David, Diana and Lydia are the sole partners and workers in a company that produces fine clocks. David and Diana are each available to work a maximum of 40 hours per week at the company, while Lydia is available to work a maximum of 20 hours per week. The company makes two different types of clocks: a grandfather clock and a wall clock. To make a clock, David (a mechanical engineer) assembles the inside mechanical parts of the clock while Diana (a woodworker) produces the hand carved wood casings. Lydia is responsible for taking orders and shipping the clocks. The amount of time required for each of these tasks is shown below. Task Time required Grandfather clock Wall clock Assemble clock mechanism 6 hours 4 hours Carve wood casing 8 hours 4 hours Ship finished clock 3 hours 3 hours Each grandfather clock built and shipped yields a profit of $300, while each wall clock yields a profit of $200. The three partners now want to determine how many clocks of each type should be produced per week to maximize the total profit using linear programming. They have asked for your help
(a) Formulate a linear programming model for this problem. In doing so, consider the following:
What are the decision variables for this problem?
What is the objective for this problem? Using your decision variables, formulate the objective function.
What are the constraints in this problem? Using your decision variables, formulate these constraints.