In this module, the development of the Simplex Algorithm is presented. Students will learn about unboundedness and how to utilize the Simplex Tableau in solving Linear Programming Problems effectively, preparing them for more complex techniques.
This module introduces the concept of Linear Programming Problems (LPP), emphasizing its significance in optimization. Students will learn how to identify and formulate real-world problems as LPPs, setting the foundation for further exploration of linear optimization techniques.
This module covers vector space concepts, focusing on linear independence and dependence. Students will explore the significance of a basis in vector spaces, enhancing their understanding of the geometric and algebraic dimensions of linear programming.
In this module, students will learn how to transition from one basic feasible solution to another, emphasizing optimality criteria. The discussions will enhance their understanding of how various solutions relate to each other within an LPP framework.
This module delves into the concepts of basic feasible solutions, including their existence and derivation. Students will explore various methods to derive feasible solutions, setting a foundation for advanced optimization techniques.
Students will investigate convex sets and the dimension of a polyhedron, along with the concept of faces. This module will also provide examples of polytopes to illustrate these foundational geometric concepts in linear programming.
This module focuses on the direction of a polyhedron and the correspondence between basic feasible solutions (bfs) and extreme points. Students will learn to identify and analyze these relationships, which are critical in optimization.
This module discusses the representation theorem, which states that every solution of an LPP is a basic feasible solution. Students will engage in exercises, enhancing their skills through Assignment 1, reinforcing the concepts learned.
In this module, the development of the Simplex Algorithm is presented. Students will learn about unboundedness and how to utilize the Simplex Tableau in solving Linear Programming Problems effectively, preparing them for more complex techniques.
This module explores the Simplex Tableau and algorithm in detail, addressing issues like cycling and introducing Bland's anti-cycling rules. Additionally, it discusses Phase I and Phase II of the Simplex method for comprehensive understanding.
This module examines the Big-M method and graphical solutions for linear programming problems. Students will explore adjacent extreme points and basic feasible solutions, enhancing their problem-solving toolkit for optimization challenges.
In this module, students will tackle Assignment 2, focusing on the progress of the Simplex algorithm on a polytope and bounded variable LPPs. This practical approach helps reinforce the theoretical concepts discussed earlier.
This module covers bounded variable LPPs, introducing the Revised Simplex Algorithm and Duality theory. Students will explore the weak duality theorem, gaining insights into the relationship between primal and dual problems.
This module focuses on the weak duality theorem, providing economic interpretations of dual variables. Students will learn how to analyze these interpretations in practical optimization scenarios.
This module presents examples of writing the dual in practical scenarios. Furthermore, it covers the complementary slackness theorem, which is crucial for understanding the relationship between primal and dual solutions.
This module addresses the complementary slackness conditions and introduces the Dual Simplex algorithm. Students will engage with Assignment 3, enhancing their understanding through practical applications of these concepts.
This module introduces the Primal-dual algorithm, a powerful method in optimization that simultaneously considers both primal and dual problems. Students will learn how to utilize this approach for effective problem-solving.
In this module, students will work on the problems discussed in the previous lecture and explore starting dual feasible solutions. The focus will be on the Shortest Path Problem, a common application in optimization.
This module continues the exploration of the Shortest Path Problem, introducing the Primal-dual method through practical examples. Students will see how these methods can be applied to real-life scenarios.
In this module, students will examine the complexity of the Shortest Path Problem, focusing on the interpretation of dual variables. This understanding is essential for advanced analysis in optimization.
This module includes Assignment 4, where students will engage in post-optimality analysis. They will explore changes in the vector b and how to add new constraints, enhancing their problem-solving skills.
In this module, students will delve into parametric LPP, focusing on right-hand side vector changes. The discussions will enhance their understanding of how variations impact the solution space.
This module addresses parametric cost vector LPPs, highlighting how changes in cost vectors affect optimal solutions. Students will learn to analyze and respond to these variations effectively.
This module introduces the Min-cost flow problem, providing a framework for understanding cost-efficient transportation in networks. Students will engage in practical examples to solidify their comprehension.
This module focuses on the Transportation Problem, exploring its complexities and offering strategies for effective solutions. Students will learn to apply optimization techniques in transportation scenarios.
In this module, students will examine the issues of degeneracy and cycling in the Transportation Problem. They will learn techniques to identify and navigate these challenges in optimization.
This module focuses on sensitivity analysis in linear programming, helping students understand how changes in parameters affect optimal solutions. Practical applications will enhance their analytical skills.
This module continues the sensitivity analysis discussions, providing deeper insights into how different factors impact solutions. Students will engage with real-world scenarios to solidify their understanding.
In this module, students will explore the Bounded Variable Transportation Problem. They will learn strategies to address constraints effectively while minimizing costs in transportation scenarios.
This module continues with the Min-cost flow problem, enabling students to analyze and optimize flow in various contexts. Practical examples will enhance their understanding of network optimization.
Students will learn about starting feasible solutions in this module, focusing on the Lexicographic method for preventing cycling in optimization problems. This technique is crucial for ensuring efficient problem-solving.
This module features Assignment 6, focusing on the Shortest Path Problem and how to find the shortest path between any two nodes in a network. Students will apply learned techniques to solve practical problems.
In this module, students will analyze sensitivity analysis for the Min-cost flow and Shortest Path problems. They will learn how to assess impacts of changing parameters on solutions.
This module examines how changes in arc capacities affect the Min-cost flow problem. Students will learn about the Max-flow problem and engage with Assignment 7 to reinforce their understanding.
In this module, students will tackle Problem 3 from Assignment 7, focusing on the Min-cut Max-flow theorem. They will also learn about the labeling algorithm, essential for solving network flow problems.
This module discusses the concept of critical capacity of an arc in Max-flow problems. Students will also explore starting solutions for the Min-cost flow problem, enhancing their understanding of network flows.
In this module, students will learn about the Improved Max-flow algorithm, focusing on enhancing efficiency in network flow problems. This practical approach will prepare students for real-world applications.
This final module covers the Critical Path Method (CPM), providing students with tools to analyze project scheduling effectively. They will learn how to identify critical tasks and optimize project timelines.
This module addresses the Programme Evaluation and Review Technique (PERT), emphasizing its role in project management. Students will explore techniques for managing uncertainty in project durations and optimizing schedules.
This module provides a real-world example illustrating why the Simplex Algorithm is not polynomial time. Students will analyze this example to deepen their understanding of algorithm efficiency in optimization.
In the final module, students will be introduced to Interior Point Methods, exploring alternative approaches to solving linear programming problems. They will compare these methods with the Simplex Algorithm, enriching their optimization toolkit.