## Abstract

This paper concerns with a stochastic version of the discrete-time Mitra–Wan forestry model defined as follows. Consider a system composed by a large number of N trees of the same species, classified according to their ages ranging from 1 to s. At each stage, all trees have a common nonnegative probability of dying (known as the mortality rate). Further, there is a central controller who must decide how many trees to harvest in order to maximize a given reward function. Considering the empirical distribution of the trees over the ages, we introduce a suitable stochastic control model M_{N} to analyze the system. However, due N is too large and the complexity involved in defining an optimal steady policy for long-term behavior, as is typically done in deterministic cases, we appeal to the mean field theory. That is we study the limit as N→ ∞ of the model M_{N} . Then, under a suitable law of large numbers we obtain a control model M , the mean field control model, that is deterministic and independent of N, and over which we can obtain a stationary optimal control policy π^{∗} under the long-run average criterion. It turns out that π^{∗} is one of the so-called normal forest policy, which is completely determined by the mortality rate. Consequently, our goal is to measure the deviation from optimality of π^{∗} when it is used to control the original process in M_{N} .

Original language | English |
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Pages (from-to) | 169-203 |

Number of pages | 35 |

Journal | Mathematical Methods of Operations Research |

Volume | 98 |

Issue number | 2 |

DOIs | |

State | Published - Oct 2023 |

### Bibliographical note

Publisher Copyright:© 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

## Keywords

- Average optimality
- Forestry model
- Golden rule
- Mean field theory