1. 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

2. 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

3. Hybrid Metaheuristics
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. Embedding metaheuristics search within other optimization solution methods
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

4. Parallel Metaheuristics
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, reconcile the solutions periodically
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

5. 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
pair-wise exchange operator
Insertion operator
Exchange operator
Inversion operator

6. Implementation - Chapter 14 -15
Summary

7. 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)
constraints

8. 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

9. Population-based metaheuristics
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
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

10. Summary of Other Heuristics
From Text
See webpage and Text

11. Next Sem – Dynamic Programming
Next Fall – Approximate DP
Thank YOU!