# Objective Function: Definition, Principle & Example

Instructor: Martin Gibbs

Martin has 16 years experience in Human Resources Information Systems and has a PhD in Information Technology Management. He is an adjunct professor of computer science and computer programming.

This lesson will define the objective function, the principle behind its use, and how it is used in business. We will provide examples of how to use the objective function.

## The Objective Function

Yes, you do use algebra in the real world. Mathematics, in particular linear equations, are prevalent when performing analysis or designing a business strategy. One of these linear functions is the objective function.

The objective function is a means to maximize (or minimize) something. This something is a numeric value: in the real world it could be the cost of a project, a production quantity, profit value, or even materials saved from a streamlined process. With the objective function, you are trying to arrive at a target for output, profit, resource use, etc.

We need to look at the relationships between constraints and any limitations within the business itself. These can include production capacity limits, resource availability, or even technology.

From a mathematical perspective, the technical representation of the objective function is as follows:

It looks like a scary function, but let's break it down and take a look at its individual components, giving an example of maximizing profit.

• ci is the coefficient that matches the ith variable
• Xi is the ith decision variable

If you are even more confused, don't worry. Think about it like this: if we want to maximize profit, Xi is a possible activity in the project. The i just indicates which activity it is; either the first or the 100th. Think of i as a slot in a list of items.

Next, ci is the net value that activity i generates (again this could be the first or the 100th).

Finally, the entire scary-looking sigma symbol (the big E-looking thing) is telling us to add everything together. That is, all activities and net value that activity provides. Not all activities will provide value or contribute, so they'd just have a coefficient of zero (better yet, don't add them into the formula!). Once we understand that, things are a little easier to work with. We can take that and map out a formula.

### Example: Products and Profits

We will continue with the goal of profit. Let's say that we manage a company that produces two product types: green tea and white tea blends. Further, our company has a toasting operation, for toasting leaves, and a packaging operation.

The following is true of these products:

Green Tea White Tea
Fixed Costs \$83 per day for both
Profit \$2.35/pound \$1.63/pound
Toasting Capacity 250 pounds/day 200 pounds/day

How many of each blend can we produce each day to maximize profit? Remember that we have to consider the constraints; in our case we are limited to production limits of 250 and 200 pounds per day for each product.

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