Structural quantization of vagueness in linguistic expert opinions in an evaluation programme

Fuzzy Sets and Systems 119 (2001) 171–186www.elsevier.com/locate/fss

Structural quantization of vagueness in linguistic expertopinions in an evaluation programme

Debjani ChakrabortyVinod Gupta School of Management, Indian Institute of Technology, Kharagpur-721302, India

Received March 1998; received in revised form October 1998

Abstract

This paper aims to develop a methodology for the construction of a membership function for linguistic expert opinionsin a realistic evaluation problem. It contains a PC-aided procedure of information acquisitions and modelling of datafor favour of an evaluator’s aid. It assumes that the evaluator need not be knowledgeable in fuzzy mathematics but usecommon linguistic terms in the making of his expert comments and uses his subjectivity and common reasoning ability(and sincerity) in conversation with the system. For this purpose, a follow-up model is supplemented which assumesexistence of a single expert only. c© 2001 Elsevier Science B.V. All rights reserved.

Keywords: Linguistic expert opinions; Acquisition of information and processing; Fuzzy sets; Averaging operator

1. Introduction

Evaluation for selection and evaluation for ranking are the two closely related and common facets ofhuman decision making activities in practice. In the working of an academic institution, business organization,awarding=rewarding=funding agencies, etc., most of the decision activities involve a process of evaluation –sometimes meant for the selection of one or few out of the many and for rest of the cases evaluation purportsto serve a process of ranking of ‘the subjects’ facing evaluation. Depending on the organizational behaviourand its overall as well as speci�c need, criteria of evaluation (or queries) change, whom=what to be evaluatedchange but what remains unaltered is the basic process of evaluation, in general, which is a resonance of a setof mental reactions which takes place in the evaluators brain. For a set of speci�c queries with di�erentiableimportance an evaluator or an expert, who is a knowledgeable person in the relevant �eld has to examinethe quali�able informations in favour of a subject (which may be a candidature, a person, an item or aproposal, whatever be it, henceforth, in general, will be termed as a ‘subject’) and has to satisfy himself withappropriate responses. Single or multiple experts may involve in this process. In an evaluation programme allsuch experts’ comments are collated to de�ne overall merit levels for individual subjects.In reality, neither all of the queries nor the corresponding expert comments are clear and unambiguous. In

most of the cases, expert opinions rather come in linguistic forms containing a lot of subjectivity, vagueness

0165-0114/01/$ - see front matter c© 2001 Elsevier Science B.V. All rights reserved.PII: S 0165 -0114(99)00044 -5

172 D. Chakraborty / Fuzzy Sets and Systems 119 (2001) 171–186

and ambiguity too. The main problem in such cases is the problem of information acquisition and modellingthem with proper stress [3]. But at this stage of development of Operations Research methodology, when weare remarkably well-equipped with the theories and applications to deal with many forms of imprecision anduncertainty, development of an object-oriented framework to support acquisition, quantization and aggregationof subjective expert opinions has not yet received considerable attention as it merits. Only a few works canbe referred in this regard. Zimmermann and Zysno [16] showed the empirical and practical relevance of thetheory of fuzzy sets to human decision making and developed an evaluation framework for ‘creditworthinessrating’ from a decision theoritic point of view. In this respect our work is concerned with rather a micro-levelproblem which considers how an individual evaluator’s linguistic responses can quantitatively be structuredin term of fuzzy sets and thus aggregated and compared towards making a decision. In this regard, this workapproaches towards making of an evaluator’s Decision Support System (DSS). Orlowska [9] and Zhang [14]compared methods on structural and functional quantization of vagueness without discussing their applicabilityin any real problem set-up. In a recent paper, Biswas [2] introduced a fuzzy evaluation method of students’answer scripts. This paper did not consider how fuzzy mark can be generated e�ectively and it uses alsovery simpli�ed procedure. In another recent paper, Chen [4] considers evaluation of three tactical missilesystems in which linguistic responses and other crisp data are ranked by integer numbers and then fuzzi�edby symmetric triangular fuzzy numbers which do not re ect vagueness in the true sense and also does notconsider the possibility of generation of a fuzzy number of assymetric shape and it has not dealt with complexranking problem.As a matter of fact, the problem of evaluation, whatever be the level of sophistication of the organization,

the exalted purpose of evaluation, the process as a whole still remains somewhat unorganized and arbitrary.This is what that initiates the present work.In compliance with the above objective, the present work approaches to develop an evaluator’s (PC-based)

DSS incorporating two initial main phases of decision making, viz., information acquisition and modelling.Section 2 contains discussion on the source of vaguenesses and on the elimination of ambiguity in anyrealistic evaluation problem, relevant assumptions and de�nitions and an example. Section 3 contains a PC-based procedure of information acquisition and quantization thereof. In Section 4, we develop (a single expert)evaluation model and aggregation strategy in terms of fuzzy set theoretic operations. The �nal section containsthe concluding remarks.

2. How and in which form vagueness may appear in expert opinion – a discussion

In any real evaluation process, an evaluator �rst has to �x (or it may be assumed that he is given) somespeci�c queries of discriminating importances (weightages) on the basis of these he thoroughly goes throughthe data in favour of the subjects and endorses his comments. To a speci�c query we have nothing to sayabout the ‘what’ of an expert opinion. Here, we are interested only in the type of expression used in an expertopinion, we are interested in knowing whether to a speci�c query an evaluator’s response is clear or not, ifit is not how should one approach so that he can quantize such expression [1].When a query to an expert is not-clear (vague and=or ambiguous), but the data structure in favour of

the ‘subjects’ (on the basis of which an evaluator judges the merit of the subject against the query andmakes his response) is clear, and the reverse case, when the query is clear but the available data structurecontains vagueness and ambiguity, an evaluator may feel in expressing his opinion in linguistic terms, say, not-satisfactory, satisfactory, more or less satisfactory, excellent, good, very good, no-merit, poor, low, medium,high, recommended, may be recommended, etc. In any real evaluation process one is frequently encounteredwith a set of such linguistically expressed expert opinions. Ranking or selection of subjects necessarily requires(as is required by any real decision making problem [6]) acquisition, quantization, aggregation and comparisonof the meaning of such linguistically expressed assessments.

D. Chakraborty / Fuzzy Sets and Systems 119 (2001) 171–186 173

On the other hand, when a query (de�ned and conceived by an evaluator) is clear and the available datastructure is also clear it is not rational to believe (and hence, let us drive out the possibility) that the evaluator’sresponse is not-clear. Here the expression, response is not-clear, as we conceive, includes two kinds of lackof clarity:The �rst is vagueness: The term used to mark the response to a query has some de�nite representation on

a given numerical domain but it does not have any sharp boundary.The other is ambiguity: Here the response includes multiplicity of meaning.In speci�cation of ‘vagueness’, lack of clarity is implicit. Hence, quantization of a vague information

requires the expert’s insight, subjectivity and comprehensibility to be translated on a numerical domain. Onthe other hand, in any ambiguous response, lack of clarity is involved explicitly. For the purpose of evaluation,a properly designed linguistic evaluation sheet, on which an evaluator has to mark his response, may be ableto decline ambiguity in evaluator’s linguistic response.Let us take an example, an evaluator is asked to mark his response in the following set of term-di�erentials:

Option 1 not-satisfactory= satisfactory

If the evaluator put a tick (√) on one of the terms, we have no other way to understand beyond thatthe evaluator is using a bivalent logic comfortably. We can understand only that, to the evaluator, whetherthe given data are quali�able or not. There may be some question about the boundary of the terms butit can be unequivocally said that the linguistic term ‘satisfactory’ is a higher-ordered term than the term‘not-satisfactory’.Take another example, here the evaluator is asked to mark his response in any one of the following sets

of term-di�erentials:

Option 1 not-satisfactory= satisfactory

Option 2 poor= satisfactory= good

Option 3 no-merit= poor= satisfactory= good= excellent

Here we �nd the term ‘satisfactory’ in all of the sets of term-di�erentials; the same linguistic term carriesmultiplicity of meaning as well as non-uniqueness in the matter of its rank among the corresponding setof terms. If an evaluator selects Option 2 and puts a √ on ‘satisfactory’, the message is conveyed that heis choosing approximately relatively worst position and rank than would he have marked that in Option 1.Another sort of interpretation may be made on this issue: The term ‘satisfactory’ in Option 2 is relativelymore imprecise than that in Option 3.Now, let us discuss the content of the above example from another view-point: An evaluator writes a

linguistic term ‘good’ as his opinion. As human beings, we are capable to (fuzzily) perceive and accessit in our memory. But for the sake of mathematical model building, its quantization is required and thenwe face a two-way problem: �rst, about the speci�cation of fuzziness contained in the expression ‘good’,and second, about the speci�cation of the position of the term good on a given numerical domain, say onthe interval [0, 100]. Whereas, choice from the set of term-di�erentials certainly eliminates ambiguity in alinguistic response. Term di�erentials are generally quantized by non-fuzzy ranks or by fuzzy partitions ofa numerical domain in relevant fuzzy literatures. But, to any speci�c query for all subjects, the successiveresponses are quantized by fuzzy numbers of equal extent of fuzziness [4,8]. This, in our opinion, is anoversimpli�ed assumption. Here, on this issue we wish to prescribe a more creditable remedy: The evaluatorwill be asked to select his response from one of the sets of the term-di�erentials.Therefore, whatever be the nature of the query clear or vague, evaluator’s response may be made unam-

biguous by designing a suitable answering pattern. A DSS [11] using an e�ective user-interface or dialogsubsystem and an e�cient knowledge-based-management-system (KBMS) considerably can reduce ambiguity


Fig. 1.

in linguistic expert opinion. At the same time, additional informations on the extent of vagueness can also beaccumulated and represented by convex (normalized) fuzzy sets.From the above discussion, we can conclude that the source of imprecision in an expert opinion may be

detected in two ways,when a query is vague and=or given a query; when; an evaluator’s perception reasoning and expressionare vague.For generality we are not bypassing natural possibilities of the existence of clear queries as well as clear

responses from an expert. Fig. 1 may describe the possible association of clarity and vagueness in query-response tie-ups in a reaslistic evaluation programme.To support applicability of this discussion let us consider a speci�c problem of evaluation as follows:To promote scienti�c=technological=educational interest, a government organization of Science & Technol-ogy, each year calls for submission of proposals to provide research grants to some prospective youngscientists. From among a number of proposals the authority selects a few on the basis of its own expertevaluation reports.

Aiming ful�llment of the above requirement, let us investigate:(1) where and in which form vagueness exists in this problem,(2) for each detected ‘relevant vague information’ how the problem of quantization may be triggered o�, and(3) whatabout on aggregation of relevant informations.Scope of this work does not include group decision making or decision making by concensus. Here, we

assume existence of only one expert. First part of this work will aid the expert with quantization of hislinguistic opinions arised from various queries.According to his own experience and requirement, an evaluator has to �x (or, he is given) some speci�c

queries with discriminating importances (weightages) on the basis of which the expert thoroughly goes throughthe submitted proposals and endorses his comments. For favour of the �rst investigatory, let us frame a simpleand realistic example of a set of queries along with their relative importance for the concerning problem.Among the set of queries some are de�nitely clear, some de�nitely vague and some require evaluators

personal interpretation and judgement to be determined to be clear or vague. For example, we do not haveany doubt with the clarity of query Q22 and the vagueness of Q21, whereas about query Q12 an evaluator


Table 1

Query serial Query description Weightagea Nature of querya

State of the art and objective of the project:Q11 Aim of the project: How will it bene�t 0.2 V

Scienti�c=Technological=Educationalinterest?

Q12 Are you con�dent that the proposed study= 0:2(K%)b Cmethodology can attain the proposedstandard

Expertise available with the proposer:Q21 Academic Excellence 0:05 VQ22 Research Experience (in no. of years) 0:075 CQ23 No. of publications in well refereed journals 0:075 CQ24 Work environment (repute of the proposed 0:05 V

place of work)Q25 Present status of the proposer (can he 0:05 V

provide considerable infrastructural facilitytime and attention as will be required by theproposed work?)

Q26 Laurels and distinctions 0:05 VQ27 Other achievements 0:05 V

Budget proposed:Q31 Cost-e�ectiveness 0:2 VQ32 If you feel that with Q31 you are not No weight is

satis�ed and that the proposed project attachedproposal is pretty satisfactory, please suggestrevision in proposed budget as you requireto consider Q31 to be excellent

a De�ned by an expert.b See Section 3.1 for detail.

may have an opinion that as it requires quantization of his belief, it should be interpreted as a vague query(Table 1). Similarly, another opinion may also be there: as it requires a normal response in either ‘Yes’ orin ‘No’ or in ‘may be’ type answer (which he can easily rank as well), it should be de�ned as a clear query.Therefore, to an evaluator de�nition of the nature of a query is a matter of his perception and reasoning overthe issue. Here, an evaluator’s convenience may also be regarded to be a supportive point in de�ning thenature of the query.Hence, we would like to dint into the matter of the �rst and second investigations by the following two

assumptions:

Assumption 2.1. Before any attempt in studying an evaluator’s linguistic response, the query in question (onthe basis of which the evaluator responds) is to be identi�ed whether it demands crisp or vague reasoning.

Assumption 2.2. By an evaluator if a query is identi�ed to be a clear one; while responding, his mode ofreasoning will follow crisp (formal) logic. On the other hand; if a query is identi�ed to be a vague one; whileresponding to the query the evaluator’s mode of reasoning will follow fuzzy logic.

2.1. Coding of nature of responses

If a query is clearEvaluator may choose his response in the following form:


either NM(Numerical marking)

or LC(Clear linguistic expression: by putting

√on selected set of term di�erentials)

If a query is vagueEvaluator may choose his response in the following form:either LC

(Clear linguistic expression: by putting√on selected set of term di�erentials)

or LP(Vague linguistic expression: by a moving pointer on a selected scale of term di�erentials)

or LD(by de�ning a linguistic expression: by plotting expert’s levels of satisfaction, 1.0 being the fullsatisfaction level and 0.0 being the level of zero satisfaction, on di�erent grades on a gradeddomain [0, 100])

Therefore, response to a query to a subject may be coded as follows:

1 2 3 4 5 6 7 8 9 10 11 12S = Q = R

where, columns 2 and 3 indicate ‘subject’ No. 6 and 7, query No. 11 and 12 indicate whether C is in 8thcolumn then evaluator chooses either NM or LC, and whether V is in 8th column then evaluator chooseseither LC or, LP or, LD and column 8 indicates the nature of query (either C or V ).

3. Acquisition and quantization of expert opinions

A response (from an evaluator for a speci�c query) is said to be clear and unambiguous when it is madeeither through a numerical marking or by selection of one out of given multiple options. On the other hand,a vague response is characterized by its unsharp boundary. A linguistic response may be ambiguous if itincludes multiplicity of optional meaning but possibility of existence of ambiguity can be eliminated by usingan e�cient man–machine interface (dialog).

3.1. Distribution of weightage among the queries vs. speci�cation of a numerical domain on whichresponses are to be quantized

Assumption 3.1. An evaluator assigns numerically de�ned weights to the queries according to his reasoningability; preferential judgement and the overall objective of evaluation.

If there are m queries and if w1; w2; w3; : : : ; wm are the numerical weights attached to the queries, thenw1 + w2 + w3 + · · ·+ wm = 1.For example (see Table 1) Q12 is dependent on Q11. While distributing weights, the evaluator may assume:

how much of the ‘aim is beni�ciary’ that much importance will be assigned to Q12.Let us refer a simple defuzzi�cation method – determination of the centre of gravity [14] which may be

used in determining the approximate value of the linguistic response of Q11. If the value is K , then the weightof Q12 will be 0:2(K%).

Proposition 3.2. For proper re ection of weightages attached to each query; all responses are de�ned andquantized on the same numerical domain.


For the example stated in Section 2, let us take the domain as [0; 100] on which all expert opinions are tobe explained.Truth of the above proposition can easily be established by the following example: For a performance

evaluation two tests are taken and numerical marks are given to the candidates:

Written Oral(Full marks – 70) (Full marks – 30)

Candidate I 45 25Candidate II 50 20

Both of the candidates achieve 70 marks out of 100 marks. Here candidates’ marks are aggregated consid-ering a distribution of weightage over the tests as 7 : 3.If both of the tests were taken as equally weighted then their individual and aggregated achievement will

result as follows:

Written Oral Total(in percentage) (in percentage) (in percentage)

Candidate I 64.29 83.33 73.81Candidate II 71.44 66.66 69.05

The main intention of this work is to highlight some issues involving ‘computation with words’, theproposition incorporates such a to-be-remembered issue for the sake of accuracy and e�ectiveness of knowledgerepresentation.

3.2. Recording of expert opinions: A PC-aided procedure of information acquisition

Five categories of responses are recorded as expert opininons. See Section 2.1. For a subject, if the natureof query is C, for response two categories are open to the evaluator:

(1) RNM ; (2) RLC :

If the nature of query is V , for response three categories are open to the evaluator:

(1) RLC ; (2) RLP; (3) RLD:

3.2.1. Standardization of responsesThese are coded as

S / Q C / R N M

Step 1: Enter responses for all of the subjects in favour of a speci�c query.Step 2: For a speci�c query, display all subjectwise responses.Step 3: Compute (pi=q)100 for ith subject, ∀i where pi6q and pi is the numerical value assigned for ith

subject and q is the maximum attainable numerical value.Explanation: Let us take an example of =Q22C=RNM . Suppose the ith candidate has 7 years of research

experience and the evaluator �nds 8 years of research experience as the maximum among the available set


of proposals. Here pi = 7 and q may be equal to 8 or more according to the evaluators choice. If it occursthat according to the evaluator’s choice, p ¿ q, then the ratio (p=q)100 may be substituted by 100.

3.2.2. Acquisitions and processing of linguistic responsesThese are coded as follows

S / Q C / R L C

Here, we describe the procedure considering the problem stated in Section 2 and Table 1. Let us describethe procedure of acquisition and processing of linguistic response for the 1st subject against Q12 coded asSO1=Q12C=RLC .

Step 1: Evaluator writes his linguistic opinion on the space provided (Figs. 2 and 3):Step 2: It is the �rst query of its kind and the �rst subject is being evaluated. To create the knowledge-base

(Kb), we need this step. See Fig. 4.Step 3: System searches the RLC databaseStep 3.1: Locates the corresponding option and the corresponding position is (column) chosen.Step 3.2: Searches columnwise whether the written linguistic term is there or not.Step 3.3: If it matches, system displays the corresponding set of term-di�erentials and requires relevant

clari�cation from the evaluator. Otherwise the system displays the �rst set of term-di�erentials replacing theterms of corresponding position by the term written by evaluator (see Step 1) and requires further clari�cationfrom the evaluator.

For example, Let the evaluators write ‘highly satisfactory’, and choose Opt. 2, [66:67; 100]. Let RLCdatabase not contain ‘highly satisfactory’, [66.67, 100] in Opt. 2, then it will display (Fig. 5).Step 4: Evaluator may or may not be satis�ed with the other terms of the displayed term set.Step 5: Creation of New Kb.

Example. If

No = √ Yes

then the new set of term di�erential will be preserved in NEW1.RLC.Kb. If√ No = Yes

then the evaluator is asked to replace other levels according to his requirement and satisfaction. Then the newset of term di�erentials will be preserved in NEW1.RLC.Kb.

Step 6: Pass to the next query=subject.To record response coded as SO2=Q12C=RLC , New Kb created for the response code SO1=Q12C=RLC

helps the evaluator to comment on Q12C= for SO2.Step 1∗: Evaluator is asked to select his response.

Example. (This continues the previous one, see Fig. 6.)

Step 2∗: If the evaluator marks his response on the existing set of term di�erentials, then entering ofresponse codes as SO2=Q12C=RLC ends. Then pass to the next query=subject. Otherwise go to the next step.Step 3∗: Evaluator is asked to write his linguistic opinion and everything will be followed as was done in re-

sponding SO1=Q12C=RLC. Then a new set of term di�erentials will be created and added to NEW1.RLC.Kb.NEW1.RLC.Kb will be upgraded to NEW2.RLC.KbNEW2.RLC.KB will help the evaluator to comment on Q12C for the next subject.


Fig. 2.

Fig. 3. RLC database.

3.2.3. Acquisition and processing of linguistic responsesThis is coded as

S = Q V = R L C

This is also a ‘choice from option’ category. Here, acquisition of expert opinions can be performed ex-actly in the same way as is done in Section 3.2.2. Even it uses the same primary database (RLC.database,


Fig. 4. Linguistic opinion written in Step 1 (Fig. 2) appears on “∗ ∗ ∗ ∗” marked space.

Fig. 5.

Fig. 3), same DBMS and user interface. It only di�ers in quantitative representations of a linguisticform.Here, a linguistic term is represented by a convex normalized fuzzy sets (Assumption 2.2) (see Fig. 7).


Fig. 6.

Fig. 7.

Fig. 8.

3.2.4. Acquisition and meaning representation of linguistic responsesThis is coded as

S = Q V = R L P

Step 1: Evaluator is asked to write his linguistic opinion.Step 2: Evaluator is asked to select a suitable scale of term-di�erentials from the scales with three or four

or �ve polar adjectives (terms).Step 3: Evaluator is asked to mark a point in favour of the meaning of the linguistic term used by a pointer

moving along the horizontal line segment. (See Fig. 8. It cites that evaluator selects scale 2 for making infavour of the linguistic term, say, good enough.)


Fig. 9. Fig. 10.

If the No. of terms of the selected scale is K then by the position of marking, we de�ne a fuzzy set (whoselabel is mentioned by the evaluator in his linguistic response) with mean value at the point of marking andextent of left and right fuzziness equal to(

length of the numerical domainK − 1

):

In this paper, length of domain =100. One thing must be noted that membership value of the newly de�nedfuzzy set will be ignored beyond the elementary domain.

3.2.5. Acquisition and meaning representation of linguistic responsesThis is coded as

S = Q V = R L D

Step 1: Evaluator is �rst asked to write his linguistic response.Step 2: For creation of Kb, he is then asked certain questions.(1) Showing the scale as in Fig. 9, he is asked to choose any grade on which his satisfaction is maximum.(2) If the evaluator marks on tth grade where t ∈ {E; E+; : : : ; A; A+} another two queries will be placed

to him.If the subject is awarded grade (t + 1), how will you be satis�ed?If the subject is awarded grade (t − 1), how will you be satis�ed?

Please, mark your response in Fig. 10.

3.2.5.1. Obtaining the membership function. From the data acquired in the table (see Fig. 10), a polygon canbe de�ned as is done in common statistical method and this may be considered as the membership functionof the general fuzzy number representing evaluator’s linguistic response codes as V=RLD.

4. (A single expert) evaluation model

This paper aims to develop a methodology for construction of membership function for linguistic expertopinions in an evaluation programme (see Table 2). For this purpose, a follow-up model is supplementedwhich assumes the existence of a single expert only. Generally as is done in aggregation with crisp data inevaluation of students performances in academic institutions, we develop a process of weighted averaging ofgeneralized fuzzy data.


Table 2

Quantized expert opinions for di�erent ‘subjects’

S1 S2 S3 · · · Sn

m queries with di�erentiable weights Query 1 11 21 31 n1w1; w2; w3; : : : ; wn Query 2 12 22 32 n2

......

Query m 1m 2m 3m nm

Aggregated assessments R1 R2 R3 : : : Rn

4.1. Aggregation of expert opinions

Expert opinions as modelled in the previous section may be of four types:(1) real numbers from C=RNM ,(2) interval numbers from C=RLC,(3) triangular fuzzy numbers from V=RLC and V=RLP,(4) generalized (convex normalized) fuzzy sets from V=RLD.To represent all four types of quantitative representation of expert opinions, we need a generalized

scheme.

De�nition 4.1. A fuzzy number is a convex normalized fuzzy subset on the real line R.A fuzzy interval A with membership function �A(x) may be speci�ed as the quadruple (a1; a2; a3; a4)

representable as in Fig. 11.(1) a1¡a26a3¡a4.(2) �A(x) is zero outside (a1; a4) and is one on [a2; a3].(3) �A(x) is continuous and monotonically increasing from 0 to 1 on [a1; a2] and decreasing from 1 to 0

on [a3; a4].(4) A is trapezoidal in shape if a2¡a3 and triangular if a2 = a3.If a1 = a2 and a3 = a4 then A=A= [aL; aR] is a closed real interval (interval number) where aL and

aR are respectively the left and right limit of interval A. If a1 = a2 = a3 = a4 = a then A= a is a realnumber.

De�nition 4.2. A generalized fuzzy number A can be de�ned and speci�ed by the family of its �-levelsets

A�= [x�L =fL(�); x�R = fR(�)] ∀�∈ [0; 1];

where fL(�1)6fL(�2)6fR(�2)6fR(�1); ∀ �1; �2 ∈ [0; 1] and �1¡�2.

A generalized fuzzy number X on R can be speci�ed by (Fig. 12)

X =(xL�=0; xL�=0:1; x

L�=0:2; : : : ; x

L�=1; x

R�=1; x

R�=0:9; : : : ; x

R�=0:1; x

R�=0)


Fig. 11. Fig. 12.

or may be denoted as

X =(xL00; xL01; xL02; : : : ; xL10; xR10; xR09; : : : ; xR01; xR00):

Arithmetic of fuzzy addition: If ∃ another general fuzzy number Y on R then X ⊕ Y = Z ⊂R where

Z =(zL00; zL01; : : : ; zL10; zR10; : : : ; zR01; zR00)

such that

zL00 = xL00 + yL00

zL01 = xL01 + yL01

zR10 = xR10 + xR10

and so on.Scalar multiplication

� : X = X ′ ⊂R

where X ′

{(�xL00; �xL01; : : : ; �xL10; �xR10; : : : ; �xR00) if � ¿ 0;

(�xR00; �xR01; : : : ; �xR10; �xL10; : : : ; �xL00) if � ¡ 0:

Averaging operation: If normalized non-negative weights, w1; w2; : : : ; wm where w1 + w2 + · · · + wm = 1,are attached to m linguistic terms represented as fuzzy sets X 1; X 2; : : : ; X m⊂R then aggregated (averaged)opinion is de�ned as

X =w1 : X 1 ⊕ w2 : X 2 ⊕ · · · ⊕ wm : X m⊂R:


How to be back to membership function representation: If X = (xL�=0; xL�=0:1; x

L�=0:2; : : : ; x

L�=1; x

R�=1 : : : x

R�=0) then

its (normal) membership function representation is as follows:

�X (x) =

0 when x6xL�=0;

0 +x − xL�=0

xL�=0:1 − xL�=0(0:1− 0) when xL�=06x6x

L�=0:1;

0:1 +x − xL�=0:1

xL�=0:2 − xL�=0:1(0:2− 0:1) when xL�=0:16x6x

L�=0:2;

0:2 +x − xL�=0:2

xL�=0:3 − xL�=0:2(0:3− 0:2) when xL�=0:26x6x

L�=0:3;

...

0:9 +x − xL�=0:9xL�=1 − xL�=0:9

(1− 0:9) when xL�=0:96x6xL�=1;

1 when xL�=16x6xR�=1;

0:9 +x − xR�=1

xR�=0:9 − xR�=1(0:9− 1) when xR�=16x6x

R�=0:9;

...

0:2 +x − xR�=0:2

xR�=0:1 − xR�=0:2(0:1− 0:2) when xR�=0:26x6x

R�=0:1;

0:1 +x − xR�=0:1xR�=0 − xR�=0:1

(0− 0:1) when xR�=0:16x6xR�=0;

0 when x¿xR�=0:

Thus the problem of construction of membership function of the fuzzy sets representing aggregated assess-ments for all subjects can be solved. Finally, comes the problem of ranking of the aggregated assessments.If all the aggregated assessments come in the form of interval numbers, for ranking of them Sengupta et al.[10] may be suggested. And if the assessments come in fuzzy forms, a very simple ranking procedure maybe suggested – Yager’s First ranking index [12]. However, it has a drawback; this index does not considerdi�erence in the extent of fuzziness for the sets with the same centroid. To eliminate the aforementioneddrawback and furthermore, to develop a generalized PC-aided ranking system for all the aggregated overallmerits of the subjects being evaluated further investigation is needed which is now being carried out.

5. Conclusion

This paper aims to describe a PC-aided procedure of information acquisition and modelling of data forfavour of an evaluator’s aid. It assumes that the concerning evaluator need not be an expert in fuzzy logicand technology. It assumes that the evaluator may use common linguistic terms in the making of his expertcomments and use his subjectivity and common reasoning ability (and sincerity) in conversation with thesystem and the system in return will help him in completing the rest of the job.


As this paper describes only the primary two phases of decisions making, i.e., information acquisition andmodelling of data, multiexpert extensions and the problem of ranking of the subjects therefrom are in the futurescope of this work. This work also may be extended to many allied industrial applications, say, evaluation andselection of comparable investment proposals, organizational performance evaluation, organization’s credibilityrating, �nancial credit rating, etc.

Acknowledgements

The author gratefully acknowledges the �nancial support provided by Department of Science & Technology,Government of India. (Sanction No. HR=SY=M-01=96).

References

[1] K. Asai, Fuzzy Systems for Information Processing, Ohmsha, Tokyo, 1995.[2] R. Biswas, An application of fuzzy sets in students’ evaluation, Fuzzy Sets and Systems 74 (1995) 187–194.[3] D. Chakraborty, Optimization in impressive and uncertain environment, Ph.D. Thesis, Dept. of Mathematics, IIT Kharagpur, 1975.[4] S.M. Chen, Evaluating weapon systems using fuzzy arithmetic operations, Fuzzy Sets and Systems 77 (1996) 265–276.[5] D. Dubois, H. Prade, Fuzzy Sets and Systems: Theory and Applications, Academic Press, New York, 1980.[6] B.R. Gaines, L.A. Zadeh, H. Zimmerman, J. Fuzzy Sets and Decision Analysis – A Perspective, in: Zimmerman, Zadeh, Gains

(Eds.), Fuzzy Sets and Decision Analysis, Elsevier Science Publishers, North-Holland, Amsterdam, 1984.[7] G.J. Klir, B. Yuan, Fuzzy Sets and Fuzzy Logic, Theory and Application, Prentice-Hall, New Delhi, 1997.[8] H.M. Lee, Applying fuzzy set theory to evaluate the rate of aggregative risk in software development, Fuzzy Sets and Systems 79

(1996) 323–336.[9] E. Orlowska, Representation of vague information, Information Systems 13(2) (1988) 167–174.[10] A. Sengupta, T.K. Pal, A-index for ordering interval numbers, Presented in Indian Science Congress, held in Delhi University,

Delhi, 1997.[11] E. Turban, Decision Support and Expert Systems, McMillan, New York, 1990.[12] R.R. Yager, A procedure for ordering fuzzy subsets of the unit interval, Inform. Sci. 24 (1981) 143–161.[13] L.A. Zadeh, Towards a theory of fuzzy systems, in: R.E. Kalman, N. De Claris (Eds.), Aspects of Network and System Theory,

Holt, Rinchart & Winston, New York, 1971, pp. 209–245.[14] L. Zhang, Structural and functional quantization of vagueness, Fuzzy Sets and Systems 55 (1993) 51–60.[15] H.J. Zimmermann, Fuzzy Set Theory and Its Applications, Allied, Delhi & Kluwer, Boston, 1991.[16] H.J. Zimmermann, P. Zysno, Decisions and evaluations by hierarchical aggregation of information, Fuzzy Sets and Systems 10

(1983) 243–260.

Documents

Structural quantization of vagueness in linguistic expert opinions in an evaluation programme