J E Beasley

OR-Notes are a series of introductory notes on topics that fall under the broad heading of the field of operations research (OR). They were originally used by me in an introductory OR course I give at Imperial College. They are now available for use by any students and teachers interested in OR subject to the following conditions.

A full list of the topics available in OR-Notes can be found here.

Linear programming tutorial solution


x1 = number of units of variety A produced
x2 = number of units of variety B produced

then the constraints are:

x1 + 2x2 <= 10 (production time)

2x1 + x2 <= 10 (space)

x1 <= 4 (demand)


x1, x2 >= 0

and the objective is

maximise 2x1 + 3x2

It is plain from the diagram below that the maximum occurs at the intersection of

x1 + 2x2 = 10 and

2x1 + x2 = 10

Solving simultaneously, rather than by reading values off the graph, we have that x1=x2=(10/3) with the value of the objective function being (50/3).