Decision Support System with Mark-Giving Method

Otilija Sedlak, Marija Cileg and Tibor Kis

Faculty of Economics Subotica, University of Novi Sad, Segedinski put 9-11, Subotica, Serbia

Keywords: Mark-Giving-Method.

Abstract: Just as in many other areas, strategic management makes use of mathematical modelling in cases such as

setting strategic goals, formulation of strategies, selection and realization of the chosen strategy, and

strategic control. Criteria and restrictions of alternatives are also encompassed in the space of uncertainty

and indeterminacy. They have multiple meanings, in addition to being incomplete and fuzzy. The ordering

method is based on the assessment, i.e. mark-giving method used by teachers in education. Values of

criteria are treated as fuzzy sets, given as marks. It can be easy programmed by fuzzy logic software.

1 INTRODUCTION

The sense, value, manner and process of decision-

making problems are determined by the cultural,

social, temporal, value, as well as logical context.

Fuzzy logic was developed more than five decades

ago. The characteristics of fuzzy logic include

operating by fuzzy notions, imprecise authentication

tables, and fuzzy inference rules. All these

characteristics of fuzzy logic are highly important,

especially if we try to exchange or supplement the

long-dominating approach of strategic decision

making with the descriptive one.

The fuzzy set theory and various mathematical

reviews, the measures of uncertainty and

information have an unlimited possibility of

application in all the fields of sciences using a lot of

information and data, like for instance, in decision-

making. The contexts of strategic management are

under the conditions of uncertainty and

indefiniteness (Dubois and Prade, 1980).

The criteria, limitations and performances of

measures of alternatives bear in themselves some

aspects of indefiniteness: in determinativeness,

multiple aspects of meaning, incompleteness and

fuzziness.

2 FUZZY DECISIONS

As the name says, the subject of the decision making

discipline is the study of how decisions are really

made, and how they can be made better and

more successful.

The predominant focus of this discipline was in

the area of business decision making, where the

decision-making process is of key importance for

functions such as investment, new product

development, resource allocation, and many others.

Fuzzy systems approximate those equations.

Fuzzy systems enable us to make optimum

approximations of the non-linear universe. If it is

possible to build a mathematical model, we shall use

it. Fuzzy systems enable us to model the universe in

linguistic terms, rather than forcing us to write a

mathematical model of the universe. The technical

term for it is model-free function approximation.

The Fuzzy Approximation Theorem claims that a

graph can always be covered with a finite number of

fuzzy patches. The more uncertain the rule, the

larger the fuzzy patch. According to the Fuzzy

Approximation Theorem, a fuzzy system can

approximate a continuous system to a sufficient

degree of accuracy. This includes almost all systems

studied by science. Fuzzy systems can model

dynamic systems changing over time.

Viewed geometrically, every portion of human

knowledge, each rule “if A then B” defines a patch

on a graph. A fuzzy system is a large set of fuzzy “if

then” rules, representing “a large set of patches”.

The more knowledge, the more rules. The more

rules, the more sets. If the rules are more indefinite,

i.e. uncertain, the patches are larger. If the rules are

more definite, the patches are smaller. If the rules

are so precise that they are not fuzzy, then patches

are reduced to points.

The Fuzzy Approximation Theorem says more

190

Sedlak O., Cileg M. and Kis T..

Decision Support System with Mark-Giving Method.

DOI: 10.5220/0004287303380342

In Proceedings of the 2nd International Conference on Operations Research and Enterprise Systems (ICORES-2013), pages 338-342

ISBN: 978-989-8565-40-2

Copyright

c

2013 SCITEPRESS (Science and Technology Publications, Lda.)

than that. Theoretically, all equations can be

translated into rule patches. Fuzzy systems

approximate systems in physics, communication,

physiology, etc. Fuzzy systems can be applied

wherever the brain is used.

It is hard to deny that modern-day knowledge is

fuzzy. Meanings of statements are undoubtedly

fuzzy. Knowledge has always been regarded in

terms of rules. If knowledge is fuzzy, then rules are

fuzzy as well. Fuzzy rules connect fuzzy sets. Fuzzy

knowledge comprises fuzzy rules, and “if A then B”

rule. Fuzzy patches cover the

system graph. It is the

Fuzzy Approximation Theorem and fuzzy patches

that explain the functioning of fuzzy systems.

2.1 Fuzzy Linear Programming

Operational research offers optimization models

aimed at finding an activity programme that will

yield the best possible results. The models use

precisely determined and known data. Constraints

are also precisely determined, and the goal function

is clearly defined, so that it can be formulated easily

and simply.

Reality, however, is different: very often we lack

precise information on the value of individual input

parameters, or the values of coefficients in constraint

and goal functions, and imprecise formulation of

limitations themselves is possible as well (Maier,

2008).

Fuzzy sets can be introduced into the existing

decision making models in several ways. As an

economic institution, a company bases its existence

on the environment, both from the aspect of

providing input and from the aspect of achieving and

valorising input. Miscellaneous knowledge and

experience, and also decision making in the areas of

investment, market operations, financial function,

production function or research and development,

can be considered more fully and exactly applying

fuzzy sets. Under the existing circumstances

containing fuzzy characteristics, there is a wish to

achieve radical improvements of the production

management and decision making.

The need arises for choosing an appropriate

corporate goal out of the available possible

alternative goals. When accomplishing and

executing the alternatives, the company achieves

different levels of increase in sales (because,

although the subject issue is decision making on

production, one must bear in mind that the ultimate

goal of production is sale of the produced

commodities).

In addition to many constraints under the given

conditions, one must particularly bear in mind

limitations, i.e. constraints such as:

that the selected alternative (goal) is to be

accomplished in the shortest possible period;

that investment in accomplishing the selected

alternative should not be excessive.

The goal of decision making is a large number of

sold products. The decision must best meet the goal

and constraints of the given problem.

The nature of the problem displays the

characteristics of uncertainty and vagueness. The

need for fuzzification, i.e. fuzzy decision making

systems from the fact that the decision maker is

faced with a large number of scenarios and sub-

scenarios out of which the optimum must be chosen,

and the imprecision of input data results from

subjective approach in interpreting per se vague

information.

2.2 The Mark Giving Method

The basic prerequisite to apply fuzzification for

obtaining more effective instruments for using

different kinds of uncertainty, as well as for using

the natural language in modelling decision-making,

in the field of business decision-making of

hierarchical level, faces a whole range of problems

which cannot be solved by the methods of classical

quantitative analysis.

Above all, we would point to the following

problems:

ambivalence of aims,

variability of factors,

subjectivity of sight,

linguistic description of variables.

In practice, we often meet models where multiple

criteria take part in decision-making simultaneously.

This article is an attempt to prepare a decision by the

use of the fuzzy method of ordering alternatives (i.e.

aims), and to set priorities among some alternatives

and criteria, in the decision-making situations where

there are multiple decision-makers, multiple criteria,

and in the multiple time periods. The applied method

of evaluating in this article is based on the usual

assessment, i.e. marking method used in education.

The mark-giving method, very similar to R.

Jain's ordering method, is based on the weighted

aggregation of marks. As mark processing can be

described by many rules, the method forms a fuzzy

set of extra marks by the aggregation on the basis of

rules, and it can also be programmed as a fuzzy

system. The values of criteria, which describe

DecisionSupportSystemwithMark-GivingMethod

191

alternatives, are given as marks. An extra mark is

assigned to every alternative, aggregating fuzzy sets

of marks which describe alternatives. Alternatives

are ordered on the basis of extra marks. The mark-

giving method based on examples can be generally

applied for ordering.

The method is applicable if the values of criteria

can be treated as marks (or if they can be

transformed into them).

Let us assume x={a

1

,a

2

,...,a

n

} is the final set of

alternatives, and then take K={k

1

,k

2

,...,km} as the

final set of fuzzy criteria. Let g

1

,g

2

,...g

m

be the

weights belonging to criteria, where the maximal

value of the weight is 1.

Let every K

j

fuzzy criteria be over x a linguistic

variable (1jm), also letting K

1

={S

1

,S

2

,...,S

p1

}

where S

1

, S

2

,…, S

p1

are the values of the linguistic

variable. The functions of belonging () S

1

, S

2

..., S

p1

to fuzzy sets are determined on the basis of marks:

S1(x) supp S1 = 0,4;1,6

(1)

S2(x) supp S2 = 1,4;2,6

(2)

Sp

j

(x) ) suppSp

j

p

j

-0,6;p

j

+0,6

(3)

Let every function of belonging be over the sets of

the same form of a triangular fuzzy set. The degree

of marking (p) can be any whole number, but the

exactness and possibilities of expression differ from

case to case (Figure 1 represents the fuzzy sets of the

criteria K, in the case p=5). The alternative a

1

(IS i

S

n

) with S

1

,S

2

,... S

p1

fuzzy sets of criteria can be

evaluated.

0,5 1 1,5 2 2,5 3 3,5 4 4,5 5 5,5

1

x

Figure 1: (Kosko, 1992).

The mark-giving method assigns every alternative a

1

one fuzzy set R

i

, i.e. extra marks, which will appear

in one E fuzzy set of results. The set E will enable

the set R

i

to be compared, as well the set R not to be

defined. The set E is a fuzzy set identical with the

set of criteria:

E = {S

1

,S

2

,...,S

p

}

(4)

where K = max P

j

(j=1,2,...m), and every R set will

be formed on the basis of partly activated subsets of

the E set.

Copying and aggregations of fuzzy sets are

necessary for forming R

1

sets. In the program

package of fuzzy logic, which is applicable, these

operations can be performed only with the help of

such program blocks which the program package

treats as Kosko's FAM (fuzzy associative memories)

(Kosko, 1992).

One simple FAM system copies n dimensional

fuzzy sets into m dimensional with K parallel FAM

rules and their simultaneous use (A

1

, B

1

), (A

2

, B

2

),…

(A

k

, B

k

). Every A-input information activates rule of

FAM system in a way every. (A

i

, B

i

) is FAM rule

and has the form:

IF C = A

i

THEN D = B

i

(5)

(where C, D are linguistic variables, and A

i

, B

i

are

their possible values). Input information A is copied

into the part of B

1

set, which is partly activated into

B

i

. The B set is produced from the whole FAM

system, which is the weighted sum of partly

activated B

1

, B

2

, … B

k

fuzzy sets:

B = w

1

· B

1

+ w

2

· B

2

+...w

k

· B

(6)

where W

i

values in the interval (0,1) designates the

weights of FAM rules. One procedure of

defuzzification is directly connected to the FAM

system, which assigns one sole number to the B

fuzzy set (Table 2). The focus point in the B set is

given by the COG (Centre of Gravity) Method.

If A

1

Then B

1

If A

2

Then B

2

If A

k

Then B

k

B

1

B

2

B

k

w

1

w

2

w

k

B Defuzzificator yXA

Figure 2: (Kosko, 1992).

Input data of the program block are the marks of the

criteria: o

1

, o

2

,… o

m

. Any o

j

mark of the given values

of the criteria S

1

, S

2

,…, S

p

. partly activates one or

two neighbouring ones, and FAM rules copy these

ICORES2013-InternationalConferenceonOperationsResearchandEnterpriseSystems

192

partly activated sets into the E set. The R

i

set is a

weighted sum, even more times, of partly activated

S

1

, S

2

,… S

pi

sets.

In the course of functioning, the FAM system,

one series of marks o

1

, o

2

…o

m

, belonging to one a

i

alternative, partly activates S

1

, S

2

,…, S

pj

sets, which

are located in the part of the conditions of the FAM

system. In the same way, the rule activates the same

set in the part of consequences. With every copying

into the set E, the given triangular number is

multiplied with the CF value.

CF (Certainty factor) gives the degree of rule

security by which the FAM system automatically

multiplies the result of the rules. This CF value is

determined by the square of criteria weights. After

copying, the sets partly activated by the operator of

the algebraic sum are aggregated and added, and

finally, the centre of gravity of the aggregated set is

formed by defuzzification.

Kosko uses the term “fuzzy associative memory”

to describe how a fuzzy system works. The system

activates all the rules in parallel and to a degree.

Computers use direct memory. Associative memory

searches the entire memory. (Kosko, 1992)

3 THE MARK-GIVING METHOD,

JAIN'S METHOD AND

YAGER'S METHOD

The formal similarities between Jain's method and

the mark-giving method are used for comparing

(formerly applied signs are used in comparing).

Steps of Jain's method:

1. One R

1

fuzzy set is formed for every a

i

alternative

in the form:

1

m

ijij

j

Rgr

(7)

where g

i

is the fuzzy set of weights, r

ij

is the fuzzy

value K

i

of criteria in case of a: alternative (signed

operations mean the multiplication and addition of

fuzzy sets).

2. A union of multiples of Ri sets is formed:

i

1

sup R

n

i

S

(8)

and one 'maximized' M fuzzy set is defined in the set

S:

max

() [ ]

M

rrr

(9)

with the function of belonging, where r

max

=sup S

and is a natural number (the set M gives the upper

limit for the values

)(r

i

R

.

3. A fuzzy set Rio is formed from M and Ri sets

with the functions belonging to:

0

() min{ (), ()},( )

ii

RRM

rrrrS

(10)

4. One Yi value is assigned to every alternative:

S)(r(r), max

io

R

i

y

(11)

Many have criticized Jain's method as it does not

give any help in forming the set M (choice ), and

Y

i

, which is assigned the alternative a

i

, represents

only one maximum value (the other ones are not

taken into consideration in ordering).

Comparing to Jain's method, the steps of this

method are the following:

1. Like in Jain’s method, one Ri fuzzy set is formed

for every ai alternative in the form:

1

m

ijij

j

Rgr

(12)

where the values of the weight gj can range

within the interval (0,1) of real numbers, the

values rij are special, and the fuzzy sets of marks

are the same for every criterion (the degree of

marks can be different depending on the criteria).

2-3. The method does not limit the values of

functions of belonging to the sets Ri, it is not

necessary to define M, nor form Rio sets.

Instead, the sets Ri are compared in the mutual E

set.

4. The value yi, which joins the alternative ai,

representing the centre of gravity, is formed

taking into consideration all the values of

criteria. The value y i, shows the ordinal number

of alternatives.

We can conclude that the mark-giving method,

compared to Jain's method, represents a different

principle of problem solving.

Taking into consideration every value of the

"possibility of realization", Yager's method assigns

the value Y

i

to the alternative a

i

(Philips, 1995).

DecisionSupportSystemwithMark-GivingMethod

193

j

kij

max min ( (a )t )

i

y

(13)

It also orders every K

j

(j m) , as well as

alternatives on the basis of the value Y

i

.

Yager's method does not always differentiate

between alternatives with approximately the same

weight, so it assigns the same numerical values to

the groups of alternatives. With the mark-giving

method we notice quite the opposite: it assigns a

different numerical value to almost every alternative.

According to this, the mark-giving method points

more to the difference between alternatives than

Yager's method.

4 CONCLUSIONS

The demonstrated fuzzy ordering method, which is

based on marking, enables the ordering such

alternatives where fuzzy criteria can be described by

marks or where the values of criteria can be

considered to be marks. The results are similar to

results achieved by other ordering methods.

The mark-giving-method treats criteria as a fuzzy

system with the rules of aggregation. It can be easy

programmed by fuzzy logic software. The method is,

in some points, similar to Jain's method of

alternative ordering, but an ordering on the basis of

weights, to assigned alternatives is a different

principle in relation to Jain's method.

REFERENCES

Dubois, D., Prade, H., 1980. Fuzzy Sets and Systems:

Theory and Applications. Academic Press, New York.

Kosko, B. 1992. Neural Networks and Fuzzy Systems.

Englewood Cliffs: Prentice – Hall.

Philips, L. 1995. Just Decision Using Multiple Criteria or:

Who Gets the Porsche?, An Application of Ronald R.

Yager's Fuzzy-Logic Method, 5

th

International

Conference on AI and Law, Maryland ACM.

Maier, R. 2008. Knowledge Management Systems,

information and Communication Technologies for

Knowledge Management, Springer-Verlag, Berlin.

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