Linear programming is used to obtain optimal solutions for operations research. Using linear programming allows researchers to find the best, most economical solution to a problem within all of its limitations, or constraints. Many fields use linear programming techniques to make their processes more efficient. These include food and agriculture, engineering, transportation, manufacturing and energy.

#### TL;DR (Too Long; Didn't Read)

Linear programming provides a method to optimize operations within certain constraints. It is used to make processes more efficient and cost-effective. Some areas of application for linear programming include food and agriculture, engineering, transportation, manufacturing and energy.

## Linear Programming Overview

Using linear programming requires defining variables, finding constraints and finding the objective function, or what needs to be maximized. In some cases, linear programming is instead used for minimization, or the smallest possible objective function value. Linear programming requires the creation of inequalities and then graphing those to solve problems. While some linear programming can be done manually, quite often the variables and calculations become too complex and require the use of computational software.

## Food and Agriculture

Farmers apply linear programming techniques to their work. By determining what crops they should grow, the quantity of it and how to use it efficiently, farmers can increase their revenue.

In nutrition, linear programming provides a powerful tool to aid in planning for dietary needs. In order to provide healthy, low-cost food baskets for needy families, nutritionists can use linear programming. Constraints may include dietary guidelines, nutrient guidance, cultural acceptability or some combination thereof. Mathematical modeling provides assistance to calculate the foods needed to provide nutrition at low cost, in order to prevent noncommunicable disease. Unprocessed food data and prices are needed for such calculations, all while respecting the cultural aspects of the food types. The objective function is the total cost of the food basket. Linear programming also allows time variations for the frequency of making such food baskets.

## Applications in Engineering

Engineers also use linear programming to help solve design and manufacturing problems. For example, in airfoil meshes, engineers seek aerodynamic shape optimization. This allows for the reduction of the drag coefficient of the airfoil. Constraints may include lift coefficient, relative maximum thickness, nose radius and trailing edge angle. Shape optimization seeks to make a shock-free airfoil with a feasible shape. Linear programming therefore provides engineers with an essential tool in shape optimization.

## Transportation Optimization

Transportation systems rely upon linear programming for cost and time efficiency. Bus and train routes must factor in scheduling, travel time and passengers. Airlines use linear programming to optimize their profits according to different seat prices and customer demand. Airlines also use linear programming for pilot scheduling and routes. Optimization via linear programming increases airlines' efficiency and decreases expenses.

## Efficient Manufacturing

Manufacturing requires transforming raw materials into products that maximize company revenue. Each step of the manufacturing process must work efficiently to reach that goal. For example, raw materials must past through various machines for set amounts of time in an assembly line. To maximize profit, a company can use a linear expression of how much raw material to use. Constraints include the time spent on each machine. Any machines creating bottlenecks must be addressed. The amount of products made may be affected, in order to maximize profit based on the raw materials and the time needed.

## Energy Industry

Modern energy grid systems incorporate not only traditional electrical systems, but also renewables such as wind and solar photovoltaics. In order to optimize the electric load requirements, generators, transmission and distribution lines, and storage must be taken into account. At the same time, costs must remain sustainable for profits. Linear programming provides a method to optimize the electric power system design. It allows for matching the electric load in the shortest total distance between generation of the electricity and its demand over time. Linear programming can be used to optimize load-matching or to optimize cost, providing a valuable tool to the energy industry.

References

- University College London: EdExcel Decision Mathematics 1: Linear Programming
- TU Delft Publications: Aerodynamic Shape Optimization by means of Sequential Linear Programming Techniques
- PLOS One: Use of Linear Programming to Develop Cost-Minimized Nutritionally Adequate Health Promoting Food Baskets
- University of Regina Math Central: Linear Programming
- The University of Texas at Austin: Product Mix Problem

About the Author

J. Dianne Dotson is a science writer with a degree in zoology/ecology and evolutionary biology. She spent nine years working in laboratory and clinical research. A lifelong writer, Dianne is also a content manager and science fiction and fantasy novelist. Dianne features science as well as writing topics on her website, jdiannedotson.com.