Extra Credit
Problem 5 Page 76
Gothams Finest
Group 3
Saud Aldegaiter
Daniel Bonneville
Tamara Vail
October 18, 2011
IE 416
Dr. Parisay
Problem Statement:
Each day, workers at the Gotham City Police Department work two 6-hour shifts 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 / Time / Workers Needed1 / 12 am to 6 am / 15
2 / 6 am to 12 pm / 5
3 / 12 pm to 6 pm / 12
4 / 6 pm to 12 am / 6
Consecutive Shifts = $12 an hour
Not Consecutive Shifts = $18 an hour
Dr. Parisay’s comments are in red.
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)
Formulation:
Shift 1 / Shift 2 / Shift 3 / Shift 4X1 / X1
X2 / X2
X3 / X3
X4 / X4
Y1 / Y1
Y2 / Y2
15 / 5 / 12 / 6
X1+X4+Y1=15 / X1+X2+Y2=5 / X2+X3+Y1=12 / X3+X4+Y2=6
Objective Function: Minimize Z = 12*12(X1+X2+X3+X4) + 12*18(Y1+Y2)
The coefficients of OF in the following table should be updated
Win QSB Input:
Win QSB Solution:
Then the total cost will be 276*12 and no change in solution.
From this Win QSB output we created this table to better explaining the solution.
Shift / Number of Workers / Total CostX1 / 5 / $60
X2 / 0 (it is fine here, but not in report to the manager) / $0
X3 / 4 / $48
X4 / 2 / $24
Y1 / 8 / $144
Y2 / 0 / $0
Zmin = / $276
This table shows the number of workers that needs to work each shift. Then it shows the total cost for each shift worked. The summation of all these costs will give us the minimum cost.
Sensitivity Analysis (Objective Function):
After getting the minimum cost of the solution we decided to perform some sensitivity analysis on this problem. We noticed that non consecutive shift 2 was the only NBV with a reduced cost was $12/hr, so we decided to perform sensitivity analysis on this shift by reducing the wage of the non consecutive shifts from $18/hr to $6/hr. Solving with a cost of $6 produced the following solution.
By doing this sensitivity analysis we were able to have people work during non consecutive shift 2. However, it is not practical to reduce a wage by $12/hr. (good observation) This would not happen in the real world. So even though it does help reduce the overall minimum cost, we cannot even consider using this alternative.
Perform SA on a BV.
Sensitivity Analysis (RHS):
From the solution, we noticed that shift 1 and shift 3 had a shadow price of 9. Meaning that if we were able to decrease the number of people needed for either one of these shifts, then the cost would go down by 9 times the number of units it was decreased. Instead of trying to decrease both shifts, we just decided to decrease the number in shift 3. The motivation behind this is because of the allowable minimum value. The range for shift 3 goes all the way down to 4. This means that shift 3 can decrease by up to 8 people. Shift 1 can only be reduced by 5 people. (Good observation). Looking at this, we can see that shift 3 has more room to decrease the amount of people they need. So we feel like it is the logical choice since it would be easier to decrease the people.
By decreasing the amount of people we need in shift 3 from 12 to 11 we can see that this dropped the overall minimum cost. The minimum cost dropped $9 dollars. This is consistent with the fact that we only decreased the amount of people by 1.
Report to Manager:
Dear Manager,
As we know there is a daily work-force demand for each shift of the Gotham City Police Department. We know that the hourly wage for consecutive versus not consecutive shifts are not the same and therefore contribute differently to the overall cost. We used a program called WinQSB Linear Programming in order to minimize the cost of meeting the daily work-force demand. The minimum total daily cost will be $276 if we follow the schedule below.
Shift / Time of Shift / Number of Workers / Total CostConsecutive Shift 1 / 12am-12pm / 5 / $60
Consecutive Shift 3 / 12pm-12am / 4 / $48
Consecutive Shift 44 / 6pm-6am / 2 / $24
Not Consecutive Shift 1 / 12am-6am & 12pm-6pm / 8 / $144
If we wanted to have workers start at shift 2 and not work consecutive shifts (Y2) the hourly wage would have to be changed from $18 to $6 per hour. We realize that this is not a practical solution to consider.
Shift / Workers Needed1 (12am-6am) / 15
2 (6am-12pm) / 5
3 (12pm-6pm) / 12
4 (6pm-12am) / 6
If we reduce the demand for workers on shift 3 for each worker from 12 to any value down to 4 workers, the minimum total cost will reduce by $9. Also for shift 1 if we reduce the demand from 15 workers to any value down to 10 workers, for each worker the minimum total cost will reduce by $9.
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