Anglers' fishing problem

TL;DR

This paper models the angler's fishing problem as a marked renewal-reward process, aiming to find two optimal stopping times to maximize satisfaction within a fixed fishing period using dynamic programming.

Contribution

It introduces a novel application of renewal processes and dynamic programming to determine optimal fishing strategies with two stopping times.

Findings

Derived explicit formulas for optimal stopping times.

Quantified maximum expected satisfaction for the angler.

Demonstrated the effectiveness of the model through theoretical analysis.

Abstract

The considered model will be formulated as related to "the fishing problem" even if the other applications of it are much more obvious. The angler goes fishing. He uses various techniques and he has at most two fishing rods. He buys a fishing ticket for a fixed time. The fishes are caught with the use of different methods according to the renewal processes. The fishes' value and the inter arrival times are given by the sequences of independent, identically distributed (i.i.d.) random variables with the known distribution functions. It forms the marked renewal--reward process. The angler's measure of satisfaction is given by the difference between the utility function, depending on the value of the fishes caught, and the cost function connected with the time of fishing. In this way, the angler's relative opinion about the methods of fishing is modelled. The angler's aim is to have as much satisfaction as possible and additionally he has to leave the lake before a fixed moment. Therefore his goal is to find two optimal stopping times in order to maximize his satisfaction. At the first moment, he changes the technique of fishing, e.g. by excluding one rod and intensifying on the rest. Next, he decides when he should stop the expedition. These stopping times have to be shorter than the fixed time of fishing. The dynamic programming methods have been used to find these two optimal stopping times and to specify the expected satisfaction of the angler at these times.