Библиотека. Зенкевич К.А. The use of Fuzzy Logic in Decision-Making Systems.

FOUNDATIONS OF CIVIL AND ENVIRONMENTAL ENGINEERING

No. 5 2004

Marcin GAJZLER Poznan University of Technology Institute of Structural Engineering

THE USE OF FUZZY LOGIC IN DECISION-MAKING

SYSTEMS

In the article the autor presents the problem of choice of the concrete construction repair systems. As far as the material choice is concerned, fuzzy logic was used. The article’s idea was to show the possibilities of the fuzzy logic, especially the possibilities of changing qualitative into quantitative data.

Key words: decision making, fuzzy logic, artificial intelligence methods, repair systems of concrete constructions

1. INTRODUCTION

The use of computer studies and artificial intelligence methods in construction is already well known and has been getting more and more common. One could enumerate management information systems, decision support systems, expert systems as well as systems based on artificial neural networks and genetic algorithms. It would be quite difficult to dispense with any of the mentioned elements. The great quantity of information reaching the recipient, noise which can interfer with the transmission, and finally the necessity of selection of certain information, processing it and making decisions based upon it – all these factors require the development of systems which facilitate and support the difficult conclusion and decision making process [3]. What comes to aid are some artificial intelligence methods. Feingenbaum describes artificial intelligence as “the branch of computer studies which concerns the methods and techniques of symbolic conclusion-making via computers as well as the symbolic representation of knowledge used in such reasoning”. Artificial intelligence methods are used to attempt to describe the reality in a way which imitates human reasoning.

Marcin Gajzler

There have been various efforts to overcome flaws of traditional computer algorithms, which often fail in situations easily solved by human beings. In AI programs, in contrast to traditional programs, we encounter symbolic processing, declarative data recording and significant data base usage. Unfortunately, there is a certain problem connected with formalization of qualitative phenomena. Artificial intelligence methods are not able to cope with the natural lack of precision of natural world phenomena and objects. Fuzzy Logic is an extremely precious instrument which allows to pass from qualitative to normative data. It also allows to define phenomena which are characterized by significant lack of definition precision. This lack of precision is usually called fuzziness [5].

2. FUZZY SETS

The fuzzy set notion, which the whole fuzzy logic is based on, was introduced by Lotfi A. Zadeh, Fuzzy Sets, „Informatic Control” in 1965 [6]. He defined the fuzzy set A as a pair:

{X,fi_A} (2.1)

where ц_A:X^> [0,1] is a function, which defines for every element X to what extent it belongs to the set A. The function ц_A is called a set membership function and it takes values of [0, 1] range. The membership function is continuous in a given range. If:

M_A(X) = 1 (2.2)

we can talk about a complete membership

Ma(x) = 0 (2.3)

we can talk about a complete lack of membership.

It should be stressed that the membership function is extremely vital for the usage of fuzzy logic, as it is through the function that the fuzzy set is defined. Both fuzzy and precise sets are subjects to classic set operations. The product, sum and complement are defined membership function operations. Except for classic fuzzy set operations there also exist some extremely vital modifiers, which change the fuzzy set shape. These modifiers are not based on mathematical theories. Each of these modifiers corresponds to a meaning, which is to transform the fuzzy set shape. One of the most common transformations is a fuzzy set concentration. It is supposed to intensify a modeled value. The opposite of concentration is fuzzy area extension. These two operations are associated with contrast modifiers, which change the fuzzy set shape and make it less (concentration) or more (extension) fuzzy. Modifiers of contrast intensification either

The use of fuzzy logic in decision-making systems

increase or decrease the membership function values, depending on the membership degree of the elements which belong to the set. Another group of modifiers consists of limitations to the fuzzy areas, which are connected with defining the order in the set of all the fuzzy subsets of the given area. As it had already been mentioned, fuzzy logic is a perfect instrument able to describe phenomena and elements characterized by small definition precision [7]. It is extremely useful as far as various tests, the results of which are descriptive, or fuzzy are concerned (e.g. questionnaires). It also supports other tests. As an instrument as such, fuzzy logic is widely used in systems supporting the decision-making process.

3. RATING OF OPTIONS – MATERIAL SELECTION (A USAGE EXAMPLE)

It has been repeatedly shown in practice that it is necessary to be supported by decision-making systems. Current economic conditions make it even more essential for decision-makers to comply with rationality rules and perform well-thought-out actions. Nevertheless, there are many obstacles and difficulties which may hinder the process of working-out of a system which supports the decision-making process. One of such obstacles may be the lack of precision in phenomena and environment description. In this article the author presents the possibility to use fuzzy logic when it comes to rating various options while choosing the repair system of concrete constructions [4]. On the basis of the carried out research [1] and using the possibilities that fuzzy reasoning offers, an attempt to create a system which assists decision-making in the construction industry has been made. Situations will be looked upon from the potential decision-contractor’s point of view – he is choosing the repair material among a few available. In the analysed case, the author is going to present the client’s point of view. It is quite significant as, depending on the decison-maker’s character, there are various preferences and limitations, as well as criteria importance. To solve the problem we will use the multicriteria approach, which consists in isolating some essential attributes. Using these attributes, we will assess the tested materials and then we will define relative attractiveness. There is also another useful tool – an element of the expert system, such as the rule base, which will serve to illustrate the preferences and possible limitations of decision-makers. The first phase is the identification of attribute, which are used in the usage of repair system assessment. These attributes may be obtained on the basis of an experienced, objective and independent expert’s specifications. In our case we can accept the following criteria: material type (MT), bending durability (BD), adhesion (A), processing time (PT), delivery conditions (DC). The presented criteria do not have to constitute a complete evaluation criteria set, some of them were deliberately ommited so as not to complicate the system as such. The aim of this article

10_____________________________Marcin Gajzler

is to show the possibilities of using fuzzy logic in the system assisting decision-making process in the contruction industry. The next phase consists of dividing the area of considered values. The typical way to do so is to evenly divide the area, which means to divide it into a specific number of fuzzy sets of identical width. Uneven area division of variable value is also possible. In this way membership functions in connection with original terms are created. Typical membership functions are triangle- or gauss-shaped.

The first thing to consider is the material type MT. The area was divided into three primary terms: A–spatula mortar, B–fine-grained mortar, C–coarse-grained mortar. Each of these terms corresponds to the value of the variable MT, which was defined in millimeters of aggregate grain thickness in the mortar.

0.0 0.2 0.4 [mm] MT

Fig. 1. Values of the variable MT

The next variable is the bending durability. Durability is a measurable value and it can be precisely defined. Still, also in this case the division of the value area had already been done, which served to separate 3 primary terms connected with the value of the variable BD: A–low durability, B–medium durability, C–high durability. The same was done as far as adhesion A is concerned. The only things changed were the ranges of the variable A, which was marked with a different font.

The use of fuzzy logic in decision-making systems

Fig. 2. Values of the variable BD; Values of the variable A

Still another variable exposed to division is the processing time, which means the time during which it is possible to use the material. In this case, more values were separated: A–very short, B–short, C–medium, D–long, E–very long (Fig. 3).

Fig. 3. Values of the variable PT

12_____________________________Marcin Gajzler

In case of the last variable, delivery conditions DC, values will be more symbolical than numeral in their nature. Variable DC is a qualitative variable and it describes the material distributor’s ability to cater for the decision-maker’s (or potential material receiver’s) needs and requirements. As the area of the considered objects for fuzzy values abstract area of [0,1] range was taken. The variable DC includes such components as delivery on time, waiting time, transport abilities of the distributor, etc. The variable DC will take the three following values: A–not very attractive, B–average, C–attractive (Fig. 4).

0.0 0.5 1.0 DC

Fig. 4. Values of the variable DC

The knowledge necessary to build the system should be obtained on the basis of needs analysis as well as decision- makers’ expectations and preferences pertaining to the materials. In this case, we can assume that the following pieces of information were obtained:

1. spatula mortar is unacceptable, otherwise it does not matter what kind of material is used

2. the ideal mortar should have high bending durability and high adhesion as well as long processing time

3. if the mortar’s bending durability is medium it must be characterized by at least average adhesion, long processing time and attractive delivery conditions in order to be attractive for the decision-maker

4. unattractive delivery conditions, in spite of high durability, adhesion and very long processing time make the material unattractive for the decision-maker

5. low durability or adhesion make the material unattractive for the decision-maker

The use of fuzzy logic in decision-making systems

6. the longer the processing time is, the more attractive the material gets, as

long as the durability and adhesion are at least average

The presented opinions, which include the decision-maker’s preferences and limitations, are subjective in their nature, while the described evaluation criteria are treated as having the character of importance. The evaluation criteria equivalence has originally been assumed, still in some conditions the criteria importance may adopt extreme values of the <0,1> range. For statement 1. in case of spatula mortar the criterium importance RM = 1, while in other cases (coarsegrained and and fine-grained mortar) criterium importance RM = 0, which practically means the possibility of ommiting this criterium. Still, at this stage, for the sake of reasoning clarity, the criteria equivalence was suggested.

On the basis of these statements, it is possible to construct an uncomplicated knowledge base, which is a set of rules for the decision-making system in process. On the basis of rules number 2 and 4 we can conclude that: IF:

- MT is not a spatula mortar

- BD is high

- A is high

- PT is very long

- DC are attractive THEN material attractiveness is very high.

This rule describes the best situation from the decision-maker’s point of view. It is enough for the DC to be unattractive and the whole material attractiveness will be very low. It can be shown in the following way: IF:

- MT is not a spatula mortar

- BD is high

- A is high

- PT is very long

- DC are not attractive THEN the material attractiveness is very low.

Acting this way we can create the model framework. Next rules are created by filling out the gaps between the knowledge base points.

5 material variants were analyzed and rated. They were described as follows: ABM:

- MT – 1/coarse-grained mortar

- BD – 1/high

- A – 1/high

- PT – 1/long

- DC – 1/average

14__________________________Marcin Gajzler

Table 1. Knowledge base rules (part)


	MT	BD	A	PT	DC	Attractiveness
1	coarse-grained	high	high	very long	attractive	very high
2	fine-grained	high	high	very long	attractive	very high
3	spatula	high	high	very long	attractive	very low
4	coarse-grained	high	high	very long	average	high
5	coarse-grained	high	high	very long	unattractive	very low
6	coarse-grained	high	high	long	attractive	very high
7	coarse-grained	high	high	long	average	high
8	coarse-grained	high	high	medium	average	average
9	coarse-grained	high	medium	medium	average	average
10	coarse-grained	medium	medium	medium	average	average
11	coarse-grained	medium	medium	short	average	low
12	coarse-grained	medium	low	short	average	very low
13	coarse-grained	low	medium	short	average	very low
14	coarse-grained	low	low	short	average	very low
15	coarse-grained	high	medium	long	average	average
16	spatula	high	medium	long	average	very low
17	spatula	medium	medium	long	average	very low
18	coarse-grained	high	medium	very long	average	high
19	coarse-grained	high	medium	long	attractive	high
20	coarse-grained	high	medium	very long	attractive	very high

AFS

- MT – 1/spatula mortar

- BD – 8 MPa

- A – 1/medium

- PT – 1/long

- DC – 1/average

SKM

- MT – the maximum grainess of 4 mm

- BD – 10 MPa

- A – medium

- PT – 0.6/long + 0.4/very long

- DC – 0.5/attractive + 0.5/average

SFM

- MT – 1/fine-grained mortar

- BD – high, although not very high

- A – 5 MPa

- PT – 45 minutes

- DC – attractive

The use of fuzzy logic in decision-making systems

SFF

- MT - 1/spatula mortar

- BD - 7 MPa

- A - 3.5 MPa

- PT - 1/long

- DC - 1/average

As it can be seen here, the values of input variables can be given in various ways and thanks to the usage of fuzzy system it is possible to standardize them. Even the BD and A values, when given in number, can be (and usually are) referential in their nature, although one can expect precise values in this case. After the definition of the input values as well as the division of the variable area and determination of the membership function, it is possible to move to the conclusion phase. In case of our example, which is based on the expert system using the fuzzy logic, all the rules of the knowledge base are activated. For every rule one determines the degree of the predecessor’s authenticity on the basis of the degree of individual entrances to the rule conditions adjustment. If one goes further, on the basis of the correlation of the predecessor with the follower, a fuzzy set, which is the result of the rule operation, is found. The results are finally consolidated into one initial set. In our case the minimal operations will be used to realize the conjunction of meeting the conditions in the rule, then the minimal correlation of the follower with the predecessor and the maximal operation for the integration of rule operating.

At the rules activation stage, the author proposes to take the criteria importance into consideration. It was impossible earlier for the sake of data qualitative character. The author suggests, while using the importance of the <0,1> range, to increase or decrease the function membership degree for each of the criteria, using the product in the process. It has to be remembered, though, that the maximum value of membership function is 1.0. In the example, the evaluation criteria equivalence was assumed.

For ABM we check the rule number 1.

If:

MT is coarse-grained and - membership in the 1 degree

BD is high and - membership in the 1 degree

A is high and - membership in the 1 degree

PT is very long and - membership in the 0 degree

DC is attractive - membership in the 0 degree

THEN ATTRACTIVENESS is very high.

As it is visible, the degree of authenticity of the predecessor of rule number 1 is:

T₁=min(1/1/1/0/0) = 0 (3.1)

If one then calculates the resulting fuzzy set for rule number 1, we have

Marcin Gajzler

ju_B ₁ (x) = min(ju_veryhigh (x),0) = 0 (3.2)

The result of the rule number 1 activation is an empty set, where the membership function is constantly 0.

Rule number 7 activation gives us a result distinct from 0. If:

MT is coarse-grained and - membership in the 1 degree

BD is high and - membership in the 1 degree

A is high and - membership in the 1 degree

PT is long and - membership in the 1 degree

DC is average - membership in the 1 degree

THEN ATTRACTIVENESS is high.

r₇=min(1/1/1/1/1) = 1.0 (3.3)

If one uses the maximal operation for the integration of the rule operation, then we have:

lu_B(x) = max(_/u_B ₁(x),..._/u_Bn(x)) = fi_B7(x) = fi_high(x) (3.4)

If one analyzes another AFS variant, it is possible to conclude that MT is a spatula mortar. On the basis of several statements, a knowledge base was worked out. One of these statements was as following: “spatula mortar is unacceptable...”. On the basis of this statement, a series of rules had been built. Nevertheless, each activation of these rules and meeting their conditions by AFS causes the following: the result of the concluding is a fuzzy set ATTRACTIVENESS very low. One can prove it by activating rules number 16 and 17. Taking it into consideration, the SFF variant was consistently omitted. If one looks at the next part of the previously quoted statement, it is possible to omit the material type MT in further consideration, as “...it does not matter what kind of material is used”.

Another analyzed variant is SKM. If one looks at the way the variables are defined, it is possible to say that they are defined in a different way. One should pay attention especially to the PT and DC variables, which are defined by fuzzy sets, e.g. for PT 0.6/long + 0.4/v. long. In practice it turns out that for SKM variant, on the basis of PT and DC definition, at least 4 rules’ results will differ from 0. It should also be remembered that in further analysis the MT variable was omitted, as it had been already been justified. Therefore: Rule number 15 If:

BD is high and - membership in the 1 degree

A is medium and - membership in the 1 degree

The use of fuzzy logic in decision-making systems

PT is long and DC is average

membership in the 0.6 degree membership in the 0.5 degree

THEN ATTRACTIVENESS is average.

τ₁₅ = min(1/1/1/0.6/0.5)

0.5

(3.5) (3.6)

µ_B₁₅(x) = min(µ_average(x)/1.0) = 0.5

Rule number 18

If:

BD is high and

A is medium and

PT is very long and

DC is average

THEN ATTRACTIVENESS is high.

membership in the 1 degree membership in the 1 degree membership in the 0.4 degree membership in the 0.5 degree

T₁₈=min(1/1/1/0.6/0.5) = 0.5 fi_B ₁₈(x) = min(//_high (x)/1.0) = 0.5

(3.7) (3.8)

Rule number 19

If:

BD is high and

A is medium and

PT is long and

DC is attractive

THEN ATTRACTIVENESS is high.

membership in the 1 degree membership in the 1 degree membership in the 0.6 degree membership in the 0.5 degree

T₁₉=min(1/1/1/0.6/0.5) = 0.5

fi_B ₁₉(x) = min(fi_high(x)/1.0) = 0.5

Rule number 20

If:

(3.9) (3.10)

BD is high and

A is medium and

PT is very long and

DC is attractive

membership in the 1 degree membership in the 1 degree membership in the 0.4 degree membership in the 0.5 degree

THEN ATTRACTIVENESS is very high.

r₂₀=min(1/1/1/0.4/0.5)

0.4

(3.11) (3.12)

µ_B₂₀ (x) = min(µ_very_.high (x)/1.0) = 0.4

Similarly as in the earlier case of using the maximal operation, we consolidate first the results of operations 18 and 15 and we get the 0.5/high set.

Marcin Gajzler

When we finally consolidate the results of all the rules’ operations we get for SKM a result which is the following = 0.5/medium + 0.5/high + 0.4/very high.

The final of the analyzed variants if SFM, which is one of the fine-grained mortars. This information allows us not to discard this variant and to analyze it further. The next considered criterion (after MT) is BD. In case of SFM it is defined rather not in a standard and precise way. BD is defined as high but not very high. According to the fuzzy sets’ properties, the function word very can be expressed by the square function.

Mve_ry _. _high( x ) = M_h _igh(x) (3.13)

and if we further use the complement of the fuzzy set we get:

Mnot__very._high ( x )= 1 " Agh (x) (3.14)

According to this, the BD variable value is a product of two fuzzy sets: high set and not very high set. While using the minimal operator as a product, we get:

Mhigh_but_not_very. high(x) = min(/^(x),1 - _M²high(x)) (3.15)

While performing the operation we get the resulting set:

BD = 0.5/medium +0.65/high For the next criteria

A = 0.5/medium + 0.5/high (the values taken from the A membership function graph for 5 MPa value)

PT = 1/long (the value taken from the PT membership function graph for 45 minutes value)

DC = 1/attractive

For such defined SFM variant, one should activate the following rules that will give the result differing from 0: 6, 24, 29, 31. After activating them, we consolidate the rule operations by using (as in earlier cases) the maximal operation.

ц_B(x) = max(//_B₁(x),...//_Bn(x)) = Mb₂₄/29/31(x) = Ц_high(x) (3.16)

ц_B (x) = max(//_B₁ (x),...fi_Bn (x)) = ц_B6 (x) = n_very_. _high (x) (3.17)

Finally the resulting set for the SFM variant is as the following:

ATTRACTIVENESS = 0.5/high + 0.5/very high

The use of fuzzy logic in decision-making systems

4. CONCLUSIONS

The final resulting sets for individual variants could be shown as following:

Variant ATTRACTIVENESS

ABM 1.0/high

AFS 0.0 (spatula mortar)

SKM 0.5/medium + 0.5/high + 0.4/very high

SFM 0.5/high + 0.5/very high

SFF 0.0 (spatula mortar)

As it can be seen from the above, the results are not unambiguous. They classify individual variants into one or several attractiveness groups and it happens in different degrees. The advantage of the fuzzy set is a possibility to go from the symbolic to numeric form of information – the resulting sets. In the discussed case the fuzzy sets are given in the tabled form, which allows one to use the centroid formula. The formula comes from the defuzzycation method, which means making the results more precise – COA (Center of Area) method. The method consists in finding the gravity center of the fuzzy set.

(4.1)

Using the given formula we get the following results:

ABM (1.0/high)=0.750 (3.)

AFS = 0.0

SFM = 0.0

SFM (0.5/high + 0.5/very high)= 0.875 (1.)

SKM (0.5/medium + 0.5/high + 0.4/very high) = 0.866 (2.)

The obtained results in numeric form suggest the already defined attractiveness of individual variants. One should remember that the results are affected by the limitations introduced by the decision-makers as well as their preferences.

The goal of the article was to show the mechanism of operating on fuzzy sets, their potential abilities, without going into much detail concerning the decision-making system, where fuzzy sets are used. One can unambiguously state that fuzzy logic:

- is a very good instrument useful to define various phenomena

- allows to describe not very precise and qualitative values and get quantitative values based on them

- can cooperate with various systems supporting the decision-making process

- is used while solving different problems (rating, forecasting, modelling).

Marcin Gajzler

REFERENCES

1. Gajzler M: Efektywnosc modyfikacji polimerami zapraw i betonów cementowych. Praca dyplomowa, Politechnika Poznańska, Wydział Budownictwa, Architektury i Inżynierii Środowiska 2001.

2. Goldberg D. E.: Algorytmy genetyczne i ich zastosowania, Warszawa, WNT 1995.

3. Kapliński O.: Modelling of construction processes. A managerial approach, Warszawa 1997.

4. Pasławski I, Gajzler M: Idea hierarchicznego systemu decyzyjnego przy wyborze mo-dyfikatora polimerowego, in: Konferencja Naukowo-Techniczna „Technologia i organi-zacja budownictwa u progu XXI wieku”, Puławy 21-24 czerwca 2001, 366-374.

5. Rutkowska D., Piliński M., Rutkowski L.: Sieci neuronowe, algorytmy gene-tyczne i systemy rozmyte, Warszawa, PWN 1997.

6. Zadeh L. A.: Fuzzy sets, Informatic Control, 8 (1965) 338-353.

7. Zieliński I: Inteligentne systemy w zarządzaniu, Warszawa, PWN 2000.

M. Gajzler

WYKORZYSTANIE LOGIKI ROZMYTEJ W SYSTEMACH DECYZYJNYCH

S t r e s z c z e n i e

W artykule zaprezentowano problem wybora materiału naprawczego konstrukcji betonowych. Jest to potencjalny problem decyzyjny. W wyborze materiału wykorzysta-no wnioskowanie rozmyte. lde% artykułu było pokazanie możliwości wykorzystanie w praktyce wnioskowania rozmytego, a zwłaszcza możliwości zmiany danych jakościo-wych na dane ilościowe.

Received 8.08.2003



	FOUNDATIONS OF CIVIL AND ENVIRONMENTAL ENGINEERING

	No. 5 2004 Marcin GAJZLER Poznan University of Technology Institute of Structural Engineering

	THE USE OF FUZZY LOGIC IN DECISION-MAKING SYSTEMS

	In the article the autor presents the problem of choice of the concrete construction repair systems. As far as the material choice is concerned, fuzzy logic was used. The article’s idea was to show the possibilities of the fuzzy logic, especially the possibilities of changing qualitative into quantitative data. Key words: decision making, fuzzy logic, artificial intelligence methods, repair systems of concrete constructions

	1. INTRODUCTION The use of computer studies and artificial intelligence methods in construction is already well known and has been getting more and more common. One could enumerate management information systems, decision support systems, expert systems as well as systems based on artificial neural networks and genetic algorithms. It would be quite difficult to dispense with any of the mentioned elements. The great quantity of information reaching the recipient, noise which can interfer with the transmission, and finally the necessity of selection of certain information, processing it and making decisions based upon it – all these factors require the development of systems which facilitate and support the difficult conclusion and decision making process [3]. What comes to aid are some artificial intelligence methods. Feingenbaum describes artificial intelligence as “the branch of computer studies which concerns the methods and techniques of symbolic conclusion-making via computers as well as the symbolic representation of knowledge used in such reasoning”. Artificial intelligence methods are used to attempt to describe the reality in a way which imitates human reasoning. © Publishing House of Poznan University of Technology, Poznań 2004 ISSN 1642-9303



	8	Marcin Gajzler

	There have been various efforts to overcome flaws of traditional computer algorithms, which often fail in situations easily solved by human beings. In AI programs, in contrast to traditional programs, we encounter symbolic processing, declarative data recording and significant data base usage. Unfortunately, there is a certain problem connected with formalization of qualitative phenomena. Artificial intelligence methods are not able to cope with the natural lack of precision of natural world phenomena and objects. Fuzzy Logic is an extremely precious instrument which allows to pass from qualitative to normative data. It also allows to define phenomena which are characterized by significant lack of definition precision. This lack of precision is usually called fuzziness [5].

	2. FUZZY SETS The fuzzy set notion, which the whole fuzzy logic is based on, was introduced by Lotfi A. Zadeh, Fuzzy Sets, „Informatic Control” in 1965 [6]. He defined the fuzzy set A as a pair: {X,fi_A} (2.1) where ц_A:X^> [0,1] is a function, which defines for every element X to what extent it belongs to the set A. The function ц_A is called a set membership function and it takes values of [0, 1] range. The membership function is continuous in a given range. If: M_A(X) = 1 (2.2) we can talk about a complete membership Ma(x) = 0 (2.3) we can talk about a complete lack of membership. It should be stressed that the membership function is extremely vital for the usage of fuzzy logic, as it is through the function that the fuzzy set is defined. Both fuzzy and precise sets are subjects to classic set operations. The product, sum and complement are defined membership function operations. Except for classic fuzzy set operations there also exist some extremely vital modifiers, which change the fuzzy set shape. These modifiers are not based on mathematical theories. Each of these modifiers corresponds to a meaning, which is to transform the fuzzy set shape. One of the most common transformations is a fuzzy set concentration. It is supposed to intensify a modeled value. The opposite of concentration is fuzzy area extension. These two operations are associated with contrast modifiers, which change the fuzzy set shape and make it less (concentration) or more (extension) fuzzy. Modifiers of contrast intensification either



	10_____________________________Marcin Gajzler

	is to show the possibilities of using fuzzy logic in the system assisting decision-making process in the contruction industry. The next phase consists of dividing the area of considered values. The typical way to do so is to evenly divide the area, which means to divide it into a specific number of fuzzy sets of identical width. Uneven area division of variable value is also possible. In this way membership functions in connection with original terms are created. Typical membership functions are triangle- or gauss-shaped. The first thing to consider is the material type MT. The area was divided into three primary terms: A–spatula mortar, B–fine-grained mortar, C–coarse-grained mortar. Each of these terms corresponds to the value of the variable MT, which was defined in millimeters of aggregate grain thickness in the mortar.



	0.0 0.2 0.4 [mm] MT Fig. 1. Values of the variable MT The next variable is the bending durability. Durability is a measurable value and it can be precisely defined. Still, also in this case the division of the value area had already been done, which served to separate 3 primary terms connected with the value of the variable BD: A–low durability, B–medium durability, C–high durability. The same was done as far as adhesion A is concerned. The only things changed were the ranges of the variable A, which was marked with a different font.



	12_____________________________Marcin Gajzler

	In case of the last variable, delivery conditions DC, values will be more symbolical than numeral in their nature. Variable DC is a qualitative variable and it describes the material distributor’s ability to cater for the decision-maker’s (or potential material receiver’s) needs and requirements. As the area of the considered objects for fuzzy values abstract area of [0,1] range was taken. The variable DC includes such components as delivery on time, waiting time, transport abilities of the distributor, etc. The variable DC will take the three following values: A–not very attractive, B–average, C–attractive (Fig. 4).



	0.0 0.5 1.0 DC Fig. 4. Values of the variable DC The knowledge necessary to build the system should be obtained on the basis of needs analysis as well as decision- makers’ expectations and preferences pertaining to the materials. In this case, we can assume that the following pieces of information were obtained: 1. spatula mortar is unacceptable, otherwise it does not matter what kind of material is used 2. the ideal mortar should have high bending durability and high adhesion as well as long processing time 3. if the mortar’s bending durability is medium it must be characterized by at least average adhesion, long processing time and attractive delivery conditions in order to be attractive for the decision-maker 4. unattractive delivery conditions, in spite of high durability, adhesion and very long processing time make the material unattractive for the decision-maker 5. low durability or adhesion make the material unattractive for the decision-maker



	The use of fuzzy logic in decision-making systems	15

	SFF - MT - 1/spatula mortar - BD - 7 MPa - A - 3.5 MPa - PT - 1/long - DC - 1/average As it can be seen here, the values of input variables can be given in various ways and thanks to the usage of fuzzy system it is possible to standardize them. Even the BD and A values, when given in number, can be (and usually are) referential in their nature, although one can expect precise values in this case. After the definition of the input values as well as the division of the variable area and determination of the membership function, it is possible to move to the conclusion phase. In case of our example, which is based on the expert system using the fuzzy logic, all the rules of the knowledge base are activated. For every rule one determines the degree of the predecessor’s authenticity on the basis of the degree of individual entrances to the rule conditions adjustment. If one goes further, on the basis of the correlation of the predecessor with the follower, a fuzzy set, which is the result of the rule operation, is found. The results are finally consolidated into one initial set. In our case the minimal operations will be used to realize the conjunction of meeting the conditions in the rule, then the minimal correlation of the follower with the predecessor and the maximal operation for the integration of rule operating. At the rules activation stage, the author proposes to take the criteria importance into consideration. It was impossible earlier for the sake of data qualitative character. The author suggests, while using the importance of the <0,1> range, to increase or decrease the function membership degree for each of the criteria, using the product in the process. It has to be remembered, though, that the maximum value of membership function is 1.0. In the example, the evaluation criteria equivalence was assumed. For ABM we check the rule number 1. If: MT is coarse-grained and - membership in the 1 degree BD is high and - membership in the 1 degree A is high and - membership in the 1 degree PT is very long and - membership in the 0 degree DC is attractive - membership in the 0 degree THEN ATTRACTIVENESS is very high. As it is visible, the degree of authenticity of the predecessor of rule number 1 is: T₁=min(1/1/1/0/0) = 0 (3.1) If one then calculates the resulting fuzzy set for rule number 1, we have



	The use of fuzzy logic in decision-making systems	19

	4. CONCLUSIONS The final resulting sets for individual variants could be shown as following: Variant ATTRACTIVENESS ABM 1.0/high AFS 0.0 (spatula mortar) SKM 0.5/medium + 0.5/high + 0.4/very high SFM 0.5/high + 0.5/very high SFF 0.0 (spatula mortar) As it can be seen from the above, the results are not unambiguous. They classify individual variants into one or several attractiveness groups and it happens in different degrees. The advantage of the fuzzy set is a possibility to go from the symbolic to numeric form of information – the resulting sets. In the discussed case the fuzzy sets are given in the tabled form, which allows one to use the centroid formula. The formula comes from the defuzzycation method, which means making the results more precise – COA (Center of Area) method. The method consists in finding the gravity center of the fuzzy set. (4.1) Using the given formula we get the following results: ABM (1.0/high)=0.750 (3.) AFS = 0.0 SFM = 0.0 SFM (0.5/high + 0.5/very high)= 0.875 (1.) SKM (0.5/medium + 0.5/high + 0.4/very high) = 0.866 (2.) The obtained results in numeric form suggest the already defined attractiveness of individual variants. One should remember that the results are affected by the limitations introduced by the decision-makers as well as their preferences. The goal of the article was to show the mechanism of operating on fuzzy sets, their potential abilities, without going into much detail concerning the decision-making system, where fuzzy sets are used. One can unambiguously state that fuzzy logic: - is a very good instrument useful to define various phenomena - allows to describe not very precise and qualitative values and get quantitative values based on them - can cooperate with various systems supporting the decision-making process - is used while solving different problems (rating, forecasting, modelling).



	20	Marcin Gajzler

	REFERENCES 1. Gajzler M: Efektywnosc modyfikacji polimerami zapraw i betonów cementowych. Praca dyplomowa, Politechnika Poznańska, Wydział Budownictwa, Architektury i Inżynierii Środowiska 2001. 2. Goldberg D. E.: Algorytmy genetyczne i ich zastosowania, Warszawa, WNT 1995. 3. Kapliński O.: Modelling of construction processes. A managerial approach, Warszawa 1997. 4. Pasławski I, Gajzler M: Idea hierarchicznego systemu decyzyjnego przy wyborze mo-dyfikatora polimerowego, in: Konferencja Naukowo-Techniczna „Technologia i organi-zacja budownictwa u progu XXI wieku”, Puławy 21-24 czerwca 2001, 366-374. 5. Rutkowska D., Piliński M., Rutkowski L.: Sieci neuronowe, algorytmy gene-tyczne i systemy rozmyte, Warszawa, PWN 1997. 6. Zadeh L. A.: Fuzzy sets, Informatic Control, 8 (1965) 338-353. 7. Zieliński I: Inteligentne systemy w zarządzaniu, Warszawa, PWN 2000.

	M. Gajzler WYKORZYSTANIE LOGIKI ROZMYTEJ W SYSTEMACH DECYZYJNYCH S t r e s z c z e n i e W artykule zaprezentowano problem wybora materiału naprawczego konstrukcji betonowych. Jest to potencjalny problem decyzyjny. W wyborze materiału wykorzysta-no wnioskowanie rozmyte. lde% artykułu było pokazanie możliwości wykorzystanie w praktyce wnioskowania rozmytego, a zwłaszcza możliwości zmiany danych jakościo-wych na dane ilościowe.

	Received 8.08.2003