You work for a small clinic that provides annual wellness checkups for children and adults. The clinic is well liked by the community, which results in the clinic always booking the entire 8 hour day full of appointments. You have been asked to help determine how many appointments should be reserved for children and how many should be reserved for adults each day. As a business, you would like to maximize your profit. Each child appointment provides the clinic with a $20 profit while an adult appointment provides you with $45 profit. However, adult appointments take 20 minutes each while children appointments only take 10 minutes. Since many schools in the area start their fall sports season soon, there is a high demand for children appointments (during which a sports physical is performed). Because of this, your clinic has decided that it wants to make sure that at least twice as many children appointments are available as compared to adult appointments.
(a) Formulate an integer program to maximize the clinic’s profit. Clearly identify the sets, parameters and decision variables. Also, describe the objective function and each of the constraints in words.
(b) Use Excel Solver to find the optimal solution and profit.
(c) Which constraints are binding at the optimal solution? What are the practical implications of this?
(d) Due to the high demand for children appointments, the clinic is considering charging a bit more for those appointments. This would result in a profit of $25 per child appointment instead of a $20 profit.
i. How would this change affect the optimal solution and profit?
ii. What effect on the graphical solution does this change have with regard to the feasible region and optimal solution?
(e) After you realize what the optimal solution would be if you raise the price of a child appoint- ment, you figure that the community will not happy if you only implement this change. You are now considering adding a restriction that would not allow the clinic to schedule more than 3 times as many children appointments as adult appointments. You think this will allow you to increase the cost of the child’s appointment to $25 but still keep the community happy.
i. How should you change your model to accommodate this situation?
ii. What is the optimal solution and profit?
iii. What effect on the graphical solution does this change have with regard to the feasible region and optimal solution?
iv. Solve the linear programming relaxation (i.e. allow the variables to be non-integer) of the problem described in part (e) thus far. What is the optimal solution and the resulting objective function value? How does it compare to the integer solution found in part (e-ii)?
v. Will the optimal solution of part (e-ii) change if you choose to not allow the clinic to schedule more than 4 times as many children appointments as adult appointments instead of 3? If so, indicate how and provide the associated objective function value.
(f) Would you recommend the clinic make one of the changes proposed in part (e) or stay with their current plan as indicated in part (a)? Why? If you choose a change from part (e), identify which change you would choose.