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Python Code || Path Planning with Grey Wolf Optimization (GWO) ~xRay Pixy

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Learn how to implement an obstacle-avoiding path planning for a robot using the Grey Wolf Optimization (GWO) in a static environment. #optimization #algorithm #metaheuristic #robotics #deeplearning #ArtificialIntelligence #MachineLearning #computervision #research #projects #thesis #Python
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Solving Transportation Problem ||Operations Research and Optimization|| ...

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Unit 1  - INTRODUCTION TO OPTIMIZATION ALGORITHMS

Video Chapters: Optimization Techniques 00:00 Introduction 02:39 Optimization Statement 05:27 Constrained Optimization Problems 06:06 Constrained Optimization Methods 07:38 Transportation Problem 14:40 Unconstrained Optimization Problems 20:15 Conclusion

1. What is Optimization?

    Optimization is the process of finding the best solution or set of solutions from a range of possible options, with the aim of maximizing or minimizing a particular objective function. The objective function is a mathematical expression that represents the quantity to be optimized, such as maximizing profit, minimizing cost, or achieving the highest performance.

There are two main types of optimization problems:

  • Maximization Problems: These involve finding the maximum value of an objective function. Examples: maximizing profit, revenue, efficiency, or any other desirable outcome.

  • Minimization Problems: These involve finding the minimum value of an objective function. Examples: minimizing cost, energy consumption, errors, or any other undesirable outcome.

Key Concepts in Optimization:

  • Decision Variables: These are the variables that decision-makers can control or adjust to optimize the objective function. For example, in a manufacturing process, decision variables might include production levels of different products.

  • Objective Function: This is the function that needs to be optimized. It quantifies the goal that is being pursued. For instance, in a transportation network, the objective function might be to minimize travel time or cost.

  • Constraints: Constraints are conditions or limitations that must be satisfied in the optimization process. These can be restrictions on the decision variables, reflecting real-world limitations.

  • Feasible Solution Space: This is the set of all possible solutions that satisfy the constraints. The goal of optimization is to find the best solution within this feasible solution space.

  • Optimal Solution: The optimal solution is the combination of decision variables that yields the best value of the objective function while satisfying all the constraints.

  • Local and Global Optima: Optimization problems can have local and global optima. A local optimum is a solution that is optimal within a specific region of the feasible solution space, while a global optimum is the best solution across the entire feasible solution space.

2. Optimization Problem Statement

    An optimization problem involves finding the best solution from a set of feasible solutions, where the "best" solution is typically defined based on certain criteria or objectives. A general statement of an optimization problem can be structured as follows:

  • Objective Function: Define an objective function that needs to be either maximized or minimized. This function represents the quantity to be optimized and is typically denoted as Minimize f(x)  or Maximize f(x).

  • Decision Variables: Specify the decision variables, which are the variables that determine the solution.

  • Constraints: Introduce any constraints that the solution must satisfy. They can be equalities or inequalities.

The formal formulation of the optimization problem:

3. Constrained Optimization Problem 

     Constrained optimization is the process of finding the minimum or maximum of a function subject to certain constraints on the decision variables. These constraints can be equality or inequality conditions and are essential to represent real-world limitations, requirements, or restrictions on the decision variables.

Key Concepts in Constrained Optimization

  • Objective Function

  • Decision Variable

  • Constraints (Equality and Inequality)

  • Feasible Solution Space

  • Optimal Solution

  • unchecked

    Analytical Methods: Lagrange Multipliers 

  • unchecked

    Numerical Methods: Interior Point Method, Penalty Function Methods, Sequential Quadratic Programming (SQP).

  • unchecked

    Heuristic Methods: Simulating Annealing, Genetic Algorithms

Challenges in Constrained Optimization:

  • Feasibility 

  • Optimality and Local Minima

  • Computational Complexity

4. Classical Optimization Problem: Transportation Problem

    In this problem, the objective is to minimize the cost of transporting goods from multiple suppliers to multiple consumers and satisfy certain constraints. transportation Cost per unit (in $) from each Supplier to each Consumer.

Cost per unit in ($)

  • Decision Variables: x_ij is the decision variable representing the number of goods to be transported from the supplier (i) to consumer j. The decision variables are the amounts shipped between each pair of suppliers and consumers.

x_ij >= 0 for all i and j

  • Objective Function: The objective is to minimize the total transportation cost, which is the sum of the products of the quantity shipped and the cost per unit to transport from supplier (i) to consumer (j).

x_ij: Quantity Shipped  & c_ij: cost per unit to transport from (i to j) 

  • Two Constraints: Supply Constraints: The total quantity supplied by each supplier (i) must not exceed the available supply (s_i). 

Each Supplier should not exceed its supply.

  • ​Demand Constraints: The total quantity demanded by each consumer (j) must be satisfied and is equal to the demand (d_j) 

Each Consumer should receive its demand 

​

Consider the following shipping plan:

  • Calculate the total transportation Cost

Z = (4×25) + (6×5) + (8×0) + (3×0) + (2×25) + (5×25) + (7×0) + (4×0) + (2×20)

Z = 345

  • Contribution of each supplier to the total transportation cost

Supplier 1 (S1)

S1 to C1= 100 & S1 to C2 = 30

Total contribution from S1= 130

Supplier 2 (S2)

S2 to C2= 50 & S2 to C3 = 125

Total contribution from S2= 175

Supplier 3 (S3)

S3 to C3 = 40 

Total contribution from S3= 40

5. Unconstrained Optimization Problem 

  Unconstrained optimization refers to the process of finding the minimum or maximum of a function without any constraints on the decision variables.

Key Concepts in Unconstrained Optimization

  • Objective Function

  • Decision Variable

  • Optimal Solution

Methods for Unconstrained Optimization

  • Analytical Methods: Calculus

  • Numerical Methods: Gradient Descent, Newton’s Method, conjugate Gradient Method, and Quasi-Newton Methods.

  • Heuristic Methods: Simulating Annealing, Genetic Algorithms

Challenges in Unconstrained Optimization:

  • Local Optima 

  • Computational Complexity

6. Multimodel Optimization Problem 

  Multimodal optimization refers to the process of finding multiple optimal solutions in a search space, as opposed to finding just a single optimal solution.

Key Concepts in Multimodal Optimization:

  • Objective Function Landscape

  • Modes

Methods for Multimodel Optimization

  • Evolutionary Algorithms: Genetic Algorithms, PSO

  • Swarm Intelligence: Ant Colony Optimization (ACO)

  • Niching Techniques: Crowding, Island Models

  • Hybrid Models: Combining Local and Global Search

Multimodal Optimization Applications

  • Function Optimization

  • Machine Learning

  • Robotics

  • Network Design

Types of Optimization Problems

  • Linear Programming (LP)

  • Nonlinear Programming (NLP)

  • Integer Programming (IP)

  • Quadratic Programming (QP)

  • Convex Optimization


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