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Multi-Objective Optimization
• NP-Hard
• Conflicting objectives
– Flow shop with both minimum makespan and tardiness objective
– TSP problem with minimum distance, time and cost objective
– Container management – balancing volume, weight and value
• Has no single solution but a set of solutions called Pareto
Optimal Solutions
– A solution is Pareto optimal if it not possible to improve a single
objective without deteriorating another objective
• The objective is to find the Pareto optimal set and the Pareto
front
• Metaheuristics can be used to approximate the Pareto
optimal set
– Both S and P – metaheuristics are used
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Metaheuristics for Multiobjective Optimization
• Fitness assignment – assign a scalar value to the quality of the
solution
• Diversity preserving – generate a diverse set of solutions
• Elitism – Select the best set of solutions at every step
General strategies
• Aggregation – use an aggregation method to covert the problem
into mono-objective
• Weighted Metric – preselect a reference value of the objective
function and measure the distance of the other solutions from this
reference and minimize this distance
• Parallel approach- treat each objective individually. Then crossover
and mutate the solutions from each objective to find a compromise
• Sequential approach- search in a preference order of objectives
• Dominance based- search using a dominant criteria set by the final
user
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Hybrid Metaheuristics
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Combining S and P or a S and S metaheuristics
Combining with other math programming methods
Metaheuristics and AI
Main classification
– Relay - sequential
– Teamwork – cooperative search
– Example
– Branch and bound – the upper bound of a node can be obtained using
metaheuristic which also yields a partial solution upto the given node
– Dynamic programming- if the state-action space is large,
metaheuristics can reduce the action space by performing a local
search among a set of all possible actions for a state
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Parallel Metaheuristics
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Speed up search
Improve quality
Solve large NP hard problems
Parallel designs
– Algorithmic level – Independent or cooperative self-contained
metaheuristics approaches are used in parallel
– Iterative level – At an iteration search is done in several
neighborhoods by different computers to speed up search
– Solution level- the generation of the objective function value and the
check for any constraint violations is done in parallel for a set of
solutions generated by one search
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Elements of the Heuristic Approach
• Representation of the solution space
– Vector of Binary values – 0/1 Knapsack, 0/1 IP problems
– Vector of discrete values- Location , and assignment problems
– Vector of continuous values on a real line – continuous, parameter
optimization
– Permutation – sequencing, scheduling, TSP
• Defining the neighborhood and the neighbors
– Flip operator – binary or over a range of numbers (+1 or -1 as in
knapsack)
– Permutation operator
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pair-wise exchange operator
Insertion operator 12345 14235
Exchange operator 12345 14325
Inversion operator 123456 154326
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Elements of the Heuristic Approach
• Defining the initial solution
– Random or greedy
• Choosing the method (algorithm for iterative search)
– Off-the shelf or tailor made heuristic
– Single-start or multistart (still single but several independent singles) or
population (solutions interact with one another)
– Strategies for escaping local optima
– Balance diversification and intensification of search
• Objective function evaluation
– Full or partial evaluation
– At every iteration or after a set of iterations
• Stopping criteria
– Number of iterations
– Time
– Counting the number of non-improving solutions in consecutive iterations.
Remember: there is a lot of flexibility in setting up the above. Optimality cannot be proved.
All you are looking for is a good solution given the resource (time, money and computing power)
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constraints
Single-Metaheuristics
• 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
• 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
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Population-based metaheuristics
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Nature-inspired
Initialize a population
A new population of solutions is generated
Integrate the new population into the current one using one these
methods – by replacement which is a selection process from the new and
current solutions
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Evolutionary Algorithms – genetic algorithm
Estimation of distribution algorithm (EDA)
Scatter search
Evolutionary programming- genetic programming
Swarm Intelligence
• Ant colony
• Particle swarm optimization (PSO)
• Bee colony
– Artificial Immune system AIS
• Continue until a stopping criteria is reached
• The generation and replacement process could be memoryless or some
search memory is used
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Summary of Other Heuristics
• See webpage and Text
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• Next Sem – Dynamic Programming
• Next Fall – Approximate DP
Thank YOU!
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