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
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:
and the second time is
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
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:
Meaning x1=0.59 for a 200 000$ missile but only 0.83 for a 400 000$ missile as an example
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.