TY - JOUR

T1 - Stochastic Mitra–Wan forestry models analyzed as a mean field optimal control problem

AU - Higuera-Chan, Carmen G.

AU - Laura-Guarachi, Leonardo R.

AU - Minjárez-Sosa, J. Adolfo

N1 - Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2023/10

Y1 - 2023/10

N2 - 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 MN 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 MN . 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 MN .

AB - 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 MN 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 MN . 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 MN .

KW - Average optimality

KW - Forestry model

KW - Golden rule

KW - Mean field theory

UR - http://www.scopus.com/inward/record.url?scp=85166417830&partnerID=8YFLogxK

U2 - 10.1007/s00186-023-00832-1

DO - 10.1007/s00186-023-00832-1

M3 - Artículo

AN - SCOPUS:85166417830

SN - 1432-2994

VL - 98

SP - 169

EP - 203

JO - Mathematical Methods of Operations Research

JF - Mathematical Methods of Operations Research

IS - 2

ER -