Anglers’ fishing problem

Abstract

The model considered here will be formulated in relation to the “fishing problem,” even if other applications of it are much more obvious. The angler goes fishing, using various techniques, and has at most two fishing rods. He buys a fishing pass for a fixed time. The fish are caught using different methods according to renewal processes. The fish’s value and the interarrival times are given by the sequences of independent, identically distributed random variables with known distribution functions. This 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 fish 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 modeled. The angler’s aim is to derive as much satisfaction as possible, and additionally he must leave the lake by a fixed time. Therefore, his goal is to find two optimal stopping times to maximize his satisfaction. At the first moment, he changes his technique, e.g., by discarding one rod and using the other one exclusively. Next, he decides when he should end his outing. These stopping times must be shorter than the fixed time of fishing. Dynamic programming methods are used to find these two optimal stopping times and to specify the expected satisfaction of the angler at these times.