Generalized morphisms, a new tool for comparative evaluation of performance of fuzzy implications,...

Fuzzy Sets and Systems 117 (2001) 297–315www.elsevier.com/locate/fss

Generalized morphisms, a new tool for comparative evaluationof performance of fuzzy implications, t-norms and co-norms

in relational knowledge elicitationEunjin Kim, Ladislav J. Kohout ∗

Department of Computer Science, Florida State University, 203 Love Building, Tallahassee, FL 32306-4530, USA

Received August 1995; received in revised form January 1999

Abstract

This paper describes a technique for identi�cation of the knowledge structures implicit in scienti�c data and furtherdevelops the supporting methodology for analysis of fuzzy relational structures using the mathematical theory of FuzzyRelations and Generalized Morphism. Initial knowledge elicited using repertory grid with semiotic descriptors is transformedto fuzzy relational knowledge structures which capture semantic relationships in a problem domain by means of BK-product.In analysis, the fuzzy relational structures computed in the di�erent sized domains or those computed by means of variousfuzzy logics can be analyzed using generalized morphism; namely (global) structure embedding, or structure preservingas well as meta analysis of fuzzy relational structures. The result of meta analysis is represented as the Hasse Diagramstructure of meta relations by classifying equivalent classes of fuzzy relational structures. The paper is concluded with a casestudy using real-life engineering data from the aviation industry. c© 2001 Elsevier Science B.V. All rights reserved.

Keywords: Fuzzy relations; Knowledge elicitation; Semiotic descriptors; MVL connectives; Generalized morphism; Metaanalysis; Generalized lattice; (Global) structure embedding

1. Introduction

The crisp theory of mathematical relations formsan important extension of crisp set theory. Indeed,Alfred Tarski in his famous paper on the calculus ofrelations [37] pointed out that (crisp) mathematicalrelations extend the set theory having the power ofset theory with added features of a group theory.

∗ Corresponding author.E-mail address: [email protected] (E. Kim), [email protected].

edu (L.J. Kohout).

Relational algebras of Tarski and his co-workers area signi�cant extension of relational calculus for thecrisp case.

The fuzzy relation theory which is an important ex-tension of fuzzy sets was introduced by Zadeh in [41].This can be generalized to the many-valued theoriesof relations based on t-norms, and residuated implica-tion operators. A excellent logic tool for doing this isH�ajek’s BL-logic [14,18,23].

The use of fuzzzy relational theory also helps tobridge the gap between heuristic, conceptual analysismethods and crisp mathematical logic methods in arti-�cial intelligence. Fuzzy relational theory can be used

0165-0114/01/$ - see front matter c© 2001 Elsevier Science B.V. All rights reserved.PII: S 0165 -0114(99)00047 -0

298 E. Kim, L.J. Kohout / Fuzzy Sets and Systems 117 (2001) 297–315

to discover meaningful structures implicit in scienti�cdata.

In particular, fuzzy relational products and al-gorithms for computing of fuzzy closures andinteriors, when applied to the real world scien-ti�c, management or engineering data, (e.g. clinicalmedicine, psychology, or urban planning, etc.) makeit possible to discover and further analyze structuralrelationships between individual items of data. Thisis very important when we deal with data which can-not always be adequately analyzed by the methodsbased on probability theory. The knowledge struc-tures extracted from such data can often be analyzedin terms of their implications and dependencies ratherthan symmetry [8]. The fuzzy relational approach thatcould be used for this sort of analysis of data has toprovide the adequate theory of fuzzy relations, thecomputing tools and also supporting methodology.

Since the late 1970s, Bandler and Kohout have beendeveloping one such practically useful relational ap-proach consisting of the mathematical theory, method-ology and epistemology as well as of the supportingcomputational algorithms. The identi�cation of struc-tures and properties of real life scienti�c data [2–4,10] is done in this approach by means of BK-FuzzyRelational Triangle Products and Fast Fuzzy Rela-tional Algorithm [3,11]. More recently, BK-productsbased relational systems based on t-norm logics havebeen conveniently formalized by means of BL-logic[18,19].

This further extends the repertory from which afuzzy logic may be chosen for the computation ofFuzzy Relational Products. The use of di�erent fuzzylogics may, however, yield di�erent fuzzy relationalstructures based on formally di�erent extensions oftheories grounded in the BL-logic. Comparative stud-ies of di�erent fuzzy relational structures are thereforerequired with respect to the application of relationalanalysis to di�erent knowledge domains and di�erentproblems.

Our paper contributes towards these ends. InSection 2, we overview methods of knowledge elic-itation suited for obtaining data for relational anal-ysis. In Section 3, we review the basic notions ofBK-products together with the methods of relationalanalysis, in order to make this paper self-su�cient.Section 4 presents an overview of generalized mor-phisms introduced by Bandler and Kohout in 1977

[1]. The description in this section is more detailedbecause the main article [1] in which these resultswere published is not easily available. Section 5presents the mathematical methodology of struc-ture analysis by means of generalized morphismsin terms of object language of a relational calcu-lus. Because in the fuzzy case the analysis yieldsdi�erent structures for the same sets of data whendi�erent many-valued connectives are employed,some methods of comparison are required. These arefurnished in Section 6. The main concern of thissection is meta-analysis and analysis of structurepreservation. Section 7 presents detailed case studywhich demonstrates the utility of our approach on thereal world engineering data.

2. Knowledge elicitation

One important application of relations is in the elic-itation of latent knowledge structures possessed byhuman experts. In this endeavor, we have to employmethods of knowledge elicitation, in order to obtainsuitable material for further relational processing bywhich the relational structural representation of theelicited knowledge is obtained.

Knowledge elicitation is extraction of expert’sdeclarative and procedural knowledge. The knowl-edge elicitation process is concerned with identi�-cation and formation of the domain concepts andstructural domain knowledge. Various methods canbe used to abstract internal and causal knowledgepossessed by experts.

In our study, we use a method of Repertory Gridtechniques to elicit the initial information as well assemiotic descriptors.

2.1. Semiotic descriptors

Fuzzy relations can manipulate semantic informa-tion that is carried by linguistic labels. The conditionsimposed on logic operations applied to fuzzy rela-tional systems are strictly mathematically de�ned. Butthe logic semantic is not su�cient to deal satisfacto-rily with the meaning of linguistic labels that carrythe conceptual meaning of applications. It has to besupplemented by some semiotic notions that can be

E. Kim, L.J. Kohout / Fuzzy Sets and Systems 117 (2001) 297–315 299

expressed as algebraic restrictions over the basic fuzzyrelational system.

Semiotic descriptors are obtained by exploratoryknowledge elicitation [31]. We use repertory grids toelicit the meaning used by human experts [17]. Oncethe relevant semiotic descriptors are identi�ed, the se-mantic relationships between them can be captured byrepertory grids using bi-polar attributes, each attributeconsisting of a pair of semiotic descriptors.

After the repertory grids are applied to a particularproblem, and the semantic relationship captured by thegrids, the grids are transformed into fuzzy relationswhich relate the semiotic descriptors.

2.2. Repertory Grid

The Repertory Grid is a tool for eliciting a relationbetween some entities. It refers to three kinds of semi-otic descriptors: objects, attributes and respondents.

Objects consist of a name and a physical=functionaldescription of items or elements such as diseases ina medical system, components to be assessed in apropulsion system or the places in the urban environ-ment where the inhabitants live, go, eat, etc., in UrbanStudies models [35].

Attributes are properties or discriminants whichcharacterize objects, such as signs or symptoms in amedical system [8], factors which impact the cost ineach process in a propulsion system, or psychologicalconstructs attached to places in urban study. Attributesare made up of two opposing concepts called poles;the emphasized, positive or favored pole is alleged tobe the primary pole.

Respondents are participants of the study. For ex-ample, medical doctors assessing patients, technolo-gists involved in a high technology design, the set ofindividual inhabitants of the urban area whose opinionis elicited by the repertory grid.

Through a repertory grid, an individual respondentmeasures or estimates the degree to which the rela-tion between objects and attributes holds (semantichedges) in terms of a bipolar graded scale. Such de-gree describes not only the level of emphasis placed ona pole but also which pole is being stressed. Thus, theobjects are evaluated with respect to the attributes andopinions of respondents. The aim of the techniques isto capture not only technological but also human fac-tors involved in the technological process.

Initial knowledges elicited by repertory grid is con-verted to a fuzzy relational matrix form, further pro-cessed by means of fuzzy relational BK-products andforms a knowledge structure of fuzzy relational formdescribed by semiotic descriptors. In the next section,we describe the mathematical de�nitions, seman-tics and properties of fuzzy relational BK-products.Table 1 shows a sample of repertory grid with semi-otic descriptors further used in our case study inSection 7 of this paper.

3. Formation of fuzzy relational structure

The essential role in extracting the latent structuresof concepts implicitly contained in empirical data isplayed by non-associative and psuedoassociative BK-products (compositions) of mathematical relations.In order to capture the conceptual meaning repre-sented by relational models, semantic descriptors areintroduced that map the types of linguistic labels intodi�erent relational sorts. This makes it possible tointerpret relational properties and structural charac-teristics of relational models in terms of conceptualfeatures of an investigated knowledge domain.

3.1. A brief overview of BK-products

3.1.1. Mathematical de�nitionsWhere R is a relation from X to Y , and S a relation

from Y to Z , a product relation R∗S is a relation fromX to Z , determined by R and S. There are several typesof product used to produce product-relations [10,22].Each product type performs a di�erent logical actionon the intermediate sets, as each logical type of theproduct enforces a distinct speci�c meaning on theresulting product-relation R ∗ S. In the following de�-nitions of the products, Rij; Sjk represent the fuzzy de-grees to which the respective statements xiRyj; yjSzkare true.

Product type Set-based de�nitionCircle product: x(R ◦ S)z⇔ xR intersects Sz

Triangle subproduct: x(R / S)z⇔ xR ∼⊆ SzTriangle superproduct: x(R . S)z⇔ xR ∼⊇ SzSquare product: x(R S)z ⇔ xR∼= Sz


Table 1Repertory grid of forging process

Pr. Primary pole Secondary pole

Description Range 3 2 1 0 −1 −2 −3 Description Range

C10 Conservative L=D 2 : 1 Risky L=D ¿2:5 : 1C11 Low cost=piece $2000 High cost=piece $20 000C12 Low variation in cost ± 5% High variation in cost ± 25%C1 High analytical modeling ¿5 Low analytical modeling...

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

...C17 Easy forgeability Di�cult forgeabilityC4 Easy lubricity 100 cycles Di�cult lubricity 1–3 forge cycleC2 Large processing window ± 25◦F Small processing window ± 2–5◦FC3 Low temperature 1400◦F High temperature 2300◦F...

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

...C26 Good �ll= ash Unacceptable �ll= ashC27 Near-net-shape, forged shape Non-near-net shape, forged shape

Product type Many-valued logic formulaCircle product: (R ◦ S)ik =

∨j (Rij ∧ Sjk)

Triangle subproduct: (R / S)ik =∧j (Rij→ Sjk)

Triangle superproduct: (R . S)ik =∧j (Rij← Sjk)

Square product: (R S)ik =∧j (Rij ≡ Sjk)

The table of de�nitions given above contains twodi�erent notational forms: (1) the notation using theconcept of set inclusion and equality [2,3]. (2) Many-valued logic (MVL)-based notation, which uses thelogic connectives

∧; → or ≡. These two di�erent

forms of relational compositions are algebraicallyequivalent, producing the same mathematical results.Distinguishing these forms is, however, importantwhen constructing fast and e�cient computationalalgorithms.

The logical symbols for the logic connectivest-norm, t-co-norm, both implications and the equiva-lence in the above formulas represent the connectivesof some many-valued logic, chosen according to theproperties of the products required.

Given a general formula (R ∗ S)ik :=⊕j(Rij@Sjk),

it yields two types of fuzzy relational products, to re-place the outer connective

⊕with

∧(de�ned above)

or with (1=|J |)∑;

(R ∗ S)ik :=∧j

(Rij @ Sjk): Harsh product:

(R ∗ S)ik :=1|J |

∑j

(Rij @ Sjk): Mean product:

In particular, by choosing appropriate many-valuedlogic operations for the logic connectives, theBoolean (crisp) case extends to a wider variety ofmany-valued logic based (fuzzy) relational systems[22,4,8,9,25,22]. While we used the classical min andmax for t-norm and t-conorm, respectively, we ap-plied various MVL implication operators for the com-putation of BK-products. Table 2 shows an exampleof MVL implication operators that we applied in ourstudy. The details of choice of the appropriate many-valued connectives are discussed in [4,8,9,25,22].


Table 2Fuzzy implication operators [4]

No. Notation Name De�nition

1. S# Standard sharp a1→ b=

{1 i� a 6= 1 or b= 10 otherwise

2. S Gaines–Rescher a2→ b=

{1 a6b0 otherwise

3. S∗ G�odel a3→ b=

{1 a6bb otherwise

4. G43 Goguen a4→ b= min(1; b=a)

4′. G43′ Modi�ed Gaines 43 a4′→ b= min(1; b=a; (1 − a)=(1 − b))

5. L Lukasiewicz a5→ b= min(1; 1 − a + b)

5.5. KDL Reichenbach a5:5→ b= min(1; 1 − a + ab)

6. KD Kleene–Dienes a6→ b= (1 − a)∨ b

7. EZ Early Zadeh a7→ b= (a∧ b)∨ (1 − a)

= (a6→ b)∧ ka; ka= (1 − a)∨ a

8. W Willmott a8→ b= ((1 − a)∨ b)∧ (a∨ (1 − b)∨ (b∧ (1 − a)))

= (a7→ b)∧ kb

= (a7→ b)∧ ka∧ kb

3.2. Linguistic labels of relational products

Each relational product de�ned above must possessa clearly de�ned meaning giving it a concrete practi-cal linguistic interpretation within the domain of itsapplication. This interpretation is provided by meansof interpretable linguistic labels, semiotic descrip-tors. The assignment of semiotic descriptors bothselect the appropriate concepts from the domain andalso partially determines the linguistic meaning ofthe composed relation computed by the relationalproduct. A simple but useful general relational modelrelates semiotic descriptors of two kinds: objects andattributes.

Let us look at a simple example from the medi-cal domain. If R is the relation between (concrete)objects and attributes such as patients and individualsymptoms, and S a relation between attributes and(abstract) objects, symptoms and diseases, R ∗ S willbe a relation between patients and diseases. The di-

agnostic clinical interpretation of each distinct logicaltype of these product-relations has a distinct clinicalmeaning [20]:

x(R ◦ S)z: degree to which patient xhas at least one symptom of illness z:

x(R / S)z: degree to which x’s symptoms areamong those which characterize z:

x(R . S)z: degree to which x’s symptoms includeall those which characterize z:

x(R S)z: degree to which x’s symptoms areexactly those of illness z:

3.3. Investigating relational properties

BK-relational product can be used to derive rela-tional structures on a particular domain. If R is anyrelation (perhaps itself a product of other relations)from X to Y , its transposed relation RT is a relationfrom Y to X . When R is composed with its transpose


RT, R ∗RT is a relation from X to X , so generating afuzzy relational structure on X while RT ∗R a relationfrom Y to Y generating a structure on Y .

Relations so constructed might exhibit someimportant relational properties that reveal importantcharacteristics and interrelationships of the source ofinformation. Hence, methods for detecting variousrelational properties of given relations are important.

Relational properties, such as re exivity, symme-try and transitivity, and classes such as tolerances,equivalences and partial orders have been systemat-ically studied since the 1940s and are well knownfor crisp relations. In 1971 Zadeh extended these tofuzzy relations in his paper “Similarity relations andfuzzy orderings” [41]. From the point of view of ap-plicability of relational methods to analysis of struc-tures, this is a very signi�cant extension, as it allowsfor approximation of the crisp structure by fuzzyones. Thus one may identify approximate similaritiesin data, approximate equivalences and orders. Suchapproximations are paramount in the context of thepresent applications – in situations when only incom-plete, partial information about a new technology isavailable.

In Zadeh’s extension, the properties are namelyglobal, and must be shared by all the elements ofa relation. To deal with the properties shared by asubset of elements, who participate in the relation,Bandler and Kohout was to provide an adequate de�-nition of locality for both crisp (non-fuzzy) and fuzzyrelations [5,10,11]. The essence of this extension isto split the property of re exivity into two separateproperties of local re exivity and covering [11]. In therelational structures constructed below, we examineboth local property and global property. In addition,we approximate the product by computing its fuzzyclosures or its fuzzy interiors in order to deal withthe relational properties which are not revealed in thestructure of the product.

3.4. Structure formation

Fast fuzzy relational algorithms [11] are used totest relational properties and extract equivalences, pre-orders and other mutual dependencies between ob-jects or attributes. It is essential for a relation to havethe properties of re exivity, transitivity, antisymme-try to be an ordering. We compute local preorder clo-

sure of a relation, say R, which is the least inclusiverelation which has the property of (local) re exiv-ity and transitivity, and includes R, and, we computethe degree of approximation of the original relationto the closure. When the degree is su�ciently high,the closure is taken as the relation used in furtheranalysis.

3.4.1. Alpha (�)-cutsBy taking proper �-cuts, crisp structures are ex-

tracted from fuzzy relations. Four di�erent levels of�-cuts are taken from the local preorder closure ofthe product with respect to the distance from the max-imum, mean and the minimum values of the productcomputed: Height(H), Half-Upper(HU), Mean(M)and Half-Lower(HL) cuts. These �-cuts are used foradditional analysis and interpretation in a variety ofcrisp perspectives.

3.4.2. Hasse diagram (HD)The order relationships of properties are represented

in the use of HD and equivalence classes. The graphi-cal representation of the implicit hierarchy on the crispmatrix of each �-cut is the HD which shows ordering.An algorithm for computing a Hasse diagram from lo-cal preorders can be found in [11]. The hierarchy ofattributes gives us insight into the opinions of a re-spondent’s view of her=his modeling system.

4. Generalized morphism

4.1. Comparing relational structures

Problems of compatibility of relations on two setscan be investigated by means of homomorphisms.Generalizations of homomorphisms, however, allowus to solve problems of compatibility of relations ofa more general nature than those amenable to anal-ysis by homomorphisms. According to Bandler andKohout, these generalizations are also of interest ingeneral systems studies because correspondence be-tween systems and their parts do not entirely satisfythe classical conditions of homomorphisms [6,7].Generally, the fuzzy relational structures computed byBK-products are generated by composing two fuzzyrelations in the generalized set so that it is moreappropriate to study the compatibility of such fuzzy


relational structures in terms of generalized mor-phisms. We apply generalized morphisms (GM) tostudy compatibility of FRSs under two scenarios.

All the de�nitions and theorems given in this sec-tion hold not only for the non-fuzzy (crisp) case butalso for general fuzzy BL-logics based on residuatedcontinuous t-norms [18]. This is because the proofsdepend only on the properties of the continuous t-norm and its residuum, many-valued implication →.All fuzzy relations in the sequel are de�ned in the in-terval [0; 1]. R(X Y ) denotes the lattice of all fuzzyrelations from X to Y ; ⇔ represents the meta-logicalstatement “if and only if ”.

Fig. An arrow diagram for generalized morphisms.

4.2. Mathematical de�nitions

De�nition 1 (Bandler and Kohout [7]). Let F; R; G;S be relations between the sets A; B; C; D suchthat F ∈R(A C), R∈R(A B), G ∈R(B D),S ∈R(C D).1. The conditions that ∀a∈A, ∀b∈B, ∀c∈C, ∀d∈D, (aFc& aRb& bGd)→ cSd will be expressedin any of the following ways:(a) FRG; S are compatible forward or forward-

compatible;(b) F;G respect R; S forwards;(c) R; S absorb F;G forwards;(d) F;G are generalized homomorphisms from R

to S.2. The conditions that ∀a∈A, ∀b∈B, ∀c∈C, ∀d∈D, (aFc& cSd& bGd)→ aRb will be expressedin any of the following ways:(a) FRG; S are compatible backward or backward-

compatible;(b) F;G respect R; S backwards;(c) R; S absorb F;G backwards;(d) F;G are generalized proteromorphisms from

R to S.

3. The simultaneous ful�llment of the conditions in(1) and (2) will be expressed as follows:(a) FRG; S are compatible both ways or bothway-

compatible;(b) F;G respect R; S both ways;(c) R; S absorb F;G both ways;(d) F;G are generalized amphimorphisms from R

to S.

De�nition 2 (Bandler and Kohout [7]). For arbi-trary fuzzy relations in [0; 1], R from the set X to Y ,S from Y to Z de�ne:1. R ◦ S = (∀x)(∀z)(∃y)(xRy&ySz);2. R / S = (∀x)(∀z)(∀y)(xRy→ySz);3. R . S = (∀x)(∀z)(∀y)(xRy←ySz).

Theorem 3. Compatibility Theorem Bandler andKohout [7]1: FRG: S are forward compatible⇔ FT ◦R ◦Gv S⇔RvF / S .GT.

2: FRG: S are backward compatible⇔ F ◦ S ◦GTvR⇔ S vFT / R .G.

FRG: S are both-ways compatible i� they areboth forward and backward compatible. The R’sof forward compatibility constitute a lower ideal,while those of backward compatibility constitutean upper ideal or �lter; the bothways problem hasa solution i� their intersection is non-empty. Theconventional homomorphism is a special case ofboth-ways compatibility. The generalized morphismspresented here are relevant not only theoretically,but have also an important practical use in solv-ing systems of relational inequalities and equa-tions. This has been further generalized by Kohout[18] to continuous t-norms based logics using theBL-logic predicate calculus of H�ajek as a prooftechnique.

5. Analysis of fuzzy relational structures by meansof generalized morphism

This section describes how generalized morphisms(GM) are applied to study fuzzy relational structures(FRSs) generated under two scenarios.


5.1. Comparisons of fuzzy relational structuresbetween in the full domain and in the sub domain

First, we apply generalized morphism to study thecompatibility of fuzzy relational structures generatedin di�erent sized domains, where one domain is a sub-domain of the other.

De�nition 4 (Bandler and Kohout [1]). Let a rela-tion R be R∈R(A→B)1. The fore-restriction of R to A′⊆A is

A′|R∈R(A′→B), given byaA′|Rb⇔ a∈A′ and aRb, that is, byA′|RS =RS ∩ (A′×B).

2. The after-restriction of R to B′⊆B isR|B′ ∈R(A→B′), given byaR|B′b⇔ b∈B′ and aRb, that is, by(R|B′)S =RS ∩ (A×B′).

3. The fore- and after-restriction of R to A′ and B′

is given by(A′|R|B′)S =RS ∩ (A′×B′).

Remark. Clearly the fore- and after-restrictionoperations commute with each other, so that(A′|R)|B′ = A′|(R|B′) = A′|R|B′ .

Algorithm 5 shows the analysis processes of com-parisons of two fuzzy relational structures in di�erentsized domains.

Algorithm 5.1. Selection of two fuzzy relations R and R′ (concep-

tual decision and mathematical restriction of theknowledge domain):R : A→O, where A is a set of attributes and O isa set of observers.R′ : A′→O, where A′⊆A.Therefore, R′ = A′|R.

2. Computation of fuzzy relational triangle subprod-ucts (FRTSubP) (from the de�nition of /):R′ / R′T : A′→A′, and R /RT : A→A.⇒R′ / R′T = A′|R /RT

|A′ .3. Computation of their local preorder closures: (see

[11] for detailed description of mathematical pro-cedures and proof)U ′ = local preorder closure of R′ / R′T, andU = local preorder closure of R /RT.Generally, U ′ 6= A′|U|A′ .

4. Performing k-level �-cuts on structures, U ′ andA′|U|A′ in 3 (k ∈{H;HU;m;HL}), yielding U ′

�k andA′|U�k |A′ as the input data for step 5.

5. Application of generalized morphism (based onthe de�nitions in Section 4.2 above):Let F be relations from the domain of U ′ to thedomain of A′|U|A′ .Let G be relations from the target of U ′ to thedomain of A′|U|A′ .Apply U ′

�k to R, while A′|U�k |A′ is applied to S.≡: (RvF / S .GT) ∧ (S vFT / R .G) –

(Global) structure embedding⇔ F U ′

�k G; A′|U�k |A′ are bothway com-patible.

⇔ U ′�k ≡ A′|U�k |A′ .

⇔ The structure of U ′�k in the sub-domain has

the exact contextual match with the struc-ture of U�k in the full domain.

⇔ The structure of U ′�k is structurally embed-

ded in the structure of U�k .⇔ U ′

�k is a (global) substructure of U�k .⇔ The structure of the sub-domain is

(globally) embedded in the structure inthe full domain.NOTE: If A′ =A, thenU ′

�k is a substructureof itself, i.e. two structures are congruent.

@: RvF / S .GT – Structure preserving⇔ F U ′

�kG; A′|U�k |A′ are forward compatible.⇔ All the relations of U ′

�k are sub relationsof A′|U�k |A′ .

⇔ All the relations of U ′�k are forward com-

patible with the relations of A′|U�k |A′ .⇔ The relations of U ′

�k in the sub-domain iscompatible with the relations of U�k in thefull domain.

⇔ The structure of U ′�k in the sub-domain is

preserved in the structure of U�k in the fulldomain as subrelations.

C(A): S vFT / R .G – Structure con ict⇔ F U ′

�k G; A′|U�k |A′ are backward com-patible.

⇔ All the relations of U ′�k are super rela-

tions of A′|U�k |A′ .⇔ Relations in U ′

�k contain more relationsas well as all the relations in A′|U�k |A′ .

⇔ The relations of U ′�k in the sub-domain

is not compatible with the relationsof U�k in the full domain because


it loses some relations in the fulldomain.Therefore, the structure of U ′

�k in thesub-domain is incompatible with thestructure of U�k in the full domain.

C(A@): ” (RvF / S .GT) ∧ ” (S vFT / R .G) –Structure con ict⇔ F U ′

�k G; A′|U�k |A′ are neither forwardcompatible nor backward compatible.

⇔ The relations of U ′�k is not compatible

with the relations of A′|U�k |A′ since thereexist some relations r1 and r2 such thatr1 ∈U ′

�k ; r2 ∈ A′|U�k |A′ but r1 =∈ A′|U�k |A′and r2 =∈U ′

�k .⇔ The relations ofU ′�k in the sub-domain is

incompatible with the relations of U�k inthe full domain since there are both gainsand loss of relations in both domains.Therefore, the structure U ′

�k in the sub-domain is incompatible with the struc-ture U�k in the full domain.

Step 5 above gives the equivalent conditions and de-termines the conceptual meaning of the operations forcomparisons of structures: structure embedding (≡),structure preserving (@), and two kinds of structurecon ict: C(A) and C(A@).

5.2. Analytic processes for comparisons of fuzzyrelational structures computed by di�erent fuzzylogics

Generalized morphism may be also applied tostudy the compatibility of fuzzy relational structurescomputed by means of di�erent MVL implicationoperators. This is essential in many-valued logic(MVL)-based relational structures. Indeed, di�erentMVL yields a di�erent relational theory. A di�erentrelational theory in turn extracts di�erent structuralfeatures from empirical data.

Algorithm 6.1. Selection of a fuzzy relation R:R : A→O, where A is a set of attributes and O isa set of observers.

2. Computation of fuzzy relational triangle sub-products (FRTSubP) by means of di�erent fuzzylogics:

Ri/ RT : A→A – a fuzzy logic i is applied.

Rj/ RT : A→A – a fuzzy logic j is applied.

3. Computation of their local preorder closures:

Ui = local preorder closure of Ri/ RT.

Uj = local preorder closure of Rj/ RT.

4. Performing k-level �-cuts on structures. Ui and Uj

in 3 (k ∈{H;HU;m;HL}), yielding Ui�k and Uj

�k asthe input data for step 5.

5. Application of generalized morphism:Let F be relations from the domain of Ui to thedomain of Uj.Let G be relations from the target of Ui to thedomain of Uj.Apply Ui

�k to R, while Uj�k is applied to S.

≡: (RvF / S .GT) ∧ (S vFT / R .G) – Con-gruence (bothway global structure embed-ding)⇔ F Ui

�k G; Uj�k are bothway compatible.

⇔ Ui�k ≡Uj

�k .⇔ The structure of Ui

�k computed by the op-erator i and the structure of Uj

�k computedby the operator j are (globally) structureembedded bothways.

⇔ The structure of Ui�k computed by the op-

erator i and the structure of Uj�k computed

by the operator j are bothway (global) sub-structure.The structure of Ui

�k computed by the op-erator i is equivalent (or congruent) withthe structure of Uj

�k computed by the op-erator j.

@: RvF / S .GT – Forward structurepreserving⇔ F Ui

�k G; Uj�k are forward compatible.

⇔ All the relations of Ui�k are sub-relations

of Uj�k .

⇔ All the relations of Ui�k computed by the

operator i are forward compatible withthe relations of Uj

�k computed by theoperator j.


erator i is preserved in forward with thestructure of Uj

�k computed by the opera-tor j as subrelations.

A: S vFT / R .G – Backward structurepreserving


⇔ F Ui�k G; Uj

�k are backward compatible.⇔ All the relations of Ui

�k are super relationsof Uj

�k .⇔ All the relations of Ui

�k computed bythe operator i are backward compatiblewith the relations of Uj

�k computed by theoperator j.


erator i is preserved in backward with thestructure of Uj

�k computed by the opera-tor j as super relations.

C(A@): ” (RvF / S .GT) ∧” (S vFT / R .G) –Structure con ict⇔ F Ui

�k G; Uj�k are neither forward com-

patible nor backward compatible.⇔ The relations of Ui

�k is not compatiblewith the relations of Uj

�k since thereexist some relations r1 and r2 suchthat r1 ∈Ui

�k ; r2 ∈Uj�k but r1 =∈Uj

�k andr2 =∈Ui

�k .⇔ The relations of Ui�k computed by the op-

erator i is incompatible with the rela-tions of Uj

�k computed by the operator jsince there are both gains and loss of re-lations in both structures. Therefore, thestructure Ui

�k computed by the operatori is incompatible with the structure Uj

�kcomputed by the operator j at k-level �-cut.

Algorithm 6 yields comparisons of two distinct re-lational structures in two di�erent relational theories.Hence, it provides a mechanism for comparing the ef-fect of logic connectives based on di�erent t-norms.

6. Further analysis

6.1. Meta analysis

The various fuzzy relational structures computed bymeans of di�erent fuzzy logics could be compared bymeans of generalized morphism in Section 5.2. Its re-sult may be further described in terms of meta analysis.We represent such a result as a Hasse Diagram struc-ture of meta relations; namely meta relational struc-ture (MRS). A meta relational structure is a structure

of FRSs computed by means of various fuzzy logics.Therefore, the MRS tells us the structural relationshipof FRSs with respect to their contextual matches anddependencies. The meta relational structure (MRS)is generated in three phases: construction of meta re-lational matrix (MRM), classi�cation of meta equiv-alent classes and hassi�cation of MRM into HasseDiagram (HD) structure of meta relations.

6.1.1. Construction of meta relational matrixAt each k-level �-cut, �H ; �HU ; �m and �HL, a meta

relational matrix, MRMk is constructed for a pair ofFRSs (FRSi; FRSj) which were computed by meansof MVL implication operator i and j, respectively.Based on the analysis result by means of generalizedmorphism in Section 5.2, the meta relational matrixis generated by the algorithm below:

Algorithm 7.1. If (FRSi; FRSj) is both forward and backward com-

patible by generalized amphimorphism at k-level�-cut, i.e. FRSi≡FRSj, then

MRMk [i; j] = 1 and MRMk [j; i] = 1:

2. If (FRSi; FRSj) is forward compatible by meansof generalized homomorphism, i.e. FRSi@FRSj,then

MRMk [i; j] = 1 whereas MRMk [j; i] = 0:

3. If (FRSi; FRSj) is backward compatible by meansof proteromorphism, i.e. FRSiAFRSj, then

MRMk [i; j] = 0 whereas MRMk [j; i] = 1:

4. If (FRSi; FRSj) is neither in forward nor in back-ward compatible, i.e. FRSi A@ FRSj, then

MRMk [i; j] = 0 and MRMk [j; i] = 0:

The meta relational matrix (MRM) at each k-level �-cut is �lled with the relational values 1 or 0with respect to the compatibility of FRSs, being pa-rameterized by the name of MVL implication operatorused for their computations. Since every FRSi is com-patible with itself bothways, meta-relations in MRMkare re exive at any of k-level �-cuts. In addition, aMRMk has the property of transitivity, because FRSiis de�nitely forward=backward compatible with FRSl


if FRSi is forward=backward compatible with FRSj,and FRSj is also forward=backward compatible withFRSl, respectively. Thus, MRMk has the property ofpreorderness. Therefore, the meta-relations in MRMmay be represented as a meta relational structure inHasse Diagram in the same way that fuzzy relationmatrix with the preorderness at each �-cut has beenrepresented as Fuzzy Relational Structure in HasseDiagram.

6.1.2. Classi�cation of equivalent classesClassi�cation of equivalent classes is a process

to group the FRSs. The FRSs which are bothway-compatible by generalized amphimorphism aregrouped into an equivalent class. When FRSi; FRSjand FRSl are mutually bothway compatible, all ofthem are classi�ed as equivalent. The MRMk gener-ated at each k-level �-cut is used to classify the equiv-alent classes. A classi�cation table of meta equivalentclasses is generated as the result. The classi�cationtable is used in the next sub-phase of hassi�cation ofmeta relations for generating the MRS.

6.1.3. Hassi�cation of the meta relationsAfter classifying equivalent classes, meta-relations

inMRMk are hassi�ed to represent them as a meta re-lational structure by means of Hasse Diagram (HD).If FRSi is compatible with FRSj in forward, FRSi isput at the lower level and is linked to FRSj. Simi-larly, if FRSi is compatible with FRSj in backward,FRSi is put at the upper level and is linked from FRSj.Through such processes, the hierarchy of MRS is gen-erated in HD structure by keeping the transitivity be-tween FRSs. In such a HD structure of meta-relations,the FRS in the lower level implies the FRS in the upperlevel if there is a link from the lower level into the up-per level. Similarly, the FRS in the upper level is im-plied by the FRS in the lower level. Equivalent FRSsclassi�ed are put within the same block. Through thesehassi�cation processes, the meta relational structure(MRS) of fuzzy relational structures computed bymeans of di�erent MVL implication operators is gen-erated in a HD structure at each k-level �-cut.

6.1.4. �-cut of �-value 1When a fuzzy relational structure is computed

by means of MVL implication operators such as

S; S∗; G43; G43′ and L, their �-cuts taken with aspeci�c �-value 1 shows a distinctive feature.

Theorem 8. Fuzzy relational structure of BK-triangle subproduct; computed by means of S; S∗;G43; G43′ of L; with a harsh or a mean criteria arealways equivalent at the �-cut of value 1.

Proof. 1. Harsh product:For a fuzzy relational product R / S, its �-cut

with �-value 1 (R / S)�=1 means that (R / S)�ik = 1if (R / S)ik = 1 or (R / S)�ik = 0 otherwise, since∀(R / S)ik ∈ [0; 1].

(Rp/ S)ik = 1; ∀p∈{S; S∗; G43; G43′; L}

⇔ ∀p; ∀j∈ J; ∧j(Rijp→ Sjk) = 1.

⇔ ∀p; ∀j; minj(Rijp→ Sjk) = 1.

⇔ ∀p; ∀j; {Ri1 p→ S1k = 1; Ri2p→ S2k = 1; : : : ; Rim

p→Smk = 1}.

⇔ ∀p; ∀j; {Ri16S1k ; Ri26S2k ; : : : ; Rim6Smk}, bythe de�nitions of operators p.

Hence, (RS/ S)ik = 1⇔ (R

S∗/ S)ik = 1⇔ (R

G43/ S)ik = 1

⇔ (RG43′/ S)ik = 1⇔ (R

L/ S)ik = 1:

Thus, (RS/ S)�=1 ≡ (R

S∗/ S)�=1 ≡ (R

G43/ S)�=1 ≡

(RG43′/ S)�=1≡ (R

L/ S)�=1:

2. Mean product:

Similarly, (Rp/ S)ik = 1; ∀p∈{S; S∗; G43; G43′; L}

⇔ ∀p; ∀j∈ J; ∑j(Rijp→ Sjk)=|J |= 1.

⇔ ∀p; ∀j; {Ri1 p→ S1k = 1; Ri2p→ S2k = 1; : : : ; Rim

p→Smk = 1}, since ∀(Rij p→ Sjk)∈ [0; 1].

⇔ ∀p; ∀j; {Ri16S1k ; Ri26S2k ; : : : ; Rim6Smk}, bythe de�nitions of operators p.

Hence, (RS/ S)ik = 1⇔ (R

S∗/ S)ik = 1⇔(R

G43/ S)ik = 1

⇔ (RG43′/ S)ik = 1⇔ (R

L/ S)ik = 1:

Thus, (RS/ S)�=1 ≡ (R

S∗/ S)�=1 ≡ (R

G43/ S)�=1 ≡

(RG43′/ S)�=1 ≡ (R

L/ S)�=1:

Therefore, fuzzy relational structures at the �-cut ofthe value 1, computed by means of S; S∗; G43; G43′

or L are always equivalent both of harsh product andof mean product.

Corollary 9. Fuzzy relational structure of BK-triangle super product; computed by means of


S; S∗; G43; G43′ or L; with a harsh or a mean criteriaare always equivalent at the �-cut of value 1.

Corollary 10. Fuzzy relational equivalence classesof BK-triangle square product; computed by meansof S; S∗; G43; G43′ or L; with a harsh or a mean cri-teria are always equivalent at the �-cut of value 1.

For the fuzzy relational structures (or equivalenceclasses) generated from BK-super product (or squareproduct), the above theorem holds and it may beproven similarly.

7. Case study

In their previous papers Kohout and Bandler andtheir associates have described the application of fuzzyrelational analysis to a variety of data obtained frommedical studies, psychology and cognitive science, ar-chitecture and urban studies, and business. In this pa-per we use for our case study data from an industrialengineering application that is directed at integratingengineering, business and marketing data in cost anal-ysis models. This may be of wide interest not only tothe engineering community but also to the businesscommunity and everyone concerned with optimizationof costs.

7.1. Description of data

The data to be used in our case study is one of thesamples used in the development of new propulsiontechnologies. In the aeronautic industry, early cost-driven decisions are critical to low cost developmentand production of jet engines. For �rms that are at theleading edge of high technology, it often involves de-termining the essential cost factors of a design thathas never before been manufactured. This creates aformidable problem, because technological and busi-ness decisions are based on incomplete, uncertain in-formation about the product that is to be designed andmanufactured [17,13].

The application of fuzzy relational knowledgerepresentation schemes allows us to incorporatemethods for dealing with incompleteness, uncer-tainty, con icts and use the data the precision ofwhich is limited at the early decision stages. The

use of fuzzy attributes for cost models and a�ord-ability applications addresses the problem of thelimited (or unavailable) data in the situation ofmaterials with limited characterization data, pro-cesses with limited empirical data, and manufac-turing processes with little or no manufacturingbase. Our case study has the following scenario[32,26].

A low pressure turbine (LPT) cover plate is to bemanufactured, using new material, namely, gammatitanium. Prior to any production characterization,the part is to be costed out, using the expert knowl-edge concerning manufacturing processes and avail-able cost estimation that is available for other small -titanium parts. This knowledge has been extrapo-lated by experts–engineers of LPT Cover Plate that isto be made. The mental model of this extrapolationpossessed by the experts is elicited by means of reper-tory grids specially designed to deal with the contextof this engineering problem. The values captured bythe repertory grids are normalized and converted intofuzzy relations. Fuzzy relations are further processesby TRISYS to produce Hasse diagrams of processparameters.

There are �ve main processing steps that will beinvolved in manufacturing the low pressure turbinecover plate: ingot, extrusion, forging, heat treatmentand machining. In this paper, we use a sample of datain forging process for the fuzzy relational study of itscost attributes. For each process there we have de-signed a speci�c repertory grid. Further details of thismanufacturing application may be found in the paperspreviously published [17,13,34,26,30]. Here we con-centrate only on the details necessary to demonstratehow the structural analysis is done in practice usingthe approach described in the previous sections of thispaper.

7.2. Fuzzy relational model

For our relational study, initial knowledge is elicitedfor each process, using the repertory grid which wasdescribed in Section 2.2.

Our model relation M to be used in the sequelM ∈R(C × (O×G)×P); relates the following threelists (sets) of entities:• the set C of cost attributes in each process;


Fig. 1. Fuzzy relational structures in the full domain.

• the Cartesian product O×G of the set O of ob-servers, assessors or measuring sensors; with the setG of process identi�ers;• the set P of parts or components.

Applying the usual selection and projection opera-tors the ternary relation is decomposed into a familyof 2-ary relations in R(C × (O×G)), indexed by theset P. The relation R from the set of process attributesC to the set (O×G) is composed with its transposeRT by means of triangle subproduct and the local pre-

order closure computed by the TRYSIS system. Theresult of this computation is a relation from the costattributes C to C in a process of the component. Thisrelation shows the dependencies and equivalences ofcost attributes in a process represented as a preorderrelation.

A sample result displaying dependences and equiv-alences of attributes in the process of forging is shownin Fig. 1. These �gures show the Hasse diagram (HD)structures displaying the preorders of cost attributes


computed from the fuzzy relational triangle sub-product over processes, by means of di�erent MVLimplication operators. The set of cost attributes for theprocess of forging used to elicit the relevant relationwere displayed in Table 1.

7.3. Examples of meta analysis

We �rst analyze fuzzy relational structures (FRS)of cost attributes in the full domain of 27 attributes.

Fuzzy relational triangle subproductsRi/ RT were com-

puted by applying each di�erent MVL implicationoperator (i) with harsh criterion in the full domain.Despite the use of di�erent implication operators,common features are observed in FRSs.

Due to the e�ect of �-value 1, FRSs computedby means of S; S∗; G43; G43′ and L shows equiva-lent structures at the height cut, shown in Fig. 1. Notonly at the high cut but also at the half-upper cut, thesame structure was generated for such implication op-erators; forming two equivalence classes, {c2; c5} and{c10; c19}.

On the other hand, fuzzy relational structures com-puted by means of KD, EZ or W operators revealinteresting common features of their own, however,di�er from FRSs computed by S, etc. One of the com-mon feature is their equivalent classes: {c8; c20} and{c3; c16; c17; c18} in all of the three structures. Such at-tributes, {c3; c16; c17; c18}; or {c8; c20} also formedrelations in chain (c16→ c3→ c17 & c16→ c3→ c18

or c20→ c8) in the FRS computed by S operator, etc.They, however, did not generate an equivalent classbecause of the lack of symmetry. Second feature isthat only one �-cut could be taken because the restof � cuts (half-upper, mean and half-lower) yields thesame �-cut with the height cut due to only two val-ues of their fuzzy products. Third feature is that FRScomputed by W operator (FRSW) yields subrelationsof FRSEZ which also yields subrelations of FRSKD. Inparticular, FRSW is locally preordered and is a sub-structure both of FRSKD and of FRSEZ.

The FRSKDL shows a local preordered structurewithout any equivalence classes.

Such di�erent fuzzy relational structures thathave been generated are further analyzed by meansof generalized morphism in terms of their for-ward or backward compatibility, as described in

Section 5.2. Using its result, the meta analysis ondi�erent fuzzy relational structures is made by �rstconstructing the meta relational matrix. Fig. 2 showsan example of meta relational matrix, meta equiva-lent classes and the meta relational structure at theheight cut.

7.4. Analysis of fuzzy relational structures betweenin the sub-domain and in the full domain

The structure analysis algorithm described inSection 5.1 is also applied to our sample data offorging process to compare fuzzy relational structurescomputed in the sub-domain with those in the fulldomain. In particular, nine attributes are selected forthe sub-domain, which are common attributes be-tween extrusion process and forging process. A harshcriterion is applied for their computations and �-cutsare taken at the height (�H), as they were in the fulldomain. Fig. 3 depicts resulting fuzzy relational struc-tures (FRS) in the sub-domain computed by means ofvarious MVL implication operators. Those FRSs maybe compared with the corresponding FRSs in the fulldomain (Fig. 1), with respect to a MVL implicationoperator, by means of the analysis processes usinggeneralized morphism.

The result contains two qualitatively di�erent casesat the height cut: (global) substructure and structurecon ict. The most dominant result is for the FRSscomputed by means of S; S∗; G43; G43′ and L inthe sub-domain. These are equivalent as they were inthe full domain. But, in addition they are also (glob-ally) structure embedded to those in the full domain.Hence, they are (global) substructures of those in thefull domain. Similarly, the FRSs computed by meansof KD, EZ or W in the sub-domain are also (global)substructures of those in the full domain, respectively.The FRS computed by means of Reichenbach, how-ever, shows a big con ict between the FRS of thesub-domain and the FRS of the full domain; only are exive relation c3→ c3 is compatible on both struc-tures. Most of the FRSs computed by di�erent MVLimplication operators in the sub-domain are structureembedded to those in the full domain so that they are(global) substructures of those in the full domain. Thisis so not only at the �-cuts with high � values such asa height cut and a half-upper cut but also at the mean


Fig. 2. Meta analysis.

cut. On the other hand, a FRS computed by means ofG43′ in the sub-domain is structure preserved in FRSof the dull domain at the mean cut rather than beingstructured embedded. The former is a subrelation ofthe latter; two relations such as c2→ c4 and c5→ c4

are lacking in the sub-domain while c4 is grouped toan equivalent class with c2 and c5 in the full domain.Though the former is not a substructure but subrela-tions of the latter, it shows a very high degree (94.7%)of similarity with the FRS in the full domain. InFig. 1, common nine attributes and their relations aredepicted in bold character and lines, so that it is easyfor the reader to compare FRSs of the sub-domain(Fig. 3) with those of the full domain, with respect totheir implication operator.

On the other hand, Fig. 4 shows an example ofthe typical structure preserving: two fuzzy relational

structure computed by means of G43′ and taken at themean cut, in the sub-domain and in the full domain,respectively. The relations c2→ c4 and c5→ c4 allowc4 to join in the equivalent class with {c2; c5} in thefull domain, while c4 depends on the equivalent classof {c2; c5} in the sub-domain; therefore, a hierarchyof c2; c4 and c5 is modi�ed in the full domain.

8. Concluding remarks

It is well known for the work in fuzzy controllers[36], medical arti�cial intelligence [12,38,39], inevaluation of medical clinical data [16,15,21,24],analysis of data in engineering and business studies[13,32,26,33,27] as well as from the work on provid-ing strategies of inference for automated reasoning


Fig. 3. Fuzzy relational structures in the sub domain and global substructure: global structure embedding.

Fig. 4. Subrelations: structure preserving.


and resolution-based theorem provers [28,40,29] thatthe choice of the appropriate many-valued base logicis determined by the nature of the problem. In appli-cations of fuzzy logics, sets and relations, the choiceof connectives determine the mathematical characterof these theories. The mathematical properties of thisapparatus must match the logical properties of theempirical data in order to make the application offuzzy mathematics successful.

The selection of the type of many-valued connec-tives (often t-norms, co-norms and correspondingimplications) is usually based on direct empiricalcomparison of results without grounding it a rigorousformal methodology for investigating and comparingthe structures and meaning of empirical results. Ourpaper, however, has provided a rigorous methodologythat facilitates empirical evaluation of appropriatenessand performance of diverse connectives for a varietyof applications by analyzing the structural similaritiesand di�erences.

In our opinion, the merit of our approach rests onthe fact that it rigorously combines in a uni�ed frame-work, the theoretical and empirical facets of meta-analysis of choice-decisions for selecting appropriatelogics for each particular application. Recently, wehave also developed a computational tool based on thetheory of the Bandler–Kohout theory of generalizedmorphisms called Gmorph [16] which usefully com-plements the earlier tool for relational analysis andtesting of local and global relational properties calledTrysis [11].

Acknowledgements

This work has been partially supported by theNSF grants No. DMI 952-5991 and DMI 972-6027,“Decision-Making with Incomplete Information inan Integrated Product and Process DevelopmentEnterprise – A Management Decision Tool for CostModeling and A�ordability Analysis”.

We would like to thank Dr. Barbara DuBroskyof Pratt & Whitney for close collaboration with usthrough the duration of the NSF supported project,for valuable advice on selecting the suitable compo-nents for a�ordability analysis and for collecting thedata by means of repertory grids, used for computingthe relational structures in Section 7.

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