Consider this scenario: your school is planning to make toques and mitts to sell at the winter festival as a fundraiser. The school’s sewing classes divide into two groups – one group can make toques, the other group knows how to make mitts. The sewing teachers are also willing to help out. Considering the number of people available and time constraints due to classes, only 150 toques and 120 pairs of mitts can be made each week. Enough material is delivered to the school every Monday morning to make a total of 200 items per week. Because the material is being donated by community members, each toque sold makes a profit of $2 and each pair of mitts sold makes a profit of $5.
In order to make the most money from the fundraiser, how many of each item should be made each week? It is important to understand that profit (the amount of money made from the fundraiser) is equal to the revenue (the total amount of money made) minus the costs: Proft = Revenue - Cost. Because the students are donating their time and the community is donating the material, the cost of making the toques and mitts is zero. So in this case, profit ≡ revenue.
If the quantity you want to optimize (here, profit) and the constraint conditions (more on them later) are linear, then the problem can be solved using a special organization called linear programming. Linear programming enables industries and companies to find optimal solutions to economic decisions. Generally, this means maximizing profits and minimizing costs. Linear programming is most commonly seen in operations research because it provides a “best” solution, while considering all the constraints of the situation. Constraints are limitations, and may suggest, for example, how much of a certain item can be made or in how much time.
Creating equations, or inequalities, and graphing them can help solve simple linear programming problems, like the one above. We can assign variables to represent the information in the above problem. x = the number of toques made weekly y = the number of pairs of mitts made weekly
Then, we can write linear inequalities based on the constraints from the problem. x ≤ 150 and y ≤ 120 | The students can only make up to 150 toques and up to 120 pairs of mitts each week. This is one restriction. | x + y ≤ 200 |
The total number of mitts and toques made each week cannot exceed 200. This is the material restriction. |
We may also want to consider that x ≥ 0 and y ≥ 0. This means that we cannot make -3 toques.