The DuFour Bottling Company has two machines for its bottle pro- duction. The problem each year is to devise a maintenance schedule. Maintenance of each machine lasts two months. In addition, only half the work force is available in July and August, so that only one ma- chine can be used during that period. Monthly demands for bottles are dt, t = 1,...,12. Machine k, k = 1,2, produces bottles at the rate of ak bottles per month but can produce less. There is also a labor constraint. Machine k requires lk labor days to produce ak, and the total available labor days per month are Lt, t = 1, . . . , 12.

(a) Formulate the problem of finding a feasible maintenance sched- ule in which all demands are satisfied.

(b) Modify your formulation to handle the objective: minimize the sum of the monthly fluctuations in labor utilization.

(c) Modify your formulation to handle the objective: minimize the largest monthly fluctuation in labor utilization.