Optimal hostage rescue problem where an action can only be taken once – case where its effectiveness vanishes thereafter

In this paper, we propose a mathematical model for an optimal hostage rescue problem. Suppose that a person is taken as a hostage and that a decision has to be made from among three alternatives: storm for rescue, wait up to the next point in time for an opportunity to present itself or take an action of negotiation which might save the situation. Here, it is assumed that the action of negotiation can only be taken once and its effectiveness completely vanishes thereafter. The objective is to find an optimal decision rule so as to maximize the probability of the hostage not being killed.