6. A forest fire is burning down a narrow valley 3 miles wide at a speed of 40 f
ID: 3129706 • Letter: 6
Question
6. A forest fire is burning down a narrow valley 3 miles wide at a speed of 40 feet per minute. The fire can be contained by cutting a firebreak through the forest across the valley. It takes 30 seconds for one person to clear one foot of the firebreak. The value of lost timber is $4,000 per square mile. Each person hired is paid $12 per hour, and it costs $30 to transport and supply each person with the appropriate equipment. A. Develop a model for determining how many people should be sent to contain the fire and for determining the best location for the firebreak. B. Implement you model on a spreadsheet and find the optimal solution using Solver.
Explanation / Answer
Solution: 1 mile=5280 foot.
Therefore 3mile=15840 foot
we could hire 7920 firefighters and have them each work for 1 minute and the firebreak would be complete. It takes a certain amount of creativity, or at least willingness to go beyond the stated facts, to recognize that there must be some principle for establishing a more reasonable upper limit to the number of firefighters to employ in this task. This principle is that each additional unit of a resource to be used to solve a problem should not cost more than the value of the damage that would be averted by adding that unit of the resource. In the context of the present problem, this means that additional firefighters should be hired to help cut the firebreak as long as the cost of each additional firefighter does not exceed the value of the timber that would be saved by hiring him/her. Recognizing this principle allows us to establish a constraint on the number of firefighters that should be hired.
The full set-up for the solution to this problem consequently appears as follows:
A
B
1
Objective Function
=B12/B4
(minutes needed to cut firebreak)
2
3
Decision variables:
4
Number of firefighters:
2
(starting value = 2)
5
6
Relevant Variables:
7
Feet of firebreak to cut:
15840
8
Cost per foot to cut:
0.1
(=$12.00/120)
9
Hourly pay cost to cut firebreak:
1584
(= b8 * b7)
10
Cost per firefighter:
30
11
Cost per minute of burn:
$90.91
12
Person minutes to cut firebreak:
7920
(= 15840/2) loss of timber would equal 7920 * $90.91 = $720,000 if one firefighter were used
13
Total cost of firefighters:
=(30*B4) + B9
14
Total cost of burn:
=(B12/B4) * B11
15
Prev. Value of Timber Saved
=720000 –
((7920/(B4 -1)) * 90.91)
16
Current value of timber saved:
=720000 –
((7920/(B4)) * 90.91)
17
Incremental Value of Timber Saved:
=B16 – B15
18
Cost of last firefighter hired:
= 30 + ((B7/(B4 * 2)) *0.2)
19
20
Constraints:
LHS
Constraint
RHS
21
B18
<=
B17
22
B4
EQ
Integer
23
B4
>=
2
24
25
Where to cut firebreak:
=ROUNDUP(B14,0) * 40
feet from current position of fir
This problem illustrates not only that it is often necessary to use one's creativity to devise reasonable constraints on the solution to an LP problem, but also that it is necessary equally often to utilize the given data in creative ways to produce the numbers that actually implement the constraints that one has devised.
A
B
1
Objective Function
=B12/B4
(minutes needed to cut firebreak)
2
3
Decision variables:
4
Number of firefighters:
2
(starting value = 2)
5
6
Relevant Variables:
7
Feet of firebreak to cut:
15840
8
Cost per foot to cut:
0.1
(=$12.00/120)
9
Hourly pay cost to cut firebreak:
1584
(= b8 * b7)
10
Cost per firefighter:
30
11
Cost per minute of burn:
$90.91
12
Person minutes to cut firebreak:
7920
(= 15840/2) loss of timber would equal 7920 * $90.91 = $720,000 if one firefighter were used
13
Total cost of firefighters:
=(30*B4) + B9
14
Total cost of burn:
=(B12/B4) * B11
15
Prev. Value of Timber Saved
=720000 –
((7920/(B4 -1)) * 90.91)
16
Current value of timber saved:
=720000 –
((7920/(B4)) * 90.91)
17
Incremental Value of Timber Saved:
=B16 – B15
18
Cost of last firefighter hired:
= 30 + ((B7/(B4 * 2)) *0.2)
19
20
Constraints:
LHS
Constraint
RHS
21
B18
<=
B17
22
B4
EQ
Integer
23
B4
>=
2
24
25
Where to cut firebreak:
=ROUNDUP(B14,0) * 40
feet from current position of fir
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