A LINEAR PROGRAMMING MODEL FOR VEGETATION ALLOCATION

Transcription

1 13 NZOR Volume 12 Number 1 January 1984 A LINEAR PROGRAMMING MODEL FOR VEGETATION ALLOCATION FREDERICK K, MARTINSON BUREAU OF LAND MANAGEMENT, DENVER FEDERAL CENTER DENVER COLORADO SUMMARY This paper describes the linear programming model used by the Bureau of Land Management for the allocation of vegetation to consumptive users. The model seeks to maximize the utilization of range forage production subject to plant maintenance constraints, resources constraints and management imposed constraints. Optimization is reached at the grazing allotment level. The allotment is made up of basic or homogeneous units and the LP problem solved exhibits a "multidivisional" structure with "divisions" corresponding to the basic units. The model is designed to operate in a man-machine interactive mode. This permits sensitivity analysis and "what if" questions. 1. BACKGROUND INFORMATION The Federal Land Policy and Management Act of 1976, passed in the closing hours of the 94th Congress, is one of the most significant pieces of land legislation enacted within this century. This Act specifies that goals and objectives be established as guidelines for public land use planning, that land management be on the basis of multiple use and sustained yield, and that public lands be managed in a manner that will protect the quality of historical, ecological and environmental values. The Bureau of Land Management (BLM) manages 173 million acres of public lands under the principles of multiple use and sustained yield. The BLM is required by the 1976 Act to make an inventory of the resources of these lands and to prepare comprehensive land use plans according to the management goals set forth in the act. In addition, the BLM is required by the National Environmental Policy Act of 1969 to prepare environmental impact statements on the effects of existing and proposed management actions upon the resources of particular areas of the public lands. To date, the BLM has inventoried 55 million acres of grazing lands and has prepared over 30 environmental impact statements within its grazing management program. Paper presented at the 1982 ORSA/TIMS Joint National Meeting, San Diego, California, October 25-27, Manuscript submitted May 1983, accepted July 1983.

2 14 2. VEGETATION ALLOCATION The vegetation growing on the public rangelands administered by the BLM is a highly significant renewable resource and the key to the benefits and products obtained for the public rangelands. Because of its significance, decisions regarding the allocation of vegetation are the most important rangeland management decisions m ade. Vegetation allocation is the apportionment of range resources among competing users. Although the users can be consumptive or nonconsumptive, this paper deals only with consumptive herbivores. Proper allocation of rangelands implies therefore a balance of forage supply with forage demand and the maintenance of a desirable ecosystem for multiple-use purposes. Specifically, the forage allocation problem concerns the determination of the optimal number and species of animals to be grazed on a given rangeland such that forage use is maximized. The forage supply is dictated by the amount of vegetation available, which must be inventoried using descriptors which characterized its growth and its maintenance requirements. The BLM's inventory method (BLM, 1979) collects data on the composition of the plant community, production by species, phenology and availability. The demand side of the problem is more complex because each consumptive user must be characterized by a set of descriptors that define its relationship with the vegetation supply. These descriptors are the total forage intake requirements, the dietary requirements by season, and the dietary preferences by season and plant species. In contrast to the supply side of the problem, only a limited amount of data can be gathered on user requirements. Dietary preferences of large herbivores for different plant species are neither readily available nor easily determinable and tend to be site-specific. Grazing behavior patterns tend to change seasonally and dietary botanical compositions are a complex function of vegetation availability and palatability and the herbivores' preference for given plant species. 3. THE BASIC MODEL The forage allocation problem can be formulated as a mathematical programming optimization model. The model seeks to maximize the utilization of range forage production subject to plant maintenance constraints, resource constraints and management-imposed constraints. Let us define the variables needed to describe the basic model: x^ : Number of animals of animal type i r^ : Pounds of forage intake per month by animal type i itk : Number of months of grazing by animal type i AU F. : Allowable use factor of plant species j over the -1 months of grazing (allowable use factor is maximum level of use of plant growth based on plant maintenance considerations)

3 f.. : Fraction of growth of plant species j consumed by O animals over the m^ months of grazing f. : Fraction of plant j in the diet of animal type i 15 Vj : Pounds of available annual production of plant species j DRF : Dietary range factor (deviation from target diet) RPV.. : Relative preference value of animal type i for plant 1-^ species j The objective function seeks to maximize the amount of forage available for grazing Subject to Max Z r. m. x. 1 1 x l Z f.. < A U F. for each j * JO - D Z f!. = 1 for each i j 13 (1-DRF) RPV.. < f!. < (1+DRF) RPV.. for each i and j ID ~ ID " ID Since the total consumption of plant j by xi animals during mi can be expressed as (x^) (f{j) (m^) (r^) and also as the product (fij) (Vj), we can set up the equation and solving for f!. ID x. f!. m. r. = f.. V. l I D i i I D D f.. V. f. = 1 ij x.r.m. The constraint set can now be expressed in terms of the structural variables Xi and fij: Z f.. < AUF. for each j (1) i I D - D L. - - r ^ v ; r - L. Z J. f.. V. v. for each i (2) i ( m i ) ( r 7 ) j i d D f.. V. < (1+DRF)(RPV..) m. r. x. for each i and j ID D 13 l l l (3) f.. V. > (1-DRF)(RPV..) m. r. x. for each i and j ID D - 13 l l l (4) Constraint (1) ensures that the total consumption of the annual growth of plant species j by all animal types does not exceed the allowable use factor of the plant. Constraint (2) computes the number of animals of type i, based on the total forage intake by

4 16 the given animal type and the fractional consumption of annual plant growth of each of the given plant species. Constraints (3) and (4) enable the decision maker to set a range about the relative preference values (RPVs). The term "relative preference value" indicates the dietary composition of plant j in the diet of animal i. The RPVs are determined from dietary sources. To set a range about the RPVs the decision maker uses the dietary range factor (DRF). The DRF attempts to capture in a subjective way the degree of uncertainty that the decision maker attaches to the estimate of the diet. The DRF can be made to vary from 0 to p and thereby cause a fluctuation about the expected diet from 0 to p><10 0 %. 4. THE BLM MODEL The BLM seeks to optimize forage allocation at the grazing allotment level. The grazing allotment is made up of basic or homogeneous units which are referred to, in BLM terminology, as Site Write-up Areas (SWA). The number of SWAs in an allotment can vary from one to as many as The resulting LP problem to be solved exhibits the special structure pictorically displayed in Figure 1. Objective Function SWA #l SWA #2 SWA #K Figure 1. Allotment Linkage Rows Structure of the LP Model. The objective function will then be expressed as Max r. m. x.. k i i i ik where Xi^ is number of animals of type i in SWA k. The constraint set for each SWA submatrix will be the constraint set of the basic model. The allotment linkage rows link all the SWA submatrices in the allotment and establish upper and lower bounds at the allotment level for each animal type: Ex., < U. for each i k Ik - 1 Ex., > L. for each i k xk - x where Ui is the upper bound of animal type i at the allotment level, and L^ is the lower bound of animal type i at the allotment

5 17 level. The Ui and values are subject to management control and can be made to vary by the decision maker. The AUF values that appear in the SWA submatrices are composite values that are specific to the grazing dates of each herbivore. These values are determined by scanning all the herbivores' grazing dates, selecting the earliest starting date and the latest finishing date of the non-overlapping calendar grazing season, and weighting the standard AUF calendar season values by the number of days grazed per calendar grazing season. 5. THE SYSTEM The inventory field data, together with supporting information (i.e., land surveys, stratification delineations) are processed through a series of editing COBOL programs to generate selected output reports for various departments within the BLM and to create the input data files to the vegetation allocation model. Individual allotment input data files are stored as permanent disk files on removable disk packs. The allotment input file contains the following data elements for each of the SWAs that make up the allotment: forage intake and seasons of use for the animal types present, annual production and allowable use factors for the plant species inventoried, and dietary botanical information for the given animal types. The allocation model is an overlaid FORTRAN program that runs on 35K words of main storage in a Honeywell DPS-3. It operates on a time-sharing mode and is fully interactive and conversational. The program reads the input data file, generates the LP matrix, invokes the solution algorithm, translates the results back to the BLM's vernacular and prints the answers at the terminal in a compact format. Initially, the Honeywell Mathematical Programming System (MPS) was used to solve the LP problem. The program generated internally all the MPS agenda control language and matrix inputs, called the MPS system, and fed the MPS solution results back into the program's mainstream. Because MPS only runs in the batch subsystem and batch jobs are queued for execution, solution times tended to be erratic and dependent on computer loads. In addition, even when a job spawned to the batch world was executed right away, solution times for large problems tended to be too long for useful man-machine interactivity. To speed up execution and improve man-machine interactivity the LP allocation problem has been reformulated as a generalized network with side constraints, and a special structure network code has been developed to solve the network problem (Glover et al., 1982). The gains in speed of execution are spectacular, and the larger the problem the faster the relative speed. Problems containing matrices of the order of 100 x 50 are solved 10 to 20 times faster; larger problems with matrices of the order of 4000 x 2000 are solved 100 to 200 times faster. These gains result from the simpler computational aspects of network logic, the network algorithm used, and the incorporation of the network code into the allocation program as a subroutine, as opposed to accessing MPS in the batch subsystem.

6 18 6. MODEL USAGE AND GENERAL REMARKS The LP allocation system is being used extensively by the BLM to prepare grazing environmental impact statements. BLM computer records indicate that the LP allocation system has been accessed on average of 1,000 times per month in the past two years. The perceived value of the model is not its ability to generate a prescriptive optimum but its capability to examine the impacts of different data elements on the carrying capacity of rangelands. Time-sharing interactivity permits the user to evaluate the initial results, make whichever changes desired via an array of options offered by the system for the handling and manipulation of the input data, and reevaluate the results again in a sort of crude but easy-to-understand sensitivity analysis fashion. It is widely recognized that most of the existing diet data tend to be soft and imprecise. On this account, an "optimal" solution is of little value to the managers. The usefulness of mathematical programming models to the grazing land manager lies on their ability to answer "what if" type questions and explore dietary trade-offs. The mathematical programming approach provides an easy way to obtain alternatives rather than a simple answer. For a detailed account of the operational capabilities of the model, the reader is referred to the Forage Allocation Users Manual (BLM, 1982). REFERENCES b l m, 1979, BLM Manual Soil-Vegetation Inventory Method. U.S. Dept, of the Interior, Bureau of Land Management. BLM, 1982, SVIM Forage Allocation Users Manual. U.S. Dept, of the Interior, Bureau of Land Management. Glover, F.W., R. Glover, and F.K. Martinson, 1982, The U.S. Bureau of Land Management's New Netform Vegetation Allocation System. University of Colorado, Division of Information Science Research. Mathematical Programming System (MPS) User Guide, 1979, Honeywell Information System, DG10.