WHEN TO CUT A STAND OF TREES?

This paper revisits the debate over the economic optimality of different timber harvest rules. The traditional Faustmann and Maximum Sustained Yield rotation determinations are confined to a deterministic world. Once stochasticity is introduced into the model formulation and the additional rental and management costs due to postponement of harvest are taken into account, we find that the optimal stopping time becomes random and varies in response to changes in the underlying price and growth processes. As a result, this stochastic optimal stopping time is bounded by the Faustmann cutting age from below if variabilities of the stochastic processes diminish to zero, but not necessarily by the MSY rotation from above.     

A BIOECONOMIC APPROACH TO THE FAUSTMANN–HARTMAN MODEL: ECOLOGICAL INTERACTIONS IN MANAGED FOREST

Abstract This paper develops a bioeconomic forestry model that makes it possible to take ecosystem services that are independent of the age structure of trees into account. We derive the Faustmann–Hartman optimal harvesting strategy as a special case. The bioeconomic model is then extended to account for the fact that forest harvesting decisions impact on other ecological resources, which provide benefits for the wider community. The paper focuses on impacts associated with disturbance caused by logging operations and habitat destruction due to tree removal. This enables us to explore the interactions between forest management and the dynamics of ecological resources. The optimal rotation rule is obtained as a variation on the traditional Faustmann–Hartman equation, where an additional term captures the potential benefits derived from the growth of the ecological resource valued at its shadow price. The steady‐state solutions to the problem and sensitivity to model parameter are identified using numerical analysis.     

GLOBAL CLIMATE CHANGE AND OPTIMAL FOREST MANAGEMENT

A policy question of current interest is how to cope with climate change. One suggestion is to use forests to offset carbon emissions, and therefore, reduce the threat of global warming. This study develops a rigorous model of the relationship between optimal forest harvesting regimes and carbon sequestration. The theoretical analysis integrates the carbon sequestration life cycle into the Faustmann framework and develops optimal cutting rules when carbon sequestration benefits are considered. The carbon life cycle includes both the sequestration of carbon and its ultimate re-release into the atmosphere. A case study of Douglas fir applies the theoretical framework.     

MULTIPLE USE VALUES AND OPTIMAL STEADY STATE AGE DISTRIBUTIONS

Forest management today, generally, focuses not only on wood values but also on the many other amenities and services provided by growing forests. The significance of these multiple use values was recognized by Hartman [1976] who derived a formula for the optimal rotation for a single stand when the services provided by the stand throughout its life are considered in addition to the value of the final harvest. Some more recent work has focused on the case of multiple stands where the amenity values at a point in time depend on the age distribution of the stands at that time. One approach to harvesting multiple stands for wood values alone is the forestry maximum principle developed by Heaps [1984] and Wan [1985]. It will be shown here how the forestry maximum principle can be modified to incorporate the amenity services provided by the growing forest. The optimal steady state age distributions for the multiple stand forest can then be identified and described with the help of Hartman's rotation formula.     

CLIMATE CHANGE AND OPTIMAL ROTATION IN A FLAMMABLE FOREST

ABSTRACT. This paper builds a Faustmann‐based model to study the effects of increased climate‐induced fire risk on the optimal forest rotation period. Simulations using species prevalent in North American forests indicate that both the commercial and socially optimal rotation ages decline as the risk increases. The reduced carbon absorbed by the standing timber can then create a positive feedback effect. This has potentially important policy implications. The Kyoto ratification agreement reached in the autumn of 2001 was dependent on allowing the ‘Umbrella Group’ of countries to use their forests' carbon‐absorbing ability to offset their need for fossil fuel emission reductions. This carbon‐absorbing ability will decline if rotation ages decrease with increased fire risk, weakening the force of the argument for allowing these countries to use their carbon ‘sinks’ to avoid reducing anthropomorphic emissions.     

Numerical Optimal Harvesting for an Age-Dependent Prey-Predator System

In this article, we develop a numerical study of an optimal harvesting problem for age-dependent prey-predator system. Here, the rates of growth and decay as well as the interaction effect between species are assumed to be depending on age, time and space. Existence, uniqueness, and necessary conditions for the optimal control are assured in case of a small final time T. The discrete parabolic nonlinear dynamical systems are obtained by using a finite difference semi-implicit scheme. Then a numerical algorithm is developed to approximate the optimal harvesting effort and the optimal harvest. Results of the numerical tests are given.     

Transient and steady-state analysis of a manufacturing system with setup changes

This paper deals with the optimal scheduling of a one-machine two-product manufacturing system with setup, operating in a continuous time dynamic environment. The machine is reliable. A known constant setup time is incurred when switching over from a part to the other. Each part has specified constant processing time and constant demand rate, as well as an infinite supply of raw material. The problem is formulated as a production flow control problem. The objective is to minimize the sum of the backlog and inventory costs incurred over a finite planning horizon. The global optimal solution, expressed as an optimal feedback control law, provides the optimal production rate and setup switching epochs as a function of the state of the system (backlog and inventory levels). For the steady-state, the optimal cyclic schedule (Limit Cycle) is determined. This is equivalent to solving a one-machine two-product Lot Scheduling Problem. To solve the transient case, the system's state space is partitioned into mutually exclusive regions such that with each region is associated an optimal control policy. A novel algorithm (Direction Sweeping Algorithm) is developed to obtain the optimal state trajectory (optimal policy that minimizes the sum of inventory and backlog costs) for this last case.     

Optimal harvesting policy for stochastic Logistic population model

In this paper, we consider some optimal harvesting policies for single population models, in which the harvest effort and the intrinsic growth rate are disturbed by environment noises. We choose the maximum sustainable yield and the maximum retained profits as two management objectives, and obtain the optimal harvesting policies, respectively. For the two objectives, we give the optimal harvest effort that maximizes the sustainable yield (or retained profits), the maximum of expectation of sustainable yield (or retained profits) and the corresponding variance. Their explicit expressions are determined by the coefficients of equation and the disturbance intensity.     

The structure of optimal solutions for harvesting a renewable resource

We consider the problem of optimal harvesting of a renewable resource whose dynamics are governed by logistic growth and whose payoff is proportional to the harvest. We consider both the case of a finite and an infinite time horizon and analyse the structure of the optimal solutions and their dependence on the parameters of the model. We show that the optimal policy can only have one of three structures: (1) maximal harvesting effort until the resource is depleted, (2) zero harvesting during an initial time interval followed by a subsequent switch to maximal harvesting effort, or (3) a singular solution, which corresponds to an intermediate level of harvesting, accompanied by the most rapid approach path. All three scenarios emerge, with minor variations, with finite and infinite time horizons, depending on the particular combination of parameters of the system. We characterize the conditions under which the singular solution is optimal and present suggestions for designing an optimal and sustainable harvesting strategy. Recommendations for Resource Managers :

• We have rigorously explored a standard optimal harvesting model and its steady states.
• We show that three different types of solutions may emerge: (i) maximal harvesting eventually leading to a complete depletion of the stock; (ii) maximal harvesting with a potential period of idleness leading to a positive stock; (iii) an initial phase of either no or full harvesting followed by a period of intermediate harvesting intensity leading to a positive stock (singular solution).
• With some modifications, similar results hold for a finite planning horizon.
• Which of these three scenarios emerges in the finite horizon case depends not only on the parameter values but also on the length of the planning horizon.

     

ENVIRONMENTAL AND FINANCIAL SUSTAINABILITY OF FOREST MANAGEMENT PRACTICES

One of the guiding themes for forest management policy throughout much of North America is sustained yield. The basic premise behind this theme is that a constant or nondeclining flow of services from the forest is socially desirable. Unfortunately, the act of capturing the benefits of this service (timber harvesting) often has detrimental effects on the timber-productive capacity of a forest site. This paper presents a dynamic program that is used to determine the optimal harvest system choice for a timber stand described by average piece size, stand density, a measure of site quality, and stumpage value. The harvest systems are defined by logging costs, reforestation and rehabilitation costs, and the impact of the system on the productivity of the site. An application of the model is presented for lodgepole pine in Alberta. We conclude that, at high discount rates, soil conservation is not economically rational. At lower discount rates, some degree of soil conservation is desirable on the more productive sites. At lower discount rates, there also appears to be an incentive for more intensive forest management. Limitations on acceptable harvest practices can have a large impact on optimal rotation age and the volume harvested. There is a large opportunity cost resulting from a requirement for sustainable volume production because of the impact of harvesting on soil productivity.     

AN OPTIMIZATION APPROACH TO IDENTIFYING TIMBER SUPPLY FUNCTION COEFFICIENTS

This paper examines an optimization approach to identifying short-run timber supply function coefficients when the form of the supply function is known. By definition, a short-run timber supply function is a functional relationship between the optimal harvest level in each period (e.g., each year) and the actual forest-market state in the same period. The short-run timber supply function represents the optimal harvest decision policy, and therefore, the problem of optimal harvesting can be formulated as a problem of determining this function. When the form of the supply function is known, the problem becomes one of identifying the coefficients of the supply function. If the management objective is to maximize the expected present value of net revenues from timber harvesting over an infinite time horizon, and the timber price process is, in a sense, stationary, the supply function coefficients correspond to the optimal solution to an anticipative optimization problem. In this case, the supply function coefficients can be determined by maximizing the expected present value of the net revenues from timber harvesting, where periodic harvest levels are determined using the supply function. Numerical results show that the short-run supply functions determined using this approach gives good approximations of the true supply function.     

Single item lot-sizing problems with backlogging on a single machine at a finite production rate

In this paper we consider a single item lot-sizing problem with backlogging on a single machine at a finite production rate. The objective is to minimize the total cost of setup, stockholding and backlogging to satisfy a sequence of discrete demands. Both varying demands over a finite planning horizon and fixed demands at regular intervals over an infinite planning horizon are considered. We have characterized the structure of an optimal production schedule for both cases. As a consequence of this characterization, a dynamic programming algorithm is proposed for the computation of an optimal production schedule for the varying demands case and a simpler one for the fixed demands case.     

ON THE OPTIMAL POLICY FOR COMBINING HARVESTING OF PREDATOR AND PREY

An optimal policy is sought for maximizing present value from the combined harvest of two ecologically dependent species, which would coexist as predator and prey in the absence of harvesting. Harvest rate for each species is assumed proportional to both stock level and effort. For a large class of biological productivity functions, it is established that the optimal equilibrium point in the phase-plane of stock levels must be a saddle-point. For quadratic productivity functions, a combination of analytical reasoning and numerical experiment is used to show that first and second order necessary conditions for optimality are satisfied by a unique approach to equilibrium, which must therefore be optimal. The corresponding control law is given, and an apparent suggestion of two previous authors concerning the policy is shown to be inappropriate. The optimal policy enables an estimate to be made of the true loss of resource value due to a catastrophic fall in stock level.     

A MIXED CONTROL PROBLEM OF THE MANAGEMENT OF NATURAL RESOURCES

In this paper, we study the optimal solutions of a model of natural resource management which allows for both impulse and continuous harvesting policies. This type of model is known in the literature as mixed optimal control problem. In the resource management context, each type of control represents a different harvesting technology, which has a different cost. In particular, we want to know when the following conjecture made by Clark is an optimal solution to this mixed optimal control problem: if the harvesting capacity is unlimited, it is optimal to jump immediately to the steady state of the continuous time problem and then to stay there. We show that under a particular relationship between the continuous and the impulse profit function, the conjecture made by Clark is true. In other cases, however, it is either better to use only continuous control variables or to jump to resource levels which are smaller than the steady state and then let the resource grow back to the steady state. These results emphasize the importance of the cost functions in the modeling of natural resource management.     

The optimal harvesting problem under price uncertainty

In this paper we study the exploitation of a one species forest plantation when timber price is governed by a stochastic process. The work focuses on providing closed expressions for the optimal harvesting policy in terms of the parameters of the price process and the discount factor, with finite and infinite time horizon. We assume that harvest is restricted to mature trees older than a certain age and that growth and natural mortality after maturity are neglected. We use stochastic dynamic programming techniques to characterize the optimal policy and we model price using a geometric Brownian motion and an Ornstein–Uhlenbeck process. In the first case we completely characterize the optimal policy for all possible choices of the parameters. In the second case we provide sufficient conditions, based on explicit expressions for reservation prices, assuring that harvesting everything available is optimal. In addition, for the Ornstein–Uhlenbeck case we propose a policy based on a reservation price that performs well in numerical simulations. In both cases we solve the problem for every initial condition and the best policy is obtained endogenously, that is, without imposing any ad hoc restrictions such as maximum sustained yield or convergence to a predefined final state.     

A new strong optimality criterion for nonstationary Markov decision processes

This paper deals with a new optimality criterion consisting of the usual three average criteria and the canonical triplet (totally so-called strong average-canonical optimality criterion) and introduces the concept of a strong average-canonical policy for nonstationary Markov decision processes, which is an extension of the canonical policies of Herna′ndez-Lerma and Lasserre [16] (pages: 77) for the stationary Markov controlled processes. For the case of possibly non-uniformly bounded rewards and denumerable state space, we first construct, under some conditions, a solution to the optimality equations (OEs), and then prove that the Markov policies obtained from the OEs are not only optimal for the three average criteria but also optimal for all finite horizon criteria with a sequence of additional functions as their terminal rewards (i.e. strong average-canonical optimal). Also, some properties of optimal policies and optimal average value convergence are discussed. Moreover, the error bound in average reward between a rolling horizon policy and a strong average-canonical optimal policy is provided, and then a rolling horizon algorithm for computing strong average ε(>0)-optimal Markov policies is given.     

Dynamics of the discrete Seno population model: Combined effects of harvest timing and intensity on population stability

The paper contains new results on the impact of harvesting times and intensities on the stability properties of Seno population models. It is proved that sufficiently high harvest intensities are stabilizing for any harvesting time in the sense that they create a positive equilibrium that attracts all positive solutions. Moreover, in the special case that the nonlinearity in the Seno model is a Ricker function, we derive a global stability result independent of timing and valid for low to medium harvesting efforts. The proof is based on a characterization of those harvesting intensities which guarantee a negative Schwarzian derivative for all harvesting times. Finally, we rigorously show that timing can be stabilizing as well as destabilizing by itself. In particular, a recent conjecture formulated by Cid et al. (2014) [1] is shown to be false.     

Optimal harvesting policy for general stochastic Logistic population model

We consider some optimal harvesting policies for a general stochastic Logistic population model. For two management objectives, that are maximum sustainable yield and the maximum retained profits, the optimal harvesting policies are obtained. Meanwhile, the optimal harvest effort, the maximum of expectation of sustainable yield (or retained profits) and the corresponding variance are given.     

The value of information in energy harvesting sensor networks

We study optimal data collection in energy harvesting sensor networks. The sensed data-stream constitutes a sequence of stationary ergodic random variables and, assuming that timely data is valued more, the value of the data is discounted over time such that frequent collection is required to avoid a loss in value. However, unlimited data collection is constrained by both the cost of data collection and the availability of energy.     

Optimal impulsive harvesting policy for single population

In this paper, we established the exploitation of impulsive harvesting single autonomous population model by Logistic equation. By some special methods, we analysis the impulsive harvesting population equation and obtain existence, the explicit expression and global attractiveness of impulsive periodic solutions for constant yield harvest and proportional harvest. Then, we choose the maximum sustainable yield as management objective, and investigate the optimal impulsive harvesting policies respectively. The optimal harvest effort that maximizes the sustainable yield per unit time, the corresponding optimal population levels are determined. At last, we point out that the continuous harvesting policy is superior to the impulsive harvesting policy, however, the latter is more beneficial in realistic operation.