A Type-2 Data Mining Optimization for Predicting
Pistachio Global Market
S. MalekMohamadi Golsefid
Department of Industrial
Engineering
Amirkabir University of Technology
Tehran, Iran
samira@aut.ac.ir
I.B. Turksen
Department of Industrial
Engineering
TOBB University of Economics and
Technology
Sogutozu, Ankara, Turkey
turksen@mie.utoronto.ca
Abstract— In this paper, a type-2 fuzzy TSK expert system is
developed for optimizing the global market prediction. Interval
type-2 fuzzy logic system permits us to model rule uncertainties
and every membership value of an element is interval itself. The
proposed type-2 fuzzy model applies the variables which indicate
the export trade trend during the specified period. The PCM
algorithm is employed to partitions international market into a
collection of clusters, and the clusters are converted into a rule
base of type-2 fuzzy TSK IF–THEN rule. The proposed model is
implemented for forecasting export value of international market
segment of pistachio from Iran. This model can be used for
selecting proper segment of international market by predicting
export value for every product.
Keywords; Type-2 fuzzy modeling; Type-2 TSK; Prediction;
global market; international market segmentation
I.
INTRODUCTION
The purpose of this study is to develop a type-2 fuzzy TSK
optimizing mechanism for predicting export value of a product
in a different segment of the global market. With this
knowledge, added emphasis can be placed on potential market
and keep actual market to increase market share in target
country and permits to identify and reduce efforts in areas
where export products do not have market.
This paper proceeds as follows: next section gives
background information on export predicting model and
variables and also reviews the Type2-Fussy logic system. In
section 3 the design approach of a rule base of type-2 fuzzy
TSK system is presented. Section 4 presents the proposed
interval type-2 fuzzy for prediction of Iran export value of each
market segment of Pistachio. Finally, Section 5 gives an
overview of the obtained result.
II.
BACKGROUND
A. Analyzing Export Trend
The trends of globalization sweeping the world offer
unparalleled opportunities and threats for industrial marketers.
While globalization efforts attract inward investment from
foreign nations, it simultaneously opens the local economy to
global competition. Therefore, evaluation and predicting the
potential of export into international market becomes an ever
more important subject in international marketing.
M. H. F. Zarandi
Department of Industrial
Engineering
Amirkabir University of Technology
Tehran, Iran
zarandi@aut.ac.ir
The theoretical model that are used in most studies in this
area is mainly based on regression models [1,2,3,4,5]. Some
soft computing methods are also applied in analyzing and
predicting export value and trend. Reference [6] adopted the
methodology GIKDE (General Intervalized Kernel Density
Estimator) to predict the amount of future exports. The fuzzy
time series method is applied for forecasting the amount of
export and he indicates that this method is more effectiveness
that ARIMA time series method[7].
Quite often, the knowledge that is used to construct the
rules in a fuzzy logic system (FLS) is uncertain. Three ways in
which such rule uncertainty can occur are: (1) the words that
are used in antecedents and consequents of rules can mean
different things to different people; (2) consequents obtained
by polling a group of experts will often be different for the
same rule because the experts will not necessarily be in
agreement; and (3) noisy training data [8]. Antecedent or
consequent uncertainties translate into uncertain antecedent or
consequent membership functions. Type-1 FLSs, whose
membership functions are type-1 fuzzy sets, are unable to
directly handle rule uncertainties. Type-2 FLSs, the subject of
this paper, in which antecedent or consequent membership
functions are type-2 fuzzy sets, can handle rule uncertainties. In
this study, we used type-2 fuzzy sets to develop the rule-based
fuzzy logic systems for prediction potential value of export to
the target market. By applying type-2 fuzzy sets the effects of
uncertainties is minimized. Type-2 fuzzy logic is very useful
when it is difficult to determine the exact membership
functions of fuzzy sets. The fuzzy type-2 rule base system that
is developed.
B. International market Segmentation
Market segmentation analyses have been especially
powerful in identifying segments deserving different levels of
marketing treatment and developing strategies to target the
identified markets [9].They have made extensive use of
various segmentation tools and methods [10]. International
segmentation aims to structure heterogeneity that exists among
countries and consumers by identifying relatively homogenous
segments of countries and/or consumers. Several methods are
available for identifying international market segments.
International market segmentation methods are classified into
Recency
Rule 1
Predict Function 1
Frequency
PCM
Algorithm
Monetary
Number of cluster (K)
Continuity
Kwon index
Trend
Cluster Label
Rule K
Predict Function K
Market Segmentation
Type-2 FLS
Type-2 TSK
Variables
Predict
Export
Value
Figure 3. The framework of the proposed type-2 TSK model
heuristic methods (Q- or R-factor analysis), cluster analysis
and model-based methods[11]. The most popular methods for
international market segmentation is cluster analysis [11,12].
A key issue with both grouping and estimation models is
the indicator used to measure market similarity [13].
Similarities and dissimilarities may be definable based on
market demand, consumer needs, preferences, and behavior
[14].
The basis of segmentation generally includes various
valuables such as demographics, socio-economic factors,
geographic location, and product related behavioral
characteristics such as purchase behavior, consumption
behavior and attitudes towards and preference for attractions,
experiences and services [15]. In some studies market is
divided based on similar purchasing behaviors [16, 10, 17, 18]
by using RFM model. RFM models represent customer
dynamic behavior and are used for solving the targeting and
the prediction problems in direct marketing [19], measuring
importance degrees of customers [20], clarifying customer
behavior patterns and identifying valuable customer and
ranking customer to concentrate promotional effort on loyal
customer to increase profit [21].
The aim of this research is to develop a of type-2 fuzzy
TSK which is capable to setting a rule base expert system to
predicting export value of a product in a different segment of
the global market. To achieve so, first of all, we defied input
and output variables based on extended RFM model. We
generate the rules of fuzzy system by using the noise-rejection
fuzzy clustering algorithm [22]. After setting fuzzy rule base
system, since Type-1 FLSs, whose membership functions are
type-1 fuzzy sets, are unable to directly handle rule
uncertainties, we design type-2 fuzzy sets which can model
and minimize the effects of uncertainties in rule-based fuzzy
logic systems. In this study, after comparing different
parametric inference engine, we selected Yu inference engine
to model the system and developing the prediction function.
We develop parametric rules for predicting the upper and
lower export value and also deffuzify this interval value to
predict an export value. The parameters associated with the
system are tuned by GA algorithm.
C. Type2-Fussy logic system
A type-2 fuzzy set, Ã , is characterized by a type-2
membership function, μà (x, u) , where xϵX and u ϵ Jx  [0,1] [23]:
̃ = {((x, u), μà (x, u))| x ∈ X, u ϵ Jx  [0,1]}
A
(1)
̃ can also express as:
in which 0 ≤ μà (x, u) ≤ 1 . Thus, A
̃=∫
A
∫
μà (x, u)/(x, u) Jx  [0,1]
(2)
x∈X u ϵ Jx
where , ∬ denotes union over all admissible x and u.
̃ , where Jx  [0,1] for
Jx is called primary membership of A
x ∈ X .The footprint of uncertainty (FOU) is a bounded region
that is indicated uncertainty in the primary memberships of a
type-2 fuzzy set à .
The upper membership function (UMF) and lower
membership function (LMF) of à are two T1 MFs that bound
the FOU. That is [23]:
̃=∫
A
∫
1/(x, u) Jx  [0,1]
(3)
x∈X u ϵ Jx
̃ ) x ∈ X
μ ̃ (x) ≡ FOU(A
{ A
̃ ) x ∈ X
μà (x) ≡ FOU(A
(4)4)
The type-2 compare to type-1 fuzzy sets can be handling the
uncertainty in a better way. The structure of a type-2 fuzzy
logic system (FLS) is very similar to of a type-1 FLS. The
inference engine computes the type-1 output set corresponding
to each rule and then a crisp output is computed by the
defuzzifer. Since in a type-2 FLS, the antecedent and/or
consequent sets are type-2, so that each rule output set is type2. Therefore the ‘‘Extended’’ versions of type-1 defuzzification
methods first reduces type-2 rule output sets to a type-1 set and
then, defuzzify the type reduced set to obtain a crisp output for
the type-2 FLS[25,26].
III.
DESIGNING THE TYPE-2 FLS
There are two very different approaches for selecting the
parameters of a type-2 FLS [27]. One is the partially
dependent approach, where a best possible type-1 FLS is
designed first, and then used to initialize the parameters of a
type-2 FLS. The other method is a totally independent
approach, where all the parameters of the type-2 FLS are
tuned from scratch without the aid of an existing type-1
design.
One advantage offered by the partially dependent approach
is smart initialization of the parameters of the type-2 FLS.
Since the baseline type-1 fuzzy sets impose constraints on the
type-2 sets, fewer parameters need to be tuned and the search
space for each variable is smaller. Therefore, the
computational cost is less than the totally independent
approach. So design flexibility is traded for a lower
computational burden. Type-2 FLSs designed via the partially
dependent approach are able to outperform the corresponding
type-1 FLSs [28], although both the FLSs have the same
number of MFs (resolution). However, the type-2 FLS has a
larger number of degrees of freedom because the fuzzy set is
more complex. The additional mathematical dimension
provided by the type-2 fuzzy set enables a type-2 FLS to
produce more complex input–output map without the need to
increase the resolution [29].
The framework of this study is based on partially dependent
approach. For developing the FLS, first a type-1 fuzzy system
is designed and then a type-2 fuzzy rule base is tuned to
increase the robustness of the system. The rules and the
number of fuzzy sets are the same as the type-1 FLS with the
only difference that the antecedent and consequent sets are
type-2.
The procedures of development of the proposed system are
as follows:
 Determination and calculation of input variables
 Clustering the input space and determination of the
number of rules
 Generating interval type-2 fuzzy rules
 Developing type 2 TSK model
 Tuning the parameters of the system by using genetic
algorithm (GA).
A. Determination and calculation of input variables of the
system
In the first step of system modeling the input variables are
identified. Determining the most relevant variables is required
studying the problem domain and negotiation with the domain
experts. Although, there are an infinite number of possible
candidates, the variables should be restricted to certain
numbers.
In this study, the input variables are extracted from trade
time series. The model variables are defined based on extended
RFM model. After data preparation, we calculate the Recency,
Frequency, Monetary [10], Continuity and trade trend variables
as input variables of system.
 Recency: measures the interval between the most
recent time and the analyzing time.
 Frequency: measures the export frequency within a
specified period.
 Monetary: measures the total monetary value within a
specified period.
 Continuity: measures the longest continues period
during the analyzing time
 Trend: measures the slop of trade trend of the export
time series.
The scores can vary depending on the types of applications
and scoring approaches. The scores calculate and normalize
for clustering purposes. Therefore, the R(𝐺𝑖 ), F(𝐺𝑖 ) , M(𝐺𝑖 ), C(𝐺𝑖 )
and T(𝐺𝑖 )scores can be redefined for ith country as follows:
𝑅
𝑅(𝐺𝑖 ) = (𝑄𝑖𝑅 )/(𝑄𝑀𝑎𝑥
)
𝐹
𝐹(𝐺𝑖 ) = (𝑄𝑖𝐹 )/(𝑄𝑀𝑎𝑥
)
𝑀
𝑀(𝐺𝑖 ) = (𝑄𝑖𝑀 )/(𝑄𝑀𝑎𝑥
)
𝐶
𝐶
𝐶(𝐺𝑖 ) = (𝑄𝑖 )/(𝑄𝑀𝑎𝑥 )
𝑇
)
{ 𝑇(𝐺𝑖 ) = (𝑄𝑖𝑇 )/(𝑄𝑀𝑎𝑥
Where QRi , QFi ,QMi ,QCi
(5)
and QTi represent the original values for
ith country according to the definition of R,F,M ,C and T. QRMax ,
C
T
QFMax , QM
Max ,Q Max and Q Max represent the maximum values of the
same. Therefore the input variable for each country is defined
as the R(Gi ), F(Gi ) , M(Gi ), C(Gi ) and T(Gi ) .
B. Clustering the input space and determination of the
number of rules
In this step, the system rules are generated based on PCM
clustering algorithm and also the optimum number of rule is
determine based on Kwon validity index[30]. For encoding
the variables, input space is clustered and the primary
membership grades of the input clusters are generated. To
achieve so, we consider PCM clustering algorithm for process
of encoding.
Clustering is the process of grouping a set of objects into
classes of similar objects. A cluster is a collection of data
objects that are similar to one another within the same cluster
and are dissimilar to the objects in other clusters [31]. There
are many tools for data partitioning like Fuzzy C-Means
(FCM). Since the FCM algorithm objective function is defied
based on the sum of squared errors, this algorithm may fail
completely in identifying outliers. Krishnapuram introduced
Possibilistic clustering algorithm (PCM) that is more robust
than the original FCM algorithm in the presence of noise[22].
PCM objective function involves unconstrained weights that
decrease with the distance from the cluster centers while it still
suffers from the same drawbacks of the original FCM
clustering. To cluster input, we use the PCM approach which
indicates in Figure 2.
(𝑡−1)
Step 1. Initialize the possibilistic C-partition 𝑈𝑡−1 = [𝑢𝑖𝑗
to cluster 𝛽𝑖
For 1 ≤ 𝑖 ≤ 𝐶 , 1 ≤ 𝑗 ≤ 𝑁 such that :
𝑢𝑖𝑗 ∈ [0,1]𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖 , 𝑗,
] (initially , 𝑡 − 1)of 𝑥𝑗 belonging
𝑁
0 ≤ ∑ 𝑢𝑖𝑗 ≤ 𝑁 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖 ,
𝑗=1
max 𝑢𝑖𝑗 > 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑗
𝑖
Step 2. Estimate the distance between xi and the center of 𝛽𝑖 using:
(a)
Compute 𝑐𝑖 is the center of cluster 𝛽𝑖 using :
𝑚
∑𝑁
𝑗=1 𝑢𝑖𝑗 𝑥𝑗
𝑐𝑖 = 𝐾 𝑁
𝑚
∑𝑗=1 𝑢𝑖𝑗
(b)
Compute 𝐹𝑖 is the fuzzy covariance matrix of cluster 𝛽𝑖 using :
𝑚
𝑇
∑𝑁
𝑗=1 𝑢𝑖𝑗 (𝑥𝑗 − 𝑐𝑖 )(𝑥𝑗 − 𝑐𝑖 )
𝐹𝑖 = 𝐾
𝑚
∑𝑁
𝑗=1 𝑢𝑖𝑗
(c)
Compute d2ij is the scaled Mahalanobis distance between xi and ci
using :
2
𝑑𝑖𝑗
= |𝐹𝑖 |1/𝑛 (𝑥𝑗 − 𝑐𝑖 )𝑇 𝐹𝑖 −1 (𝑥𝑗 − 𝑐𝑖 )
Step 3. Estimate the average fuzzy intera-cluster distance of cluster 𝛽𝑖 using:
𝑚 2
∑𝑁
𝑗=1 𝑢𝑖𝑗 𝑑𝑖𝑗
𝑖 = 𝐾 𝑁 𝑚
∑𝑗=1 𝑢𝑖𝑗
(𝑡)
Step 4. Update the prototype 𝑈𝑡 = [𝑢𝑖𝑗 ] by the following procedure. For each 𝑥𝑗 , 1 ≤ 𝑗 ≤ 𝑁,
(a)
Compute 𝑈𝑡+1using :
1
𝑢𝑖𝑗 =
1
𝑚−1
𝑑2
𝑖𝑗
1+(  )
𝑖
(b)
Increment 𝑡
Step 5. If ‖𝑈𝑡 − 𝑈𝑡−1 ‖ ≤ 𝜀 then stop; otherwise go to step 2.
Figure 3. The Possiblestic Clustering Algorithm (PCM)
The clustering method needs a validation index to define
the number of clusters (c). In this study we use Kwon index ,
which is represented by Eq. (9):
𝒄
𝒏
𝟐
𝑽𝑲 = (∑ ∑ 𝒖𝒎
𝒊𝒋 ‖𝒙𝒋 − 𝒗𝒊 ‖ +
𝒊=𝟏 𝒋=𝟏
𝒄
𝟏
2
∑ ‖𝒗𝒊 − 𝒗‖𝟐 ) / min‖vi − vj ‖
i≠j
𝒄
𝒊=𝟏
(6)
where v = ∑nj=1 xj /n.
The first term of the numerator in Eq. (6) measures the
intraclass similarity and indicates the cluster compactness. The
more similar (compact) the classes, the smaller the first term is.
It is independent of the number of patterns. The second term in
the numerator in Eq. (6) is used to eliminate the decreasing
tendency when the number of clusters c becomes very large
and close to the number of patterns n. The denominator in Eq.
(6), which is the minimum distance between cluster centroids,
measures the interclass difference. A larger value of it indicates
that every cluster is well-separated. The optimum number is
found by solving min Vk (c) to produce the best clustering
2≤c≤cmax
performance for the data set 𝑋 [32].
C. Fuzzy Type-2 TSK Modeling
The Takagi–Sugeno–Kang (TSK) model is one of the most
influential fuzzy reasoning models [33]. In this model, the
consequent part of each fuzzy rule is expressed as a linear
function of the input variables, instead of a fuzzy set, reducing
the number of required fuzzy rules [34]. The TSK model
consists of a set of IF…THEN rules with fuzzy implications
and first-order functional consequence parts, which are proved
to be a universal approximator [35].
In this study, we develop a type-2 TSK rule by converting
each cluster to form a fuzzy rule base. To model the multi
input–single output type-2 TSK assume that the system has n
inputs, x1 , x2 ,…, xn and one output y. Suppose we have j
clusters when all the training patterns have been considered.
Each cluster is converted to a type-2 TSK fuzzy rule and have
a set of j rules, R1 , R 2 , . . . , R𝑗 , each having the following
form:
𝑚
(7)
R : IF 𝑥 𝑖𝑠 𝐴̃̅ 𝑎𝑛𝑑 𝑥 𝑖𝑠 𝐴̃̅ 𝑎𝑛𝑑 … . 𝑥 𝑖𝑠 𝐴̃̅ THEN 𝑦̅̃ = ∑ 𝑏̅̃ 𝑥 + 𝑏̅̃
𝑗
1
𝑗1
2
𝑗2
𝑚
𝑗𝑚
𝑗
𝑖𝑗 𝑖
0𝑗
𝑖=1
Where x1 , x2 , … , xm are input variables, y̅̃j (j = 1, … , n) are
output variables, b̅̃ denotes the type -2 parameters in the
ij
̃ (i = 1, … , m , j = 1, … , n) is
̅
consequent part of the rules. A
ji
type-2 membership function for jth rule of ith input. For
generating interval type-2 fuzzy rule bases, first A Gaussian
̃
̃
̅ is such that for every ∈ A
̅ ,μ ̃̅ (x) is a Gaussian
type-2 set A
A
type-1 set; and, a type-2 FLS is tuned by GA in which all the
antecedent sets are Gaussian type-2 sets.Therefore each
membership function of the antecedent part is represented by
an upper and a lower membership functions as following:
μA̅̃ji (xi ) = [ μA̅̃ji (xi ), μA̅̃ (xi )]
(8)
ji
The product t-norm operator for the lower and upper
membership function are calculated by following equation
respectively:
wj = μÃ̅ j1 (x1 ) μÃ̅ j2 (x2 )  … μÃ̅ jm (xm )
wj = μA̅̃ (x1 ) μA̅̃ (x2 )  … μA̅̃ (xm )
j1
j2
(9)
jm
In this study we compare the performance of Dombi,
Hamacher, Schweizer & Sklar, Yager, Dubois & Prade, Weber
and Yu inference engines which are indicated in Table 1[36].
We select the optimum operator by using output error. To
calculate the error of the output, the upper and lower bounds
are defined for each cluster as βj and βj respectively. So the
error of xi which is belong to the cluster jth are determined as:
𝑒𝑖 = {
0
1
𝛽𝑗 ≤ y𝑖 ≤ 𝛽𝑗
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
where 𝛽𝑗 and 𝛽𝑗
(10)
are tuned by GA algorithm. The total error
of system is calculated as:
𝑚
(11)
𝐸 = ∑ 𝑒𝑖
𝑖=1
The final output of type-2 TSK is determined as:
y=
∑n
j=1 wj yj
∑n
j=1 wj
,y=
∑n
j=1 𝑤𝑗 y𝑗
(12)
∑n
j=1 𝑤𝑗
The following equation is used for defuzzification of the
result:
𝑦 = 𝛼𝑦 + (1 − 𝛼) 𝑦
(13)
Where , 0 ≤ 𝛼 ≤ 1 and determine the degree of sharing the
lower and upper bound of predicted value.
TABLE I.
Refrence
SOME OF INFERENCE ENGINES
Parameter
range
t-norm
1 −1

 
1
1
{1 + [( − 1) + ( − 1) ] }
𝑎
𝑏
Dombi
𝑎𝑏
𝑟 + (1 − 𝑟)(𝑎 + 𝑏 − 𝑎𝑏)
Hamacher
Schweizer & Sklar 1
Schweizer & Sklar 2
Schweizer & Sklar 3
Schweizer & Sklar 4
Yager
Dubois & Prade
Weber
Yu
1
1]𝑝
[max(0, 𝑎𝑝 + 𝑏𝑝 −
1 − [(1 − 𝑎)𝑝 + (1 − 𝑏)𝑝 − (1 − 𝑎)𝑝 (1 − 𝑏)𝑝 ]1/𝑝
exp(−(|𝑙𝑛𝑎|𝑝 + |𝑙𝑛𝑏|𝑝 )1/𝑝 )
𝑎𝑏
[𝑎𝑝 + 𝑏𝑝 − 𝑎𝑝 𝑏𝑝 ]1/𝑝
[(1
1 − 𝑚𝑖𝑛{1,
− 𝑎)𝑤 + (1 − 𝑏)𝑤 ]1/𝑤 }
𝑎𝑏
max(𝑎, 𝑏, 𝛼)
𝑎 + 𝑏 + 𝑎𝑏 − 1
𝑚𝑎𝑥 (0,
)
1+
𝑚𝑎𝑥[0, (1 + )(𝑎 + 𝑏 − 1) − 𝑎𝑏]
>0
r>0
p≠0
p>0
p>0
p>0
w>0
𝑎 ∈ [0,1]
 > −1
 > −1
D. Tuning the parameters of the system
In this research we implement genetic algorithms (GAs)
for tuning the main parameters of the fuzzy system. GA is
theoretically and empirically proven to provide a robust search
in complex spaces, thereby offering a valid approach to
problems requiring efficient and effective searches [37,38].
Genetic algorithm is a heuristic for the function optimization,
where the extreme of the function (i.e., minimal or maximal)
cannot be analytically established. A population of potential
solution is refined iteratively by employing a strategy inspired
by Darwinist evolution or natural selection. Genetic
algorithms promote “survival of the fittest”. This type of
heuristic has been applied in many different fields, including
construction of neural networks and multi-disorder diagnosis.
In GA, first a population of chromosomes is formed. Each
chromosome represents a possible solution to the problem.
The population will undergo operations similar to genetic
evolution, namely reproduction, crossover, and mutation. In
this paper, we use GA for tuning the parameters of the type-2
fuzzy TSK system.
IV.
IMPLEMENTATION OF THE PROPOSED MODEL IN EXPORT
VALUE FORECASTING
To demonstrate the performance of the proposed rule based
expert system to predict export potential in target market, the
study uses the Pistachio export data from Iran to the
international Market. To observe countries export behavior
from Iran, the monetary values of HS tariffs retrieve which is
related to Pistachio from international trade data bases during
the specified 9 years period ending 2010. According to
retrieved information from trade databases, 4925 transactions
generated jointly by 107 foreign customers in transaction data.
(a)
(b)
(c)
Figure 4. International pistachio market segmentation (a) cluster1 (b) cluster2 (c) cluster3
Recency
Frequency
Monetary
Continuity
Trend
Figure 5. Interval type-2 rule base of international market of Pistachio
In this section, we present a type-2 fuzzy TSK model for data
analysis of international market of Pistachio.
To observe the behavior of Pistachio trade trend, first of all
five variables including Recency, Frequency, Monetary,
Continuously and Trend are determined from export time series
using equation (8) and are considered as input variables of the
system
To setting rules of fuzzy system, we segment input
variables space by using PCM clustering algorithm which is
describe in Figure 1. Table 2 summarized the result of
calculating Kwon validity index for 2 ≤ c ≤ 9 . By comparing
the clustering result, the optimum number of clusters is
obtained 3 clusters.
TABLE II.
. RESULT OF KWON VALIDITY INDEX FOR 2 ≤ 𝑐 ≤ 𝑐𝑚𝑎𝑥
number of cluster
2 Cluster
3 Cluster
4 Cluster
5 Cluster
6 Cluster
7 Cluster
8 Cluster
9 Cluster
Kwon validity index
31.68
10.02
41.29
567.02
221.50
132.02
1,898.46
541.70
Table 3 indicates the specification of the each market
segments and Figure 3 illustrates the segmentation of the
international market. In other words, each rule describes the
characteristics of the customers as following: cluster 1
including most valuable customers that have long trade
relationship with Iran since now. So it is important to keep this
loyal customer. The second cluster members are valuable no
as well as first segment but they are rather new customers and
they have potential to increase export to them. The members
of cluster 3 are the short term customer and this segment is not
valuable in global trade of Pistachio.
TABLE III.
INPUT VALUE OF CLUSTER CENTER
Cluster
cluster 1
cluster2
cluster 3
RN
1.000
0.988
0.650
FN
1.000
0.953
0.337
MN
0.800
0.780
0.562
Cn
1.000
0.929
0.287
Tn
0.685
0.666
0.432
number
4
56
47
The segments of international Iranian Pistachio market are
converted into a rule base of type-2 fuzzy TSK IF-THEN rule.
For generating antecedent sets, we use Gaussian type-2 sets. In
Figure 4 the interval type-2 rule base of international market of
Pistachio is shown. As shown in Figure 4, there are five inputs
(Recency, Frequency, Monetary, Continuity and trade Trend)
and three rules.
TABLE IV.
COMPARING THE MSE OF DIFFERENT INFERENCE ENGINES
Parameter
Refrence
Upper
0.750
0.899
0.261
0.029
0.291
0.931
0.397
-2.027
0.133
0.613
Dombi
Hamacher
Schweizer & Sklar 1
Schweizer & Sklar 2
Schweizer & Sklar 3
Schweizer & Sklar 4
Yager
Dubois & Prade
Weber
Yu
Lower
0.588
0.178
0.959
0.455
0.887
0.000
0.238
0.505
-0.158
0.366
error
0.895
0.377
0.372
0.931
0.075
0.143
0.966
1.000
0.060
0.056
Table 4 summarized the result of implementing the
different inference engine including parameter and error of
system. Based on error calculation, the Yu inference engine
with  = 0.366 and  = 0.613 has minimum error. By assuming
β1 = β2 and β2 = β3 , the optimum bounds of cluster are
indicated in table 5.
TABLE V.
Lower
Upper
OPTIMUM CLUSTER BOUNDS
𝛽1
0
0.111
𝛽2
0.111
0.826
𝛽3
0.826
0.955
By using equation (10) for j = 1, . . ,3 the type-2 TSK model
can be defined as equation (16) with Yu inference engine. The
error of the system is 0.056 and it is less compare to other
inference engines.
𝑦1 = 0.731 + 0.298𝑥1 + 0.797𝑥2 + 0.878𝑥3 + 0.416𝑥4 + 0.497𝑥5
(14)
𝑦1 = 0.256 + 0.032𝑥1 + 0.742𝑥2 + 0.269𝑥3 + 0.079𝑥4 + 0.505𝑥5
𝑦̅2 = 0.803 + 0.853𝑥1 + 0.262𝑥2 + 0.755𝑥3 + 0.942𝑥4 + 0.871𝑥5
𝑦2 = 0.453 + 0.944𝑥1 + 0.955𝑥2 + 0.998𝑥3 + 0.390𝑥4 + 0.743𝑥5
𝑦3 = 0.447 + 0.322𝑥1 + 0.876𝑥2 + 0.334𝑥3 + 0.590𝑥4 + 0.857𝑥5
y3 = 0.112 + 0.767x1 + 0.546x2 + 0.371x3 + 0.963x4 + 0.676x5
In these equations by determining input variables 𝑥1 , 𝑥2 ,
𝑥3 , 𝑥4 , 𝑥5 as recency, frequency, monetary, continuity and
trade trend, the upper and lower value of the market export
trade are calculated by using equation(14). This interval value
can be defuzzified using equation (15).
V.
CONCLUSION
In this paper, a type-2 fuzzy TSK system for export value
prediction was presented. The inputs of this system are
defined based on international trade trend time series. Indirect
approach is used to fuzzy system modeling by implementing
the PCM clustering algorithm on input space for generating
rules and the using Kwon validity index to determine optimum
number of rules. The inputs of system are considered as the
interval Gaussian type-2 fuzzy sets. The model is implemented
based on Yu inference engine which has better performance in
modeling the system. The TSK fuzzy expert system is
developed with 3 rules and 5 inputs which indicate the
predicted export value. The fuzzy parameters are tuned by
Genetic algorithms to obtain optimum result.
The proposed system shows its superiority with respect to
robustness, flexibility and error minimization. The system may
be used by international trader for selecting the target market
and increasing the export performance.
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