Wichitas famous Sethi Restaurant is open 24 hours a day Serv

Wichita\'s famous Sethi Restaurant is open 24 hours a day. Servers report for duty at 3 A.M., 7 A.M., 11 A.M., 3 PM., 7 P.M., or 11 P.M., and each works an 8-hour shift. The following table shows the minimum number of workers needed during the 6 periods into which the day is divided Decision vairable: x-number of workers reporting for the start of working period , where i-1, 2, 3, 4, 5, or 6. Period Number of Servers Required 4 12 15 10 13 4 Time 3 A.M. -7 A.M 2 3 PM.-7 P.M. 7 P.M-11 P.M 11 P.M.-3 A.M. 4 Owner Avanti Sethi\'s scheduling problem is to determine how many servers should report for work at the start of each time period in order to minimize the total staff required for one day\'s operation. (Hint: Let Xi equal the number of servers beginning work in time period i, where i 1, 2, 3, 4, 5, 6.) Objective function Minimize Z = X, +X2+X3+X4 +X5+X6

Solution

GIVEN : Minimize Z = X1 + X2 + X3 +X4 + X5 + X6

Let us take X1 + X2 = 12

X2 + X3 = 15

X3+ X4 = 10

X4 + X5 = 13

X5 + X6 =4

X6 + X1 = 4

Convert all terms in X1 Form:

X1 + X2 = 12

X2 = 12- X1

X2 + X3 = 15

Substitute X2 value in above equation

12 - X1 + X3 = 15

X3 = 3 + X1

X3 + X4 = 10

X4 = 7 - X1

X4 + X5 = 13

7 - X1 + X5 = 13

X5 = 6 +X1

X5 + X6 = 4

6 + X1 + X6 = 4

X6 = -2 - X1

Substitute all the Xi in minimize formula

Minimize Z = X1 + X2 + X3 + X4 + X5 + X6

= X1 +12 - X1 + 3 + X1 +7 -X1 +6 +X1 - 2 - X1

= 26 ( All the X1\'s will be canceled)

so , The optimal Solution results in total workers = 26