Crashing CPM | Optimization Methods – Linear Programming | Engineering Essentials

Optimization Methods – Linear Programming | Engineering Essentials
This playlist provides a comprehensive academic introduction to linear programming and related optimization techniques in operations research. Linear programming seeks the best outcome in a system modelled by linear equations or inequalities. It involves optimizing a linear objective function subject to linear constraints, yielding a feasible solution space that is a convex polytope.
This is aimed at advanced undergraduate and graduate students in engineering, management science, mathematics, and statistics (IITs, MIT, Stanford, IIMs, etc.) and covers evergreen concepts fundamental to analytics and resource optimization. This playlist highlights applications across industries (transportation, energy, manufacturing, telecommunications) and problem domains (planning, routing, scheduling, assignment, network flows). By studying these videos, learners will gain deep insight into modelling and solving optimization problems in science, engineering and management without reference to temporal details – ensuring the content remains relevant over time.
Core Topics
• Linear Programming Fundamentals: Definition of an LP problem with a linear objective and linear constraints. Formulating decision variables, objective function and feasible region. Discussion of convex polytope, feasible solutions, optimum points, and duality.
• Graphical Solution Method: Solving LP problems with two variables using geometric interpretation. Plotting constraints as lines, identifying the feasible polygon, and locating the optimal vertex. Use of objective contour lines to visualize optimization.
• Canonical and Standard Form: Transforming LPs into standard/canonical form for algorithmic solution. Representing all variables as nonnegative and writing constraints as equalities. Introduction of artificial variables (if needed) and matrix notation for compact formulation.
• Slack and Surplus Variables: Introducing slack variables for “≤” constraints and surplus variables for “≥” constraints to convert inequalities into equalities. Explanation of how adding a nonnegative slack variable to a “≤” constraint yields an equivalent equality, enabling systematic solution methods.
• Simplex Method: Dantzig’s simplex algorithm for linear programming. Explanation of tableau form, basic and non basic variables, pivoting operations, and iterating through basic feasible solutions. Discussion of optimality conditions and convergence.
• Constraint Inequalities: Role of constraints and inequalities in defining the feasible region. Discussion of binding vs non-binding constraints via slack variable values. Characterization of feasible region as the intersection of half-spaces (a convex polytope).
• Network Flow Models: Optimization in graph networks (e.g. shortest path, maximum flow, minimum-cost flow).
• Critical Path Method (CPM): Scheduling and project planning. CPM as an algorithm for determining the longest sequence of dependent project activities, which defines the project duration. Computing earliest and latest event times, identifying the critical path, and understanding float/slack in project schedules.
• Project Crashing (CPM Crashing): Techniques to reduce project completion time at increased cost. Formulating project crashing as an optimization trade-off (shortening critical activities subject to budget and resource constraints).
• Transportation Models and Algorithms: The transportation problem in logistics and supply-chain optimization - methods for finding initial feasible solutions (e.g. Northwest Corner, Vogel’s approximation, Least-Cost Method) and then optimizing (Stepping-Stone or MODI). Discussion of network structure and degeneracy in transportation models.
• Assignment Models and Hungarian Algorithm: The assignment problem for one-to-one task allocation (special case of transportation/min-cost flow). Formulating it as a binary integer LP. Introduction to the Hungarian (Munkres) algorithm – a combinatorial optimization method to find the minimum-cost assignment in polynomial time.
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