Novel operations research methods for efficiently determining irreplaceable sites for conservation

We examine the irreplaceability of sites in the context of the species set covering problem and the maximal covering species problem. We show that a succession of 0-1 programming problems can be solved to quickly determine the set of irreplaceable sites. For the widely available Oregon data set, we find a general lack of trend in the number of irreplaceable sites with the number of sites available for selection used. Moreover, we observe that irreplaceability at one level of resource may not be a predictor of irreplaceability at a higher or a lower level of resource. Finally, we investigate the economic value of irreplaceable sites by trial removals of those sites. This paper is dedicated to the memory of Dr. Charles S. ReVelle.     

Designing a conservation reserve network with minimal fragmentation: A linear integer programming approach

In the biological conservation literature, the optimum reserve site selection problem has often been addressed by using the prototype set covering and maximal covering formulations, assuming that representation of species is the only criterion in site selection. This approach usually results in a small but highly fragmented reserve, which is not useful for practical conservation planning. To improve the chances of species' persistence, it may be desirable to reduce habitat fragmentation. This paper presents a linear integer programming formulation to minimize spatial gaps between selected sites in a reserve network, which is applied to a data set on breeding birds. The authors express their willingness to share the database used in this study. Those readers who wish to have access to the data may contact Robert A. Briers at r.briers@napier.ac.uk.     

Representational success: A new paradigm for achieving species protection by reserve site selection

Models for designing habitat reserve networks have focused on minimizing the number of sites necessary to cover each species one or more times. A solution to this problem is usually one from among a large number of alternative optimal configurations of sites. This paper develops an iterative method for building reserve networks that produces an optimal solution to the species set covering problem (SSCP) and also maximizes the number of species covered two or more times, three or more times, and so on, conditional on the solution to the previous iteration. We refer to this as representational success. Thus, a pareto optimal species set covering is achieved that is preferable to an arbitrary optimal solution to the SSCP.     

The SITES reserve selection system: A critical review

Numerous models have been put forth to help with the growing demand for the establishment of biodiversity reserves. One site selection model that has been used in several recent studies is SITES [S.J. Andelman, I. Ball, F.W. Davis and D.M. Stoms, SITES V 1.0: an analytical toolbox for designing ecoregional conservation portfolios, Unpublished manual prepared for the nature conservancy, 1999, 1–43. (available at )]. SITES includes two heuristic solvers: based on Greedy and Simulated Annealing. We discuss the formulation of the SITES model, present a new formulation for that problem, and solve a number of test problems optimally using off-the-shelf software. We compared our optimal results with the SITES Simulated Annealing heuristic and found that SITES frequently returns significantly suboptimal solutions. Our results add further support to the argument, started by Underhill [L.G. Underhill, Optimal and suboptimal reserve selection algorithms, Biol. Conserv. 70 (1994) 85–87], continuing through Rodrigues and Gaston [A.S.L. Rodrigues and K.J. Gaston, Optimization in reserve selection procedures – why not?, Biol. Conserv. 107 (2002) 123–129], for greater integration of optimal methods in the reserve design/selection literature.     

Delineating protected wildlife corridors with multi‐objective programming

Protected wildlife corridors can help counteract habitat fragmentation and link isolated reserve “islands” into connected reserve systems. The need for wildlife corridors will grow as expanding human populations place increasing pressure on remaining undeveloped land. A two‐objective zero–one programming model is formulated for the problem of selecting land for a system of wildlife corridors that must connect a known set of existing reserves or critical habitat areas. This problem is modeled as a network Steiner tree problem, under the objectives of minimizing corridor land costs and minimizing the amount of unsuitable land within the corridor system. Linear programming is used to find exact solutions with little or no branching and bounding, and the multi‐objective weighting method is used to generate non‐inferior alternatives. Two hypothetical examples demonstrate the model and solution procedure. Results can help inform planning and decision making for protected area land acquisition and habitat restoration.     

Analysis of the Threshold and Expected Coverage Approaches to the Probabilistic Reserve Site Selection Problem

Two approaches to formulating the reserve site selection problem when species occurrence data is probabilistic were solved for terrestrial vertebrates in a small set of potential reserve sites in Oregon. The expected coverage approach, which maximizes the sum of the occurrence probabilities, yielded solutions that covered more species on average in Monte Carlo simulations than the threshold approach, which maximizes the number of species for which the occurrence probability exceeds some threshold.     

Counterpart Models in Facility Location Science and Reserve Selection Science

Five classes of zero–one programming models for discrete facility location problems are compared to counterpart models for the selection of conservation reserves. The basic problem of siting facilities to cover demand for services is analogous to the problem of selecting reserves to support species diversity. The classes of models include the set covering and maximal covering models, as well as models for backup and redundant coverage. Issues of reliability and uncertainty are addressed by chance constrained covering models and maximal expected covering models. Exact and heuristic solution approaches are discussed. Multi-objective and economic issues are considered.     

Optimizing the removal of small fish passage barriers

Removing small artificial barriers that hinder upstream migrations of fish is a major problem in riparian habitat restoration. Because of budgetary limitations, it is necessary to prioritize barrier removal and repair decisions. These have usually been based on scoring and ranking procedures, which, although simple to use, can be very inefficient in terms of increasing the amount of accessible instream habitat. We develop a novel decision-making approach, based on integer programming techniques, which optimizes repair and removal decisions. Results show based on real datasets of barrier culverts located in Washington State that scoring and ranking is over 25% below the optimum on average and a full 100% below in the worst case, producing no net habitat gain whatsoever. This is compared to a dynamic programming method that was able to find optimal solutions in less than a second, even for problems with up to several hundred variables, and a heuristic method, which found solutions with less than a 1% average optimality gap in even less time.     

Integrating Land Market Feedbacks into Conservation Planning—A Mathematical Programming Approach

Nature reserves are often designated ad hoc. Despite increasing conservation efforts, loss of biodiversity is still accelerating. Considering land scarcity and demand for alternative uses, efficiency in conservation strongly correlates with efficiency in land allocation. Systematic conservation planning can effectively prioritize conservation activities. Previous studies minimize opportunity costs for given conservation targets. However, these studies assume constant marginal costs of habitat protection. We extend this cost minimization approach by also considering a dynamic representation of marginal costs. The more land is allocated to nature reserves, the higher are opportunity costs, i.e., costs of forgone agricultural production. This increase in costs results from changes in the prices of agricultural commodities. We employ a deterministic, spatially explicit mathematical optimization model to allocate species habitats by minimizing opportunity costs for setting aside land for conservation purposes. The model is designed as a mixed integer programming problem and solved with GAMS/CPLEX. Our results show the need for integrating land market feedbacks into conservation planning. We find that ignoring land rent adjustments can lead to highly cost-ineffective solutions in reserve selection.     

Mathematical Methods for Spatially Cohesive Reserve Design

The problem of designing spatially cohesive nature reserve systems that meet biodiversity objectives is formulated as a nonlinear integer programming problem. The multiobjective function minimises a combination of boundary length, area and failed representation of the biological attributes we are trying to conserve. The task is to reserve a subset of sites that best meet this objective. We use data on the distribution of habitats in the Northern Territory, Australia, to show how simulated annealing and a greedy heuristic algorithm can be used to generate good solutions to such large reserve design problems, and to compare the effectiveness of these methods.     

Marine Reserve Design with Ocean Currents and Multiple Objectives

There is a growing tendency to consider marine reserves as a management and conservation tool. We investigate a spatial bio-economic model to determine fishery profits under conservation efforts. Rather than imposing a marine reserve on our model, we ask, “When and where should a marine reserve be implemented ?” For one-dimensional habitat, we determine conditions under which marine reserves emerge as a part of the optimal policy. Depending upon the size of the habitat, the optimal strategy is either to avoid fishing or to fish at maximum rate. The effect of ocean currents is analyzed through numerical simulations. We find that in the presence of strong currents, a marine reserve may become ineffective. If the currents are at low rate, a marine reserve could emerge as a variable management tool.     

A method for monitoring ground water quality near waste storage facilities

A practical optimization approach developed in this paper derives effective monitoring configurations for detecting contaminants in ground water. The approach integrates numerical simulation of contaminant transport and mathematical programming. Well sites identified by the methodology can be monitored to establish the occurrence of a contaminant release before a plume migrates to a regulatory compliance boundary. Monitoring sites are established along several horizons located between the downgradient margin of a contaminant source and a compliance boundary. A horizon can form an effective line of defense against contaminant migration to the compliance boundary if it is spanned (covered) by a sufficient number of sites to yield a well spacing that is equal to or less than a maximum value established by numerical modeling. The objective function of the integer programming model formulation expresses the goals of: (1) covering a maximum number of siting horizons, and (2) allocating wells to the single most effective horizon. The latter is determined from well spacing requirements and the width of the zone of potential contaminant migration traversed by the horizon. The methodology employs a highly tractable linear programming model formulation, and the user is not required to predefine a set of potential well sites. These attributes can facilitate its implementation in practice.     

Time-Consistent Equilibria in Common Access Resource Games with Asymmetric Players Under Partial Cooperation

Given a differential game, if agents have different time preference rates, cooperative (Pareto optimum) solutions obtained by applying Pontryagin’s maximum principle become time inconsistent. We derive a set of dynamic programming equations in continuous time whose solutions are time-consistent equilibria for problems in which agents differ in their utility functions and also in their time preference rates. The solution assumes cooperation between agents at every time. Since coalitions at different times have different time preferences, equilibrium policies are calculated by looking for Markov (subgame perfect) equilibria in a (noncooperative) sequential game. The results are applied to the study of a cake-eating problem describing the management of a common property exhaustible natural resource. The extension of the results to a simple common property renewable natural resource model in infinite horizon is also discussed.     

A Superiority-Inferiority-Based Inexact Fuzzy Stochastic Programming Approach for Solid Waste Management Under Uncertainty

A superiority–inferiority-based inexact fuzzy stochastic programming (SI-IFSP) model was developed for planning municipal solid waste management systems under uncertainty. The SI-IFSP approach represents a new attempt to tackle multiple uncertainties in objective function coefficients which are beyond the capabilities of existing inexact programming methods. Through introducing the concept of fuzzy random boundary interval, SI-IFSP is capable of reflecting multiple uncertainties (i.e., interval values, fuzzy sets, probability distributions, and their combinations) in both the objective function and constraints, leading to enhanced system robustness. The developed SI-IFSP method was applied to a case study of long-term municipal solid waste management. Useful solutions were generated. A number of decision alternatives could be generated based on projected applicable conditions, reflecting the compromise between system optimality and reliability as well as the tradeoffs between economic and environmental objectives. Moreover, the consequences of system violations could be quantified through introducing a set of economic penalties, reflecting the relationships between system costs and constraint violation risks. The results suggest that the proposed SI-IFSP method can explicitly address complexities in municipal solid waste management systems and is applicable to practical waste management problems.     

A Hierarchical Approach to Designing Compact Ecological Reserve Systems

A number of mathematical models and solution techniques have been developed to design systems of reserve sites to protect species and their natural habitats. This paper presents two optimization models for obtaining ecological reserve systems that are spatially compact. Compact reserve systems will increase species persistence relative to more fragmented systems and can reduce the costs associated with reserve boundaries. These optimization models employ a hierarchical approach that incorporates both the boundary length of all clusters and the sum of within-cluster distances. Unlike previous approaches, we argue that measuring the distance between sites within clusters (and not between clusters) is more appropriate. Numerical experiments are conducted on synthetic grid systems and on a real-world hexagonal data set. These experiments confirm that the models produce reasonably compact clusters of reserve sites that respect species availability and budget constraints. Simplifications to the models are developed that significantly reduce the computational effort, while still identifying reasonable sets of reserve clusters. Sensitivity of the results to changes in model parameters (number of allowable sites, number of possible clusters, and rarity of species covered) is also explored.     

A Model for Landscape Planning Under Complex Spatial Conditions

We present a new mathematical programming framework that is adaptable to a variety of spatially explicit landscape problems in environmental investment, conservation, and land-use planning, transport planning, and agriculture. As part of capturing spatial interdependencies, the framework considers decision variables at two levels, finely spaced grid cells and landholdings. We applied the framework to an environmental investment problem using objective functions representing biodiversity and carbon sequestration. We also tested the model to optimize the path of a road through part of the landscape. Using the Nambucca case study in eastern Australia, we applied a hybrid greedy randomised adaptive search procedure (GRASP) to find solutions to the model.     

Models for preserving species diversity with backup coverage

The problem of selecting land for preservation of species has been a rich and active area of research over the past two decades. Typically, reserve selection models have tried to maximize species diversity by preserving areas that contain the greatest number of species. However, several studies have shown that seldom do these species-rich areas contain the rarest species most in need of protection. Most reserve selection models seek to maximize diversity by choosing parcels so that all species are covered by or represented in at least one parcel. This approach would usually be expected to result in coverage by a single parcel for the rarest species, especially for those that do not coincide with more abundant species. It is precisely these rare species, however, that would be lost or whose survivability would be most challenged, if the single parcels in which they are represented became unavailable due to some unforeseen event. In this paper, we introduce to reserve selection models the concept of secondary, or backup, coverage of species. Briefly stated, a species is said to have backup representation in the system of reserves if it is covered by, or represented in, two or more parcels. Having backup coverage guarantees that every species is still covered in the event that a natural or man-made catastrophe makes a given parcel uninhabitable. The results show that backup coverage can be obtained at little additional cost (as expressed by the number of parcels selected). Bi-objective formulations that trade-off primary with backup coverage show that backup coverage can be guaranteed for larger numbers of species with little reduction in primary coverage.     

Interval-parameter Fuzzy-stochastic Semi-infinite Mixed-integer Linear Programming for Waste Management under Uncertainty

An interval-parameter fuzzy-stochastic semi-infinite mixed-integer linear programming (IFSSIP) method is developed for waste management under uncertainties. The IFSSIP method integrates the fuzzy programming, chance-constrained programming, integer programming and interval semi-infinite programming within a general optimization framework. The model is applied to a waste management system with three disposal facilities, three municipalities, and three periods. Compared with the previous methods, IFSSIP have two major advantages. One is that it can help generate solutions for the stable ranges of the decision variables and objective function value under fuzzy satisfaction degree and different levels of probability of violating constraints, which are informative and flexible for solution users to interpret/justify. The other is that IFSSIP can not only handle uncertainties through constructing fuzzy and random parameter, but also reflect dynamic features of the system conditions through interval function of time over the planning horizon. By comparing IFSSIP with interval-parameter mixed-integer linear semi-infinite programming and parametric programming, the IFSSIP method is more reasonable than others.     

Spatial attributes and reserve design models: A review

A variety of decision models have been formulated for the optimal selection of nature reserve sites to represent a diversity of species or other conservation features. Unfortunately, many of these models tend to select scattered sites and do not take into account important spatial attributes such as reserve shape and connectivity. These attributes are likely to affect not only the persistence of species but also the general ecological functioning of reserves and the ability to effectively manage them. In response, researchers have begun formulating reserve design models that improve spatial coherence by controlling spatial attributes. We review the spatial attributes that are thought to be important in reserve design and also review reserve design models that incorporate one or more of these attributes. Spatial modeling issues, computational issues, and the trade-offs among competing optimization objectives are discussed. Directions for future research are identified. Ultimately, an argument is made for the development of models that capture the dynamic interdependencies among sites and species populations and thus incorporate the reasons why spatial attributes are important.