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0-1線型計画法による小班別収穫予定In-place timber harvest scheduling by zero-one integer program木平, 勇吉The purposes of this paper are: (1) to identify the individual forest compartments that are scheduled for cutting in each planning period so that harvest scheduling plans can be clear and concrete, (2) to find an efficient method for harvest scheduling that can regulate the spacial order of forest compartment allocation. Soundness and function of forests can be increased when individual forests are well allocated and an ideal spacial order is attained. The only means whereby spacial order for the forest management is realized is the harvest. Therefore, the harvest scheduling must indicate not only the cutting volume or the cutting area as a whole, but also each individual forest location scheduled for cutting. On the assumption that a compartment is the unit of harvest, scheduling problems can be defined as the problem of deciding the cutting period for each compartment, and it can be treated as a zero-one integer linear program. Because, once the cutting period of each compartment is determined, the total cutting volume and area can be easily calculated by summing up the individual compartments. Further, this type of plan itself gives extreamly concrete information to the planners, because the location of individual forests are already clarified. This method is called in-place timber harvest scheduling. This forest management problem is formulated as a mathematical problem and a solution is sought using a zero-one integer program: Find the set of binary variables x_<ij> corresponding to the cutting of compartment i in planning period j. If x_<ij> is 1, compartment i should be cut in period j, and if x_<ij> is 0, compartment i should not be cut in period j. The objective function: total yield over the whole planning period should be maximized. To find the most desirable yield plan under some managerial constraints such as the limit of cutting volume or area in each period, cutting age of each species, forest road construction and allocation of compartments, the following zero-one integer program is used: Find x_<ij> { i=1, 2,……, n. n: number of compartment j=1, 2,…… , m. m: number of planning period} such that Z=n Σ i=1 m Σ j=1 v_<ij>・x_<ij> max Subject to (1) x_<ij>=0 or 1 (2) Limit of total cutting volume in each period V_j≦n Σ i=1 v_<ij>・x_<ij>≦V'_j (V_j, V'_j: lower and upper limit of total cutting volume) (3) Limit of total cutting area in each period A_j≦n Σ j=1 a_i・x_<ij>≦A'_j (A_j, A'_j: lower and upper limit of total cutting area) (4) Each compartment should be cut only once during a rotation m Σ j=1 x_<ij>=1 (5) All coefficents: v_<ij> (Volume of compartment i in periodj), ai (: Area of compartment i) are nonnegative 0≦v_<ij>, 0≦a_i (6) Stands grow and the volume does not decrease. v_<ij>≦v_<j, j+1> An application of a zero-one integer algorithm to the forestry problem was reported by Egon Balas in 1964. The zero-one integer algorithm is clear in logic and the number of solutions are limited, therefore an optimal solution can be solved by enumeration procedure. However, as the number of variables x_<ij> become numerous, it is very difficult to find an optimal solution because of the tremendous computation load, and thus it has not been applied to practical problems yet. An efficient algorithm appliable to forestry problems was developed and is explained in this paper. A forestry oriented algorithm (FOA) which is based on Balas' additive algorithm is designed for solving harvest scheduling zero-one problems mentioned above, using its unique characteristics. That is, FOA can accelerate the computation time on the assumption that (1) every compartment should be cut only once during a rotation, (2) all coefficients are nonnegative, and (3) stands grow and the volume does not decrease. A systematic enumeration procedure is repeated successively according to the nodes shown on the solution tree. When any evidence is found on any node that no feasible or optimal solution exists, that node is abandoned, and the computation is abbreviated. This procedure is called implicit enumeration. The characteristics of FOA which differ from Balas' additive algorithm are: (1) the solution tree itself is smaller in size and each node has different possibilities for optimality and feasiblity of the solution according to the location on the tree, therefore, (2) the enumeration sequence of nodes can be determined automatically, (3) the three standards that may become the evidence for abbreviation of enumeration are established so that the number of exhaustive enumerations (enumeration that is procedured in practice) is eliminated. Before applying this program, meaningless variables must be rejected to reduce the size of zero-one problems, and subcompartments must be arranged in the decreasing order of their area in order to accelerate the computation. This algorithm was translated into a computer program and the computation time is examined by using numerical examples. It becomes clear that the computation time lengthens in proportion to the problem's size, however, (1) only a short time is needed when the constraints are lax, that is, there is a wide range of allowable cutting volume or cutting area in each period, and (2) on the contrary, when the constraints are limited, the computation time becomes notably longer. Further, it is clear that the computation time is strictly proportioned to the number of exhaustive enumerations. Define the ratio r: r=(the number of exhaustive enumerations)/(the number of all nodes on the solution tree), the ratio r can be seen as the efficiency of FOA program. Although the ratio r changes mainly depend on the constraints as mentioned above, r is between 0.099～0.00007% by the numerical examples as shown in the table. That is, from 99.9% to 99.99993% of whole nodes are abandoned and the computation is abbreviated. 〓As a result, FOA can become practical and efficient for forestry oriented zero-one problems. It can be concluded that the timber harvest scheduling problem by individual compartments, although the number of compartments are limited, can be solved by a zero-one integer linear program. This FOA program was applied to Kiso national district forest to predict a sustained yield plan, and a plan was formed. The forest is divided into three working sections, and in each section, the cutting period of each compartment is scheduled so that logging and cilviculture can be sustained continually for 20 years. The optimal solution indicates not only the total cutting volume and area, but also clarifies the cutting period of the individual compartments. Therefore, each compartment is evaluated as to whether it can be logged or not in each of the planning periods. The cutting sequence or neighbouring relationship of each cutting compartment can be discussed. As an element of harvest scheduling, forest road planning is formulated to the constraints in a zero-one problem and some examples are tested. A forest road plan is drawn on a map and the roads are difinded as to which compartments they intersect. When schedule of road construction is recieved, then the compartments which are not ready to transport are rejected from the immediate cutting schedule. This harvesting schedule was applied to a forest of 20 compartments and the results follow: By comparing the problem with road planning and the problem without road planning, it becomes clear that (1) if the limits of the cutting volume or cutting area in each period are wide, feasible and optimal solutions can be found in both problems. (2) On the contrary, when the limits are narrow, a feasible solution does not exist in the problem with road planning. Forest allocation in the future are visualized and discussed. A disirable cutting period of each compartment indicated by the optimal solution is drawn on the map, and the spacial order is evaluated from managerial and/or environmental view points. Requirements of any compartment that need its cutting schedule changed, are added to the constraints of the zero-one problem. Then the revised problem is recalculated and the new optimal solution is shown on the map. By this repetition, a final optimal solution that can satisfy the spacial order intended by planners is found. In the examples the results show that the total cutting volume decreases when the constraints are limited, that is, the demands of the individual compartments and the demands of optimization of the whole are opposed to each other. It can be concluded that to attain an ideal spacial order in a management forest, a strict scheduling of cutting compartments is required. The zero-one problem discussed in this paper can contribute to this requirement. The auther believes an optimal solution of a mathematical problem is always information that may assist in the better decision making of planners, and the in-place harvest scheduling method surely gives useful information that was not previously available to the forest management planner.Article信州大学農学部演習林報告 19: 1-66(1982)信州大学農学部附属演習林1982-10-15jpndepartmental bulletin paperhttp://hdl.handle.net/10091/14007https://soar-ir.repo.nii.ac.jp/records/112550559-8613AN00121330信州大学農学部演習林報告19166https://soar-ir.repo.nii.ac.jp/record/11255/files/Agri_Forests-19-01.pdfapplication/pdf3.1 MB2015-09-25