An objective function is a mathematical expression in linear programming that maximizes or minimizes some quantity (often profit or cost, but any goal may be used). Objective function: Minimize Z= X, + X2 + X3 + X4 + X5 + X6 Constraints are restrictions that limit the degree to which a manager can pursue an objective. Subject to: (7 A.M. - 11 A.M.) (11 A.M. - 3 P.M.) (3 P.M. - 7 P.M.) x + X 214 X2 + X3 2 15 X3 + X 210 X + X 2 12 Xg+ X 25 X + X, 24 (7 P.M. - 11 P.M.) (11 P.M.- 3 A.M.) (3 A.M. - 7 A.M.) For all X,20 A linear programming software shows the optimal solution as follows: X, = 4 (11 P.M. - 3 A.M.) Xg+ X 25 x + X, 24 (3 A.M. - 7 A.M.) For all X 20 A linear programming software shows the optimal solution as follows: X = 4 X = 12 X=3 X = 7 Xg=5 X=0 To find the minimum total staff required for one day's operation, substitute the optimal solution into the objective function. Z = 4 + 12 + 3 + 7 +5+0 =(enter your response as a whole number). 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 P.M., 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 i, where i = 1, 2, 3, 4, 5, or 6. Period 1 2 3 4 Time 3 A.M. - 7 A.M 7 A.M. - 11 A.M. 11 A.M. - 3 P.M. 3 P.M. - 7 P.M. 7 P.M. - 11 P.M. 11 P.M. - 3 A.M. Number of Servers Required 4 12 15 10 13 4 6 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 X, equal the number of servers beginning work in time period i, where i = 1, 2, 3, 4, 5, 6.) Objective function: Minimize Z =