(1994), and Snyder and Revelle (1996).. As noted by Weintraub et al. (1994) ..... Kapple Douglas; Hoganson, H. 1995. Close-to-market intensification of forest ...
MANAGEMENT SCHEDULING MODELS FOR INTEGRATING OBJECTIVES Howard M. Hoganson1
ABSTRACT.—The University of Minnesota has a long history of developing forest management planning models. Specialized approaches can now decompose large problems into manageable-size parts and utilize an understanding of the problem within the mathematical solution process. One system served as a basis for analyses for the Minnesota Generic Environmental Impact Statement (GEIS) on timber harvesting. More recently, it has been applied to look at statewide issues regarding reforestation investments, reserve areas, short-rotation intensive culture, and extended-rotation forestry. Most recent model developments can address complex spatial aspects of management. The newest system, developed initially to address adjacency constraints, utilizes linkages with GIS information to value spatial measures like interior space and forest edge production directly within the management scheduling process.
More is expected from our forests than ever before. Increased demands raise concerns about sustaining timber production and a diverse forest environment. Higher timber prices have made feasible a new array of management activities. These activities range from uneven-aged management to short-rotation intensive silviculture. Increased awareness about the environment also points to the importance of viewing decisions from a spatial perspective. Clearly, forest managers are faced with a challenging task to design forest-wide plans that best integrate the many options available. Few if any good guidelines are available to help. Value judgments are also involved, making an understanding of tradeoffs important. Computer models offer potential to help managers develop and learn about possible management schedules. Models can help managers examine the situation in a very systematic way. Computers make it possible to examine almost countless combinations of site-level management options. Models can also be linked with geographic information systems to take advantage of the enormous amount of information available and to help portray results in a format easier to understand. In this paper we will first look at a very simple example to help explain the importance of looking at management decisions from a forest-wide or a “systems” perspective. We will also consider the importance of using economic analysis and emphasize how economics can deal with decisionmaking involving all forest values. Then we will look briefly at each of three management scheduling
1
University of Minnesota, Department of Forest Resources, 1861 Highway 169 East, Grand Rapids, MN 55744.
models developed at the University of Minnesota. These models are unique in that each uses optimization techniques and builds an understanding of the forestry model into its mathematical solution process. Examples will be given to illustrate how each has been used in practice. ECONOMIC ANALYSIS AT THE SYSTEMS LEVEL “Economic analysis” is often not an appreciated term in forestry. Some equate economic analysis with financial analysis and assume a bias towards timber production. Others often see economics as just a requirement to justify a budget, whether it be in the public or private sector. Forest management and forest economics are difficult to distinguish. An introductory economics text (Spencer 1974) defines economics as a social science concerned chiefly with the way society chooses to employ its limited resources, to produce goods and services for present and future consumption. Key to this definition is that economics is about decisionmaking involving the allocation of limited resources over time. These resources need not be traded in the marketplace. Economic analyses at the forest-wide or systems level can lead to quite different conclusions than an economic analysis at the site level. A simple example can help illustrate this. Consider a reforestation option that costs $400/acre and would yield 80 cords/acre at rotation 80 years later. If we assume timber is worth $100 per cord and the real discount rate is 4%, then a cash-flow analysis would estimate net present value (NPV) of the first rotation as a negative $53/acre. But let’s consider the value of this alternative if there is a forest-wide even-flow policy that limits timber harvest to a maximum sustainable yield level. Investing in this alternative may increase the sustainable level, and thus most of the benefits would be realized much sooner. For example, if we determined
that the sustainable harvest level could increase by the average annual growth rate for this option, then for every acre reforested, the sustainable harvest level would increase by one cord. Assuming real timber values are constant over time, this would translate into an NPV of $1,932/acre, a very high value compared to even the value of prime agricultural land! This system-wide impact of an allowable cut policy on the value of forestry investments is referred to as an allowable cut effect or ACE (Schweitzer et al. 1972). Although quite basic, it is important to realize that sustainable yield levels can vary with the level of investment in management. System-wide impacts need not be just in terms of timber impacts. For example, we could consider the value of the reforestation investment above for the situation where it is assumed that the objective of management is to sustain timber harvest levels at some projected “desirable” level and reserve as many acres as possible for their environmental values. Although forest-wide harvest levels would not change with the reforestation investment, it could still be quite valuable under this objective. The key question in the economic analysis would be the number of additional acres that could be placed in reserves with the investment. Clearly as we take more of a systems approach to management, economic analysis needs to take on a systemwide perspective. Typically these analyses take the form of maximizing some objective subject to a set of forest management policies or constraints. These problem situations fit well with the problem-solving tools of operations research. Typically the desired model output is not just a management schedule. Emphasis is also on better understanding the tradeoffs involved between alternative forest-wide policies and objectives. DUALPLAN Hoganson and Rose (1984) developed an approach for forest management scheduling that decomposes the problem into subproblems and uses an economic interpretation of the associated dual problem to help determine an optimal solution. The DUALPLAN model implements this approach for large problems (Hoganson and Rose (1989). The approach is a form of lagrangian relaxation (Fisher 1981), with specialized techniques to search for the values of the dual variables. Key to the process is the similarity between a forest-wide scheduling model and basic economic analyses of individual projects. This similarity has also been emphasized by others (Paredes and Brodie 1989) and is key in interpreting model results. Each DUALPLAN subproblem is an economic cash-flow analysis for a specific management unit. For each unit, shadow prices (dual variables) associated with the forestwide constraints are used to value the impact of site-level decisions on forest-wide constraints. An iterative process
is used to search for shadow prices that make the overall schedule within tolerable limits of the forest-wide constraints. A forest management text (Davis and Johnson, 1987, p. 672) gives a positive review of this approach. The approach works well for several reasons. First, impacts of changes in forest values are fairly predictable, as harvest timings can usually be adjusted to take advantage of periods of high product prices and avoid periods of low prices. Shadow prices on timber products are thus expected to be correlated over time. Second, the forestry problem is not as precisely defined as one might interpret from a strict mathematical formulation; for most situations, some minor violations of the forest-wide constraints are acceptable. More common primal solution methods emphasize the need to maintain strict feasibility, regardless of computation cost. Like linear programming models, DUALPLAN provides estimates of the value of all options for each management unit. Such information is valuable in implementation when managers consider whether factors not recognized in the model are reason to select another alternative. DUALPLAN solutions also do not split management units by assigning more than one alternative to each unit. With more traditional methods, this requires the use of mixed integer programming techniques. Recent work has shown that DUALPLAN can address additional forest-wide constraints involving management inputs or outputs other than harvest amounts. Hoganson and Kapple (1995) used the model to examine various policies on extended rotation forestry for the Minnesota Department of Natural Resources in northeastern Minnesota. The area of older forest each decade was treated as a modeled forest output. Results showed that the opportunity cost of sustaining older forest conditions varied substantially by forest cover type. Of importance is the role that thinning can be integrated into the extended rotation management prescriptions. DTRAN DTRAN is an expansion of the DUALPLAN approach that recognizes competition between timber markets in a region (Hoganson and Kapple 1991). Timber supply situations can vary by market, yet each cannot be modeled separately because of the interactions (competition) between markets. Using DTRAN, it is assumed that timber markets are perfectly competitive with perfect information. Essentially the model mimics the case where the perfectly competitive market system finds the efficient (optimal) solution. By utilizing maps of wood procurement zones for each market directly within the solution process, DTRAN overcomes the combinatorial nature of integrating harvest-timing decisions and wood-
shipping decisions. An optimal shipping schedule is identified for each management unit by simply referencing its location on maps relative to the procurement zones. In terms of an LP formulation of the problem, it is the marginal values for the different products in each period at each market location that define the zone boundaries. Boundaries change on the maps in the solution process as estimates of the marginal value of each product change for each market. The model thus overcomes the need to utilize separate decision variables to enumerate all combinations of plausible shipping options and harvesttiming options. DTRAN served as the forest management scheduling model for the $900,000 Minnesota Generic Environmental Impact Statement (GEIS) on Timber Harvesting and Management (Jaakko Pöyry Consulting 1994). Statewide analyses for the GEIS involved over 10,000 analysis areas and up to four timber product groups for each of six major market centers. Considerable effort went into this application. The data collected provided the basis for several additional studies. One shortcoming of the GEIS analysis is that it used too short of a planning horizon to fully recognize the benefits of reforestation investments. Hoganson and McDill (1993) extended the Minnesota GEIS applications over a substantially longer planning horizon to look specifically at reforestation investment opportunities. Multiple model runs were used to look at assumptions on environmental costs of clearcutting and environmental benefits of reserving additional acres of forestland from harvesting. Recognizing environmental costs of clearcutting suggested longer rotation ages. Impacts of reservation values were more pronounced, with reservation values of even $50/ acre suggesting substantially more investments in reforestation. For every additional acre managed more intensively, at least several acres of existing forestland could be reserved, while sustaining harvest target levels at the levels used for the medium scenario from the Minnesota GEIS. The Minnesota GEIS suggested concerns about aspen timber supply. According to the GEIS analyses, unless a substantial portion of the current aspen demand shifts to other species, temporary aspen supply shortfalls can be expected. DTRAN was used to examine the role that short-rotation intensive-silviculture investments might play in helping to overcome the aspen supply problem (Kapple and Hoganson 1995). Results suggested enormous potential benefits from such investments, with benefits not just temporary in nature because of the potential to reserve other older forest areas from harvesting. Results were most sensitive to the total area considered biologically and environmentally feasible for hybrid aspen management. Nearly all cases suggested that hybrid aspen investments be made up to this desirable
limit. A substantial portion of the returns from implementing hybrid aspen would come in the form of reduced timber transport costs. Hybrid aspen investments on 500,000 acres in northern Minnesota could reduce transport costs by 10%, or approximately 2.2 million dollars annually—approximately 1.5 million truck miles. Concentration of timber production on accessible lands close to markets would also reduce the need to gain access to more distant areas, thereby reducing road construction and maintenance costs. DTRAN applications have demonstrated that this specialized technique can be used to help model an enormous amount of detail at even a statewide level. Furthermore, these techniques offer “room” to incorporate substantially more detail. A logical next step is the explicit recognition of additional spatial interrelationships. THE MOVING-WINDOWS SPATIAL MODEL With the large and increased interest in ecosystem management, methods for addressing spatial relationships within forest management scheduling models have received considerable attention in the forestry literature. Emphasis has been on adjacency concerns. Examples include: Jones et al. (1991), Nelson and Errico (1993), Weintraub et al. (1994), and Snyder and Revelle (1996).. As noted by Weintraub et al. (1994), most studies are theoretical with little consideration of how such methods would be implemented. Methods generally involve either integer programming formulations or simulation approaches. With simulation, stands are scheduled sequentially with spatial implications associated with stands scheduled earlier eliminating alternatives for stands scheduled later. The sequencing of stands for analysis is thus important. With integer programming, model size is limiting because of numerous additional constraints needed to represent spatial limitations. Simulation approaches leave open the question of performance in terms of optimality, and the need for considering other forest-wide constraints is generally ignored or greatly simplified. The USDA Forest Service has used a revised version of SNAP (Sessions and Sessions 1988), a simulation model, to help address spatial concerns. SNAP emphasizes site-level constraints but is not tied to forestwide objectives. LANDIS (Mladenoff et al. 1996), an ecosystem model for the Lake States, is also a simulation model that links directly with a GIS and has routines for tracking various indices of landscape pattern. It does not have a scheduling routine for harvests and depends on user input for such decisions. Dynamic programming (DP), when applicable, is usually far more efficient than linear or integer programming. Model efficiency is critical when there is desire to expand models to recognize important detail. Hoganson and Borges (1998) developed a DP approach for addressing
adjacency concerns. In the DP formulation, each stage of the problem corresponds directly to the management decision for a specific stand (management unit).. State variables relate to the schedule implied for adjacent stands addressed earlier in the DP network. Each arc represents a management alternative for the associated stage (stand). Objective function values associated with each arc are the net returns for with the corresponding alternative. For arcs that violate adjacency constraints, either a large cost can be assumed or the arc can be eliminated. The recursive nature of DP links the value of all stands represented in the network. Key to the DP formulation is the definition of the state variables. For each stage, there is an associated “front” that separates the “scheduled” stands from the “unscheduled” stands. “Scheduled” stands are those stands represented in earlier stages of the network (the schedules selected for them are determined until the optimal solution is traced through the network after all stages are addressed in the solution process). Each “scheduled” stand along this front at each stage represents one state dimension for the stage. Each dimension tracks the status of its corresponding stand over time in terms of adjacency concerns. By tracking conditions for all stands along the front, it is possible to identify which alternatives for the current stand (stage) would violate any adjacency constraints. Key is the fact that the formulation does not grow in size like a decision tree. State dimensions are no longer needed for a stand once it is no longer along the front. The order in which stands are sequenced in the DP network will influence model size. A situation in which model size would be smallest is one where stands in the forest are all in a single row with no stand adjacent to more than two others. With this arrangement, one could start the DP formulation at one end of the forest and move sequentially across it with the “front” always only one stand wide. The number of stands in the forest would not be critical, because only the number of stages would increase with increasing stand numbers. Expanding the concept to a forest two stands wide (stands of similar size and shape), the DP formulation would have two to three stands along the front depending on how stands are arranged. In general, assuming that the number of alternative types does not vary substantially between stands, DP formulations will be smaller for narrow forests such that the stands can be sequenced “lengthwise” to keep the “front” narrow. To address realistic-size problems, Hoganson and Borges (1998) developed an overlapping, moving-windows approach where the overlap of the windows links the subproblems. The approach is defined by five steps:
(1)
Start at one edge of the forest and select a strip of the forest that can be solved using DP. Make the strip long, generally running the entire length of the forest and narrow enough to be manageable in terms of a DP formulation for stands within the strip (window).
(2)
Formulate and solve the DP problem assuming the only stands in the forest are the stands within the strip.
(3)
If the entire forest has been included in a DP window, then all stands yet to be scheduled can be scheduled according to the optimal DP solution found in step 2. Otherwise, for stands along the outside edge of the strip (subproblem) in step 2, schedule those stands according to the optimal DP solution.
(4)
Re-define the strip by: (a) eliminating from the latest strip, all stands scheduled in step 3, and (b) adding to the strip, a set of stands adjacent to the strip that have not yet been analyzed.
(5)
Return to step 2.
The approach is comparable to the use of moving windows in geographic information systems. Windows in this case are long in one direction. Window widths influence the number of states in each stage of the DP and must be matched with model size limitations. In effect, each pass of the window schedules stands along the outside edge of the window. It is assumed that the optimal schedule for these stands are not influenced by adjacency constraints involving stands not yet considered in a window. The ability of the approach to find near-optimal solutions thus depends on what is lost by considering only a portion of the stands in the model (window) at any one time. A hypothetical 1,000-stand forest was used to examine performance of the “moving windows” approach for large problems (Hoganson and Borges 1998). Each stand was one cell of a rectangular grid of stands. Up to seven management alternatives were considered for each stand. Information for each stand was generated randomly to mimic age and site quality conditions of the aspen forest type in the Lake States. For each of three test cases, the approach was applied varying both the side (edge) of the forest on which to start the process and the window size. For the case of an older forest with more stands financially mature, like those typical of Minnesota, the lowestcost solution had adjacency constraints costing $45.70 per acre. This cost was approximately 9.5% of the net present value of the forest under the unconstrained case. For each of the three test cases, the approach was compared to the
approach proposed by Weintraub et al. (1994). A window size only two stands wide outperformed the Weintraub algorithm in all cases. Solutions showed no sensitivity to the side of the forest used to start the process. All formulations involving less than 16,000 states at any stage (window size of 4 to 5) were solved in less than 3 minutes using a 50 MHZ 486DX2 computer. Considering forests with irregular-shaped stands complicates the implementation as formulating the DP efficiently is not straightforward. Borges et al. (1998) implemented the DP approach to 20 test cases involving forests with 800 to 3,200 stands of irregular shapes and sizes. Formulations with as many as 8 periods and 16 management alternative types were considered. Model performance was especially encouraging. Tests suggested the approach can come very close to optimal solutions. With irregularshaped stands, the approach still easily outperformed the Weintraub approach and greatly outperformed simpler heuristics. Over large areas, savings from a model like this could be enormous. Case studies mimicked relatively low-valued, pulp-producing, forest lands in northern Minnesota. One would expect greater gains in areas involving higher valued forest products. Results from test case applications suggested some potentially important implications for management. Returns were not overly sensitive to how management units were defined. If true, this could have substantial implications. It would suggest it is appropriate to focus primarily on environmental concerns when designing management units. Not as surprising, results also suggest that costs of adjacency constraints will vary tremendously depending on whether sustainable flow constraints are also imposed. Without sustainability constraints, opportunity costs of adjacency can be especially high. Test case results showed that when considering forestwide sustainability objectives, the costs of adjacency constraints are short term in nature. Unless timber prices could also rise in the short term, forest-wide harvest levels with adjacency constraints would drop by nearly 10% in the first decade before returning to long-term sustainable levels. Similar to stand age class imbalances related to basic forest regulation concepts in the traditional temporal dimension, age distributions are also not likely to be balanced initially in the spatial dimension; thus there is also likely a “conversion period” over which spatial conditions need to adjust. Test case results also add insight to the impact of adjacency constraints on overall spatial conditions. Results showed that even with adjacency constraints imposed, conditions like the amount and type of forest edge and interior space will vary tremendously over time. This
clearly suggests that improved problem formulations are needed to better address other spatial concerns. The DP approach has recently been enhanced to examine other spatial measures. It can be used to recognize explicitly both forest edge and forest interior space as modeled forest “outputs” over time. Modeling these measures is similar to modeling adjacency constraints, because the measures also depend directly on how neighboring stands are managed. Modeling spatial conditions as outputs helps shift attention towards the objectives of management rather than assuming constraints like adjacency constraints achieve management objectives when applied strictly across the entire forest. With increased attention to the need for diversity in terms of harvest patch sizes, it seems oversimple to assume that adjacency constraints are desirable in their strictest sense. CONCLUSIONS Forest management situations are complicated by the wide range of management alternatives available at the site level and the need to coordinate site-level decisions to help maximize broader forest-wide objectives related to both timber production and forest environmental conditions over time. Computer models offer enormous potential to explore an almost limitless set of potential management combinations. We have enormous potential to combine our understanding of the forestry problem and operations research to design specific modeling approaches to make practical application possible. And much of what is combined need not be all that complicated. Major gains can likely be made simply by utilizing maps in solution processes. Mathematical modeling and common sense can mix to help lead us to better forest management decisions. LITERATURE CITED Borges, Jose; Hoganson, H.; Rose, D. (in press). Combining a decomposition strategy with dynamic programming to solve spatially constrained forest management scheduling problems. Forest Science. Davis, Lawrence; Johnson, K. 1987. Forest management. New York: McGraw-Hill. 790 p. Fisher, Marshall. 1981. The Lagrangian relaxation method for solving integer programming problems. Management Science. 27(1): 1-18. Hoganson, Howard; Borges, J. 1998. Using dynamic programming and overlapping subproblems to address adjacency in large harvest scheduling problems. Forest Science. 44(4): 526-538.
Hoganson. Howard; Kapple, D. 1995. Estimating impacts of extended rotation forestry. Final research report submitted to the Minnesota Department of Natural Resources. St. Paul, MN: University of Minnesota Department of Forest Resources. 170 p. Hoganson, Howard; Kapple, D. 1991. DTRAN 1.0: a multi-market timber supply model. Staff Pap. Ser. Rep. 82. St. Paul, MN: College of Natural Resources and Agricultural Experiment Station, Department of Forest Resources, University of Minnesota Department of Forest Resources. 66 p. Hoganson, Howard; McDill, M. 1993. Relating reforestation investments in northern Minnesota with forest industry needs, nontimber values and mitigation strategies. Research report submitted to the Charles K. Blandin Foundation. St. Paul, MN: University of Minnesota Department of Forest Resources. 232 p.
Kapple Douglas; Hoganson, H. 1995. Close-to-market intensification of forest management. Final report submitted to the University of Minnesota Center for Transportation studies. 123 p. Mladenoff, David; Host, G.; Boeder, J.; Crow, T. 1996. A spatial model of forest landscape disturbance, succession and management. In: Goodchild, M.; et al. eds. GIS and Environmental Modeling: progress and research issues. Ft. Collins, CO: GIS World Books: 175-179. Nelson, J.D.; Errico, D. 1993. Multiple pass harvesting and spatial constraints: an old technique applied to a new problem. Forest Science. 39(1): 1-15. Paredes Gonzalo; Brodie, J. 1989. Land value and the linkage between stand and forest level analyses. Land Economics. 65(2): 158-166.
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Sessions John; Sessions, J. 1988. Scheduling and network analysis program (SNAP). User’s Guide. Ft. Collins, CO: Department of Forest Management, Oregon State University. Snyder, Stephanie; ReVelle, C. 1996. The grid packing problem: selecting a harvest pattern in an area with forbidden regions. Forest Science. 42(1): 27-34. Weintraub, Andres; Barahona, F.; Epstein, R. 1994. A column generation algorithm for solving general forest planning problems with adjacency constraints. Forest Science. 40(1): 142-161.
