Spatial Analysis and Fuzzy Relation Equations

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 [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 𝑎𝑎₁₁₂₁∧𝑥₁∨⋯∨𝑎_1𝑛∧𝑥_𝑛=𝑏₁,𝑎_𝑚1∧𝑥₁∨⋯∨⋮𝑎_2𝑛∧𝑥_𝑛=𝑏₂,∧𝑥₁∨⋯∨𝑎_𝑚𝑛∧𝑥_𝑛=𝑏_𝑚.(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., [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,…,𝑏_𝑚} of (1). The values of the coefficients 𝑎_𝑖𝑗 are set by the expert and the whole set of solutions (𝑥₁,…,𝑥_𝑛) of (1) is determined as well.(iv)A fuzzy partition of the domain [0,1] is created for the output variables 𝑜₁,…,𝑜_𝑘; every fuzzy set of the partition corresponds to a determined value 𝑥_𝑗. (v)The output data 𝑜₁,…,𝑜_𝑘 are extracted. A partition of fuzzy sets corresponds to each output variable 𝑜_𝑗 (𝑗=1,…,𝑘); in this phase the linguistic label of the most appropriate fuzzy set is assigned to the output variable 𝑜_𝑗.

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 𝑎_𝑖𝑗 (𝑖=1,…,𝑛 and 𝑗=1,…,𝑚) represent the impact of the 𝑗th cause 𝑥_𝑗 to the production of the 𝑖th symptom 𝑏_𝑖, where the value of 𝑏_𝑖 is the membership degree in the corresponding fuzzy set and let 𝐵=[𝑏₁,…,𝑏_𝑚]. In another subzone the input data vector 𝐵 and the matrix 𝐴 can vary. For example, we consider the equation:0.8∨∧𝑥0₁.3∨∧𝑥0₅.2∧𝑥₂∨0.0∧𝑥₃∨0.8∧𝑥₄∨0.0∧𝑥₆=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”.

(a)

(b)

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 𝑥_𝑖 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 𝑏_𝑖 (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 𝑋_max(1),…,𝑋_max(𝑇). In order to describe the extraction process of the solutions, let 𝑋_max(𝑡), 𝑡∈{1,…,𝑇}, be a maximal interval solution given below, where 𝑋^low is a minimal solution and 𝑋^gr is the greatest solution. Our aim is to assign the linguistic label of the most appropriate fuzzy sets corresponding to the unknown {𝑥_𝑗1,𝑥_𝑗1,…,𝑥_𝑗𝑠} related to an output variable 𝑜_𝑠, 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 𝑥_𝑗, 𝑗=𝑗₁,…,𝑗_𝑠. We can write their membership functions 𝜇_𝑗1,𝜇_𝑗2,…,𝜇_𝑗ℎ as follows:𝜇_𝑗1=⎧⎪⎪⎨⎪⎪⎩1,ifINF𝑗¹SUP𝑗₁≤𝑥≤MEAN𝑗₁,−𝑥SUP𝑗₁−MEAN𝑗₁,ifMEAN𝑗¹𝜇0,otherwise,<𝑥≤SUP𝑗₁,(3)_𝑗=⎧⎪⎪⎨⎪⎪⎩𝑥−INF(𝑗)MEASNU(P𝑗()𝑗−)−IN𝑥F(𝑗),ifINF(𝑗)≤𝑥≤MEAN(𝑗),0S,UP(𝑗)−MEAN(𝑗),iofthM𝑗e∈rEwAisN𝑗e₂,(𝑗)<𝑥≤SUP(𝑗),𝜇,…,𝑗_𝑠−1,(4)_𝑗𝑠=⎧⎪⎪⎨⎪⎪⎩𝑥−INF𝑗^𝑠MEAN𝑗_𝑠−INF𝑗_𝑠,ifINF𝑗^𝑠1,ifMEAN𝑗≤_𝑠𝑥≤MEAN𝑗_𝑠,0,otherwise.<𝑥≤SUP𝑗_𝑠,(5)

Table 1 

TFNs values for the fuzzy sets.

If 𝑋Min_𝑡(𝑗) (resp., 𝑋Max_𝑡(𝑗)) is the min (resp., max) value of every interval corresponding to the unknown 𝑥_𝑗, we can calculate the arithmetical mean value 𝑋Mean_𝑡(𝑗) of the 𝑗th component of the above maximal interval solution 𝑋_max(𝑡) as𝑋Mean_𝑡(𝑗)=𝑋Min^𝑡(𝑗)+𝑋Max_𝑡(𝑗)2,(6) and we get the vector column 𝑋Mean_𝑡=[𝑋Mean_𝑡(1),…,𝑋Mean_𝑡(𝑛)]⁻¹ (cf. Table 2). The value given from max{𝑋Mean_𝑡(𝑗₁),…,𝑋Mean_𝑡(𝑗_𝑠)} obtained for the unknowns 𝑥_𝑗1,…,𝑥_𝑗𝑠 corresponding to the output variable 𝑜_𝑠, is the linguistic label of the fuzzy set assigned to 𝑜_𝑠 and it is denoted by score_t (𝑜_𝑠), defined also as reliability of 𝑜_𝑠 in the interval solution 𝑡. In our example, we have that “𝑜₁ = mean state of buildings = scanty” and “𝑜₂ = mean soil permeability = medium”, hence score_𝑡(𝑜₁)=0.70 and score_𝑡(𝑜₂)=0.55. For the output vector 𝑂=[𝑜₁,…,𝑜_𝑘], we define the following reliability index in the interval solution 𝑡 asRel_𝑡(𝑂)=1𝑘⋅^𝑘_𝑠=1score_𝑡𝑜_𝑠(7) and then as final reliability index of 𝑂, the number Rel(𝑂)=max{Rel_𝑡(𝑂)∶𝑡=1,…,𝑇}.

Table 2 

TFNs mean values.

In our example, we have Rel_𝑡(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:𝐴∘𝑋=𝐵,(8) where 𝐴=(𝑎_𝑖𝑗) is the matrix of coefficients, 𝑋 = (𝑥₁,𝑥₂,…,𝑥_𝑛)⁻¹ is the column vector of the unknowns, and 𝐵 = (𝑏₁,𝑏₂,…,𝑏_𝑚)⁻¹ is the column vector of the known terms, being 𝑎_𝑖𝑗,𝑥_𝑗,𝑏_𝑖∈[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 𝑋∈Ω is called a minimal solution if 𝑋≤𝑋 for some 𝑋∈Ω implies 𝑋=𝑋, 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 𝑋^gr=(𝑥^gr₁,𝑥^gr₂,…,𝑥^gr_𝑛)⁻¹ if Ω≠∅ [10]. A matrix interval 𝑋_interval of the following type:𝑋_interval=⎛⎜⎜⎜⎜⎜⎜⎝𝑎¹𝑎₂,𝑏₁[…𝑎,,…_𝑛𝑏₂],𝑏_𝑛⎞⎟⎟⎟⎟⎟⎟⎠,(9) where [𝑎_𝑗,𝑏_𝑗]⊆[0,1] for each 𝑗=1,…,𝑛, is called an interval solution of the SFRE (8) if every 𝑋=(𝑥₁,𝑥₂,…,𝑥_𝑛)⁻¹ such that 𝑥_𝑗∈[𝑎_𝑗,𝑏_𝑗] for each 𝑗=1,…,𝑛, belongs to Ω. If 𝑎_𝑗 is a membership value of a minimal solution and 𝑏_𝑗 is a membership value of 𝑋^gr for each 𝑗=1,…,𝑛, then 𝑋_interval is called a maximal interval solution of the SFRE (8), and it is denoted by 𝑋_max(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≥⋯≥𝑏_𝑚. The time computational complexity to reduce an SFRE in a normal form is polynomial [6, 8]. Now we consider the matrix 𝐴^∗=(𝑎^∗𝑖𝑗) so defined:𝑎^∗𝑖𝑗=⎧⎪⎨⎪⎩0,if𝑎^𝑖𝑗𝑏_𝑖<𝑏_𝑖,1,,iiff𝑎𝑎_{𝑖𝑗𝑖𝑗}=𝑏_𝑖,>𝑏_𝑖,(10) where 𝑖=1,…,𝑚 and 𝑗=1,…,𝑛. The linguistic description of 𝑎^∗𝑖𝑗 as S-type coefficient (Smaller) if 𝑎_𝑖𝑗<𝑏_𝑖, E-type coefficient (Equal) if 𝑎_𝑖𝑗=𝑏_𝑖, and G-type coefficient (Greater) if 𝑎_𝑖𝑗>𝑏_𝑖 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 𝐴^∗𝑗={𝑎^∗1𝑗,…,𝑎^∗𝑚𝑗}. If 𝐴^∗(𝑗) contains G-type coefficients and 𝑘∈{1,…,𝑚} is the greatest index of row such that 𝑎^∗𝑘𝑗=1, then the following coefficients in 𝐴^∗(𝑗) are called selected:(i)𝑎^∗𝑖𝑗 for 𝑖∈{1,…,𝑘} with 𝑎^∗𝑖𝑗≥𝑏_𝑖=𝑏_𝑘,(ii)𝑎^∗𝑖𝑗 for 𝑖∈{𝑘+1,…,𝑚} with 𝑎^∗𝑖𝑗=𝑏_𝑖.

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 𝑎^∗𝑟𝑗=𝑏_𝑟, then any 𝑎^∗𝑖𝑗=𝑏_𝑖 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 productPN2=^𝑚_𝑖=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 𝑋^gr with component 𝑥^gr_𝑗=𝑏_𝑘 if the jth column 𝐴^∗(𝑗) of 𝐴^∗ contains selected G-type coefficients 𝑎^∗𝑘𝑗 and 𝑥^gr_𝑗=1 otherwise. (ii)The complexity time function for computing 𝑋^gr is 𝑂(𝑚⋅𝑛).

A help matrix 𝐻=(ℎ_𝑖𝑗), with 𝑖=1,…,𝑚 and 𝑗=1,…,𝑛, is defined as follows:ℎ_𝑖𝑗=𝑏^𝑖0,,iofth𝑎e^∗𝑖𝑗rwisissee.lected,(12)

Let |𝐻_𝑖| be the number of coefficients ℎ_𝑖𝑗 in the 𝑖th equation of the SFRE (8). Then the number of potential minimal solutions cannot exceed the valuePN1=^𝑚_𝑖=1||𝐻_𝑖||,(13) where PN2≤PN1.

Definition 5. Let ℎ_𝑖=(ℎ_𝑖1,ℎ_𝑖2,…,ℎ_𝑖𝑛) and ℎ_𝑘=(ℎ_𝑘1,ℎ_𝑘2,…,ℎ_𝑘𝑛) be the 𝑖th and the 𝑘th rows of the help matrix 𝐻. If for each 𝑗=1,…𝑛, ℎ_𝑖𝑗≠0 implies both ℎ_𝑘𝑗≠0 and ℎ_𝑘𝑗≤ℎ_𝑖𝑗, 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:ℎ^∗𝑖𝑗=⎧⎪⎨⎪⎩ℎ0,iafntohthee𝑖rtheqeuqautaiotino,_𝑖𝑗nisdominatedby,otherwise.(14)

For each 𝑖=1,…,𝑚, now we set |𝐻^∗𝑖| as the number of coefficients ℎ^∗𝑖𝑗=𝑏_𝑖≠0 in the 𝑖th row of the dominance matrix 𝐻^∗. When this value is 0, we set |𝐻^∗𝑖|=1. Then the number of potential minimal solutions of the SFRE cannot exceed the value PN3=^𝑚_𝑖=1||𝐻^∗𝑖||,(15) where PN3≤PN2≤PN1. In [6, 8, 20], the authors use the symbol ⟨𝑏_𝑖/𝑗⟩ to indicate the coefficients ℎ^∗𝑖𝑗=𝑏_𝑖≠0. We have ℎ^∗𝑖𝑗∧𝑥_𝑗=𝑏_𝑖 if 𝑥_𝑗∈[𝑏_𝑖,1] and 𝑥_𝑗=𝑏_𝑖 is the 𝑗th component of a minimal solution. A solution of the 𝑖th equation can be written as𝐻_𝑖=^𝑛_𝑗=1𝑏𝑗^𝑖.(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=^𝑚_𝑖=1𝐻_𝑖=^𝑚_𝑖=1^𝑛_𝑗=1𝑏𝑗^𝑖.(17)

The following properties hold:(i)commutativity: 𝑏𝑗^𝑖1₁𝑏𝑗^𝑖2₂=𝑏𝑗^𝑖2₂𝑏𝑗^𝑖1₁,(18)(ii)associativity: 𝑏𝑗^𝑖1₁𝑏𝑗^𝑖2₂𝑏𝑗^𝑖3₃=𝑏𝑗^𝑖1₁𝑏𝑗^𝑖2₂𝑏𝑗^𝑖3₃,(19)(iii)distributivity with respect to the addition: 𝑏𝑗^𝑖1₁𝑏𝑗^𝑖2₂+𝑏𝑗^𝑖3₃=+𝑏𝑗₁𝑏^𝑖1_𝑖1𝑏𝑗^𝑖2₂𝑗₁𝑏𝑗^𝑖3₃,(20)(iv)absorption for multiplication: 𝑏𝑗^𝑖1₁𝑏𝑗^𝑖2₂=⎧⎪⎨⎪⎩𝑏^𝑖1∧𝑏_𝑖2𝑗,if𝑗¹unchanged,other=wi𝑗s₂e,=𝑗,(21)(v)absorption for addition: 𝑏𝑗=^𝑖1₁⎧⎪⎨⎪⎩𝑏𝑏𝑗^𝑖2_2𝑖1⋯𝑏𝑗^𝑖𝑚_𝑛+𝑏𝑗^𝑘1₁𝑏𝑗^𝑘2₂⋯𝑏𝑗^𝑘𝑚_𝑛𝑗₁𝑏𝑗^𝑖2₂⋯𝑏𝑗^𝑖𝑚_𝑛,if𝑏^𝑖ℎunchanged,ℎoth∈er{w=1i,s𝑏…e._𝑘ℎ,,𝑚},(22)

We can determine the minimal solutions 𝑋^low(𝑡)=(𝑥^low1(𝑡),𝑥^low2(𝑡),…,𝑥^{low𝑛(𝑡)})⁻¹, 𝑡∈{1,…,PN(3)}, with components 𝑥^{low𝑗(𝑡)}=𝑏^𝑖𝑡0,,oifth𝑏e_𝑖𝑡r≠wi0s,e.(23)

The above definitions shall be clarified in the following example of an SFRE with 4 equations and 6 unknown:1.0∨∧𝑥0₁.9∨∧𝑥0₄.0∧𝑥₂∨0.0∧𝑥₃0.5∧𝑥₁∨0.2∧𝑥₅∨0.0∧𝑥₆=0.1,∨0.5∨∧𝑥0₄.3∧𝑥₂∨0.4∧𝑥₃0.7∧𝑥₁∨0.3∧𝑥₅∨0.4∧𝑥₆=0.3,∨0.7∨∧𝑥0₄.4∧𝑥₂∨0.2∧𝑥₃0.4∧𝑥₁∨0.4∧𝑥₅∨0.2∧𝑥₆=0.3,∨0.4∨∧𝑥0₄.7∧𝑥₂∨0.2∧𝑥₃∨0.7∧𝑥₅∨0.2∧𝑥₆=0.3.(24)

We have𝐴=⎛⎜⎜⎜⎜⎜⎜⎝1000....05740000....03470000....04220000....95740000....23470000....0422⎞⎟⎟⎟⎟⎟⎟⎠,𝐵=⎛⎜⎜⎜⎜⎜⎜⎝0000....1333⎞⎟⎟⎟⎟⎟⎟⎠.(25)

By using the normal form, we obtain that 𝐴=⎛⎜⎜⎜⎜⎜⎜⎝0001....57400000....34700000....42200000....57490000....34720000....4220⎞⎟⎟⎟⎟⎟⎟⎠,𝐵=⎛⎜⎜⎜⎜⎜⎜⎝0000....3331⎞⎟⎟⎟⎟⎟⎟⎠.(26)

Now we compute the matrix 𝐴^∗ and the vector IND as follows:𝐴^∗=⎛⎜⎜⎜⎜⎜⎜⎝1111....00000110....30001000....00001111....00000111....30001000....0000⎞⎟⎟⎟⎟⎟⎟⎠,IND=⎛⎜⎜⎜⎜⎜⎜⎝3113⎞⎟⎟⎟⎟⎟⎟⎠.(27)

The SFRE is consistent because each component of IND is not null. The greatest solution is given by𝑋^gr=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝000000......133113⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.(28)

Now we calculate the help matrix 𝐻 and the dominant matrix 𝐻^∗ as follows:𝐻𝐻=⎛⎜⎜⎜⎜⎜⎜⎝0000....00010000....33300000....30000000....00010000....00010000....3000_∗⎞⎟⎟⎟⎟⎟⎟⎠,=⎛⎜⎜⎜⎜⎜⎜⎝0000....00010000....00300000....00000000....00010000....00010000....0000⎞⎟⎟⎟⎟⎟⎟⎠.(29)

Then we have |𝐻^∗1|=|𝐻^∗2|=|𝐻^∗3|=1, |𝐻^∗1|=3 and hence PN3=3. By using the properties (18)–(23), we have that 𝑊=0.3=02.101.1+04.1+05.1+10.302.3+02.304.1205.1.(30)

The three minimal solutions are given by𝑋^low(1)=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝000000......130000⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝑋^low(2)=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝000000......030100⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝑋^low(3)=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝000000......030010⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.(31) The three maximal interval solutions are given by𝑋_max(1)=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝[[[[[[000000𝑋......130000,,,,,,000000......133113]]]]]]⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝑋_max(2)=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝[[[[[[000000......030100,,,,,,000000......133113_max(3)]]]]]]⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝[[[[[[000000......030010,,,,,,000000......133113]]]]]]⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.(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 𝑏_𝑖, 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 𝑎_𝑖𝑗 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 𝑥_𝑗, the mean interval solution 𝑋Max_𝑀,𝑡(𝑗) with (6). The linguistic label Rel_𝑡(𝑜_𝑠) is assigned to the output variable 𝑜_𝑠. Then he calculates the reliability index Rel_𝑡(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 associate 𝑘 facts 𝑖₁,…,𝑖_ℎ to every subzone.(ii)For each input fact, a fuzzy partition in 𝑚_𝑓 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+⋯+𝑚_𝑘. The input vector 𝐵=[𝑏₁,…,𝑏_𝑚] is set, where each component 𝑏_𝑖 (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 𝑎_𝑖𝑗 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)maxRel_𝑡∶=0// (the maximal reliability is initialized to 0).(v)For each maximal interval solution 𝑋_max,𝑡, with 𝑡=1,…,𝑇, we define the vector column 𝑋Mean_𝑡 via formula (6).(vi)Rel_𝑡∶=0.(vii)For each output variable 𝑜_𝑠, with 𝑠=1,…,𝑘, if 𝑥_𝑗1,…,𝑥_𝑗𝑠 are the unknown associated to 𝑜_𝑠, let score_𝑡(𝑠)=max{𝑋Mean_𝑡(𝑗₁),…,𝑋Mean_𝑡(𝑗_𝑠)}.(viii)Rel_𝑡∶=Rel_𝑡+score_𝑡(𝑜_𝑠).(ix)Next 𝑠.(x)Rel_𝑡∶=Rel_𝑡/𝑘// (the reliability index is calculated via formula (7)).(xi)If Rel_t > maxRel_t, then the linguistic label of the fuzzy set corresponding to the unknown with maximum mean solution is assigned to the output vector 𝑂=[𝑜₁,…,𝑜_𝑘].(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.

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.

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 𝑏_𝑖. In order to form the SFRE (1) in each subzone, the expert defines the equations by setting the impact values 𝑎_𝑖𝑗 by basing over the most significant symptoms.

Table 5 

TFNs values for the input domains.

Table 6 

TFNs for the symptoms 𝑏₁ ÷ 𝑏₁₂.

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).

Table 8 

Final linguistic labels for the output variables in the district Barra.

The matrix 𝐴 of the impact values 𝑎_𝑖𝑗 has dimensions 10×12 and the vector 𝐵 of the symptoms 𝑏_𝑖 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 𝑋_max(𝑡) (𝑡=1,…,4) and we calculate the vector column 𝑋Mean_𝑡 on each maximal interval solution. Hence we associate to the output variable 𝑜_𝑠 (𝑠=1,…,4), the linguistic label of the fuzzy set with the higher value calculated with formula (6) obtained for the corresponding unknowns 𝑥_𝑗1,…,𝑥_𝑗𝑠 and given in Table 8. For determining the reliability of our solutions, we use the index given by formula (7). We obtain that Rel_𝑡(𝑜₁)=Rel_𝑡(𝑜₂)=Rel_𝑡(𝑜₃)=Rel_𝑡(𝑜₄)=0.6025 for 𝑡=1,…,4 and hence Rel(𝑂)=max{Rel_𝑡(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=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝0001000000..........53205312151000000000..........05723715420000000000..........0220131240𝐵0000000000..........=4428631114⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝10000000000001000001....................0573472443936046154086300009100000000000⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠..........2421131140,0000000000..........23286222140000000000..........76724715530000000000..........32221311500000000001..........10031113200000000000..........30000237410000000000..........2000013350⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,(33)𝑋𝑋_max(1)=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝[[[[[[[[[[[[000000000100............430000004000060000001000,,,,,,,,,,,,001010100100............430303034011060606061000]]]]]]]]]]]]⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝑋_max(2)=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝[[[[[[[[[[[[000000000100............400300004000000600001000,,,,,,,,,,,,001010100100............430303034011060606061000_max(3)]]]]]]]]]]]]⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑋[[[[[[[[[[[[000000000100M............400003004000000006001000e,,,,,,,,,,,,a001010100100n............₁430303034011060606061000]]]]]]]]]]]]⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝑋_max(4)=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝[[[[[[[[[[[[000000000100............400300034000000600061000,,,,,,,,,,,,001010100100............430303034011060606061000]]]]]]]]]]]]⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝑋Mean₃=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝000000000100............435151514000060808081055⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝑋Mean₂=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝000000000100............415351514000080608081055⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝000000000100............415153511000080806088055⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝑋Mean₄=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝000000000100............410351534000085608061055⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.(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 𝑎_𝑖𝑗 has dimension 11×12 and the vector 𝐵 of the symptoms 𝑏_𝑖 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 𝑋_max(𝑡) (𝑡=1,…,12), and we calculate the vector column 𝑋Mean_𝑡 on each maximal interval solution. The relevant quantities are given below. 𝐴=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝00000000000...........5223432104111100000000...........0007574911200000000000...........02236361501𝐵00000000000...........=42233331181⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝1110000000000000000100...................0007574925299638172...0733937735500000000000⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠...........22236361531,00000000000...........2222322215100100000000...........7907574823200000000000...........3223636251100000000000...........1001010207300000000000...........3002021813600000000000...........20021122402⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,(35)𝑋𝑋_max(1)=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝[[[[[[[[[[0000[[00000000..........3172000000..007355,,,,,,,,,,,,000000010000......111121......3031723333537355]]]]]]]]]]]]⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝑋_max(2)=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝[[[[[[[[[[0000[[00000000..........3172000000..007355,,,,,,,,,,,,000000010000......111121......303172333353_max(3)7355]]]]]]]]]]]]⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝑋=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝[[[[[[[[[[0000[[00000000..........3712000000..007535,,,,,,,,,,,,000000010000......111121......3037123333537535]]]]]]]]]]]]⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝑋_max(4)=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝[[[[[[[[[[0000[[00000000..........3712000000..007535,,,,,,,,,,,,000000010000......111121......303712333353_max(5)7535]]]]]]]]]]]]⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝑋=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝[[[[[[[[[[0000[[00000000..........3712000000..007535,,,,,,,,,,,,000000010000......111121......3037123333537535]]]]]]]]]]]]⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝑋_max(6)=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝[[[[[[[[[[0000[[00000000..........3712000000..007535,,,,,,,,,,,,000000010000......111121......303712333353_max(7)7535]]]]]]]]]]]]⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝑋=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝[[[[[[[[[[0000[[00000000..........3712000000..007535,,,,,,,,,,,,000000010000......111121......3037123333537535]]]]]]]]]]]]⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝑋_max(8)=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝[[[[[[[[[[0000[[00000000..........3712000000..007535,,,,,,,,,,,,000000010000......111121......303712333353_max(9)7535]]]]]]]]]]]]⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝑋=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝[[[[[[[[[[0000[[00000000....3712........000000007535,,,,,,,,,,,,000000010000............1111213037123333537535]]]]]]]]]]]]⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝑋_max(10)=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝[[[[[[[[[[0000[[00000000..........3712000000..007535,,,,,,,,,,,,000000010000......111121......303712333353_max(11)7535]]]]]]]]]]]]⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,=𝑋⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝[[[[M[[[[[[0000[[00000000....e3721........000000a007553,,,,,,n,,,,,,000000₁010000............1111123037213333357553]]]]]]]]]]]]⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝑋_max(12)=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝[[[[[[[[[[0000[[00000000..........3721000000..007553,,,,,,,,,,,,000000010000......111112......3037213333357553]]]]]]]]]]]]⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝑋Mean₃=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝00000000000.....000001......311720.66662575355555555⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝑋Mean₂=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝000000000000......000010......31172566662675355555555⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝑋Mean₅=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝00000000000......0000010.....31712.666626575535555555⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝑋Mean₄=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝000000000000......000010......31712566662675535555555⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠𝑋Mean₇=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝00000000000.....000001......317120.66662575535555555⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝑋Mean₆=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝000000000000.....00001.......31712056666275535555555⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝑋Mean₉=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝00000000000......0000010.....31712.666626575535555555⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝑋Mean₈=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝000000000000......000010......31712566662675535555555⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝑋Mean₁₁=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝000000000000...........00001317120.56666275535555555⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝑋Mean₁₀=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝000000000000.....00001.......31712056666275535555555⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝000000000000...........00000131721.566666275553555555⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝑋Mean₁₂=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝000000000000......000001......31721566666275553555555⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.(36) For determining the reliability of our solutions, we use the index given by formula (7). We obtain Rel(𝑂_𝑘) = 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.

Figure 6 

Thematic map for output variable 𝑜1 (Economic prosperity).

Figure 7 

Thematic map of the output variable 𝑜2 (Transition into the job).

Figure 8 

Thematic map for the output variable 𝑜3 (Social Environment).

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].

Figure 10 

Histogram of the reliability index Rel(𝑂) for the four subzones.