Optimization Problems

1. Optimization Problems

2. Agenda • Introduction • Example of Optimization Problem • Multi-Objective Optimization Problem (MOP) • Pareto-Optimal Solutions • Goals of MOO 2

3. Introduction • An optimization problem is the problem of finding the best solution from all feasible solutions. • A decision problem is in P if there is a known polynomial-time algorithm to get that answer. • A decision problem is in NP if there is a known polynomial-time algorithm for a non-deterministic machine to get the answer. Source:http://en.wikipedia.org/wiki/Optimization_problem

4. Introduction • Single Objective Optimization • Optimization problem involves only one objective function, the task of finding the optimal solution is called single-objective optimization. • Example: Find out a CAR with Minimum cost.

5. Introduction • A single objective optimization problem (SOOP)

6. Example of Optimization Problem Traveling Salesman Problem (TSP) A Salesman wishes to travel around a given set of cities, and return to the beginning, covering the smallest total distance. To find the shortest possible route that visits each city exactly once and returns to the origin city.

7. Example of Optimization Problem • Example Applications of Travelling Salesman Problem • Computer Wiring: connecting together computer components using minimum wire length • Job Sequencing: sequencing jobs in order to minimise total set-up time between jobs

8. 3am-5am 10am-1pm 7am-8am 4pm-7pm 6pm-7pm 8am-10am 6am-9am 2pm-3pm Major Practical Extension of the TSP Vehicle Routing - Meet customers demands within given time windows using lorries of limited capacity Depot Much more difficult than TSP

9. Multi-Objective Optimization Problem (MOP) • An advantage of the multi-objective optimization is that the decision-making becomes easier and less subjective. • The single-objective optimization is a degenerated case of multi-objective optimization because it can find only one solution.

10. Multi-Objective Optimization Problem (MOP) • Example: we need to fly on a long trip: • with the cheapest ticket (more connections) or shortest travel time (more expensive) • A MOO problem (MOP) with constraints will have many solutions in the feasible region.

11. Pareto-Optimal Solutions • If we compare tickets A & B, we can’t say that either is superior without knowing the relative importance of Travel Time vs. Price. • However, comparing tickets B & C shows that C is better than B in both objectives, so we can say that C “dominates” B. • So, as long as C is a feasible option, there is no reason we would choose B.

12. Pareto-Optimal Solutions • If we finish the comparisons, we also see that D is dominated by E. • The rest of the options (A, C, & E) have a trade-off associated with Time vs. Price, so none is clearly superior to the others. • We call this the “non-dominated” set of solutions become none of the solutions are dominated.

13. Introduction: Swarm Intelligence • A group of simple or complexindividuals can exhibit very complex emergent behavior • collective behavior • applies to many processes in nature, creating a useful concept in many contexts • collectively migrating bacteria • insects or birds, or phenomena where groups of organisms or non-living objects synchronize their signals or motion

14. Introduction • A collective behavior shows a seeming intelligence that far transcends the abilities of the members, named “swarm intelligence” • Decentralized system • requires multiple agents to make their own independent decisions • Self-organized system • it is not directed or controlled by any agent or subsystem inside or outside of the system

15. Decentralized System • There is no single centralized authority that makes decisions on behalf of all the agents. • Agent makes local autonomous decisions towards its individual goals which may possibly conflict with those of other agents. • Agents directly interact with each other and share information or provide service to other agents. http://www.isr.uci.edu/projects/pace/decentralization.html

16. Self-organized System • “In biological systems self-organization is a process in which pattern at the global level of a system emerges solely from numerous interactions among the lower-level components of the system. Moreover, the rules specifying interactions among the system's components are executed using only local information, without reference to the global pattern” Camazine, Deneubourg, Franks, Sneyd, Theraulaz, Bonabeau, Self-Organization in Biological Systems, Princeton University Press, 2003.

17. Self-organized System • Examples • Coordination of human movement • Flocking behaviour • Creation of structures by social animals

18. Nature of Swarm • The word swarmconjures up images of large groups of small insects in which each member performs a simple role, but the action produces complex behavior as a whole. • Termites swarm to build colonies • Ants swarm to find food sources • Bees swarm to reproduce • Bird swarms • each bird tries to find another to fly with • flies slightly higher to one side to reduce drag, with the birds eventually forming a flock.

19. Categorizing Collective Behaviors • Coordination • Interactions between individuals generate synchronized and oriented movements of the individuals toward a specific goal. • Coordination is at work in most of the building activities in insect colonies. • Nest building in certain species of social insects http://science.howstuffworks.com/environmental/life/zoology/insects-arachnids/termite3.htm

20. Categorizing Collective Behaviors • Cooperation • Occurs when individuals achieve together a task that could not be done by a single one. • The individuals must combine their efforts in order to successfully solve a problem that goes beyond their individual abilities. • Bringking food back to the nest http://deepintoscripture.com/2012/05/01/in-which-there-are-ants-and-a-news-reporter-and-tournament-results/

21. Categorizing Collective Behaviors • Deliberation • Deliberation refers to mechanisms that occur when a colony faces several opportunities. • These mechanisms result in a collective choice for at least one of the opportunities. • Ants have discovered several food sources with different qualities or richness, or several paths that lead to a food source, they generally select only one of the different opportunities. • In this case, the deliberation is driven by the competition between the chemical trails leading to each opportunity http://www.responsiblepestcontrolmesa.com/argentine-ant-pest-control-exterminating/

22. Assignment • Title: Swarm Intelligence (SI) for Optimization Problem • Multi-objective Optimization Problems • Case Study • Buy a new Notebook • Buy a mobile phone • How to apply the algorithm to the selected Case Study • General Ideas • Overview of Case Study • Criteria/Objectives: Minimize or Maximize • Possible Parameters or Variables • Apply the algorithm to find the optimal solutions (DETAIL=>depend on the selected case study) • Paper Report • Presentation

23. References • Satchidananda Dehuri, Multi-objective Optimization Using Particle Swarm Optimization, Department of Information and Communication Technology, Fakir Mohan University, INDIA. • Multi-Objective Optimization, www.polymtl.ca • Abdullah Konak, David W. Coit, Alice E. Smith, Multi-objective optimization using genetic algorithms: A tutorial, Available online at: www.sciencedirect.com • Sunantha Sodsee, A Multi-objective Bisexual Reproduction Genetic Algorithm for Computer Network Design, Master Thesis, Information Technology, KMUTNB, 2004 • H.P. Williams , The Travelling Salesman Problem, London School of Economics