By Keith Jackson and the Mind Tools Team
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# Linear Programming

BartCo

When resources are limited, you need to allocate them wisely.

If you've only got limited resources at your disposal, then it's helpful to calculate how best to maximize those resources – whether that's time, money, or space.

Let's say, for example, that you have 50 square feet of office space to use for storage.

Your budget is \$200, and there are a variety of cabinet types and sizes from which to choose.

How do you optimize the space you have available, and stay within the allotted budget?

Or suppose you have three delivery trucks, and 10 drop-off points.

How do you plan the most efficient route and schedule for these trucks?

Or consider that you manufacture three products using the same basic raw materials. However, as each product uses different amounts of material, some are more expensive to produce than others. A few of the materials are perishable, and need to be used quickly. How much of each product should you manufacture to minimize your cost? And which combination produces the least waste?

Questions like these may seem very complex. With so many variables and constraints to take into consideration, how do you decide what to do? The answer is to use linear programming.

Linear programming is a mathematical technique that determines the best way to use available resources. Managers use the process to help make decisions about the most efficient use of limited resources – like money, time, materials, and machinery.

#### Note:

You can use linear programming only if there is a linear relationship between the variables you're looking at. The relationship is linear if a unit change in one is connected to a constant amount of change in the other. On a graph, a linear relationship is shown as a straight line.

## The Linear Programming Technique

To help you understand linear programming, we'll work through an example. Let's say your company makes two products – Joops and Goops. The capacity of your manufacturing unit is...