Problem 5 Page 75
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
1
2
3
4
Time
12 am to 6 am
6 am to 12 pm
12 pm to 6 pm
6 pm to 12 am
Workers Needed
15
5
12
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
X1
Shift 2
X1
X2
X4
Y1
Shift 3
Shift 4
X2
X3
X3
X4
Y1
15
Y2
5
12
Y2
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:
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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
X1
X2
X3
X4
Y1
Y2
Number of Workers
Total Cost
5
$60
0 (it is fine here, but not in
$0
report to the manager)
4
$48
2
$24
8
$144
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
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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
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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,
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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
Consecutive Shift 1
Consecutive Shift 3
Consecutive Shift 44
Not Consecutive Shift 1
Time of Shift
12am-12pm
12pm-12am
6pm-6am
12am-6am & 12pm-6pm
Number of Workers
5
4
2
8
Total Cost
$60
$48
$24
$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
1 (12am-6am)
2 (6am-12pm)
3 (12pm-6pm)
4 (6pm-12am)
Workers Needed
15
5
12
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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