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Locational issues in forest management
Location Science 6 (1998) 137±153
Locational issues in forest management Richard L. Church
a,* ,
Alan T. Murray b, Andres Weintraub
c
a
National Center for Geographic Information and Analysis and the Department of Geography, University of California at Santa Barbara, Santa Barbara, CA 93106-4060, USA b Department of Geographical Sciences and Planning, University of Queensland, Brisbane, Queensland 4072, Australia c Departamento de Ingenieria Industrial Universidad de Chile, Santiago, Chile
Abstract Modeling to support forest management is a complex task. In the public sector, agencies such as the US Forest Service manage millions of acres of land. In the private sector, large forest product companies also own and manage millions of acres. Many countries, like Ireland and Finland, utilize large-scale models to optimize the present value of forest activities and outputs. Forest planning can be viewed as a multi-level management problem. The highest or strategic level involves identifying feasible outputs and goals for long-term operations over decades. The middle or tactical level of analysis is associated with determining speci®c levels of activities on large tracts of land. At the lowest or operational level, speci®c stands of timber are slated for harvesting, roads are scheduled to be built or improved, and harvesting systems are designed to minimize the cost of extracting trees across the terrain. Each level of analysis often involves numerous locational decisions. Our objective is to provide a review of forest modeling that highlights many of the locational issues found in forest management. Ó 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction Forestry is an important primary industry in a number of countries including Canada, The United States, Finland, Chile, Russia, Indonesia, as well as many others. Many types of location decisions are faced within this industry, of which many have been the subject of modeling; see, for example, Weintraub and Bare (1996) and Weintraub and Navon (1986). Finding the best place for a lumber mill or a pulping plant can involve hundreds of millions of dollars in plant investment
*
Corresponding author. E-mail: [email protected].
0966-8349/99/$ ± see front matter Ó 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 6 - 8 3 4 9 ( 9 8 ) 0 0 0 5 1 - 5
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as well as implied costs of raw material and ®nished product transportation. Such a problem clearly ®ts the de®nition of the classic Weber location model. The purpose of this paper is to describe the types of location problems that have been faced in forest management and discuss some of the recent problems of interest. We will concentrate our review on forest management and operations problems. Many countries consider their forest lands to be renewable resources that must be managed for many uses, including forest products, recreation, and biodiversity. Both governmental agencies and private companies are involved in the management and operation of forest lands. For example, in the United States, over 190 million acres of forest and grasslands are managed by the US Forest Service. Major forest holdings are also owned and managed by large private forest product companies as well. Whether private or public management, a number of forest management and operation problems have been analyzed using location model constructs, such as: · · · · ·
Strategic Modeling Tactical Modeling Operation Modeling Production, Processing, and Transfer Biodiversity Protection and Reserve Design.
The objective of this paper is to describe the basic types of models used in the forest industry which involve some type of locational component. Many of the models that we will describe have imbedded model structures which are related to classic location models. Forestry models can be large and dicult to solve optimally and some require signi®cant temporal and spatial data. Much of that data involves inventory information about the type, distribution and age-class of trees. Areas are divided into stand units or blocks, which are groups of contiguous stands. Each basic unit is represented by the type and age-class of its vegetation (called strata). Geographical Information Systems (GIS) are now being used to store and manage such data and most models now utilize data that is exported from the GIS into the optimization model. Results are often depicted on maps that are generated by GIS systems as well. As this industry matures, simulation and optimization models are becoming important tools in planning ecient operations. The remainder of this paper is organized around the above topics, but an emphasis will be placed upon the areas of operations, production and processing, and biodiversity protection.
2. Strategic management and tactical forest planning Many of the most productive forest lands come under the control and management of large corporations or governmental agencies. When making manage-
R.L. Church et al./Location Science 6 (1998) 137±153
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ment decisions for large land holdings, it is reasonable to attempt to optimize the return on investment, present value of net bene®ts, or some other measure of productivity. Perhaps the most well-known forest management models are FORPLAN and its successor SPECTRUM, which were developed under the support of the US Forest Service for making large-scale strategic decisions in forest management. This model development was an outgrowth of earlier models like Timber RAM, Navon (1971). FORPLAN has probably had more impact on the management of forests throughout the world than any other optimization model. In application, the model attempts to optimize prescriptions and timing choices for each planning unit of a forest. Speci®cally, the idea is to manage a forest to provide a number of goods and services, typically called multiple use, without impacting the long-term viability of the forest. To do this, it is necessary to determine which strata should be treated or harvested in a given time period. Often such models are used to optimize land use management over 150 years, represented as 15 decadal periods. Because the number of prescriptions (cutting, thinning, etc.) and timing choices are large, it is necessary to keep the representation of strata types to a relatively small number. Principal constraints prevent the violation of environmental restrictions and guideline standards, while other constraints maintain a non-declining level of production over a long period of time; see, for example, Bare and Field (1987), Johnson et al. (1986), Kent et al. (1991) or Hof and Baltic (1991). Strategic models are designed to specify which general strategies can be applied and which outputs should be produced over a long planning horizon for a large area of land. Its application is based on aggregating spatial data to the extent that it is often dicult to establish realistic constraints on operations in smaller spatial units. Without tracking spatial detail, it is impossible to maintain a host of environmental conditions like limiting sediment loading in streams, limiting habitat disruption in an area, preventing a viewshed from being impacted too heavily, ensuring that open forage areas are provided for certain animals, etc., see Church and Barber (1992). Most conditions are maintained as constraints on operations across spatially de®ned units. Since FORPLAN and other strategic management models specify major outputs based on managing strata (Weintraub and Cholaky, 1991), they do not specify exactly where such operations take place. It is the goal of tactical models to translate strategic goals and prescription levels to speci®c units of land, such as a small watershed. Good examples of tactical level models include the work of Weintraub and Cholaky (1991), the RELMdss models of Church et al. (1996a,b), the hierarchical model of Davis and Martell (1993), and the work of Bare and Liermann (1994) and Nelson et al. (1991). Such models address smaller units of land in greater detail in order to identify an optimal plan for a shorter planning horizon. They do dier in the focus; Opticort (Weintraub and Cholaky, 1991) addresses an industrial perspective with few constraints whereas RELMdss focuses on identifying feasible solutions to a sizable set of environmental conditions. In a location context, consider a manufacturing company that plans to locate several new warehouses for product storage in the United States; one question may be how many. If we attempt to determine which cities should be chosen for warehouse sites,
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this might well be considered a tactical level decision. We may not use speci®c site information for a given city, but generic information about the city and its relative location to demand and other cities. Once cities have been chosen for warehouses, we may then focus attention on identifying the best site within each chosen city. This last decision can be considered an operational problem. We often do not detail such dierences in the location science literature. At the tactical level in forestry, we attempt to identify those areas that will be part of a harvesting plan at some future time. Then, the exact locations of harvest units are determined in the next level, operations.
3. Forest operations and location science Forest operations problems represent short-term issues such as harvesting, planting, pest control, ®re management, and road building. For private forest product companies as well as public agencies, operational level problems involve short-term planning, usually a few decades for daily operations. Suppose that a region has been divided into a set of harvest blocks, each comprised of one or more stands of trees. Each harvest block is a relatively contiguous area that is not larger than a pre-speci®ed size. Let i be an index of harvest units or areas; t be an index for time periods; xit 1; if harvest unit i is harvested in time period t, xit 0, otherwise; Vit the present value in harvesting unit i in time period t; and A f i; j j units i and j are adjacentg: Harvest units are often de®ned such that they may be harvested by clear cutting. The size of such units is limited, so that the opening caused by clear-cutting never exceeds a certain amount; for example, clear cutting may increase erosion, visual impacts, and disruption of habitats. By limiting the size of an opening, it is possible to limit such impacts (Barahona et al., 1992). This means that two adjacent units cannot be harvested in the same time period, or in adjacent time periods in more restricted settings. The main objective is to maximize the present value of the timber harvested by determining which units to harvest in each time period, subject to the condition that no two adjacent units are harvested in the same time period. This can be modeled as: Maximize Z
XX i
Vit xit
t
subject to: 1. Cannot harvest any two adjacent blocks in the same time period xit xjt 6 1
8 i; j 2 A; and t:
2. Each unit cannot be harvested more than once
R.L. Church et al./Location Science 6 (1998) 137±153
X
141
xit 6 1 8i:
t
3. Integer restrictions on decision variables xit 0; 1 8i; t: If the above model structure is used for only one time period, then the model would be equivalent to the anti-covering location model (Moon and Chaudhry, 1984; Murray and Church, 1996b). Thus, one can consider the above model to be a general multi-time period anti-covering location model. It is interesting to note that this problem is an important forest management model structure and that there has been considerable eort made to tighten the pairwise adjacency constraints. Examples of this work include: Meneghin et al. (1988), Jones et al. (1991), Murray and Church (1996a), Yoshimoto and Brodie (1994a,b), Torres and Brodie (1990) and Snyder and ReVelle (1996). This has been an important area in forest model research and is perhaps the best example of the applicability of the anti-covering location problem. It is also important to note that some of the techniques developed to improve the structure of the above model have been applied to the anti-covering model as well. The spatial optimization model given above is only part of the general operational management model. First, each harvest unit will produce a volume of timber based upon the time period in which it is scheduled to be harvested. The idea is that most operations need to have an even ¯ow of timber being produced. One common condition is to make sure that at each time period, the harvest is at least as great as in the previous time period. This is called the non-declining yield constraint. Second, harvesting cannot take place without some form of log transport. Sometimes logs are transported by helicopter to the nearest road, but most often a network of logging roads is built to reach harvest units. If logging in an area takes place over three time periods (each a decade long), then both road layout and timing can be important. Consider then, the additional notation and mode: j index of road links; V^it the undiscounted revenue generated from harvesting unit i in period t; vit the volume of unit i if harvested in period t; cjt the discounted cost to build road link j in period t; c^jt the undiscounted cost to build road link j in period t; Ht the upper bound on total volume of harvest in period t; Lt the lower bound on total volume of harvest in period t; LRt the lower bound on total undiscounted revenue generated in period t; p the harvest exclusion period length; Ni the set of harvest units adjacent to unit i; ni the coecient necessary to impose adjacency restrictions around unit i; Mj the set of road links that must be built in order to reach link j; Si the set of road links any one of which can be used to access unit i. The decision variables are: rjt
1
if road link j is built in time period t:
0
otherwise:
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Operation forest planning problem formulation: Maximize
Z
XX i
Vit xit ÿ
t
XX j
cjt rjt
t
subject to: 1. Limit harvest to at most once in the planning interval t ÿ p to t p tp X
xil 6 1
8i; t 2 p 1; T ÿ p:
ltÿp
2. Limit construction of each road link to at most one time period X
rjt 6 1
8j:
t
3. Cannot build road link j unless link j^ has been built rjt 6
t X l1
8j; t; j^ 2 Mj :
rjl^
4. Adjacency restrictions to prevent simultaneous harvesting of neighboring units ni xit
X ^i2Ni
x^it 6 ni
8i; t:
5. Cannot harvest a unit unless necessary access roads are built to unit xit 6
t X
rjl
8i; j 2 Si ; t:
l1
6. Upper and lower bounds on harvest volume in each time period a
X
vit xit P Lt
8t;
vit xit 6 Ht
8t:
i
b
X i
7. Undiscounted revenue bound requirement in each time period X i
V^it xit ÿ
X j
c^jt rjt P LRt
8t:
R.L. Church et al./Location Science 6 (1998) 137±153
143
8. Integer requirements xit 0; 1 8i; t; rit 0; 1 8i; t: The above operational model schedules/locates which harvest units will be cut in each time period, meets adjacency restrictions, and ensures that no unit is harvested without a completed road route that can reach that unit; see, for example, Kirby et al. (1986), Nelson and Brodie (1990) and Sessions and Sessions (1993). The objective is cast in terms of maximizing the present value of return on the harvested units as well as minimizing the present cost of road building. This problem represents a type of combination of the network design models described by Current and Marsh (1993) and the anti-covering model. In a way, one can think of this type of problem as not just a generalized version of network optimization and anti-covering, but cast in a temporal form. It is easy to see that if the simpler problems are somewhat dicult to solve, this general forest operations model would be as well. Researchers have approached this problem using a wide variety of techniques including dual ascent, Lagrangian heuristics, Monte-Carlo integer programming, simulated annealing, and tabu search just to name a few; see Lockwood and Moore (1993), Nelson and Brodie (1990), Murray and Church (1995), Weintraub et al. (1994, 1995). Because of the importance of this problem in practice, there will be a continuing interest to increase the capabilities for solving larger problems. Perhaps one of the most interesting forest operations problems involves the design of a harvesting plan (Dykstra and Riggs, 1977). Suppose that we have a large unit that is to be harvested during a season or two. For this problem, assume that the area has been divided into a number of grid cells. At each cell, we know attributes like ground slope, harvestable timber volume and aspect. For the sake of explanation, assume that there are two types of harvesting techniques, towers with cables and skidders. Harvesting by cable is required for those areas that have a high slope. Harvesting a cell is accomplished by a tower that is placed at a higher elevation and within line-of-sight of the cell. If the area is relatively ¯at, then it can be harvested by a skidder. Skidder operations are cheaper and are thus preferred. One of the objectives is to place towers in such a manner that the majority of cells can be reached that must be logged by cable. Moving a tower costs time and money, so we would like to design a logging operation that does not need too many tower locations. Skidder operations need some type of designated loading area. The costs of harvesting from a given tower or skidder loading area are a function of distance and slope from the loading and tower locations. The last element of the problem involves the location of a road network, where each skidder and tower must be connected to a system of roads. If a road is to be used during a wet season, then there can be a dierence in cost, as materials such as gravel might be required to make the road passable.
144
· · · · · · · · · · · · · · · · · ·
R.L. Church et al./Location Science 6 (1998) 137±153
De®ne i; j as the indices used to reference cells k the index of equipment type, e.g. skidder or tower Oj the amount of timber that is available to be harvested in cell j K1 the upper limit on ¯ow along a road (in cubic meters) xki 1, if harvesting equipment type k is located in cell i, xki 0, otherwise zij 1, if a road link between cells i and j is constructed, zij 0 otherwise, wkij 1, if cell j is harvested by equipment type k in cell i, wkij 0 otherwise, fij the amount of timber transported on road segment from cell i to cell j T {i | cell i is a potential site for some type of harvesting equipment} Ti {k | harvesting equipment type k can be allocated to cell i} Tk {i | cell i is a potential site for equipment k} Tjk {i | cell j can be harvested from cell i using equipment k and i 2 T k } NT {i | cell i is a potential road link node but not suitable for harvesting equipment} Cij the cost to build road link (i, j) Cik the cost to locate harvesting equipment type k at cell i Hijk the cost of harvesting cell j from cell i using harvesting equipment type k tij the cost per cubic meter of hauling timber on road link connecting i and j Vjk the value of the timber harvested from cell j using equipment k, and RL as the set of potential road links (i, j).
We can now formulate a harvesting operation design model in the following manner: Maximize : Z
XXX j
ÿ
k
i2Tjk
XX i2T k2Ti
Vjk wkij ÿ
Cik xki ÿ
X
Cij zij
i;j2RL
XXX j
k
i2Tjk
Hijk wkij ÿ
X
tij fij tji fji :
i;j2RL
subject to: 1. A cell cannot be harvested more than once XX k
i2Tjk
wkij 6 1
8j:
2. A cell can only be harvested when the appropriate equipment is allocated wkij 6 xki
8j; k; and i 2 Tjk :
3. Only one type of harvesting equipment can be allocated to a given cell
R.L. Church et al./Location Science 6 (1998) 137±153
X k2Ti
xki 6 1
145
8i 2 T :
4. Harvested timber must be transported on the road network XX k
j
Oj wkij
X
fli ÿ
X
fij 0
8i 2 T :
j2Ni
l2Ni
5. Timber ¯ow balance at intermediate nodes X l2N^i
fl^i ÿ
X l2N^i
8^i 2 NT :
f^il 0
6. Timber transport occurs only on constructed road links fij fji 6 K1 zij
8 i; j 2 RL:
7. Restrictions on decision variables xki 0; 1
8k; i 2 T k
wkij 0; 1 8j; k; i 2 Tjk Zij 0; 1 8 i; j 2 RL fij P 0
8 i; j 2 RL
The above model involves the location of harvesting equipment in such a manner as to harvest the landscape, represented as grid cells (Epstein et al., 1995). The distance, slope, and aspect of the landscape dictate whether a given cell can be harvested by equipment of a given type from another cell. The road network is optimized to reach all equipment locations. Further, the cost of hauling the timber along the road network is also minimized. Thus, a network may be somewhat longer than necessary when it results in substantial reductions in transport costs. It is possible that the best solution does not harvest all cells. This would occur if the cost to harvest and build access roads exceed the value of the timber that would be harvested. The above model represents a hybrid of a location model and a network design problem. Accomplishing the data handling aspects requires functions that are typically provided by GIS. An example of one application of this model is the PLANEX system, which is closely linked to ARC/INFO and utilizes a specialized heuristic to solve the equipment placement problem (Epstein et al., 1995).
146
R.L. Church et al./Location Science 6 (1998) 137±153
4. Production, processing and transfer Once timber has been harvested, it is transported to several types of facilities. They include log sorting and grading yards, storage and staging areas, mills (saw and pulp), and ports for export. For example, logs may be brought to a sorting yard, where high grade logs are separated and taken to a port for export and lower grade logs are taken to a lumber mill. Whatever the case, there are often intermediate processing and storage facilities, such as the transfer yard. A transfer yard usually involves some preprocessing of the logs and a change in transport mode, e.g. truck to rail. An example of this type of problem is given by Sessions and Paredes (1987): · · · · · · · · · · ·
i is the index for harvest areas for logs; j the index for ®nal destinations (e.g. mills, ports, etc.); k the index of potential transfer yard locations; yk 1, if site k is selected for a transfer yard, yk 0, otherwise; xij the volume of logs to be shipped from source node i to ®nal destination node j; xik the volume of logs to be shipped from source node i to transfer yard k; xkj the volume of logs to be shipped from transfer yard k to ®nal destination j; ai the volume of logs that must be shipped from source node i; cij the cost to ship logs from source node i to ®nal destination j (per unit volume); cik the cost to ship logs from source node i to transfer yard k (per unit volume); ckj the cost to ship logs from transfer yard k to ®nal destination j (per unit volume); · dj the demand for logs at destination j; · fk the ®xed cost to establish a transfer yard at node k; · Capk the potential capacity of the transfer yard at node k. The log transfer yard location model can then be formulated as: Minimize Z
XX i
cij xij
j
XX i
k
cik xik
XX k
ckj xkj
j
subject to: 1. Log volumes must be transported from source areas X
xik
X
xij ai
8i:
j
k
2. There must be no long term storage at the transfer yard X i
xik ÿ
X j
xkj 0
8k:
X k
fk yk
R.L. Church et al./Location Science 6 (1998) 137±153
147
3. Demand for logs at ®nal destinations must be met X X xij xkj dj 8j: i
k
4. Capacity restrictions at transfer yards must not be exceeded X xik 6 Capk yk 8k: i
5. Integer and nonnegative restrictions of the variables yk 0; 1 8k xij ; xjk ; xik P 0: The above model follows the structure of the well-known ®xed-charge location model. Modal changes in transportation are often made to decrease long distance haul costs. Such transfers and mode changes are common with other bulk materials. For example, most grain is hauled over short distances by truck to storage silos, which are located along railroad tracks. When the grain is taken from storage, it is then shipped via the railroad for long distance bulk transport; since rail is considerably more ecient than truck, overall costs are minimized. Another example of such a bulk commodity is the transportation of solid waste, which is often collected in small, somewhat inecient collection trucks. The waste is then taken to a transfer facility, where it is commonly shredded and further compacted into a large truck or rail car for transport to a land®ll or recycle center. As an example, much of the waste generated in Seattle, WA is transported to Oregon for land®lling (over 250 miles away). Thus, the transfer yard location problem found in forestry operations is related to this general phenomena. ReVelle et al. (1970) structured a similar type of model to optimize the location of solid waste transfer stations and developed a special branch-and-bound network ¯ow algorithm to solve the problem. Such a model can also be approached by the use of a ®xed charge network optimization problem. This model structure has also been extended to handle the sorting of logs; see, for example, Sessions and Paredes (1987) and Broad (1989). The structure of the log sorting and yard location problem is equivalent to a multi-commodity intermediate facilities location problem. It is interesting to note that in some areas, harvesting varies by season. This is due to the fact that some roads cannot be used all year, but only during the drier months. To feed mills at a more constant and even ¯ow, it is necessary to harvest and store logs over seasons. This requires an optimization model where both location and seasonal transport and storage is optimized. 5. Location modeling, biodiversity and reserve design World-wide deforestation, with resulting loss of species and native habitats, has focused concern for protecting biodiversity. For example, the Sierra±Nevada
148
R.L. Church et al./Location Science 6 (1998) 137±153
Region of the United States spans approximately 300 miles from North to South, comprises portions of 23 river basins and is contained almost wholly in California. About 400 species occur only in the Sierra Nevada including 3 tree species, 20 shrubs, and several hundred herbaceous plants. Of this total, 218 are considered rare or endangered. The Sierra region has been managed principally for forest products, grazing, mining, and water management (US Forest Service, 41%; the National Park Service, 6%; the Bureau of Land Management, 11%; and approximately 37% is privately held). A major issue is how such lands should be managed in order to protect biodiversity; see, for example, Thompson et al. (1973), Yoshimoto and Brodie (1994a), Hof and Joyce (1992, 1993) and Hof et al. (1994). Often, management of biodiversity involves special indicator or keystone species, such as, the spotted owl of the Northwest, the leadbetter possum in Australia, and the cockcaded woodpecker in the southeast United States. It is thought that providing suitable and sucient connected habitat area for a keystone species will provide general protection of a whole family of species which utilize the same habitat. In order to plan for the viability of the spotted owl in the Northwestern United States, a presidential commission recommended setting aside most of the remaining oldgrowth forest, resulting in substantial decreases in harvesting over the next several decades. Trading o the impacts of harvesting with species protection is dicult for many countries, especially when forest products can bring in needed currency. To approach this type of problem, one might ask how we could design a reserve system so that as many species as possible are protected with the least amount of land. Many researchers have de®ned the basic planning unit as a small watershed unit, large enough to be considered as a viable reserve site. Assume that we have divided a region into such watershed planning units. Consider then the following notation: p is the number of planning units to be selected for reserve status; j the index of planning units or watersheds; i the index of species that should be represented in the reserve system; yi 1, if species i is not present among the sites selected for the reserve, and 0, otherwise; xj 1, if planning unit i is selected for the reserve system, and 0, otherwise; Ni fj j site j contains species i in sufficient quantitiesg. We can now de®ne a simple reserve site selection model as (Church et al., 1996a): Minimize
Z
X
yi
i
subject to: 1. De®ne whether a species is included in the reserve system X
xj y i P 1
8 i:
j2Ni
2. Use exactly p-sites or planning units
R.L. Church et al./Location Science 6 (1998) 137±153
X
149
xj p:
j
3. De®ne decision variables xj 0; 1
8j;
yi 0; 1
8i:
This model selects a ®xed number of sites in such a manner that the combined system has the least number of species that are not represented, which is equivalent to selecting the set of sites that houses the largest number of species or genetic elements. Essentially, this is the Maximal Covering Location Problem of Church and ReVelle (1974). The Location Set Covering model has also been suggested as a potential reserve selection model as well (Underhill, 1994). Extended forms of the maximal covering model have also been developed for reserve design by Williams and ReVelle (1996). Camm et al. (1996) have suggested a general form of the above model where the objective is to cover each species a pre-speci®ed number of times. Some of the multiple cover, maximal covering models appear to be quite dicult to solve (Camm et al., 1996), which means that further research on solution approaches is needed. Species presence is not the only factor of interest in protecting biodiversity in forested areas. The amount of protected area of a given habitat is also an important indicator of reserve goodness; for example, how much area should be protected in some level of management status in order to enhance long term biodiversity. To address this, researchers of the Sierra Nevada Ecosystem Project de®ned the Biodiversity Management Area Selection Model (BMAS) (Church et al., 1996b), BMAS involves selecting small watershed units in order to minimize the area selected and maximize the amount of protection given for certain threatened species. Consider the following notation: · k is an index to represent species or elements that are considered at risk or in jeopardy; · aj the area of watershed or planning unit j; · xj 1, if planning unit j is selected for a Biodiversity Management Area, xj 0, otherwise; · Sj the suitability of planning unit j being selected as a biodiversity management area; · mink minimum area containing element k that needs to be brought under biodiversity management in order to remove element k from jeopardy; · ajk the area of element k in unit j that is impacted by current or planned activity; · ws the importance weight attached to optimizing suitability; · wa the importance weight attached to minimizing area. We can now de®ne BMAS as Church et al. (1996a,b)
150
R.L. Church et al./Location Science 6 (1998) 137±153
Minimize
Z wa
X j
aj xj ws
X
Sj x j
j
subject to: 1. Set aside enough area that contains species or element k X
ajk xj P min
j
k
8 k:
2. De®ne integer restrictions on decision variables xj 0; 1 8j: This model can be transformed into an equivalent knapsack problem and therefore it falls into the class of NP-hard. The BMAS model was used to study the impact on land use change that is needed to protect various habitats and species at risk. The application of the BMAS model in the Sierra Nevada involves over 2700 planning units and 200 elements. Exact solutions have not been generated in many cases, as the model appears to be dicult, if not impossible, to solve optimally. This should be a fertile area for new research, as such models will likely be used in many dierent planning contexts.
6. Conclusions We have attempted to describe a number of types of forest management problems, many of which are variations of existing location models. These models are typically dicult to solve, have problem instances that are considered quite large, and have been the subject of considerable research. We have discussed some of the main features of such models and similarities with existing location models. It is hoped that this paper will be bene®cial to forest analysts as a basis of appreciation for the ®eld of location science in their work. It is also hoped that this paper will demonstrate to location scientists that the ®eld of forestry presents a wide variety of location problems of sucient complexity and detail to warrant interest, encourage new research, and serve as a potential application arena.
Acknowledgements We would like to thank personnel from regions 5 and 6 of the USFS, who have assisted us, especially Klaus Barber and Richard Dyrland. We also wish to thank Phil Aune and the Paci®c South West Laboratory for research support.
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References Barahona, F., Weintraub, A., Epstein, R., 1992. Habitat dispersion in forest planning and stable set problem. Operations Research 40 Supp. No. 1, S14±S21. Bare, B., Field, R., 1987. An evaluation of FORPLAN from an operations research perspective. In: FORPLAN: an evaluation of a forest planning tool, USDA Forest Service General Technical Report RM-140, Rocky Mountain Forest Range Experiment Station, Fort Collins, CO. Bare, B., Liermann, M., 1994. Incorporating spatial relationships in forest planning models: the development of a modeling strategy. Final Report to USDA Forest Service, PNW Forest Experiment Station, Portland, OR, p. 66. Broad, L., 1989. Note on log sort yard location problem. Forest Science 35, 640±645. Camm, J.D., Polasky, S., Solow, A., Csuti, B., 1996. Biological Conservation (in press). Church, R., Barber, K., 1992. Disaggregating forest management plans to treatment areas. Stand Inventory Technologies 92, Portland, Oregon, 13±17 September, pp. 287±295. Church, R., Murray, A., Barber, K., 1995a. Designing a hierarchical planning model for USDA Forest Service planning. In: Proceedings of the Sixth Symposium on Systems Analysis and Management Decisions in Forestry, 6±9 September, 1994, Paci®c Grove, California. Church, R., Murray, A., Barber, K., Dyrland, D., 1995b. A spatial decision support system for doing hierarchical habitat planning. In: Proceedings of the analysis in support of ecosystem management: Analysis Workshop III, Ft. Collins, CO, pp. 283±289. Church, R., ReVelle, C., 1974. The maximal covering location problem. Papers of the Regional Science Association 32, 101±118. Church, R., Stoms, D., Davis, F., 1996a. Reserve selection as a maximal covering location problem. Biological Conservation 76, 105±112. Church, R., Stoms, D., Davis, F., Okin, W., 1996b. Planning activities to protect biodiversity with a GIS and an integrated optimization model. In: Proceedings of the Third International Conference/ Workshop on Integrating GIS and Environmental Modeling. Santa Fe. Current, J., Marsh, M., 1993. Multiobjective transportation network design and routing problems taxonomy and annotation. European Journal of Operational Research 65, 4±19. Davis, R., Martell, D., 1993. A decision support system that links short-term silvicultural operating plans with long-term forest-level strategic plans. Canadian Journal of Forest Research 23, 1078± 1095. Dykstra, D., Riggs, J., 1977. An application of facilities location theory to the design of forest harvesting areas. AIIE Transactions 9, 270±277. Epstein, R., Weintraub, A., Sapunar, P., Neito, E., Sessions, J., Sessions, B., 1995. PLANEX: a system for optimal assignment and harvesting. In: Proceedings of the LIRO Conference on Harvest Planning, Nelson, New Zealand, Paper 2a. Hof, J., Baltic, T., 1991. A multilevel analysis of production capabilities of the National Forest system. Operations Research 39, 543±552. Hof, J., Bevers, M., Joyce, L., Kent, B., 1994. An integer programming approach for spatially and temporally optimizing wildlife populations. Forest Science 40, 177±191. Hof, J., Joyce, L., 1992. Spatial optimization for wildlife and timber in managed forest ecosystems. Forest Science 38, 489±508. Hof, J., Joyce, L., 1993. A mixed integer linear programming approach for spatially optimizing wildlife and timber in managed forest ecosystems. Forest Science 39, 816±834. Johnson, K.N., Stuart, T., Crim, S., 1986 FORPLAN Version 2: An overview. USDA Forest Service, Land Management Planning, Washington, DC. Jones, J.G., Meneghin, B.J., Kirby, M.W., 1991. Formulating adjacency constraints in linear optimization models for scheduling projects in tactical planning. Forest Science 37, 1283±1297. Kent, B., Bare, B.B., Field, R., Bradley, G., 1991. Natural resource land management planning using large-scale linear programming: the USDA Forest Service experience with FORPLAN. Operations Research 39, 13±27.
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Kirby, M., Hager, W., Wong, P., 1986. Simultaneous planning of wildland management and transportation alternatives. TIMS Studies in the Management Sciences 21, 371±387. Lockwood, C., Moore, T., 1993. Harvest scheduling with spatial constraints: a simulated annealing approach. Canadian Journal of Forest Research 23, 468±478. Meneghin, B.J., Kirby, M.W., Jones, J.G., 1988. An algorithm for writing adjacency constraints eciently in linear programming models. In: The 1988 Symposium on systems analysis in forest resources. US Forest Service Rocky Mountain Forest and Range Experiment Station General Technical Report RM161, pp. 46±53. Moon, I.D., Chaudhry, S., 1984. An analysis of network location problems with distance constraints. Management Science 30, 290±307. Murray, A., Church, R., 1995. Heuristic solution approaches to operational forest planning problems. OR Spektrum 17 (2/3), 193±203. Murray, A., Church, R., 1996a. Analyzing cliques for imposing adjacency restrictions in forest models. Forest Science 42. Murray, A., Church, R., 1996b. Using proximity restrictions for locating undesirable facilities. Studies in Locational Analysis (to appear). Navon, D., 1971. Timber RAM, a long range planning method for commercial timber lands under multiple-use management. USDA Forest Service Research Paper PSW-70, Berkeley, CA. Nelson, J.D., Brodie, J.D., 1990. Comparison of random search algorithm and mixed integer programming for solving area-based forest plans. Canadian Journal of Forest Research 20, 934± 942. Nelson, J.D., Brodie, J.D., Sessions, J., 1991. Integrating short-term, area-based logging plans with longterm harvest schedules. Forest Science 37, 101±122. ReVelle, C., Marks, D., Liebman, J., 1970. An analysis of private and public sector location problems. Management Science 16, 692±707. Sessions, J., Paredes, G., 1987. A solution procedure for the sort yard location problem in forest operations. Forest Science 33, 750±762. Sessions, J., Sessions, J.B., 1993. Scheduling and Network Analysis Program (SNAP II) UserÕs Guide, version 2.07. Snyder, S., ReVelle, C., 1996. The grid packing problem: selecting a harvesting pattern in an area with forbidden regions. Forest Science 42, 27±34. Thompson, E.F., Halterman, B.G., Lyon, T.J., Miller, R.L., 1973. Integrating timber and wildlife management planning. Forestry Chronicle 49, 247±250. Torres, R.J.M., Brodie, J.D., 1990. Adjacency constraints in harvest scheduling: an aggregation heuristic. Canadian Journal of Forest Research 20, 978±986. Underhill, L.G., 1994. Optimal and suboptimal reserve selection algorithms. Biological Conservation 70, 85±87. Weintraub, A., Bare, B., 1996. New issues in forest land management from an operations research perspective. Interfaces 26, 9±25. Weintraub, A., Barahona, F., Epstein, R., 1994. A column generation algorithm for solving general forest planning problems with adjacency constraints. Forest Science 40, 142±161. Weintraub, A., Cholaky, A., 1991. A hierarchical approach to forest planning. Forest Science 37, 439± 460. Weintraub, A., Jones, G., Meacham, M., Magendzo, A., Magendzo, A., Malchuk, D., 1995. Heuristic procedures for solving mixed-integer harvest scheduling-transportation planning models. Canadian Journal of Forest Research 25, 1618±1626. Weintraub, A., Navon, D., 1986. Mathematical programming in large scale forestry modeling and applications. TIMS Studies in the Management Sciences 21, 337±351. Williams, J., ReVelle, C., 1996. A multicriteria heuristic approach to selecting land for conservation. Presented at the 43rd North American Meetings of the Regional Science Association, Arlington VA, 14±17 November.
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Yoshimoto, A., Brodie, J.D., 1994a. Short-and long-term impacts of spatial restrictions on harvest scheduling with reference to riparian zone planning. Canadian Journal of Forest Research 24, 1617± 1628. Yoshimoto, A., Brodie, J.D., 1994b. Comparative analysis of algorithms to generate adjacency constraints. Canadian Journal of Forest Research 24, 1277±1288.
Locational issues in forest management Richard L. Church
a,* ,
Alan T. Murray b, Andres Weintraub
c
a
National Center for Geographic Information and Analysis and the Department of Geography, University of California at Santa Barbara, Santa Barbara, CA 93106-4060, USA b Department of Geographical Sciences and Planning, University of Queensland, Brisbane, Queensland 4072, Australia c Departamento de Ingenieria Industrial Universidad de Chile, Santiago, Chile
Abstract Modeling to support forest management is a complex task. In the public sector, agencies such as the US Forest Service manage millions of acres of land. In the private sector, large forest product companies also own and manage millions of acres. Many countries, like Ireland and Finland, utilize large-scale models to optimize the present value of forest activities and outputs. Forest planning can be viewed as a multi-level management problem. The highest or strategic level involves identifying feasible outputs and goals for long-term operations over decades. The middle or tactical level of analysis is associated with determining speci®c levels of activities on large tracts of land. At the lowest or operational level, speci®c stands of timber are slated for harvesting, roads are scheduled to be built or improved, and harvesting systems are designed to minimize the cost of extracting trees across the terrain. Each level of analysis often involves numerous locational decisions. Our objective is to provide a review of forest modeling that highlights many of the locational issues found in forest management. Ó 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction Forestry is an important primary industry in a number of countries including Canada, The United States, Finland, Chile, Russia, Indonesia, as well as many others. Many types of location decisions are faced within this industry, of which many have been the subject of modeling; see, for example, Weintraub and Bare (1996) and Weintraub and Navon (1986). Finding the best place for a lumber mill or a pulping plant can involve hundreds of millions of dollars in plant investment
*
Corresponding author. E-mail: [email protected].
0966-8349/99/$ ± see front matter Ó 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 6 - 8 3 4 9 ( 9 8 ) 0 0 0 5 1 - 5
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as well as implied costs of raw material and ®nished product transportation. Such a problem clearly ®ts the de®nition of the classic Weber location model. The purpose of this paper is to describe the types of location problems that have been faced in forest management and discuss some of the recent problems of interest. We will concentrate our review on forest management and operations problems. Many countries consider their forest lands to be renewable resources that must be managed for many uses, including forest products, recreation, and biodiversity. Both governmental agencies and private companies are involved in the management and operation of forest lands. For example, in the United States, over 190 million acres of forest and grasslands are managed by the US Forest Service. Major forest holdings are also owned and managed by large private forest product companies as well. Whether private or public management, a number of forest management and operation problems have been analyzed using location model constructs, such as: · · · · ·
Strategic Modeling Tactical Modeling Operation Modeling Production, Processing, and Transfer Biodiversity Protection and Reserve Design.
The objective of this paper is to describe the basic types of models used in the forest industry which involve some type of locational component. Many of the models that we will describe have imbedded model structures which are related to classic location models. Forestry models can be large and dicult to solve optimally and some require signi®cant temporal and spatial data. Much of that data involves inventory information about the type, distribution and age-class of trees. Areas are divided into stand units or blocks, which are groups of contiguous stands. Each basic unit is represented by the type and age-class of its vegetation (called strata). Geographical Information Systems (GIS) are now being used to store and manage such data and most models now utilize data that is exported from the GIS into the optimization model. Results are often depicted on maps that are generated by GIS systems as well. As this industry matures, simulation and optimization models are becoming important tools in planning ecient operations. The remainder of this paper is organized around the above topics, but an emphasis will be placed upon the areas of operations, production and processing, and biodiversity protection.
2. Strategic management and tactical forest planning Many of the most productive forest lands come under the control and management of large corporations or governmental agencies. When making manage-
R.L. Church et al./Location Science 6 (1998) 137±153
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ment decisions for large land holdings, it is reasonable to attempt to optimize the return on investment, present value of net bene®ts, or some other measure of productivity. Perhaps the most well-known forest management models are FORPLAN and its successor SPECTRUM, which were developed under the support of the US Forest Service for making large-scale strategic decisions in forest management. This model development was an outgrowth of earlier models like Timber RAM, Navon (1971). FORPLAN has probably had more impact on the management of forests throughout the world than any other optimization model. In application, the model attempts to optimize prescriptions and timing choices for each planning unit of a forest. Speci®cally, the idea is to manage a forest to provide a number of goods and services, typically called multiple use, without impacting the long-term viability of the forest. To do this, it is necessary to determine which strata should be treated or harvested in a given time period. Often such models are used to optimize land use management over 150 years, represented as 15 decadal periods. Because the number of prescriptions (cutting, thinning, etc.) and timing choices are large, it is necessary to keep the representation of strata types to a relatively small number. Principal constraints prevent the violation of environmental restrictions and guideline standards, while other constraints maintain a non-declining level of production over a long period of time; see, for example, Bare and Field (1987), Johnson et al. (1986), Kent et al. (1991) or Hof and Baltic (1991). Strategic models are designed to specify which general strategies can be applied and which outputs should be produced over a long planning horizon for a large area of land. Its application is based on aggregating spatial data to the extent that it is often dicult to establish realistic constraints on operations in smaller spatial units. Without tracking spatial detail, it is impossible to maintain a host of environmental conditions like limiting sediment loading in streams, limiting habitat disruption in an area, preventing a viewshed from being impacted too heavily, ensuring that open forage areas are provided for certain animals, etc., see Church and Barber (1992). Most conditions are maintained as constraints on operations across spatially de®ned units. Since FORPLAN and other strategic management models specify major outputs based on managing strata (Weintraub and Cholaky, 1991), they do not specify exactly where such operations take place. It is the goal of tactical models to translate strategic goals and prescription levels to speci®c units of land, such as a small watershed. Good examples of tactical level models include the work of Weintraub and Cholaky (1991), the RELMdss models of Church et al. (1996a,b), the hierarchical model of Davis and Martell (1993), and the work of Bare and Liermann (1994) and Nelson et al. (1991). Such models address smaller units of land in greater detail in order to identify an optimal plan for a shorter planning horizon. They do dier in the focus; Opticort (Weintraub and Cholaky, 1991) addresses an industrial perspective with few constraints whereas RELMdss focuses on identifying feasible solutions to a sizable set of environmental conditions. In a location context, consider a manufacturing company that plans to locate several new warehouses for product storage in the United States; one question may be how many. If we attempt to determine which cities should be chosen for warehouse sites,
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this might well be considered a tactical level decision. We may not use speci®c site information for a given city, but generic information about the city and its relative location to demand and other cities. Once cities have been chosen for warehouses, we may then focus attention on identifying the best site within each chosen city. This last decision can be considered an operational problem. We often do not detail such dierences in the location science literature. At the tactical level in forestry, we attempt to identify those areas that will be part of a harvesting plan at some future time. Then, the exact locations of harvest units are determined in the next level, operations.
3. Forest operations and location science Forest operations problems represent short-term issues such as harvesting, planting, pest control, ®re management, and road building. For private forest product companies as well as public agencies, operational level problems involve short-term planning, usually a few decades for daily operations. Suppose that a region has been divided into a set of harvest blocks, each comprised of one or more stands of trees. Each harvest block is a relatively contiguous area that is not larger than a pre-speci®ed size. Let i be an index of harvest units or areas; t be an index for time periods; xit 1; if harvest unit i is harvested in time period t, xit 0, otherwise; Vit the present value in harvesting unit i in time period t; and A f i; j j units i and j are adjacentg: Harvest units are often de®ned such that they may be harvested by clear cutting. The size of such units is limited, so that the opening caused by clear-cutting never exceeds a certain amount; for example, clear cutting may increase erosion, visual impacts, and disruption of habitats. By limiting the size of an opening, it is possible to limit such impacts (Barahona et al., 1992). This means that two adjacent units cannot be harvested in the same time period, or in adjacent time periods in more restricted settings. The main objective is to maximize the present value of the timber harvested by determining which units to harvest in each time period, subject to the condition that no two adjacent units are harvested in the same time period. This can be modeled as: Maximize Z
XX i
Vit xit
t
subject to: 1. Cannot harvest any two adjacent blocks in the same time period xit xjt 6 1
8 i; j 2 A; and t:
2. Each unit cannot be harvested more than once
R.L. Church et al./Location Science 6 (1998) 137±153
X
141
xit 6 1 8i:
t
3. Integer restrictions on decision variables xit 0; 1 8i; t: If the above model structure is used for only one time period, then the model would be equivalent to the anti-covering location model (Moon and Chaudhry, 1984; Murray and Church, 1996b). Thus, one can consider the above model to be a general multi-time period anti-covering location model. It is interesting to note that this problem is an important forest management model structure and that there has been considerable eort made to tighten the pairwise adjacency constraints. Examples of this work include: Meneghin et al. (1988), Jones et al. (1991), Murray and Church (1996a), Yoshimoto and Brodie (1994a,b), Torres and Brodie (1990) and Snyder and ReVelle (1996). This has been an important area in forest model research and is perhaps the best example of the applicability of the anti-covering location problem. It is also important to note that some of the techniques developed to improve the structure of the above model have been applied to the anti-covering model as well. The spatial optimization model given above is only part of the general operational management model. First, each harvest unit will produce a volume of timber based upon the time period in which it is scheduled to be harvested. The idea is that most operations need to have an even ¯ow of timber being produced. One common condition is to make sure that at each time period, the harvest is at least as great as in the previous time period. This is called the non-declining yield constraint. Second, harvesting cannot take place without some form of log transport. Sometimes logs are transported by helicopter to the nearest road, but most often a network of logging roads is built to reach harvest units. If logging in an area takes place over three time periods (each a decade long), then both road layout and timing can be important. Consider then, the additional notation and mode: j index of road links; V^it the undiscounted revenue generated from harvesting unit i in period t; vit the volume of unit i if harvested in period t; cjt the discounted cost to build road link j in period t; c^jt the undiscounted cost to build road link j in period t; Ht the upper bound on total volume of harvest in period t; Lt the lower bound on total volume of harvest in period t; LRt the lower bound on total undiscounted revenue generated in period t; p the harvest exclusion period length; Ni the set of harvest units adjacent to unit i; ni the coecient necessary to impose adjacency restrictions around unit i; Mj the set of road links that must be built in order to reach link j; Si the set of road links any one of which can be used to access unit i. The decision variables are: rjt
1
if road link j is built in time period t:
0
otherwise:
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Operation forest planning problem formulation: Maximize
Z
XX i
Vit xit ÿ
t
XX j
cjt rjt
t
subject to: 1. Limit harvest to at most once in the planning interval t ÿ p to t p tp X
xil 6 1
8i; t 2 p 1; T ÿ p:
ltÿp
2. Limit construction of each road link to at most one time period X
rjt 6 1
8j:
t
3. Cannot build road link j unless link j^ has been built rjt 6
t X l1
8j; t; j^ 2 Mj :
rjl^
4. Adjacency restrictions to prevent simultaneous harvesting of neighboring units ni xit
X ^i2Ni
x^it 6 ni
8i; t:
5. Cannot harvest a unit unless necessary access roads are built to unit xit 6
t X
rjl
8i; j 2 Si ; t:
l1
6. Upper and lower bounds on harvest volume in each time period a
X
vit xit P Lt
8t;
vit xit 6 Ht
8t:
i
b
X i
7. Undiscounted revenue bound requirement in each time period X i
V^it xit ÿ
X j
c^jt rjt P LRt
8t:
R.L. Church et al./Location Science 6 (1998) 137±153
143
8. Integer requirements xit 0; 1 8i; t; rit 0; 1 8i; t: The above operational model schedules/locates which harvest units will be cut in each time period, meets adjacency restrictions, and ensures that no unit is harvested without a completed road route that can reach that unit; see, for example, Kirby et al. (1986), Nelson and Brodie (1990) and Sessions and Sessions (1993). The objective is cast in terms of maximizing the present value of return on the harvested units as well as minimizing the present cost of road building. This problem represents a type of combination of the network design models described by Current and Marsh (1993) and the anti-covering model. In a way, one can think of this type of problem as not just a generalized version of network optimization and anti-covering, but cast in a temporal form. It is easy to see that if the simpler problems are somewhat dicult to solve, this general forest operations model would be as well. Researchers have approached this problem using a wide variety of techniques including dual ascent, Lagrangian heuristics, Monte-Carlo integer programming, simulated annealing, and tabu search just to name a few; see Lockwood and Moore (1993), Nelson and Brodie (1990), Murray and Church (1995), Weintraub et al. (1994, 1995). Because of the importance of this problem in practice, there will be a continuing interest to increase the capabilities for solving larger problems. Perhaps one of the most interesting forest operations problems involves the design of a harvesting plan (Dykstra and Riggs, 1977). Suppose that we have a large unit that is to be harvested during a season or two. For this problem, assume that the area has been divided into a number of grid cells. At each cell, we know attributes like ground slope, harvestable timber volume and aspect. For the sake of explanation, assume that there are two types of harvesting techniques, towers with cables and skidders. Harvesting by cable is required for those areas that have a high slope. Harvesting a cell is accomplished by a tower that is placed at a higher elevation and within line-of-sight of the cell. If the area is relatively ¯at, then it can be harvested by a skidder. Skidder operations are cheaper and are thus preferred. One of the objectives is to place towers in such a manner that the majority of cells can be reached that must be logged by cable. Moving a tower costs time and money, so we would like to design a logging operation that does not need too many tower locations. Skidder operations need some type of designated loading area. The costs of harvesting from a given tower or skidder loading area are a function of distance and slope from the loading and tower locations. The last element of the problem involves the location of a road network, where each skidder and tower must be connected to a system of roads. If a road is to be used during a wet season, then there can be a dierence in cost, as materials such as gravel might be required to make the road passable.
144
· · · · · · · · · · · · · · · · · ·
R.L. Church et al./Location Science 6 (1998) 137±153
De®ne i; j as the indices used to reference cells k the index of equipment type, e.g. skidder or tower Oj the amount of timber that is available to be harvested in cell j K1 the upper limit on ¯ow along a road (in cubic meters) xki 1, if harvesting equipment type k is located in cell i, xki 0, otherwise zij 1, if a road link between cells i and j is constructed, zij 0 otherwise, wkij 1, if cell j is harvested by equipment type k in cell i, wkij 0 otherwise, fij the amount of timber transported on road segment from cell i to cell j T {i | cell i is a potential site for some type of harvesting equipment} Ti {k | harvesting equipment type k can be allocated to cell i} Tk {i | cell i is a potential site for equipment k} Tjk {i | cell j can be harvested from cell i using equipment k and i 2 T k } NT {i | cell i is a potential road link node but not suitable for harvesting equipment} Cij the cost to build road link (i, j) Cik the cost to locate harvesting equipment type k at cell i Hijk the cost of harvesting cell j from cell i using harvesting equipment type k tij the cost per cubic meter of hauling timber on road link connecting i and j Vjk the value of the timber harvested from cell j using equipment k, and RL as the set of potential road links (i, j).
We can now formulate a harvesting operation design model in the following manner: Maximize : Z
XXX j
ÿ
k
i2Tjk
XX i2T k2Ti
Vjk wkij ÿ
Cik xki ÿ
X
Cij zij
i;j2RL
XXX j
k
i2Tjk
Hijk wkij ÿ
X
tij fij tji fji :
i;j2RL
subject to: 1. A cell cannot be harvested more than once XX k
i2Tjk
wkij 6 1
8j:
2. A cell can only be harvested when the appropriate equipment is allocated wkij 6 xki
8j; k; and i 2 Tjk :
3. Only one type of harvesting equipment can be allocated to a given cell
R.L. Church et al./Location Science 6 (1998) 137±153
X k2Ti
xki 6 1
145
8i 2 T :
4. Harvested timber must be transported on the road network XX k
j
Oj wkij
X
fli ÿ
X
fij 0
8i 2 T :
j2Ni
l2Ni
5. Timber ¯ow balance at intermediate nodes X l2N^i
fl^i ÿ
X l2N^i
8^i 2 NT :
f^il 0
6. Timber transport occurs only on constructed road links fij fji 6 K1 zij
8 i; j 2 RL:
7. Restrictions on decision variables xki 0; 1
8k; i 2 T k
wkij 0; 1 8j; k; i 2 Tjk Zij 0; 1 8 i; j 2 RL fij P 0
8 i; j 2 RL
The above model involves the location of harvesting equipment in such a manner as to harvest the landscape, represented as grid cells (Epstein et al., 1995). The distance, slope, and aspect of the landscape dictate whether a given cell can be harvested by equipment of a given type from another cell. The road network is optimized to reach all equipment locations. Further, the cost of hauling the timber along the road network is also minimized. Thus, a network may be somewhat longer than necessary when it results in substantial reductions in transport costs. It is possible that the best solution does not harvest all cells. This would occur if the cost to harvest and build access roads exceed the value of the timber that would be harvested. The above model represents a hybrid of a location model and a network design problem. Accomplishing the data handling aspects requires functions that are typically provided by GIS. An example of one application of this model is the PLANEX system, which is closely linked to ARC/INFO and utilizes a specialized heuristic to solve the equipment placement problem (Epstein et al., 1995).
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R.L. Church et al./Location Science 6 (1998) 137±153
4. Production, processing and transfer Once timber has been harvested, it is transported to several types of facilities. They include log sorting and grading yards, storage and staging areas, mills (saw and pulp), and ports for export. For example, logs may be brought to a sorting yard, where high grade logs are separated and taken to a port for export and lower grade logs are taken to a lumber mill. Whatever the case, there are often intermediate processing and storage facilities, such as the transfer yard. A transfer yard usually involves some preprocessing of the logs and a change in transport mode, e.g. truck to rail. An example of this type of problem is given by Sessions and Paredes (1987): · · · · · · · · · · ·
i is the index for harvest areas for logs; j the index for ®nal destinations (e.g. mills, ports, etc.); k the index of potential transfer yard locations; yk 1, if site k is selected for a transfer yard, yk 0, otherwise; xij the volume of logs to be shipped from source node i to ®nal destination node j; xik the volume of logs to be shipped from source node i to transfer yard k; xkj the volume of logs to be shipped from transfer yard k to ®nal destination j; ai the volume of logs that must be shipped from source node i; cij the cost to ship logs from source node i to ®nal destination j (per unit volume); cik the cost to ship logs from source node i to transfer yard k (per unit volume); ckj the cost to ship logs from transfer yard k to ®nal destination j (per unit volume); · dj the demand for logs at destination j; · fk the ®xed cost to establish a transfer yard at node k; · Capk the potential capacity of the transfer yard at node k. The log transfer yard location model can then be formulated as: Minimize Z
XX i
cij xij
j
XX i
k
cik xik
XX k
ckj xkj
j
subject to: 1. Log volumes must be transported from source areas X
xik
X
xij ai
8i:
j
k
2. There must be no long term storage at the transfer yard X i
xik ÿ
X j
xkj 0
8k:
X k
fk yk
R.L. Church et al./Location Science 6 (1998) 137±153
147
3. Demand for logs at ®nal destinations must be met X X xij xkj dj 8j: i
k
4. Capacity restrictions at transfer yards must not be exceeded X xik 6 Capk yk 8k: i
5. Integer and nonnegative restrictions of the variables yk 0; 1 8k xij ; xjk ; xik P 0: The above model follows the structure of the well-known ®xed-charge location model. Modal changes in transportation are often made to decrease long distance haul costs. Such transfers and mode changes are common with other bulk materials. For example, most grain is hauled over short distances by truck to storage silos, which are located along railroad tracks. When the grain is taken from storage, it is then shipped via the railroad for long distance bulk transport; since rail is considerably more ecient than truck, overall costs are minimized. Another example of such a bulk commodity is the transportation of solid waste, which is often collected in small, somewhat inecient collection trucks. The waste is then taken to a transfer facility, where it is commonly shredded and further compacted into a large truck or rail car for transport to a land®ll or recycle center. As an example, much of the waste generated in Seattle, WA is transported to Oregon for land®lling (over 250 miles away). Thus, the transfer yard location problem found in forestry operations is related to this general phenomena. ReVelle et al. (1970) structured a similar type of model to optimize the location of solid waste transfer stations and developed a special branch-and-bound network ¯ow algorithm to solve the problem. Such a model can also be approached by the use of a ®xed charge network optimization problem. This model structure has also been extended to handle the sorting of logs; see, for example, Sessions and Paredes (1987) and Broad (1989). The structure of the log sorting and yard location problem is equivalent to a multi-commodity intermediate facilities location problem. It is interesting to note that in some areas, harvesting varies by season. This is due to the fact that some roads cannot be used all year, but only during the drier months. To feed mills at a more constant and even ¯ow, it is necessary to harvest and store logs over seasons. This requires an optimization model where both location and seasonal transport and storage is optimized. 5. Location modeling, biodiversity and reserve design World-wide deforestation, with resulting loss of species and native habitats, has focused concern for protecting biodiversity. For example, the Sierra±Nevada
148
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Region of the United States spans approximately 300 miles from North to South, comprises portions of 23 river basins and is contained almost wholly in California. About 400 species occur only in the Sierra Nevada including 3 tree species, 20 shrubs, and several hundred herbaceous plants. Of this total, 218 are considered rare or endangered. The Sierra region has been managed principally for forest products, grazing, mining, and water management (US Forest Service, 41%; the National Park Service, 6%; the Bureau of Land Management, 11%; and approximately 37% is privately held). A major issue is how such lands should be managed in order to protect biodiversity; see, for example, Thompson et al. (1973), Yoshimoto and Brodie (1994a), Hof and Joyce (1992, 1993) and Hof et al. (1994). Often, management of biodiversity involves special indicator or keystone species, such as, the spotted owl of the Northwest, the leadbetter possum in Australia, and the cockcaded woodpecker in the southeast United States. It is thought that providing suitable and sucient connected habitat area for a keystone species will provide general protection of a whole family of species which utilize the same habitat. In order to plan for the viability of the spotted owl in the Northwestern United States, a presidential commission recommended setting aside most of the remaining oldgrowth forest, resulting in substantial decreases in harvesting over the next several decades. Trading o the impacts of harvesting with species protection is dicult for many countries, especially when forest products can bring in needed currency. To approach this type of problem, one might ask how we could design a reserve system so that as many species as possible are protected with the least amount of land. Many researchers have de®ned the basic planning unit as a small watershed unit, large enough to be considered as a viable reserve site. Assume that we have divided a region into such watershed planning units. Consider then the following notation: p is the number of planning units to be selected for reserve status; j the index of planning units or watersheds; i the index of species that should be represented in the reserve system; yi 1, if species i is not present among the sites selected for the reserve, and 0, otherwise; xj 1, if planning unit i is selected for the reserve system, and 0, otherwise; Ni fj j site j contains species i in sufficient quantitiesg. We can now de®ne a simple reserve site selection model as (Church et al., 1996a): Minimize
Z
X
yi
i
subject to: 1. De®ne whether a species is included in the reserve system X
xj y i P 1
8 i:
j2Ni
2. Use exactly p-sites or planning units
R.L. Church et al./Location Science 6 (1998) 137±153
X
149
xj p:
j
3. De®ne decision variables xj 0; 1
8j;
yi 0; 1
8i:
This model selects a ®xed number of sites in such a manner that the combined system has the least number of species that are not represented, which is equivalent to selecting the set of sites that houses the largest number of species or genetic elements. Essentially, this is the Maximal Covering Location Problem of Church and ReVelle (1974). The Location Set Covering model has also been suggested as a potential reserve selection model as well (Underhill, 1994). Extended forms of the maximal covering model have also been developed for reserve design by Williams and ReVelle (1996). Camm et al. (1996) have suggested a general form of the above model where the objective is to cover each species a pre-speci®ed number of times. Some of the multiple cover, maximal covering models appear to be quite dicult to solve (Camm et al., 1996), which means that further research on solution approaches is needed. Species presence is not the only factor of interest in protecting biodiversity in forested areas. The amount of protected area of a given habitat is also an important indicator of reserve goodness; for example, how much area should be protected in some level of management status in order to enhance long term biodiversity. To address this, researchers of the Sierra Nevada Ecosystem Project de®ned the Biodiversity Management Area Selection Model (BMAS) (Church et al., 1996b), BMAS involves selecting small watershed units in order to minimize the area selected and maximize the amount of protection given for certain threatened species. Consider the following notation: · k is an index to represent species or elements that are considered at risk or in jeopardy; · aj the area of watershed or planning unit j; · xj 1, if planning unit j is selected for a Biodiversity Management Area, xj 0, otherwise; · Sj the suitability of planning unit j being selected as a biodiversity management area; · mink minimum area containing element k that needs to be brought under biodiversity management in order to remove element k from jeopardy; · ajk the area of element k in unit j that is impacted by current or planned activity; · ws the importance weight attached to optimizing suitability; · wa the importance weight attached to minimizing area. We can now de®ne BMAS as Church et al. (1996a,b)
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Minimize
Z wa
X j
aj xj ws
X
Sj x j
j
subject to: 1. Set aside enough area that contains species or element k X
ajk xj P min
j
k
8 k:
2. De®ne integer restrictions on decision variables xj 0; 1 8j: This model can be transformed into an equivalent knapsack problem and therefore it falls into the class of NP-hard. The BMAS model was used to study the impact on land use change that is needed to protect various habitats and species at risk. The application of the BMAS model in the Sierra Nevada involves over 2700 planning units and 200 elements. Exact solutions have not been generated in many cases, as the model appears to be dicult, if not impossible, to solve optimally. This should be a fertile area for new research, as such models will likely be used in many dierent planning contexts.
6. Conclusions We have attempted to describe a number of types of forest management problems, many of which are variations of existing location models. These models are typically dicult to solve, have problem instances that are considered quite large, and have been the subject of considerable research. We have discussed some of the main features of such models and similarities with existing location models. It is hoped that this paper will be bene®cial to forest analysts as a basis of appreciation for the ®eld of location science in their work. It is also hoped that this paper will demonstrate to location scientists that the ®eld of forestry presents a wide variety of location problems of sucient complexity and detail to warrant interest, encourage new research, and serve as a potential application arena.
Acknowledgements We would like to thank personnel from regions 5 and 6 of the USFS, who have assisted us, especially Klaus Barber and Richard Dyrland. We also wish to thank Phil Aune and the Paci®c South West Laboratory for research support.
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