FUZZY MATRICES: AN APPLICATION IN AGRICULTURE Ximing Sun**, David A. Thomas** and Peter W. Eklund*, nique can be applied, we demonstrate how to perform comprehensive land evaluation for selecting suitable cropping options in agriculture. The domain is interesting because decision makers base their choices on a wide range of
[email protected] erations, e.g. biophysical land capability, environmental impacts, socio-economic and political ABSTRACT considerations, local infrastructure etc. Land evaluation needs a synthesis of this information This paper describes a simple (yet powerful) to guide objectives which are easily constructed methodology for decision making based on fuzzy and viewed in a structure called an interaction sets. In an example to demonstrate the method- matrix. ology we show how to determine a preferential grain crop for given conditions and grower cir2. Comprehensive evaluation cumstances. The paper proposes a multi-level fuzzy evalthe object set C = C1; C2; : : :; Cn uation function [4, 6] which will totally order a Assume the object's attribute set X = X1; X2; : : : ; Xm number of crop alternatives. The model can be and where Xi is a fuzzy subset of C . Given the grade used to consider various input conditions and al- membership of each object to each attribute, lows for the exible treatment of such issues as soil degradation, erosion and pollution. These X (Ci) = ij [0; 1] ; for i [1; m] and j [1; n] : issues and others are graded subjectively in terms of a membership function and partitioned into a matrix which is crossed with an aggregated is; then the fuzzy relationship matrix of C to weighted index for crop yield. *Department of Computer Science, **Department of Environmental Science & Rangeland Management The University of Adelaide, Adelaide 5005, Australia
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The resulting multi-levelcomprehensive fuzzy evaluation function considers environmental issues, management skills and economic bene ts with biophysical suitability and allows us to make recommendations for land usage.
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1. Introduction
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If weightings (ai) are given to each attribute X , The technique described in this paper provides i a normalized fuzzy set or a fuzzy vector by A = (a1; a2; : : :; am) where, a mechanism for integrating multi-dimensional on Pm a is=given 1 and 0 ai < 1, i.e. there is more variables, determined from both empirical and i=1 i expert knowledge, to give a overall numeric as- than a single weighting. sessment of interactions between variables. The The comprehensive evaluation result B is: technique has a wide range of applications in B = A R; (2) many elds. As an example of how the techX
where B is a fuzzy set on C , B = (b1; b2; : : : ; bn). The preferential object in the fuzzy set B is B (B ) = max(b1; b2; : : :; bn). There are a number of ways we can consider combining the matrices A and R. They can be crossed using a number of algebraic variants[6];
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(3)
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(4)
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(5)
(ai ij ):
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considered. Such factors may also exist at several levels of in uence. The diculty with a rst level comprehensive evaluation model, is that weightings, given each factor, are often dicult to assign. In addition, the weighting set A must satisfy the unity sum condition Pmi=1 ai = 1 (although this is not true of a - composition operator it is true of (5) and (6). The compound operation in the fuzzy matrix may be given by taking a max or min value. When these are used and the number of factors is large, weightings will tend to have extremal in uence and dominate grade membership in the matrix. To counter this, a multi-level comprehensive evaluation model can be formulated [4]. The principle is to divide properties into factor sets. The rst level of evaluation in each set is to obtain the fuzzy vectors (Ai) for each X . These vectors are then combined into a relationship matrix Ri for each group of factors. These vectors and matrices are then used to operate over the weighting set to obtain a high-level comprehensive evaluation B . The process of formulating a multi-level evaluation model is: ? >
(3) and (4) are called primary factor determination types: the primary factor determines the result. (5) and (6) highlight the primary factors but also consider secondary factors. (4) and (6) accord weightings by considering all factors with equal in uence. Note that these naturally are not the only composition operators. Generally we can use a - composition where and are T-conorm and T-norm respectively but only a subclass of T-conorms and T-norms are associative. Note that (3) and (4) always factors set give a bP [0 ; 1], i.e. B is a fuzzy subset of C j X = X1; X2; : : :; Xm ; card(X ) = m even if mi=1 ai = 1 with P ai [0; 1]. (5) and (6) m can only be used since i=1 ai = 1 otherwise groups of factors set, bj [0; 1] and B would not be a fuzzy subset of C. X=C = 1; 2; : : :; p ; with p [1; m] : The four operators can be used to compute an evaluation under varying domain circumstances. for every i [1; p], every i is de ned by (6) will be suitable in most domains but (4) i =PXl X , with card( i ) = ki such should be used for Markovian processes for inthat pi=1 ki = m. stance. (6) will resume to a convex weighting of the m fuzzy subsets of Xi[5]. The more the In the example presented we have; agreement between the models, the greater the con dence in the result. X = X 1 ; X2 ; : : : ; X 8 ; m = 8 with X as biophysical factors;X as economic 3. Fuzzy multi-level comprehen- factors;1 X3 as management skills;2 X4 as crop preference; X5 as social status; X6 as soil degrasive evaluation dation; X7 as soil erosion and X8 as polution. In any complex system many factors may in u- The following four (4) groups of factors (p = 4) ence a nal outcome and each factor needs to be are subsequentially constructed; ?
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3.1 Biophysical evaluation
group biophysical factors; = X1 ; k1 = 1;
The biophysical evaluation vector can be obtained through our A (aggregated weighting index) from interaction matrix[3]. We then ask group economic factors: the user for the desired yield for each crop. This = X ; k = 1; information can be taken as the optimal value 2 2 2 for the membership function for each crop and the membership function of biophysical suitabilgroup social factors: ity for each crop can be expressed, 8 3 = X3 ; X4 ; X5 ; k3 = 3; > < 11 c A + 1) A=3 < A < A ; (2c Ci (A ) = 1=10 2 3 >0 group environmental factors: : (9) 4 = X6 ; X7 ; X8 ; k4 = 3: where A is the aggregated weighted index for P5 the users desired yield for each crop Ci. Where We can verify that x=1 ki = 8. A in this case is the biophysical aggregated A rst level comprehensive evaluation on each weighted index for a speci c parcel of land. c1 i with ki factors proceeds as follows. and c2 are constants for a given production reSuppose the weighing assigned to i factors gion. is a vector Ai, the evaluation relationship matrix The membership function can be obtained is Ri, i.e., by linear conversion of the curve of yield against A which is generated from the interaction maAi Ri = Bi = (bi1; bi2; : : :; bin) (7) trix for each crop [3]. From these considerations, the biophysical evaluation nal vector The multi-level comprehensive evaluation modelBB = AB RB can be obtained (7). B is given as; 2A R 3 3.2 Economic (cost-bene t) evaluation 1 1 66 : : : 77 B = A R = A 666 Ai Ri 777 = (b1; b2; : : :; bn); The economic evaluation (or cost bene t) vec4 ::: 5 tor can be obtained from membership function Ap R p CB . As the estimated yield of each crop can (8) be obtained using A . where A: overall weighting set and R: relaThe price return of the yield minus the xed tionship matrix. and variable costs can be used to estimate the For example, in an agricultural domain, the gross margin for each crop. A satisfactory gross most suitable crop for any one parcel of land margin for each crop can be obtained by the should be selected by considering, the biophysi- user. Suppose GM is the satisfactory gross cal, economic (cost-bene t), social and environ- margin for the user then the membership funcmental impacts of a given crop. Suppose we tion (CB ) of economic evaluation is: have ve crops to choose from, X1
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1 if GM GM (10) CB (GM) = GM=GM if GM < GM
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We can now evaluate the suitability of these crops in terms of the above considerations. where GM is the gross margin for each crop and GM is the user's satisfactory gross margin. 0
The rst-level economic evaluation vector BC = AC RC can be directly calculated from (7).
3.3. Social evaluation Suppose the social suitability of these ve crops is determined by three factors: management skills; preference; social status etc. Suppose the weightings assigned in these three factors are,
AS = (as1 as2 as3): The fuzzy relationship matrix is (from (7)):
RS =
X1 X2 X3
2 C ::: C 3 1 5 66 1;1 : : : 1;5 77 64 : : : 75 2;1 2;5
3;1 : : : 3;5
(11)
Figure 1: Intensity index based on factor sets.
ij in the fuzzy relationship matrix can be Then the rst-level social evaluation vector determined: (i) for E1 by cultivation intensity BS is; and nutrient removal; (ii) for E2 by empirical data; (iii) for E3 by fertilizer input and level of 2 3 pesticide use ( gure 1).
1;1 : : : 1;5 Then the rst-level environmental impact vecBS = AS RS = (aS1; aS2; aS3) 64 2;1 : : : 2;5 75 :
3;1 : : : 3;5 tor BE is, (12)
2 3
1;1 : : : 1;5 3.4 Environmental impact evaluation BE = AE RE = (aE1; aE2; aE3) 64 2;1 : : : 2;5 75 :
3;1 : : : 3;5 Suppose there are three aspects which deter(13)
mine the environmental impacts: soil degradaFinally, the multi-level fuzzy comprehensive tion (E1); soil erosion (E2); pollution (E3). The evaluation is B , the fuzzy subset of C , where weighting set for these three aspects is AE , bi is the degree of membership of Ci; AE = (aE1; aE2; aE3): 3 2 AB R B AE can be obtained from the intensity index 7 6 C 7 ( gure 1) which is based on factor sets. B = A R = A 664 AAC R 7 = (b1 ; : : :; b5): (14) S RS 5 AE RE
The most suitable crop is the one which has a degree of membership equal to max(b1; : : : ; b5).
4. Discussion The methodology described gives a simple and e ective means of incorporating interdisciplinary
knowledge, both empirical and human, in one comprehensive model. The advantage of the methodology is that the user, by way of weightings, can estimate how the interplay of various practices will a ect the end result. The users' active participation in discussion and a comparison of the results obtained is of great help in learning and understanding the decision process. Continual dynamic adjustment of weightings are used in the training phase and are set by a domain expert. Extensions to the work include automating the weighting assignment through symbolic and subsymbolic machine learning[1, 2]. The use of aggregated knowledge with the incorporation of fuzzy membership constructions in models has been used successfully. It signi cantly increases the predictive value of crop yield estimates over conventional techniques[3].
5. Conclusion This paper describes a methodology based on fuzzy sets which allows us to determine a preferential grain crop for a given set of circumstances. This methodology is based on a multilevel fuzzy evaluation function which produces a total order crop alternatives under consideration. The resulting multi-levelcomprehensive fuzzy evaluation function considers environmental issues, management skills and economic bene ts with biophysical suitability and allows us to make recommendations for land usage. The paper demonstrated, by way of a simple example, how degradation, erosion and pollution, management skills and economic bene ts could be reconciled with crop yields for optimal land use.
Acknowledgement This research was supported by the Australian Research Council and The University of Adelaide. Thanks also to Dr. Zyed Zalila for detailed comments and suggestions for improvements on an earlier draft of this paper.
References [1] P. W. Eklund and A. Sahim. Experiments with the automated acquisition of classi cation heuristics from g.i.s. In Conf. on Land Information Management and GIS, University of New South Wales, 1993. [2] P. W. Eklund, S.Kirkby, and A. Sahim. A framework for incremental knowledge-base update from additional data coverages. 7th Australiasian Conf. on Remote Sensing, 1994. [3] Sun Xi Ming, D.A. Thomas, and P.W. Eklund. Aggregated knowledge and fuzzy membership construction. In Australian and New Zealand Conference on Intelligent Information Systems, 1993. [4] Chen Yong Yi. A comprehensive evaluation model. Journal of Fuzzy Mathematics, 1, 1983. [5] Z Zalila. XX. Phd thesis, BP 649, Universite de Technologie de Compiegne, 1993. [6] Wang Pei Zhuang. Fuzzy sets and their application. Technical Press, 1983.
