22897_fuzzymultipleobjectiveprogtechniquesmodeling

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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 Fuller

Department of Computer Science, Eotvos Lorand 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 dif-

ficult their management. In fact they are a typical example of a joint produc-

tion (market and social goods) and therefore every management policy must

face tradeoffs between forest use and forest preservation. In [6] Steuer andSchuler 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 num-

bers) 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 EUFIT94 Conference, September 20-23, 1994, Aachen, Germany, Verlagder 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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1 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 evalua-

tion criteria and often conflictual expectations; the solution can be unstable under

small changes in the imprecise data; tradeoffs between forest preservation and for-est 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, analysisand 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 non-

continuous planning models (Multi Attribute Decision Making models). Due tothe peculiarities and limits of both models, it is usually more promising to follow a

mixedapproach based both on MODM methods (to determine efficient alterna-

tives) and on MADM methods (to determine the most suitable alternative): there

are some technical rules being able to convertthe effecient alternatives obtained

in a mathematical programming model in a multiattribute multiperson decision

scheme.

In the decision phase we evaluate the decision makerspreferences and deter-

mine an alternative with the highest consensus degree.

In this paper we focus our attention on the modeling part of forest manage-

ment support systems and demonstrate the applicability fuzzy multiple objective

programming techniques for the resource allocation problem.

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2 Formulation of the resource allocation problem

The basic problem is to allocate acres and budget monies to alternative manage-

ment options to meet the best a set of objectives usually specified in goal attainment

terms.

We can formulate the forestry problem as follows

maximize{(C1x , . . . ,C 5x)|Ax b} (1)

whereCi = (Ci1, . . . , C in)is a vector of fuzzy numbers,A is a crisp matrix,bis avector in Rm andx Rn is the vector of crisp decision variables.

Suppose that for each objective function of (1) we have two reference fuzzy

numbers, denoted bymi and Mi, which represent undesired and desired levels for

thei-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 Cix is as close

as possible to the desired point Mi, and it is as far as possible from the undisered

pointmi for eachi.

In multiple objective programs, application functions are established to mea-

sure the degree of fulfillment of the decision makers 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 offinding good compro-

misesolutions.

Now we should find an x Rn such thatCix is as close as possible to the

desired pointMi, and it is as far as possible from the undisered point mi for each

i.

Let didenote the maximal distance between the -level sets ofmiand Mi, and

let mibe the fuzzy number obtained by shifting miby the value2diin the direction

ofMi. Then we considermias the reference level for the biggest acceptable value

for thei-th objective function.

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m1 1 C1x*m'1

m1 M1C1x*

C1x is too far fromM1.

It is clear that good compromise solutions should be searched betweenMiandmi, and we can introduce weigths measuring the importance ofcloseness and

farness.

C1x is close toM1, but not far enough from m1.

Let [0, 1] be the grade of importance ofcloseness to the disered leveland then1denotes the importance offarnessfrom the undisered level.

We can use the following family of application functions [1]

Hi(x) = 1

1 + d(Mi(), Cix)

where Mi() = Mi+(1)miand d is a metric in the family of fuzzy numbers.

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M1

C1

x*'1M1+ (1-)m'1

A good compromise solution

Then (1) turns into the following problem

max{(H1(x), . . . H 5(x))| Ax b}. (2)

And (2) can be transformed into single objective problem by using the mini-

mum operator for the interpretation of the logical andoperator

max{min(H1(x), . . . , H 5(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.Fuller, Fuzzy reasoning for solving fuzzy multiple ob-

jective linear programs, in: R.Trappl ed., Cybernetics and Systems 94,

Proceedings of the Twelfth European Meeting on Cybernetics and Sys-

tems Research, World Scientific Publisher, London, 1994, vol.1, 295-

301.

[2] L.S.Davis and G.Liu, Integrated forest planning across multiple owner-

ships 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 sys-

tem for environmental planning, IEEE Transactions on Systems, Man

and Cybernetics20(1990) 777-790.

[6] R.E.Steuer and A.T.Schuler, An interactive multiple-objective linear pro-

gramming approach to a problem in forest management,Operations Re-

search, 26(1978) 254269.

[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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