Forest management models and combinatorial algorithms: analysis of state of the art

Linear Programming and Mixed Integer Linear Programs have been used for forest planning since the 60's to support decision making on forest harvesting and management. In particular, during the last two decades of forest management there has been an increased interest in spatial issues. Further, new environmental concerns, such as resource sustainability and wildlife protection, impose that increased attention be paid to activities carried out on the ground. Road building needed for access also requires spatial definiton. As a result, more complex models must be used. We discuss the issues which have led to the combinatorial nature of some main forest management problems and the solution algorithms that have been proposed for these problems, including local search heuristics, random search approaches, strengthening of mixed integer model formulations and Lagrangian relaxation. In this survey, we discuss which of the proposed approaches have been used succesfully, the advantages and shortcomings of each and what are still open research problems. This revised version was published online in June 2006 with corrections to the Cover Date.     

Forestry management under uncertainty

The forest harvest and road construction planning problem consists fundamentally of managing land designated for timber production and divided into harvest cells. For each time period the planner must decide which cells to cut and what access roads to build in order to maximize expected net profit. We have previously developed deterministic mixed integer linear programming models for this problem. The main contribution of the present work is the introduction of a multistage Stochastic Integer Programming model. This enables the planner to make more robust decisions based on a range of timber price scenarios over time, maximizing the expected value instead of merely analyzing a single average scenario. We use a specialization of the Branch-and-Fix Coordination algorithmic approach. Different price and associated probability scenarios are considered, allowing us to compare expected profits when uncertainties are taken into account and when only average prices are used. The stochastic approach as formulated in this work generates solutions that were always feasible and better than the average solution, while the latter in many scenarios proved to be infeasible.     

Fuel treatment planning: Fragmenting high fuel load areas while maintaining availability and connectivity of faunal habitat

Reducing the fuel load in fire-prone landscapes is aimed at mitigating the risk of catastrophic wildfires but there are ecological consequences. Maintaining habitat for fauna of both sufficient extent and connectivity while fragmenting areas of high fuel loads presents land managers with seemingly contrasting objectives. Faced with this dichotomy, we propose a Mixed Integer Programming (MIP) model that can optimally schedule fuel treatments to reduce fuel hazards by fragmenting high fuel load regions while considering critical ecological requirements over time and space. The model takes into account both the frequency of fire that vegetation can tolerate and the frequency of fire necessary for fire-dependent species. Our approach also ensures that suitable alternate habitat is available and accessible to fauna affected by a treated area. More importantly, to conserve fauna the model sets a minimum acceptable target for the connectivity of habitat at any time. These factors are all included in the formulation of a model that yields a multi-period spatially-explicit schedule for treatment planning. Our approach is then demonstrated in a series of computational experiments with hypothetical landscapes, a single vegetation type and a group of faunal species with the same habitat requirements. Our experiments show that it is possible to fragment areas of high fuel loads while ensuring sufficient connectivity of habitat over both space and time. Furthermore, it is demonstrated that the habitat connectivity constraint is more effective than neighbourhood habitat constraints. This is critical for the conservation of fauna and of special concern for vulnerable or endangered species.     

SUSTAINABLE FOREST MANAGEMENT FOR OPTIMIZING MULTISPECIES WILDLIFE HABITAT: A COASTAL DOUGLAS-FIR EXAMPLE

Wildlife species viability optimization models are developed to convert a given set of initial forest conditions, through a combination of natural growth and management treatments, to a forest system which addresses the joint habitat needs of multispecies populations over time. A linear model of forest cover and wildlife populations is used to form a system of forest management control variables for wildlife habitat modification. The paper examines two objective functions coupled to this system for optimizing sustainable joint species viability. The first maximizes the product of periodic joint viabilities over all time periods, focusing management resources on long-term equilibria, with less emphasis on conversion strategy. The second iteratively maximizes the minimum periodic joint viability over all time periods. This focuses management resources on the most limiting time periods, typically the conversion phase periods. Both objective functions resulted in either point or cyclic equilibria, with cycle lengths equal to minimum forest treatment ages. A third objective, based on maximizing the minimum individual species periodic viability is used to examine single species emphasis. Examples are developed through a case study of 92 vertebrate species found in coastal Douglas-fir stands of northwestern California.     

Forest management optimisation models and habitat diversity: a goal programming approach

This paper proposes an analytical framework, based upon Goal Programming, to incorporate an operational measure of habitat diversity into a forest management optimisation model. First of all, the trade-off curve between the proposed measure of habitat diversity and financial returns is determined. Then, the measure of habitat diversity is integrated in conjunction with other relevant criteria into a compact forest management model. The way in which both models work is explained with the help of a simple but illustrative case study.     

IRREVERSIBLE INVESTMENT AND OPTIMAL FOREST EXPLOITATION

A forest harvest scheduling model which includes as activities the level of investment in harvest capacity and the accumulated harvest capacity in each period, is presented. The inclusion of these activities, in addition to the harvest activities, allows for the removal of harvest-flow constraints found in more typical Model II formulations of the harvest scheduling problem. The optimal harvest and investment policy can be determined by linear programming or quadratic programming methods, depending on whether prices are constant or supply-dependent. The new model better reflects economic reality than existing models, and provides a method for determining the optimal economic development of a forest industry, and the optimal draw-down of old growth forest. Numerical examples are given.     

A Constraint Programming model for fast optimal stowage of container vessel bays

Container vessel stowage planning is a hard combinatorial optimization problem with both high economic and environmental impact. We have developed an approach that often is able to generate near-optimal plans for large container vessels within a few minutes. It decomposes the problem into a master planning phase that distributes the containers to bay sections and a slot planning phase that assigns containers of each bay section to slots. In this paper, we focus on the slot planning phase of this approach and present a Constraint Programming and Integer Programming model for stowing a set of containers in a single bay section. This so-called slot planning problem is NP-hard and often involves stowing several hundred containers. Using state-of-the-art constraint solvers and modeling techniques, however, we were able to solve 90% of 236 real instances from our industrial collaborator to optimality within 1 second. Thus, somewhat to our surprise, it is possible to solve most of these problems optimally within the time required for practical application.     

Piece-wise linear approximation of functions of two variables

The goal of increasing computational efficiency is one of the fundamental challenges of both theoretical and applied research in mathematical modeling. The pursuit of this goal has lead to wide diversity of efforts to transform a specific mathematical problem into one that can be solved efficiently. Recent years have seen the emergence of highly efficient methods and software for solving Mixed Integer Programming Problems, such as those embodied in the packages CPLEX, MINTO, XPRESS-MP. The paper presents a method to develop a piece-wise linear approximation of an any desired accuracy to an arbitrary continuous function of two variables. The approximation generalizes the widely known model for approximating single variable functions, and significantly expands the set of nonlinear problems that can be efficiently solved by reducing them to Mixed Integer Programming Problems. By our development, any nonlinear programming problem, including non-convex ones, with an objective function (and/or constraints) that can be expressed as sums of component nonlinear functions of no more than two variables, can be efficiently approximated by a corresponding Mixed Integer Programming Problem.     

Production planning via scenario modelling

Several Linear Programming (LP) and Mixed Integer Programming (MIP) models for the production and capacity planning problems with uncertainty in demand are proposed. In contrast to traditional mathematical programming approaches, we use scenarios to characterize the uncertainty in demand. Solutions are obtained for each scenario and then these individual scenario solutions are aggregated to yield a nonanticipative or implementable policy. Such an approach makes it possible to model nonstationarity in demand as well as a variety of recourse decision types. Two scenario-based models for formalizing implementable policies are presented. The first model is a LP model for multi-product, multi-period, single-level production planning to determine the production volume and product inventory for each period, such that the expected cost of holding inventory and lost demand is minimized. The second model is a MIP model for multi-product, multi-period, single-level production planning to help in sourcing decisions for raw materials supply. Although these formulations lead to very large scale mathematical programming problems, our computational experience with LP models for real-life instances is very encouraging.     

Constructing integer programming models by the predicate calculus

A modelling language for Integer Programming (IP) based on the Predicate Calculus is described. This is particularly suitable for building models with logical conditions. Using this language a model is specified in terms of predicates. This is then converted automatically by a series of transformation rules into a normal form from which an IP model can be created. There is also some discussion of alternative IP formulations which can be incorporated into the system as options. Further practical considerations are discussed briefly concerning implementation language and incorporation into practical Mathematical Programming Systems.     

A stochastic model for operating room planning with elective and emergency demand for surgery

This paper describes a stochastic model for Operating Room (OR) planning with two types of demand for surgery: elective surgery and emergency surgery. Elective cases can be planned ahead and have a patient-related cost depending on the surgery date. Emergency cases arrive randomly and have to be performed on the day of arrival. The planning problem consists in assigning elective cases to different periods over a planning horizon in order to minimize the sum of elective patient related costs and overtime costs of operating rooms. A new stochastic mathematical programming model is first proposed. We then propose a Monte Carlo optimization method combining Monte Carlo simulation and Mixed Integer Programming. The solution of this method is proved to converge to a real optimum as the computation budget increases. Numerical results show that important gains can be realized by using a stochastic OR planning model.     

Stronger upper and lower bounds for a hard batching problem to feed assembly lines

In this paper we present an Integer Programming reformulation for a hard batching problem encountered in feeding assembly lines. The study was motivated by the real process to feed the production flow through the shop floor in a leading automobile industry in Brazil. The problem consists of deciding the assignment of items to containers and the frequency of moves from the storage area to the line in order to meet demands with minimum cost. Better lower and upper bounds were obtained by a branch-and-bound algorithm based on the proposed reformulation. We also present valid inequalities that may improve such algorithm even further.     

An Equivalence Theorem on Minimum Sheltering Speed for Non-convex Habitats

The paper is devoted to a new family of variational problems for differential inclusions, motivated by the protection of human and wildlife habitats when an invasive environmental disaster occurs. Indeed, the mathematical model consists of a differential inclusion describing the expansion of invaded region over time, and an artificial barrier that cannot be penetrated by the invasive agent is erected to shield the habitat, which serves as the control strategy and is characterized as a one-dimensional rectifiable set. We consider an isotropic case that the disaster spreads with uniform speed in all directions, and develop an equivalence result on the minimum construction speed required for implementing a sheltering strategy determined by a rectifiable Jordan curve. By measuring the invaded portion of barriers over time, it is shown that each connected non-convex habitat admits an equivalent habitat that contains the given habitat and their minimum speeds are equal. In particular, the boundary of an equivalent habitat can be partitioned into two arcs: an arc is locally convex, and the other is part of the boundary of illuminated area. This leads to a corollary on the existence of admissible sheltering strategies for non-convex habitats.     

Home Care optimization: impact of pattern generation policies on scheduling and routing decisions

In Home Care optimization, operators have to be assigned to patients by taking into account compatibility skill constraints, and patient visits have to be scheduled in a given planning horizon. Moreover, operator tours have to be determined. Integer Linear Programming models have been proposed which use the concept of patterns, i.e. a priori scheduling profiles, to combine the diverse decision levels. Computational results on real instances show that pattern generation policies are crucial to address scheduling and routing in large Home Care instances.     

MODELING HARVEST SCHEDULING IN MULTIFUNCTIONAL PLANNING OF FORESTS FOR LONGTERM WATER YIELD OPTIMIZATION

In this study, wood production and hydrologic functions of forests were accommodated within a planning procedure for separate working circles (areas dedicated to certain forest functions) that were delineated according to an Ecosystem‐Based Functional Planning approach. Mixed integer goal programming was used as the optimization technique. The timing and scheduling of a maintenance cutting (partial harvest) was the decision variable in the modeling effort, and an original formulation was developed as a multiobjective planning procedure. Four sample planning strategies were developed and model outputs were evaluated according to these strategies. Spatial characteristics of stands were considered, and used to prohibit the regeneration of adjacent stands during the same time period. Because of the positive relationship between qualified water production and standing timber volume in the forest, the model attempts to maximize qualified water production levels by increasing standing volume stocks in the forest through the delay of regeneration activities.     

An integer linear programming approach for bilinear integer programming

We introduce a new Integer Linear Programming (ILP) approach for solving Integer Programming (IP) problems with bilinear objectives and linear constraints. The approach relies on a series of ILP approximations of the bilinear IP. We compare this approach with standard linearization techniques on random instances and a set of real-world product bundling problems.     

A duality theorem for linear congruences

An analogous duality theorem to that for Linear Programming is presented for systems of linear congruences. It is pointed out that such a system of linear congruences is a relaxation of an Integer Programming model (for which the duality theorem does not hold). Algorithms are presented for both the resulting primal and dual problems. These algorithms serve to give a constructive proof of the duality theorem.     

基于DNA遗传算法的随机整数规划期望值模型求解

为了求解随机整数规划问题,提出了随机整数规划期望值模型的概念,分析了利用DNA遗传算法求解此类问题的优点,并设计了求解算法,最后通过报童问题,验证了算法的可行性和有效性.