Linear programming problem. Show your work! A post office requires different num

Question

Linear programming problem. Show your work!

A post office requires different numbers of full-time employees on different days of the week The number of full-time employees required on each day is given in the table below Union rules state that each full-time employee must work five consecutive days and then receive two days off. For example, an employee who works Monday to Friday must be off on Saturday and Sunday. The post office wants to meet its daily requirements using only full-time employees. Formulate a model that the post office can use to minimize the number of full-time employees that must be hired. In the post office problem above, 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 part-time employee works 4 hours a day for five consecutive days. A full-time employee costs the post office $15 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 a model to minimize the post office's weekly labor costs.

Explanation / Answer

2. Decision variables: x1, x2,....x7

Let x1 be the number of workers that work from monday till friday.

x2 be the number of workers who work from tuesday till saturday.

x3 from wednesday to sunday, x4 from thurday to monday, and so on.

Objective Function:

Minimize x1+x2+x3+x4+x5+x6+x7

Constraints:

X1+X4+X5+X6+X7>=17

X1+X2+X5+X6+X7>=13

X1+X2+X3+X6+X7>=15

X1+X2+X3+X4+X7>=19

X1+X2+X3+X4+X5>=14

X2+X3+X4+X5+X6>=16

X3+X4+X5+X6+X7>=11

X1,X2,X3,X4,X5,X6,X7>=0

3. Here, Xi represent number of full ytime employees, and Yi represents number of part time employees, i=1 to 7.

Objective Function: Minimize [15(X1+X2+X3+X4+X5+X6+X7)+10(Y1+Y2+Y3+Y4+Y5+Y6+Y7)]

Constraints:

8*X1+4*Y1+8*X4+4*Y4+8*X5+4*Y5+8*X6+4*Y6+8*X7+4*Y7>=136

8*X1+4*Y1+8*X2+4*Y2+8*X5+4*Y5+8*X6+4*Y6+8*X7+4*Y7>=104

8*X1+4*Y1+8*X2+4*Y2+8*X3+4*Y3+8*X6+4*Y6+8*X7+4*Y7>=120

8*X1+4*Y1+8*X2+4*Y2+8*X3+4*Y3+8*X4+4*Y4+8*X7+4*Y7>=152

8*X1+4* Y1+8*X2+4*Y2+8*X3+4*Y3+8*X4+ 4* Y4+8*X5+4*Y5>= 112

8*X2+4*Y2+8*X3+4* Y3+8*X4+4*Y4+8*X5+4*Y5+8*X6+4*Y6>=128

8*X3+4*Y3+8*X4+4*Y4+8*X5+4*Y5+8*X6+4*Y6+8*X7+4*Y7>=88

X1,X2,X3,X4,X5,X6,X7,Y1,Y2,Y3,Y4,Y5,Y6,Y7>=0

4*(Y1+Y2+Y3+Y4+Y5+Y6+Y7)<=210 {25% OF TOTAL REQUIRED HRS PER WEEK=840 HRS}

MON X1 X4 X5 X6 X7 17 TUE X1 X2 X5 X6 X7 13 WED X1 X2 X3 X6 X7 15 THU X1 X2 X3 X4 X7 19 FRI X1 X2 X3 X4 X5 14 SAT X2 X3 X4 X5 X6 16 SUN X3 X4 X5 X6 X7 11

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