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Advances in Fuzzy Systems
Volume 2011 (2011), Article ID 429498, 14 pages
http://dx.doi.org/10.1155/2011/429498
Research Article

Spatial Analysis and Fuzzy Relation Equations

Dipartimento di Costruzioni e Metodi Matematici in Architettura, Università degli Studi di Napoli Federico II, Via Monteoliveto 3, 80134 Napoli, Italy

Received 29 June 2011; Accepted 24 July 2011

Academic Editor: Irina G. Perfilieva

Copyright © 2011 Ferdinando Di Martino and Salvatore Sessa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We implement an algorithm that uses a system of fuzzy relation equations (SFRE) with the max-min composition for solving a problem of spatial analysis. We integrate this algorithm in a Geographical Information System (GIS) tool, and the geographical area under study is divided in homogeneous subzones (with respect to the parameters involved) to which we apply our process to determine the symptoms after that an expert sets the SFRE with the values of the impact coefficients. We find that the best solutions and the related results are associated to each subzone. Among others, we define an index to evaluate the reliability of the results.

1. Introduction

A Geographical Information System (GIS) is used as a support decision system for problems in a spatial domain; in many cases, we use a GIS to analyze spatial distribution of data, spatial relations, the impact of event data on spatial areas; simple examples of this analysis are the creation of thematic maps, the geoprocessing operators, the buffer analysis, and so forth. Often the expert analyzes spatial data in a decision making process with the help of a GIS which involves integration of images, spatial layers, attributes information and an inference mechanism based on these attributes. The diversity and the inhomogeneity between the individual layers of spatial information and the inaccuracy of the results can lead to uncertain decisions, so that one needs the use of fuzzy inference calculus to handle these uncertain information. Many authors [15] propose models to solve spatial problems based on fuzzy relational calculus. In this paper, we propose an inferential method to solve spatial problems based on an algorithm for the resolution of a system of fuzzy relation equations (shortly, SFRE) given in [6] (cf. also [7, 8]) and applied in [9] to solve industrial application problems. Here we integrate this algorithm in the context of a GIS architecture. Usually an SFRE with max-min composition is read as 𝑎11𝑥1𝑎1𝑛𝑥𝑛=𝑏1,𝑎21𝑥1𝑎2𝑛𝑥𝑛=𝑏2,𝑎𝑚1𝑥1𝑎𝑚𝑛𝑥𝑛=𝑏𝑚.(1)

The system (1) is said consistent if it has solutions. In his pioneering paper [10], the author determines the greatest solution in case of max-min composition. After these results, many researchers have found algorithms which determine minimal solutions of max-min fuzzy relation equations (cf., e.g., [1118]). In [6, 7] a method is described for the consistence of the system (1), and moreover it calculates the complete set of the solutions. This method is schematized in Figure 1 and described below.(i)Input extraction: the input data are extracted and stored in the dataset.(ii)The input variable is fuzzified. A fuzzy partition of the input domain is created; the corresponding membership degree of every input data is assigned to each fuzzy set.(iii)The membership degrees of each fuzzy set determine the coefficients {𝑏1,,𝑏𝑚} of (1). The values of the coefficients 𝑎𝑖𝑗 are set by the expert and the whole set of solutions (𝑥1,,𝑥𝑛) of (1) is determined as well.(iv)A fuzzy partition of the domain [0,1] is created for the output variables 𝑜1,,𝑜𝑘; every fuzzy set of the partition corresponds to a determined value 𝑥𝑗. (v)The output data 𝑜1,,𝑜𝑘 are extracted. A partition of fuzzy sets corresponds to each output variable 𝑜𝑗 (𝑗=1,,𝑘); in this phase the linguistic label of the most appropriate fuzzy set is assigned to the output variable 𝑜𝑗.

429498.fig.001
Figure 1: Resolution process of an SFRE.

This process has been applied to a real spatial problem in which the input data vary for each subzone of the geographical area. We have the same input data, and the expert applies the same SFRE (1) on each subzone. The expert starts from a valuation of input data, and he uses linguistic labels for the determination of the output results for each subzone. The input data are the facts or symptoms; the parameters to be determined are the causes. For example, let us consider a planning problem. A city planner needs to determine in each subzone the mean state of buildings (𝑥1) and the mean soil permeability (𝑥2), knowing the number of collapsed building in the last year (𝑏1) and the number of flooding in the last year (𝑏2). In Figure 2, we suppose to create for each symptom’s and cause’s variable domain a fuzzy partition of three fuzzy sets (generally, one is faced with trapezoidal or triangular fuzzy number, this last one is denoted in the sequel shortly with the acronym TFN). The expert creates the SFRE (1) for each subzone by setting the impact matrix A, whose entries 𝑎𝑖𝑗 (𝑖=1,,𝑛 and 𝑗=1,,𝑚) represent the impact of the 𝑗th cause 𝑥𝑗 to the production of the 𝑖th symptom 𝑏𝑖, where the value of 𝑏𝑖 is the membership degree in the corresponding fuzzy set and let 𝐵=[𝑏1,,𝑏𝑚]. In another subzone the input data vector 𝐵 and the matrix 𝐴 can vary. For example, we consider the equation:0.8𝑥10.2𝑥20.0𝑥30.8𝑥40.3𝑥50.0𝑥6=𝑏3=0.9.(2) The expert sets for the symptom 𝑏3 = “collapsed building in the last year = high” = 0.9, an impact 0.8 of the variable “mean state of buildings = scanty”, an impact 0.2 of the variable “mean state of buildings = medium”, an impact 0.0 of the variable “mean state of buildings = high”, an impact 0.8 of the variable “mean soil permeability = low”, an impact 0.3 of the variable “mean soil permeability = medium”, or an impact 0.0 of the variable “mean soil permeability = high”.

fig2
Figure 2: Examples of trapezoidal fuzzy numbers used for symptoms and causes.

We can determine the maximal interval solutions of (1). Each maximal interval solution is an interval whose extremes are the values taken from a minimal solution and from the greatest solution. Every value 𝑥𝑖 belongs to this interval. If the SFRE (1) is inconsistent, it is possible to determine the rows for which no solution is permitted. If the expert decides to exclude the row for which no solution is permitted, he considers that the symptom 𝑏𝑖 (for that row) is not relevant to its analysis, and it is not taken into account. Otherwise, the expert can modify the setting of the coefficients of the matrix 𝐴 to verify if the new system has some solution. In general, the SFRE (1) has T maximal interval solutions 𝑋max(1),,𝑋max(𝑇). In order to describe the extraction process of the solutions, let 𝑋max(𝑡), 𝑡{1,,𝑇}, be a maximal interval solution given below, where 𝑋low is a minimal solution and 𝑋gr is the greatest solution. Our aim is to assign the linguistic label of the most appropriate fuzzy sets corresponding to the unknown {𝑥𝑗1,𝑥𝑗1,,𝑥𝑗𝑠} related to an output variable 𝑜𝑠, 𝑠=1,,𝑘. For example, assume that the three fuzzy sets 𝑥1, 𝑥2, 𝑥3 (resp., 𝑥4, 𝑥5, 𝑥6) are related to 𝑜1 (resp., 𝑜2) and are represented from the TFNs given in Table 1, where INF(𝑗), MEAN(𝑗), and SUP(𝑗) are the three fundamental values of the generic TFN 𝑥𝑗, 𝑗=𝑗1,,𝑗𝑠. We can write their membership functions 𝜇𝑗1,𝜇𝑗2,,𝜇𝑗 as follows:𝜇𝑗1=𝑗1,ifINF1𝑗𝑥MEAN1,𝑗SUP1𝑥𝑗SUP1𝑗MEAN1𝑗,ifMEAN1𝑗<𝑥SUP1,𝜇0,otherwise,(3)𝑗=𝑥INF(𝑗)MEAN(𝑗)INF(𝑗),ifINF(𝑗)𝑥MEAN(𝑗),SUP(𝑗)𝑥𝑗SUP(𝑗)MEAN(𝑗),ifMEAN(𝑗)<𝑥SUP(𝑗),0,otherwise,𝑗2,,𝑗𝑠1,𝜇(4)𝑗𝑠=𝑗𝑥INF𝑠𝑗MEAN𝑠𝑗INF𝑠𝑗,ifINF𝑠𝑗𝑥MEAN𝑠,𝑗1,ifMEAN𝑠𝑗<𝑥SUP𝑠,0,otherwise.(5)

tab1
Table 1: TFNs values for the fuzzy sets.

If 𝑋Min𝑡(𝑗) (resp., 𝑋Max𝑡(𝑗)) is the min (resp., max) value of every interval corresponding to the unknown 𝑥𝑗, we can calculate the arithmetical mean value 𝑋Mean𝑡(𝑗) of the 𝑗th component of the above maximal interval solution 𝑋max(𝑡) as𝑋Mean𝑡(𝑗)=𝑋Min𝑡(𝑗)+𝑋Max𝑡(𝑗)2,(6) and we get the vector column 𝑋Mean𝑡=[𝑋Mean𝑡(1),,𝑋Mean𝑡(𝑛)]1 (cf. Table 2). The value given from max{𝑋Mean𝑡(𝑗1),,𝑋Mean𝑡(𝑗𝑠)} obtained for the unknowns 𝑥𝑗1,,𝑥𝑗𝑠 corresponding to the output variable 𝑜𝑠, is the linguistic label of the fuzzy set assigned to 𝑜𝑠 and it is denoted by scoret (𝑜𝑠), defined also as reliability of 𝑜𝑠 in the interval solution 𝑡. In our example, we have that “𝑜1 = mean state of buildings = scanty” and “𝑜2 = mean soil permeability = medium”, hence score𝑡(𝑜1)=0.70 and score𝑡(𝑜2)=0.55. For the output vector 𝑂=[𝑜1,,𝑜𝑘], we define the following reliability index in the interval solution 𝑡 asRel𝑡1(𝑂)=𝑘𝑘𝑠=1score𝑡𝑜𝑠(7) and then as final reliability index of 𝑂, the number Rel(𝑂)=max{Rel𝑡(𝑂)𝑡=1,,𝑇}.

tab2
Table 2: TFNs mean values.

In our example, we have Rel𝑡(𝑂)=(0.7+0.55)/2=0.625. Therefore, the higher the reliability of our solution, the closer the final reliability index Rel(𝑂) to 1. In Section 2, we give an extended and articulated overview on how to determine the whole set of the solutions of an SFRE, and in Section 3 we show how the proposed algorithm is applied in spatial analysis. Section 4 contains the results of our simulation.

2. SFRE: An Extended Overview

In this paper, we investigate the solutions of the SFRE (1), which is abbreviated in the following known form:𝐴𝑋=𝐵,(8) where 𝐴=(𝑎𝑖𝑗) is the matrix of coefficients, 𝑋 = (𝑥1,𝑥2,,𝑥𝑛)−1 is the column vector of the unknowns, and 𝐵 = (𝑏1,𝑏2,,𝑏𝑚)−1 is the column vector of the known terms, being 𝑎𝑖𝑗,𝑥𝑗,𝑏𝑖[0,1] for each 𝑖=1,,𝑚 and 𝑗=1,,𝑛. We have the following definitions and terminologies: the whole set of all solutions 𝑋 of the SFRE (8) is denoted by Ω. If Ω, then the SFRE (8) is called consistent, otherwise it is called inconsistent. A solution 𝑋Ω is called a minimal solution if 𝑋𝑋 for some 𝑋Ω implies 𝑋𝑋=, where “≤” is the partial order induced in Ω from the natural order of [0,1]. If the minimal solution is unique, then it is the least (or minimum) solution of the SFRE (8). We also recall that the system (8) has the unique greatest (or maximum) solution 𝑋gr=(𝑥gr1,𝑥gr2,,𝑥gr𝑛)1 if Ω [10]. A matrix interval 𝑋interval of the following type:𝑋interval=𝑎1,𝑏1𝑎2,𝑏2[]𝑎,𝑛,𝑏𝑛,(9) where [𝑎𝑗,𝑏𝑗][0,1] for each 𝑗=1,,𝑛, is called an interval solution of the SFRE (8) if every 𝑋=(𝑥1,𝑥2,,𝑥𝑛)1 such that 𝑥𝑗[𝑎𝑗,𝑏𝑗] for each 𝑗=1,,𝑛, belongs to Ω. If 𝑎𝑗 is a membership value of a minimal solution and 𝑏𝑗 is a membership value of 𝑋gr for each 𝑗=1,,𝑛, then 𝑋interval is called a maximal interval solution of the SFRE (8), and it is denoted by 𝑋max(𝑡), where 𝑡 varies from 1 to the number of minimal solutions. The SFRE (8) is said to be in normal form if 𝑏1𝑏2𝑏𝑚. The time computational complexity to reduce an SFRE in a normal form is polynomial [6, 8]. Now we consider the matrix 𝐴=(𝑎𝑖𝑗) so defined:𝑎𝑖𝑗=0,if𝑎𝑖𝑗<𝑏𝑖,𝑏𝑖,if𝑎𝑖𝑗=𝑏𝑖,1,if𝑎𝑖𝑗>𝑏𝑖,(10) where 𝑖=1,,𝑚 and 𝑗=1,,𝑛. The linguistic description of 𝑎𝑖𝑗 as S-type coefficient (Smaller) if 𝑎𝑖𝑗<𝑏𝑖, E-type coefficient (Equal) if 𝑎𝑖𝑗=𝑏𝑖, and G-type coefficient (Greater) if 𝑎𝑖𝑗>𝑏𝑖 is often used. 𝐴 is called augmented matrix, and the system 𝐴𝑋=𝐵 is said associated to the SFRE (8). Without loss of generality, from now on we suppose that the system (8) is in normal form. We also obtained the following definitions and results from [6, 8, 19, 20].

Definition 1. Let the SFRE (8) be consistent and 𝐴𝑗={𝑎1𝑗,,𝑎𝑚𝑗}. If 𝐴(𝑗) contains G-type coefficients and 𝑘{1,,𝑚} is the greatest index of row such that 𝑎𝑘𝑗=1, then the following coefficients in 𝐴(𝑗) are called selected:(i)𝑎𝑖𝑗 for 𝑖{1,,𝑘} with 𝑎𝑖𝑗𝑏𝑖=𝑏𝑘,(ii)𝑎𝑖𝑗 for 𝑖{𝑘+1,,𝑚} with 𝑎𝑖𝑗=𝑏𝑖.

Definition 2. If 𝐴(𝑗) does not contain G-type coefficients, but it contain E-type coefficients and 𝑟{1,,𝑚} is the smallest index of row such that 𝑎𝑟𝑗=𝑏𝑟, then any 𝑎𝑖𝑗=𝑏𝑖 in 𝐴(𝑗) for 𝑖{𝑟,,𝑚} is called selected.

Theorem 3. Consider an SFRE (8). Then the following occurs.(i)The SFRE (8) is consistent if and only if there exist at least one selected coefficient for each 𝑖th equation, 𝑖=1,,𝑚.(ii)The complexity time function for determining the consistency of the SFRE (8) is 𝑂(𝑚𝑛).

Consequently, when an SFRE (8) is inconsistent, the equations for which no element is a selected coefficient could not be satisfied simultaneously with the other equations having at least one selected coefficient. Furthermore, a vector IND=(IND(1),,IND(𝑚)) is defined by setting IND(𝑖) equal to the number of selected coefficients in the 𝑖th equation for each 𝑖=𝑙,,𝑚. If IND(𝑖)=0, then all the coefficients in the 𝑖th equation are not selected and the system is inconsistent. The system is consistent if IND(𝑖)0 for each 𝑖=𝑙,,𝑚 and the productPN2=𝑚𝑖=1IND(𝑖),(11) gives the upper bound of the number of the eventual minimal solutions.

Theorem 4. Let the SFRE (8) be consistent. Then the following occurs.(i)The SFRE has a unique greatest solution 𝑋gr with component 𝑥gr𝑗=𝑏𝑘 if the jth column 𝐴(𝑗) of 𝐴 contains selected G-type coefficients 𝑎𝑘𝑗 and 𝑥gr𝑗=1 otherwise. (ii)The complexity time function for computing 𝑋gr is 𝑂(𝑚𝑛).

A help matrix 𝐻=(𝑖𝑗), with 𝑖=1,,𝑚 and 𝑗=1,,𝑛, is defined as follows:𝑖𝑗=𝑏𝑖,if𝑎𝑖𝑗isselected,0,otherwise.(12)

Let |𝐻𝑖| be the number of coefficients 𝑖𝑗 in the 𝑖th equation of the SFRE (8). Then the number of potential minimal solutions cannot exceed the valuePN1=𝑚𝑖=1||𝐻𝑖||,(13) where PN2PN1.

Definition 5. Let 𝑖=(𝑖1,𝑖2,,𝑖𝑛) and 𝑘=(𝑘1,𝑘2,,𝑘𝑛) be the 𝑖th and the 𝑘th rows of the help matrix 𝐻. If for each 𝑗=1,𝑛, 𝑖𝑗0 implies both 𝑘𝑗0 and 𝑘𝑗𝑖𝑗, then the 𝑖th row (resp., equation) is said dominant over the 𝑘th row in 𝐻 (resp., equation) or that the 𝑘th row (resp., equation) is said dominated by the 𝑖th row (resp., equation).

In other terms, if the 𝑖th equation is dominant over the 𝑘th equation in (8), then the 𝑘th equation is a redundant equation of the system. By using Definition 5, we can build a matrix of dimension 𝑚×𝑛, called dominance matrix, with components:𝑖𝑗=0,ifthe𝑖thequationisdominatedbyanotherequation,𝑖𝑗,otherwise.(14)

For each 𝑖=1,,𝑚, now we set |𝐻𝑖| as the number of coefficients 𝑖𝑗=𝑏𝑖0 in the 𝑖th row of the dominance matrix 𝐻. When this value is 0, we set |𝐻𝑖|=1. Then the number of potential minimal solutions of the SFRE cannot exceed the value PN3=𝑚𝑖=1||𝐻𝑖||,(15) where PN3PN2PN1. In [6, 8, 20], the authors use the symbol 𝑏𝑖/𝑗 to indicate the coefficients 𝑖𝑗=𝑏𝑖0. We have 𝑖𝑗𝑥𝑗=𝑏𝑖 if 𝑥𝑗[𝑏𝑖,1] and 𝑥𝑗=𝑏𝑖 is the 𝑗th component of a minimal solution. A solution of the 𝑖th equation can be written as𝐻𝑖=𝑛𝑗=1𝑏𝑖𝑗.(16)

In [6, 8] the concept of concatenation 𝑊 is introduced to determine all the components of the minimal solutions and it is given by𝑊=𝑚𝑖=1𝐻𝑖=𝑚𝑖=1𝑛𝑗=1𝑏𝑖𝑗.(17)

The following properties hold:(i)commutativity: 𝑏𝑖1𝑗1𝑏𝑖2𝑗2=𝑏𝑖2𝑗2𝑏𝑖1𝑗1,(18)(ii)associativity: 𝑏𝑖1𝑗1𝑏𝑖2𝑗2𝑏𝑖3𝑗3=𝑏𝑖1𝑗1𝑏𝑖2𝑗2𝑏𝑖3𝑗3,(19)(iii)distributivity with respect to the addition: 𝑏𝑖1𝑗1𝑏𝑖2𝑗2+𝑏𝑖3𝑗3=𝑏𝑖1𝑗1𝑏𝑖2𝑗2+𝑏𝑖1𝑗1𝑏𝑖3𝑗3,(20)(iv)absorption for multiplication: 𝑏𝑖1𝑗1𝑏𝑖2𝑗2=𝑏𝑖1𝑏𝑖2𝑗,if𝑗1=𝑗2=𝑗,unchanged,otherwise,(21)(v)absorption for addition: 𝑏𝑖1𝑗1𝑏𝑖2𝑗2𝑏𝑖𝑚𝑗𝑛+𝑏𝑘1𝑗1𝑏𝑘2𝑗2𝑏𝑘𝑚𝑗𝑛=𝑏𝑖1𝑗1𝑏𝑖2𝑗2𝑏𝑖𝑚𝑗𝑛,if𝑏𝑖=𝑏𝑘,{1,,𝑚},unchanged,otherwise.(22)

We can determine the minimal solutions 𝑋low(𝑡)=(𝑥low1(𝑡),𝑥low2(𝑡),,𝑥low𝑛(𝑡))1, 𝑡{1,,PN(3)}, with components 𝑥low𝑗(𝑡)=𝑏𝑖𝑡,if𝑏𝑖𝑡0,0,otherwise.(23)

The above definitions shall be clarified in the following example of an SFRE with 4 equations and 6 unknown:1.0𝑥10.0𝑥20.0𝑥30.9𝑥40.2𝑥50.0𝑥6=0.1,0.5𝑥10.3𝑥20.4𝑥30.5𝑥40.3𝑥50.4𝑥6=0.3,0.7𝑥10.4𝑥20.2𝑥30.7𝑥40.4𝑥50.2𝑥6=0.3,0.4𝑥10.7𝑥20.2𝑥30.4𝑥40.7𝑥50.2𝑥6=0.3.(24)

We have𝐴=1.00.00.00.90.20.00.50.30.40.50.30.40.70.40.20.70.40.20.40.70.20.40.70.2,𝐵=0.10.30.30.3.(25)

By using the normal form, we obtain that 𝐴=0.50.30.40.50.30.40.70.40.20.70.40.20.40.70.20.40.70.21.00.00.00.90.20.0,𝐵=0.30.30.30.1.(26)

Now we compute the matrix 𝐴 and the vector IND as follows:𝐴=31131.00.31.01.00.31.01.01.00.01.01.00.01.01.00.01.01.00.01.00.00.01.01.00.0,IND=.(27)

The SFRE is consistent because each component of IND is not null. The greatest solution is given by𝑋gr=0.10.30.30.10.10.3.(28)

Now we calculate the help matrix 𝐻 and the dominant matrix 𝐻 as follows:,𝐻𝐻=0.00.30.30.00.00.30.00.30.00.00.00.00.00.30.00.00.00.00.10.00.00.10.10.0=.0.00.00.00.00.00.00.00.00.00.00.00.00.00.30.00.00.00.00.10.00.00.10.10.0(29)

Then we have |𝐻1|=|𝐻2|=|𝐻3|=1, |𝐻1|=3 and hence PN3=3. By using the properties (18)–(23), we have that 𝑊=0.320.11+0.14+0.15=0.110.32+0.320.14+0.320.15.(30)

The three minimal solutions are given by𝑋low(1)=0.10.30.00.00.00.0,𝑋low(2)=0.00.30.00.10.00.0,𝑋low(3)=0.00.30.00.00.10.0.(31) The three maximal interval solutions are given by𝑋max(1)=[][][][][][]0.1,0.10.3,0.30.0,0.30.0,0.10.0,0.10.0,0.3,𝑋max(2)=[][][][][][],𝑋0.0,0.10.3,0.30.0,0.30.1,0.10.0,0.10.0,0.3max(3)=[][][][][][].0.0,0.10.3,0.30.0,0.30.0,0.10.1,0.10.0,0.3(32)

In order to determine if an SFRE is consistent, hence its greatest solution and minimal solutions, we have used the universal algorithm of [6, 8] based on the above concepts. For brevity of presentation, here we do not give this algorithm which has been implemented and tested under C++ language. The C++ library has been integrated in the ESRI ArcObject Library of the tool ArcGIS 9.3 for a problem of spatial analysis illustrated in Section 3.

3. SFRE in Spatial Analysis

We consider a specific area of study on the geographical map on which we have a spatial data set of “causes” and we want to analyze the possible “symptoms”. We divide this area in P subzones (see, e.g., Figure 3), where a subzone is an area in which the same symptoms are derived by input data or facts, and the impact of a symptom on a cause is the same one as well. It is important to note that even if two subzones have the same input data, they can have different impact degrees of symptoms on the causes. For example, the cause that measures the occurrence of floods may be due to different degrees of importance to the presence of low porous soils or to areas subjected to continuous rains. Afterwards the area of study is divided in homogeneous subzones, hence the expert creates a fuzzy partition for the domain of each input variable and, for each subzone, he determines the values of the symptoms 𝑏𝑖, as the membership degrees of the corresponding fuzzy sets (cf. input fuzzification process of Figure 1). For each subzone, then the expert sets the most significant equations and the values 𝑎𝑖𝑗 of impact of the 𝑗th cause to the 𝑖th symptom creating the SFRE (1). After the determination of the set of maximal interval solutions by using the algorithm of Section 2, the expert for each interval solution calculates, for each unknown 𝑥𝑗, the mean interval solution 𝑋Max𝑀,𝑡(𝑗) with (6). The linguistic label Rel𝑡(𝑜𝑠) is assigned to the output variable 𝑜𝑠. Then he calculates the reliability index Rel𝑡(𝑂), given from formula (7), associated to this maximal interval solution 𝑡. After the iteration of this step, the expert determines the reliability index (7) for each maximal interval solution, by choosing the output vector 𝑂 for which Rel(𝑂) assumes the maximum value. Iterating the process for all the subzones, the expert can show the thematic map of each output variable. We schematize the whole process in Figure 4.

429498.fig.003
Figure 3: Subdivision in homogeneous subzones.
429498.fig.004
Figure 4: Flux diagram of the resolution problem.

We suppose to subdivide the area of study in P subzones. The steps of the process are described below.(i)In the spatial dataset, we associate 𝑘 facts 𝑖1,,𝑖 to every subzone.(ii)For each input fact, a fuzzy partition in 𝑚𝑓 fuzzy sets is created for every 𝑓=1,,. To each fuzzy set, the expert associates a linguistic label. After the fuzzification process, the expert determines the 𝑚 most significant equations, where 𝑚𝑚1+𝑚2++𝑚𝑘. The input vector 𝐵=[𝑏1,,𝑏𝑚] is set, where each component 𝑏𝑖 (𝑖=1,,𝑚) is the membership degree to the 𝑖th fuzzy set of the corresponding input fact. To create the fuzzy partitions, we use TFNs (cf. formulae (3), (4), (5)). The expert sets the impact of the 𝑚 symptoms to the 𝑛 causes by defining the impact matrix 𝐴 with entries 𝑎𝑖𝑗 with 𝑖=1,,𝑚, 𝑗=1,,𝑛.(iii)An SFRE (1) with 𝑚 equations and 𝑛 unknowns is created. We use the algorithm from [8] to determine all the solutions of (1). Thus we determine 𝑇 maximal interval solutions.(iv)maxRel𝑡=0// (the maximal reliability is initialized to 0).(v)For each maximal interval solution 𝑋max,𝑡, with 𝑡=1,,𝑇, we define the vector column 𝑋Mean𝑡 via formula (6).(vi)Rel𝑡=0.(vii)For each output variable 𝑜𝑠, with 𝑠=1,,𝑘, if 𝑥𝑗1,,𝑥𝑗𝑠 are the unknown associated to 𝑜𝑠, let score𝑡(𝑠)=max{𝑋Mean𝑡(𝑗1),,𝑋Mean𝑡(𝑗𝑠)}.(viii)Rel𝑡=Rel𝑡+score𝑡(𝑜𝑠).(ix)Next 𝑠.(x)Rel𝑡=Rel𝑡/𝑘// (the reliability index is calculated via formula (7)).(xi)If Relt > maxRelt, then the linguistic label of the fuzzy set corresponding to the unknown with maximum mean solution is assigned to the output vector 𝑂=[𝑜1,,𝑜𝑘].(xii)Next 𝑡 with 𝑡=1,,𝑇.(xiii)Next 𝑝 with 𝑝=1,,𝑃.

At the end of the process, the user can create a thematic map of a specific output variable over the area of study and also a thematic map of the reliability index value obtained for the output variable. If the SFRE related to a specific subzone is inconsistent, the expert can decide whether or not eliminate rows to find solutions: in the first case, he decides that the symptoms associated to the rows that make the system inconsistent are not considered and eliminates them, so reducing the number of the equations. In the second case, he decides that the correspondent output variable for this subzone remains unknown and it is classified as unknown on the map.

4. Simulation Results

Here we show the results of an experiment in which we apply our method to census statistical data agglomerated on four districts of the east zone of Naples (Italy) (Figure 5). We use the year 2000 census data provided by the ISTAT (Istituto Nazionale di Statistica). These data contain information on population, buildings, housing, family, employment work for each census zone of Naples. Every district is considered as a subzone with homogeneous input data given in Table 4.

429498.fig.005
Figure 5: Area of study: four districts at east of Naples (Italy).

In this experiment, we consider the following four output variables: “𝑜1= Economic prosperity” (wealth and prosperity of citizens), “𝑜2= Transition into the job” (ease of finding work), “𝑜3= Social Environment” (cultural levels of citizens), and “𝑜4= Housing development” (presence of building and residential dwellings of new construction). For each variable, we create a fuzzy partition composed by three TFNs called “low”, “mean”, and “high” presented in Table 3.

tab3
Table 3: Values of the TFNs low, mean, high.
tab4
Table 4: Input data obtained for the four subzones.

Moreover, we consider the following seven input parameters: 𝑖1 = percentage of people employed = number of people employed/total work force, 𝑖2 = percentage of women employed = number of women employed/number of people employed, 𝑖3 = percentage of entrepreneurs and professionals = number of entrepreneurs and professionals/number of people employed, 𝑖4 = percentage of residents graduated = numbers of residents graduated/number of residents with age >6 years, 𝑖5 = percentage of new residential buildings = number of residential buildings built since 1982/total number of residential buildings, 𝑖6 = percentage of residential dwellings owned = number of residential dwellings owned/total number of residential dwellings, and 𝑖7 = percentage of residential dwellings with central heating system = number of residential dwellings with central heating system/total number of residential dwellings. In Table 4, we show these input data for the four subzones.

For the fuzzification process of the input data, the expert indicates a fuzzy partition for each input domain formed from three TFNs labeled “low”, “mean”, and “high”, whose values are reported in Table 5. In Tables 6 and 7, we show the values obtained for the 21 symptoms 𝑏1,,𝑏21; moreover, we report the input variable and the linguistic label of the correspondent TFN for each symptom 𝑏𝑖. In order to form the SFRE (1) in each subzone, the expert defines the equations by setting the impact values 𝑎𝑖𝑗 by basing over the most significant symptoms.

tab5
Table 5: TFNs values for the input domains.
tab6
Table 6: TFNs for the symptoms 𝑏1 ÷ 𝑏12.
tab7
Table 7: TFNs for the symptoms 𝑏13 ÷ 𝑏21.

Now we illustrate this procedure for each subzone.

4.1. Subzone “Barra”

The expert chooses the significant symptoms 𝑏2, 𝑏4, 𝑏5, 𝑏7, 𝑏10, 𝑏11, 𝑏15, 𝑏17, 𝑏18, 𝑏19, by obtaining an SFRE (1) with 𝑚=10 equations and 𝑛=12 unknowns (Table 8).

tab8
Table 8: Final linguistic labels for the output variables in the district Barra.

The matrix 𝐴 of the impact values 𝑎𝑖𝑗 has dimensions 10×12 and the vector 𝐵 of the symptoms 𝑏𝑖 has dimension 10×1 and both are given below. The SFRE (1) is inconsistent and eliminating the rows for which the value IND(𝑗) = 0, we obtain four maximal interval solutions 𝑋max(𝑡) (𝑡=1,,4) and we calculate the vector column 𝑋Mean𝑡 on each maximal interval solution. Hence we associate to the output variable 𝑜𝑠 (𝑠=1,,4), the linguistic label of the fuzzy set with the higher value calculated with formula (6) obtained for the corresponding unknowns 𝑥𝑗1,,𝑥𝑗𝑠 and given in Table 8. For determining the reliability of our solutions, we use the index given by formula (7). We obtain that Rel𝑡(𝑜1)=Rel𝑡(𝑜2)=Rel𝑡(𝑜3)=Rel𝑡(𝑜4)=0.6025 for 𝑡=1,,4 and hence Rel(𝑂)=max{Rel𝑡(𝑂)𝑡=1,,4}=0.6025 where 𝑂={𝑜1,𝑜4}. We note that the same final set of linguistic labels associated to the output variables 𝑜1 = “high”, 𝑜2 = “mean”, 𝑜3 = “low”, and 𝑜4 = “low” is obtained as well. The relevant quantities are given below. ,,𝐴=0.51.00.00.41.00.20.20.70.30.10.30.20.30.50.20.40.50.40.30.60.20.00.00.00.20.70.20.20.70.20.20.70.20.00.00.01.00.20.00.80.30.10.80.20.20.30.00.00.50.30.10.60.40.10.60.40.10.10.00.00.30.70.30.30.70.30.20.70.30.10.20.10.10.10.10.10.20.10.20.10.10.10.30.30.20.50.20.10.40.10.20.50.10.30.70.30.10.40.40.10.40.40.10.50.50.20.40.50.50.20.00.40.30.00.40.30.01.00.10.0𝐵=0.980.360.631.000.400.600.100.590.411.00(33)𝑋max(1)=[][][][][][][][][][][][]0.40,0.400.36,0.360.00,1.000.00,0.360.00,1.000.00,0.360.00,1.000.00,0.360.41,0.411.00,1.000.00,0.100.00,0.10,𝑋max(2)=[][][][][][][][][][][][],𝑋0.40,0.400.00,0.360.00,1.000.36,0.360.00,1.000.00,0.360.00,1.000.00,0.360.41,0.411.00,1.000.00,0.100.00,0.10max(3)=[][][][][][][][][][][][]0.40,0.400.00,0.360.00,1.000.00,0.360.00,1.000.36,0.360.00,1.000.00,0.360.41,0.411.00,1.000.00,0.100.00,0.10,𝑋max(4)=[][][][][][][][][][][][],0.40,0.400.00,0.360.00,1.000.36,0.360.00,1.000.00,0.360.00,1.000.36,0.360.41,0.411.00,1.000.00,0.100.00,0.10𝑋Mean1=0.400.360.500.180.500.180.500.180.411.000.050.05,𝑋Mean2=,0.400.180.500.360.500.180.500.180.411.000.050.05𝑋Mean3=0.400.180.500.180.500.360.500.180.181.000.050.05,𝑋Mean4=.0.400.180.050.360.500.180.500.360.411.000.050.05(34)

4.2. Subzone “Poggioreale”

The expert chooses the significant symptoms 𝑏2, 𝑏5, 𝑏8, 𝑏11, 𝑏12, 𝑏14, 𝑏15, 𝑏17, 𝑏18, 𝑏19, 𝑏20, by obtaining an SFRE (1) with 𝑚=11 equations and 𝑛=12 unknowns (Table 9). The matrix 𝐴 of the impact values 𝑎𝑖𝑗 has dimension 11×12 and the vector 𝐵 of the symptoms 𝑏𝑖 has dimension 11×1 which are given below. The SFRE (1) is inconsistent and eliminating the rows for which the value IND(j) = 0, we obtain 12 maximal interval solutions 𝑋max(𝑡) (𝑡=1,,12), and we calculate the vector column 𝑋Mean𝑡 on each maximal interval solution. The relevant quantities are given below. ,,𝐴=0.51.00.00.41.00.20.20.70.30.10.30.20.21.00.20.21.00.20.20.90.20.00.00.00.21.00.20.21.00.20.21.00.20.00.00.00.30.70.30.30.70.30.20.70.30.10.20.20.40.50.60.30.50.60.30.50.60.00.00.10.30.70.30.30.70.30.20.70.30.10.20.10.20.40.60.30.40.60.20.40.60.00.10.20.10.90.10.10.90.10.20.80.20.20.80.20.00.10.50.10.20.50.10.20.50.00.10.40.40.10.00.80.50.30.50.30.10.70.30.00.10.20.10.10.20.10.10.20.10.30.60.2𝐵=0.930.991.00.630.370.70.30.870.130.750.25(35)𝑋max(1)=[][][][][][][][][][][][]0.37,0.370.0,0.30.13,0.130.75,0.750.0,0.130.0,0.130.0,1.00.0,0.130.0,0.130.25,0.250.0,0.250.0,0.13,𝑋max(2)=[][][][][][][][][][][][],𝑋0.37,0.370.0,0.30.13,0.130.75,0.750.0,0.130.0,0.130.0,1.00.0,0.130.0,0.130.0,0.250.25,0.250.0,0.13max(3)=[][][][][][][][][][][][]0.37,0.370.0,0.30.0,0.130.75,0.750.13,0.130.0,0.130.0,1.00.0,0.130.0,0.130.25,0.250.0,0.250.0,0.13,𝑋max(4)=[][][][][][][][][][][][],𝑋0.37,0.370.0,0.30.0,0.130.75,0.750.13,0.130.0,0.130.0,1.00.0,0.130.0,0.130.0,0.250.25,0.250.0,0.13max(5)=[][][][][][][][][][][][]0.37,0.370.0,0.30.0,0.130.75,0.750.13,0.130.0,0.130.0,1.00.0,0.130.0,0.130.25,0.250.0,0.250.0,0.13,𝑋max(6)=[][][][][][][][][][][][],𝑋0.37,0.370.0,0.30.0,0.130.75,0.750.13,0.130.0,0.130.0,1.00.0,0.130.0,0.130.0,0.250.25,0.250.0,0.13max(7)=[][][][][][][][][][][][]0.37,0.370.0,0.30.0,0.130.75,0.750.0,0.130.0,0.130.0,1.00.13,0.130.0,0.130.25,0.250.0,0.250.0,0.13,𝑋max(8)=[][][][][][][][][][][][],𝑋0.37,0.370.0,0.30.0,0.130.75,0.750.0,0.130.0,0.130.0,1.00.13,0.130.0,0.130.0,0.250.25,0.250.0,0.13max(9)=[][][][][][][][][][][][]0.37,0.370.0,0.30.0,0.130.75,0.750.0,0.130.0,0.130.0,1.00.0,0.130.13,0.130.25,0.250.0,0.250.0,0.13,𝑋max(10)=[][][][][][][][][][][][],𝑋0.37,0.370.0,0.30.0,0.130.75,0.750.0,0.130.0,0.130.0,1.00.0,0.130.13,0.130.0,0.250.25,0.250.0,0.13max(11)=[][][][][][][][][][][][]0.37,0.370.0,0.30.0,0.130.75,0.750.0,0.130.0,0.130.0,1.00.0,0.130.0,0.130.25,0.250.0,0.250.13,0.13,𝑋max(12)=[][][][][][][][][][][][],0.37,0.370.0,0.30.0,0.130.75,0.750.0,0.130.0,0.130.0,1.00.0,0.130.0,0.130.0,0.250.25,0.250.13,0.13𝑋Mean1=0.370.150.130.750.0650.0650.50.0650.0650.250.1250.05,𝑋Mean2=,0.370.150.130.750.0650.0650.50.0650.0650.1250.250.065𝑋Mean3=0.370.150.0650.750.130.0650.50.0650.0650.250.1250.065,𝑋Mean4=0.370.150.0650.750.130.0650.50.0650.0650.1250.250.065𝑋Mean5=0.370.150.0650.750.0650.130.50.0650.0650.250.1250.05,𝑋Mean6=,0.370.150.0650.750.0650.130.50.0650.0650.1250.250.05𝑋Mean7=0.370.150.0650.750.0650.0650.50.130.0650.250.1250.065,𝑋Mean8=,0.370.150.0650.750.0650.0650.50.130.0650.1250.250.065𝑋Mean9=0.370.150.0650.750.0650.0650.50.0650.130.250.1250.05,𝑋Mean10=,0.370.150.0650.750.0650.0650.50.0650.130.1250.250.05𝑋Mean11=0.370.150.0650.750.0650.0650.50.0650.0650.250.1250.13,𝑋Mean12=.0.370.150.0650.750.0650.0650.50.0650.0650.1250.250.13(36) For determining the reliability of our solutions, we use the index given by formula (7). We obtain Rel(𝑂𝑘) = 0.4675 for 𝑘=1,,12. Then we obtain two final sets of linguistic labels associated to the output variables: 𝑜1 = “low”, 𝑜2 = “low”, 𝑜3 = “low”, 𝑜4 = “low”, and 𝑜1 = “low”, 𝑜2 = “low”, 𝑜3 = “low”, 𝑜4 = “mean”, with a same reliability index value 0.4675. The expert prefers to choose the second solution: 𝑜1 = “low”, 𝑜2 = “low”, 𝑜3 = “low”, 𝑜4 = “mean” because he considers that in the last two years in this district the presence of building and residential dwellings of new construction has increased although marginally. We obtain four final thematic maps shown in Figures 6, 7, 8, 9 for the output variable 𝑜1, 𝑜2, 𝑜3, 𝑜4, respectively.

tab9
Table 9: Final linguistic labels for the output variables in the district Poggioreale.
429498.fig.006
Figure 6: Thematic map for output variable 𝑜1 (Economic prosperity).
429498.fig.007
Figure 7: Thematic map of the output variable 𝑜2 (Transition into the job).
429498.fig.008
Figure 8: Thematic map for the output variable 𝑜3 (Social Environment).
429498.fig.009
Figure 9: Thematic map for the output variable 𝑜4 (Housing development).

The results show that there was no housing development in the four districts in the last 10 years, and there is difficulty in finding job positions. In Figure 10, we show the histogram of the reliability index Rel(𝑂) for each subzone, where 𝑂=[𝑜1,𝑜2,𝑜3,𝑜4].

429498.fig.0010
Figure 10: Histogram of the reliability index Rel(𝑂) for the four subzones.

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