New Post

Nash Equilibrium In Game Theory ~xRay Pixy

-
Image
 Video Link  CLICK HERE... Learn Nash Equilibrium In Game Theory Step-By-Step Using Examples. Video Chapters: Nash Equilibrium  00:00 Introduction 00:19 Topics Covered 00:33 Nash Equilibrium  01:55 Example 1  02:30 Example 2 04:46 Game Core Elements 06:41 Types of Game Strategies 06:55  Prisoner’s Dilemma  07:17  Prisoner’s Dilemma Example 3 09:16 Dominated Strategy  10:56 Applications 11:34 Conclusion The Nash Equilibrium is a concept in game theory that describes a situation where no player can benefit by changing their strategy while the other players keep their strategies unchanged.  No player can increase their payoff by changing their choice alone while others keep theirs the same. Example : If Chrysler, Ford, and GM each choose their production levels so that no company can make more money by changing their choice, it’s a Nash Equilibrium Prisoner’s Dilemma : Two criminals are arrested and interrogated separately. Each has two ...
Post a Comment

Objective Function Evaluation | Greedy Method | Knapsack Problem Example...

-

Knapsack Problem using Greedy Method


Algorithm Design Techniques
  • Divide and Conquer
  • Greedy Method
  • Dynamic Programming
  • Back Tracing
  • Branch and Bound
Divide and Conquer: Many algorithms are recursive in structure. To solve any problem, they call themselves recursively again and again [one or more times]. Three steps are followed by divide and conquer algorithms.

1.) Divide the problem into the number of sub-problems.
2.) Conquer the sub-problems by solving them recursively.
3.) Combine the solution to the sub-problems into the solution for the original problem.

The greedy method is the Straight design technique. It can be applied to a wide variety of problems. Obtain a subset that satisfies the same constraints.  Feasible Solution: If any subset satisfies these constraints. 
Our GOAL: Find a feasible solution that either Maximize or Minimize the given Objective Function. A feasible solution that does this is known as OPTIMAL SOLUTION.  A feasible Solution is any subset that satisfies these constraints.

Greedy Method Example : KNAPSACK PROBLEM
SUPPOSE: We have 
        n  = Objects and a Knapsack.
𝑤_𝑖 = Object i has weight 
 m = Knapsack Capacity

IF a fraction 𝑥_𝑖, of object i is placed into the knapsack. 0 ≤ 𝑥_𝑖 ≤ 1 than Profit Earned.
Objective: Obtain filling of Knapsack and Gain maximum profit.


n = 3;                         //Objects
m = 20;                                 //Knapsack Capacity
𝑤1,𝑤2,𝑤3 = 18, 15,10; //Objects Weight
𝑃1,𝑃2,𝑃3 = 25, 24, 15; //Profits

Comments

Post a Comment

Popular Post

PARTICLE SWARM OPTIMIZATION ALGORITHM NUMERICAL EXAMPLE

-
Image
 PARTICLE SWARM OPTIMIZATION ALGORITHM NUMERICAL EXAMPLE PSO is a computational method that Optimizes a problem. It is a Population-based stochastic search algorithm. PSO is inspired by the Social Behavior of Birds flocking. n Particle Swarm Optimization the solution of the problem is represented using Particles. [Flocking birds are replaced with particles for algorithm simplicity]. Objective Function is used for the performance evaluation for each particle / agent in the current population. PSO solved problems by having a Population (called Swarms) of Candidate Solutions (Particles). Local and global optimal solutions are used to update particle position in each iteration. Particle Swarm Optimization (PSO) Algorithm step-by-step explanation with Numerical Example and source code implementation. - PART 2 [Example 2] 1.) Initialize Population [Current Iteration (t) = 0] Population Size = 4; 𝑥𝑖 : (i = 1,2,3,4) and (t = 0) 𝑥1 =1.3; 𝑥2=4.3; 𝑥3=0.4; 𝑥4=−1.2 2.) Fitness Function u...
Read more

Cuckoo Search Algorithm for Optimization Problems

-
Image
 Cuckoo Search Algorithm - Metaheuristic Optimization Algorithm What is Cuckoo Search Algorithm? Cuckoo Search Algorithm is a Meta-Heuristic Algorithm. Cuckoo Search Algorithm is inspired by some Cuckoo species laying their eggs in the nest of other species of birds. In this algorithm, we have 2 bird Species.  1.) Cuckoo birds   2.) Host Birds (Other Species) What if Host Bird discovered cuckoo eggs? Cuckoo eggs can be found by Host Bird.  Host bird discovers cuckoos egg with Probability of discovery of alien eggs.  If Host Bird Discovered Cuckoo Bird Eggs. The host bird can throw the egg away. Abandon the nest and build a completely new nest. Mathematically, Each egg represent a solution and it is stored in the host bird nest. In this algorithm Artificial Cuckoo Birds are used. Artificial Cuckoo can lay one egg at a time. We will replace New and better solutions with less fit solutions. It means eggs that are more similar to host bird has opportunity to de...
Read more

Particle Swarm Optimization (PSO)

-
Image
Particle Swarm Optimization (PSO) is a p opulation-based stochastic search algorithm. PSO is inspired by the Social Behavior of Birds flocking. PSO is a computational method that Optimizes a problem. PSO searches for Optima by updating generations. It is popular is an intelligent metaheuristic algorithm.  In Particle Swarm Optimization the solution of the problem is represented using Particles. [Flocking birds are replaced with particles for algorithm simplicity]. Objective Function is used for the performance evaluation for each particle / agent in the current population. After a number of iterations agents / particles will find out optimal solution in the search space. Q. What is PSO? A. PSO is a computational method that Optimizes a problem. Q. How PSO will optimize? A. By Improving a Candidate Solution. Q. How PSO Solve Problems? A. PSO solved problems by having a Population (called Swarms) of Candidate Solutions (Particles). Local and global optimal solutions are used to ...
Read more

PSO (Particle Swarm Optimization) Example Step-by-Step

-
Image
 Particle swarm optimization (PSO) What is meant by PSO? PSO is a computational method that Optimizes a problem. It is a Population-based stochastic search algorithm. PSO is inspired by the Social Behavior of Birds flocking. n Particle Swarm Optimization the solution of the problem is represented using Particles. [Flocking birds are replaced with particles for algorithm simplicity]. Objective Function is used for the performance evaluation for each particle / agent in the current population. PSO solved problems by having a Population (called Swarms) of Candidate Solutions (Particles). Local and global optimal solutions are used to update particle position in each iteration. How PSO will optimize? By Improving a Candidate Solution. How PSO Solve Problems? PSO solved problems by having a Population (called Swarms) of Candidate Solutions (Particles). The population of Candidate Solutions (i.e., Particles). What is Search Space in PSO? It is the range in which the algorithm computes th...
Read more

how is the LBP |Local Binary Pattern| values calculated? Step-by-Step with Example

-
Image
 Local Binary Pattern Introduction to Local Binary Pattern (LBP) Q. What is Digital Image? A. Digital images are collections of pixels or numbers ( range from 0 to 255).  Q. What is Pixel? A. Pixel is the smallest element of any digital image. Pixel can be categorized as Dark Pixel and Bright Pixel. Dark pixels contain low pixel values and bright pixels contain high pixel values. Q. Explain Local Binary Pattern (LBP)? A. Local binary pattern is a popular technique used for image processing. We can use the local binary pattern for face detection and face recognition. Q. What is LBP Operator? A. LBP operator is an image operator. We can transform images into arrays using the LBP operator. Q. How LBP values are computed? A. LBP works in 3x3 (it contain a 9-pixel value ). Local binary pattern looks at nine pixels at a time. Using each 3x3 window in the digital image, we can extract an LBP code. Q. How to Obtain LBP operator value?  A. LBP operator values can be obtained by ...
Read more

PSO Python Code || Particle Swarm Optimization in Python || ~xRay Pixy

-
Image
Particle Swarm Optimization Implementation in Python Video Chapters: 00:00 Introduction 02:01 Code 05:55 Position Initialization 08:06 PSO Main Loop 08:42 Velocity Calculation 10:02 Position Update 10:36 Fitness Evaluation 13:21 Objective Function 17:44 Result 19:00 Conclusion .....................................................SOURCE CODE......................................................................... import random import numpy as np from tkinter import messagebox #Define Class Particles class Particle: def __init__ (self,position): self.position=position self.velocity=np.zeros_like(position) self.best_position=position self.best_fitness=float('inf') def PSO(ObjF,Pop_Size,D,MaxT): swarm_best_position=None swarm_best_fitness=float('inf') particles=[] #Posotion Initialization position=np.random.uniform(-0.5,0.5,D) particle=Particle(position) particles.append(particle) #Fit...
Read more

Grey Wolf Optimization Algorithm

-
Image
 Grey Wolf Optimization Algorithm  (GWO) Grey Wolf Optimization Grey Wolf Optimization Algorithm is a metaheuristic proposed by Mirjaliali Mohammad and Lewis, 2014. Grey Wolf Optimizer is inspired by the social hierarchy and the hunting technique of Grey Wolves. What is Metaheuristic? Metaheuristic means a High-level problem-independent algorithmic framework (develop optimization algorithms). Metaheuristic algorithms find the best solution out of all possible solutions of optimization. Who are the Grey Wolves? Wolf (Animal): Wolf Lived in a highly organized pack. Also known as Gray wolf or Grey Wolf, is a large canine. Wolf Speed is 50-60 km/h. Their Lifespan is 6-8 years (in the wild). Scientific Name: Canis Lupus. Family: Canidae (Biological family of dog-like carnivorans). Grey Wolves lived in a highly organized pack. The average pack size ranges from 5-12.  4 different ranks of wolves in a pack: Alpha Wolf, Beta Wolf, Delta Wolf, and Omega Wolf. How Grey Wolf Optimiza...
Read more

Bat algorithm Explanation Step by Step with example

-
Image
There are about 1000 species of Bats. Bat Algorithm is based on the echolocation behavior of Micro Bats with varying pulse rates of emission and loudness. All bats use echolocation to sense distance and background barriers.  Microbats are small to medium-sized flying mammals. Micro Bats used a Sonar that is known as Echolocation to detect their prey. Bats fly randomly with the velocity at the position with a fixed frequency and loudness for prey.  Q. Whats is Frequency? A. Frequency is the number of waves that pass a fixed point in unit time. Wavelength is the minimum distance between two nearest particles which are in the same phase. Here, Sound waves are used by microbats to detect prey.  Q. What is Position? A. A place where something or someone is located.  Q. What is Velocity? A. Speed of something in a given direction.  Q. What is loudness. A. Loudness refers to how soft or loud sound seems to listeners.  Q. What is pulse rate? ...
Read more

Grey Wolf Optimization Algorithm Numerical Example

-
Image
 Grey Wolf Optimization Algorithm Numerical Example Grey Wolf Optimization Algorithm Steps 1.) Initialize Grey Wolf Population. 2.) Initialize a, A, and C. 3.) Calculate the fitness of each search agent. 4.) 𝑿_𝜶 = best search agent 5.) 𝑿_𝜷 = second-best search agent 6.) 𝑿_𝜹 = third best search agent. 7.) while (t<Max number of iteration) 8.) For each search agent       update the position of the current search agent by the above equations end for 9.) update a, A, and C 10.) Calculate the fitness of all search agents. 11.) update 𝑿_𝜶, 𝑿_𝜷, 𝑿_𝜹 12.) t = t+1 end while 13.) return 𝑿_𝜶 Grey Wolf Optimization Algorithm Numerical Example STEP 1.  Initialize the Grey wolf Population [Initial Position for each Search Agent] 𝒙_(𝒊  )  (i = 1,2,3,…n)     n = 6 // Number of Search Agents  [ -100, 100] // Range Initial Wolf Position      3.2228     4.1553    -3.8197     4.2330  ...
Read more

Whale Optimization Algorithm Code Implementation || WOA CODE || ~xRay Pixy

-
Image
Whale Optimization Algorithm Code Implementation Whale Optimization Algorithm Code Files function obj_fun(test_fun) switch test_fun     case 'F1'         x = -100:2:100; y=x;     case 'F2'         x = -10:2:10; y=x; end end function [LB,UB,D,FitFun]=test_fun_info(C) switch C     case 'F1'         FitFun = @F1;         LB = -100;          UB = 100;         D = 30;     case 'F2'         FitFun = @F2;         LB = -10;         UB = 10;         D = 30; end % F1 Test Function     function r = F1(x)         r = sum(x.^2);     end % F2 Test Function     function r = F2(x)         r = sum(abs(x))+prod(abs(x));     end end function Position = initialize(Pop_Size,D,UB,LB) SS_Bo...
Read more