Consider the Following Problem

Malcolm's Glass Shop

Consider the following problem:

Malcolm owns a glass-molding machine and two sets of dies: one to make six-ounce juice glasses, and one for ten-ounce cocktail glasses. He wants to know how many juice glasses and how many cocktail glasses he ought to make each week in order to make the greatest profit.

Malcolm can make 100 cases of six-ounce glasses in six hours, and he can make 100 cases of ten-ounce glasses in five hours. On every case of six-ounce glasses, he makes a profit of $5.00; on every case of ten-ounce glasses he makes $4.50. Unfortunately there is only one customer for six-ounce glasses, and she will buy no more than 800 cases per week.

Malcolm works in his shop 60 hours every week and makes deliveries at the end of the week. Therefore he needs to store the whole week’s production in his 15,000 cubic foot warehouse. Each case of six-ounce glasses occupies 10 cubic feet, and a case of ten-ounce glasses occupies 20 cubic feet.

Your assignment: Express Malcolm’s problem mathematically, in the form of variables, objectives, and constraints. Try to approach the problem in three steps:

(1) Define the decision variables (those things that Malcolm has a choice about). For example, Malcolm has a choice regarding the number of cases of 6-oz glasses to produce. We could define a variable called X1, and let it represent the number of cases of 6-oz glasses in 100s. So if Malcolm decides to produce 100 cases of 6-oz glasses, then X1 =1.

(2) Define the objective (a mathematical expression for what Malcolm is trying to do). In this case Malcolm wants to maximize his profit, so come up with an expression that represents his profit in mathematical terms.

(3) Define the constraints (those facts which somehow limit Malcolm’s range of possible choices). For example, Malcolm can only sell 800 cases of 6-oz glasses, so

You don’t have to solve the problem, just express it as a mathematical model.

B60.23501Prof. Juran