# Encyclopedia of Operations Research and Management Science

2001 Edition
| Editors: Saul I. Gass, Carl M. Harris

# Linear programming

Reference work entry
DOI: https://doi.org/10.1007/1-4020-0611-X_545

Linear programming is one of the most widely used techniques of operations research and management science. Its name means that planning (programming) is being done with a mathematical model (called a linear programming model) where all the functions in the model are linear functions.

## LINEAR PROGRAMMING MODELMODELS

Linear programming models come in a variety of forms. To illustrate one common form, consider the problem of determining the most profitable mix of products for a manufacturer. Let n be the number of possible products. For each product j (j = 1, 2, ..., n), a decision variable xj is introduced to represent the decision on its production rate. Let cj be the profit per unit of product j produced, and let Z be the total rate of profit resulting from the choice of product mix. This choice is constrained by the limited capacities of the production facilities available for these products. Let m be the number of different types of facilities needed. For each type i (i = 1, 2, ..., m...
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