I\'m mostly having trouble with part b of this question. Any help would be great

Question

I'm mostly having trouble with part b of this question. Any help would be greatly appreciated!

4. (25 pts) At the post office on the Kentucky Street, each employee works exactly for 5 consecutive days per week. To provide a satisfactory customer service, the post office needs the following number of employees each day: (a) Provide an LP formulation to minimize the number of employees. (b) Suppose some employees are willing to do overtime and work for one more day right after their 5 day regular schedule. Suppose that overtime labor rate is 50% more than regular labor rate. Provide an LP formulation to minimize the labor costs.

Explanation / Answer

let xi be the number of workers hired for working in i th day

objective function:

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

constraints

x1 + x4 + x5 + x6 + x7 >= 9

x1 + x2 + x5 + x6 + x7 >= 6

x1 + x2 + x3 + x6 + x7 >= 5

x1 + x2 + x3 + x4 + x7 >= 8

x1 + x2 + x3 + x4 + x5 >= 11

x2 + x3 + x4 + x5 + x6 >=13

x3 + x4 + x5 + x6 + x7 >=4

b)

let xi be the number of workers hired for i th day

let xi* be the number of workers working for day i and doing one more day over time

let dailar labor rate for one employee is 1 unit , so for 5 days its 5 units and for overtime day its 1.5 units

objective function:

min 5*(x1 + x2 + x3 + x4 + x5 + x6 + x7) + 1.5 * (x1* + x2* + x3* + x4* + x5* + x6* + x7*)

constraints

x1 + x4 + x5 + x6 + x7 +x3* >= 9

x1 + x2 + x5 + x6 + x7 +x4* >= 6

x1 + x2 + x3 + x6 + x7 + x5*>= 5

x1 + x2 + x3 + x4 + x7 +x6* >= 8

x1 + x2 + x3 + x4 + x5 +x7* >= 11

x2 + x3 + x4 + x5 + x6 +x1* >=13

x3 + x4 + x5 + x6 + x7 +x2*>=4

xi > xi*

xi,xi*>0

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