off curve or production possibility frontier is obtained. This curve will inform us of the clash between the best solutions for both criteria. In other words, this curve will
determine the opportunity cost of diversity in terms of financial returns. An application of this approach is shown below.
D. A forest management optimization model with multiple criteria
• By solving model 1–10, the optimal final ages Fi for the m harvest units are obtained from a habitat diversity viewpoint.
• These final ages are then incorporated as inputs or target levels into a forest management optimization model in combination with other relevant criteria, such as
the above mentioned net present value of harvested timber, volume control and ending forest volume inventory. To accomplish this task the following additional
parameters and variables are defined:
Hij = volume harvested from the ith unit in the jth period.
Hj = total volume harvested at the jth cutting period.
I
ij
=volume of ending inventory of the ith unit when it is harvested in the jth cutting period.
I
f i
= total volume of ending inventory of the ith unit. I
i
= total volume of initial forest inventory. • The harvest scheduling model has the following structure in terms of constraints and
goals:
Constraints: Feasible domain of final ages and maturity condition:
Goals: Final ages of harvest units from the habitat diversity viewpoint:
Net present value of harvested timber:
Volume control:
Ending forest volume inventory:
Universitas Sumatera Utara
• Although the structure and role of goals 15–20 is self-explanatory, the following clarifications can be useful for an easy understanding. Thus, given that the target
value of equation 16 is an ideal value, the minimization of the negative deviation variable nNV will imply the maximization of the net present value. Volume control
equations 17–18 impose strict even-flow of timber volume harvested over the planning horizon.
• The inclusion of positive pjH and negative njH deviation variables rules out the possibility of obtaining infeasible schedules. As far the ending forest volume
inventory, if the forest is considered to be adequately stocked at the beginning of the planning horizon, then both deviation variables of equation 20 are unwanted. On the
other hand, if the initial forest inventory is considered to be insufficient, the negative deviation variable is unwanted, and vice-versa.
• To obtain a satisfying or best-compromise schedule, a function of the unwanted deviation variables has to be minimized:
• Two ways of minimizing 21 are suggested. One corresponds to a Weighting Goal Programming WGP model and the other to a MINMAX GP model.20,21 The
structure of the WGP model is the following:
Achievement function:
• subject to: constraints and goals 14–20 where parameters r are normalizing factors and parameters w represent preferential weights, i.e. the relative importance
that the decision maker attaches to the achievement of every formulated goal. As it is well illustrated in the literature, a WGP model implies the maximization of a separable
additive utility function of the goals considered. In short, the harvest schedule provided by model 22 implies the maximum aggregate achievement maximum
efficiency for the considered goals Tamiz et al., 1998.
• The structure of the MINMAX GP model for our forest management optimization model is as follows:
Achievement function:
• Constraints and goals 14–20 where D is the maximum deviation. As it is well illustrated in the literature, the MINMAX GP model implies the maximization of a utility
Universitas Sumatera Utara
function where the maximum deviation D is minimized. In short, the harvest schedule provided by model 23 implies the most balanced solution between the achievement
of the different goals maximum equity Tamiz et al., 1998. • The harvest schedules provided by models 22 and 23 represent two opposite
poles. Thus, the first solution can be extremely biased towards the achievement of some of the goals, whereas the MINMAX solution can provide poor aggregated
performance for the total of goals. In order to obtain the best-compromises between these two opposite views of optimizing i.e. efficiency versus equity the following
Extended GP model can be formulated Romero, 2001.
Achievement function: Minimize
• Subject to: Constraints and goals of model 23. • Parameter m weighs the importance attached to the minimization of the weighed sum
of normalized unwanted deviation variables. Thus, for μ = 1, we have the WGP
model, and for μ = 0, the MINMAX GP model. For other values of parameter μ, an
intermediate solution, if any, between those of the two GP models considered can be obtained.
IV. Results and Discussion A. Computational procedures: some comments
• Model 1–10 is of combinatorial nature, as a solution will consist in some combination of binary and continuous variables. Moreover, in the Theory of
Computational Complexity it is said to be a NP-hard problem, for which no polynomial-time algorithm is known for solving them. On the other hand, the size of
the model is computationally manageable. Thus, in terms of variables, we have m + 4L + 2C continuous variables: F
i
, n
i,
p
i
, u
i
, v
i
, α
j
; β
j
; h+1m binary variables: X
ij
. • In terms of equations, the total number of rows for the model is 2L+ 2m + C + 1. In
this way, in order to guarantee the optimality of the solutions, we resorted to specialized software equipped with GP speed up techniques such as IGPSYS Jones
et al, 1998 LINGO 4.0 LINGO, 1998 and CPLEX CPLEX, 1994. However, besides the computational complexity the structure of the block of goals 6 is problematic,
since it solely consists of deviation a variable which usually causes computational difficulties Ignizio and Cavalier, 1994. Consequently, the above general-purpose
codes turn inefficient for solving the problem posed at a reasonable computer time. However, after several hours of running time they provided very similar solutions after
reaching a stabilization phase. Thus, we can assume that good near-optimal solutions are obtained.
• Taking into account that to derive the trade-off curve between net present value and the diversity index, the authors need to solve the model several times in a parametric
fashion, computer time had to be sped up. For this reason, in order to find good solutions at a reasonable computational cost, we resorted to more efficient
approaches such as metaheuristics. In this sense, we used the general-purpose optimizer known as OptQuest which operates as an Add-in function to Excel and
whose algorithm is based on metaheuristics such as Tabu Search and Scatter Search OptQuest, 1998 and Glover and Laguna, 1997. Scatter Search is a
population-based metaheuristic that operates on a population of solutions and Tabu Search provides the optimizer with several memory-based functions to guide the
search.
• Model 14–20 is computationally easier to solve to optimality, provided that the structure of all the goals and constraints is well-behaved and consequently
commercial software like LINGO was capable of solving big instances in a few
Universitas Sumatera Utara
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
