Extra Credit Problem 5 Page 76 Gothams Finest

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 �Win. QSB 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 Shift i 1 2 3 4 Time 12 am to 6 am to 12 pm to 6 pm to 12 am Workers Needed 15 5 12 6 Consecutive Shifts = $12 an hour Not Consecutive Shifts = $18 an hour

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

Formulation of Problem Shift 1 X 4 Y 1 15 Constraints : X 1+X 4+Y 1=15 Shift 2 X 1 X 2 Y 2 5 Shift 3 X 2 X 3 Y 1 12 Shift 4 X 3 X 4 Y 2 6 X 1+X 2+Y 2=5 X 2+X 3+Y 1=12 X 3+X 4+Y 2=6 Objective Function: Minimize Z = 12(X 1+X 2+X 3+X 4) + 18(Y 1+Y 2)

Win. QSB 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

Win. QSB Solution

Solution Summary Shift X 1 X 2 X 3 X 4 Y 1 Y 2 Number of Workers 5 0 4 2 8 0 Total Cost $60 $0 $48 $24 $144 $0 Zmin = $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.

Sensitivity Analysis of O. F. �Although the Zmin 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 Shift Consecutive Shift 1 Consecutive Shift 2 Consecutive Shift 3 Consecutive Shift 44 Not Consecutive Shift 1 Not Consecutive Shift 2 Time of Shift 12 am-12 pm 6 am-6 pm 12 pm-12 am 6 pm-6 am 12 am-6 am & 12 pm 6 am-12 pm & 6 pm 12 am Number of Workers 5 0 4 2 8 Total Cost $60 $0 $48 $24 $144 0 $0 �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

QUESTIONS?