User:Jftsang/Bomber problem

In operations research, the bomber problem asks the following: Suppose that a bomber must travel a further t discrete time steps before reaching its target to drop its bombs. At each step, there is a probability p that it will encounter an enemy fighter. The bomber carries n air-to-air missiles. If it fires k missiles when it encounters a fighter, then it has a probability $1-\alpha ^{k}$ of surviving the encounter (each missile has a probability $\alpha$ of missing the fighter). If the bomber encounters a fighter at a given state $(n,t)$, let $k(n,t)$ be the optimal number of missiles to fire in order to maximise the probability Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle V(n, t)} that the bomber will reach its target (where $V(n,0)=1$).

Conjecture

Richard R. Weber has proved that $k(n,t)$ is nonincreasing in n and that $n-k(n,t)$ is nondecreasing in n. He has also conjectured and attempted to prove that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle k(n,t)} is also nondecreasing in n.