# A sequential approach in mathematical programming to include spatial aspects of biodiversity in long range forest management planning

von Kerkvoorde, Michiel

## Abstract

In the discussion about forest management the maintenance of biodiversity is coming more and more to the fore. Like 120 other countries, Sweden committed itself to a sustainable use of for ests at the convention of Rio de Janeiro. Sweden has a long tradition of forest management fo cusing on woodproduction. This implies that almost all the forest land is managed and that the area of natural forests is very small. The maintenance of biodiversity should therefor not be limited to reserved areas but it should be incorporated into the management of the total forest area. There is a need for improved methods to balance the economic and ecological benefits of forest management. In this study I designed an algorithm for mathematical programming that includes the determination of a contiguous area of old forest into a long range forest manage ment plan that strives for the sustainable production of wood as well as the maintenance of biodiversity. The boreal forests of Sweden were characterised by two main disturbance patterns. In dry and mesic forests, fires determined the structure of the forest. This resulted in large scale pattern where Scots pine and broad-leaved trees dominated. In the wetter forests, dominated by Nor way spruce, small scale disturbances like windthrow brought about a heterogeneous forest structure with a long continuity. With the management of the forests for woodproduction, some characteristic features disappeared. The amount of old-growth forests, deciduous trees and coarse woody debris decreased significantly and large scale clearcuts resulted in a loss of re tained trees, characteristic for an area after forest fire. Especially these features are of impor tance for the maintenance of biodiversity. Rare and sensitive species are largely dependent on structures determined by natural disturbance patterns. The objective function in the long range forest management plan is a maximisation of the Net Present Value of the woodproduction. To ensure a sustainable supply of wood in the future this objective function is bounded by even flow and ending stock constraints. For the maintenance of biodiversity, constraints are implemented as well. These constraints stem from the assump tion that the biodiversity benefits by the establishment of certain features abundant in natural forests and scarce in managed forests. Special attention is paid to a certain contiguous area of old forest. This is a conversion of the current attention for spatial aspects in biodiversity. A contiguous area of old forest ensures a core area without any edge effects. Many red-listed species depend on this. In natural resource management, Linear Programming (LP) is most common among the mathematical programming techniques. It gives an optimal solution and is efficient. The model described above can largely be solved with LP. The constraint on the contiguous area of old forest will however cause problems. The way this constraint is formulated is known as a Quad ratic Assignment Problem (QAP). This kind of problem is hard to solve and LP is not suitable. Other exact solution methods like the Branch and Bound method can only solve very small problems. Heuristics are a good alternative. Although they do not guarantee an optimal solu tion, they are able to solve large problems in an acceptable amount of time. Simulated Anneal ing (SA) is a heuristic method that gives high quality solutions because it has the ability to overcome local optima and convergence to a high value. I have designed a sequential approach. Herein the strong points of both methods are united. First the QAP is solved using SA. After that, the rest of the problem is solved by LP. This part is bounded by the outcome of the SA. In the objective function for the SA, next to a measure for the contiguous area of forest, the Net Present Value is added. A weightfactor determines the relative importance of these two ele ments in relation to each other. With this the SA solution will fit as effective as possible into the final solution for the whole problem, that is at the least costs. The algorithm is tested on validity, reliability and efficiency. Validity relates to the question if the algorithm does what it is perceived to do. The test revealed some points that need extra attention. The determination of the parameters for the so called cooling down of the SA is some what troublesome and needs to be regarded from case to case. Focusing strongly on either the contiguous area or the NPV in the SA-objective function gives results that are poor in relation to the results of an objective function wherein the two are balanced. The inclusion of the NPV in the SA-objective function does make the SA-solution effective in terms of the final outcome. In general the performance of the algorithm is as expected. The algorithm turned out to be very reliable, that means that a repetition gives about the same value all the time. For a sample for est of sixteen stands with more than 900 possible management regimes the algorithm could produce a solution within a minute, indicating that it is efficient. In a case study the sequential approach algorithm is applied on the Brattaker area, near Umea in Northern Sweden. The Simulated Annealing created contiguous old forest areas. The results are some more or less contiguous areas of between 100 and 150 hectares. The outcome is lim ited by the actual situation in the area. Not all forest can be old in fifty years. Next to that the algorithm shows a preference for small stands, causing a large part of the area, a part consist ing of large stands, to be without old forest. The costs for contiguous old forest are relatively small compared to non contiguous old forest. The Simulated Annealing algorithm has the ca pability to form contiguous old forest in an efficient way. If the contiguous old forest is worth the additional costs, compared to non contiguous old forest, is a question for the decision mak ers. The Linear Programming produces a long range forest management plan. In this plan the features of importance for biodiversity are all established. With this it is assumed that biodi versity is maintained the coming fifty years. The sustainable provision of wood is guaranteed for this period. But this is at the expanse of the further future because the age structure is se verely distorted. The main reason for this seems to be the combination of interest rate, initial age structure and the requirement for old forest. The used interest rate caused a strong prefer ence for a present income to a future income. Because of that, and the initial age structure, the cut volume in the first period is extraordinary high, which has its effects on the age structure after 50 years. The requirement for old forest is in conflict with the age structure, which might cause problems for the provision of old forest after the planning horizon of 50 years. The main conclusion is that the algorithm can give good results. It shows some of the possibili ties to link a heuristic method with Linear Programming. An improvement could be the use of a shape index for the formation of contiguous old forest. It became clear that the situation and parameters in a case study have a major influence. The interest rate and an already unbalanced age structure limited the possibilities to test the algorithm. To generate data that is of use in the decision making process the algorithm needs more refinement. The site characteristics should be taken into account and a situation should be created that will ensure the sustainable provi sion of wood and biodiversity also after the planning horizon.

## Keywords

forest management; planning; biodiversity

## Published in

Arbetsrapport / Sveriges lantbruksuniversitet, Institutionen för skoglig resurshushållning

1996,

Publisher: Institutionen för skoglig resurshushållning, Sveriges lantbruksuniversitet

## UKÄ Subject classification

Forest Science

## Permanent link to this page (URI)

https://res.slu.se/id/publ/125929