Variable Definition

Employee Type  Shift Worked   Variable name

Full time          9:00am to 5:00pm   x1

Full time          9:00am to 6:00pm x2

Full time           9:00am to 7:00pm x3

part time          9:00am to 1:00pm x4

part time          9:00am to 2:00pm x5

part time          9:00am to 3:00pm x6

part time          9:00am to 4:00pm x7

part time          10:00am to 2:00pm x8

part time          10:00am to 3:00pm x9

part time          10:00am to 4:00pm x10

part time          10:00am to 5:00pm x11

part time          11:00am to 3:00pm x12

part time          11:00am to 4:00pm x13

part time          11:00am to 5:00pm x14

part time          11:00am to 6:00pm x15

part time          Noon to 4:00 pm x16

part time          Noon to 5:00 pm x17

part time          Noon to 6:00 pm x18

part time          Noon to 7:00 pm x19

part time          1:00pm to 5:00pm x20

part time          1:00pm to 6:00pm x21

part time          1:00pm to 7:00pm x22

part time          2:00pm to 6:00pm x23

part time          2:00pm to 7:00pm x24

part time          3:00pm to 7:00pm x25

ModelObjective Function:Minimize (x1+x2+x3)*8*10.11+(x2+2x3)*8.08+7.82*(4x4+5x5+6x6+7x7+4x8+5x9+6x10+7x11+4x12+5x13+6x14+7x15+4x16+5x17+6x18+7x19+4x20+5x21+6x22+4x23+5x24+4x25)

Subject toConstraintx1+x2+x3+x4+x5+x6+x7>=14x1+x2+x3+x4+x5+x6+x7+x8+x9+x10+x11>=250.5*x1+0.5*x2+0.5*x3+x4+x5+x6+x7+x8+x9+x10+x11+x12+x13+x14+x15>=260.5*x1+0.5*x2+0.5*x3+x4+x5+x6+x7+x8+x9+x10+x11+x12+x13+x14+x15+x16+x17+x18+x19 >=38x1+x2+x3+x5+x6+x7+x8+x9+x10+x11+x12+x13+x14+x15+x16+x17+x18+x19+x20+x21+x22 >=55x1+x2+x3+x6+x7+x9+x10+x11+x12+x13+x14+x15+x16+x17+x18+x19+x20+x21+x22+x23+x24 >= 60x1+x2+x3+x7+x10+x11+x13+x14+x15+x16+x17+x18+x19+x20+x21+x22+x23+x24+x25 >= 51x1+x2+x3+x11+x14+x15+x17+x18+x19+x20+x21+x22+x23+x24+x25 >= 29x2+x3+x15+x18+x19+x21+x22+x23+x24+x25 >= 14x3+x19+x22+x24+x25 >= 9

(x4+x5+x6+x7+x8+x9+x10+x11+x12+x13+x14+x15+x16+x17+x18+x19+x20+x21+x22+x23+x24+x25)<= 0.4*(x1+x2+x3+x4+x5+x6+x7+x8+x9+x10+x11+x12+x13+x14+x15+x16+x17+x18+x19+x20+x21+x22+x23+x24+x25)xi>=0xi are integer.

Solution Using Excel.:Employee Type  Shift Worked   Variable name

Full time          9:00am to 5:00pm   x1 17

Full time          9:00am to 6:00pm x2 0

Full time           9:00am to 7:00pm x3 9

part time          9:00am to 1:00pm x4 0

part time          9:00am to 2:00pm x5 0

part time          9:00am to 3:00pm x6 0

part time          9:00am to 4:00pm x7 0

part time          10:00am to 2:00pm x8 0

part time          10:00am to 3:00pm x9 0

part time          10:00am to 4:00pm x10 0

part time          10:00am to 5:00pm x11 0

part time          11:00am to 3:00pm x12 9

part time          11:00am to 4:00pm x13 4

part time          11:00am to 5:00pm x14 0

part time          11:00am to 6:00pm x15 0

part time          Noon to 4:00 pm x16 16

part time          Noon to 5:00 pm x17 0

part time          Noon to 6:00 pm x18 0

part time          Noon to 7:00 pm x19 0

part time          1:00pm to 5:00pm x20 0

part time          1:00pm to 6:00pm x21 0

part time          1:00pm to 7:00pm x22 0

part time          2:00pm to 6:00pm x23 5

part time          2:00pm to 7:00pm x24 0

part time          3:00pm to 7:00pm x25 0

                              Total Cost   $3343.12 per day

2. What are the limitations of the model used to answer question 1?

Limitation: We assume that the variables have linear relationships, which may not be correct. While

adding more workers the extra worker added may have diminishing productivity for many reasons.

The productivity of full time and part time workers may not be same for reasons such as motivation, benefits etc.

The part time workers may have differential wages based on what period of the day they are working.

3. Costs might be reduced by relaxing the constraint that no more than 40% of the day’s requirement be met by part-timers. Would changing the 40% to a higher value significantly reduce costs?

Yes. Since part time workers are relatively cheaper as compared to full time workers cost can be reduced by eliminating the constraint. But for the reasons given in above quation, it may not be practicable or there may be other qualitative issues. Mathematically it is possible, if we relax the condition to 50%, the cost can be reduced to 3149.60

4. What’s you conclusion and recommendation for the overall case study?

Overall: A combination of full time and part time workers would leads to better management of workers and would reduce the cost. The costs can be further reduced by relaxing the constraint on part time workers.