Fuzzy multiple objective programming
techniques in modeling forest planning ∗
Claudia Anderle, Mario Fedrizzi and Silvio Giove
Dipt. di Informatica e Studi Aziendali, University of Trento,
via Inama 5-7, I-38100 Trento, Italy
Robert Fullér †
Department of Computer Science, Eötvös Loránd University,
Muzeum krt. 6-8, H-1088 Budapest, Hungary
Abstract
In many cases it is under legislative mandate to manage publicly owned
forest resources for multiple uses (i.e., timber production, hunting, grazing).
Forest resources have some particular characteristics which make rather difficult their management. In fact they are a typical example of a joint production (market and social goods) and therefore every management policy must
face tradeoffs between forest use and forest preservation. In [6] Steuer and
Schuler presented a case study of an attempt to apply multiple objective linear
programming techniques in management of the Mark Twain National Forest
in Missouri. More often than not, accurate market values are not available
for some forest products (e.g. dispersed recreation) and, therefore, instead
of exact coefficients we have to deal with their approximations (fuzzy numbers) in the modeling phase. In this paper we demonstrate the applicability
of fuzzy multiple objective programming techniques for resource allocation
problems in forest planning.
Keywords: Forest planning, fuzzy multiple objective program, decision support
system
∗
in: Proceedings of EUFIT’94 Conference, September 20-23, 1994, Aachen, Germany, Verlag
der Augustinus Buchhandlung, Aachen, 1994 1500-1503.
†
Presently visiting professor at Dipt. di Informatica e Studi Aziendali, University of Trento, Italy.
Partially supported by the Hungarian National Scientific Research Fund OTKA under the contracts
T 4281, T 14144, 816/1991, I/3-2152 and T 7598
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Introduction
Forest planning is a very complex activity because there are many goals which
should be achieved simultaneously and a lot of components and elements which
must be considered.
In fact, it is a typical example of a multicriteria multiperson decision making
problem: flexible models must therefore be determined and utilized in order to
evaluate the present and future potentialities of the territory and the efficiency of
the possible solutions.
Furthermore, in forest management one encounters a lot of difficulties e.g.,
most of the data are not measurable exactly (uncertain or fuzzy); different evaluation criteria and often conflictual expectations; the solution can be unstable under
small changes in the imprecise data; tradeoffs between forest preservation and forest use must always be considered (see [2, 4, 5]).
It is why one should develop a decision support system, which determines a
solution that satisfies these requirements as far as possible.
In the forest planning process each decision model distinguishes three main
phases: information, analysis and decision.
In the information phase the goals, the evaluation indexes (technical and/or
logical) of the technically possible alternatives (sylvicolture and/or infrastructure)
and the territory potentialities have to be identified.
In the analysis phase Pareto-optimal alternatives are searched for by using
continuous planning models (Multi Objective Decision Making models) and noncontinuous planning models (Multi Attribute Decision Making models). Due to
the peculiarities and limits of both models, it is usually more promising to follow a
”mixed” approach based both on MODM methods (to determine efficient alternatives) and on MADM methods (to determine the most suitable alternative): there
are some technical rules being able to ”convert” the effecient alternatives obtained
in a mathematical programming model in a multiattribute multiperson decision
scheme.
In the decision phase we evaluate the decision makers’ preferences and determine an alternative with the highest consensus degree.
In this paper we focus our attention on the modeling part of forest management support systems and demonstrate the applicability fuzzy multiple objective
programming techniques for the resource allocation problem.
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Formulation of the resource allocation problem
The basic problem is to allocate acres and budget monies to alternative management options to meet the best a set of objectives usually specified in goal attainment
terms.
We can formulate the forestry problem as follows
maximize{(C1 x, . . . , C5 x)|Ax ≤ b}
(1)
where Ci = (Ci1 , . . . , Cin ) is a vector of fuzzy numbers, A is a crisp matrix, b is a
vector in Rm and x ∈ Rn is the vector of crisp decision variables.
Suppose that for each objective function of (1) we have two reference fuzzy
numbers, denoted by mi and Mi , which represent undesired and desired levels for
the i-th objective, respectively.
Table 1: Objectives, desired and undisered goal levels of attainment
Objective
Desired goal levels
Undisered goal levels
Timber production
M1
m1
Dispersed recreation
M2
m1
Hunting forest species
M3
m3
Hunting open land species
M4
m4
Grazing
M5
m5
We now can state (1) as follows: find an x∗ ∈ Rn such that Ci x∗ is as close
as possible to the desired point Mi , and it is as far as possible from the undisered
point mi for each i.
In multiple objective programs, application functions are established to measure the degree of fulfillment of the decision maker’s requirements (achievement
of goals, nearness to an ideal point, satisfaction, etc.) about the objective functions
(see e.g. [3, 7]) and are extensively used in the process of finding ”good compromise” solutions.
Now we should find an x∗ ∈ Rn such that Ci x∗ is as close as possible to the
desired point Mi , and it is as far as possible from the undisered point mi for each
i.
Let di denote the maximal distance between the α-level sets of mi and Mi , and
let mi be the fuzzy number obtained by shifting mi by the value 2di in the direction
of Mi . Then we consider mi as the reference level for the biggest acceptable value
for the i-th objective function.
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m1
1
m'1 C1x*
C1 x∗ is too far from M1 .
It is clear that good compromise solutions should be searched between Mi and
and we can introduce weigths measuring the importance of ”closeness” and
”farness”.
mi ,
m1
C1x*
M1
C1 x∗ is close to M1 , but not far enough from m1 .
Let Ω ∈ [0, 1] be the grade of importance of ”closeness” to the disered level
and then 1 − Ω denotes the importance of ”farness” from the undisered level.
We can use the following family of application functions [1]
Hi (x) =
1
1 + d(Mi (Ω), Ci x)
where Mi (Ω) = ΩMi +(1−Ω)mi and d is a metric in the family of fuzzy numbers.
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M1
ΩM 1 + (1-Ω)m'1
C1x*
'1
A good compromise solution
Then (1) turns into the following problem
max{(H1 (x), . . . H5 (x)) | Ax ≤ b}.
(2)
And (2) can be transformed into single objective problem by using the minimum operator for the interpretation of the logical ”and” operator
max{min(H1 (x), . . . , H5 (x)) | Ax ≤ b}
(3)
It is clear that the bigger the value of the objective function of problem (3) the
closer the fuzzy functions to their desired levels.
References
[1] C.Carlsson and R.Fullér, Fuzzy reasoning for solving fuzzy multiple objective linear programs, in: R.Trappl ed., Cybernetics and Systems ’94,
Proceedings of the Twelfth European Meeting on Cybernetics and Systems Research, World Scientific Publisher, London, 1994, vol.1, 295301.
[2] L.S.Davis and G.Liu, Integrated forest planning across multiple ownerships and decision makers, Forest Science, 37(1991) 200-226.
[3] M.Delgado,J.L.Verdegay and M.A.Vila, A possibilistic approach for
multiobjective programming problems. Efficiency of solutions, in:
R.Slowinski and J.Teghem eds., Stochastic versus Fuzzy Approaches to
Multiobjective Mathematical Programming under Uncertainty, Kluwer
Academic Publisher, Dordrecht, 1990 229-248.
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[4] P.Kourtz, Artificial intelligence: a new tool for forest management,
Canadian Journal of Forest Research, 20(1990) 428-437.
[5] M.Kainuma, Y.Nakamori and T.Morita, Integrated decision support system for environmental planning, IEEE Transactions on Systems, Man
and Cybernetics 20(1990) 777-790.
[6] R.E.Steuer and A.T.Schuler, An interactive multiple-objective linear programming approach to a problem in forest management, Operations Research, 26(1978) 254–269.
[7] H.-J.Zimmermann, Fuzzy programming and linear programming with
several objective functions, Fuzzy Sets and Systems, 1(1978) 45-55.
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