The IVFN-IOWAD operator

A special case of the Quasi IVFN-IOWAD is the IVFN-IOWAD operator, pre- sented next.

Definition 15 ( [210]). An IVFN-IOWAD operator of dimension n is a mapping f : Rn_×IVFNn_×IVFNn_{→ R that has an associated weighting vector W of dimension}

f (hu1,A1,B1i,hu2,A2,B2i,...,hun,An,Bni) = n

∑

j=1

wjDj, (5.6)

where Dj is the d(Ai,Bi) value of the triplet hui,Ai,Bii having the jth largest ui,

where uiis the order inducing variable and Ai,Biare the argument variable repre-

sented in the form of IVFN’s.

Theorem 5.2.2 ( [210]). Based on the theorem provided for the Quasi IVFN- IOWAD operator, if f is a IVFN-IOWAD operator, then it is commutative, mono- tone, idempotent, and bounded.

5.2.5 Examples

To better illustrate how the presented definitions can be used, this Section will present a numerical example calculating the OWA-distance of triangular-shaped IVFN‘s and secondly some explanations on how OWAD operators can be em- ployed for fuzzy ontologies are presented.

Numerical example

To illustrate the OWAD concept, we will calculate the OWA-distance of triangular- shaped IVFN’s (the upper and lower fuzzy numbers are triangular fuzzy numbers) choosing g(x) = x, which is a special case of the definition, an IVFN-IOWAD op- erator. In the example we will use the following six triangular IVFN’s:

AL 1= (6,3,2), AU1 = (6,4,3), AL 2= (8,5,4), AU2 = (8,7,6), AL 3= (2,2,4), AU3 = (3,3,6), BL 1= (6,3,3), BU1 = (6,4,4), BL 2= (7,4,3), BU2 = (7,5,4), BL 3= (2,1,1), BU3 = (2,3,3),

The corresponding order inducing variables and weights are defined as: u1=4,u2=1,u3=7.

W = (0.1,0.5,0.4). The aggregation can be calculated as

f (h4,A1,B1i,h1,A2,B2i,h7,A3,B3i)

=_0.1|E(A3− B3)| + 0.5|E(A1− B1)| + 0.4|E(A2− B2)|

OWAD Operators for Fuzzy Ontologies

For this example, it is assumed that the relation specifying different relationships between individuals and concepts takes IVFN’s as values. I.e. by implementing IVFN’s in a fuzzy ontology, they can represent different types of relationships be- tween individuals and concepts (e.g., ’belongs-to’, ’has-a’). The fuzzy quantities range in the [0,1] interval, 1 and 0 indicate a strong and weak relationship, respec- tively.

The information for the fuzzy ontolgy can be obtained from experts. Instead of using the set of IVFN’s with support in the [0,1] interval as the range of the relation, the experts could utilize a reasonable set of linguistic descriptions to assess the value of the relationships in the ontology. For this purpose, linguistic variables represented by trapezoidal IVFN’s can be employed.

When a fuzzy ontology is employed as a decision support system, the first step is to create a subset of the concepts used to describe specific cases or situations. Based on the descriptions and the ontology relations, we can choose the object which provides the most satisfactory solution.

The process of using a fuzzy ontology and the OWAD operators as decision support tools can be summarized in the following steps:

• Step 0.Creating the ontology: using expert knowledge the individuals, I = {i1,i2, . . . ,in}, and the concepts, C = {c1,c2, . . . ,cm}, are defined using fuzzy

relations to specify the relationships between the individuals and objects. • Step 1. Specifying the context: defining a subset of the concepts, Cl =

{cl1,cl2, . . . ,clk}, it can sufficiently describe a given case. Experts can spec-

ify the connection between the case and the concepts using IV FN’s; using the same set of linguistic variables used in previous examples (Table 5.1). Trapezoidal shaped upper and lower fuzzy numbers are implemented. If we have more experts, the opinions will be aggregated by employing an OWA operator.

• Step 2. Defining the importance of the concepts: the decision maker is able to specify importance weights associated with every element in the de- fined subset Cl. These values will be used as order inducing variables in the

aggregation step.

• Step 3. Calculating the distance: using a case description and the order in- ducing variables provided by the experts together with a set of OWA weights, the distance of the case from the individuals in the ontology employing the Quasi IVFN-IOWAD operator can be calculated.

• Step 4. Choosing the closest solution: when the distances are obtained, the individuals that have the smallest distance from the case description will be

chosen as possible solutions. These solutions can then be presented for the decision maker to e.g. evaluate or confirm.

Scenarios illustrating these definitions will be presented in Chapter 7.

5.3 Summary

This Chapter has introduced different generalisations of the OWA operator, apply- ing interval-valued fuzzy numbers as arguments of the aggregation process. Fur- thermore, novel generalisations of the induced OWAD operators have been intro- duced, where the operator is applied on interval-valued fuzzy numbers based on the mean value of IVFN’s. Further, it has been proven that the different introduced ex- tensions of the OWA operator satisfy important properties, such as commutativity and monotonicity.

A lot of decision making problems tend to become unmanageable due to com- plexity. By utilizing fuzzy numbers and linguistic variables one can create suitable description of complicated situations. I.e. the process of aggregating imprecise in- formation described by fuzzy variables plays a crucial role in the decision making process. Developing new variations of the OWA operator is therefore a necessary task.

To show the usability of these novel definitions, some examples are presented to demonstrate the usability of the new OWA definitions as well as validate their produced benefits, for instance by employing the OWA operators in fuzzy ontolo- gies for aggregating information.

The definitions developed in this Chapter is utilized from a more practical per- spective in both Chapter 6 and Chapter 7, where they are applied for aggregating the imprecise information needed for producing different types of advice.

Chapter 6

Fuzzy Ontology Applications

In a fast-paced business environment, there is an need for distributing and sharing the collective knowledge that exists and is being created in organisations. Previous chapters in this thesis have presented methods for utilizing and storing imprecise and tacit knowledge. The development of new mobile technologies, e.g. for the Se- mantic Web, has opened the door for new possibilities to distribute this information effectively [251].

The mobilisation of knowledge will change the business processes of today, as users will be able to receive illustrative, real time advice and support regardless of where they are currently operating. As an example, maintenance personnel could wear a pair of eyeglasses that are connected to the organisations database and the user could receive step-by-step instructions on how to perform the reparation of a broken product; this could work even if the person wearing the classes has no previous experience and knowledge about that particular problem and solution. The expertise of the user in combination with detailed information and instructions received through the eyeglasses will make the whole process possible.

In this chapter, we show how information technologies in combination with fuzzy ontology can utilize and mobilise tacit knowledge; this is demonstrated by presenting different versions of novel web platform and Android applications.

6.1 The Structure of the Applications

This Section introduces the initial application structure developed for mobilising knowledge with the help of fuzzy ontologies, presenting the basic building stones of the server and the graphical user interface (GUI). This structure was used as a basis for the further advancements presented later on in this Chapter.

As the context for the developed applications, a dinner setting is used, where the participants should decide what wine they want to drink with their food. The Fuzzy Wine Ontology 4.1 was used as the main source of knowledge.

Figure 6.1: Technical structure of the application