# A Critique On Forest Management Optimization Models And Habitat Diversity: A Goal Programming Approach

**KARYA TULIS **

**Journal Report **

**on Application of Goal Programming **

**A Critique On **

**FOREST MANAGEMENT OPTIMIZATION MODELS AND **

**HABITAT DIVERSITY: A GOAL PROGRAMMING APPROACH **

**BY: **

**RAHMAWATY **

**DEPARTEMEN KEHUTANAN **

**FAKULTAS PERTANIAN **

**UNIVERSITAS SUMATERA UTARA **

**2010 **

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**KATA PENGANTAR **

### Puji syukur penulis panjatkan kepada Tuhan Yang Maha Esa, yang telah memberikan segala

### rahmat dan karunia-Nya sehingga KARYA TULIS ini dapat diselesaikan. Judul yang dipilih adalah

**“**

### Forest management optimization models and habitat diversity: a goal programming approach

**”.**

### Tulisan ini merupakan suatu kritik mengenai Aplikasi GOAL PROGRAMMING DI Bidang Kehutanan,

### Penulisnya adalah Author : M Bertomeu and C Romero dan bersumber dari

*Journal of the Operational *

*Research Society (2002) 53, 1175–1184 #2002 Operational Research Society Ltd.*

### Kami menyadari bahwa karya tulis ini masih jauh dari sempurna, oleh karena itu kami

### mengharapkan saran dan kritik yang bersifat membangun untuk lebih menyempurnakan karya tulis ini.

### Akhir kata kami ucapkan semoga karya tulis ini dapat bermanfaat.

### Medan, April 2010

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**DAFTAR ISI **

### I

### Title of the Study

### 1

### II

### Rationale, Background and Objectives of the Study

### 1

### III Research Procedure

### 2

### A. Goal Programming

### 2

### 1. Definition

### 2

### 2. History

### 2

### 3. Variants

### 2

### 4. Procedure for the development of a goal programming model

### 3

### B. A proposal of habitat diversity characterization

### 3

### C. Modeling the trade-off between habitat diversity and financial returns

### 3

### D. A forest management optimization model with multiple criteria

### 6

### IV Results and Discussion

### 8

### A. Computational procedures: some comments

### 8

### B. Numerical illustrations

### 9

### V Conclusion

### 11

### VI General Comments on the Paper

### 11

### A. Strengths

### 12

### B. Weakness /Limitations

### 12

### C. Suggestion

### 12

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**A Critique on **

**Forest management optimization models and habitat diversity: **

**a goal programming approach **

Author : M Bertomeu and C Romero*

*Journal of the Operational Research Society (2002) 53, 1175–1184 #2002 Operational Research Society Ltd. *

**I. Title of the Study **

• As a researcher, the title provided by the authors has the element of simplicity, brevity, specificity and location and subject matter focused. The reader can easily determine what the study is all about and what it tries to investigate (Forest management optimization models and habitat diversity) and what mathematical model? a goal programming approach. Brief title but very informative. It would be better if the author mention the research location on the title to make more informative.

• Just going through the title, one can easily understand that the concern of the research study is related to optimization model in forestry. The keywords used such as “Forest management optimization models and habitat diversity: a goal programming approach” clearly indicate that the subject matter is in the field of forestry.

**II. Rationale, Background and Objectives of the Study **

**A. Rationale of the Study: **

• The authors looked into the forest management optimization models and habitat diversity: a goal programming approach. Goal Programming is concerned where a decision maker needs to consider multiple criteria in arriving at the overall best decision.

**B. Background of the Study: **

• According the authors, the modern societies demand from forests not only private goods sold in well-established markets but also public goods and services for which there are no defined markets. Among the set of public goods and services demanded by society, the protection of biodiversity has become of paramount importance in the past few years.

• Basically, biodiversity can be considered at three different levels of organization: genes, species and habitats (Westman, 1990). Most managers who aim at the protection of biological diversity of forest ecosystems specifically focus on species and habitat diversity. In the last years, many efforts have been undertaken in order to incorporate biodiversity into forest management planning processes.

• The structural diversity is measured by the Shannon index or the so called normalized absolute deviation index for different characteristics (e.g. the species diversity, the basal area and/or diameter distribution diversity, the foliage height diversity).

• Finally, many other papers (Mealey *et al*., 1982; Carter *et al*., 1997; Bettinger *et al*.,
1997; and Bevers and Hof, 1999) focus on the optimization of the spatial arrangement
of forest stands age classes, provided that the habitat requirements of species with
respect to the amount of edges, the juxtaposition of different habitats, and other
aspects related to the spatial distribution of habitats are previously known.

**C. Objectives of the Study: **

• The aim of this paper, as cited by the author is to propose a different analytical formulation for the incorporation of edge contrast as an operational measure of habitat diversity proposed in the forest ecology field (Harris, 1984 and Hunter, 1990). The starting point is a mathematical programming model recently formulated (Bertomeu and

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Romero (2001) in which a harvest schedule holding some desirable properties from the habitat diversity viewpoint was determined.

• In this paper, the trade-off curve between the proposed measure of habitat diversity and financial returns from harvested timber is determined. Then, a zero-one goal programming model that integrates the mentioned habitat diversity index and the economic criterion in conjunction with other relevant forest management criteria such as volume control over the planning horizon and ending forest volume inventory is formulated. From this model, several best compromise or satisfying harvest schedules are obtained and interpreted in utility terms.

**III. Research Procedure **
**A. Goal Programming **
**1. Definition **

• Goal Programming is a fancy name for a very simple idea: the line between objectives and constraints is not completely solid. In particular, when there are a number of objectives, it is normally a good idea to treat some or all of them as constraints instead of objectives (Trick, 1996).

• Goal Programming is a procedure based on linear programming that allows several goals to be considered instead of just one single objective. Goal programming is a branch of multiple objective programming, which in turn is a branch of multi-criteria decision analysis (MCDA), also known as multiple-criteria decision making (MCDM). It can be thought of as an extension or generalization of linear programming to handle multiple, normally conflicting objective measures. Each of these measures is given a goal or target value to be achieved. Unwanted deviations from this set of target values are then minimized in an achievement function. This can be a vector or a weighted sum dependent on the goal programming variant used. As satisfaction of the target is deemed to satisfy the decision maker(s), an underlying satisfying philosophy is assumed (Wikipedia encyclopedia, 2006)

**2. History **

• Goal programming was first used by Charnes, *et al*. (1955), although the actual name
first appear in a 1961 (Charnes and Cooper, 1961). Seminal works by Lee (1972),
Ignizio (1976), Ignizio and Cavalier (1994), and Romero (1991) followed. Scniederjans
(1995) gives in a bibliography of a large number of pre 1995 articles relating to goal
programming and Jones and Tamiz (2002) give an annotated bibliography of the period
1990-2000.

• The first engineering application of goal programming, due to Ignizio in 1962, was the design and placement of the antennas employed on the second stage of the Saturn V. This was used to launch the Apollo space capsule which landed the first men on the moon.

**3. Variants **

• The original goal programming formulations ordered the unwanted deviations into a number of priority levels, with the minimization of a deviation in a higher priority level being of infinitely more importance than any deviations in lower priority levels. This is known as lexicographic or pre-emptive goal programming. Ignizio (1976) gives an algorithm showing how a lexicographic goal programme can be solved as a series of linear programmes.

• It is important to recognize that deviations measured in different units cannot be summed directly due to the phenomenon of incommensurability. Hence each unwanted deviation is multiplied by a normalization constant to allow direct comparison. Popular choices for normalization constants are the goal target value of the corresponding objective (hence turning all deviations into percentages) or the range of the

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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 = T/t. *

M = number of harvest units.

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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 = m/C.

C = desired number of age classes, ie C = R/t.

L = number of pairwise adjacencies among the m harvest units considered.

M = arbitrary large value.

rF; rc = 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.

NVij =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.

*Sc* = 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: *

*Fi * =final age of the ith harvest unit.

Xij =binary (0/1) decision variables, so that Xij = 1 if the ith unit is cut in the j*th* period, otherwise Xij

= 0. It should be noted that an artificial cutting period h + 1 is introduced to consider that the i*th*

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.

bi =binary decision variables needed to avoid that variables ni and pl take non-negative values

simultaneously.

*Deviation variables: *

ni , pi , ui and vi =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: **

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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

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trade-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 F*i 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 i*th* unit in the j*th* period.
Hj = total volume harvested at the jth cutting period.

Iij =volume of ending inventory of the i*th* unit when it is harvested in the
j*th* cutting period.

Ifi = total volume of ending inventory of the i*th* unit.
Ii = 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:

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• 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

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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: Fi, ni, pi, ui , vi, αj; βj; (h+1)*m* binary variables: Xij .
• 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

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minutes of computer time. In terms of variables, it consists of 4m + 3h + 2 continuous variables: Fi, Hj , I

f

i ; binary

variables: Xij. 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 =100/20 = 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 = 20/10 = 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 X22 = 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 X22, 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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• More appropriate using a goal programming approach than other mathematical model (like: integer programming, linier programming, and dynamic programming), because there are multiple objectives (with trade-offs) and deviations from constrains are penalized.

• This paper used a model, so this study more effective because no need go to the field to all the measurement. Using a goal programming approach, forest management optimization models and habitat diversity can be analyzed and assessed. The linkages with the field of forestry that need the long time to research, so this model was very helpful.

• A major strength of goal programming is its simplicity and ease of use. This accounts for the large number of goal programming applications in many and diverse fields (Jones and Tamiz, 2002). As weighted and Chebyshev goal programmes can be solved by widely available linear programming computer packages, finding a solution tool is not difficult in most cases. Lexicographic goal programmes can be solved as a series of linear programming models, as described by Ignizio and Cavalier (1994). Goal programming can hence handle relatively large numbers of variables, constraints and objectives.

• This paper had been answering all the following questions investigated (on the objective this paper).

**B. Weakness /Limitations **

• The model gave the guide for us to take the decision in the forestry, although is not the end decision; at least this paper gave information and is useful as suggest for decision maker.

• A debated weakness is the ability of goal programming to produce solutions that are not Pareto efficient. This violates a fundamental concept of decision theory that is no rational decision maker will knowingly choose a solution that is not Pareto efficient. However, techniques are available according to Hannan (1980); Romero (1991); Tamiz, Mirrazavi, and Jones (1999) to detect when this occurs and project the solution onto the Pareto efficient solution in an appropriate manner.

• The setting of appropriate weights in the goal programming model is another area that has caused debate, with some authors Gass (1987) suggesting the use of the

Analytic Hierarchy Process or interactive methods for this purpose.
**C. Suggestion **

• I think there is still a need further explain that:

1. Not all goal programming problems involve multiple priority levels. For problems with one priority level, only one linear program needs to be solved to obtain the goal programming solution. One simply minimizes the weighted deviations from the goals. Trade-offs is permitted among the goals since they are all at the same priority level. 2. The goal programming approach can be employed when one is confronted with an

infeasible solution to an ordinary linear program. By reformulating some of the constraints as goal equations with deviation variables, a solution can be found that minimizes the weighted sum of the deviation variables. Often, this approach will suggest a reasonable solution.

3. The approach have utilized to solve goal programming problems with multiple priority levels is to solve a sequence of linear programs. These linear programs are closely related so that complete reformulation and solution are not necessary. By changing the objective function and adding a constraint, we can go from one linear program to the next.

(16)

### 13

**REFERENCES **

Bertomeu, M and C. Romero. 2002. Forest management optimization models and habitat
diversity: a goal programming approach.* Journal of the Operational Research Society *
*(2002) 53, 1175–1184. Operational Research Society Ltd. *

Charnes, A., WW. Cooper. and R . Ferguson. 1955. Optimal estimation of executive compensation by linear programming, Management Science, 1, 138-151.

Charnes, A and WW. Cooper. 1961. Management models and industrial applications of linear programming, Wiley, New York.

Flavell, B. 1976. A new goal programming formulation, Omega, 4, 731-732.

Gass, S.I. 1987 A process for determining priorities and weights for large scale linear goal programmes, Journal of the Operational Research Society, 37, 779-785.

Hannan, E.L. 1980. Non-dominance in goal programming, INFOR, 18, 300-309

Ignizio, JP. 1976. Goal programming and extensions, Lexington Books, Lexington, MA. Ignizio, JP ., and M. Cavalier. 1994. Linear programming, Prentice Hall.

Jones, DF and M. Tamiz. 2002. Goal programming in the period 1990-2000, in Multiple Criteria Optimization: State of the art annotated bibliographic surveys, M. Ehrgott and X.Gandibleux (Eds.), 129-170. Kluwer

Lee, SM. 1972. Goal programming for decision analysis, Auerback, Philadelphia.

Romero, C. 1991. Handbook of critical issues in goal programming, Pergamon Press, Oxford.

Scniederjans, MJ. 1995. Goal programming methodology and applications, Kluwer publishers, Boston.

Tamiz, M., SK. Mirrazavi, DF Jones. 1999. Extensions of Pareto efficiency analysis to integer goal programming, Omega, 27, 179-188.

Trick, M. A. 1996. Goal Programming.

http://mat.gsia.cmu.edu/mstc/multiple/node5.html#SECTION00050000000000000000 Wikipedia Encyclopedia. 2006. http://en.wikipedia.org/wiki/Goal_programming"

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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: Fi, ni, pi, ui , vi, *αj; βj; (h+1)m binary variables: Xij .
• 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

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minutes of computer time. In terms of variables, it consists of 4m + 3h + 2 continuous variables: Fi, Hj , I

f

i ; binary

variables: Xij. 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 =100/20 = 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 = 20/10 = 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 X22 = 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 X22, 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.

(5)

• More appropriate using a goal programming approach than other mathematical model (like: integer programming, linier programming, and dynamic programming), because there are multiple objectives (with trade-offs) and deviations from constrains are penalized.

• This paper used a model, so this study more effective because no need go to the field to all the measurement. Using a goal programming approach, forest management optimization models and habitat diversity can be analyzed and assessed. The linkages with the field of forestry that need the long time to research, so this model was very helpful.

• A major strength of goal programming is its simplicity and ease of use. This accounts for the large number of goal programming applications in many and diverse fields (Jones and Tamiz, 2002). As weighted and Chebyshev goal programmes can be solved by widely available linear programming computer packages, finding a solution tool is not difficult in most cases. Lexicographic goal programmes can be solved as a series of linear programming models, as described by Ignizio and Cavalier (1994). Goal programming can hence handle relatively large numbers of variables, constraints and objectives.

• This paper had been answering all the following questions investigated (on the objective this paper).

**B. Weakness /Limitations **

• The model gave the guide for us to take the decision in the forestry, although is not the end decision; at least this paper gave information and is useful as suggest for decision maker.

• A debated weakness is the ability of goal programming to produce solutions that are not Pareto efficient. This violates a fundamental concept of decision theory that is no rational decision maker will knowingly choose a solution that is not Pareto efficient. However, techniques are available according to Hannan (1980); Romero (1991); Tamiz, Mirrazavi, and Jones (1999) to detect when this occurs and project the solution onto the Pareto efficient solution in an appropriate manner.

• The setting of appropriate weights in the goal programming model is another area that has caused debate, with some authors Gass (1987) suggesting the use of the Analytic Hierarchy Process or interactive methods for this purpose.

**C. Suggestion **

• I think there is still a need further explain that:

1. Not all goal programming problems involve multiple priority levels. For problems with one priority level, only one linear program needs to be solved to obtain the goal programming solution. One simply minimizes the weighted deviations from the goals. Trade-offs is permitted among the goals since they are all at the same priority level. 2. The goal programming approach can be employed when one is confronted with an

infeasible solution to an ordinary linear program. By reformulating some of the constraints as goal equations with deviation variables, a solution can be found that minimizes the weighted sum of the deviation variables. Often, this approach will suggest a reasonable solution.

3. The approach have utilized to solve goal programming problems with multiple priority levels is to solve a sequence of linear programs. These linear programs are closely related so that complete reformulation and solution are not necessary. By changing the objective function and adding a constraint, we can go from one linear program to the next.

(6)

## 13

**REFERENCES **

Bertomeu, M and C. Romero. 2002. Forest management optimization models and habitat
diversity: a goal programming approach. Journal of the Operational Research Society
*(2002) 53, 1175–1184. Operational Research Society Ltd. *

Charnes, A., WW. Cooper. and R . Ferguson. 1955. Optimal estimation of executive compensation by linear programming, Management Science, 1, 138-151.

Charnes, A and WW. Cooper. 1961. Management models and industrial applications of linear programming, Wiley, New York.

Flavell, B. 1976. A new goal programming formulation, Omega, 4, 731-732.

Gass, S.I. 1987 A process for determining priorities and weights for large scale linear goal programmes, Journal of the Operational Research Society, 37, 779-785.

Hannan, E.L. 1980. Non-dominance in goal programming, INFOR, 18, 300-309

Ignizio, JP. 1976. Goal programming and extensions, Lexington Books, Lexington, MA. Ignizio, JP ., and M. Cavalier. 1994. Linear programming, Prentice Hall.

Jones, DF and M. Tamiz. 2002. Goal programming in the period 1990-2000, in Multiple Criteria Optimization: State of the art annotated bibliographic surveys, M. Ehrgott and X.Gandibleux (Eds.), 129-170. Kluwer

Lee, SM. 1972. Goal programming for decision analysis, Auerback, Philadelphia.

Romero, C. 1991. Handbook of critical issues in goal programming, Pergamon Press, Oxford.

Scniederjans, MJ. 1995. Goal programming methodology and applications, Kluwer publishers, Boston.

Tamiz, M., SK. Mirrazavi, DF Jones. 1999. Extensions of Pareto efficiency analysis to integer goal programming, Omega, 27, 179-188.

Trick, M. A. 1996. Goal Programming.

http://mat.gsia.cmu.edu/mstc/multiple/node5.html#SECTION00050000000000000000 Wikipedia Encyclopedia. 2006. http://en.wikipedia.org/wiki/Goal_programming"