Finding the Efficient Frontier of a Bi-Criteria, Spatially Explicit, Harvest Scheduling Problem Sa ´ndor F. To ´th, Marc E. McDill, and Stephanie Rebain Abstract: This article evaluates the performance of five traditional methods and one new method of generating the efficient frontier for a bi-criteria, spatially explicit harvest scheduling problem. The problem is to find all possible efficient solutions, thus defining the trade-offs between two objectives: (1) maximizing the net present value of the forest and (2) maximizing the minimum area over the planning horizon in large, mature forest patches. The methods for generating the efficient frontier were tested using a hypothetical forest consisting of 50 stands. The methods were compared based on the number of efficient solutions each method can identify and on how quickly the solutions were identified. The potential to generalize these algorithms to 3- or n-criteria cases is also assessed. Three of the traditional approaches, the - constraining; the triangles method, the decomposition algorithm based on the Tchebycheff metric; and the new, proposed method are capable of generating all or most of the efficient solutions. However, the triangles and the new method far outperformed the other approaches in terms of solution time. The new method, called alpha-delta, appears to be the simplest to generalize to the tri-criteria case. FOR.SCI. 52(1):93–107. Key Words: Multicriteria optimization, wildlife habitat, trade-off analysis, 0 –1 programming. S OCIETY EXPECTS MORE from its forest resources than merely timber production. Increasingly, values such as wildlife habitat, recreation, water quality, esthet- ics, and spiritual values are also recognized. In accordance with these expectations, the Multiple-Use Sustained-Yield Act (1960) requires the national forests of the United States to be managed for the multiple uses of water, timber, wildlife, fish, recreation, and range (Fedkiw 1997). The emerging field of multiple-objective forest planning reflects this diverse nature of forest resources management (Pukkala 2002). Sustaining large patches of mature forests (forest stands that are older than a certain age) throughout the planning horizon can contribute to fulfilling many of the multiple uses demanded by society (Rebain and McDill 2003a). In addition, adjacency constraints, which limit the size of harvest openings, have been promoted as contribut- ing to these objectives (e.g., Thompson et al. 1973, Jones et al. 1991, Murray and Church 1996a, b, Snyder and ReVelle 1996a, b, 1997a, b, Carter et al. 1997, Murray 1999). How- ever, adjacency constraints tend to work against the goal of developing and preserving large, mature patches of forest (Harris 1984, Franklin and Forman 1987, Rebain and Mc- Dill 2003a). As adjacency constraints are intended to pre- vent large clearcuts, they tend to disperse harvesting activ- ities across the forest in relatively small patches. Large, contiguous tracts of mature forests are not likely to be maintained this way. One way of tackling this problem is to include con- straints that require the models to maintain a minimum total area in mature patches meeting both a minimum age and a minimum size requirement, while maximizing the net present value (NPV) of the forest (Rebain and McDill 2003a, c). However, it might be difficult to identify an appropriate total area of large, mature patches that will adequately meet conservation goals but not be overly re- strictive. Nevertheless, single-objective models have often been applied to forest planning problems with multiple objectives where the minimum or maximum level of other outputs or values are defined by constraints (Leuschner et al. 1975, Mealy and Horn 1981, Cox and Sullivan 1995, Bettinger et al. 1997). A priori methods, such as goal programming (Field 1973, Kao and Brodie 1979, Field et al. 1980, Arp and Lavigne 1982, Hotvedt 1983, Mendoza 1987, Rustagi and Bare 1987, or Davis and Lui 1991) also suffer from the limitation that the decision-maker (DM) is required to identify his or her preferences before the solution process. Expecting the DM to specify the desired level of achieve- ment or to specify his or her preferences for the various objectives without knowing what is possible is not only unrealistic, but might also lead to poor management deci- sions. An interactive method, where the DM helps drop certain regions of the feasible solution set by comparing and ranking a limited number of alternative solutions, is a fea- sible approach that might remedy this shortcoming. With an interactive approach, at each iteration the DM progressively articulates his or her preferences and the focus of the search becomes more confined. This way, the search converges toward a solution that maximizes the DM’s utility—the best Sa ´ndor F. To ´ th, Research Associate, Penn State School of Forest Resources, 214B Ferguson Bldg., University Park, PA 16802—Phone: (814) 865-1602; Fax: (814) 865-3725; sft108@psu.edu. Marc E. McDill, Associate Professor, Penn State School of Forest Resources, 214B Ferguson Bldg., University Park, PA 16802—Phone: (814) 865-1602; mem14@psu.edu. Stephanie A. Rebain, Forester, USDA Forest Service, Forest Management Service Center, 2150A Centre Avenue, Suite 341A, Ft. Collins, CO 80526-1891—Phone: (970) 295-5793; sarebain@fs.fed.us. Acknowledgments: The authors thank the Associate Editor and two anonymous reviewers for their helpful comments. Thanks also to the Pennsylvania Bureau of Forestry for providing financial support for this research. Manuscript received October 29, 2004, accepted October 31, 2006 Copyright © 0 by the Society of American Foresters Forest Science 52(1) 2006 93