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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