The staffing needs of a 24 hour cafeteria are tabled below: Each staff member wo

Question

The staffing needs of a 24 hour cafeteria are tabled below:

Each staff member works eight consecutive hours and may be required to start at any hour. (They won't get lunch. They eat on the job). Those whose starting time falls between midnight and 6am (inclusive) receive a 10% shift bonus. Determine how many employees must start at which times of day to meet these requirements with minimum cost. Make sure that the number of employees starting at each hour is integer valued.

Time of Day Minimum Staff on Duty 2-6 4 6-10 8 10-14 10 14-18 7 18-22 12 22-2 4

Explanation / Answer

Let xibe the number of workers starting work at shift time i

Minimize Z= 1.1*(x1 + x2 ) + C(x3+x4+x5+x6)

subject to

x1+x2 >=8

x2+x3>=10

x3+x4>=7

x4+x5>=12

x5+x6>=4

x6+x1>=4

xi >=0 ,for all i={1,2,3,4,5,6}

xi are integers for all values of i={1,2,3,4,5,6}

constraints are added this way because and worker starting work at shift i , will be working for shift i and i+1 both since he has to work for 8 continuous hours.

So at any shift , say shift i, the workers who are present will be the ones who started work at shift i-1 and at shift i.

optimal solution to this integer program is x1=0, x2=8, x3=2, x4=12, x5=0, x6=4

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