Optimising probability of hitting a target

[Lobotomy]

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,x2)=(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 exampleThe 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.
edit: this may be more of a calculus problem than probability problem... move it to the correct forum if desired

 

[viraltux]

I think you need to tell us first what government you're working for

 

[Stephen Tashi]

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

If you use that model, you make x4 a function of x3. Is it supposed to be?

 

[Lobotomy]

If you use that model, you make x4 a function of x3. Is it supposed to be?

not sure...im not sure if I have modeled it correctly, because there is not suppose to be 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?

 

[Lobotomy]

I think you need to tell us first what government you're working for

north korea

 

[Stephen Tashi]

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?

It would be contradictory, if by "relates" you mean "is a function of".

A typical military model would model the delivery accuracy of the warhead and the probability of damaging the target as a function of both the delivery accuracy of the warhead and the effect of the warhead. I don't know any abstract principles that tell you how the factors of accuracy and war head effect are related to cost and weight. In an actual analysis, you'd have to get data about real missile systems or hypothesized missile systems, including their costs. Then you might be able to empirically fit a function describing X1 as a function of both X2 and X3.