minutes of computer time. In terms of variables, it consists of 4m + 3h + 2 continuous variables: F
i
, H
j
, I
f i
; binary
variables: X
ij
. In terms of equations, the total number of rows is 4m + 2h + 2.
B. Numerical illustrations
• In order to illustrate the functioning of the models formulated in the preceding sections, we present a case study with data from Nelson and Brodie 1990 and
Yoshimoto and Brodie 1994. The problem concerns a forest area consisting of 20 stands of Douglas-fir Pseudotsuga menziesii Mirb Franco characterised by the age
class distribution and acreage shown in Table 1.
Table 1 Harvest units acreage and initial age
• In our example, the stands will be the harvest units. Figure 1 shows the spatial arrangement of the harvest units the forest area comprises.
• By direct observation of Figure 1, we can determine that the number of pairwise adjacencies L is 35. Planning horizon T is 100 y and the time length of cutting periods
t is 20 y, thus the planning horizon will encompass h =10020 = 5 cutting periods and all harvests are assumed to occur at the midpoint of the periods. The minimum
specified age for cutting a unit maturity age is 70 y. According to the habitat diversity objective, the desired rotation age for every forest stand at the end of the planning
horizon will be assumed to be R = 200 y. Therefore, the provision of all the forest succession stages will be attained by having stands in every age class that R
comprises: 1–20, 21–40, 41- 60; . . . ; 181- 200. In addition, given the similarity of area of the harvest units, the balance of age classes will be achieved by allocating a
number of q = 2010 = 2 harvest units to each age class.
• Table 2 about Stand age, volume, and price see the journal shows the timber volume yield and net present value per hectare corresponding to each stand age
class. This way, if harvest unit 2 is cut in the second period, ie X
22
= 1, the age of the stand at that moment will be of 100 y. Hence, if a discount rate of 4 is used, then
the net present value NV22 attached to variable X
22
, will be equal to: 38. 41,985 . 1 + 0.04
-30
=491 903 dollars. • It is important to indicate that given the planning horizon 100 y, the time length of
periods 20 y and the maturity age 70 y then in our example, the maximum number of possible cuts will be two, specifically in cutting periods 1 and 5. Hence, for this
particular example, we can easily extend the model and allow a maximum of two cuts by only incorporating variables Xi;h+2 in equations 2–5 of the first model and in
equations 16, 17 and 19 of the second model. In this sense, Xi;h+2 = 1, if the ith unit is cut twice within the planning horizon i.e. the first cut in the first period and the
second cut in the fifth period, otherwise Xi;h+2 = 0. This is quite an ad hoc procedure, but a general procedure to allow multiple cuts within the planning horizon
can be found in the Appendix.
• The first step in our analysis consists in determining the pay-off matrix for the two criteria involved—the diversity indicator and net present value. The first row of the
pay-off matrix is obtained by maximizing 1 subject to 2–10 and by substituting the optimum values for Xij in accounting row 12.
• The second row of the pay-off matrix is obtained by maximizing subject to 2–5 and by substituting the optimum values of decision variables in
accounting row 13. • Table 3 shows the pay-off matrix. From this table it is easy to capture the significant
degree of conflict between the two criteria considered.
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Table 3. Pay-off matrix bold figures represent the edge contrast and underlined figures the balance of age classes
• The interpretation of the above figures is straightforward. Thus, efficient point D represents a level of fulfillment of 67 in terms of edge contrast and of 72 in terms
of balanced age classes. • The actual values of the trade-offs i.e. the opportunity costs between net present
value and diversity are represented by the slopes of the straight lines connecting the efficient points shown in Figure 2.
• Figure 2 Trade-off curves between net present value and habitat diversity index. • It is important to note that in the portion of the trade-off curve near point G, the
opportunity cost of improving the diversity index in terms of reduction of net present value is very low. However, when we move towards point A, the mentioned
opportunity costs increase significantly. Near point A, the trade-offs between both criteria achieve very high values. Once the trade-offs between net present value and
diversity have been illustrated, the GP models presented in the
• Preceding section will then help us to determine the best compromise or satisfying solutions between diversity, net present value and other forest management criteria.
Again as a first step, the pay-off matrix for the four criteria considered will be determined. This task is accomplished in the usual way. That is, each criterion
function is optimized in turn and then the corresponding optimum is substituted in the three other criterion functions.
• Table 4 shows the pay-off matrix. Again from the examination of this table, the existence of a significant degree of conflict between the four criteria is found.
Moreover, from the analysis of the table, it is also easy to conclude that no solution generated by the single optimization of any criterion i.e. any row of the pay-off
matrix would be considered acceptable since for any of these solutions the achievement of the criteria is very unbalanced. Hence, it is essential to look for some
best compromise or satisfying solutions between the criteria considered.
• In order to determine the best-compromise or satisfying solutions, the GP models proposed in the preceding section will be implemented. As a first step, we need to
normalize the four criteria considered since they are measured in different units dollars, cubic meters, etc. Among the different established normalisation methods,
we chose as normaliser weights r—the ranges for each criterion—ie the absolute value of the difference between the ideal and antiideal values. A justification of this
normalizing system Diaz-Balteiro and Romero, 1998.
• Table 5 shows the results obtained in the criteria space, for the WGP formulation see model 22 and for the MINMAX GP formulation see model 23, for a scenario of
equal preferential weights. A natural extension of the model will consist in studying the influence of the preferential weights w in the schedules obtained. To undertake
this task there are two possible approaches. One consists in implementing a sensitivity analysis with the values of the weights. The other approach will consist in
eliciting the weights through a formalised interactive dialogue with the decision-maker Diaz-Balteiro and Romero, 1998.
• Nevertheless, the application of these approaches is beyond the scope of this paper. In order to assess the two GP solutions, the corresponding normalized degrees of
closeness are shown in Table 6. • From a review of Tables 5 and 6, the following conclusions are derived:
1. There is a high degree of closeness between the two solutions. This means, that for this particular problem, the schedules of maximum efficiency WGP
model and maximum balance MINMAX GP are very similar. It is rather obvious that this coincidence makes it easier to choose a harvest schedule.
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2. The best-compromise or satisfying harvest schedules shown in Table 5 seem easier to implement in the real world than the schedules derived from the
single optimization solutions shown in Table 4. There is a strong degree of discrepancy between the criteria considered and most specifically between
net present value and the habitat diversity measure. However the two GP solutions shown in Table 5 represent balanced solutions for the four criteria
considered. Thus, Table 6 shows how for the WGP schedules, the most unbalanced criteria are habitat diversity and net present value with a
maximum level of disagreement of 0.33 and 0.44, respectively, and for the MINMAX GP schedules these two criteria present a maximum disagreement
of 0.37. This kind of unbalance, although significant, is clearly inferior to the maximum unbalance underlying the schedules shown in Table 4.
3. Due to the marked similarity between the WGP and the MINMAX GP solutions, general model 24 has no practical interest in this case study. In
fact, there are no efficient schedules significantly different from the two solutions shown in Table 5.
• As pointed out, the model for habitat diversity presents some computational limitations when the number of harvest units that the forest comprises is large.
However, the use of metaheuristics can provide us with good solutions. In addition, a real scenario could involve maximizing edge contrast in some part of the forest area
while trying to encourage habitat for interior species in other forest tracts. Finally, it should be noted that due to extent limitations, the corresponding harvest schedules in
the decision variable space Xij are not presented in the paper. Details on this type of solutions can be found in Bertomeu 2001.
V. Conclusions
• Based on the study, the authors concluded as follows: The results presented in the paper show how an operational measure of habitat diversity taken from the forest
ecology field, can be integrated in conjunction with other criteria into a robust mathematical programming model for forest management optimization.
• The first model proposed allows the establishment of the trade-off curve between diversity and financial returns. This first model presents some computational
difficulties. For this reason and thinking of larger models, it is advisable to resort to metaheuristic approaches such as Tabu Search and Scatter Search. In this sense,
the use of the optimizer OptQuest provided satisfactory results.
• The second model let us determine some best compromise or satisfying harvest schedules when the following criteria are considered: habitat diversity, financial
returns, volume control and ending forest volume inventory. In this case, the model is well-behaved and consequently does not present computational difficulties and
consequently optimum solutions can be found in relatively short computer time with the help of commercial software like LINGO.
• In summary, although the proposed analytical approach is still tentative, it seems to be a promising and efficient way of incorporating a measure of habitat diversity in
conjunction with other relevant criteria into a forest management optimization model. VII. General Comments on the Paper
A. Strengths
• On the whole, this paper is very good, because of giving information to us about the forest management optimization models and habitat diversity using a goal
programming approach. • The reader can easy to understand that this paper related with Optimization Model in
Forestry, because this paper using a goal programming approach for forestry. This method was applied in various cases was linked with forestry, especially to
investigate the forest management and habitat diversity.
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