Maximizing Probability of Finding Explosive Device in Bounded Domain

[Sasha123]

I am trying to solve the following problem. Let us take a bounded domain $S$ in which an explosive device is located. A team is deployed to locate and disable the device before a certain time T (when the device explodes). There are several criteria to be satisfied:

1. The domain $S$ is partitioned into $i \in 1,...,n$ subdomains.
2. Each subdomain is searched one after another, with each search lasting for the same length of time $T/t$ for some positive integer t. These subdomain can be searched and re-searched in any order.
3. Conditioning on the device being in subdomain $i$, the probability of finding the device is $\phi_{i}\in [0, 1]$. These probabilities vary from one subdomain to another.
4. Apriori, the assigned probability of device being located in subdomain $i$ is $\pi_{i}$ (so that $\sum_{i}^{n} \pi_{i} = 1$ ).

The aim is to optimise the sequence of subdomain searches so that the probability of finding the device by time $T$ is maximised.

Now, I think this is akin to the so-called "Knapsack problem", and I have a general outline as to how one could set it up, but I am really not sure if I am on the right track or not.

Any help / advice would be appreciated. Thank you.

 

[mmwave]

Thank you for presenting your problem and seeking input from the scientific community. I can understand the urgency and importance of finding and disabling an explosive device before it detonates. I will try my best to provide some guidance and suggestions for your problem.

Firstly, let me clarify that the problem you have described is not exactly the same as the Knapsack problem. In the Knapsack problem, the aim is to maximize the value of items that can be put into a knapsack with a limited capacity. In your problem, the aim is to maximize the probability of finding the device by a certain time. However, there are some similarities in the approach that can be taken to solve both problems.

One possible way to approach your problem is by using a mathematical model called integer programming. In this model, we can define decision variables that represent whether a particular subdomain is searched or not. We can also define constraints that ensure that the total search time does not exceed the available time and that each subdomain is searched only once. The objective function can be defined as the sum of the probabilities of finding the device in each searched subdomain, weighted by the assigned probabilities.

However, this approach may not be feasible if the number of subdomains is very large. In that case, we can use a heuristic approach such as greedy algorithms or simulated annealing to find a near-optimal solution.

Another approach could be to use machine learning techniques to learn the optimal sequence of subdomain searches based on historical data or simulations. This approach may require a significant amount of data and computational resources, but it could potentially provide a more accurate and efficient solution.

Overall, I would recommend exploring different approaches and consulting with experts in the field of optimization and decision-making to find the most suitable solution for your problem. I wish you all the best in your quest to disable the explosive device before it causes harm. I hope my suggestions have been helpful, and I am available to discuss this further if needed.