NYNEX must schedule round-the-clock coverage for its telephone operators. To keep the number of different shifts down to a manageable level, it has only four different shifts

Operators work eight-hour shifts and can begin work at either midnight, 8 a.m., noon, or 4 p.m. Operators are needed according to the following demand pattern, given in four-hour time blocks.

Time Period Operators Needed
midnight to 4 a.m. 4
4 a.m. to 8 a.m. 6
8 a.m. to noon 90
Noon to 4 p.m. 85
4 p.m. to 8 p.m. 55
8 p.m. to midnight 20

Formulate this scheduling decision as a linear programming problem, defining fully your decision variables and then giving the objective function and constraints.

Let X1 = the number of telephone operators starting their shift at midnight.
X2 = the number of telephone operators starting their shift at 8 a.m.
X3 = the number of telephone operators starting their shift at noon.
X4 = the number of telephone operators starting their shift at 4 p.m.
Min: X1 + X2 + X3 + X4
subject to X1 > 4 Midnight to 4 a.m.
X1 > 6 4 a.m. to 8 a.m.
X2 > 90 8 a.m. to noon
X2 + X3 > 85 noon to 4 p.m.
X3 + X4 > 55 4 p.m. to 8 p.m.
X4 > 20 8 p.m. to midnight
X1, X2, X3, X4 > 0