Lecture 13

Introduction to Linear Programming

The maximum flow and minimum cut problems are examples of a general class of problems called linear programming. Many other optimization problems fall into this class, including minimum spanning trees and shortest paths, as well as several common problems in scheduling, logistics, and economics. Linear programming was used implicitly by Fourier and Jacobi in the early 1800s, but it was first formalized and applied to problems in economics in the 1930s by Leonid Kantorovich. Kantorivich’s work was hidden behind the Iron Curtain (where it was largely ignored) and therefore unknown in the West. Linear programming was rediscovered and applied to shipping problems in the late 1930s by Tjalling Koopmans. The first complete algorithm to solve linear programming problems, called the simplex method, was published by George Dantzig in 1947. Koopmans first proposed the name “linear programming" in a discussion with Dantzig in 1948.
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Sensitivity Analysis and Shadow Prices

In linear programming, sensitivity analysis investigates how changes to the model's parameters (like objective function coefficients or constraint coefficients) affect the optimal solution, helping determine the robustness of the solution and identify critical parameters
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What Is Linear Programming? Meaning, Methods, and Examples

Linear programming is a technique in algebra that uses linear equations to determine how to arrive at the optimal situation (maximum or minimum) as an answer to a mathematical problem, assuming the finiteness of resources and the quantifiable nature of the end optimization goal. Linear programming (LP) uses many linear inequalities pertaining to a given scenario to determine the “optimal” value one can obtain under those constraints. A classic example would be calculating the “optimal” production levels to maximize profits, given the restrictions of supplies and personnel. In the “real world,” linear programming is an essential subfield of mathematics known as optimization methods. This area of research (or at least its applicable findings) is used in resource allocation and management. These “real-world” systems may have dozens or even hundreds of variables. In algebra, however, you will only see the basic (and graphable) linear example with two variables.
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Using Excel to Solve Linear Programming Problems

Technology can be used to solve a system of equations once the constraints and objective function have been defined. Excel has an add-in called the Solver which can be used to solve systems of equations or inequalities.There are two methods – one uses tables within the worksheet while the other uses only the constraints. The second method is preferred when we know the constraints since it is much faster!
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Introduction to Shadow Pricing

Shadow pricing is the practice of assigning a monetary value to an item, commodity, or service that is not ordinarily bought and sold in any marketplace. Shadow pricing is the practice of assigning a monetary value to an item, commodity, or service that is not ordinarily bought and sold in any marketplace.
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Linear Programming Optimization: Foundations

Linear programming is a powerful optimization technique that is used to improve decision making in many domains. This is the first part in a multi-part series that will cover important topics having to do with linear programming. This first article will be the most simple and is meant to cover the basics before getting into more advanced linear programming topics.

In this article, we will discuss what constitutes a linear programming problem, go over how linear programming works and why it is powerful and run a linear programming example in Python.