ISE 410 Heuristics in Optimization
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ISE 410 Heuristics in Optimization Particle Swarm Optimization Swarm Intell igence. Origins in Artificial Life (Alife) Research ALife studies how computational techniques can help when studying biological phenomena

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Swarm intell igence

ISE 410 Heuristics in OptimizationParticle Swarm Optimization

Swarm intell igence

Swarm Intelligence

  • Origins in Artificial Life (Alife) Research

    • ALife studies how computational techniques can help when studying biological phenomena

    • ALife studies how biological techniques can help out with computational problems

  • Two main Swarm Intelligence based methods

    • Particle Swarm Optimization (PSO)

    • Ant Colony Optimization(ACO)

Swarm intelligence

Swarm Intelligence

  • Swarm Intelligence (SI) is the property of a system whereby

    the collective behaviors of (unsophisticated) agents

    interacting locally with their environment

    cause coherent functional global patterns to emerge.

  • SI provides a basis with which it is possible to explore collective (or distributed) problem solving without centralized control or the provision of a global model.

  • Leverage the power of complex adaptive systems to solve difficult non-linear stochastic problems

Swarm intelligence1

Swarm Intelligence

  • Characteristics of a swarm:

    • Distributed, no central control or data source;

    • Limited communication

    • No (explicit) model of the environment;

    • Perception of environment (sensing)

    • Ability to react to environment changes.

Swarm intelligence2

Swarm Intelligence

  • Social interactions (locally shared knowledge) provides the basis for unguided problem solving

  • The efficiency of the effort is related to but not dependent upon the degree or connectedness of the network and the number of interacting agents

Swarm intelligence3

Swarm Intelligence

  • Robust exemplars of problem-solving in Nature

    • Survival in stochastic hostile environment

    • Social interaction creates complex behaviors

    • Behaviors modified by dynamic environment.

  • Emergent behavior observed in:

    • Bacteria, immune system, ants, birds

    • And other social animals

Particle swarm optimization pso

Particle Swarm Optimization(PSO)

  • History

  • Main idea and Algorithm

  • Comparisons with GA

  • Advantages and Disadvantages

  • Implementation and Applications

Particle swarm optimization pso1

Particle Swarm Optimization(PSO)

  • History

  • Main idea and Algorithm

  • Comparisons with GA

  • Advantages and Disadvantages

  • Implementation and Applications

Swarm intell igence

Origins and Inspiration of PSO

  • Population based stochastic optimization technique inspired by social behaviour of bird flocking or fish schooling.

  • Developed by Jim Kennedy, Bureau of Labor Statistics, U.S. Department of Labor and Russ Eberhart, Purdue University

  • A concept for optimizing nonlinear functions using particle swarm methodology

Swarm intell igence

  • Inspired by simulation social behavior

  • Related to bird flocking, fish schooling and swarming theory

    - steer toward the center

    - match neighbors’ velocity

    - avoid collisions

  • Suppose

    • a group of birds are randomly searching food in an area.

    • There is only one piece of food in the area being searched.

    • All the birds do not know where the food is. But they know how far the food is in each iteration.

    • So what's the best strategy to find the food? The effective one is to follow the bird which is nearest to the food.

What is pso

What is PSO?

  • In PSO, each single solution is a "bird" in the search space.

  • Call it "particle".

  • All of particles have fitness values

    • which are evaluated by the fitness function to be optimized, and

  • have velocities

    • which direct the flying of the particles.

  • The particles fly through the problem space by following the current optimum particles.

Pso algorithm

PSO Algorithm

  • Initialize with randomly generated particles.

  • Update through generations in search for optima

  • Each particle has a velocity and position

  • Update for each particle uses two “best” values.

    • Pbest: best solution (fitness) it has achieved so far. (The fitness value is also stored.)

    • Gbest: best value, obtained so far by any particle in the population.

Swarm intell igence

  • PSO algorithm is not only a tool for optimization, but also a tool for representing sociocognition of human and artificial agents, based on principles of social psychology.

  • A PSO system combines local search methods with global search methods, attempting to balance exploration and exploitation.

Swarm intell igence

  • Population-based search procedure in which individuals called particles change their position (state) with time.

     individual has position

    & individual changes velocity

Swarm intell igence

  • Particles fly around in a multidimensional search space. During flight, each particle adjusts its position according to its own experience, and according to the experience of a neighboring particle, making use of the best position encountered by itself and its neighbor.

Swarm intell igence

Particle Swarm Optimization (PSO) Process

  • Initialize population in hyperspace

  • Evaluate fitness of individual particles

  • Modify velocities based on previous best and global (or neighborhood) best positions

  • Terminate on some condition

  • Go to step 2

Pso algorithm1

PSO Algorithm

  • Update each particle, each generation

    v[i]= v[i] + c1 * rand() * (pbest[i] - present[i]) + c2 * rand() * (gbest[i] - present[i])and

    present[i] = persent[i] + v[i]

    where c1 and c2 are learning factors (weights)



Pso algorithm2


Personal influence

Social (global) influence

PSO Algorithm

  • Update each particle, each generation

    v[i] = v[i] + c1 * rand() * (pbest[i] - present[]) + c2 * rand() * (gbest[i] - present[i])and

    present[i] = present[i] + v[i]

    where c1 and c2 are learning factors (weights)



Swarm intell igence

PSO Algorithm

  • Inertia Weight

    d is the dimension, c1 and c2 are positive constants, rand1and rand2 are random numbers, and w is the inertia weight

    Velocity can be limited to Vmax

Particle swarm optimization pso2

Particle Swarm Optimization(PSO)

  • History

  • Main idea and Algorithm

  • Comparisons with GA

  • Advantages and Disadvantages

  • Implementation and Applications

Pso and ga comparison

PSO and GA Comparison

  • Commonalities

    • PSO and GA are both population based stochastic optimization

    • both algorithms start with a group of a randomly generated population,

    • both have fitness values to evaluate the population.

    • Both update the population and search for the optimium with random techniques.

    • Both systems do not guarantee success.

Pso and ga comparison1

PSO and GA Comparison

  • Differences

    • PSO does not have genetic operators like crossover and mutation. Particles update themselves with the internal velocity.

    • They also have memory, which is important to the algorithm.

    • Particles do not die

    • the information sharing mechanism in PSO is significantly different

      • Info from best to others, GA population moves together

Swarm intell igence

  • PSO has a memory

    not “what” that best solution was, but “where” that best solution was

  • Quality: population responds to quality factors pbest and gbest

  • Diverse response: responses allocated between pbest and gbest

  • Stability: population changes state only when gbest changes

  • Adaptability: population does change state when gbest changes

Swarm intell igence

  • There is no selection in PSO

    all particles survive for the length of the run

    PSO is the only EA that does not remove candidate population members

  • In PSO, topology is constant; a neighbor is a neighbor

  • Population size: Jim 10-20, Russ 30-40

Swarm intell igence

PSO Velocity Update Equations

  • Global version vs Neighborhood version

     change pgd to pld .

    where pgd is the global best position

    and pld is the neighboring best position

Swarm intell igence

Inertia Weight

  • Large inertia weight facilitates global exploration, small on facilitates local exploration

  • w must be selected carefully and/or decreased over the run

  • Inertia weight seems to have attributes of temperature in simulated annealing

Swarm intell igence


  • An important parameter in PSO; typically the only one adjusted

  • Clamps particles velocities on each dimension

  • Determines “fineness” with which regions are searched

    if too high, can fly past optimal solutions

    if too low, can get stuck in local minima

Pso pros and cons

PSO – Pros and Cons

  • Simple in concept

  • Easy to implement

  • Computationally efficient

  • Application to combinatorial problems?

     Binary PSO

Swarm intell igence

Books and Website

  • Swarm Intelligence by Kennedy, Eberhart, and Shi, Morgan Kaufmann division of Academic Press, 2001.






Ant colony optimization

Ant Colony Optimization

Aco concept

ACO Concept

  • Ants (blind) navigate from nest to food source

  • Shortest path is discovered via pheromone trails

    • each ant moves at random

    • pheromone is deposited on path

    • ants detect lead ant’s path, inclined to follow

    • more pheromone on path increases probability of path being followed

Aco system

ACO System

  • Virtual “trail” accumulated on path segments

  • Starting node selected at random

  • Path selected at random

    • based on amount of “trail” present on possible paths from starting node

    • higher probability for paths with more “trail”

  • Ant reaches next node, selects next path

  • Continues until reaches starting node

  • Finished “tour” is a solution

Aco system cont

ACO System, cont.

  • A completed tour is analyzed for optimality

  • “Trail” amount adjusted to favor better solutions

    • better solutions receive more trail

    • worse solutions receive less trail

    • higher probability of ant selecting path that is part of a better-performing tour

  • New cycle is performed

  • Repeated until most ants select the same tour on every cycle (convergence to solution)

Aco system cont1

ACO System, cont.

  • Often applied to TSP (Travelling Salesman Problem): shortest path between n nodes

  • Algorithm in Pseudocode:

    • Initialize Trail

    • Do While (Stopping Criteria Not Satisfied) – Cycle Loop

      • Do Until (Each Ant Completes a Tour) – Tour Loop

      • Local Trail Update

      • End Do

      • Analyze Tours

      • Global Trail Update

    • End Do

Aco background

ACO Background

  • Discrete optimization problems difficult to solve

  • “Soft computing techniques” developed in past ten years:

    • Genetic algorithms (GAs)

      • based on natural selection and genetics

    • Ant Colony Optimization (ACO)

      • modeling ant colony behavior

Aco background cont

ACO Background, cont.

  • Developed by Marco Dorigo (Milan, Italy), and others in early 1990s

  • Some common applications:

    • Quadratic assignment problems

    • Scheduling problems

    • Dynamic routing problems in networks

  • Theoretical analysis difficult

    • algorithm is based on a series of random decisions (by artificial ants)

    • probability of decisions changes on each iteration

What is aco as optimization tech

What is ACO as Optimization Tech

  • Probabilistictechnique for solvingcomputational problems which can be reduced to finding good paths through graphs

  • They are inspired by the behavior of ants in finding paths from the colonyto food.



  • Can be used for both Static and Dynamic Combinatorial optimization problems

  • Convergence is guaranteed, although the speed is unknown

    • Value

    • Solution

The algorithm

The Algorithm

  • Ant Colony Algorithms are typically use to solve minimum cost problems.

  • We may usually have N nodes and A undirected arcs

  • There are two working modes for the ants: either forwards or backwards.

  • Pheromones are only deposited in backward mode. (so that we know how good the path was to update its trail)

The algorithm1

The Algorithm

  • The ants memory allows them to retrace the path it has followed while searching for the destination node

  • Before moving backward on their memorized path, they eliminate any loops from it. While moving backwards, the ants leave pheromones on the arcs they traversed.

The algorithm2

The Algorithm

  • 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.

The aco algorithm

The ACO Algorithm

  • At 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:

  • where is the neighborhood of ant k when in node i.

The algorithm3

The Algorithm

  • When the arc (i,j) is traversed , the pheromone value changes as follows:

  • By using this rule, the probability increases that forthcoming ants will use this arc.

The algorithm4

The Algorithm

  • After each ant k has moved to the next node, the pheromones evaporate by the following equation to all the arcs:

  • where is a parameter. An iteration is a complete cycle involving ants’ movement, pheromone evaporation, and pheromone deposit.

Steps for solving a problem by aco

Steps for Solving a Problem by ACO

  • Represent the problem in the form of sets of components and transitions, or by a set of weighted graphs, on which ants can build solutions

  • Define the meaning of the pheromone trails

  • Define the heuristic preference for the ant while constructing a solution

  • If possible implement a efficient local search algorithm for the problem to be solved.

  • Choose a specific ACO algorithm and apply to problem being solved

  • Tune the parameter of the ACO algorithm.








Efficiently Solves NP hard Problems

  • Routing

    • TSP (Traveling Salesman Problem)

    • Vehicle Routing

    • Sequential Ordering

  • Assignment

    • QAP (Quadratic Assignment Problem)

    • Graph Coloring

    • Generalized Assignment

    • Frequency Assignment

    • University Course Time Scheduling



  • Scheduling

    • Job Shop

    • Open Shop

    • Flow Shop

    • Total tardiness (weighted/non-weighted)

    • Project Scheduling

    • Group Shop

  • Subset

    • Multi-Knapsack

    • Max Independent Set

    • Redundancy Allocation

    • Set Covering

    • Weight Constrained Graph Tree partition

    • Arc-weighted L cardinality tree

    • Maximum Clique



  • Other

    • Shortest Common Sequence

    • Constraint Satisfaction

    • 2D-HP protein folding

    • Bin Packing

  • Machine Learning

    • Classification Rules

    • Bayesian networks

    • Fuzzy systems

  • Network Routing

    • Connection oriented network routing

    • Connection network routing

    • Optical network routing

Ant colony algorithms

Ant Colony Algorithms

  • Let um and lm be the number of ants that have used the upper and lower branches.

  • The probability Pu(m) with which the (m+1)th ant chooses the upper branch is:

Traveling salesperson problem

Traveling Salesperson Problem

  • Famous NP-Hard Optimization Problem

  • Given a fully connected, symmetric G(V,E) with known edge costs, find the minimum cost tour.

  • Artificial ants move from vertex to vertex to order to find the minimum cost tour using only pheromone mediated trails.

Traveling salesperson problem1

Traveling Salesperson Problem

  • The three main ideas that this ant colony algorithm has adopted from real ant colonies are:

    • The ants have a probabilistic preference for paths with high pheromone value

    • Shorter paths tend to have a higher rate of growth in pheromone value

    • It uses an indirect communication system through pheromone in edges

Traveling salesperson problem2

Traveling Salesperson Problem

  • Ants select the next vertex based on a weighted probability function based on two factors:

    • The number of edges and the associated cost

    • The trail (pheromone) left behind by other ant agents.

  • Each agent modifies the environment in two different ways :

    • Local trail updating: As the ant moves between cities it updates the amount of pheromone on the edge

    • Global trail updating: When all ants have completed a tour the ant that found the shortest route updates the edges in its path

Traveling salesperson problem3

Traveling Salesperson Problem

  • Local Updating is used to avoid very strong pheromone edges and hence increase exploration (and hopefully avoid locally optimal solutions).

  • The Global Updating function gives the shortest path higher reinforcement by increasing the amount of pheromone on the edges of the shortest path.

Empirical results

Empirical Results

  • Compared Ant Colony Algorithm to standard algorithms and meta-heuristic algorithms on Oliver 30 – a 30 city TSP

  • Standard: 2-Opt, Lin-Kernighan,

  • Meta-Heuristics: Tabu Search and Simulated Annealing

  • Conducted 10 replications of each algorithm and provided averaged results

Comparison to standard algorithms

Comparison to Standard Algorithms

  • Examined Solution Quality – not speed; in general, standard algorithms were significantly faster.

  • Best ACO solution - 420

Comparison to meta heuristic algorithms

Comparison to Meta-Heuristic Algorithms

  • Meta-Heuristics are algorithms that can be applied to a variety of problems with a minimum of customization.

  • Comparing ACO to other Meta-heuristics provides a “fair market” comparison (vice TSP specific algorithms).

Other application areas

Other Application Areas

  • Scheduling : Scheduling is a widespread problem of practical importance.

  • Paul Forsyth & Anthony Wren, University of Leeds Computer Science department developed a bus driver scheduling application using ant colony concepts.

Advantages and disadvantages

Advantages and Disadvantages

Advantages and disadvantages1

Advantages and Disadvantages

  • For TSPs (Traveling Salesman Problem), relatively efficient

    • for a small number of nodes, TSPs can be solved by exhaustive search

    • for a large number of nodes, TSPs are very computationally difficult to solve (NP-hard) – exponential time to convergence

  • Performs better against other global optimization techniques for TSP (neural net, genetic algorithms, simulated annealing)

  • Compared to GAs (Genetic Algorithms):

    • retains memory of entire colony instead of previous generation only

    • less affected by poor initial solutions (due to combination of random path selection and colony memory)

Advantages and disadvantages cont

Advantages and Disadvantages, cont.

  • Can be used in dynamic applications (adapts to changes such as new distances, etc.)

  • Has been applied to a wide variety of applications

  • As with GAs, good choice for constrained discrete problems (not a gradient-based algorithm)

Advantages and disadvantages cont1

Advantages and Disadvantages, cont.

  • Theoretical analysis is difficult:

    • Due to sequences of random decisions (not independent)

    • Probability distribution changes by iteration

    • Research is experimental rather than theoretical

  • Convergence is guaranteed, but time to convergence uncertain

Advantages and disadvantages cont2

Advantages and Disadvantages, cont.

  • Tradeoffs in evaluating convergence:

    • In NP-hard problems, need high-quality solutions quickly – focus is on quality of solutions

    • In dynamic network routing problems, need solutions for changing conditions – focus is on effective evaluation of alternative paths

  • Coding is somewhat complicated, not straightforward

    • Pheromone “trail” additions/deletions, global updates and local updates

    • Large number of different ACO algorithms to exploit different problem characteristics



  • Dorigo, Marco and Stützle, Thomas. (2004) Ant Colony Optimization, Cambridge, MA: The MIT Press.

  • Dorigo, Marco, Gambardella, Luca M., Middendorf, Martin. (2002) “Guest Editorial,” IEEE Transactions on Evolutionary Computation, 6(4): 317-320.

  • Thompson, Jonathan, “Ant Colony Optimization.”, accessed April 24, 2005.

  • Camp, Charles V., Bichon, Barron, J. and Stovall, Scott P. (2005) “Design of Steel Frames Using Ant Colony Optimization,” Journal of Structural Engineeering, 131 (3):369-379.

  • Fjalldal, Johann Bragi, “An Introduction to Ant Colony Algorithms.”, accessed April 24, 2005.



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