Mathematical Modeling of Forest Fire Fighting Why does the number of team members increase when the

If there is a fire in the forest, how many firefighters will be sent to fight the fire? If there are a large number of people, the fire loss will be small but the rescue cost may be large, and vice versa, the fire loss will be large and the rescue cost may be small.

Problem analysis: The loss fee includes the area burned by the forest, which is related to the duration of the fire, and thus the number of firefighters, the more people, the faster the fire is extinguished. The rescue fee is related to the number of firefighters and also to the time it takes to extinguish the fire.

The time of fire loss is t= 0, the time of starting fire fighting is t= t1, and the time of extinguishing the fire is t= t2If the area burned by the forest at time t is b(t), then the area burned by the forest in the fire is b(t2).

It is advisable to make the fire area a circular area with a radius of r, and when there is no one to put out the fire, r is proportional to t, and db dt is proportional to r, so db dt is proportional to t (0 t t1).After starting the fire (t> t1), the fire is under control and db dt decreases, and by t=t2, dbdt=0

The rescue fee is divided into two parts: one part is the consumption of equipment and the salary of firefighters, which is directly proportional to the number of people and time, and the other part is a one-time investment, which is only related to the number of people.

Model Assumption 1The cost of loss is proportional to the area of forest burned B, and the proportional factor is C1 (cost of loss per unit area).

2.In the time period (0 t t1), db dt = t, 0db dt reaches a maximum value t1 at t = t1 and is denoted as b

3.Send x firefighters, after starting to fight the fire (t> t1), the fire is under control, db dt begins to drop, db dt = b+ (x)(t?t1), where < x

When t = t2, db dt = 0

4.The cost per unit of time for each firefighter is C2, and there is also a one-time cost C3 for each team member, so that the total cost of rescue is [C2(T2?t1) +c3]x.

The cost per unit time of rescue increases, and in order to shorten the rescue time, it is necessary to increase the number of team members.

Refer to "Mathematical Model (Third Edition) Exercise Reference Answers", edited by Jiang Qiyuan, Xie Jinxing, Ye Jun, etc., upstairs is the answer to this question in the book.

x is the number of people; y is the loss; m is the speed of extinguishing the fire; b is the fire. So there is:

x=a（ym/b）+c

where a, c are constants.

This is from Jiang Qiyuan's "Mathematical Model", Chapter 3 Simple Optimization Model, 66 pages!

A must be greater than B for the fire to be extinguished.

To put it simply, the inverse proportional function should be relatively close, and there is no other data.

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