corresponding objective between the best and the worst possible values, hence mapping all deviations onto a zero-one range Onal, 1997.
• For decision makers more interested in obtaining a balance between the competing objectives, Chebyshev goal programming should be used. Introduced by Flavell 1976,
this variant seeks to minimize the maximum unwanted deviation, rather than the sum of deviations. This utilizes the
Chebyshev distance metric, which emphasizes justice and
balance rather than ruthless optimization.
4. Procedure for the development of a goal programming model
1. Identify the goals and any constraints that reflect resource capacities or other restrictions that may prevent achievement of the goals.
2. Determine the priority level of each goal, starting with the highest level Priority 1. 3. Define the decision variables.
4. Formulate the constraints in the usual linear programming fashion. 5. For each goal, develop a goal equation, with the RHS specifying the target value for
the goals. Deviation variables di+ and di- are included in each goal equation to reflect the possible deviations above or below the target value.
6. Write the objective function in terms of minimizing a prioritized function of the deviation variables.
B. A proposal of habitat diversity characterization
• This is how the authors did their briefly summarize the basic points on which our harvest scheduling model for habitat diversity focuses, as follows:
1. The creation of biologically mature old stands, provided that they constitute a key habitat for many wildlife species.
2. The provision of all stand age classes that the desired biological forest rotation age encompasses.
3. The maintenance of a balance of age classes, in order to achieve the well-known condition of area regulation.
4. The maximization of the edge contrast between any two adjacent stands that the forest area comprises.
• Point 4 relies on the assumption that some wildlife species’ diversity and abundance are larger near edges Leopold, 1933; Giles, 1978; Harris and McElveen, 1981.
• This condition obviously implies that the age difference between adjacent stands should be equal to half of the rotation age. Nevertheless, it is important to note that
many other authors have pointed out that maximizing edge contrast may increase habitat fragmentation resulting in a negative effect of edges on interior species.
• In this paper, the authors will work on the basis that if the size and shape of harvest units we adequate for interior species, then the negative impact of edges could be
diminished. C. Modeling the trade-off between habitat diversity and financial returns
• The starting point of the analysis is a zero-one Goal Programming GP model proposed in Bertomeu and Romero 2001 In this research, the original model has
been expanded with a parametric constraint regarding the net present value attached to harvest schedules and two accounting rows computing the values of harvest
schedules in terms of net present value and habitat diversity.
• The following notation will be used: Constants:
t = time length of cutting periods. T = planning horizon.
R = desired rotation age. H = number of cutting periods, ie h = Tt.
M = number of harvest units. I
i
= initial age of the ith harvest unit.
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I
max
= initial age of the oldest harvest unit. q
= desired number of harvest units on each age class, ie q = mC. C
= desired number of age classes, ie C = Rt. L
= number of pairwise adjacencies among the m harvest units considered. M
= arbitrary large value. r
F
; r
c
= normalising factors, calculated as the ranges of variation, ie nadir minus ideal values, for the sum of deviation variables referring to the edge contrast and to the balance of age classes
respectively. Thus, both aspects of the habitat diversity index are made commensurable, as they are measured in different units and achieve very different absolute values.
NV
ij
=net present value attached to the harvest of the ith unit in the jth cutting period. NV
=ideal value for the net present value. This figure is obtained by maximising Subject to constraints 2–5.
NV =anti-ideal value for the net present value; this figure is obtained by substituting the optimum
corresponding to model 1–10 in accounting row 12.
Index sets:
Μ = index set of pair of values i; j that implies cutting a unit below its maturity age.
P = index set of pairwise adjacent harvest units.
S
c
= index set of harvest units at age class c at the end of the planning horizon. Specifically, in equation 10 c makes reference to final age classes h þ 1, h þ 2; . . . ; C, which can be
obtained if harvest units remain unharvested over the planning horizon, ie Xi;h+1 = 1. On the other hand, equation 9 takes account of final age classes c: 1; 2; . . . ; h, which correspond
with cutting periods: h; h - 1; . . . ; 1 respectively.
Variables:
F
i
=final age of the ith harvest unit. X
ij
=binary 01 decision variables, so that Xij = 1 if the ith unit is cut in the jth period, otherwise Xij = 0. It should be noted that an artificial cutting period h + 1 is introduced to consider that the ith
unit remains unharvested over the planning horizon T i.e. Xi;h+1 = 1. B
=habitat diversity index. NV
=net present value resulting from harvest schedules. b
i
=binary decision variables needed to avoid that variables n
i
and pl take non-negative values simultaneously.
Deviation variables: ni , pi , u
i
and v
i
=negative and positive deviation variables for the edge contrast. αj and βj
=negative and positive deviation variables for the balance of age classes. The structure of the model is the following for a more detailed explanation see Bertomeu and
Romero 2001. Achievement function:
Constraints: Accounting rows for final ages of harvest units:
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Goals: Maximum edge contrast between adjacent harvest units:
Balance of age classes:
Net present value parametric constraint:
Net present value and habitat diversity index accounting rows:
• Constraints 2–5 guarantee that the final ages of the m harvest units are logically feasible. Constraint 4 secures that no harvest unit is cut before it reaches a
minimum harvest age or maturity age in order to avoid potential harvests of immature timber. Goals 6 and 7 imply that the age difference between adjacent units be
equal to half the final forest rotation age Bertomeu and Romero 2001.
• In block 8 bi are binary variables and M represents an arbitrary large value. In this way, it is precluded that both deviation variables will be simultaneously in the basis.
Therefore, goals 6 and 7 plus block of auxiliary constraints 8 in conjunction with the first term of achievement function 1 imply the maximization of the edge contrast.
• Finally, goals 9 and 10 together with the second term of the achievement function guarantee, as much as possible, that the number of harvest units belonging to each
desired age class is the same, as well as the provision of all the stand age classes that R encompasses.
• lt should be noted that constraints 2 and 3 allow a maximum of one cut over the planning horizon for each harvest unit. However, the model can be extended to a
more general context where the number of harvests per unit within the planning horizon can he more than one. In fact, our case study has been solved considering
the more realistic scenario of multiple cuts within the planning horizon. Details on possible extensions of the model can be found in the Appendix.
• it is important to point out that goals 9 and 10 will not provide us with the same area on every age class at the end of the planning horizon unless all harvest units are
strictly of equal area. In this sense, we have assumed that the acreage of forest harvest units is similar enough to attain the desired condition of area regulation by
means of goals 9 and 10. Nevertheless, the proposed model can be straightforwardly modified so that the area regulation condition is fully satisfied.
• The trade-off curve between net present value and the habitat diversity measure can be obtained by applying the constraint method19 to the above model. Thus, through
parametric variations of the right hand side l of constraint 11 the commented trade-
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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
