Soft ComputDOI 10.1007/s00500-014-1263-3

METHODOLOGIES AND APPLICATION

Performance evaluation of industrial enterprises via fuzzyinference system approach: a case study

Gözde Ulutagay · Fatih Ecer · Efendi Nasibov

© Springer-Verlag Berlin Heidelberg 2014

Abstract The aim of this study is not only to give self-contained and methodological steps of data mining with itsareas of applications, but also to provide a compact sourceof reference for the researchers who want to use data min-ing and fuzzy inference in their area of work. We constructa fuzzy inference system to predict the profit of the major500 industrial enterprises of Turkey. For this aim, we usemost of the data mining tools. First, we use fuzzy c-meansclustering algorithm and obtain the linguistic terms of thevariables. Having used decision tree technique, fuzzy rulesare revealed. Eventually, we compare various defuzzificationstrategies to obtain crisp prediction values of our fuzzy infer-ence system. We can conclude that the prediction results ofthe smallest of maxima defuzzification strategy-based fuzzyinference system has circa 40 % smaller sum square errorthan that of classical regression model.

Communicated by T. Allahviranloo.

G. Ulutagay (B)Department of Industrial Engineering, Faculty of Engineering,Izmir University, Gursel Aksel Blv. No. 14, Uckuyular,35350 Izmir, Turkeye-mail: gozde.ulutagay@gmail.com; gozde.ulutagay@izmir.edu.tr

F. EcerDepartment of International Trade and Finance,Faculty of Economics and Administrative Sciences,Afyon Kocatepe University, Afyon, Turkeye-mail: fatih.ecer@aku.edu.tr

E. NasibovDepartment of Computer Science, Faculty of Science, Dokuz EylülUniversity, Izmir, Turkeye-mail: efendi.nasibov@deu.edu.tr

E. NasibovInstitute of Cybernetics, Azerbaijan National Academy of Sciences,Baku, Azerbaijan

Keywords Data mining · Soft computing · Fuzzy inferencesystem · Fuzzy clustering · Decision tree

1 Introduction

With the rapid growth of databases, data mining has turnedout to be an increasingly crucial technology for statistical dataanalysis performed in many modern enterprises. In most ofthe cases, a large database, which probably contains noisy,inconsistent or missing data, is of concern for data min-ing application. Data preprocessing deals with the above-mentioned issues one by one and accomplishes preparatorytasks like data cleaning, data integration, data transformationand data reduction. The exploratory data mining comprisesdiscovering patterns in the data using summary statistics andvisualization when it is performed in the early stage of theprocess (Olafsson et al. 2008).

Data mining has been used for knowledge discovery dueto the prevalent usability of a large amount of data and theinstance of transforming such kind of data into meaningfulinformation. By analyzing this huge and new reserve, sci-entists, engineers, practitioners and business people makescientific discoveries, discover financially significant pat-terns, optimize production control and provide business man-agement. To achieve these, they have both modified well-known algorithms from statistics, machine learning, neuralnetworks, and databases and developed new methods tar-geted at large data mining problems. Since traditional datamining techniques deal with data consisting of verbal andnumerical objects, a combination of fuzzy and neural tech-niques, soft computing, is needed.

Soft computing, with the integration of data mining hasbeen an innovative approach to artificial intelligence for

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recent years (Jang et al. 1997; Lin and Lee 1996; Olmo etal. 2012). It deals with imprecise and uncertain problems andhence primarily includes probabilistic reasoning, fuzzy logicand neural networks.

Fuzzy inference system has been one of the most effec-tive application area of fuzzy set theory, fuzzy if–then rulesand fuzzy reasoning (Jang et al. 1997; Lin and Lee 1996;Kreinovich et al. 1998). It has successful applications in dataclassification, robotics, decision analysis, industrial processcontrol, automatic control, medical diagnosis and patternrecognition (Ulutagay 2012; Korol and Korodi 2011). Thebasic idea behind fuzzy inference system is to integrate expertexperience of a human in the design of the controller by usingthree conceptual components: a rule base, a database and areasoning mechanism. In developing any type of fuzzy sys-tem, the basic problem is to determine the shape and thenumber of fuzzy sets which represent the inputs and out-puts.

The main objective of our study is to predict the profit ofthe firms by using fuzzy inference system. Profit is the differ-ence between revenue from selling output and the cost acquir-ing the factors necessary to produce it (Liu 2007; Carden2009). It is also a signal which tells entrepreneurs whetherthey are successful or not; thus it is the ultimate goal forfirms (Carden 2009; Delis et al. 2009). Profits encourageproduction and innovation, provide powerful incentive andreward entrepreneurs for successfully adjusting the struc-ture of production to better suit the desires of consumers(Carden 2009). According to the neo-classical microeco-nomic theory, the firm behavior is usually characterized byprofit maximization (Kuosmanen et al. 2010). By using superslacks-based model of DEA, Duzakin (2007) performed 500major industrial enterprises in Turkey. They used net assetsand average number of employees as inputs and gross valueadded, profit before tax and export revenues as outputs in themodel.

As mentioned above, soft computing methods are gener-ally successful in prediction issues. In this study, we usedfuzzy inference system, one of the most efficient tools ofsoft computing, to predict the success of firms in termsof profit. Fuzzy clustering and decision tree techniques areused to form fuzzy inference system term sets and rules,respectively. To transform the fuzzy values into a single crispvalue, various defuzzification methods are compared and themethod with minimum sum square of error (SSE) is deter-mined.

In the next section, the basic instruments to constructa fuzzy inference system, such as fuzzy term determi-nation, rule determination and defuzzification, are given.In Sect. 3, an application of the fuzzy inference systemto industrial enterprises’ data is performed. The study isconcluded by giving the highlights of the study in Sect.4.

2 Fuzzy inference system

The fuzzy inference system, also known as fuzzy expert sys-tem, fuzzy rule-based system, fuzzy model or fuzzy asso-ciative memory, represents the induced tree as a series ofmultiple input, single output if–then rules by using linguis-tic labels. In particular, the if part, the antecedent, of arule defines a fuzzy region in the input space, while thethen part, the consequent, specifies the output in the fuzzyregion.

Relying on the specific structure of the consequent part,many fuzzy inference system model structures have been pro-posed. In this study, we will use the fuzzy rule base structureof Zadeh (1965):

R : cAlsoi=1

[If

nAND

j=1(x j ∈ X j A ji )THEN y ∈ Y isr Bi

](1)

In statement (1), c represents the number of rules in themodel; x j , j = 1, . . . , n is the j th input variable, n is thetotal number of variables, X j is the domain of x j , A ji isthe linguistic term of the related input variable, y is the out-put linguistic variable, and Bi is the linguistic term of therelated output variable in the i th rule which has a mem-bership function μBi (y) : Y → [0, 1]. Note that isr isintroduced by Zadeh and represents that the assignmentis not fuzzy, rather than crisp (Celikyilmaz and Turksen2009).

To specify the operation of the above-mentioned fuzzyinference system completely, t-norm or s-norm functionsshould be assigned to AND, OR, implication, aggrega-tion and defuzzification operators (Jang et al. 1997).Note that in statement (1), AND and OR operators arelogical connective, used for aggregating the membershipvalues of input variables, and logical implication con-nective, used for aggregating model outputs of the fuzzyrules, respectively. The selection of t-norm and s-norm com-bination depends on the kind of problem (Novak et al.2004; Krzysztof 2014). In general, to compute fast, minand max functions are used for t-norm and s-norm, respec-tively:

Tmin(a, b) = min(a, b) = a ∧ b (2)

Smax(a, b) = max(a, b) = a ∨ b. (3)

Our aim is to make a decision by using a fuzzy infer-ence system. To construct fuzzy inference system, theterms and the rules must be determined. Although thereare many ways to determine both the terms and the rules,we decided to use clustering and decision tree, respec-tively.

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2.1 Term determination

2.1.1 Fuzzy c-means (FCM) algorithm

In many practical situations, it is informative to cluster thegiven data. The aim of clustering is to partition the data setinto homogenous groups with respect to proper similaritymeasures in such a way that patterns are similar to eachother in the same cluster, and as dissimilar as possible indifferent clusters. The data structure could be representedwell by using clustering methods in the formation of mem-bership functions, which is the fundamental component of afuzzy inference system.

FCM algorithm is a simple and the most preferred fuzzyclustering approach among all fuzzy clustering methods(Bezdek 1974). In FCM algorithm, a collection of n vec-tors (X = {x1, x2, . . . , xn} , xi ∈ �p, i = 1, . . . , n) is parti-tioned into a priori known number of clusters, c, such that theweighted within-group sum of squared error objective func-tion is minimized. The objective function and constraintsof FCM clustering algorithm are given in Eqs. (4) and (5),respectively:

min : J (X; U, V ) =c∑

i=1

n∑j=1

(μi j )md2(x j , νi )A (4)

subject to : 0 ≤ μi j ≤ 1,∀i, jc∑

i=1

μi j = 1,∀ j

0 <

c∑i=1

μi j < n,∀i (5)

In Eq. (4), μi j is the membership of the j th data point inthe i th cluster, vi is the i th cluster center, and d(vi , x j ) is thedistance between the j th object and the i th cluster center. Thegeneral formula of the distance measure is given as follows:

d2(νi , x j ) = (ν−i x j )

T Ai (ν−i x j ) (6)

where Ai is a positive definite symmetric matrix which rep-resents a particular norm for each cluster, and it is used asan optimization parameter. Fuzzy exponent, m ∈ (1,∞),determines the degree of overlapping of the clusters, i.e., itrepresents the degree of fuzziness of the FCM algorithm.In general, the larger m is, the fuzzier are the membershipassignments of the clustering.

By combining Eqs. (4) and (5), FCM becomes a constraintoptimization problem. Hence, it should be minimized toobtain optimum results. Such a constraint optimization prob-lem can be solved by the Lagrange multipliers method; by

this way, the model is reduced to an unconstrained optimiza-tion problem with one objective function. By differentiatingJ (X; U, V ) with respect to cluster centers, V , and member-ship values, U , the necessary conditions for J (X; U, V ) toreach its minimum are

νti =

∑ nj=1(μ

(t)i j )m x j∑ n

j=1(μ(t)i j )m

(7)

and

μ(t)i j =

[c∑

l=1

(d(x j , v

(t−1)i )

d(x j , v(t−1)l )

)2

(m−1)

]−1, ∀i = 1, . . . , c.

(8)

Fuzzy c-means clustering algorithm (FCM):Initialization: Given unlabeled data vectors X =

{x1, . . . , xn}, the number of clusters, c, degree of fuzziness,m, and termination constant, ε > 0, initialize the partitionmatrix, U , randomly.

Step 1: Find the initial cluster centers, using membershipvalues of initial partition matrix in Eq. (7) as inputs.

Step 2: Start iteration t = 1.Step 3: Update membership values of each input data

object j in cluster i , μ(t)i j , using Eq. (8), where x j are the

input data objects as vectors and v(t−1)i are cluster centers

from the (t − 1)th iteration.Step 4: Update cluster center of each cluster i at iteration t ,

v(t)i , using Eq. (7), where the inputs are the input data matrix,

x j , and the membership values of iteration t , μ(t)i j .

Step 5: If the termination condition,∥∥v(t) − v(t−1)

∥∥ ≤ ε,is satisfied, stop. Otherwise, assign t = t +1 and go to step 3.

Since the assumption of a priori known or a fixed numberof clusters is not relevant for many data analysis problems,cluster validity index methods are used to decide for the opti-mal number of clusters (Ulutagay and Nasibov 2012). Somewell-known cluster validity indices are discussed and usedthroughout this study.

2.1.1.1 Partition coefficient cluster validity index Bezdekproposed a validity function to measure how similar a fuzzypartition is to a crisp one (Bezdek 1974). This measure usesthe following quantity as a measure of how well x j has beenclassified:

s j =c∑

i=1

u2i j . (9)

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The partition coefficient is the average of s j ’s over the dataset X and expressed as follows:

VPC = 1

n

c∑i=1

n∑j=1

u2i j . (10)

The partition coefficient measures the quality or non-fuzziness degree of a fuzzy partition and the closer VPC isto one, the better the partition is.

2.1.1.2 Xie–Beni cluster validity index Xie and Beni(1991) proposed a cluster validity index inspired by the objec-tive function J (X; U, V ) of the FCM algorithm. The consid-ered validity functional is a combination of the inadequacymeasure and inter-prototype distances. The point is that notonly membership degrees are taken into account to expressthe quality of the fuzzy data clustering. Xie and Beni con-sidered the VXB index as the ratio of total cluster variation,σ(U, V ), and the separation sep(V ) of the cluster centers asfollows:

VXB = σ(U, V )

sep(V )= 1

n

c∑i=1

n∑j=1

μ2i j d

2(vi , x j )

mini =l

d2(vi , vl). (11)

The best fuzzy partition, which is the indicator of compact-ness and separation of all classes, is considered to minimizethe functional VXB.

2.1.1.3 Fukuyama–Sugeno cluster validity index TheFukuyama–Sugeno cluster validity measure integrates theobjective function and the deviations of cluster centers,vi , i = 1, . . . , c, from the mean vector m X , where m X =M(X) is the mean vector of the data set X (Fukuyama andSugeno 1989).

Fukuyama–Sugeno validity function is defined as follows:

VFS =c∑

i=1

n∑j=1

μmi j [d2(x j , vi ) − d2(m X , vi )]

= Jm(P, v) −c∑

i=1

d2(m X , vi )

n∑j=1

μmi j . (12)

Let us denote

Km(P, v) =c∑

i=1

d2(m X , vi )

n∑j=1

μmi j (13)

which can be considered as the geometrical compactness ofthe representation v. Thus, the following can be written:

VFS = Jm(P, v) − Km(P, v). (14)

The best fuzzy partition is considered to minimize the func-tional VFS.

2.2 Rule determination

2.2.1 Decision trees

For the purpose of rule extraction of fuzzy inference system,decision or classification trees are widely used. A decisiontree is a simple structure which is used as a classifier. For theproblem of learning from a set of independent instances, adivide-and-conquer approach leads to a style of representa-tion. A decision tree is a flowchart-like tree structure. Eachnode of the tree refers to a test on an attribute, each branchindicates an outcome of the test and leaf nodes denote classes(Witten 2005). The uppermost node in the tree is accepted asthe root node.

The objective of decision tree rule induction is to acquirea set of rules to classify objects from a set of objects whoseattribute values and classes are known. The architectures con-sist of a root node, branches, internal nodes and leaf nodes.Initially, a root node is selected by a test. Then, the data setis split according to the value of these test attributes. Then,the process is repeated. At the end of the process, each pathis formed by branches, and each leaf node set at the end ofthe way denotes classes. Each path can be converted intoa decision rule for the classification problem. It means thatthese rules can be used to predict the class of a new sample(Quinlan 1993).

2.2.1.1 Classification and regression trees (CART) Hav-ing contained exactly two branches for each decision node,the decision trees obtained by classification and regressiontree algorithm is strictly binary (Breiman et al. 1984). CARTpartitions the objects of the training set into subsets of objectsrecursively with close values for the target variable. TheCART algorithm expands the tree by conducting each node.It searches all appropriate variables and all possible splittingvalues, and selects the optimal split with respect to a prioridetermined criteria (Kennedy et al. 1995).

Let �(s|t) be a measure of the goodness of a candidatesplit s at node t , where

�(s|t) = 2PL PR

# of classes∑j=1

|P( j |tL) − P( j |tR)| (15)

and where tL and tR are left and right child nodes of nodet , respectively; PL and PR are the ratios of the number ofrecords at tL and tR, respectively, to the number of recordsin the training set; P( j |tL) and P( j |tR) are, respectively, theratio of the number of j class records at tL and tR to thenumber of records at t . Then the optimal split is whicheversplit that maximizes measure �(s|t) over all possible splitsat node t (Larose 2005).

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2.2.1.2 C4.5 algorithm The C4.5 algorithm is Quinlan’sextension of his previously proposed identification tree, ID3,algorithm to obtain a decision tree (Quinlan 1993). Both ID3and C4.5 are greedy algorithms for the induction of deci-sion trees. Similar to CART, C4.5 algorithm recursively turnsover each decision node and selects the optimal split until nofurther splits are possible. C4.5 model is successful in deal-ing with continuous variables and missing data (Abellán andMasegosa 2010). Note that C4.5 algorithm uses the conceptof information gain or entropy reduction to select the optimalsplit and chooses the attributes according to the maximuminformation gain ratio in the split. If there is a candidate splitS, which partitions the training data set T into several subsetsT1, T2, . . . , Tk , the average information requirement is com-puted as the weighted sum of the entropies for the individualsubsets as given below:

HS(T ) =k∑

i=1

Pi HS(Ti ), (16)

where Pi indicates the ratio of the records in subset i . Then,the information gain is calculated asgain(S) = H(T ) −HS(T ), i.e., the increase in information produced by parti-tioning the training data T according to this candidate split S.

2.3 Defuzzification

In the last stage of the fuzzy inference system, membershipfunction is transformed into a single crisp output. To achievethis aim, defuzzification methods are used. Defuzzificationis a mapping from a space of fuzzy control actions definedover an output universe of discourse into a space of crispcontrol actions (Nasibov 2003). Since crisp control action isnecessary for decision making in many practical applications,defuzzification strategy, whose aim is to produce a non-fuzzycontrol action that could represent the possibility distributionof an inferred fuzzy control action in the best way, is required.Unfortunately, choosing the best defuzzification strategy isnot definite (Kreinovich et al. 1998). Among many defuzzifi-cation methods, we handle the most widely used ones belowand give a brief explanation of each strategy.

Let A be a fuzzy set, and Z be a universe of discourse.

2.3.1 Center of area (COA)

Center of area is the most widely used defuzzification strat-egy. COA is similar to the calculation of the expected valuesof probability distributions as given below (Jang et al. 1997):

zCOA =∫

Z μA(z)zdz∫Z μA(z)dz

. (17)

where μA is the aggregated output membership function.

2.4 Bisector of area (BOA)

The zBOA satisfies

zBOA∫α

μA(z)dz =β∫

zBOA

μA(z)dz, (18)

where α = min{z|z ∈ Z} and β = max{z|z ∈ Z}. Thatis, the vertical line z = zBOA partitions the region betweenz = α, z = β, y = 0 and y = μA(z) into two regions withthe same area.

2.4.1 Mean of maxima (MOM)

The zMOM is the average of the z at which the membershipfunction reaches the maximum μ∗:

zMOM =∫

Z ′ zdz∫Z ′ dz

. (19)

where Z ′ = {z|μA(z) = μ∗}. In particular, if μA(z) has thesingle maximum at z = z∗, then zMOM = z∗. Moreover, ifμA(z) reaches its maximum whenever z ∈ [zleft, zright], thenzMOM = (zleft + zright)/2.

2.4.2 Smallest of maximum (SOM)

In terms of magnitude, zSOM is the minimum of the maxi-mizing z:

zSOM = min{z|μA(z) = μ∗}, (20)

where μ∗ = max μA(z). − ∞ < z < ∞.

2.4.3 Largest of maximum (LOM)

In terms of magnitude, zLOM is the maximum of the maxi-mizing z:

zLOM = max{z|μA(z) = μ∗}. (21)

where μ∗ = max μA(z). − ∞ < z < ∞.

3 Application

In this study, we aim to predict the success of industrial enter-prises in terms of profit by using fuzzy inference systemwhich is an efficient tool of soft computing technology. Weused fuzzy clustering to obtain term sets of fuzzy clustering,

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Fig. 1 Distribution of the major 500 industrial enterprises throughout Turkey

and decision tree to obtain the rules. To transform the out-put of fuzzy inference system, several widely used defuzzi-fication methods are compared and the one with minimumprediction error is determined. For the sake of comparingthe results of our study, we applied regression analysis. As aresult, we conclude that profit prediction with fuzzy inferencesystem gives 40 % less SSE than prediction with regressionanalysis.

The steps of our application are given in the followingalgorithm.

Step 1. For each variable in the dataset:

1. Determine the optimal number of clusters according to asuitable cluster validity index.

2. Perform clustering for the optimal number of clusters.3. Transform membership functions for each cluster into the

nearest parametrical form (Gaussian, triangular, trape-zoidal, etc.) and designate each cluster with the appro-priate linguistic term.

Step 2. Run a decision tree algorithm to determine therules.

Step 3. Apply the fuzzy inference system.Step 4. Defuzzify the results using an appropriate defuzzi-

fication method which gives minimum SSE.In our study, the major industrial enterprises of Turkey are

handled to analyze their financial performance. We obtainedthe dataset from Istanbul Chamber of Industry. The order-ing is made according to their sales from production values(Ecer et al. 2011). The location of the enterprises throughoutTurkey is shown on a map (Fig. 1).

A data set of 500 major industrial enterprises of Turkeyis used including the variables sales revenue, assets, equity,export and number of employee as input and profit as pre-dicted output. After completing the preprocessing stage, i.e.,dealing with outliers, filling empty data using appropriatemethods, etc., membership functions for the variables salesrevenue, assets, equity, export, number of employee and

Table 1 Optimal number of clusters determined by PC, FS and XBcluster validity indices

Variable Optimal number of clusters

PC FS XB

Sales 2 5 2

Equity 2 4 2

Assets 2 5 2

Export 2 6 2

Employee 2 7 2

Profit 2 5 2

profit are formed by applying the FCM algorithm to each ofthe variables. The number of linguistic variables, i.e., num-ber of clusters for FCM algorithm, is determined with clustervalidity index by changing the number of clusters 2 through10. The optimal results for each variable are given in Table1 and the membership functions of the variables are given inFig. 2.

The fuzzy membership functions of the variables areassumed to be Gaussian and triangular. The Gaussian func-tion depends on two parameters, mean, μ and σ , standarddeviation:

f (x;μ; σ) = e− (x−μ)2

2σ2 . (22)

In Fuzzy Logic Toolbox of Matlab, we used gauss2mffunction, which is a combination of the parameters μ and σ .The first function, specified by μ1 and σ1, and the secondfunction, specified by μ2 and σ2, determine the shape of theleftmost and the rightmost curves, respectively. Wheneverμ1 < μ2, the gauss2mf function reaches a maximum valueof 1. Otherwise, the maximum value is less than 1. Its syntaxis given as:

y = gauss2mf(x .[σ1μ1σ2μ2]) (23)

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Sales_norm

Mem

ber

ship

deg

ree

1,00,80,60,40,20,0

1,0

0,8

0,6

0,4

0,2

0,0

Variable

mediumhightoo_high

too_lowlow

Membership Function of Sales

Equity_norm

Mem

ber

ship

deg

ree

1,00,80,60,40,20,0

1,0

0,8

0,6

0,4

0,2

0,0

Variable

normal

high

too_low

low

Membership Function of Equity

Assets_norm

Mem

ber

ship

deg

ree

1,00,80,60,40,20,0

1,0

0,8

0,6

0,4

0,2

0,0

Variable

mediumhightoo_high

too_lowlow

Membership Function of Assets

Export_normM

emb

ersh

ip d

egre

e1,00,80,60,40,20,0

1,0

0,8

0,6

0,4

0,2

0,0

Variable

normalhighvery_highvery_very_high

too_lowlow

Membership Function of Export

Employee_norm

Mem

ber

ship

deg

ree

1,00,80,60,40,20,0

1,0

0,8

0,6

0,4

0,2

0,0

Variable

lownormalhightoo_highextremely_high

extremely_lowtoo_low

Membership Function of Employee

Profit_norm

Mem

ber

ship

deg

ree

1,00,80,60,40,20,0

1,0

0,8

0,6

0,4

0,2

0,0

Variable

medium

high

too_high

too_low

low

Membership Function of Profit

Fig. 2 Fuzzy membership functions of the variables

To transform Gaussian membership function into triangular,we used mf2mf function of Matlab. The values of the para-meters σ1, σ2, μ1, and μ2 for each variable, calculated withrespect to fuzzy clustering results, are given in Tables 2, 3,4, 5, 6 and 7.

The most fabulous property of a decision tree is its inter-pretability. Particularly, decision rules could be extracted eas-ily by using any given path from root through any leaf. Wecan express the decision rules in the form of IF–THEN rules.IF part, i.e., antecedent, is formed by the attribute values fromthe branches which are assembled by a specific path over thetree. THEN, i.e., consequent, part has the classification valuefor the target variable, which is obtained by a specific leafnode. We apply C4.5 algorithm to determine the rules forfuzzy inference system. The decision tree, obtained by usingSPSS Clementine 10.1, contains 37 rules. Some of the rulesare given in Table 8 as examples.

Table 2 σ1, σ2, μ1, and μ2 parameters for sales

Sales

Too_low Low Medium High Too_high

Cluster center 0.0118 0.0607 0.1547 0.3632 0.8024

Mean

Left (μ1) 0.000047 0.000125 0.000483 0.001756 0.00821

Right (μ2) 0.000097 0.000289 0.001242 0.00824 0.012478

SD

Left (σ1) 0.006834 0.011186 0.021981 0.041904 0.090611

Right (σ2) 0.009873 0.017007 0.035237 0.090773 0.111706

According to various firing strengths and defuzzificationmethods, the classification accuracy of the models is com-pared. In addition, sum square error of the results are cal-culated for two different triangular and Gaussian member-

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Table 3 σ1, σ2, μ1, and μ2 parameters for equity

Equity

Too_low Low Medium High

Cluster center 0.1068 0.1711 0.3134 0.6476Mean

Left (μ1) 0.000304 0.000254 0.001020 0.005426

Right μ2) 0.000199 0.000754 0.004184 0.031266

SD

Left (σ1) 0.017446 0.015935 0.031932 0.073665

Right (σ2) 0.014102 0.027455 0.064681 0.176821

Table 4 σ1, σ2, μ1, and μ2 parameters for assets

Assets

Too_low Low Medium High Too_high

Cluster center 0.0449 0.1127 0.2217 0.4573 0.8395

Mean

Left (μ1) 0.000331 0.000334 0.000695 0.002341 0.007515

Right (μ2) 0.000255 0.000528 0.002010 0.007802 0.008382

SD

Left (σ1) 0.018184 0.018276 0.026371 0.04838 0.086689

Right (σ2) 0.015966 0.022975 0.044837 0.088327 0.091554

ship functions, each defuzzified with five different methods(Table 9). Among these methods, the SOM method gave thesmallest sum of squares both with triangular and gaussianmembership functions as indicated in bold.

4 Conclusion

In this study, we have dealt with the prediction of profit of thetop 500 firms in Turkey by using a fuzzy inference system.After completing the preprocessing stage, i.e., dealing withoutliers, filling empty data using appropriate methods, etc.,

Table 7 σ1, σ2, μ1, and μ2 parameters for profit

Profit

Too_low Low Medium High Too_high

Cluster center 0.1784 0.3362 0.3942 0.4966 0.7082

Mean

Left (μ1) 0.008759 0.000831 0.000221 0.000529 0.002199

Right (μ2) 0.001101 0.000699 0.000365 0.001591 0.018545

SD

Left (σ1) 0.09359 0.028832 0.014856 0.022999 0.046895

Right (σ2) 0.033188 0.026447 0.019104 0.039884 0.136178

Table 5 σ1, σ2, μ1, and μ2parameters for export Export

Too_low Low Normal High Very_high Very very high

Cluster center 0.0159 0.0840 0.1681 0.2863 0.4647 0.8613

Mean

Left (μ1) 0.000134 0.000164 0.000405 0.000769 0.001821 0.006894

Right (μ2) 0.000205 0.000359 0.000573 0.001476 0.005627 0.010463

SD

Left (σ1) 0.011586 0.012811 0.020133 0.027723 0.042673 0.08303

Right (σ2) 0.014308 0.018954 0.023942 0.038418 0.075014 0.102287

Table 6 σ1, σ2, μ1, and μ2parameters for number ofemployee

Number of employee

Ext-low Too_low Low Normal High Too_high Ext-high

Cluster center 0.0251 0.0639 0.1183 0.1878 0.3044 0.5147 0.8775

Mean

Left (μ1) 0.000138 0.000082 0.000166 0.000287 0.000775 0.002071 0.005005

Right (μ2) 0.000076 0.00014 0.000215 0.000496 0.001574 0.004783 0.006452

SD

Left (σ1) 0.011732 0.009034 0.012884 0.016952 0.027834 0.045514 0.070749

Right (σ2) 0.008741 0.011833 0.014663 0.022278 0.039673 0.06916 0.080322

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Performance evaluation of industrial enterprises

Table 8 Some of the rules obtained by C4.5 algorithm

IF equity is “high” OR “normal”, THEN profit is “too high”

IF equity is “low” AND number of employee is “extremely high”, THEN profit is “high”

IF equity is “low” AND number of employee is “high” AND export is “high”, THEN profit is “high”

IF equity is “low” AND number of employee is “high” AND export is “medium”, THEN profit is “low”

IF equity is “too low” AND assets is “low” AND number of employee is “extremely high”, THEN profit is “high”

IF equity is “too low” AND assets is “low” AND number of employee is “extremely low”, THEN profit is “low”

Table 9 Sum of square errors for triangular and Gaussian membershipfunctions for various defuzzification methods

Defuzzification method Triangularmembershipfunction (x1017)

Gaussianmembershipfunction (x1017)

Center of area (zCOA) 9.8728 9.5912

Bisector of area (zBOA) 9.7941 9.6876

Mean of maxima (zMOM) 11.683 11.1688

Largest of maxima (zLOM) 19.663 19.654

Smallest of maxima (zSOM) 8.2943 8.2959

membership functions for the linguistic terms of the variablessales revenue, assets, equity, export, number of employeeand profit are formed by applying the FCM algorithm toeach of the variables. Moreover, the number of linguisticvariables, i.e., the number of clusters for FCM algorithm,is determined using various cluster validity indices. Then,we apply decision tree and determine the rules for fuzzyinference system. According to various firing strengths anddefuzzification methods, the classification accuracy of themodels is compared. We can conclude that using fuzzy infer-ence system approach for performance analysis of enterprisesfor profitability will be a renewal of financial applications,and the study on the whole will provide a compact guide forresearchers since it investigates the data mining tools step bystep.

Acknowledgments Gözde Ulutagay has been partly supported byGrant 10-183-RG/ITC/AS_C from TWAS (The Academy of Sciencesfor the Developing World)-COMSTECH (Committee on Scientific andTechnological Cooperation).

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