# Spatial Analysis and Fuzzy Relation Equations

Authors: Ferdinando Di Martino; Salvatore Sessa
Journal: Advances in Fuzzy Systems
 (2011)

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## 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.


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## Body

## 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 [1–5] 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 (1)(a11∧x1)∨⋯∨(a1n∧xn)=b1,(a21∧x1)∨⋯∨(a2n∧xn)=b2,⋮(am1∧x1)∨⋯∨(amn∧xn)=bm.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., [11–18]).  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 datao1,…,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.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:(2)(0.8∧x1)∨(0.2∧x2)∨(0.0∧x3)∨(0.8∧x4)∨(0.3∧x5)∨(0.0∧x6)=b3=0.9.
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”.Examples of trapezoidal fuzzy numbers used for symptoms and causes.
(a)(b)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,…,μjh as follows:(3)μj1={1,ifINF(j1)≤x≤MEAN(j1),SUP(j1)-xSUP(j1)-MEAN(j1),ifMEAN(j1)<x≤SUP(j1),0,otherwise,(4)μj={x-INF(j)MEAN(j)-INF(j),ifINF(j)≤x≤MEAN(j),SUP(j)-xSUP(j)-MEAN(j),ifMEAN(j)<x≤SUP(j),0,otherwise,j∈{j2,…,js-1},(5)μjs={x-INF(js)MEAN(js)-INF(js),ifINF(js)≤x≤MEAN(js),1,ifMEAN(js)<x≤SUP(js),0,otherwise.Table 1
TFNs  values for the fuzzy sets.
UnknownINF(j)MEAN(j)SUP(j)x10.00.20.4x20.30.50.7x30.60.81.0x40.00.20.4x50.30.50.7x60.60.80.1IfXMint(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) as(6)XMeant(j)=XMin⁡t(j)+XMax⁡t(j)2,
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 as(7)Relt(O)=1k⋅∑s=1kscoret(os)
and then as  final reliability index of O,  the number Rel(O)=max{Relt(O):t=1,…,T}.Table 2
TFNs mean values.
Output variableUnknown componentLinguistic labelXMint(j)XMaxt(j)XMeant(j)x1scanty0.60.80.70o1x2medium0.20.40.30x3good0.00.10.05x4low0.30.50.40o2x5medium0.40.70.55x6good0.00.30.15In our example, we haveRelt(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:(8)A∘X=B,
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≠Ø,  then the SFRE (8) is called consistent, otherwise it is called inconsistent. A solution X̂∈Ω is called a minimal solution if X≤X̂ for some X∈Ω implies X=X̂, 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=(x1gr,x2gr,…,xngr)-1 if Ω≠Ø [10]. A matrix interval Xinterval of the following type:(9)Xinterval=([a1,b1][a2,b2][…,…][an,bn]),
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 b1≥b2≥⋯≥bm. 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:(10)aij*={0,ifaij<bi,bi,ifaij=bi,1,ifaij>bi,
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 A*∘X=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 aij*≥bi=bk,(ii)
aij*  for  i∈{k+1,…,m} with aij*=bi.Definition 2.
IfA*(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(m·n).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 product(11)PN2=∏i=1mIND(i),
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 solutionXgr   with component  xjgr=bk if the  jth column  A*(j) of A* contains selected G-type coefficients akj* and xjgr=1 otherwise.(ii)
The complexity time function for computingXgr is O(m·n).A help matrixH=(hij), with  i=1,…,m and j=1,…,n, is defined as follows:(12)hij={bi,ifaij*isselected,0,otherwise.Let|Hi| be the number of coefficients hij in the ith equation of the SFRE (8).  Then the number of potential minimal solutions cannot exceed the value(13)PN1=∏i=1m|Hi|,
where PN2≤PN1.Definition 5.
Lethi=(hi1,hi2,…,hin) and hk=(hk1,hk2,…,hkn) be the ith and the kth rows of the help matrix H. If for each j=1,…n,  hij≠0 implies both hkj≠0 and hkj≤hij, 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 theith 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:(14)hij*={0,iftheithequationisdominatedbyanotherequation,hij,otherwise.For eachi=1,…,m, now we set |Hi*| as the number of coefficients hij*=bi≠0 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 (15)PN3=∏i=1m|Hi*|,
where PN3≤PN2≤PN1. In [6, 8, 20], the authors use the symbol 〈bi/j〉 to indicate the coefficients hij*=bi≠0. We have hij*∧xj=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 as(16)Hi=∑j=1n〈bij〉.In [6, 8] the concept of concatenation W is introduced to determine all the components of the minimal solutions and it  is given by(17)W=∏i=1mHi=∏i=1m(∑j=1n〈bij〉).The following properties hold:(i)
commutativity:(18)〈bi1j1〉〈bi2j2〉=〈bi2j2〉〈bi1j1〉,(ii)
associativity:(19)〈bi1j1〉(〈bi2j2〉〈bi3j3〉)=(〈bi1j1〉〈bi2j2〉)〈bi3j3〉,(iii)
distributivity with respect to the addition:(20)〈bi1j1〉(〈bi2j2〉+〈bi3j3〉)=〈bi1j1〉〈bi2j2〉+〈bi1j1〉〈bi3j3〉,(iv)
absorption for multiplication:(21)〈bi1j1〉〈bi2j2〉={〈bi1∧bi2j〉,ifj1=j2=j,unchanged,otherwise,(v)
absorption for addition:(22)〈bi1j1〉〈bi2j2〉⋯〈bimjn〉+〈bk1j1〉〈bk2j2〉⋯〈bkmjn〉={〈bi1j1〉〈bi2j2〉⋯〈bimjn〉,ifbih=bkh,h∈{1,…,m},unchanged,otherwise.We can determine the minimal solutionsXlow(t)=(x1low(t),x2low(t),…,xnlow(t))-1, t∈{1,…,PN(3)}, with components (23)xjlow(t)={bit,ifbit≠0,0,otherwise.The above definitions shall be clarified in the following example of an SFRE with 4 equations and 6 unknown:(24)(1.0∧x1)∨(0.0∧x2)∨(0.0∧x3)∨(0.9∧x4)∨(0.2∧x5)∨(0.0∧x6)=0.1,(0.5∧x1)∨(0.3∧x2)∨(0.4∧x3)∨(0.5∧x4)∨(0.3∧x5)∨(0.4∧x6)=0.3,(0.7∧x1)∨(0.4∧x2)∨(0.2∧x3)∨(0.7∧x4)∨(0.4∧x5)∨(0.2∧x6)=0.3,(0.4∧x1)∨(0.7∧x2)∨(0.2∧x3)∨(0.4∧x4)∨(0.7∧x5)∨(0.2∧x6)=0.3.We have(25)A=(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).By using the normal form, we obtain that(26)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).Now we compute the matrixA* and the  vector IND as follows:(27)A*=(1.00.31.01.00.31.01.01.00.01.01.00.01.01.00.01.01.00.01.00.00.01.01.00.0),IND=(3113).The SFRE is consistent because each component of IND is not null. The greatest solution is given by(28)Xgr=(0.10.30.30.10.10.3).Now we calculate the help matrixH and the dominant matrix H* as follows:(29)H=(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),H*=(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).Then we have|H1*|=|H2*|=|H3*|=1, |H1*|=3 and hence PN3=3. By using the properties (18)–(23), we have that (30)W=〈0.32〉(〈0.11〉+〈0.14〉+〈0.15〉)=〈0.11〉〈0.32〉+〈0.32〉〈0.14〉+〈0.32〉〈0.15〉.The three minimal solutions are given by(31)Xlow(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).
The three maximal interval solutions are given by(32)Xmax⁡⁡(1)=([0.1,0.1][0.3,0.3][0.0,0.3][0.0,0.1][0.0,0.1][0.0,0.3]),Xmax⁡⁡(2)=([0.0,0.1][0.3,0.3][0.0,0.3][0.1,0.1][0.0,0.1][0.0,0.3]),Xmax⁡⁡(3)=([0.0,0.1][0.3,0.3][0.0,0.3][0.0,0.1][0.1,0.1][0.0,0.3]).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., Figure3), 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.Figure 3
Subdivision in homogeneous subzones.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 associatek facts i1,…,ih to every subzone.(ii)
For each input fact, a fuzzy partition inmf fuzzy sets is created for every f=1,…,h. To each fuzzy set, the expert associates a linguistic label. After the fuzzification process, the expert determines the m most significant equations, where m≤m1+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 solutionXmax,t, with t=1,…,T, we define the vector column XMeant  via  formula (6).(vi)
Relt∶=0.(vii)
For each output variableos, 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)
Nexts.(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)
Nextt  with  t=1,…,T.(xiii)
Nextp  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) (Figure5). 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.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.Table 3
Values of  the TFNs  low, mean, high.
OutputLowMeanHighINFMEANSUPINFMEANSUPINFMEANSUPo10.00.30.50.30.50.80.50.81.0o20.00.30.50.30.50.80.50.81.0o30.00.30.50.30.50.80.50.81.0o40.00.30.50.30.50.80.50.81.0Table 4
Input data obtained for the four subzones.
Districti1i2i3i4i5i6i7Barra0.6040.2270.0390.0320.1110.4240.067Poggioreale0.6640.2970.0600.0510.0860.3380.149Ponticelli0.6090.2530.0390.0420.1560.3720.159S. Giovanni0.5760.2440.0410.0310.0540.3530.097Moreover, 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 Table5.  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.Table 5
TFNs values for the input domains.
Input variableLowMeanHighINFMEANSUPINFMEANSUPINFMEANSUPi10.000.400.600.400.600.800.600.801.00i20.000.100.300.100.300.400.300.501.00i30.000.040.060.040.060.100.070.201.00i40.000.020.040.020.040.070.040.071.00i50.000.050.080.050.080.100.080.101.00i60.000.100.300.100.300.600.300.601.00i70.000.100.300.100.300.500.300.501.00Table 6
TFNs for the symptomsb1 ÷ b12.
Subzoneb1: i1 = lowb2: i1= meanb3: i1 = highb4: i2 = lowb5: i2 = meanb6: i2 = highb7: i3 = lowb8: i3 = meanb9: i3 = highb10: i4 = lowb11: i4 = meanb12: i4 = highBarra0.000.980.020.360.630.001.000.000.000.400.600.00Poggioreale0.000.930.070.010.990.000.001.000.000.000.630.37Ponticelli0.000.910.050.230.760.001.000.000.000.000.930.07S. Giovanni0.120.880.000.280.720.000.950.050.000.450.550.00Table 7
TFNs for the symptomsb13 ÷ b21.
Subzoneb13: i5 = lowb14: i5 = meanb15: i5 =highb16: i6 = lowb17: i6 = meanb18: i6 = highb19: i7 = lowb20: i7 = meanb21: i7 = highBarra0.000.000.100.000.590.411.000.000.00Poggioreale0.000.700.300.000.870.130.750.250.00Ponticelli0.000.001.000.000.760.240.700.300.00S. Giovanni0.870.130.000.000.820.181.000.000.00Now we illustrate this procedure for each subzone.


### 4.1. Subzone “Barra”

The expert chooses the significant symptomsb2, b4, b5, b7, b10, b11, b15, b17, b18, b19, by obtaining an SFRE (1) with m=10 equations and n=12 unknowns (Table 8).Table 8
Final linguistic labels for the output variables in the district Barra.
Output variableScore1(os)Score2(os)Score3(os)Score4(os)o1highhighhighhigho2meanmeanmeanmeano3lowlowlowlowo4lowlowlowlowThe matrixA 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. (33)A=(0.51.00.00.41.00.20.20.70.30.10.30.20.30.50.20.40.50.40.30.60.20.00.00.00.20.70.20.20.70.20.20.70.20.00.00.01.00.20.00.80.30.10.80.20.20.30.00.00.50.30.10.60.40.10.60.40.10.10.00.00.30.70.30.30.70.30.20.70.30.10.20.10.10.10.10.10.20.10.20.10.10.10.30.30.20.50.20.10.40.10.20.50.10.30.70.30.10.40.40.10.40.40.10.50.50.20.40.50.50.20.00.40.30.00.40.30.01.00.10.0),B=(0.980.360.631.000.400.600.100.590.411.00),(34)Xmax⁡⁡(1)=([0.40,0.40][0.36,0.36][0.00,1.00][0.00,0.36][0.00,1.00][0.00,0.36][0.00,1.00][0.00,0.36][0.41,0.41][1.00,1.00][0.00,0.10][0.00,0.10]),Xmax⁡⁡(2)=([0.40,0.40][0.00,0.36][0.00,1.00][0.36,0.36][0.00,1.00][0.00,0.36][0.00,1.00][0.00,0.36][0.41,0.41][1.00,1.00][0.00,0.10][0.00,0.10]),Xmax⁡⁡(3)=([0.40,0.40][0.00,0.36][0.00,1.00][0.00,0.36][0.00,1.00][0.36,0.36][0.00,1.00][0.00,0.36][0.41,0.41][1.00,1.00][0.00,0.10][0.00,0.10]),Xmax⁡⁡(4)=([0.40,0.40][0.00,0.36][0.00,1.00][0.36,0.36][0.00,1.00][0.00,0.36][0.00,1.00][0.36,0.36][0.41,0.41][1.00,1.00][0.00,0.10][0.00,0.10]),XMean1=(0.400.360.500.180.500.180.500.180.411.000.050.05),XMean2=(0.400.180.500.360.500.180.500.180.411.000.050.05),XMean3=(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).


### 4.2. Subzone “Poggioreale”

The expert chooses the significant symptomsb2, 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. 								(35)A=(0.51.00.00.41.00.20.20.70.30.10.30.20.21.00.20.21.00.20.20.90.20.00.00.00.21.00.20.21.00.20.21.00.20.00.00.00.30.70.30.30.70.30.20.70.30.10.20.20.40.50.60.30.50.60.30.50.60.00.00.10.30.70.30.30.70.30.20.70.30.10.20.10.20.40.60.30.40.60.20.40.60.00.10.20.10.90.10.10.90.10.20.80.20.20.80.20.00.10.50.10.20.50.10.20.50.00.10.40.40.10.00.80.50.30.50.30.10.70.30.00.10.20.10.10.20.10.10.20.10.30.60.2),B=(0.930.991.00.630.370.70.30.870.130.750.25),(36)Xmax⁡⁡(1)=([0.37,0.37][0.0,0.3][0.13,0.13][0.75,0.75][0.0,0.13][0.0,0.13][0.0,1.0][0.0,0.13][0.0,0.13][0.25,0.25][0.0,0.25][0.0,0.13]),Xmax⁡⁡(2)=([0.37,0.37][0.0,0.3][0.13,0.13][0.75,0.75][0.0,0.13][0.0,0.13][0.0,1.0][0.0,0.13][0.0,0.13][0.0,0.25][0.25,0.25][0.0,0.13]),Xmax⁡⁡(3)=([0.37,0.37][0.0,0.3][0.0,0.13][0.75,0.75][0.13,0.13][0.0,0.13][0.0,1.0][0.0,0.13][0.0,0.13][0.25,0.25][0.0,0.25][0.0,0.13]),Xmax⁡⁡(4)=([0.37,0.37][0.0,0.3][0.0,0.13][0.75,0.75][0.13,0.13][0.0,0.13][0.0,1.0][0.0,0.13][0.0,0.13][0.0,0.25][0.25,0.25][0.0,0.13]),Xmax⁡⁡(5)=([0.37,0.37][0.0,0.3][0.0,0.13][0.75,0.75][0.13,0.13][0.0,0.13][0.0,1.0][0.0,0.13][0.0,0.13][0.25,0.25][0.0,0.25][0.0,0.13]),Xmax⁡⁡(6)=([0.37,0.37][0.0,0.3][0.0,0.13][0.75,0.75][0.13,0.13][0.0,0.13][0.0,1.0][0.0,0.13][0.0,0.13][0.0,0.25][0.25,0.25][0.0,0.13]),Xmax⁡⁡(7)=([0.37,0.37][0.0,0.3][0.0,0.13][0.75,0.75][0.0,0.13][0.0,0.13][0.0,1.0][0.13,0.13][0.0,0.13][0.25,0.25][0.0,0.25][0.0,0.13]),Xmax⁡⁡(8)=([0.37,0.37][0.0,0.3][0.0,0.13][0.75,0.75][0.0,0.13][0.0,0.13][0.0,1.0][0.13,0.13][0.0,0.13][0.0,0.25][0.25,0.25][0.0,0.13]),Xmax⁡⁡(9)=([0.37,0.37][0.0,0.3][0.0,0.13][0.75,0.75][0.0,0.13][0.0,0.13][0.0,1.0][0.0,0.13][0.13,0.13][0.25,0.25][0.0,0.25][0.0,0.13]),Xmax⁡⁡(10)=([0.37,0.37][0.0,0.3][0.0,0.13][0.75,0.75][0.0,0.13][0.0,0.13][0.0,1.0][0.0,0.13][0.13,0.13][0.0,0.25][0.25,0.25][0.0,0.13]),Xmax⁡⁡(11)=([0.37,0.37][0.0,0.3][0.0,0.13][0.75,0.75][0.0,0.13][0.0,0.13][0.0,1.0][0.0,0.13][0.0,0.13][0.25,0.25][0.0,0.25][0.13,0.13]),Xmax⁡⁡(12)=([0.37,0.37][0.0,0.3][0.0,0.13][0.75,0.75][0.0,0.13][0.0,0.13][0.0,1.0][0.0,0.13][0.0,0.13][0.0,0.25][0.25,0.25][0.13,0.13]),XMean1=(0.370.150.130.750.0650.0650.50.0650.0650.250.1250.05),XMean2=(0.370.150.130.750.0650.0650.50.0650.0650.1250.250.065),XMean3=(0.370.150.0650.750.130.0650.50.0650.0650.250.1250.065),XMean4=(0.370.150.0650.750.130.0650.50.0650.0650.1250.250.065)XMean5=(0.370.150.0650.750.0650.130.50.0650.0650.250.1250.05),XMean6=(0.370.150.0650.750.0650.130.50.0650.0650.1250.250.05),XMean7=(0.370.150.0650.750.0650.0650.50.130.0650.250.1250.065),XMean8=(0.370.150.0650.750.0650.0650.50.130.0650.1250.250.065),XMean9=(0.370.150.0650.750.0650.0650.50.0650.130.250.1250.05),XMean10=(0.370.150.0650.750.0650.0650.50.0650.130.1250.250.05),XMean11=(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).
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.Table 9
Final linguistic labels for the output variables in the district Poggioreale.
OutputScore1 (os)Score2 (os)Score3 (os)Score4 (os)Score5 (os)Score6 (os)Score7 (os)Score8 (os)Score9 (os)Score10 (os)Score11 (os)Score12 (os)o1lowlowlowhighlowlowlowhighlowlowlowhigho2lowlowlowmeanlowlowlowmeanlowlowlowmeano3lowlowlowlowlowlowlowlowlowlowlowlowo4lowmeanlowmeanlowmeanlowmeanlowmeanlowmeanFigure 6
Thematic map for output variableo1 (Economic prosperity).Figure 7
Thematic map of the output variableo2  (Transition into the  job).Figure 8
Thematic map for the output variableo3 (Social Environment).Figure 9
Thematic map for the output variableo4 (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 Figure10, we show the histogram of the reliability index Rel(O) for each subzone, where O=[o1,o2,o3,o4].Figure 10
Histogram of the reliability index Rel(O) for the four subzones.


## 4.1. Subzone “Barra”

The expert chooses the significant symptomsb2, b4, b5, b7, b10, b11, b15, b17, b18, b19, by obtaining an SFRE (1) with m=10 equations and n=12 unknowns (Table 8).Table 8
Final linguistic labels for the output variables in the district Barra.
Output variableScore1(os)Score2(os)Score3(os)Score4(os)o1highhighhighhigho2meanmeanmeanmeano3lowlowlowlowo4lowlowlowlowThe matrixA 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. (33)A=(0.51.00.00.41.00.20.20.70.30.10.30.20.30.50.20.40.50.40.30.60.20.00.00.00.20.70.20.20.70.20.20.70.20.00.00.01.00.20.00.80.30.10.80.20.20.30.00.00.50.30.10.60.40.10.60.40.10.10.00.00.30.70.30.30.70.30.20.70.30.10.20.10.10.10.10.10.20.10.20.10.10.10.30.30.20.50.20.10.40.10.20.50.10.30.70.30.10.40.40.10.40.40.10.50.50.20.40.50.50.20.00.40.30.00.40.30.01.00.10.0),B=(0.980.360.631.000.400.600.100.590.411.00),(34)Xmax⁡⁡(1)=([0.40,0.40][0.36,0.36][0.00,1.00][0.00,0.36][0.00,1.00][0.00,0.36][0.00,1.00][0.00,0.36][0.41,0.41][1.00,1.00][0.00,0.10][0.00,0.10]),Xmax⁡⁡(2)=([0.40,0.40][0.00,0.36][0.00,1.00][0.36,0.36][0.00,1.00][0.00,0.36][0.00,1.00][0.00,0.36][0.41,0.41][1.00,1.00][0.00,0.10][0.00,0.10]),Xmax⁡⁡(3)=([0.40,0.40][0.00,0.36][0.00,1.00][0.00,0.36][0.00,1.00][0.36,0.36][0.00,1.00][0.00,0.36][0.41,0.41][1.00,1.00][0.00,0.10][0.00,0.10]),Xmax⁡⁡(4)=([0.40,0.40][0.00,0.36][0.00,1.00][0.36,0.36][0.00,1.00][0.00,0.36][0.00,1.00][0.36,0.36][0.41,0.41][1.00,1.00][0.00,0.10][0.00,0.10]),XMean1=(0.400.360.500.180.500.180.500.180.411.000.050.05),XMean2=(0.400.180.500.360.500.180.500.180.411.000.050.05),XMean3=(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).


## 4.2. Subzone “Poggioreale”

The expert chooses the significant symptomsb2, 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. 								(35)A=(0.51.00.00.41.00.20.20.70.30.10.30.20.21.00.20.21.00.20.20.90.20.00.00.00.21.00.20.21.00.20.21.00.20.00.00.00.30.70.30.30.70.30.20.70.30.10.20.20.40.50.60.30.50.60.30.50.60.00.00.10.30.70.30.30.70.30.20.70.30.10.20.10.20.40.60.30.40.60.20.40.60.00.10.20.10.90.10.10.90.10.20.80.20.20.80.20.00.10.50.10.20.50.10.20.50.00.10.40.40.10.00.80.50.30.50.30.10.70.30.00.10.20.10.10.20.10.10.20.10.30.60.2),B=(0.930.991.00.630.370.70.30.870.130.750.25),(36)Xmax⁡⁡(1)=([0.37,0.37][0.0,0.3][0.13,0.13][0.75,0.75][0.0,0.13][0.0,0.13][0.0,1.0][0.0,0.13][0.0,0.13][0.25,0.25][0.0,0.25][0.0,0.13]),Xmax⁡⁡(2)=([0.37,0.37][0.0,0.3][0.13,0.13][0.75,0.75][0.0,0.13][0.0,0.13][0.0,1.0][0.0,0.13][0.0,0.13][0.0,0.25][0.25,0.25][0.0,0.13]),Xmax⁡⁡(3)=([0.37,0.37][0.0,0.3][0.0,0.13][0.75,0.75][0.13,0.13][0.0,0.13][0.0,1.0][0.0,0.13][0.0,0.13][0.25,0.25][0.0,0.25][0.0,0.13]),Xmax⁡⁡(4)=([0.37,0.37][0.0,0.3][0.0,0.13][0.75,0.75][0.13,0.13][0.0,0.13][0.0,1.0][0.0,0.13][0.0,0.13][0.0,0.25][0.25,0.25][0.0,0.13]),Xmax⁡⁡(5)=([0.37,0.37][0.0,0.3][0.0,0.13][0.75,0.75][0.13,0.13][0.0,0.13][0.0,1.0][0.0,0.13][0.0,0.13][0.25,0.25][0.0,0.25][0.0,0.13]),Xmax⁡⁡(6)=([0.37,0.37][0.0,0.3][0.0,0.13][0.75,0.75][0.13,0.13][0.0,0.13][0.0,1.0][0.0,0.13][0.0,0.13][0.0,0.25][0.25,0.25][0.0,0.13]),Xmax⁡⁡(7)=([0.37,0.37][0.0,0.3][0.0,0.13][0.75,0.75][0.0,0.13][0.0,0.13][0.0,1.0][0.13,0.13][0.0,0.13][0.25,0.25][0.0,0.25][0.0,0.13]),Xmax⁡⁡(8)=([0.37,0.37][0.0,0.3][0.0,0.13][0.75,0.75][0.0,0.13][0.0,0.13][0.0,1.0][0.13,0.13][0.0,0.13][0.0,0.25][0.25,0.25][0.0,0.13]),Xmax⁡⁡(9)=([0.37,0.37][0.0,0.3][0.0,0.13][0.75,0.75][0.0,0.13][0.0,0.13][0.0,1.0][0.0,0.13][0.13,0.13][0.25,0.25][0.0,0.25][0.0,0.13]),Xmax⁡⁡(10)=([0.37,0.37][0.0,0.3][0.0,0.13][0.75,0.75][0.0,0.13][0.0,0.13][0.0,1.0][0.0,0.13][0.13,0.13][0.0,0.25][0.25,0.25][0.0,0.13]),Xmax⁡⁡(11)=([0.37,0.37][0.0,0.3][0.0,0.13][0.75,0.75][0.0,0.13][0.0,0.13][0.0,1.0][0.0,0.13][0.0,0.13][0.25,0.25][0.0,0.25][0.13,0.13]),Xmax⁡⁡(12)=([0.37,0.37][0.0,0.3][0.0,0.13][0.75,0.75][0.0,0.13][0.0,0.13][0.0,1.0][0.0,0.13][0.0,0.13][0.0,0.25][0.25,0.25][0.13,0.13]),XMean1=(0.370.150.130.750.0650.0650.50.0650.0650.250.1250.05),XMean2=(0.370.150.130.750.0650.0650.50.0650.0650.1250.250.065),XMean3=(0.370.150.0650.750.130.0650.50.0650.0650.250.1250.065),XMean4=(0.370.150.0650.750.130.0650.50.0650.0650.1250.250.065)XMean5=(0.370.150.0650.750.0650.130.50.0650.0650.250.1250.05),XMean6=(0.370.150.0650.750.0650.130.50.0650.0650.1250.250.05),XMean7=(0.370.150.0650.750.0650.0650.50.130.0650.250.1250.065),XMean8=(0.370.150.0650.750.0650.0650.50.130.0650.1250.250.065),XMean9=(0.370.150.0650.750.0650.0650.50.0650.130.250.1250.05),XMean10=(0.370.150.0650.750.0650.0650.50.0650.130.1250.250.05),XMean11=(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).
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.Table 9
Final linguistic labels for the output variables in the district Poggioreale.
OutputScore1 (os)Score2 (os)Score3 (os)Score4 (os)Score5 (os)Score6 (os)Score7 (os)Score8 (os)Score9 (os)Score10 (os)Score11 (os)Score12 (os)o1lowlowlowhighlowlowlowhighlowlowlowhigho2lowlowlowmeanlowlowlowmeanlowlowlowmeano3lowlowlowlowlowlowlowlowlowlowlowlowo4lowmeanlowmeanlowmeanlowmeanlowmeanlowmeanFigure 6
Thematic map for output variableo1 (Economic prosperity).Figure 7
Thematic map of the output variableo2  (Transition into the  job).Figure 8
Thematic map for the output variableo3 (Social Environment).Figure 9
Thematic map for the output variableo4 (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 Figure10, we show the histogram of the reliability index Rel(O) for each subzone, where O=[o1,o2,o3,o4].Figure 10
Histogram of the reliability index Rel(O) for the four subzones.



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*Source: 429498-2011-10-15.xml*