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

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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

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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.

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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

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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

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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 Y2Y2 155126 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

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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

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WinQSB Solution

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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

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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

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Sensitivity Analysis of O.F.

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

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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.

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Sensitivity Analysis of RHS

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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

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QUESTIONS?

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