Advances in Fuzzy Systems

Advances in Fuzzy Systems / 2011 / Article
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Fuzzy Functions, Relations, and Fuzzy Transforms: Theoretical Aspects and Applications to Fuzzy Systems

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Research Article | Open Access

Volume 2011 |Article ID 429498 |

Ferdinando Di Martino, Salvatore Sessa, "Spatial Analysis and Fuzzy Relation Equations", Advances in Fuzzy Systems, vol. 2011, Article ID 429498, 14 pages, 2011.

Spatial Analysis and Fuzzy Relation Equations

Academic Editor: Irina G. Perfilieva
Received29 Jun 2011
Accepted24 Jul 2011
Published15 Oct 2011


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 [1โ€“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 ๎€ท๐‘Ž11โˆง๐‘ฅ1๎€ธ๎€ท๐‘Žโˆจโ‹ฏโˆจ1๐‘›โˆง๐‘ฅ๐‘›๎€ธ=๐‘1,๎€ท๐‘Ž21โˆง๐‘ฅ1๎€ธ๎€ท๐‘Žโˆจโ‹ฏโˆจ2๐‘›โˆง๐‘ฅ๐‘›๎€ธ=๐‘2,โ‹ฎ๎€ท๐‘Ž๐‘š1โˆง๐‘ฅ1๎€ธ๎€ท๐‘Žโˆจโ‹ฏโˆจ๐‘š๐‘›โˆง๐‘ฅ๐‘›๎€ธ=๐‘๐‘š.(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., [11โ€“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 {๐‘1,โ€ฆ,๐‘๐‘š} of (1). The values of the coefficients ๐‘Ž๐‘–๐‘— are set by the expert and the whole set of solutions (๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›) of (1) is determined as well.(iv)A fuzzy partition of the domain [0,1] is created for the output variables ๐‘œ1,โ€ฆ,๐‘œ๐‘˜; every fuzzy set of the partition corresponds to a determined value ๐‘ฅ๐‘—. (v)The output data ๐‘œ1,โ€ฆ,๐‘œ๐‘˜ are extracted. A partition of fuzzy sets corresponds to each output variable ๐‘œ๐‘— (๐‘—=1,โ€ฆ,๐‘˜); in this phase the linguistic label of the most appropriate fuzzy set is assigned to the output variable ๐‘œ๐‘—.

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 (๐‘ฅ1) and the mean soil permeability (๐‘ฅ2), knowing the number of collapsed building in the last year (๐‘1) and the number of flooding in the last year (๐‘2). In Figure 2, we suppose to create for each symptomโ€™s and causeโ€™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 ๐‘Ž๐‘–๐‘— (๐‘–=1,โ€ฆ,๐‘› and ๐‘—=1,โ€ฆ,๐‘š) represent the impact of the ๐‘—th cause ๐‘ฅ๐‘— to the production of the ๐‘–th symptom ๐‘๐‘–, where the value of ๐‘๐‘– is the membership degree in the corresponding fuzzy set and let ๐ต=[๐‘1,โ€ฆ,๐‘๐‘š]. In another subzone the input data vector ๐ต and the matrix ๐ด can vary. For example, we consider the equation:๎€ท0.8โˆง๐‘ฅ1๎€ธโˆจ๎€ท0.2โˆง๐‘ฅ2๎€ธโˆจ๎€ท0.0โˆง๐‘ฅ3๎€ธโˆจ๎€ท0.8โˆง๐‘ฅ4๎€ธโˆจ๎€ท0.3โˆง๐‘ฅ5๎€ธโˆจ๎€ท0.0โˆง๐‘ฅ6๎€ธ=๐‘3=0.9.(2) The expert sets for the symptom ๐‘3 = โ€œcollapsed building in the last year = highโ€ = 0.9, an impact 0.8 of the variable โ€œmean state of buildings = scantyโ€, an impact 0.2 of the variable โ€œmean state of buildings = mediumโ€, an impact 0.0 of the variable โ€œmean state of buildings = highโ€, an impact 0.8 of the variable โ€œmean soil permeability = lowโ€, an impact 0.3 of the variable โ€œmean soil permeability = mediumโ€, or an impact 0.0 of the variable โ€œmean soil permeability = highโ€.

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 ๐‘ฅ๐‘– 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 ๐‘๐‘– (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 ๐ด to verify if the new system has some solution. In general, the SFRE (1) has T maximal interval solutions ๐‘‹max(1),โ€ฆ,๐‘‹max(๐‘‡). In order to describe the extraction process of the solutions, let ๐‘‹max(๐‘ก), ๐‘กโˆˆ{1,โ€ฆ,๐‘‡}, be a maximal interval solution given below, where ๐‘‹low is a minimal solution and ๐‘‹gr is the greatest solution. Our aim is to assign the linguistic label of the most appropriate fuzzy sets corresponding to the unknown {๐‘ฅ๐‘—1,๐‘ฅ๐‘—1,โ€ฆ,๐‘ฅ๐‘—๐‘ } related to an output variable ๐‘œ๐‘ , ๐‘ =1,โ€ฆ,๐‘˜. For example, assume that the three fuzzy sets ๐‘ฅ1, ๐‘ฅ2, ๐‘ฅ3 (resp., ๐‘ฅ4, ๐‘ฅ5, ๐‘ฅ6) are related to ๐‘œ1 (resp., ๐‘œ2) and are represented from the TFNs given in Table 1, where INF(๐‘—), MEAN(๐‘—), and SUP(๐‘—) are the three fundamental values of the generic TFN ๐‘ฅ๐‘—, ๐‘—=๐‘—1,โ€ฆ,๐‘—๐‘ . We can write their membership functions ๐œ‡๐‘—1,๐œ‡๐‘—2,โ€ฆ,๐œ‡๐‘—โ„Ž as follows:๐œ‡๐‘—1=โŽงโŽชโŽชโŽจโŽชโŽชโŽฉ๎€ท๐‘—1,ifINF1๎€ธ๎€ท๐‘—โ‰ค๐‘ฅโ‰คMEAN1๎€ธ,๎€ท๐‘—SUP1๎€ธโˆ’๐‘ฅ๎€ท๐‘—SUP1๎€ธ๎€ท๐‘—โˆ’MEAN1๎€ธ๎€ท๐‘—,ifMEAN1๎€ธ๎€ท๐‘—<๐‘ฅโ‰คSUP1๎€ธ,๐œ‡0,otherwise,(3)๐‘—=โŽงโŽชโŽชโŽจโŽชโŽชโŽฉ๐‘ฅโˆ’INF(๐‘—)MEAN(๐‘—)โˆ’INF(๐‘—),ifINF(๐‘—)โ‰ค๐‘ฅโ‰คMEAN(๐‘—),SUP(๐‘—)โˆ’๐‘ฅ๎€ฝ๐‘—SUP(๐‘—)โˆ’MEAN(๐‘—),ifMEAN(๐‘—)<๐‘ฅโ‰คSUP(๐‘—),0,otherwise,๐‘—โˆˆ2,โ€ฆ,๐‘—๐‘ โˆ’1๎€พ,๐œ‡(4)๐‘—๐‘ =โŽงโŽชโŽชโŽจโŽชโŽชโŽฉ๎€ท๐‘—๐‘ฅโˆ’INF๐‘ ๎€ธ๎€ท๐‘—MEAN๐‘ ๎€ธ๎€ท๐‘—โˆ’INF๐‘ ๎€ธ๎€ท๐‘—,ifINF๐‘ ๎€ธ๎€ท๐‘—โ‰ค๐‘ฅโ‰คMEAN๐‘ ๎€ธ,๎€ท๐‘—1,ifMEAN๐‘ ๎€ธ๎€ท๐‘—<๐‘ฅโ‰คSUP๐‘ ๎€ธ,0,otherwise.(5)

UnknownINF( ๐‘— )MEAN( ๐‘— )SUP( ๐‘— )

๐‘ฅ 1
๐‘ฅ 2
๐‘ฅ 3
๐‘ฅ 4
๐‘ฅ 5
๐‘ฅ 6

If ๐‘‹Min๐‘ก(๐‘—) (resp., ๐‘‹Max๐‘ก(๐‘—)) is the min (resp., max) value of every interval corresponding to the unknown ๐‘ฅ๐‘—, we can calculate the arithmetical mean value ๐‘‹Mean๐‘ก(๐‘—) of the ๐‘—th component of the above maximal interval solution ๐‘‹max(๐‘ก) as๐‘‹Mean๐‘ก(๐‘—)=๐‘‹Min๐‘ก(๐‘—)+๐‘‹Max๐‘ก(๐‘—)2,(6) and we get the vector column ๐‘‹Mean๐‘ก=[๐‘‹Mean๐‘ก(1),โ€ฆ,๐‘‹Mean๐‘ก(๐‘›)]โˆ’1 (cf. Table 2). The value given from max{๐‘‹Mean๐‘ก(๐‘—1),โ€ฆ,๐‘‹Mean๐‘ก(๐‘—๐‘ )} obtained for the unknowns ๐‘ฅ๐‘—1,โ€ฆ,๐‘ฅ๐‘—๐‘  corresponding to the output variable ๐‘œ๐‘ , is the linguistic label of the fuzzy set assigned to ๐‘œ๐‘  and it is denoted by scoret (๐‘œ๐‘ ), defined also as reliability of ๐‘œ๐‘  in the interval solution ๐‘ก. In our example, we have that โ€œ๐‘œ1 = mean state of buildings = scantyโ€ and โ€œ๐‘œ2 = mean soil permeability = mediumโ€, hence score๐‘ก(๐‘œ1)=0.70 and score๐‘ก(๐‘œ2)=0.55. For the output vector ๐‘‚=[๐‘œ1,โ€ฆ,๐‘œ๐‘˜], we define the following reliability index in the interval solution ๐‘ก asRel๐‘ก1(๐‘‚)=๐‘˜โ‹…๐‘˜๎“๐‘ =1score๐‘ก๎€ท๐‘œ๐‘ ๎€ธ(7) and then as final reliability index of ๐‘‚, the number Rel(๐‘‚)=max{Rel๐‘ก(๐‘‚)โˆถ๐‘ก=1,โ€ฆ,๐‘‡}.

Output variableUnknown componentLinguistic label ๐‘‹ M i n ๐‘ก ( ๐‘— ) ๐‘‹ M a x ๐‘ก ( ๐‘— ) ๐‘‹ M e a n ๐‘ก ( ๐‘— )

๐‘ฅ 1 scanty0.60.80.70
๐‘œ 1 ๐‘ฅ 2 medium0.20.40.30
๐‘ฅ 3 good0.00.10.05
๐‘ฅ 4 low0.30.50.40
๐‘œ 2 ๐‘ฅ 5 medium0.40.70.55
๐‘ฅ 6 good0.00.30.15

In our example, we have Rel๐‘ก(๐‘‚)=(0.7+0.55)/2=0.625. Therefore, the higher the reliability of our solution, the closer the final reliability index Rel(๐‘‚) 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:๐ดโˆ˜๐‘‹=๐ต,(8) where ๐ด=(๐‘Ž๐‘–๐‘—) is the matrix of coefficients, ๐‘‹ = (๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›)โˆ’1 is the column vector of the unknowns, and ๐ต = (๐‘1,๐‘2,โ€ฆ,๐‘๐‘š)โˆ’1 is the column vector of the known terms, being ๐‘Ž๐‘–๐‘—,๐‘ฅ๐‘—,๐‘๐‘–โˆˆ[0,1] for each ๐‘–=1,โ€ฆ,๐‘š and ๐‘—=1,โ€ฆ,๐‘›. We have the following definitions and terminologies: the whole set of all solutions ๐‘‹ of the SFRE (8) is denoted by ฮฉ. If ฮฉโ‰ โˆ…, then the SFRE (8) is called consistent, otherwise it is called inconsistent. A solution ๎๐‘‹โˆˆฮฉ is called a minimal solution if ๎๐‘‹๐‘‹โ‰ค for some ๐‘‹โˆˆฮฉ implies ๎๐‘‹๐‘‹=, where โ€œโ‰คโ€ is the partial order induced in ฮฉ 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 ๐‘‹gr=(๐‘ฅgr1,๐‘ฅgr2,โ€ฆ,๐‘ฅgr๐‘›)โˆ’1 if ฮฉโ‰ โˆ… [10]. A matrix interval ๐‘‹interval of the following type:๐‘‹interval=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽ๎€บ๐‘Ž1,๐‘1๎€ป๎€บ๐‘Ž2,๐‘2๎€ป[]๎€บ๐‘Žโ€ฆ,โ€ฆ๐‘›,๐‘๐‘›๎€ปโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ,(9) where [๐‘Ž๐‘—,๐‘๐‘—]โŠ†[0,1] for each ๐‘—=1,โ€ฆ,๐‘›, is called an interval solution of the SFRE (8) if every ๐‘‹=(๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›)โˆ’1 such that ๐‘ฅ๐‘—โˆˆ[๐‘Ž๐‘—,๐‘๐‘—] for each ๐‘—=1,โ€ฆ,๐‘›, belongs to ฮฉ. If ๐‘Ž๐‘— is a membership value of a minimal solution and ๐‘๐‘— is a membership value of ๐‘‹gr for each ๐‘—=1,โ€ฆ,๐‘›, then ๐‘‹interval is called a maximal interval solution of the SFRE (8), and it is denoted by ๐‘‹max(๐‘ก), where ๐‘ก varies from 1 to the number of minimal solutions. The SFRE (8) is said to be in normal form if ๐‘1โ‰ฅ๐‘2โ‰ฅโ‹ฏโ‰ฅ๐‘๐‘š. The time computational complexity to reduce an SFRE in a normal form is polynomial [6, 8]. Now we consider the matrix ๐ดโˆ—=(๐‘Žโˆ—๐‘–๐‘—) so defined:๐‘Žโˆ—๐‘–๐‘—=โŽงโŽชโŽจโŽชโŽฉ0,if๐‘Ž๐‘–๐‘—<๐‘๐‘–,๐‘๐‘–,if๐‘Ž๐‘–๐‘—=๐‘๐‘–,1,if๐‘Ž๐‘–๐‘—>๐‘๐‘–,(10) where ๐‘–=1,โ€ฆ,๐‘š and ๐‘—=1,โ€ฆ,๐‘›. The linguistic description of ๐‘Žโˆ—๐‘–๐‘— as S-type coefficient (Smaller) if ๐‘Ž๐‘–๐‘—<๐‘๐‘–, E-type coefficient (Equal) if ๐‘Ž๐‘–๐‘—=๐‘๐‘–, and G-type coefficient (Greater) if ๐‘Ž๐‘–๐‘—>๐‘๐‘– is often used. ๐ดโˆ— is called augmented matrix, and the system ๐ดโˆ—โˆ˜๐‘‹=๐ต 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 ๐ดโˆ—๐‘—={๐‘Žโˆ—1๐‘—,โ€ฆ,๐‘Žโˆ—๐‘š๐‘—}. If ๐ดโˆ—(๐‘—) contains G-type coefficients and ๐‘˜โˆˆ{1,โ€ฆ,๐‘š} is the greatest index of row such that ๐‘Žโˆ—๐‘˜๐‘—=1, then the following coefficients in ๐ดโˆ—(๐‘—) are called selected:(i)๐‘Žโˆ—๐‘–๐‘— for ๐‘–โˆˆ{1,โ€ฆ,๐‘˜} with ๐‘Žโˆ—๐‘–๐‘—โ‰ฅ๐‘๐‘–=๐‘๐‘˜,(ii)๐‘Žโˆ—๐‘–๐‘— for ๐‘–โˆˆ{๐‘˜+1,โ€ฆ,๐‘š} with ๐‘Žโˆ—๐‘–๐‘—=๐‘๐‘–.

Definition 2. If ๐ดโˆ—(๐‘—) does not contain G-type coefficients, but it contain E-type coefficients and ๐‘Ÿโˆˆ{1,โ€ฆ,๐‘š} is the smallest index of row such that ๐‘Žโˆ—๐‘Ÿ๐‘—=๐‘๐‘Ÿ, then any ๐‘Žโˆ—๐‘–๐‘—=๐‘๐‘– in ๐ดโˆ—(๐‘—) for ๐‘–โˆˆ{๐‘Ÿ,โ€ฆ,๐‘š} 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 ๐‘–th equation, ๐‘–=1,โ€ฆ,๐‘š.(ii)The complexity time function for determining the consistency of the SFRE (8) is ๐‘‚(๐‘šโ‹…๐‘›).

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),โ€ฆ,IND(๐‘š)) is defined by setting IND(๐‘–) equal to the number of selected coefficients in the ๐‘–th equation for each ๐‘–=๐‘™,โ€ฆ,๐‘š. If IND(๐‘–)=0, then all the coefficients in the ๐‘–th equation are not selected and the system is inconsistent. The system is consistent if IND(๐‘–)โ‰ 0 for each ๐‘–=๐‘™,โ€ฆ,๐‘š and the productPN2=๐‘š๎‘๐‘–=1IND(๐‘–),(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 ๐‘‹gr with component ๐‘ฅgr๐‘—=๐‘๐‘˜ if the jth column ๐ดโˆ—(๐‘—) of ๐ดโˆ— contains selected G-type coefficients ๐‘Žโˆ—๐‘˜๐‘— and ๐‘ฅgr๐‘—=1 otherwise. (ii)The complexity time function for computing ๐‘‹gr is ๐‘‚(๐‘šโ‹…๐‘›).

A help matrix ๐ป=(โ„Ž๐‘–๐‘—), with ๐‘–=1,โ€ฆ,๐‘š and ๐‘—=1,โ€ฆ,๐‘›, is defined as follows:โ„Ž๐‘–๐‘—=๎‚ป๐‘๐‘–,if๐‘Žโˆ—๐‘–๐‘—isselected,0,otherwise.(12)

Let |๐ป๐‘–| be the number of coefficients โ„Ž๐‘–๐‘— in the ๐‘–th equation of the SFRE (8). Then the number of potential minimal solutions cannot exceed the valuePN1=๐‘š๎‘๐‘–=1||๐ป๐‘–||,(13) where PN2โ‰คPN1.

Definition 5. Let โ„Ž๐‘–=(โ„Ž๐‘–1,โ„Ž๐‘–2,โ€ฆ,โ„Ž๐‘–๐‘›) and โ„Ž๐‘˜=(โ„Ž๐‘˜1,โ„Ž๐‘˜2,โ€ฆ,โ„Ž๐‘˜๐‘›) be the ๐‘–th and the ๐‘˜th rows of the help matrix ๐ป. If for each ๐‘—=1,โ€ฆ๐‘›, โ„Ž๐‘–๐‘—โ‰ 0 implies both โ„Ž๐‘˜๐‘—โ‰ 0 and โ„Ž๐‘˜๐‘—โ‰คโ„Ž๐‘–๐‘—, then the ๐‘–th row (resp., equation) is said dominant over the ๐‘˜th row in ๐ป (resp., equation) or that the ๐‘˜th row (resp., equation) is said dominated by the ๐‘–th row (resp., equation).

In other terms, if the ๐‘–th equation is dominant over the ๐‘˜th equation in (8), then the ๐‘˜th equation is a redundant equation of the system. By using Definition 5, we can build a matrix of dimension ๐‘šร—๐‘›, called dominance matrix, with components:โ„Žโˆ—๐‘–๐‘—=โŽงโŽชโŽจโŽชโŽฉโ„Ž0,ifthe๐‘–thequationisdominatedbyanotherequation,๐‘–๐‘—,otherwise.(14)

For each ๐‘–=1,โ€ฆ,๐‘š, now we set |๐ปโˆ—๐‘–| as the number of coefficients โ„Žโˆ—๐‘–๐‘—=๐‘๐‘–โ‰ 0 in the ๐‘–th row of the dominance matrix ๐ปโˆ—. When this value is 0, we set |๐ปโˆ—๐‘–|=1. Then the number of potential minimal solutions of the SFRE cannot exceed the value PN3=๐‘š๎‘๐‘–=1||๐ปโˆ—๐‘–||,(15) where PN3โ‰คPN2โ‰คPN1. In [6, 8, 20], the authors use the symbol โŸจ๐‘๐‘–/๐‘—โŸฉ to indicate the coefficients โ„Žโˆ—๐‘–๐‘—=๐‘๐‘–โ‰ 0. We have โ„Žโˆ—๐‘–๐‘—โˆง๐‘ฅ๐‘—=๐‘๐‘– if ๐‘ฅ๐‘—โˆˆ[๐‘๐‘–,1] and ๐‘ฅ๐‘—=๐‘๐‘– is the ๐‘—th component of a minimal solution. A solution of the ๐‘–th equation can be written as๐ป๐‘–=๐‘›๎“๐‘—=1๎ƒก๐‘๐‘–๐‘—๎ƒข.(16)

In [6, 8] the concept of concatenation ๐‘Š is introduced to determine all the components of the minimal solutions and it is given by๐‘Š=๐‘š๎‘๐‘–=1๐ป๐‘–=๐‘š๎‘๐‘–=1๎ƒฉ๐‘›๎“๐‘—=1๎ƒก๐‘๐‘–๐‘—๎ƒข๎ƒช.(17)

The following properties hold:(i)commutativity: ๎ƒก๐‘๐‘–1๐‘—1๐‘๎ƒข๎ƒก๐‘–2๐‘—2๎ƒข=๎ƒก๐‘๐‘–2๐‘—2๐‘๎ƒข๎ƒก๐‘–1๐‘—1๎ƒข,(18)(ii)associativity: ๎ƒก๐‘๐‘–1๐‘—1๐‘๎ƒข๎‚ต๎ƒก๐‘–2๐‘—2๐‘๎ƒข๎ƒก๐‘–3๐‘—3=๐‘๎ƒข๎‚ถ๎‚ต๎ƒก๐‘–1๐‘—1๐‘๎ƒข๎ƒก๐‘–2๐‘—2๐‘๎ƒข๎‚ถ๎ƒก๐‘–3๐‘—3๎ƒข,(19)(iii)distributivity with respect to the addition: ๎ƒก๐‘๐‘–1๐‘—1๐‘๎ƒข๎‚ต๎ƒก๐‘–2๐‘—2๎ƒข+๎ƒก๐‘๐‘–3๐‘—3=๎ƒก๐‘๎ƒข๎‚ถ๐‘–1๐‘—1๐‘๎ƒข๎ƒก๐‘–2๐‘—2๎ƒข+๎ƒก๐‘๐‘–1๐‘—1๐‘๎ƒข๎ƒก๐‘–3๐‘—3๎ƒข,(20)(iv)absorption for multiplication: ๎ƒก๐‘๐‘–1๐‘—1๐‘๎ƒข๎ƒก๐‘–2๐‘—2๎ƒข=โŽงโŽชโŽจโŽชโŽฉ๎ƒก๐‘๐‘–1โˆง๐‘๐‘–2๐‘—๎ƒข,if๐‘—1=๐‘—2=๐‘—,unchanged,otherwise,(21)(v)absorption for addition: ๎ƒก๐‘๐‘–1๐‘—1๐‘๎ƒข๎ƒก๐‘–2๐‘—2๎ƒขโ‹ฏ๎ƒก๐‘๐‘–๐‘š๐‘—๐‘›๎ƒข+๎ƒก๐‘๐‘˜1๐‘—1๐‘๎ƒข๎ƒก๐‘˜2๐‘—2๎ƒขโ‹ฏ๎ƒก๐‘๐‘˜๐‘š๐‘—๐‘›๎ƒข=โŽงโŽชโŽจโŽชโŽฉ๎ƒก๐‘๐‘–1๐‘—1๐‘๎ƒข๎ƒก๐‘–2๐‘—2๎ƒขโ‹ฏ๎ƒก๐‘๐‘–๐‘š๐‘—๐‘›๎ƒข,if๐‘๐‘–โ„Ž=๐‘๐‘˜โ„Ž,โ„Žโˆˆ{1,โ€ฆ,๐‘š},unchanged,otherwise.(22)

We can determine the minimal solutions ๐‘‹low(๐‘ก)=(๐‘ฅlow1(๐‘ก),๐‘ฅlow2(๐‘ก),โ€ฆ,๐‘ฅlow๐‘›(๐‘ก))โˆ’1, ๐‘กโˆˆ{1,โ€ฆ,PN(3)}, with components ๐‘ฅlow๐‘—(๐‘ก)=๎‚ป๐‘๐‘–๐‘ก,if๐‘๐‘–๐‘กโ‰ 0,0,otherwise.(23)

The above definitions shall be clarified in the following example of an SFRE with 4 equations and 6 unknown:๎€ท1.0โˆง๐‘ฅ1๎€ธโˆจ๎€ท0.0โˆง๐‘ฅ2๎€ธโˆจ๎€ท0.0โˆง๐‘ฅ3๎€ธโˆจ๎€ท0.9โˆง๐‘ฅ4๎€ธโˆจ๎€ท0.2โˆง๐‘ฅ5๎€ธโˆจ๎€ท0.0โˆง๐‘ฅ6๎€ธ๎€ท=0.1,0.5โˆง๐‘ฅ1๎€ธโˆจ๎€ท0.3โˆง๐‘ฅ2๎€ธโˆจ๎€ท0.4โˆง๐‘ฅ3๎€ธโˆจ๎€ท0.5โˆง๐‘ฅ4๎€ธโˆจ๎€ท0.3โˆง๐‘ฅ5๎€ธโˆจ๎€ท0.4โˆง๐‘ฅ6๎€ธ๎€ท=0.3,0.7โˆง๐‘ฅ1๎€ธโˆจ๎€ท0.4โˆง๐‘ฅ2๎€ธโˆจ๎€ท0.2โˆง๐‘ฅ3๎€ธโˆจ๎€ท0.7โˆง๐‘ฅ4๎€ธโˆจ๎€ท0.4โˆง๐‘ฅ5๎€ธโˆจ๎€ท0.2โˆง๐‘ฅ6๎€ธ๎€ท=0.3,0.4โˆง๐‘ฅ1๎€ธโˆจ๎€ท0.7โˆง๐‘ฅ2๎€ธโˆจ๎€ท0.2โˆง๐‘ฅ3๎€ธโˆจ๎€ท0.4โˆง๐‘ฅ4๎€ธโˆจ๎€ท0.7โˆง๐‘ฅ5๎€ธโˆจ๎€ท0.2โˆง๐‘ฅ6๎€ธ=0.3.(24)

We haveโŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ๐ด=,๐ต=

By using the normal form, we obtain that โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ๐ด=,๐ต=

Now we compute the matrix ๐ดโˆ— and the vector IND as follows:๐ดโˆ—=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽ3113โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ,IND=.(27)

The SFRE is consistent because each component of IND is not null. The greatest solution is given by๐‘‹gr=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ

Now we calculate the help matrix ๐ป and the dominant matrix ๐ปโˆ— as follows:โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ,๐ป๐ป=โˆ—=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ .

Then we have |๐ปโˆ—1|=|๐ปโˆ—2|=|๐ปโˆ—3|=1, |๐ปโˆ—1|=3 and hence PN3=3. By using the properties (18)โ€“(23), we have that ๎‚ฌ๐‘Š=0.32๎‚ญ๎‚€๎‚ฌ0.11๎‚ญ+๎‚ฌ0.14๎‚ญ+๎‚ฌ0.15=๎‚ฌ๎‚ญ๎‚0.11๎‚ญ๎‚ฌ0.32๎‚ญ+๎‚ฌ0.32๎‚ญ๎‚ฌ0.14๎‚ญ+๎‚ฌ0.32๎‚ญ๎‚ฌ0.15๎‚ญ.(30)

The three minimal solutions are given by๐‘‹low(1)=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ,๐‘‹low(2)=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ,๐‘‹low(3)=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ The three maximal interval solutions are given by๐‘‹max(1)=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ[][][][][][]โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ 0.1,0.10.3,0.30.0,0.30.0,0.10.0,0.10.0,0.3,๐‘‹max(2)=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ[][][][][][]โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ,๐‘‹0.0,0.10.3,0.30.0,0.30.1,0.10.0,0.10.0,0.3max(3)=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ[][][][][][]โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ .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 โ€œcausesโ€ and we want to analyze the possible โ€œsymptomsโ€. 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 ๐‘๐‘–, 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 ๐‘Ž๐‘–๐‘— of impact of the ๐‘—th cause to the ๐‘–th 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 ๐‘ฅ๐‘—, the mean interval solution ๐‘‹Max๐‘€,๐‘ก(๐‘—) with (6). The linguistic label Rel๐‘ก(๐‘œ๐‘ ) is assigned to the output variable ๐‘œ๐‘ . Then he calculates the reliability index Rel๐‘ก(๐‘‚), given from formula (7), associated to this maximal interval solution ๐‘ก. After the iteration of this step, the expert determines the reliability index (7) for each maximal interval solution, by choosing the output vector ๐‘‚ for which Rel(๐‘‚) 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.

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 ๐‘˜ facts ๐‘–1,โ€ฆ,๐‘–โ„Ž to every subzone.(ii)For each input fact, a fuzzy partition in ๐‘š๐‘“ fuzzy sets is created for every ๐‘“=1,โ€ฆ,โ„Ž. To each fuzzy set, the expert associates a linguistic label. After the fuzzification process, the expert determines the ๐‘š most significant equations, where ๐‘šโ‰ค๐‘š1+๐‘š2+โ‹ฏ+๐‘š๐‘˜. The input vector ๐ต=[๐‘1,โ€ฆ,๐‘๐‘š] is set, where each component ๐‘๐‘– (๐‘–=1,โ€ฆ,๐‘š) is the membership degree to the ๐‘–th 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 ๐‘š symptoms to the ๐‘› causes by defining the impact matrix ๐ด with entries ๐‘Ž๐‘–๐‘— with ๐‘–=1,โ€ฆ,๐‘š, ๐‘—=1,โ€ฆ,๐‘›.(iii)An SFRE (1) with ๐‘š equations and ๐‘› unknowns is created. We use the algorithm from [8] to determine all the solutions of (1). Thus we determine ๐‘‡ maximal interval solutions.(iv)maxRel๐‘กโˆถ=0// (the maximal reliability is initialized to 0).(v)For each maximal interval solution ๐‘‹max,๐‘ก, with ๐‘ก=1,โ€ฆ,๐‘‡, we define the vector column ๐‘‹Mean๐‘ก via formula (6).(vi)Rel๐‘กโˆถ=0.(vii)For each output variable ๐‘œ๐‘ , with ๐‘ =1,โ€ฆ,๐‘˜, if ๐‘ฅ๐‘—1,โ€ฆ,๐‘ฅ๐‘—๐‘  are the unknown associated to ๐‘œ๐‘ , let score๐‘ก(๐‘ )=max{๐‘‹Mean๐‘ก(๐‘—1),โ€ฆ,๐‘‹Mean๐‘ก(๐‘—๐‘ )}.(viii)Rel๐‘กโˆถ=Rel๐‘ก+score๐‘ก(๐‘œ๐‘ ).(ix)Next ๐‘ .(x)Rel๐‘กโˆถ=Rel๐‘ก/๐‘˜// (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 ๐‘‚=[๐‘œ1,โ€ฆ,๐‘œ๐‘˜].(xii)Next ๐‘ก with ๐‘ก=1,โ€ฆ,๐‘‡.(xiii)Next ๐‘ with ๐‘=1,โ€ฆ,๐‘ƒ.

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.

In this experiment, we consider the following four output variables: โ€œ๐‘œ1= Economic prosperityโ€ (wealth and prosperity of citizens), โ€œ๐‘œ2= Transition into the jobโ€ (ease of finding work), โ€œ๐‘œ3= Social Environmentโ€ (cultural levels of citizens), and โ€œ๐‘œ4= Housing developmentโ€ (presence of building and residential dwellings of new construction). For each variable, we create a fuzzy partition composed by three TFNs called โ€œlowโ€, โ€œmeanโ€, and โ€œhighโ€ presented in Table 3.


๐‘œ 1
๐‘œ 2
๐‘œ 3
๐‘œ 4

District ๐‘– 1 ๐‘– 2 ๐‘– 3 ๐‘– 4 ๐‘– 5 ๐‘– 6 ๐‘– 7

S. Giovanni0.5760.2440.0410.0310.0540.3530.097

Moreover, we consider the following seven input parameters: ๐‘–1 = percentage of people employed = number of people employed/total work force, ๐‘–2 = percentage of women employed = number of women employed/number of people employed, ๐‘–3 = percentage of entrepreneurs and professionals = number of entrepreneurs and professionals/number of people employed, ๐‘–4 = percentage of residents graduated = numbers of residents graduated/number of residents with age >6 years, ๐‘–5 = percentage of new residential buildings = number of residential buildings built since 1982/total number of residential buildings, ๐‘–6 = percentage of residential dwellings owned = number of residential dwellings owned/total number of residential dwellings, and ๐‘–7 = 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 โ€œlowโ€, โ€œmeanโ€, and โ€œhighโ€, whose values are reported in Table 5. In Tables 6 and 7, we show the values obtained for the 21 symptoms ๐‘1,โ€ฆ,๐‘21; moreover, we report the input variable and the linguistic label of the correspondent TFN for each symptom ๐‘๐‘–. In order to form the SFRE (1) in each subzone, the expert defines the equations by setting the impact values ๐‘Ž๐‘–๐‘— by basing over the most significant symptoms.

Input variableLowMeanHigh

๐‘– 1 0.000.400.600.400.600.800.600.801.00
๐‘– 2
๐‘– 3
๐‘– 4
๐‘– 5
๐‘– 6
๐‘– 7

Subzone ๐‘ 1 : ๐‘– 1 = low ๐‘ 2 : ๐‘– 1 = mean ๐‘ 3 : ๐‘– 1 = high ๐‘ 4 : ๐‘– 2 = low ๐‘ 5 : ๐‘– 2 = mean ๐‘ 6 : ๐‘– 2 = high ๐‘ 7 : ๐‘– 3 = low ๐‘ 8 : ๐‘– 3 = mean ๐‘ 9 : ๐‘– 3 = high ๐‘ 1 0 : ๐‘– 4 = low ๐‘ 1 1 : ๐‘– 4 = mean ๐‘ 1 2 : ๐‘– 4 = high

S. Giovanni0.120.880.000.280.720.000.950.050.000.450.550.00

Subzone ๐‘ 1 3 : ๐‘– 5 = low ๐‘ 1 4 : ๐‘– 5 = mean ๐‘ 1 5 : ๐‘– 5 =high ๐‘ 1 6 : ๐‘– 6 = low ๐‘ 1 7 : ๐‘– 6 = mean ๐‘ 1 8 : ๐‘– 6 = high ๐‘ 1 9 : ๐‘– 7 = low ๐‘ 2 0 : ๐‘– 7 = mean ๐‘ 2 1 : ๐‘– 7 = high

S. Giovanni0.870.

Now we illustrate this procedure for each subzone.

4.1. Subzone โ€œBarraโ€

The expert chooses the significant symptoms ๐‘2, ๐‘4, ๐‘5, ๐‘7, ๐‘10, ๐‘11, ๐‘15, ๐‘17, ๐‘18, ๐‘19, by obtaining an SFRE (1) with ๐‘š=10 equations and ๐‘›=12 unknowns (Table 8).

Output variableScore1( ๐‘œ ๐‘  )Score2( ๐‘œ ๐‘  )Score3( ๐‘œ ๐‘  )Score4( ๐‘œ ๐‘  )

๐‘œ 1 highhighhighhigh
๐‘œ 2 meanmeanmeanmean
๐‘œ 3 lowlowlowlow
๐‘œ 4 lowlowlowlow

The matrix ๐ด of the impact values ๐‘Ž๐‘–๐‘— has dimensions 10ร—12 and the vector ๐ต of the symptoms ๐‘๐‘– has dimension 10ร—1 and both are given below. The SFRE (1) is inconsistent and eliminating the rows for which the value IND(๐‘—) = 0, we obtain four maximal interval solutions ๐‘‹max(๐‘ก) (๐‘ก=1,โ€ฆ,4) and we calculate the vector column ๐‘‹Mean๐‘ก on each maximal interval solution. Hence we associate to the output variable ๐‘œ๐‘  (๐‘ =1,โ€ฆ,4), the linguistic label of the fuzzy set with the higher value calculated with formula (6) obtained for the corresponding unknowns ๐‘ฅ๐‘—1,โ€ฆ,๐‘ฅ๐‘—๐‘  and given in Table 8. For determining the reliability of our solutions, we use the index given by formula (7). We obtain that Rel๐‘ก(๐‘œ1)=Rel๐‘ก(๐‘œ2)=Rel๐‘ก(๐‘œ3)=Rel๐‘ก(๐‘œ4)=0.6025 for ๐‘ก=1,โ€ฆ,4 and hence Rel(๐‘‚)=max{Rel๐‘ก(๐‘‚)โˆถ๐‘ก=1,โ€ฆ,4}=0.6025 where ๐‘‚={๐‘œ1,โ€ฆ๐‘œ4}. We note that the same final set of linguistic labels associated to the output variables ๐‘œ1 = โ€œhighโ€, ๐‘œ2 = โ€œmeanโ€, ๐‘œ3 = โ€œlowโ€, and ๐‘œ4 = โ€œlowโ€ is obtained as well. The relevant quantities are given below. โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ,โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ,๐ด=๐ต=0.980.360.631.000.400.600.100.590.411.00(33)๐‘‹max(1)=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ[][][][][][][][][][][][]โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ 0.40,0.400.36,0.360.00,1.000.00,0.360.00,1.000.00,0.360.00,1.000.00,0.360.41,0.411.00,1.000.00,0.100.00,0.10,๐‘‹max(2)=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ[][][][][][][][][][][][]โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ,๐‘‹0.40,0.400.00,0.360.00,1.000.36,0.360.00,1.000.00,0.360.00,1.000.00,0.360.41,0.411.00,1.000.00,0.100.00,0.10max(3)=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ[][][][][][][][][][][][]โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ 0.40,0.400.00,0.360.00,1.000.00,0.360.00,1.000.36,0.360.00,1.000.00,0.360.41,0.411.00,1.000.00,0.100.00,0.10,๐‘‹max(4)=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ[][][][][][][][][][][][]โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ,0.40,0.400.00,0.360.00,1.000.36,0.360.00,1.000.00,0.360.00,1.000.36,0.360.41,0.411.00,1.000.00,0.100.00,0.10๐‘‹Mean1=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ 0.400.360.500.180.500.180.500.180.411.000.050.05,๐‘‹Mean2=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ,0.400.180.500.360.500.180.500.180.411.000.050.05๐‘‹Mean3=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ 0.400.180.500.180.500.360.500.,๐‘‹Mean4=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ .0.400.180.050.360.500.180.500.360.411.000.050.05(34)

4.2. Subzone โ€œPoggiorealeโ€

The expert chooses the significant symptoms ๐‘2, ๐‘5, ๐‘8, ๐‘11, ๐‘12, ๐‘14, ๐‘15, ๐‘17, ๐‘18, ๐‘19, ๐‘20, by obtaining an SFRE (1) with ๐‘š=11 equations and ๐‘›=12 unknowns (Table 9). The matrix ๐ด of the impact values ๐‘Ž๐‘–๐‘— has dimension 11ร—12 and the vector ๐ต of the symptoms ๐‘๐‘– has dimension 11ร—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 ๐‘‹max(๐‘ก) (๐‘ก=1,โ€ฆ,12), and we calculate the vector column ๐‘‹Mean๐‘ก on each maximal interval solution. The relevant quantities are given below. โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ,โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ,๐ด=๐ต=0.930.991.00.630.370.70.30.870.130.750.25(35)๐‘‹max(1)=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ[][][][][][][][][][][][]โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ 0.37,0.370.0,0.30.13,0.130.75,0.750.0,0.130.0,0.130.0,1.00.0,0.130.0,0.130.25,0.250.0,0.250.0,0.13,๐‘‹max(2)=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ[][][][][][][][][][][][]โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ,๐‘‹0.37,0.370.0,0.30.13,0.130.75,0.750.0,0.130.0,0.130.0,1.00.0,0.130.0,0.130.0,0.250.25,0.250.0,0.13max(3)=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ[][][][][][][][][][][][]โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ 0.37,0.370.0,0.30.0,0.130.75,0.750.13,0.130.0,0.130.0,1.00.0,0.130.0,0.130.25,0.250.0,0.250.0,0.13,๐‘‹max(4)=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ[][][][][][][][][][][][]โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ,๐‘‹0.37,0.370.0,0.30.0,0.130.75,0.750.13,0.130.0,0.130.0,1.00.0,0.130.0,0.130.0,0.250.25,0.250.0,0.13max(5)=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ[][][][][][][][][][][][]โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ 0.37,0.370.0,0.30.0,0.130.75,0.750.13,0.130.0,0.130.0,1.00.0,0.130.0,0.130.25,0.250.0,0.250.0,0.13,๐‘‹max(6)=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ[][][][][][][][][][][][]โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ,๐‘‹0.37,0.370.0,0.30.0,0.130.75,0.750.13,0.130.0,0.130.0,1.00.0,0.130.0,0.130.0,0.250.25,0.250.0,0.13max(7)=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ[][][][][][][][][][][][]โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ 0.37,0.370.0,0.30.0,0.130.75,0.750.0,0.130.0,0.130.0,1.00.13,0.130.0,0.130.25,0.250.0,0.250.0,0.13,๐‘‹max(8)=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ[][][][][][][][][][][][]โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ,๐‘‹0.37,0.370.0,0.30.0,0.130.75,0.750.0,0.130.0,0.130.0,1.00.13,0.130.0,0.130.0,0.250.25,0.250.0,0.13max(9)=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ[][][][][][][][][][][][]โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ 0.37,0.370.0,0.30.0,0.130.75,0.750.0,0.130.0,0.130.0,1.00.0,0.130.13,0.130.25,0.250.0,0.250.0,0.13,๐‘‹max(10)=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ[][][][][][][][][][][][]โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ,๐‘‹0.37,0.370.0,0.30.0,0.130.75,0.750.0,0.130.0,0.130.0,1.00.0,0.130.13,0.130.0,0.250.25,0.250.0,0.13max(11)=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ[][][][][][][][][][][][]โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ 0.37,0.370.0,0.30.0,0.130.75,0.750.0,0.130.0,0.130.0,1.00.0,0.130.0,0.130.25,0.250.0,0.250.13,0.13,๐‘‹max(12)=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ[][][][][][][][][][][][]โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ,0.37,0.370.0,0.30.0,0.130.75,0.750.0,0.130.0,0.130.0,1.00.0,0.130.0,0.130.0,0.250.25,0.250.13,0.13๐‘‹Mean1=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ 0.370.150.130.750.0650.0650.50.0650.0650.250.1250.05,๐‘‹Mean2=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ,0.370.150.130.750.0650.0650.50.0650.0650.1250.250.065๐‘‹Mean3=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ 0.370.150.0650.750.130.0650.50.0650.0650.250.1250.065,๐‘‹Mean4=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ 0.370.150.0650.750.130.0650.50.0650.0650.1250.250.065๐‘‹Mean5=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ 0.370.150.0650.750.0650.130.50.0650.0650.250.1250.05,๐‘‹Mean6=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ,0.370.150.0650.750.0650.130.50.0650.0650.1250.250.05๐‘‹Mean7=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ 0.370.150.0650.750.0650.0650.50.130.0650.250.1250.065,๐‘‹Mean8=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸ