Developing Linear Programming Models for Simple Problems

Instructor: Melanie Olczak

Melanie has taught high school Mathematics courses for the past ten years and has a master's degree in Mathematics Education.

This lesson will provide instruction for how to develop a linear programming model for a simple manufacturing problem. The steps to solve such a problem will be described and a graphical representation is presented.

Linear Programming

Any time some one decides to go into business there are various parameters they must consider, such as costs of production, how much can be produced, and at what price point they will sell their goods. Did you know that mathematics and manufacturing go hand in hand? Linear programming is a mathematical process that allows us to look at all the parameters to determine a maximum or minimum value where all of the parameters can be modeled by lines.

Suppose you own a candy company and you can produce two types of candies, chocolates and sweethearts. We definitely want to maximize our profit for selling the candy. Our projections indicate that consumers will demand at least 100 lbs of chocolate and 80 lbs of sweethearts to be produced daily. Because of limitations on production capacity, no more than 200 lbs of chocolate and 170 lbs of sweethearts can be made daily. We must also satisfy a shipping contract, where a total of at least 200 lbs of candy must be shipped each day. If the profit on each pound of chocolate is \$1.75 and the profit on each pound of sweetheart is \$2.25, how many pounds of each type of candy should we sell to maximize our profit?

There is a lot of information in this paragraph so the first thing we are going to do is determine what we are talking about. Then we can translate this paragraph into mathematics. In order to do this we will follow five steps, starting with translating from words to mathematics.

A variable is a letter that represents a number or quantity that is unknown or can change. In this case the first question we ask is what are we talking about? Since we are talking about chocolate and sweethearts, those are going to be our variables. We need to be very specific when defining our variables. We can use any letters we want, but since we will be graphing this information it is easier to use x and y. We cannot simply say that x is chocolate and y is sweethearts, because I'm not sure what that means. Are we talking about the cost of chocolate and sweethearts? Or are we talking about the quantity of chocolate and sweethearts?

Since we are trying to find out how much we must produce, we are talking about the quantity of chocolate and sweethearts. Therefore to define our variables, we will let x equal the pounds of chocolate and y equal the pounds of sweethearts that will be produced.

Now that we have our variables, we can use them to write equations and inequalities.

Step 2 - Define the Objective Function

The objective function of a linear programming problem is the whole point to the problem. What are we trying to do? What is our objective? In this case we want to maximize profit. Since we don't know what that profit will be, we will use P to represent our profit.

What does it say in the problem about profit? Well, we know that each pound of chocolate gives us \$1.75 of profit and each pound of sweethearts gives us \$2.25 of profit. The total profit, P is equal to \$1.75 times the number of pounds of chocolate produced plus \$2.25 times the number of pounds of sweethearts produced.

There is still a great deal of information in the problem that we must account for. These parameters are called constraints because they tell us what limitations we have on our variables. If we didn't have any limitations, there would be no maximum profit.

Step 3 - Writing the Constraints

Each constraint will be a separate inequality. If we look back at the question we see that there are three sentences that provide some constraints on how much we can manufacture.

We need to make at least 100 lbs of chocolate and 80 lbs of sweethearts. In mathematical terms this means x must be greater than or equal to 100 and y must be greater than or equal to 80.

We can't make more than 200 lbs of chocolate or 170 lbs of sweethearts. Here we have x must be less than or equal to 200 and y must be less than or equal to 170.

We need a total of at least 200 lbs of both types of candy. This means that x plus y must be greater than or equal to 200.

We can write these inequalities in a list.

Now that we have our variables, objective function and all of our constraints, we are ready to graph.

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