Linear Programming

In industry, managers often investigate more economical ways of doing business. They must consider physical limitations, standards of quality, customer demand, availability of materials, and manufacturing expenses as restrictions, or constraints, that determine how much of an item they can produce – usually to minimize production costs and maximize profits. This method of finding a feasible region and determining a point that gives minimum or maximum values is called linear programming.

Investigation- Maximizing Profit

The Elite Pottery Shoppe makes two kinds of birdbaths: a fancy glazed and a simple glazed. An unglazed birdbath requires 0.5hr to make using a pottery wheel and 3hr. in a kiln. A glazed birdbath takes 1hr. on the wheel and 18hr in the kiln. The company’s one pottery wheel is available for at most 8 hours per day. The three kiln’s can be used a total of at most 60hr/day. The company has a standing order for 6 unglazed birdbaths per day, so it must produce at least that many. The pottery shop’s profit on each unglazed birdbath is $10 and the profit on each glazed birdbath is $40. How many of each kind of birdbath should the company produce each day in order to maximize profit?

Step 1 Organize the information into the table.

Amount per unglazed birdbath / Amount per glazed birdbath / Constraining Value
Wheel hours
Kiln hours
Profit / Maximize

Step 2

Use your table to help write inequalities that reflect the constraints given, and be sure to include any commonsense constraints. Let x represent the number of unglazed birdbaths, and let y represent the number of glazed birdbaths. Graph the feasible region to show the combinations of unglazed and glazed birdbaths the shop could produce, and label the coordinates of the vertices. (Note: profit is not a constraint; it is what you are trying to maximize)

Step 3

It will make sense to produce only whole numbers of birdbaths. List the coordinates of all the vertices of the feasible region.

Step 4.

Write the equation that will determine profit based on the number of unglazed and glazed birdbaths produced. Calculate the profit that the company would earn at each of the feasible points you found in Step 3. You may want to divide this task among the members of the group.

Step 5

What number of each kind of birdbath should the Elite Pottery Shoppe produce to maximize profit? What is the maximum profit possible?

Practice: Marco is planning a snack of graham crackers and blueberry yogurt to provide at his school’s track practice. Because he is concerned about health and nutrition, he wants to make sure that the snack contains no more than 700 calories and no more than 20gr of fat. He also wants at least 17 grams of protein and at least 30% of the daily recommended value of iron. The nutritional content of each food is listed in the table. Each serving of yogurt costs $.30 and each graham cracker costs $.06. What combination of servings should Marco provide to minimize costs?

Serving / Calories / Fat / Protein / Iron
Graham Cracker / 1 cracker / 60 / 2gr / 2gr / 6%
Blueberry Yogurt / 4.5 oz. / 130 / 2gr / 5gr / 1%