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Transactions in GIS, 2011, 15(3): 329–346
Research Article
A Comparison of Fuzzy AHP and Ideal Point Methods for Evaluating Land Suitability Mukhtar Elaalem
Alexis Comber
Department of Soil and Water Science Al-Fatah University
Department of Geography University of Leicester
Pete Fisher Department of Geography University of Leicester
Abstract This article compares two fuzzy approaches to land suitability evaluations, Analytical Hierarchy Process (AHP) and Ideal Point. The methods were evaluated using a case study which models the opportunities for wheat production under irrigation conditions in the north-western region of Jeffara Plain, Libya. A number of relevant soil and landscape criteria were identified through a review of the literature and their weights specified as a result of discussions with local experts. The results of the Fuzzy AHP showed that the majority of the study area has membership values to the set of suitability between 0.40 and 0.50, while the results of the Ideal Point approach revealed most of the study area to have membership values between 0.30 and 0.40. While the Fuzzy AHP and Ideal Point approaches accommodate the continuous nature of many soil properties and produce more intuitive distributions of land suitabilities values, the Fuzzy AHP approach was found to be better than Fuzzy Ideal Point. This was due to the latter’s tendency to be biased towards positive and negative ideal values.
1 Introduction Land resources are gradually becoming scarce as increases in population places pressure on natural resources. Population increases and urbanization has increased pressure Address for correspondence: Department of Soil and Water Science, Faculty of Agriculture, Al-Fatah University, Tripoli, Libya. E-mail: [email protected] © 2011 Blackwell Publishing Ltd doi: 10.1111/j.1467-9671.2011.01260.x
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on agricultural resources (Orhan et al. 2003). However, increased pressure on land resources may result in land degradation, particularly in countries with restricted water and other natural resources. The challenge in the next decades is to ensure global and regional food security and to increase food production to support the growing population: increases in food production are urgently required to tackle poverty and land degradation problems in developing countries (Fredrick and Julie 1997). As a result, food security is one of the top agricultural policies in developing countries and the evaluation of arable land and agricultural potential in these countries is needed in order to support current and future agricultural uses. The FAO framework for land evaluation is the most commonly used approach for land evaluation (FAO 1985) and is based on the biophysical factors and socioeconomic parameters of an area. The FAO approach evaluates the suitability of land for specific land use rather than general land use (as land capability) and seeks to match land utilization types with the land use requirements across all land units. The FAO approach requires a description of land in terms of land qualities or land characteristics. It classifies the land into four suitability classes: land suitability orders, land suitability classes, land suitability sub-classes and land suitability units. Orders indicate lands suitable for crops (S) or not suitable for crops (N). Classes show the degree of land suitability, such as (S1) highly suitable, (S2) moderately suitable, (S3) marginally not suitable, (N1) currently not suitable and (N2) permanently not suitable. The subclasses indicate the type of limitation (De La Rosa et al. 1992; FAO 1976, 2007). The FAO framework for land suitability uses a Boolean mapping approach which has been criticized by a number of authors (e.g. Burrough 1989, Hall et al. 1992, Davidson et al. 1994, McBratney and Odeh 1997, Baja et al. 2001, Delgado et al. 2009, Keshavarzi 2010) because the Boolean representations ignore the continuous nature of soil, landscape variation and uncertainties in measurement. Each of these aspects can result in areas being excluded from the set of suitable land because they fail to match strictly defined requirements, when in reality they may be quite suitable. The implicit assumption in Boolean approaches is the absence of any uncertainty or vagueness associated with the suitability model, measurement, imprecision and the concepts that are specified. In reality these assumptions may be invalid. Fuzzy set methodologies have been proposed as a method for overcoming problems related to vagueness in definition and other uncertainties. The use of fuzzy set methodologies in land suitability evaluation allows imprecise representations of vague, incomplete and uncertain information. Fuzzy land evaluations define continuous suitability classes rather than “true” or “false” as in the Boolean model (e.g. Burrough 1989, Sicat et al. 2005, Ziadat 2007, Keshavarzi 2010). Fuzzy set methodologies have the potential to provide better land evaluations compared to Boolean approaches because they are able to accommodate attribute values and properties which are close to category boundaries. A number of fuzzy Multi-Criteria Decision Analysis (MCDA) approaches have been developed for assessing suitability. These include Fuzzy Analytical Hierarchy Process (Fuzzy AHP) and TOPSIS (Technique for Order Preference by Similarity to Ideal Solutions) (Malczewski 1999). The fuzzy methods are able to address and explore the uncertainties associated with land resources, especially if they are integrated with fuzzy set models (Xiang et al. 1992, Ceballos-Silva and Lopez-Blanco 2003, Prakash 2003, Duc 2006, Moreno 2007, Chaddad et al. 2009, Chuong 2008). However fuzzy MCDA methods are still relatively unknown in land suitability evaluations for agricultural crops. © 2011 Blackwell Publishing Ltd Transactions in GIS, 2011, 15(3)
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This article compares land suitability evaluation models using two MCDA methods: Fuzzy Analytical Hierarchy Process (Fuzzy AHP) and Ideal Point methods.
2 Multi-Criterion Decision Analysis (MCDA) MCDA approaches are used to tackle real-world problems that are multi dimensional in nature. Criteria are defined, and MCDA methods are used to combine qualitative and quantitative criteria and to specify the degree and nature of the relationships between those criteria in order to support spatial decision-making. In a GIS context MCDA are used to combine layers of spatial data representing the criteria in the model. The model specifies how the layers are combined, such as the relative weighting given to each individual criterion and how the data are combined. For instance, one model is the Weighted Linear Combination (WLC) as described by Voogd (1983) providing a refinement to Boolean approaches. Overall suitability is calculated from the sum of the weighted normalized data layers representing factors in the model in the following way: n
Si = ∑ w j ⋅ xi , j
(1)
j =1
n
where
∑w
j
=1
j =1
and Si is the suitability score for site i, wj is the weight of criterion j, xij is the value of site i under criterion j, and n is the total number of criteria. Unlike Boolean approaches WLC allows low values in one criterion to be compensated for by high values in another (trade-off as described by Jiang and Eastman 2000). A good description of MCDA can be found in Malczewski (2006).
2.1 Analytical Hierarchy Process (AHP) The Analytic Hierarchy Process (AHP) was introduced and developed by Saaty (1977) and is an effective method of dealing with the context of the decision making process (Saaty 1980, Banai 1993, Eastman et al. 1993, Siddiqui et al. 1996, Wu 1998, Malczewski 1999, Mendoza et al. 1999, Zhu and Dale 2001, Eldrandaly et al. 2005, Duc 2006, Saaty 2008, Vogel 2008). Saaty (1977) and Malczewski (1999) note that the relationship among the objectives and attributes has a hierarchical structure: at the highest level, the objectives can be defined, and at lower levels, the attributes can be decomposed. Decision-makers play as an important role in making pairwise comparisons between criteria at each level of the hierarchy and to develop relative weights. The AHP process involves the following steps: • • • •
Criteria or factors contributing to the set of suitable are identified; The relative importance of each factor relative to each other factor – i.e. between pairs of criteria. This is usually done by domain experts; The consistency of the overall set of pairwise comparisons is assessed using its Consistency Ratio (CR); If the CR is greater than 0.1, then the expert revises the pairwise weights iteratively and
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M Elaalem, A Comber and P Fisher Table 1 An example pairwise comparisons scale (from Saaty 1980) Intensity of Importance
Definition
1 2 3 4 5 6 7 8 9
Equal importance Equal to moderate importance Moderate importance Moderate to strong importance Strong importance Strong to very strong importance Very strong importance Very to extremely strong importance Extreme importance
The pairwise comparison requires a scale. Saaty (1980) recommends a scale from 1/9 to 9, as shown in Table 1. The quality of the overall set of pairwise comparisons is evaluated through the Consistency Ratio (CR). In the pairwise comparison step, the eigenvector values must have a Consistency Ratio (CR) ⱕ 0.1. The CR determines the internal consistency of the weights relative to the overall solution – it is a measurement that reveals how much difference is allowed (Malczewski 1996, 1999).
2.2 Ideal Point The Ideal Point approach uses a group of separation metrics to derive the best alternatives from a range of factors by ordering them based on their distance from the ideal point. The distance is calculated as follows: 1
Si + = ⎡⎣ ∑ i w iP ( Xij − X + i ) ⎤⎦ P P
(2)
where Si+ is the separation of the alternative from the ideal point, wi is a weight assigned to the criteria, Xij is the normalized criterion value of the alternative, X+i is the ideal value for the criterion, and p is the power factor rating from 1 to •. According to Malczewski (1999) “the larger values of P reflect greater concern for minimizing the maximum separation from the ideal. If the factor is set at 1, rectangular distance is computed. For P = 2 the straight-line distance is obtained. If P = •, the minimum of the maximum separation is sought”. The Ideal Point approach identifies one of many possible points that could be used for ordering the set of feasible alternatives. For instance, one could identify the negative ideal point and calculate the separation of the alternatives from that point. The Ideal Point method assumes that the most suitable alternatives have the shortest distance from the positive ideal solution, and the longest distance from the negative ideal solution. The most popular Ideal Point approach is the Technique for Order Performance by Similarity to the Ideal Solution (TOPSIS) which was first proposed by Hwang and Yoon 1981 (Malczewski 1999). The TOPSIS method considers alternatives that are close to the ideal point to be the most suitable alternatives (Hwang and Yoon 1981). Data in the Ideal Point approach are standard© 2011 Blackwell Publishing Ltd Transactions in GIS, 2011, 15(3)
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ized and then weighted to generate the most suitable alternatives. The advantage of the use of the Ideal Point approach to the land suitability problem is that it generates complete sets of weights and ranks for each attribute. It overcomes some of the disadvantages that are associated with interdependence between criteria under approaches such as the AHP and Weighted Liner Combination approaches (Zeleny 1982, Pereira and Duckstein 1993, Malczewski 1996).
2.3 Fuzzy sets Fuzzy sets are classes without sharp boundaries; that is, the transition between membership and non-membership of a location in the class is gradual (Zadeh 1965). A fuzzy set is described by a fuzzy membership functions (MFs) that range from 0.0 to 1.0, representing a continuous increase from non-membership to complete membership. The use of fuzzy logic to land suitability evaluation for agricultural crops was first introduced by Burrough (1989), who noted that the soil information being used as inputs to land suitability evaluations were mainly defined by imprecise terms such as ‘slightly susceptible to soil erosion’, ‘poorly drained’ and so on. These were used to determine a number of clearly defined boundaries between land suitability classes (FAO 1976). As a result, Burrough explored fuzzy sets as a tool to deal with the imprecision in land suitability evaluations. Examples of using fuzzy sets models to generate Membership Function values (MFs) for different land characteristics can be seen in Burrough (1987, 1989, 1992), Davidson et al. (1994), McBratney and Odeh (1997), Baja et al. (2001), Van Ranst et al. (1996) and Moreno (2007). Figure 1 shows three models which were used to create MFs for different criteria associated with land suitability in this article. The different models in Figure 1 are as follows:
Figure 1 The Fuzzy Membership function models: (A) asymmetrical left model; (B) asymmetrical right model; and (C) symmetrical model © 2011 Blackwell Publishing Ltd Transactions in GIS, 2011, 15(3)
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Model A: An asymmetrical left model is used when decreases in the quality function of the criteria improves the suitability:
MF( xi) = ⎡⎣1 {1 + 1 d 2 ( χ − b ) }⎤⎦ 2
(3)
where d is the width of the transition zone, and b is for an ideal point level and is the value of land characteristics. Examples of using this particular model (A) to covert row data such as the Available Water Holding Capacity (AWHC) to fuzzy membership functions are given below:
MF( AWHC) = ⎡⎣1 {1 + 1 40 ( χ − 150 ) }⎤⎦ 2
2
MF ( AWHC ) = 1 for χ > 150 MF ( AWHC ) = 1 for missing values where, is the value of AWHC: mm/m. Model B: An asymmetrical right model used when increases in quality function of criteria decreases the suitability:
MF( xi) = ⎡⎣1 {1 + 1 d 2 ( χ + b ) }⎤⎦ 2
(4)
where d is the width of the transition zone, and b is for an ideal point level and is the value of land characteristics. Example of using the model (B) to covert row data such as soil salinity (abbreviated to EC) to fuzzy membership functions are given below:
MF( EC) = ⎡⎣1 {1 + 1 1.42 ( χ + 6 ) }⎤⎦ 2
MF ( EC ) = 1 for χ ≤ 6 MF ( EC ) = 1 for missing values where c is the value of EC: dS/cm. Model C: A symmetrical or optimum range model, when suitability is indicated by being within a range or plateau of values.
MF( xi) = 1 if ( b1 + d1 ) ≤ xi ≤ ( b2 − d 2 )
(5)
where d is the width of the transition zone, and b1 and b2 are for an ideal point level and are the value of land characteristics.
3 Materials and Methods 3.1 Study Area The study area is located within the north-west Jeffara Plain region, Libya between Tripoli and AZ-Zahra city; between longitudes 12° 45′ and 13° 15′ east and latitudes 31° 52′ and 32° 52′ north; and has the area about 309,396 ha (Figure 2). © 2011 Blackwell Publishing Ltd Transactions in GIS, 2011, 15(3)
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Country Jeffara Plain
N
Study area City
335
W
E S
Figure 2 The study area location
The study area is located in the zone of the Mediterranean climate. Using the United States soil pedology classification, the main soil orders in the study area are Aridsols, Entisols and Inceptsols (Selkhozpromexport 1980, Ben Mahmoud 1995). In the study area there are plans to accommodate three cash crops, barley, wheat and maize under irrigation conditions. In this article, the wheat crop was selected as the crop to study.
3.2 Data Sources For this work 120 soil profiles were available. These were derived from soil samples collected from different horizons of the profiles and through physical and chemical analysis a number of attributes relating to crop production were generated. The physical and chemical characteristics were: soil texture, rootable depth, available water holdingcapacity, soil reaction, soil organic matter, cation exchange capacity, soil salinity, soil alkalinity, and soil calcium carbonate, stones at surface, soil drainage, and infiltration rate. Soil erosion and slope steepness were also taken into consideration in the study area because these parameters limit wheat production. Climate characteristics such as mean temperature of the growing cycle and relative humidity were not included in the models, because these characteristics are not limiting factors for irrigated wheat production in the study area.
3.3 Framework of Land Suitability Decision Making 3.3.1 Fuzzy analytical hierarchy process (Fuzzy AHP) The Fuzzy AHP approach can be divided into five stages Stage 1: Hierarchical organization of the land characteristics for wheat production. The hierarchical structure employed in this study is shown in Figure 3. © 2011 Blackwell Publishing Ltd Transactions in GIS, 2011, 15(3)
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Goal
Objectives
Attributes
Out put
pH Chemica
% CaCO3 % O. M
Soil
% ESP
Land suitability
CEC
Soil Physical
AWHC
Land suitability map
EC
% Stones Drainage Depth Infiltration
Topographic
% Slope
Erosion Hazard
Soil Erosion
Figure 3 Hierarchical organization of the land characteristics for wheat production
Stage 2: Standardizing land characteristics. Asymmetric and symmetric models were used for land characteristics, as described above. Table 2 shows fuzzy set models employed to convert the selected criteria to the fuzzy numbers in this article. Stage 3: Weighting Factors. Weighting the model criteria provides relative measures of the interaction and importance of the criteria. The weights were obtained through a pairwise comparison analysis in an AHP approach in discussion with local experts. The local experts played an important role in the process of land suitability and in the iterative adjustment of weights to improve the Consistency Ratio ⱕ 0.1. Table 3 shows the pairwise comparison matrix for wheat in the study area. Stage 4: Derive the weighted criterion map layers. The weighted criterion layers are generated using the following function:
WFkn = Wi × MFi
(6)
where Wi, is the weight of the land propriety from the pairwise comparison and MFi is the Membership Function for the land propriety. © 2011 Blackwell Publishing Ltd Transactions in GIS, 2011, 15(3)
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Table 2 Fuzzy set models for the selected land characteristics in the study area Land characteristics
Fuzzy set models
Rootable depth (cm) AWHC (mm/m) % organic matter CEC (me/100g soil) Soil drainage classes (mm/h) Infiltration rate (mm/h) Soil salinity (EC) % ESP % CaCO3 in root zones % stones at surface % slope Soil texture (class) Soil erosion (class) Soil pH
Asymmetrical left
Asymmetrical right
Symmetrical model
Stage 5: Derive the overall land suitability map layers. The suitability is calculated by combining the weighted criterion layers. This function sums the weighted fuzzy maps of the different land proprieties to obtain land suitability maps at the final level:
R i = WFk1 + WFk2 + …… WFkn
(7)
where Ri, is the overall rating score for the suitability of land and WFkn is the weighted value for the different land properties. The overall land suitability maps show the overall land suitability classes with a continuous scale range from 0 to 1.
3.3.2 Ideal point approach To derive the land suitability for wheat based on Ideal Point mapping, the weighted map layers created in the Fuzzy AHP method are applied as the input data and the Ideal Point approach utilized in this article is an extension of the Fuzzy AHP approach. The stages of land suitability evaluation using the Ideal Point method are described below: Stage 1: Determine the maximum and minimum values. In this stage the maximum values (the values determine the Ideal point) and minimum values (the values determine the Negative Ideal values) form the weighted map layer for each land characteristic. Stage 2: Apply a separation measure to the positive ideal point: The distance between the ideal point and each alternative were calculated using: 2 si+ = ⎡⎣ ∑ j ( aij − a+ j ) ⎤⎦
0 .5
(8)
where si+ is the separation of the alternative, aij is the weighted map, a+j is the maximum value for the weighted map. © 2011 Blackwell Publishing Ltd Transactions in GIS, 2011, 15(3)
% Slope
1 2 2 2 2 2 2 2 2 2 2 2 2 2 0.01
Land Characteristics
% Slope Soil texture % CaCO3 % O. M % ESP AWHC EC % Stones Soil drainage Soil pH CEC Depth Infiltration rate Soil erosion Consistency ratio
1 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3
Soil texture
1 1/3 1/3 3 3 3 3 3 3 3 3 3
% CaCO3
1 1/3 3 5 2 2 2 2 2 2 2
% OM
Table 3 The pairwise comparison matrix for wheat
1 2 2 2 3 3 3 3 3 3
% ESP
1 1/3 1/3 1/3 2 1/3 1/3 1/3 1/3
AWHC
1 1/3 1/2 3 1/2 2 2 2
EC
1 2 3 2 2 2 2
% Stones
1 2 2 2 2 2
Soil drainage
1 1/3 1/3 1/3 1/3
Soil pH
1 3 1/3 3
CEC
1 1/3 3
Depth
1 3
Infiltration rate
1
Soil erosion
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Stage 3: Apply a separation measure to the negative ideal point: The distance to the negative ideal point and each alternative is determined using: 2 si− = ⎡⎣ ∑ j ( aij − a− j ) ⎤⎦
0 .5
(9)
where si- is the separation of the alternative, aij is the weighted map, a-j is the minimum values for the weighted map. Stage 4: Create maps from computing the relative closeness to the ideal point: At this stage, the closeness, Ci+, between the ideal point and the alternatives was computed and map layers for wheat created using:
Ci + =
si − si + + si −
(10)
where si+ and si- is the separation of the alternative and Ci+ is closeness between the ideal point and the alternatives. Stage 5: Calculate the final rating land suitability map layers: Land suitability maps for each crop were created as a continuous scale ranging from 0 to 1.
4 Results 4.1 Weighting Factors The results indicated that soil texture, available-water-holding capacity and soil reaction have higher weights than other criteria and therefore they are considered the most significant criteria in the study area (Table 4).
Table 4 The weights (Eigenvalues) for wheat Land characteristic
Weights / Eigenvalues
Soil texture Available-water-holding capacity Stones at surface Rootable depth Infiltration rate Soil drainage Calcium carbonate Organic matter Soil alkalinity Soil reaction Cation exchange capacity Soil salinity Slope steepness Soil erosion
0.150 0.123 0.043 0.080 0.059 0.051 0.042 0.035 0.028 0.132 0.062 0.069 0.032 0.094
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The accuracy of these weights is mainly dependent on local staff judgment. The results of the relative weights were able to be used in land suitability evaluation because the Consistency Ratios were within the established acceptable limits (0.1). The Consistency Ratios show, as noted earlier, any inconsistencies that may have arisen through the pairwise comparison analysis.
4.2 Map Comparison The distribution of land suitability maps for wheat from the Fuzzy AHP and Ideal Point approaches are illustrated in Figure 4. Whilst in some regions, the MFs are similar, a large portion of the study area have MFs in the range of 0.40 to 0.50 with the Fuzzy AHP approach and MFs in the range of 0.30 to 0.40 using the Ideal Point approach. These
A
B
Figure 4 Comparison of: (A) suitability evaluation for wheat using Fuzzy AHP; and (B) Ideal Point approaches © 2011 Blackwell Publishing Ltd Transactions in GIS, 2011, 15(3)
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Table 5 Comparison of the results of the Fuzzy AHP and Ideal Point approaches for wheat Overall suitability for wheat production Continuous
Fuzzy AHP
Ideal Point
Classification
(ha)
%
(ha)
%
0.20–0.30 0.30–0.40 0.40–0.50 0.50–0.60 0.60–0.70 0.70–0.80 No data
388 36,825 218,124 35,918 980 1,637 15,524
0.1 11.9 70.5 11.6 0.3 0.5 5.0
10,601 195,595 67,141 18,186 1,937 412 15,524
3.4 63.2 21.7 5.9 0.6 0.1 5.0
distributions of values under each approach are summarized in Table 5. The Fuzzy AHP and Ideal Point approaches illustrate that no locations in the study area have been mapped with Joint Membership Functions equal to 1. The value 1 indicates highly suitable classes and 0 less suitable classes, so few locations in the study area that are suitable for wheat production have been found.
4.3 Map Agreement In previous work, the agreement and disagreement between different measures of land suitability maps created using different approaches have not been assessed. Only the study by Moreno (2007) has derived the overall agreement between the Boolean and fuzzy maps. However, Moreno did not use an appropriate technique to compare the results: he employed a hard classification approach to compare the results which required recasting the fuzzy results to four crisp classes which were then compared. This approach essentially used alpha cuts to partition the fuzzy memberships, without justification for the threshold values. Additionally, there has been a historical difficulty in validating the results against crop yields, which may be significant: Young and Goldsmith (1977) note that “the differences in land management might lead to yield differences between farms as much as three- to five-fold in developing countries”, while in developing countries the differences are likely to be much less – for example ranging from 30 to 40% between the best and the worst management practices (Dent and Young 1981). To overcome this problem, this article used a cross-tabulation based on a soft classification analysis to derive the overall agreement between the maps. A soft cross-tabulation allows all pixels to have simultaneous partial membership of more than one class and employs three different operators: multiplication, minimum and composite. The composite operator guarantees that the matrix’s entries sum to 100%, which the minimum operator fails to do. These operators were defined by Pontius and Cheuk (2006) as follows: Multiplication operator: “The contemporary ontology envisions the classes of a pixel as located at points distributed randomly within the pixel. The randomization of points within each pixel is independent of the randomization of the points within any © 2011 Blackwell Publishing Ltd Transactions in GIS, 2011, 15(3)
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other pixel” (Pontius and Cheuk 2006, p. 10). For calculating the agreement and disagreement for the maps that are cross-tabulated using the multiplication operator the following equation is used:
Pnij = Pni ⋅ × Pn ⋅ j
(11)
According to Pontius and Cheuk (2006), the multiplication operator has many disadvantages. The main critical issue is that when a pixel is not hard-classified, the agreement between a pixel and itself is less than unity. Therefore, if the multiplication operator evaluates a map to itself, the resulting cross-tabulation matrix is not a diagonal matrix. Furthermore, it is possible to find a counter-intuitive result that the agreement between a pixel and itself is less than the agreement between the pixel and a dissimilar pixel. Minimum operator: A fuzzy ontology calls for a Minimum operator to compute both the diagonal and off-diagonal entries (Pontius and Cheuk 2006). The equation below can be used to estimate the agreement and disagreement for the cross-tabulated map using the minimum operator:
Pnij = MIN , Pni⋅, Pn ⋅ j
(12)
The minimum operator is helpful in situations where the category membership is uncertain, although it has problematic features regarding its use for multiple-resolution analysis. Consequently, if the minimum operator compares a soft-classified map layer to itself, the resulting cross-tabulation matrix is not necessarily a diagonal matrix (Pontius and Cheuk, 2006). Composite operator: “The multiple-resolution ontology calls for a two-step process in computing diagonal entries (i.e. agreement) and off-diagonal entries (i.e. disagreement). The composite rule has many attractive characteristics that the other rules lack, the most important being that it produces the identity matrix when a soft-classified image is compared to itself” (Pontius and Cheuk 2006, p. 11). For agreement the Equation (12) can be used, while for disagreement Equation (13) is employed:
⎡ Pn ⋅ j − Pnjj ⎤ ⎥ Pnij = ( Pni ⋅ − Pnii ) × ⎢ j ⎢ ∑ ( Pn ⋅ − Pnjj ) ⎥ ⎦ ⎣ j =1
For i ≠ j
(13)
where n is the pixel in the map, Pni· is the total membership function and the agreement is Pnii. For disagreement, n is the pixel in the reference map for class j, and the disagreement is Pni· – Pnjj. According to Pontius and Cheuk (2006), the composite operator, with a different scale of resolution, is better for comparing the maps because it resolves the difficulties of computing the cross-tabulation matrix derived from the use of the multiplication and minimum operators. The composite operator is also helpful in illustrating how well two layers or maps agree in terms of how the categories are clustered spatially. This operator has been used to compare the resulting maps from the use of the Fuzzy AHP and Ideal Point approaches. Figure 5 summarizes the overall agreement and disagreement between land suitability maps for wheat using the Ideal Point and Fuzzy © 2011 Blackwell Publishing Ltd Transactions in GIS, 2011, 15(3)
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Figure 5 The percentages of overall agreement and disagreement for wheat land suitability maps by using composite operator with multiple-resolution scale analysis
AHP approaches. It is that the overall agreement between the Fuzzy AHP and Ideal Point maps is around 90% and that the agreement between the maps increases and the disagreement decreases when the resolution scale increases. The implications are that to obtain less disagreement between the maps it is necessary to go to higher resolutions.
5 Discussion According to the local expert’s judgment, soil properties are the most sensitive criteria in the suitability classification for wheat in the study area. Therefore, the local experts allocated larger weights to soil criteria than to other factors (slope, erosion, etc.). In addition to this, most of the study area is sited in a plain which has few slope and erosion limitations – hence these criteria were given smaller weights by the experts. Nonetheless, the sensitivity of the results of this work is dependent on the designated weights that were given by local experts to different land characteristics (criteria). The results of the Fuzzy AHP and Ideal Point approaches showed that no locations in the study area were mapped with a degree of suitability or JMF values equal to 1. In the Fuzzy AHP model, a number of locations in specific criteria were given MFs of 1 due to the strength of support they offered in the overall assessment of wheat suitability. However, the derivation of the overall suitability using the Fuzzy AHP approach was not only based on the fuzzy membership function values that were assigned to GIS layer properties, but also the weighting values allocated to each criterion. The result was that Fuzzy AHP land suitability maps show the interaction between the fuzzy membership function values and their weights. It is evident from the results that the range of the JMF from Fuzzy AHP is almost similar to the range of the JMFs derived from Ideal Point methods. This is because the weighted fuzzy maps from the Fuzzy AHP were used as input to the Ideal Point approach. The differences in land suitability evaluation as determined by the Fuzzy AHP and Ideal Point approaches were obvious and mainly due to the Ideal Point approach taking the maximum and the minimum values of the fuzzy weighted maps into account and ignoring others. The Fuzzy AHP suitability model does not apply this function. The high overall agreement among the Fuzzy AHP and Ideal Point maps means that there were good correspondences between the pixel MFs to the © 2011 Blackwell Publishing Ltd Transactions in GIS, 2011, 15(3)
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set of land suitable for wheat. It is evident from the map agreement analysis that the overall agreement between the maps was increased when the resolution scale was increased and this may be explained as Pontius and Cheuk (2006, p. 24) note, “the Composite operator examines the agreement within the boundaries of a pixel, so when the boundaries become larger, the potential for agreement also becomes larger”.
6 Conclusions The objectives of this work were to compare the use the Fuzzy AHP and Ideal Point approaches to modelling agricultural land suitability and to explore the possibilities of using continuous classification in a land suitability evaluation model rather than traditional Boolean approaches. The most important development described in this article is the extension of the Fuzzy AHP approach to the Ideal Point approach for decision making in land suitability analyses. The Fuzzy AHP and Ideal Point approaches are both variants of fuzzy set methodologies in that they attempt to extend the concept of continuous variation of land properties from the geographic space to the attribute space. The use of Fuzzy AHP and Ideal Point techniques produced land suitability evaluations (in this case for irrigated wheat) in continuous scales and allowed subtle differences in criteria to be examined. They reveal important information relating to the limitations of crop production and strategies for overcoming them. The Fuzzy AHP and Ideal Point approaches allowed measurements of the environment to be inherently imprecise and do not attempt to limit land by the systems of data measurement made by the land or soil surveyor. The Fuzzy AHP and Ideal Point approaches produce good results because they address and accommodate the uncertainties that are associated with boundary conditions in criteria, taking into account the effects of properties which happen to have values close to category boundaries. This article asserts that that Fuzzy AHP approach to modelling land suitability evaluation is better than the Ideal Point approach, because the Ideal Point approach has some bias toward positive and negative ideal values. The results of this work provide information to decision-makers in their land planning decisions and further work should develop trial plots to groundtruth the suitability measures.
References Baja S, Chapman D M, and Dragovich D 2001 A conceptual model for defining and assessing land management units using a fuzzy modeling approach in a GIS environment. Environmental Management 29: 647–61 Banai R 1993 Fuzziness in Geographical Information Systems: Contributions from the Analytic Hierarchy Process. International Journal of Geographical Information Systems 7: 315–29 Ben-Mahmoud R 1995 Libyan Soils (First Edition). Tripoli, National Scientific Research Organization Burrough P A 1987 Mapping and mapping analysis: New tools for land evaluation. Soil Use Management 3: 20–5 Burrough P A 1989 Fuzzy mathematical methods for soil survey and land evaluation. Journal of Soil Science 40: 447–92 Burrough P A, MacMillan R A, and Van Deursen W 1992 Fuzzy classification methods for determining land suitability from soil profile observations and topography. Journal of Soil Science 43: 193–210 © 2011 Blackwell Publishing Ltd Transactions in GIS, 2011, 15(3)
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Chaddad I, Feng C B, Al-Husni M, and Jin Zhen R 2009 An application of land suitability evaluation for FTDP: A fuzzy MCDM approach. International Journal of Arab Culture, Management and Sustainable Development 1: 160–76 Chuong H V 2008 Multicriteria land suitability evaluation for crops using GIS at community level in central Vietnam: With case study in Thuy Bang-Thua Thien Hue Province. In Proceedings of the International Symposium on Geoinformatics for Spatial Infrastructure, Hanoi, Vietnam Ceballos-Silva A and Lopez-Blanco J 2003 Delineation of suitable areas for crops using a multicriteria evaluation approach and land use/cover mapping: A case study in central Mexico. Agricultural Systems 77: 117–36 Davidson D A, Theocharopoulos S P, and Bloksma R J 1994 A land evaluation project in Greece using GIS and based on Boolean and fuzzy set methodologies. International Journal of Geographic Information Systems 8: 369–84 De La Rosa D, Moreno J A, and Garica L V 1992 Micro LEIS: A microcomputer-based Mediterranean Land Evaluation Information System. Soil Use and Management 6: 7–20 Delgado G, Aranda V, Calero J, Sánchez-Marañón M, Serrano J M, Sanchez D, and Vila M A 2009 Using fuzzy data mining to evaluate survey data from olive grove cultivation. Computers and Electronics in Agriculture 65: 99–113 Dent D and Young A 1981 Soil Survey and Land Evaluation. London, George Allen and Unwin Duc T T 2006 Using GIS and AHP Technique for land use suitability analysis. In Proceedings of the International Symposium on Geoinformatics for Spatial Infrastructure, Hanoi, Vietnam Eastman J R, Kyem P A K, Toledano J, and Jin W 1993 GIS and Decision Making. Geneva, Switzerland, UNITAR Explorations in Geographic Information System Technology Vol. 4 Eldrandaly K, Eldin N, Sui D, Shouman M, and Nawara G 2005 Integrating GIS and MCDM using COM Technology. International Arab Journal of Information Technology 2: 162–67 FAO 1976 A Framework for Land Evaluation. Rome, Food and Agricultural Organization FAO 1985 Guidelines: Land Evaluation for Irrigated Agriculture. Rome: Food and Agricultural Organization of the United Nations FAO 2007 Land Evaluation: Towards a Revised Framework. Rome, Food and Agricultural Organization Fredrick N M and Julie V B 1997 Planning sustainable land management: finding a balance between user needs and possibilities. ITC Journal (1997): 3–4 Hall G B, Wang F, and Subaryono J 1992 Comparison of Boolean and fuzzy classification methods in land suitability analysis by using geographical information systems. Environment and Planning A 24: 497–516 Hwang C-L and Yoon K 1981 Multiple Attribute Decision Making: Methods and Applications. Berlin, Springer-Verlag Jiang H and Eastman R J 2000 Application of fuzzy measures in multi-criteria evaluation in GIS. International Journal of Geographical Information Science 14: 173–84 Keshavarzi A 2010 Land suitability evaluation using fuzzy continuous classification (A case study: Ziaran region). Modern Applied Science 4(7): 72–81 Malczewski J 1996 A GIS-based approach to multiple criteria group decision-making. International Journal of Geographical Information Systems 10: 955–71 Malczewski J 1999 GIS and Multicriteria Decision Analysis. New York, John Wiley and Sons Malczewski J 2006 Ordered weighted averaging with fuzzy quantifiers: GIS-based Multicriteria evaluation for land use suitability analysis. International Journal of Applied Earth Observation and Geoinformation 8: 249–68 McBratney A B and Odeh I O A 1997 Application of Fuzzy sets in soil science: Fuzzy logic, fuzzy measurements and fuzzy decisions. Geoderma. 77: 85–113 Mendoza G A, Macoun P, Prabhu R, Sukadri D, Purnomo H, and Hartanto H 1999 Guidelines for Applying Multi-criteria Analysis to the Assessment of Criteria and Indicators. Bogor, Indonesia, CIFOR C&I Toolbox Monograph No. 9 Moreno J 2007 Applicability of knowledge-based and fuzzy theory-oriented approaches to land suitability for upland rice and rubber, as compared to the farmers’ perception: A case study of Lao PDR. Unpublished M.S. Thesis, International Institute for Geo-Information Science and Earth Observation, Enschede, The Netherlands © 2011 Blackwell Publishing Ltd Transactions in GIS, 2011, 15(3)
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Orhan D, llhami B, and Mahmut Y 2003 Geographic Information System and remote sensing based land evaluation of Beypazari area soils by ILSEN model. Turkish Journal of Agriculture and Forestry 27: 145–53 Pereira J M C and Duckstein L 1993 A multiple criteria decision making approach to GIS-based land suitability evaluation. International Journal Geographical Information Systems 7: 407–24 Pontius R G Jr and Cheuk M L 2006 A generalized cross-tabulation matrix to compare softclassified maps at multiple resolutions. International Journal of Geographical Information Science 20: 1–30 Prakash J 2003 Land Suitability Evaluation for Agricultural Crops: A Fuzzy Multicriteria Decision Making Approach. Unpublished M.S. Thesis, International Institute for Geo-Information Science and Earth Observation, Enschede, The Netherlands Saaty T L 1977 A scaling method for priorities in hierarchical structures. Journal of Mathematical Psychology 15: 234–81 Saaty T L 1980 The Analytic Hierarchy Process. New York, McGraw–Hill Saaty T L 2008 Decision making with the analytic hierarchy process. International Journal of Services Sciences 1: 83–97 Selkhozpromexport 1980 Soils Study in the Western Zone of the Socialist People’s Libyan Arab Jamahiriya. Tripoli, Secretariat for Agricultural Reclamation and Land Development Sicat R S, Carranza E J M, and Nidumolu U B 2005 Fuzzy modeling of farmers’ knowledge for land suitability classification. Agricultural Systems 83: 49–75 Siddiqui M Z, Everett J W, and Vieux B E 1996 Landfill siting using geographical information systems: A demonstration. Journal of Environmental Engineering 122: 515–23 Van Ranst E, Tang H, Groenemans S, and Sinthurahat S 1996 Application of fuzzy logic to land suitability for rubber production in peninsular Thailand. Geoderma 70: 1–19 Vogel R 2008 A software framework for GIS-based multiple criteria evaluation of land suitability. In Proceedings of the Eleventh AGILE International Conference on Geographic Information Science, Girona, Spain: 1–32 Voogd H 1983 Multicriteria Evaluation for Urban and Regional Planning. London, Pion Wu F 1998 SimLand: A prototype to simulate land conversion through the integrated GIS and CA with AHP-derived transition rules. International Journal of Geographical Information Science 12: 63–82 Xiang W-N, Gross M, Fabos J, and MacDougall E B 1992 A fuzzy-group multicriteria decision making model and its application to land-use planning. Environment and Planning B 19: 61–84 Young A and Goldsmith P F 1977 Soil survey and land evaluation in developing countries. Geographical Journal 143: 67–74 Zadeh L A 1965 Fuzzy sets. Information and Control 8: 338–53 Zeleny M 1982 Multiple Attribute Decision Making. New York, McGraw Hill Zhu X and Dale A P 2001 Java AHP: A Web-based decision analysis tool for natural resource and environmental management. Environmental Modeling and Software 16: 251–62 Ziadat F M 2007 Land suitability classification using different sources of information: Soil maps and predicted soil attributes in Jordan. Geoderma 140: 73–80
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Research Article
A Comparison of Fuzzy AHP and Ideal Point Methods for Evaluating Land Suitability Mukhtar Elaalem
Alexis Comber
Department of Soil and Water Science Al-Fatah University
Department of Geography University of Leicester
Pete Fisher Department of Geography University of Leicester
Abstract This article compares two fuzzy approaches to land suitability evaluations, Analytical Hierarchy Process (AHP) and Ideal Point. The methods were evaluated using a case study which models the opportunities for wheat production under irrigation conditions in the north-western region of Jeffara Plain, Libya. A number of relevant soil and landscape criteria were identified through a review of the literature and their weights specified as a result of discussions with local experts. The results of the Fuzzy AHP showed that the majority of the study area has membership values to the set of suitability between 0.40 and 0.50, while the results of the Ideal Point approach revealed most of the study area to have membership values between 0.30 and 0.40. While the Fuzzy AHP and Ideal Point approaches accommodate the continuous nature of many soil properties and produce more intuitive distributions of land suitabilities values, the Fuzzy AHP approach was found to be better than Fuzzy Ideal Point. This was due to the latter’s tendency to be biased towards positive and negative ideal values.
1 Introduction Land resources are gradually becoming scarce as increases in population places pressure on natural resources. Population increases and urbanization has increased pressure Address for correspondence: Department of Soil and Water Science, Faculty of Agriculture, Al-Fatah University, Tripoli, Libya. E-mail: [email protected] © 2011 Blackwell Publishing Ltd doi: 10.1111/j.1467-9671.2011.01260.x
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on agricultural resources (Orhan et al. 2003). However, increased pressure on land resources may result in land degradation, particularly in countries with restricted water and other natural resources. The challenge in the next decades is to ensure global and regional food security and to increase food production to support the growing population: increases in food production are urgently required to tackle poverty and land degradation problems in developing countries (Fredrick and Julie 1997). As a result, food security is one of the top agricultural policies in developing countries and the evaluation of arable land and agricultural potential in these countries is needed in order to support current and future agricultural uses. The FAO framework for land evaluation is the most commonly used approach for land evaluation (FAO 1985) and is based on the biophysical factors and socioeconomic parameters of an area. The FAO approach evaluates the suitability of land for specific land use rather than general land use (as land capability) and seeks to match land utilization types with the land use requirements across all land units. The FAO approach requires a description of land in terms of land qualities or land characteristics. It classifies the land into four suitability classes: land suitability orders, land suitability classes, land suitability sub-classes and land suitability units. Orders indicate lands suitable for crops (S) or not suitable for crops (N). Classes show the degree of land suitability, such as (S1) highly suitable, (S2) moderately suitable, (S3) marginally not suitable, (N1) currently not suitable and (N2) permanently not suitable. The subclasses indicate the type of limitation (De La Rosa et al. 1992; FAO 1976, 2007). The FAO framework for land suitability uses a Boolean mapping approach which has been criticized by a number of authors (e.g. Burrough 1989, Hall et al. 1992, Davidson et al. 1994, McBratney and Odeh 1997, Baja et al. 2001, Delgado et al. 2009, Keshavarzi 2010) because the Boolean representations ignore the continuous nature of soil, landscape variation and uncertainties in measurement. Each of these aspects can result in areas being excluded from the set of suitable land because they fail to match strictly defined requirements, when in reality they may be quite suitable. The implicit assumption in Boolean approaches is the absence of any uncertainty or vagueness associated with the suitability model, measurement, imprecision and the concepts that are specified. In reality these assumptions may be invalid. Fuzzy set methodologies have been proposed as a method for overcoming problems related to vagueness in definition and other uncertainties. The use of fuzzy set methodologies in land suitability evaluation allows imprecise representations of vague, incomplete and uncertain information. Fuzzy land evaluations define continuous suitability classes rather than “true” or “false” as in the Boolean model (e.g. Burrough 1989, Sicat et al. 2005, Ziadat 2007, Keshavarzi 2010). Fuzzy set methodologies have the potential to provide better land evaluations compared to Boolean approaches because they are able to accommodate attribute values and properties which are close to category boundaries. A number of fuzzy Multi-Criteria Decision Analysis (MCDA) approaches have been developed for assessing suitability. These include Fuzzy Analytical Hierarchy Process (Fuzzy AHP) and TOPSIS (Technique for Order Preference by Similarity to Ideal Solutions) (Malczewski 1999). The fuzzy methods are able to address and explore the uncertainties associated with land resources, especially if they are integrated with fuzzy set models (Xiang et al. 1992, Ceballos-Silva and Lopez-Blanco 2003, Prakash 2003, Duc 2006, Moreno 2007, Chaddad et al. 2009, Chuong 2008). However fuzzy MCDA methods are still relatively unknown in land suitability evaluations for agricultural crops. © 2011 Blackwell Publishing Ltd Transactions in GIS, 2011, 15(3)
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This article compares land suitability evaluation models using two MCDA methods: Fuzzy Analytical Hierarchy Process (Fuzzy AHP) and Ideal Point methods.
2 Multi-Criterion Decision Analysis (MCDA) MCDA approaches are used to tackle real-world problems that are multi dimensional in nature. Criteria are defined, and MCDA methods are used to combine qualitative and quantitative criteria and to specify the degree and nature of the relationships between those criteria in order to support spatial decision-making. In a GIS context MCDA are used to combine layers of spatial data representing the criteria in the model. The model specifies how the layers are combined, such as the relative weighting given to each individual criterion and how the data are combined. For instance, one model is the Weighted Linear Combination (WLC) as described by Voogd (1983) providing a refinement to Boolean approaches. Overall suitability is calculated from the sum of the weighted normalized data layers representing factors in the model in the following way: n
Si = ∑ w j ⋅ xi , j
(1)
j =1
n
where
∑w
j
=1
j =1
and Si is the suitability score for site i, wj is the weight of criterion j, xij is the value of site i under criterion j, and n is the total number of criteria. Unlike Boolean approaches WLC allows low values in one criterion to be compensated for by high values in another (trade-off as described by Jiang and Eastman 2000). A good description of MCDA can be found in Malczewski (2006).
2.1 Analytical Hierarchy Process (AHP) The Analytic Hierarchy Process (AHP) was introduced and developed by Saaty (1977) and is an effective method of dealing with the context of the decision making process (Saaty 1980, Banai 1993, Eastman et al. 1993, Siddiqui et al. 1996, Wu 1998, Malczewski 1999, Mendoza et al. 1999, Zhu and Dale 2001, Eldrandaly et al. 2005, Duc 2006, Saaty 2008, Vogel 2008). Saaty (1977) and Malczewski (1999) note that the relationship among the objectives and attributes has a hierarchical structure: at the highest level, the objectives can be defined, and at lower levels, the attributes can be decomposed. Decision-makers play as an important role in making pairwise comparisons between criteria at each level of the hierarchy and to develop relative weights. The AHP process involves the following steps: • • • •
Criteria or factors contributing to the set of suitable are identified; The relative importance of each factor relative to each other factor – i.e. between pairs of criteria. This is usually done by domain experts; The consistency of the overall set of pairwise comparisons is assessed using its Consistency Ratio (CR); If the CR is greater than 0.1, then the expert revises the pairwise weights iteratively and
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Definition
1 2 3 4 5 6 7 8 9
Equal importance Equal to moderate importance Moderate importance Moderate to strong importance Strong importance Strong to very strong importance Very strong importance Very to extremely strong importance Extreme importance
The pairwise comparison requires a scale. Saaty (1980) recommends a scale from 1/9 to 9, as shown in Table 1. The quality of the overall set of pairwise comparisons is evaluated through the Consistency Ratio (CR). In the pairwise comparison step, the eigenvector values must have a Consistency Ratio (CR) ⱕ 0.1. The CR determines the internal consistency of the weights relative to the overall solution – it is a measurement that reveals how much difference is allowed (Malczewski 1996, 1999).
2.2 Ideal Point The Ideal Point approach uses a group of separation metrics to derive the best alternatives from a range of factors by ordering them based on their distance from the ideal point. The distance is calculated as follows: 1
Si + = ⎡⎣ ∑ i w iP ( Xij − X + i ) ⎤⎦ P P
(2)
where Si+ is the separation of the alternative from the ideal point, wi is a weight assigned to the criteria, Xij is the normalized criterion value of the alternative, X+i is the ideal value for the criterion, and p is the power factor rating from 1 to •. According to Malczewski (1999) “the larger values of P reflect greater concern for minimizing the maximum separation from the ideal. If the factor is set at 1, rectangular distance is computed. For P = 2 the straight-line distance is obtained. If P = •, the minimum of the maximum separation is sought”. The Ideal Point approach identifies one of many possible points that could be used for ordering the set of feasible alternatives. For instance, one could identify the negative ideal point and calculate the separation of the alternatives from that point. The Ideal Point method assumes that the most suitable alternatives have the shortest distance from the positive ideal solution, and the longest distance from the negative ideal solution. The most popular Ideal Point approach is the Technique for Order Performance by Similarity to the Ideal Solution (TOPSIS) which was first proposed by Hwang and Yoon 1981 (Malczewski 1999). The TOPSIS method considers alternatives that are close to the ideal point to be the most suitable alternatives (Hwang and Yoon 1981). Data in the Ideal Point approach are standard© 2011 Blackwell Publishing Ltd Transactions in GIS, 2011, 15(3)
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ized and then weighted to generate the most suitable alternatives. The advantage of the use of the Ideal Point approach to the land suitability problem is that it generates complete sets of weights and ranks for each attribute. It overcomes some of the disadvantages that are associated with interdependence between criteria under approaches such as the AHP and Weighted Liner Combination approaches (Zeleny 1982, Pereira and Duckstein 1993, Malczewski 1996).
2.3 Fuzzy sets Fuzzy sets are classes without sharp boundaries; that is, the transition between membership and non-membership of a location in the class is gradual (Zadeh 1965). A fuzzy set is described by a fuzzy membership functions (MFs) that range from 0.0 to 1.0, representing a continuous increase from non-membership to complete membership. The use of fuzzy logic to land suitability evaluation for agricultural crops was first introduced by Burrough (1989), who noted that the soil information being used as inputs to land suitability evaluations were mainly defined by imprecise terms such as ‘slightly susceptible to soil erosion’, ‘poorly drained’ and so on. These were used to determine a number of clearly defined boundaries between land suitability classes (FAO 1976). As a result, Burrough explored fuzzy sets as a tool to deal with the imprecision in land suitability evaluations. Examples of using fuzzy sets models to generate Membership Function values (MFs) for different land characteristics can be seen in Burrough (1987, 1989, 1992), Davidson et al. (1994), McBratney and Odeh (1997), Baja et al. (2001), Van Ranst et al. (1996) and Moreno (2007). Figure 1 shows three models which were used to create MFs for different criteria associated with land suitability in this article. The different models in Figure 1 are as follows:
Figure 1 The Fuzzy Membership function models: (A) asymmetrical left model; (B) asymmetrical right model; and (C) symmetrical model © 2011 Blackwell Publishing Ltd Transactions in GIS, 2011, 15(3)
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Model A: An asymmetrical left model is used when decreases in the quality function of the criteria improves the suitability:
MF( xi) = ⎡⎣1 {1 + 1 d 2 ( χ − b ) }⎤⎦ 2
(3)
where d is the width of the transition zone, and b is for an ideal point level and is the value of land characteristics. Examples of using this particular model (A) to covert row data such as the Available Water Holding Capacity (AWHC) to fuzzy membership functions are given below:
MF( AWHC) = ⎡⎣1 {1 + 1 40 ( χ − 150 ) }⎤⎦ 2
2
MF ( AWHC ) = 1 for χ > 150 MF ( AWHC ) = 1 for missing values where, is the value of AWHC: mm/m. Model B: An asymmetrical right model used when increases in quality function of criteria decreases the suitability:
MF( xi) = ⎡⎣1 {1 + 1 d 2 ( χ + b ) }⎤⎦ 2
(4)
where d is the width of the transition zone, and b is for an ideal point level and is the value of land characteristics. Example of using the model (B) to covert row data such as soil salinity (abbreviated to EC) to fuzzy membership functions are given below:
MF( EC) = ⎡⎣1 {1 + 1 1.42 ( χ + 6 ) }⎤⎦ 2
MF ( EC ) = 1 for χ ≤ 6 MF ( EC ) = 1 for missing values where c is the value of EC: dS/cm. Model C: A symmetrical or optimum range model, when suitability is indicated by being within a range or plateau of values.
MF( xi) = 1 if ( b1 + d1 ) ≤ xi ≤ ( b2 − d 2 )
(5)
where d is the width of the transition zone, and b1 and b2 are for an ideal point level and are the value of land characteristics.
3 Materials and Methods 3.1 Study Area The study area is located within the north-west Jeffara Plain region, Libya between Tripoli and AZ-Zahra city; between longitudes 12° 45′ and 13° 15′ east and latitudes 31° 52′ and 32° 52′ north; and has the area about 309,396 ha (Figure 2). © 2011 Blackwell Publishing Ltd Transactions in GIS, 2011, 15(3)
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Country Jeffara Plain
N
Study area City
335
W
E S
Figure 2 The study area location
The study area is located in the zone of the Mediterranean climate. Using the United States soil pedology classification, the main soil orders in the study area are Aridsols, Entisols and Inceptsols (Selkhozpromexport 1980, Ben Mahmoud 1995). In the study area there are plans to accommodate three cash crops, barley, wheat and maize under irrigation conditions. In this article, the wheat crop was selected as the crop to study.
3.2 Data Sources For this work 120 soil profiles were available. These were derived from soil samples collected from different horizons of the profiles and through physical and chemical analysis a number of attributes relating to crop production were generated. The physical and chemical characteristics were: soil texture, rootable depth, available water holdingcapacity, soil reaction, soil organic matter, cation exchange capacity, soil salinity, soil alkalinity, and soil calcium carbonate, stones at surface, soil drainage, and infiltration rate. Soil erosion and slope steepness were also taken into consideration in the study area because these parameters limit wheat production. Climate characteristics such as mean temperature of the growing cycle and relative humidity were not included in the models, because these characteristics are not limiting factors for irrigated wheat production in the study area.
3.3 Framework of Land Suitability Decision Making 3.3.1 Fuzzy analytical hierarchy process (Fuzzy AHP) The Fuzzy AHP approach can be divided into five stages Stage 1: Hierarchical organization of the land characteristics for wheat production. The hierarchical structure employed in this study is shown in Figure 3. © 2011 Blackwell Publishing Ltd Transactions in GIS, 2011, 15(3)
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Goal
Objectives
Attributes
Out put
pH Chemica
% CaCO3 % O. M
Soil
% ESP
Land suitability
CEC
Soil Physical
AWHC
Land suitability map
EC
% Stones Drainage Depth Infiltration
Topographic
% Slope
Erosion Hazard
Soil Erosion
Figure 3 Hierarchical organization of the land characteristics for wheat production
Stage 2: Standardizing land characteristics. Asymmetric and symmetric models were used for land characteristics, as described above. Table 2 shows fuzzy set models employed to convert the selected criteria to the fuzzy numbers in this article. Stage 3: Weighting Factors. Weighting the model criteria provides relative measures of the interaction and importance of the criteria. The weights were obtained through a pairwise comparison analysis in an AHP approach in discussion with local experts. The local experts played an important role in the process of land suitability and in the iterative adjustment of weights to improve the Consistency Ratio ⱕ 0.1. Table 3 shows the pairwise comparison matrix for wheat in the study area. Stage 4: Derive the weighted criterion map layers. The weighted criterion layers are generated using the following function:
WFkn = Wi × MFi
(6)
where Wi, is the weight of the land propriety from the pairwise comparison and MFi is the Membership Function for the land propriety. © 2011 Blackwell Publishing Ltd Transactions in GIS, 2011, 15(3)
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Table 2 Fuzzy set models for the selected land characteristics in the study area Land characteristics
Fuzzy set models
Rootable depth (cm) AWHC (mm/m) % organic matter CEC (me/100g soil) Soil drainage classes (mm/h) Infiltration rate (mm/h) Soil salinity (EC) % ESP % CaCO3 in root zones % stones at surface % slope Soil texture (class) Soil erosion (class) Soil pH
Asymmetrical left
Asymmetrical right
Symmetrical model
Stage 5: Derive the overall land suitability map layers. The suitability is calculated by combining the weighted criterion layers. This function sums the weighted fuzzy maps of the different land proprieties to obtain land suitability maps at the final level:
R i = WFk1 + WFk2 + …… WFkn
(7)
where Ri, is the overall rating score for the suitability of land and WFkn is the weighted value for the different land properties. The overall land suitability maps show the overall land suitability classes with a continuous scale range from 0 to 1.
3.3.2 Ideal point approach To derive the land suitability for wheat based on Ideal Point mapping, the weighted map layers created in the Fuzzy AHP method are applied as the input data and the Ideal Point approach utilized in this article is an extension of the Fuzzy AHP approach. The stages of land suitability evaluation using the Ideal Point method are described below: Stage 1: Determine the maximum and minimum values. In this stage the maximum values (the values determine the Ideal point) and minimum values (the values determine the Negative Ideal values) form the weighted map layer for each land characteristic. Stage 2: Apply a separation measure to the positive ideal point: The distance between the ideal point and each alternative were calculated using: 2 si+ = ⎡⎣ ∑ j ( aij − a+ j ) ⎤⎦
0 .5
(8)
where si+ is the separation of the alternative, aij is the weighted map, a+j is the maximum value for the weighted map. © 2011 Blackwell Publishing Ltd Transactions in GIS, 2011, 15(3)
% Slope
1 2 2 2 2 2 2 2 2 2 2 2 2 2 0.01
Land Characteristics
% Slope Soil texture % CaCO3 % O. M % ESP AWHC EC % Stones Soil drainage Soil pH CEC Depth Infiltration rate Soil erosion Consistency ratio
1 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3
Soil texture
1 1/3 1/3 3 3 3 3 3 3 3 3 3
% CaCO3
1 1/3 3 5 2 2 2 2 2 2 2
% OM
Table 3 The pairwise comparison matrix for wheat
1 2 2 2 3 3 3 3 3 3
% ESP
1 1/3 1/3 1/3 2 1/3 1/3 1/3 1/3
AWHC
1 1/3 1/2 3 1/2 2 2 2
EC
1 2 3 2 2 2 2
% Stones
1 2 2 2 2 2
Soil drainage
1 1/3 1/3 1/3 1/3
Soil pH
1 3 1/3 3
CEC
1 1/3 3
Depth
1 3
Infiltration rate
1
Soil erosion
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Stage 3: Apply a separation measure to the negative ideal point: The distance to the negative ideal point and each alternative is determined using: 2 si− = ⎡⎣ ∑ j ( aij − a− j ) ⎤⎦
0 .5
(9)
where si- is the separation of the alternative, aij is the weighted map, a-j is the minimum values for the weighted map. Stage 4: Create maps from computing the relative closeness to the ideal point: At this stage, the closeness, Ci+, between the ideal point and the alternatives was computed and map layers for wheat created using:
Ci + =
si − si + + si −
(10)
where si+ and si- is the separation of the alternative and Ci+ is closeness between the ideal point and the alternatives. Stage 5: Calculate the final rating land suitability map layers: Land suitability maps for each crop were created as a continuous scale ranging from 0 to 1.
4 Results 4.1 Weighting Factors The results indicated that soil texture, available-water-holding capacity and soil reaction have higher weights than other criteria and therefore they are considered the most significant criteria in the study area (Table 4).
Table 4 The weights (Eigenvalues) for wheat Land characteristic
Weights / Eigenvalues
Soil texture Available-water-holding capacity Stones at surface Rootable depth Infiltration rate Soil drainage Calcium carbonate Organic matter Soil alkalinity Soil reaction Cation exchange capacity Soil salinity Slope steepness Soil erosion
0.150 0.123 0.043 0.080 0.059 0.051 0.042 0.035 0.028 0.132 0.062 0.069 0.032 0.094
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The accuracy of these weights is mainly dependent on local staff judgment. The results of the relative weights were able to be used in land suitability evaluation because the Consistency Ratios were within the established acceptable limits (0.1). The Consistency Ratios show, as noted earlier, any inconsistencies that may have arisen through the pairwise comparison analysis.
4.2 Map Comparison The distribution of land suitability maps for wheat from the Fuzzy AHP and Ideal Point approaches are illustrated in Figure 4. Whilst in some regions, the MFs are similar, a large portion of the study area have MFs in the range of 0.40 to 0.50 with the Fuzzy AHP approach and MFs in the range of 0.30 to 0.40 using the Ideal Point approach. These
A
B
Figure 4 Comparison of: (A) suitability evaluation for wheat using Fuzzy AHP; and (B) Ideal Point approaches © 2011 Blackwell Publishing Ltd Transactions in GIS, 2011, 15(3)
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Table 5 Comparison of the results of the Fuzzy AHP and Ideal Point approaches for wheat Overall suitability for wheat production Continuous
Fuzzy AHP
Ideal Point
Classification
(ha)
%
(ha)
%
0.20–0.30 0.30–0.40 0.40–0.50 0.50–0.60 0.60–0.70 0.70–0.80 No data
388 36,825 218,124 35,918 980 1,637 15,524
0.1 11.9 70.5 11.6 0.3 0.5 5.0
10,601 195,595 67,141 18,186 1,937 412 15,524
3.4 63.2 21.7 5.9 0.6 0.1 5.0
distributions of values under each approach are summarized in Table 5. The Fuzzy AHP and Ideal Point approaches illustrate that no locations in the study area have been mapped with Joint Membership Functions equal to 1. The value 1 indicates highly suitable classes and 0 less suitable classes, so few locations in the study area that are suitable for wheat production have been found.
4.3 Map Agreement In previous work, the agreement and disagreement between different measures of land suitability maps created using different approaches have not been assessed. Only the study by Moreno (2007) has derived the overall agreement between the Boolean and fuzzy maps. However, Moreno did not use an appropriate technique to compare the results: he employed a hard classification approach to compare the results which required recasting the fuzzy results to four crisp classes which were then compared. This approach essentially used alpha cuts to partition the fuzzy memberships, without justification for the threshold values. Additionally, there has been a historical difficulty in validating the results against crop yields, which may be significant: Young and Goldsmith (1977) note that “the differences in land management might lead to yield differences between farms as much as three- to five-fold in developing countries”, while in developing countries the differences are likely to be much less – for example ranging from 30 to 40% between the best and the worst management practices (Dent and Young 1981). To overcome this problem, this article used a cross-tabulation based on a soft classification analysis to derive the overall agreement between the maps. A soft cross-tabulation allows all pixels to have simultaneous partial membership of more than one class and employs three different operators: multiplication, minimum and composite. The composite operator guarantees that the matrix’s entries sum to 100%, which the minimum operator fails to do. These operators were defined by Pontius and Cheuk (2006) as follows: Multiplication operator: “The contemporary ontology envisions the classes of a pixel as located at points distributed randomly within the pixel. The randomization of points within each pixel is independent of the randomization of the points within any © 2011 Blackwell Publishing Ltd Transactions in GIS, 2011, 15(3)
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other pixel” (Pontius and Cheuk 2006, p. 10). For calculating the agreement and disagreement for the maps that are cross-tabulated using the multiplication operator the following equation is used:
Pnij = Pni ⋅ × Pn ⋅ j
(11)
According to Pontius and Cheuk (2006), the multiplication operator has many disadvantages. The main critical issue is that when a pixel is not hard-classified, the agreement between a pixel and itself is less than unity. Therefore, if the multiplication operator evaluates a map to itself, the resulting cross-tabulation matrix is not a diagonal matrix. Furthermore, it is possible to find a counter-intuitive result that the agreement between a pixel and itself is less than the agreement between the pixel and a dissimilar pixel. Minimum operator: A fuzzy ontology calls for a Minimum operator to compute both the diagonal and off-diagonal entries (Pontius and Cheuk 2006). The equation below can be used to estimate the agreement and disagreement for the cross-tabulated map using the minimum operator:
Pnij = MIN , Pni⋅, Pn ⋅ j
(12)
The minimum operator is helpful in situations where the category membership is uncertain, although it has problematic features regarding its use for multiple-resolution analysis. Consequently, if the minimum operator compares a soft-classified map layer to itself, the resulting cross-tabulation matrix is not necessarily a diagonal matrix (Pontius and Cheuk, 2006). Composite operator: “The multiple-resolution ontology calls for a two-step process in computing diagonal entries (i.e. agreement) and off-diagonal entries (i.e. disagreement). The composite rule has many attractive characteristics that the other rules lack, the most important being that it produces the identity matrix when a soft-classified image is compared to itself” (Pontius and Cheuk 2006, p. 11). For agreement the Equation (12) can be used, while for disagreement Equation (13) is employed:
⎡ Pn ⋅ j − Pnjj ⎤ ⎥ Pnij = ( Pni ⋅ − Pnii ) × ⎢ j ⎢ ∑ ( Pn ⋅ − Pnjj ) ⎥ ⎦ ⎣ j =1
For i ≠ j
(13)
where n is the pixel in the map, Pni· is the total membership function and the agreement is Pnii. For disagreement, n is the pixel in the reference map for class j, and the disagreement is Pni· – Pnjj. According to Pontius and Cheuk (2006), the composite operator, with a different scale of resolution, is better for comparing the maps because it resolves the difficulties of computing the cross-tabulation matrix derived from the use of the multiplication and minimum operators. The composite operator is also helpful in illustrating how well two layers or maps agree in terms of how the categories are clustered spatially. This operator has been used to compare the resulting maps from the use of the Fuzzy AHP and Ideal Point approaches. Figure 5 summarizes the overall agreement and disagreement between land suitability maps for wheat using the Ideal Point and Fuzzy © 2011 Blackwell Publishing Ltd Transactions in GIS, 2011, 15(3)
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Figure 5 The percentages of overall agreement and disagreement for wheat land suitability maps by using composite operator with multiple-resolution scale analysis
AHP approaches. It is that the overall agreement between the Fuzzy AHP and Ideal Point maps is around 90% and that the agreement between the maps increases and the disagreement decreases when the resolution scale increases. The implications are that to obtain less disagreement between the maps it is necessary to go to higher resolutions.
5 Discussion According to the local expert’s judgment, soil properties are the most sensitive criteria in the suitability classification for wheat in the study area. Therefore, the local experts allocated larger weights to soil criteria than to other factors (slope, erosion, etc.). In addition to this, most of the study area is sited in a plain which has few slope and erosion limitations – hence these criteria were given smaller weights by the experts. Nonetheless, the sensitivity of the results of this work is dependent on the designated weights that were given by local experts to different land characteristics (criteria). The results of the Fuzzy AHP and Ideal Point approaches showed that no locations in the study area were mapped with a degree of suitability or JMF values equal to 1. In the Fuzzy AHP model, a number of locations in specific criteria were given MFs of 1 due to the strength of support they offered in the overall assessment of wheat suitability. However, the derivation of the overall suitability using the Fuzzy AHP approach was not only based on the fuzzy membership function values that were assigned to GIS layer properties, but also the weighting values allocated to each criterion. The result was that Fuzzy AHP land suitability maps show the interaction between the fuzzy membership function values and their weights. It is evident from the results that the range of the JMF from Fuzzy AHP is almost similar to the range of the JMFs derived from Ideal Point methods. This is because the weighted fuzzy maps from the Fuzzy AHP were used as input to the Ideal Point approach. The differences in land suitability evaluation as determined by the Fuzzy AHP and Ideal Point approaches were obvious and mainly due to the Ideal Point approach taking the maximum and the minimum values of the fuzzy weighted maps into account and ignoring others. The Fuzzy AHP suitability model does not apply this function. The high overall agreement among the Fuzzy AHP and Ideal Point maps means that there were good correspondences between the pixel MFs to the © 2011 Blackwell Publishing Ltd Transactions in GIS, 2011, 15(3)
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set of land suitable for wheat. It is evident from the map agreement analysis that the overall agreement between the maps was increased when the resolution scale was increased and this may be explained as Pontius and Cheuk (2006, p. 24) note, “the Composite operator examines the agreement within the boundaries of a pixel, so when the boundaries become larger, the potential for agreement also becomes larger”.
6 Conclusions The objectives of this work were to compare the use the Fuzzy AHP and Ideal Point approaches to modelling agricultural land suitability and to explore the possibilities of using continuous classification in a land suitability evaluation model rather than traditional Boolean approaches. The most important development described in this article is the extension of the Fuzzy AHP approach to the Ideal Point approach for decision making in land suitability analyses. The Fuzzy AHP and Ideal Point approaches are both variants of fuzzy set methodologies in that they attempt to extend the concept of continuous variation of land properties from the geographic space to the attribute space. The use of Fuzzy AHP and Ideal Point techniques produced land suitability evaluations (in this case for irrigated wheat) in continuous scales and allowed subtle differences in criteria to be examined. They reveal important information relating to the limitations of crop production and strategies for overcoming them. The Fuzzy AHP and Ideal Point approaches allowed measurements of the environment to be inherently imprecise and do not attempt to limit land by the systems of data measurement made by the land or soil surveyor. The Fuzzy AHP and Ideal Point approaches produce good results because they address and accommodate the uncertainties that are associated with boundary conditions in criteria, taking into account the effects of properties which happen to have values close to category boundaries. This article asserts that that Fuzzy AHP approach to modelling land suitability evaluation is better than the Ideal Point approach, because the Ideal Point approach has some bias toward positive and negative ideal values. The results of this work provide information to decision-makers in their land planning decisions and further work should develop trial plots to groundtruth the suitability measures.
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