# Math Help - Optimising probability of missile system

1. ## Optimising probability of missile system

Hello
We have a missile system and we want to optimise the probability of hitting and destroying a target.

The function we want to maximise is
f(x1)=(1-(1-x1)^x2

which is the probability of succeeding the mission, using a certain type and amount of missiles.

x1= probability of hitting and destroying the target with 1 missile
x2= number of missiles launched

Assuming x1 for a particular missile to be 0,3 the probability of succeeding the first time is:
(1-(1-0,3)^1=0,3
and the second time is
(1-(1-0,3)^2=0,51
etc

Now we have a couple of constraints when designing a new missile.

we introduce some variables.
x3= price per missile
x4=weight per missile

The limitations determined by customer are
x2*x3<1 million dollar
x2*x4<100 kg

which describes how much a customer is willing to pay in terms of cost and weight to succeed in a mission.

Furthermore the relationship between probability of 1 missile (x1) and the weight(x4) and cost(x3) of the missile is NOT linear. It is exponential (thus meaning if we want to increase probability of 1 missile just a little bit, we must change our design so that it becomes a lot more expensive and heavy).

The relationship is described with the following equations:

x1=1-(1/(e^(4,5*x3)))
Meaning x1=0.59 for a 200 000$missile but only 0.83 for a 400 000$ missile as an example

x1=1-(1/(e^(0,03*x4)))
Meaning x1=0.45 for a 20kg missile but only 0.69 for a 40kg missile as an example

The question is of course: what is the optimal combination of x1 and x2 giving the maximum probability of succeeding the mission? given the constraints and relationships between x1 and x3 and x4

I guess it can be resolved with some kind of non-linear optimisation method. I may have forgotten some aspect or misformulated the problem, please notice me if so.

2. ## Re: Optimising probability of missile system

Originally Posted by Lobotomy
The relationship is described with the following equations:

x1=1-(1/(e^(4,5*x3)))
Meaning x1=0.59 for a 200 000$missile but only 0.83 for a 400 000$ missile as an example

x1=1-(1/(e^(0,03*x4)))
Meaning x1=0.45 for a 20kg missile but only 0.69 for a 40kg missile as an example
So you can express x3 and x4 in terms of x1?

Then these constraints

Originally Posted by Lobotomy
x2*x3<1 million dollar
x2*x4<100 kg
in terms of x1 and x2, and very soon in terms of f(x1, x2), which you've called f(x1) but is apparently a function of two variables.

So that gives you two upper limits on f(x1, x2). The lower one then determines a relation between x1 and x2.

3. ## Re: Optimising probability of missile system

So you can express x3 and x4 in terms of x1?

Yeah.. but im not sure if I have modelled it correctly, because there is no linear relationship between price and weight. Some stuff can be expensive and not so heavy, while others are the opposite. it all depends. Do you have a better way to model the relationships where both price and weight relates to X1 but not to each other...or is that a contradiction in terms?

Anyhow assuming this model is correct ill do as you said.

x3=1-e^(-4,5*x1)
x4=1-e^(-0,03*x1)

Then into the constraints....

x2*1-e^(-4,5*x1)<1
x2*1-e^(-0,03*x1)<100

and very soon in terms of f(x1, x2), which you've called f(x1) but is apparently a function of two variables.

So that gives you two upper limits on f(x1, x2). The lower one then determines a relation between x1 and x2.

dont really know what to do here. should i express x2 in terms of x1 as well? like this:
x2=1/(1-e^(-4,5*x1))
x2=100/(1-e^(-0,03*x1))

which one to put into the maximising function (1-(1-x1)^x2

4. ## Re: Optimising probability of missile system

Originally Posted by Lobotomy
Anyhow assuming this model is correct ill do as you said.

x3=1-e^(-4,5*x1)
x4=1-e^(-0,03*x1)
True.

Originally Posted by Lobotomy
Then into the constraints....

x2*1-e^(-4,5*x1)<1
x2*1-e^(-0,03*x1)<100
Woah.

\displaystyle \begin{align*} x_1\ &=\ 1 - \frac{1}{e^{\frac{3}{100}x_4}} \\ \\ \Rightarrow\ \ x_4\ &=\ - \frac{100}{3} \ln (1 - x_1) \end{align*}

Plug that in. (To the second constraint.)