, Ant Colony Optimization,MIT press, 2004 (1430 NT dollars in tenlong bookstore)

and sorry no pretty girl picture today