Introduction Combinatorial Optimization problems are in many real life cases (Medical, Engineering, Computer Science and Operations Research). New combinatorial optimization problems arising in the area of highthroughput assay design . A lot of these problems are difficult to solve in large scale. If we solved them by traditional methods may take thousands of years (NP-hard ). Some of them can not be solved using traditional methods (NPcomplete ) . Using swarm intelligent modified techniques will lead to Faster Convergence. Near optimal solution with small tolerances . Reduced Time. Reduced Cost .
Combinatorial Optimization Cryptography
Optimization problems can be continuous or Access Control combinatorial . Finding an optimal object from a finite set of objects. The set of feasible solutions is discrete or can be Watermarking reduced to discrete.
Combinatorial Optimization Some common problems involving combinatorial optimization are : Traveling salesman problem ("TSP") , Minimum Spanning Tree Problem, Assignment, Transportation, Knapsack problem Timetabling problem Closure problem Constraint satisfaction problem Cutting stock problem Integer Programming … etc. It has important applications in several fields, including artificial intelligence, machine learning, mathematics, auction theory, and software engineering.
Swarm Intelligence Study of collective behavior in decentralized, selforganized systems. Originated from the study of colonies, or swarms of social organisms. Collective intelligence arises from interactions.
Swarm Intelligent Algorithms Major Swarm Intelligent algorithms including Ant Colony Optimization (ACO), Bee algorithms (BA), Particle Swarm Optimization (PSO), Harmony Search (HS), Firefly Algorithm (FA), Cuckoo Search (CS) , Bat-inspired Algorithm (BA), and Flower Pollination Algorithm(FPA)… etc.
Particle Swarm Optimization Algorithm (Kennedy & Eberhart 1995)
• Inspired by social behavior of birds and shoals of fish • Each particle is searching for the optimum • Each particle is moving and hence has a velocity. • Each particle remembers the position it was in where it had its best result
PSO Algorithm Step1 : Initialize the swarm from the solution space Step2 : Evaluate fitness of each particle Step3 : Update individual and global bests Step4 : Update velocity and position of each particle
Step5 : Go to step 2, and repeat until termination condition
Harmony Search Algorithm (Geem et al. ,2001 )
Harmony search is a music-based metaheuristic optimization algorithm. It was inspired by the observation that the aim of music is to search for a perfect state of harmony. The effort to find the harmony in music is analogous to find the optimality in an optimization process. In other words, a jazz musician’s improvisation process can be compared to the search process in optimization. A musician always intends to produce a piece of music with perfect harmony.
Harmony Search Algorithm
Harmony Search Algorithm Procedures Step 1. Prepare a Harmony Memory. Step 2. Improvise a new Harmony with Experience (HM) or Randomness (rather than Gradient). HMCR, PAR and BW Step 3. If the new Harmony is better, include it in Harmony Memory. Step 4. check if termination criterion has been met . Otherwise, Repeat Step 2 and Step 3.
Flower Pollination Algorithm (Xin-She Yang 2013)
Inspired by the flow pollination process of flowering plants are the following rules: Rule 1: Biotic and cross-pollination can be considered as a process of global pollination process, and pollen-carrying pollinators move in a way that obeys Le'vy flights. Rule 2: For local pollination, a biotic and self-pollination are used. Rule 3: Pollinators such as insects can develop flower constancy, which is equivalent to a reproduction . Rule 4: The interaction or switching of local pollination and global pollination can be controlled by a switch probability p[0,1].
Procedures of Flower Pollination Algorithm
Taxonomy Using Swarm Intelligent Techniques for Solving Combinatorial Optimization Problems
Flower Pollination Improved FPA
Solution Techniques
Problem Identification
Swarm Algorithms
Combinatorial Optimization
Harmony Search Improved HS
Improved HS with FPA
PSO Improved FPA with PSO
Bat Algorithm Improved BA
Integer Programming
Assignment Problems
Constraint Satisfaction Problems
Sudoku Puzzles
Global optimization
Aim of the work Why we use HS, FPA,BA and PSO? Can be applied to a wide range of applications. Easy to understand. Easy to implement. Computationally efficient. Almost no parameter tuning . Faster convergence .
Chaos Chaos is a kind of common nonlinear phenomenon, which has diverse, complex and sophisticated native under apparent disorder. Chaotic motion is characterized by ergodicity, randomness, and ‘regularity’ which can traverse all status according to its own ‘rule’ without repetition
Chaos Properties Deterministic. It has deterministic rather than probabilistic underlying rules which every Nonlinear. The underlying rules are nonlinear; if they are linear, it cannot be chaos. Irregular. The behavior of the system shows sustained irregularity. Sensitive to initial conditions. Small changes in the initial state of chaotic systems can lead to radically different behavior in the final state. Long term prediction is practically impossible. Due to sustained irregularity and sensitivity to initial conditions, which can only be known to a finite precision
Chaos Directions First group aimed at global optimization by chaotic neural network introduced in many approaches. Second group searched the global optimum by the Chaos Optimization Algorithm, which utilized the nature of chaos sequences such as pseudo-randomness, ergodicity and irregularity . Third group utilized computational instabilities to solve nonlinear equations and optimization problems that resulted in the development of new methods
Chaotic Maps A chaotic map is a map (= evolution function) that exhibits some sort of chaotic behavior. Chaotic maps often occur in the study of dynamical systems Examples Logistic map, Sine map and Circle map ..etc
An Improved Flower Pollination Algorithm with Chaos ď ąA new method is developed based on the flower pollination algorithm combined with chaos theory (IFPCH) to solve definite integral. ď ą The definite integral has wide ranging applications in operation research, computer science, mathematics, mechanics, physics, and civil and mechanical engineering. ď ąNumerical simulation results show that the algorithm offers an effective way to calculate numerical value of definite integrals, and it has a high convergence rate, high accuracy and robustness.
IFPCH Algorithm
Numerical Results for IFPCH Algorithm
Thanks for your attention