Question

3. (35 points) In the post office example, suppose that each full-time employee works 8 hours per day. Thus, Monday's requirement of 17 workers may be viewed as a requirement of 8(17) = 136 hours. The post office may meet its daily labor requirements by using both full-time and part-time employees. During each week, a full-time employee works 8 hours a day for five consecutive days, and a p A full-time employee costs the post office S15 per hour, whereas a part-time employee (with reduced fringe benefits) costs the post office only $10 per hour. Union requirements limit part-time labor to 25% of weekly labor requirements. Formulate an LP to minimize the post office's weekly labor costs. art-time employee works 4 hours a day tor five consecutive days

Answer 1

Let the number of full time employees be F and let the number of part time employees be P.

We need to minimize the weekly labor costs which is the objective function. As each full-time employee works 8 hours for 5 days, total weekly hours worked by a full time employee is 8*5 = 40 hours. Hence weekly cost of one full-time employee is 15*40 = $600. As each part-time employee works 4 hours for 5 days, total weekly hours worked by a full time employee is 4*5 = 20 hours. Hence weekly cost of one full-time employee is 10*20 = $200. The objective function Z is the the total cost of F full-time employees and P part-time employees and this needs to be minimized. The objective function is as below

Minimize Z = 600*F + 200*P

Now there are 2 constraints that need to be met. First is that the employees should meet the weekly labor requirements. As the daily labor requirements is 136 hours, the weekly labor requirements is 136*5 = 680 hours. As each full time employee works 40 hours per week and each part-time employee works 20 hours per week, this constraint is modeled as below

40*F+20*P = 680

The next constraint is that a maximum of 25% of weekly labor requirement can be met by the part-time employees. The total weekly labor requirement is 680 hours, hence 25% of this is 170 hours. The total hours worked by part-time employees cannot exceed 170 hours. Hence this constraint is modeled as

20*P $\small \leq$ 170

In addition as number of employees cannot be fraction we also have a constraint that

F, P belong to integer

Hence the LP formulation is as below

Minimize Z = 600*F + 200*P

s.t. 40*F+20*P = 680

20*P $\small \leq$ 170

F, P belong to integer