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Linear Programming Essay

993 words - 4 pages


This scheduling problem can be solved most expeditiously using linear programming. Let F denote the number of full-time employ- ees. Some number, F1, of them will work one hour of overtime between 5 PM and 6 PM each day and some number, F2, of the full- time employees will work overtime between 6 PM and 7 PM. There will be seven sets of part-time employees who begin their work day at hour j=j␣1,2,...,7,withP1beingthenumberofworkers beginning at 9 AM, P2 at 10 AM, . . . , P7 at 3 PM. Note that because part-time employees must work a minimum of four hours, none can start after 3 PM because the entire operation ends at 7 PM. Similarly, some number of part-time employees, ...view middle of the document...


Which is 40% of the day’s total requirement of 321 person hours. This also leads to the objective function. The total daily labor cost, which must be minimized, is:

Z 8(10.11)F 8.08(F1 F2) 7.82(10P1 9P2 8P3 7P4 6P5 5P6 4P7 −6Q4 −5Q5 −4Q6 −3Q7 −2Q8 −Q9)

Total overtime for a full-time employee is restricted to five hours or less, an average of one hour or less per day per employee. Thus, the number of overtime hours worked per day cannot exceed the number of full-time employees:

F1 F2 <= F
Because part-time employees must work at least four hours per day,

Q4 <= P1

for those leaving at the end of the fourth hour. At the end of the fifth hour, those leaving must be drawn from the P1 −Q4 remain- ing plus the P2 that arrived at the start of the second hour:
Q5 P1 P2 −Q4

Similarly, for the remainder of the day:

Q6 P1 P2 P3 −Q4 −Q5 Q7 P1 P2 P3 P4 −Q4 −Q5 −Q6 Q8 P1 P2 P3 P4 P5 −Q4 −Q5 −Q6 −Q7 Q9 P1 P2 P3 P4 P5 P6 −Q4 −Q5 −Q6 −Q7 −Q8

To ensure that all part-timers who began at 9 AM do not work more than seven hours:

Q4 Q5 Q6 Q7 P1 Q4 Q5 Q6 Q7 Q8 P1 P2
Q4 Q5 Q6 Q7 Q8 Q9 P1 P2 P3

Finally, to ensure that all part-time employees leave at some time:
P1 P2 P3 P4 P5 P6 P7 Q4 Q5 Q6 Q7 Q8 Q9


The resulting problem has 16 variables and 22 constraints. If integer programming software with sufficient capacity is not avail- able, the linear...

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