Problem 3: Hospital Assignment
The director of Burtonville Civil Defense Agency has been
ordered to draw up a disaster plan for assigning casualties to hospitals in the event of a serious
earthquake. For simplicity, we will assume that causalities will occur at two points in the city and will
be transported to three hospitals. It is estimated that there will be 300 casualties at point A and 200
at point B. Travel times to hospitals 1, 2, and 3 are 25, 15, and 10 minutes, respectively; from point B
they are 20, 5, and 15 minutes. Hospital capacities for emergency cases are 250, 150, and 150 patients.
How should the victims be assigned to hospitals to minimize the total time lost in transporting them?
The objective is to minimize the total time lost to transport.
The variables that the policy maker controls are the number of victims sent to each hospital to each site
(2 sites X 3 hospitals = 6 variables).
The constraints are the capacity of each hospital and the number of victims at each site.
The empty solver table looks like this if Xab represents number transported from site a to hospital b:
Variables
XA1 XA2 XA3 XB1 XB2 XB3
Weights
Minutes per victim
Objective
Constraints
Site A
Site B
Hospital 1 Capacity
Hospital 2 Capacity
Hospital 3 Capacity
Equation
LT,GT,etc.
Value
The weights in the objective function are the minutes per victim for each transportation option: 25, 10,
15, 20, 5, 15:
25XA1 + 10XA2 + 15XA3 + 20XB1 + 5XB2 + 15XB3
The constraint values are given by
Site A 300, Site B 200, Hospital 1 250, Hospital 2 150, Hospital 3 150. Each variable contributes to the
hospital and site constraints of the entities at either end of the transport. These get a 1 in the table with
other entries being 0. The equations are
Site A XA1 + XA2 + XA3 = 300
Site B XB1 + XB2 + XB3 = 200
Hospital 1 Capacity XA1 + XB1 ≤ 250
Hospital 2 Capacity XA2 + XB2 ≤ 150
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Hospital 3 Capacity XA3 + XB3 ≤ 150
Variables
XA1 XA2 XA3 XB1 XB2 XB3
150
Weights
Minutes per victim
Constraints
Site A
Site B
Hospital 1 Capacity
Hospital 2 Capacity
Hospital 3 Capacity
0
150
50
150
0
Objective
25
1
0
1
0
0
15
1
0
0
1
0
10
1
0
0
0
1
20
0
1
1
0
0
5
0
1
0
1
0
15
0
1
0
0
1
Equation
300
200
200
150
150
LT,GT,etc.
Value
300
200
250
150
150
We set this up in solver for simplex-LP with the addition of integer constraints and non-negative
variables.
The results given by solver are
Variables
Weights
Minutes per victim
XA1
150
25
XA2
0
15
XA3
150
10
XB1
50
20
XB2
150
5
XB3
0
15
Objective
7000
In other words, we send 150 victims from site A to hospital 1, none from that site to hospital 2, and the
other 150 to hospital 3. From site B we send 50 to hospital 1, 150 to hospital 2 and none to hospital 3.
REPORTS
Solver generates three reports (shown below).
Binding Constraints and Slack. From the Answer report, in addition to the information already
mentioned, we see which of our constraints are binding or not. In this case, the hospital 2 and 3
capacities are binding, as are the number of victims, but the capacity of hospital 1 is not. In other words,
this resource is not maxxed out by this situation but the others are. The slack value of 50 means that at
this optimal point there are 50 capacity units at hospital 1 that are not being used.
Allowable Increase/Decrease and Shadow Prices. For each variable and constraint, Solver gives us an
allowable increase and decrease. For variables, this represents how much the coefficient could change
(up or down) without changing the values of the variables in the solution (though it might change the
value of the objective function). In this case we see that XB2 could be infinitely higher or lower without
changing the solution. This means that the optimal solution is independent of the length of the trip from
site B to hospital 2.
The increase/descrease on the constraints is one measure of how sensitive the solution is to changes in
this constraint. In this case,
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All three hospitals’ capacity can increase by any amount without changing the solution but any decrease
in 2 or 3 would change it while hospital one could have up to 50 fewer slots without changing the
solution. The number of victims cannot increase without changing the solution but the number at site B
can be as low as 50 before the solution would change.
Finally, the shadow prices indicate the marginal cost of the constraint – telling us what change we can
expect for a one unit change in the value of the constraint. In the table below what we see is that a one
unit increase in hospital 2 or 3 capacity will result in a 15 minute savings in total transport time. An
increase in one victim at either site, though, will increase the total transport time (A by 25 and B by 20
minutes).
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Microsoft Excel 14.1 Answer Report
Worksheet: [hospital-transport-solver-solution.xlsx]Hospital Transport
Report Created: 10/17/2013 1:36:49 PM
Result: Solver found a solution. All constraints and optimality conditions are satisfied.
Solver Engine
Engine: Simplex LP Solution Time: 0.571563 Seconds. Iterations:
1 Subproblems: 0
Solver Options
Max Time Unlimited, Iterations Unlimited, Precision 0.000001
Max Subproblems Unlimited, Max Integer Sols Unlimited, Integer Tolerance 1%, Assume NonNegative
Objective Cell (Min)
Cell
$F$2
XB2
Variable Cells
Cell
$F$2
XB2
Constraints
Cell
$H$10
$H$11
$H$7
$H$8
$H$9
Name
Original Value
150
Final Value
150
Name
Original Value
150
Final Value
Integer
150 Contin
Name
Cell Value
Hospital 2 Capacity Equation
Hospital 3 Capacity Equation
Site A Equation
Site B Equation
Hospital 1 Capacity Equation
Formula
150
150
300
200
200
$H$10<=$J$10
$H$11<=$J$11
$H$7=$J$7
$H$8=$J$8
$H$9<=$J$9
Status
Binding
Binding
Binding
Binding
Not Binding
Slack
0
0
0
0
50
Microsoft Excel 14.1 Sensitivity Report
Worksheet: [hospital-transport-solver-solution.xlsx]Hospital Transport
Report Created: 10/17/2013 2:18:18 PM
Variable Cells
Cell
Name
Final
Reduced
Objective
Allowable
Allowable
Value
Cost
Coefficient
Increase
Decrease
$B$2
XA1
150
0
25
5
10
$C$2
XA2
0
5
15
1E+30
5
$D$2
XA3
150
0
10
10
1E+30
$E$2
XB1
50
0
20
10
5
$F$2
XB2
150
0
5
5
1E+30
$G$2
XB3
0
10
15
1E+30
10
Constraints
Cell
Name
Final
Shadow
Constraint
Allowable
Allowable
Value
Price
R.H. Side
Increase
Decrease
$H$10
Hospital 2 Capacity Equation
150
-15
150
50
50
$H$11
Hospital 3 Capacity Equation
150
-15
150
150
50
$H$7
Site A Equation
300
25
300
50
150
$H$8
Site B Equation
200
20
200
50
50
$H$9
Hospital 1 Capacity Equation
200
0
250
1E+30
50
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