A systematic type-2 fuzzy optimization model for
global market analysis and its application
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— The aim of this paper is developing the optimization
model for global market analysis. The type-2 fuzzy model is
developed based on the variables which indicate the export trade
trend during the specified period. The proposed model is
implemented for forecasting export value of international market
segment of Parisian carpet. This model can be used for selecting
proper segment of international market by predicting export
value for every product.
Keywords- Type-2 fuzzy modeling; Fuzzy rule based systems;
Forecasting; global market; international market selection.
I.
INTRODUCTION
The purpose of this study is to develop a fuzzy modeling
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 interval type-2
fuzzy logic system is presented. Section 4 presents the
proposed interval type-2 fuzzy for prediction of export value of
each market segment of Persian carpet. Finally, in Section 5,
we discuss implication of the results and suggest further study
areas.
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.
The theoretical model that are used in the most studies in
this area is mainly based on regression models [1,2,3,4,5,6].
Some soft computing methods are also applied in analyzing
M. H. F. Zarandi
Department of Industrial Engineering
Amirkabir University of Technology
Tehran, Iran
zarandi@aut.ac.ir
and predicting export value and trend. Neural network
modeling methods is applied to the prediction of export [7] and
developing the model of international market selection [8].
Reference [9] adopts the methodology GIKDE (General
Intervalized Kernel Density Estimator) to predict the amount of
future exports. Reference [10] develops a multi-dimensional
network export performance (NEP) scale using as a basis
network theory to assess export performance. Reference [11]
applies fuzzy time series method for forecasting the amount of
export and he indicates that this method is more effectiveness
that ARIMA time series method.
The rule in a fuzzy logic system (FLS) is constructed
based on uncertain knowledge. The rule uncertainty is related
to the different reasons such as different meaning of the words
that are used in antecedents and consequents of rules for
different people, lack of consensus of experts on the same rule;
and noisy training data[12]. Type-2 FLSs, in which antecedent
or consequent membership functions, are type-2 fuzzy sets is
more powerful than Type-1 FLSs to handle rule uncertainties.
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. Since type-2 fuzzy logic is very
useful when it is difficult to determine the exact membership
functions of fuzzy sets. We develop fuzzy type-2 rule base
system to predict export value.
B. International market Segmentation
Market segmentation makes it possible to find
homogeneous smaller markets thereby helping marketers to
identify marketing opportunities and to develop products and
services in a more tailor-made manner [13]. The challenges
faced by industrial marketers, therefore, lie in the necessity to
choose specific sectors for export promotion and allocate their
limited resources among these sectors.
They have made extensive use of various segmentation
tools and methods [14]. Several methods are available for
identifying international market segments. In the review of
Reference [15], international market segmentation methods are
classified into heuristic methods (Q- or R-factor analysis),
cluster analysis and model-based methods. The most popular
methods for international market segmentation is cluster
analysis [15, 16]. A key issue with grouping is the indicator
used to measure market similarity [17]. Similarities and
dissimilarities may be definable based on market demand,
consumer needs, preferences, financial and individual behavior
[18].
Some studies divide market based on similar purchasing
behaviors [13, 19, 24] by using RFM model. RFM models
represent customer dynamic behavior and are used for solving
the targeting and the prediction problems in direct marketing
[20], measuring importance degrees of customers [21],
clarifying customer behavior patterns and identifying valuable
customer [22] and also ranking customer to concentrate
promotional effort on loyal customer to increase profit [23].
Calculation Variables
Calculate Recency
for ith country
Calculate
Frequency for ith
country
Calculate Monetary
for ith country
Calculate
Continually for ith
country
Calculate Trend for
ith country
Ri
Fi
Mi
Ci
Ti
Calculate export value for each country based on R,F,M,C and S: Vi as output variables
Clusteing
Clustering countries based on R,F,M,C and S by using fuzzy clustering noise-rejection algorithm
Jaccard Distance
Euclidian Distance
Comparing two approaches
Determining optimum number of cluster by using Cluster validity index
Determining optimum weighted exponent by using fuzzy total scatter matrix
Developing model
Generating interval type-2 fuzzy rules
Setting parametric inference engine : Yager environment (t-norm, s-norm, negation, and defuzzification)
Mamdani Approach
Logical approach
Combine approach
Type reduction
Defuzzification
Tuning fuzzy parameters by using GA algorithm
Export Value Predicting
Figure 1. the framework of the proposed type-2 fuzzy system
C. Type2-Fussy logic system
̃ , is specified by a type-2 membership
A type-2 fuzzy set, A
function, μà (x, u) , where xϵX and u ϵ Jx  [0,1] [26]:
̃ = {((x, u), μà (x, u))| x ∈ X, u ϵ Jx  [0,1]}
A
(1)
in which 0 ≤ μà (x, u) ≤ 1 . If ∬ denotes union over all
̃ can also define as:
admissible 𝑥 and 𝑢, A
̃=∫ ∫
A
μ ̃ (x, u)/(x, u) Jx  [0,1]
(2)
x∈X u ϵ Jx A
̃
Jx is the primary membership of A , where Jx  [0,1] for
x ∈ X . The footprint of uncertainty (FOU) is used to measure
the uncertainty in the primary memberships of a type-2 fuzzy
̃ . The upper membership function (UMF) and lower
set A
̃ are two T1 MFs that bound
membership function (LMF) of A
the FOU. That is[26]:
̃=∫ ∫
A
1/(x, u) Jx  [0,1]
(3)
x∈X u ϵ J
x
{
̃ ) x ∈ X
μà (x) ≡ FOU(A
̃ ) x ∈ X
μà (x) ≡ FOU(A
(4)
The union and intersection operations on type-2 sets are
performed by using t-conorm and t-norm operations between
type-1 sets based on Zadeh’s Extension Principle as follow
[27]:
̃ and B
̃ (μ ̃ (x) = ∫ fx (u)/u
(a) The union of two T2 FSs, A
A
u
and μB̃ (x) = ∫w g w (w)/w, where u, w ∈ Jx ), is
̃∪B
̃  μÃ∪B̃ (x) = μà (x) ∐ μB̃ (x) = ∫ ∫ (fx (u) ∗ g x (w))/(u w )
A
u w
(5)
̃∩B
̃  μÃ∩B̃ (x) = μà (x) ∏ μB̃ (x). = ∫ ∫ (fx (u) ∗ g x (w))/(u ∗ w )
A
u w
(6)
̃ and B
̃, is
(b) The intersection of two T2 FSs, A
̃, A
̃ is
(c) The complement of IT2 FS, A
̃  μ (x) = μà (x) = ∫ fx (u)/1 − u
A
̃
u
A
(7)
The type-2 is more capable to handle the uncertainty
compare to type-1 fuzzy sets. The antecedent and/or
consequent sets in a type-2 FLS are type-2, as result the system
output set is type-2, too. So, in type-2 FLS, the type-2 output
set is computed by the inference engine it first reduces to a
type-1 set and then, defuzzify the type reduced set to a crisp
output as final result of the type-2 FLS [28,29].
The aim of this research is developing a fuzzy modeling
mechanism which is capable to setting a rule base fuzzy system
to predict 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. The RFM
model (Recency, Frequency and Monetary) is extended to
RFMCT
by including two
additional parameter,
C(ontinuously), which indicates the length of longest
subsequent trade series and T(rend), which indicates the slope
of trade trend times series. The countries’ V(alue) in trade net
is calculated based on RFMCT model. These variables describe
the behavior of export trade trend perfectly. The rules of the
fuzzy system are generated by using noise-rejection fuzzy
clustering algorithm [25] for clustering input space and the
optimum number of rules is determined based on validity
index. To increase the performance of segmentation, we
implement clustering algorithm based on Jaccard distance
function instead of Euclidian distance. 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, the parametric inference engine is Yager
environment (t-norm, s-norm, negation, and defuzzification)
and the approach is combination of Mamdani and Logical LM.
At last step, the type 2 result of combine approach is reduced to
type 1 and deffuzified. The Yegar parameters are tuned by GA
algorithm. Figure 1 indicates the framework of study.
III.
DESIGNING THE TYPE-2 FLS
The procedures of development of the proposed system are
described in the following subsections.
A. Determination and calculation of input and output
variables of the system
Identification of input and output variables is the first step
of system modeling. Determining the most relevant variables
as input and output are 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 model variables are defined based on
extended RFM model[20]. After data preparation, we calculate
the R, F, M, C and T variables as input variables of system.
Recency measures the interval between the most recent
exporting time and the analyzing time. Frequency measures
the export frequency within a specified period. Monetary
measures the total monetary value within a specified
period[14]. In this study continuity(C) and trade trend(T) is
calculated additionally. Continuity is the longest continues
period during the analyzing time and Trend is 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. The combined
value can be defined as follow:
V(Gi ) = W R × R(Gi ) + W F × F(Gi ) + W M × M(Gi ) + W C ×
C(Gi ) + W T × T(Gi )
(8)
Where R(Gi ), F(Gi ) , M(Gi ), C(Gi ) and T(Gi ) represent the
scores for the ith country in terms of R, F, M ,C and T
respectively. W R , W F , W M , W C and W T represent the
importance weights for the same. In addition, W R + W F +
W M + W C + W T = 1. Therefore the input variable for each
country is defined as the R(Gi ), F(Gi ) , M(Gi ), C(Gi ) and
T(Gi ) and also the output variable is 𝑉(Gi ).
B. Clustering the input space and determination of the
number of rules
In this step, the system rules are generated based on
clustering algorithm and also the optimum number of rule is
determine based on cluster validity index. For encoding the
variables, input space is clustered and the primary membership
grades of the input clusters are generated. To achieve so, we
consider robust noise-rejection fuzzy clustering algorithm for
process of encoding.
Clustering is the process of grouping a set of objects into
classes of similar objects. 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. Reference [30] introduced Possibilistic
clustering algorithm (PCM) that is more robust than the
original FCM algorithm in the presence of noise. 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 robust noise-rejection
fuzzy clustering algorithm which is defined by Reference [25]
and minimized :
C
m
2
̅ 2
Scs (U, V; X) = ∑N
k=1 ∑i=1(uik ) (‖x k − Vi ‖ − ‖Vi − V‖ ) (9)
̅
The center of ith cluster is Vi and V is the fuzzy total mean
vector of the data set considering their belonging to each of the
clusters. It can be defined as:
̅ = N C1
∑N ∑C (u )m xk
V
(10)
∑
∑ (u )m k=1 i=1 ik
k=1
i=1
ik
For selecting optimum weighting exponent a fuzzy total
scatter matrix is used as follows:
C
m
̅
̅ T
s T = ∑N
(11)
k=1(∑i=1(uik ) )(x k − V)(x k − V)
The trace of the fuzzy total scatter matrix decreases
monotonically from a constant value z to zero as m varies from
one to infinity. A suitable value for weight exponent is that
which gives a trace (sT )=z/2 which z is calculated as :
1
1
N
N
T
N
N
z = trac (∑N
k=1 [(x k − ∑k=1 x k ) (x k − ∑k=1 x k ) ])
(12)
The robust noise rejection fuzzy clustering algorithm [25]
is demonstrated in Figure 2.

Estimating initial cluster centers using agglomerative hierarchical clustering algorithm (AHC) as
follow:
o Putting each of the n data vectors in an individual cluster
o Calculating dissimilarities matrix is using:
2n n
i j
dij = d(Xi , Xj ) = √n +n ‖Vhi − Vhj ‖
i





A
Identifying noise through the data points that have large values of Wj :
Determining the number of outliers ηn and calculating the percentage of good data points:
η
z = Nn , ẑ = (1 − z)
calculating resolution parameter by using :
υi =


j
 𝑉ℎ𝑖 , 𝑉ℎ𝑗 : mean vectors of the hard clusters 𝑋𝑖 , 𝑋𝑗
 𝑛𝑖 , 𝑛𝑗 : number of data in the hard cluster 𝑋𝑖 , 𝑋𝑗
Assigning a threshold Ω to trim outliers
Calculating summation of the distance of the data point xj to all cluster centers as fallowing:
Wj = ∑Ci=1‖xj − Vhi ‖
median(d2(xj ,Vi ))
χ2
0.5
Calculating the cutoff distance using:
d2cut = υi χ2ẑ
Calculating the membership matrix using:
1
uij =
1
d2 (xj , Vi ) m−1
}
1+{ υ
i
Figure 2.
A robust noise rejection fuzzy clustering algorithm
In this study, to improve the performance of clustering we
also measure all dissimilarity by using the Jaccard distance and
compare the results with the Euclidian distance. The Jaccard
distance function is:
xi  xj
dij =
(13)
xi  xj
As result, the input space is partition into homogenizes
segments based on Jaccard and Euclidean distance by using
NPCM algorithm. Consequently, the output space clusters is
obtained by ‘‘projecting’’ the input space partition onto output
variable space.
C. Fuzzy Modeling
There are two very different approaches for selecting the
parameters of a type-2 FLS [31].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. In this study for generating
interval type-2 fuzzy rule bases that the antecedent and
consequent sets are interval type-2 sets, a Gaussian primary
MF are estimated and tuned by genetic algorithm.
The construction of a fuzzy logic system in this study is
multi- input-single-output (MISO) system. This model consists
of creating the set of linguistic statements, called fuzzy rules,
where furzy subsets of input and output variables are used as
antecedents and consequents. The rule can be written as:
j1
j2
jn
R j : IF x1 isr A1 AND x2 isrA2 AND … AND xn isr An THEN u isr Aj
ji
Ai
Where
is the jth term of linguistic variable i
ji
corresponding to the membership function μi (xi ) and Aj
corresponds to the membership function μj (u) representing a
term of control action variable [33]. In this step, the fuzzy
output of the MIS0 model is obtained using Mamdani
approach, Logical approach and combination of these two
approaches.
1) Mamdani approach : first to compute the degrees of
membership of the input values in the rule antecedent.
Employing the s-norm operator as a model for the “and”, we
compute the degree of match of rule r as:
ji input
αr = mini=1,…,n {μi (xi
)}
(14)
The aggregation of all consequences is obtained by using
the t-norm (max) operator:
conseq
(u) = αr  μj (u)
μr
(15)
Finally, we compute the fuzzy set of the control action:
μconseq (u) = αr  μj (u)
(16)
2) Logical approach: the consequence fuzzy sets is
computed by using t-norm operator as following:
conseq
(u) = αr  μj (u)
μr
(17)
And the result of this evaluation process is obtained by
using the s-norm(max) operator as:
μconseq (u) = αr  μj (u)
(18)
3) Combination of Mamdani and logical approaches: To
determine the final result of the system, we combine Mamdani
and Logical approach by using following equation:
y = (1 − λ)ym + λyl
(19)
λ is the parameter of combination and indicated that the
system tend to be logical or Mamdani.
4) Defuzzification: Defuzzification is the calculation of a
crisp numerical value from a space of fuzzy control action.
Defuzzification is usually the most the consuming operation in
fuzzy processing. For defuzzification of the result, first we
reduce type-2 to type-1 sets. To achieve so, we apply type
reduction for Gaussian type-2 fuzzy logic system [29].
̃ is such that for every ∈ A
̃ ,μ ̃ (x)
A Gaussian type-2 set A
̃
A
is a Gaussian type-1 set; and, a Gaussian type-2 FLS is a type-2
FLS in which all the antecedent and consequent sets are
Gaussian type-2 sets. Consider the weighted average as
following:
y(z1 , … , zM , w1 , … , wM ) =
∑M
l=1 wl zl
(20)
∑M
l=1 wl
Where zl ∈ R and wl ∈ [0,1] for l = 1, … , M. If each zl is
̃ l ∈ [0,1], then the extension of
replaced by type-1 fuzzy set W
weighted average gives:
(21)
̃
̃1, … , W
̃ M ) = ∫ … ∫ ∫ … ∫ τM
Y(Z̃1 , … , Z̃M , W
̃ (z1 ) ⋆
l=1 μZ
τM
l=1
μW
̃ l (w1 ) |
∑M
l=1 wl zl
z1
zM
w1
wM
l
∑M
l=1 wl
Where τ and ⋆ both indicate the t-norm used (in this study
̃ l and zl ∈ Z̃l for l = 1, … , M.
Yegar), wl ∈ W
Then the Basic Defuzzification Distribution (BADD) is
used for defuzzification of the result.
y∗ =
∫ y[μF (y)]α dy
∫[μF (y)]α dy
(22)
D. Tuning the parameters of the system
In this, the main parameters of the fuzzy system are tuned
by genetic algorithms (GAs). 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 [34,35]. Genetic algorithm is a
heuristic for the function optimization, 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 parameter of FLS
membership, t-norm, s-norm, negation of Mamdani and logical
approaches, combination of both approaches and
defuzzification of the system.
IV.
IMPLEMENTATION OF THE PROPOSED MODEL IN EXPORT
VALUE FORECASTING
To demonstrate the performance of the proposed rule based
fuzzy system to predict export potential in target market, the
study uses the Persian carpet export data of the specified 9
years period ending 2010. According to retrieved information
from trade databases, Iran exports carpet to the total 146
countries and free zones. To observe countries export behavior,
the monetary values of HS tariffs retrieve which is related to
carpet from international trade data bases during the specified
period. 5811 transactions generated jointly by 146 foreign
customers in transaction data. In this section, we present a
type-2 fuzzy model for data analysis of international market of
Persian carpet. To observe the behavior of export trade, first of
all five variables including Recency, Frequency, Monetary,
Continuously and Trend are determined from time series of
export Persian carpet for each country as input variables of
system and export value which is calculated by equation (8) is
considered as output variable.
TABLE I.
Number of
Custer
2 Cluster
3 Cluster
4 Cluster
5 Cluster
6 Cluster
7 Cluster
8 Cluster
9 Cluster
10 Cluster
VALIDITY INDEX
Euclidian
Jaccard
distance
distance
-13.9606
-3.53885
-9.2351
-5.1269
-9.4728
-4.25369
-9.9229
-4.57767
-8.3614
-4.25459
-6.6510
-4.2613
-5.9581
-4.29202
-5.4066
-3.96446
-5.5176
-4.0919
To setting rules of fuzzy system, we segment input
variables space by using noise-rejection fuzzy clustering
algorithm (section IV-B). The suitable weight exponent is
selected as m =3.2 for Euclidian distance and m=2.2 for
Jaccard distance. In addition, the optimum number of clusters
obtained using the fuzzy clustering algorithm based Euclidian
and Jaccard distance are 2 and 3 clusters, respectively. Table I
summarized the result of segmentation step. By comparing the
clustering result, the optimum number of cluster is selected 3
and optimum m=2.2 based on Jaccard distance.
Table II indicates the specification of the each market
segments and Figure 3 illustrates the segmentation of the
international market. Each rule describes the characteristics of
the customers as following: cluster 1 including most valuable
customers that have long trade relationship with Iran till now.
So it is important to keep this loyal customer. The second
cluster members are valuable no as well as first segment but
For generating interval type-2 fuzzy rule bases that the
Figure 1. International Persian carpet market segmentation
Recency
Frequency
Monetary
Continuity
Trend
Export Value
Mamdani approach
Logical approach
Combination of Mamdani and Logical
Type reduction
Figure 4. Interval type-2 rule base of international market of Persian carpet
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 Persian carpets.
TABLE II.
Cluster
Cluster 1
Cluster 2
Cluster 3
INPUT AND OUTPUT VARIABLES
Recency
Frequency
Monetary
Continually
Trend
Export Value
1.000
0.903
0.301
1.000
0.515
0.163
0.609
0.374
0.176
1.000
0.403
0.150
0.793
0.776
0.791
0.823
0.551
0.258
The segments of international Persian carpet market are
modeled into a multiple-input-single-output (MISO) system.
antecedent and consequent sets are interval type-2 sets, a
Gaussian primary MF is estimated and tuned by GA.
In Figure 4 the interval type-2 rule base of international
market of Persian carpet is shown. As shown in the figure,
there are five inputs (Recency, Frequency, Monetary,
Continuity and trade Trend) , one output (Export Value) and
three rules. We use Mamdani , Logical approach and combine
these two approach to develop the system based on Yager
inference engine. Additionally, some method is used for type
reduction and defuzzification (BADD). The t-norm, s-norm
and complement in Yager environment are defined as
following, respectively [36]:
1
t w (a, b) = 1 − min {1, [(1 − a)w + (1 − b)w ]w }
sw (a, b) = min{1, [(a)w + (b)w ]1/w }
(23)
Cw (a) = [1 − (a)w ]w
(25)
(24)
1
The result of Combine Mamdani and Logical approach and
type reduction of Interval type-2 rule base of international
market of Persian carpet is also shown in figure 4.
V.
CONCLUSION
In this paper, interval type-2 fuzzy system for export value
prediction was presented. The inputs and output of this system
is defined based on international trade trend time series.
Indirect approach is used to fuzzy system modeling by
implementing the noise-rejection fuzzy clustering algorithm on
input space for generating rules and the using validity index to
determine optimum number of rules. Both input and output of
system are considered as the interval Gaussian type-2 fuzzy
sets. The model is implemented based on Yegar inference
engine (t-norm, s-norm, complement, defuzzification). The
MISO fuzzy system is developed with 3 rules and 5 inputs.
Based on combination of Mamdani and Logical, the results of
system first are reduced to type-1 and then defuzzification to
the value 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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