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

The function we want to maximiseis

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 havea 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.

Furthermorethe 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.

edit: this may be more of a calculus problem than probability problem... move it to the correct forum if desired

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# Optimising probability of hitting a target

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