Extra Credit Problem 5 Page 76 Gothams Finest Group 3 Saud Aldegaiter Daniel Bonneville Tamara Vail October 18, 2011 IE 416 Dr. Parisay

Outline Problem Statement Summary of Problem Formulation of Problem WinQSB Inputs and Solution Solution Summary Objective Function Sensitivity Analysis RHS Sensitivity Analysis Report to Manager

Problem Statement Pg. 76 Problem 5 Each day, workers at the Gotham City Police Department work two 6-hour shift chosen from 12 a.m. to 6 a.m., 6 a.m. to 12 p.m., 12 p.m. to 6 p.m., and 6 p.m. to 12 a.m. The number of workers are needed during each shift are shown at table below. Workers whose two shifts are consecutive are paid $12 per hour; workers whose shifts are not consecutive are paid $18 per hour. Formulate an LP that can be used to minimize the cost of meeting the daily work-force demands of the Gotham City Police Department.

Summary Consecutive Shifts = $12 an hour Not Consecutive Shifts = $18 an hour Shift iTimeWorkers Needed 112 am to 6 am15 26 am to 12 pm5 312 pm to 6 pm12 46 pm to 12 am6

Formulation of Problem Decision Variables: X i = Number of workers whose two shifts are consecutive and start at shift I (Shift i=1,2,3,4) Y j = Number of workers whose two shifts are not consecutive and start at shift j (Shift j=1,2) X i ≥ 0 Y j ≥ 0

Formulation of Problem Objective Function: Minimize Z = 12(X 1 +X 2 +X 3 +X 4 ) + 18(Y 1 +Y 2 ) Shift 1Shift 2Shift 3Shift 4 X1X1 X1X1 X2X2 X2X2 X3X3 X3X3 X4X4 X4X4 Y1Y1 Y1Y1 Y2Y2 Y2Y Constraints: X 1 +X 4 +Y 1 =15 X 1 +X 2 +Y 2 =5 X 2 +X 3 +Y 1 =12 X 3 +X 4 +Y 2 =6

WinQSB Input Consecutive Shift 1 = X 1 Consecutive Shift 2 = X 2 Consecutive Shift 3 = X 3 Consecutive Shift 4 = X 4 Not Consecutive Shift 1 = Y 1 Not Consecutive Shift 2 = Y 2

WinQSB Solution

Solution Summary ShiftNumber of Workers Total Cost X1X1 5$60 X2X2 0$0 X3X3 4$48 X4X4 2$24 Y1Y1 8$144 Y2Y2 0$0 Z min =$276

Sensitivity Analysis of O.F. The only possible SA that can be done to the O.F. Is the not consecutive shift 2 because it is the only variable that has an RC value. Therefore we changed the hourly rate from $18 to $6 based on RC=12

Sensitivity Analysis of O.F.

Although the Z min decreases we realize that it is not practical for the hourly wage to decrease by $12.

Sensitivity Analysis of RHS We choose to decrease the number of workers on shift 3 for a couple reasons: ◦ It has one of the largest SP (SP=9) ◦ More flexibility in the allowable range which will minimize our cost.

Sensitivity Analysis of RHS

Report to Manager Minimum cost $276 Decrease hourly wage for non consecutive shift 2 from $18 to $6 Decrease number of workers in shift 3 from 12 to 11 which is within the range ShiftTime of ShiftNumber of Workers Total Cost Consecutive Shift 112am-12pm5$60 Consecutive Shift 26am-6pm0$0 Consecutive Shift 312pm-12am4$48 Consecutive Shift 4 4 6pm-6am2$24 Not Consecutive Shift 112am-6am & 12pm-6pm8$144 Not Consecutive Shift 26am-12pm & 6pm-12am0$0

QUESTIONS?