Estimate Output of a Production Unit in Production Possibility Set

M. Adabitabar firozja, et al /IJDEA Vol.4, No.2, (2016).737-749
Available online at http://ijdea.srbiau.ac.ir
Int. J. Data Envelopment Analysis (ISSN 2345-458X)
Vol.4, No.2, Year 2016 Article ID IJDEA-00423, 9 pages
Research Article
International Journal of Data Envelopment Analysis
Science and Research Branch (IAU)
Estimate Output of a Production Unit in Production
Possibility Set with Fuzzy Inference Mechanism
Mohamad Adabitabar firozja
a*
b
, Mohamad Adabi firozjaei , Mousa Eslamian
c
(a) Department of Mathematics, Qaemshahr Branch, Islamic Azad University, Qaemshahr,
Iran.
(b) Department of Management , Babol Branch, Islamic Azad University, Babol, Iran.
(c) Phd Student of Management, Semnan Branch, Islamic Azad Universal University,
Semnan, Iran.
Received 30 June 2016, Revised 13 September 2016, Accepted 26 September 2016
Abstract
In this paper, we consider the production possibility set with n production units such that the
following four principles that governs: inclusion observations, conceivability, immensity and
convexity. Our goal is to estimate the output of a same and new production unit with existing
production possibility and amount of input is specified. So, initially we find the interval changes of
each inputs and outputs and then partitioning each interval with fuzzy numbers. We use each of the
possible to produce with specific inputs and outputs the fuzzy if-then rules and estimate the amount of
outputs with mamdany method in fuzzy inference mechanism. In the end, we present an example of
implementation.
Keywords: Production possibility set, Fuzzy number, Fuzzy inference.
*
Corresponding Author: [email protected]
957
M. Adabitabar firozja, et al /IJDEA Vol.4, No.2, (2016).957-965
1. Introduction
present the estimating most productive scale
Data Envelopment Analysis (DEA) was
size using data envelopment analysis. In this
originally proposed by Charnes et al. [7] as a
work, in section 2 we present some basic
method for evaluating the relative efficiency
definitions of DEA, fuzzy numbers and its
of Decision Making Units (DMUs) performing
property and then inference mechanism. In
essentially the same task. Units use similar
section 3, we state the the new method for
multiple inputs to produce similar multiple
estimate output of a production unit with using
outputs. DEA deals with the evaluation of the
fuzzy inference mechanism with specific
performance
a
input. In section 4, we present one example for
transformation process of several inputs to
illustration of method. Conclusion is drawn in
several outputs. There are several ways to
section 5.
evaluate the units such as CCR [7], BCC [4]
2. Background
and etc. Production units are always looking
We provide the contents of two topics, data
for the best output. Zadeh in [12], introduced
envelopment
the concept of fuzzy number. The concept of
mechanism.
fuzzy numbers and arithmetic operations with
2.1. Data envelopment analysis
fuzzy numbers were first introduce and
DEA utilizes a technique of mathematical
investigated by Chang and Zadeh [6], and
programming for evaluation of n DMUs.
others. Applications of fuzzy numbers for
Suppose
indicating uncertain and vague information in
𝐷𝑀𝑈1 , 𝐷𝑀𝑈2 , … , 𝐷𝑀𝑈𝑛 . Let the input and
decision making, linguistic controllers, expert
output data for 𝐷𝑀𝑈𝑗 be 𝑋𝑗 = (𝑥1𝑗 , … , 𝑥𝑚𝑗 )
systems, data mining, pattern recognition, etc.
and 𝑌𝑗 = (𝑦1𝑗 , … , 𝑦𝑠𝑗 ), respectively.[3]
of
DMU
performing
were stated in recently published articles
analysis
there
(DEA)
are
and
n
fuzzy
DMUs:
The set of feasible activities is called the
[1,2,11]. Systems of fuzzy IF-THEN rules are
Production Possibility Set (PPS) and is
widely used in applications of fuzzy set theory,
denoted by P with the properties: inclusion
for example in fuzzy control, identification of
observations, conceivability, immensity and
dynamic systems, prediction of dynamic
convexity, where [8]
systems, decision-making, etc. In this paper,
𝑛
𝑃 = (𝑋, 𝑌) 𝑋 ≥
we consider the fuzzy IF-THEN rules method
𝑛
𝜆𝑗 𝑋𝑗𝑡
𝑡
𝑗 =1
for problem "Estimate output of a production
𝜆𝑗 𝑌𝑗 𝑡
𝑡
,𝑌 ≤
(1)
𝑗 =1
CCR model [7] for evaluate relative efficiency
unit in production possibility set with specific
of 𝐷𝑀𝑈𝑘 is as follows,
input such that in comparison with similar
min
units is efficient". Some researchers worked on
𝑛
𝑠. 𝑡.
the estimate output such as in [5], R.D. Banker
𝜆𝑗 𝑥𝑖𝑗 ≤ 𝜃𝑥𝑖𝑘
𝑗 =1
958
𝜃
𝑖 = 1, … , 𝑚
(2)
M. Adabitabar firozja, et al /IJDEA Vol.4, No.2, (2016). 957-965
𝑛
𝜆𝑗 𝑦𝑟𝑗 ≤ 𝜃𝑦𝑟𝑘
and for 𝑇 ∈ 𝐹 𝑇 (𝑅) its membership function is
𝑟 = 1, … , 𝑠
𝑗 =1
𝜆𝑗 ≥ 0,
as follows:
0
𝑥 − 𝑎1
𝑎2 − 𝑎1
𝑇 𝑥 = 1
𝑎4 − 𝑥
𝑎4 − 𝑎3
0
𝑗 = 1, … , 𝑛
Unit DMU𝑘 is CCR efficient if the optimal
value of the objective function in model (2)
equals one (𝜃 ∗ = 1).
𝑥 ≤ 𝑎1
𝑎1 ≤ 𝑥 ≤ 𝑎2
𝑎2 ≤ 𝑥 ≤ 𝑎3
𝑎3 ≤ 𝑥 ≤ 𝑎4
𝑎4 ≤ 𝑥
2.2 Fuzzy mechanism
Where a1 , a2 , a3 , a4 ∈ R
There are many definitions for fuzzy numbers.
fuzzy number 𝑇 is as:
We denote set of all fuzzy numbers by 𝐹(𝑅)
𝑇𝛼 = 𝑇𝑙 𝛼 , 𝑇𝑟 𝛼
and define as follows.[9]
(6)
and 𝛼 −level of
= [𝑎1 + 𝑎2 − 𝑎1 𝛼 , 𝑎4
− 𝑎4 − 𝑎3 𝛼]
(7)
Triangular fuzzy numbers are also special
Definition 1. For fuzzy number 𝐴 ∈ 𝐹(𝑅) , we
cases of trapezoidal fuzzy numbers with
show the membership function by 𝐴(𝑥) which
𝑎2 = 𝑎3 .
is given by
𝐴 𝑥 =
0
𝑙𝐴 𝑥
1
𝑟𝐴 𝑥
0
𝑥 ≤ 𝑎1
𝑎1 ≤ 𝑥 ≤ 𝑎2
𝑎2 ≤ 𝑥 ≤ 𝑎3
𝑎3 ≤ 𝑥 ≤ 𝑎4
𝑎4 ≤ 𝑥
2.2.1 Inference mechanism
When we model a knowledge system, it is
(3)
often represented by the form of fuzzy rule
base. The fuzzy rule base consists of fuzzy ifthen rules. The rule base has the form of a
Where a1 , a2 , a3 , a4 ∈ R
decreasing and 𝑟𝐴 𝑥
and 𝑙𝐴 𝑥 is non-
MIMO
is non-increasing and
𝑙𝐴 𝑎1 = 0, 𝑙𝐴 𝑎2 = 1, 𝑟𝐴 𝑎3 = 1
(multiple
input
multiple
output)
system.[10]
and
𝑟𝐴 𝑎4 = 0 . For any 𝛼 ∈ 0,1 ; 𝛼 −cut of
2.2.2 Mamdani inference mechanism
fuzzy numbe 𝐴 is a crisp interval as
We assume that we have k fuzzy control rules
𝐴𝛼 = 𝑥 ∈ 𝑅: 𝐴(𝑥) ≥ 𝛼 = 𝐴𝑙 𝛼 , 𝐴𝑟 𝛼
(4)
of the form
If 𝐴, 𝐵 ∈ 𝐹(𝑅) and 𝜆 ∈ 𝑅 then (A + B)α =
𝑅𝑢𝑙𝑒1 ∶ 𝐼𝑓 𝑥 1 𝑖𝑛 𝐴11 ,· · · , 𝑥 𝑚 𝑖𝑛 𝐴 𝑚1 𝑇ℎ𝑒𝑛
𝐴𝛼 + Bα and A = B, if 𝐴𝑙 𝛼 = 𝐵𝑙 (𝛼) and
𝑦 1 𝑖𝑛 𝐵11 ,· · · , 𝑦 𝑠 𝑖𝑛 𝐵𝑠1
𝐴𝑟 𝛼 = 𝐵𝑟 𝛼 almost everywhere, 𝛼 ∈ [0,1].
⋮
(𝜆𝐴)𝛼 =
𝜆𝐴𝑙 𝛼 , 𝜆𝐴𝑟 𝛼
𝜆𝐴𝑙 𝛼 , 𝜆𝐴𝑟 𝛼
𝜆≥0
𝜆<0
𝑅𝑢𝑙𝑒 𝑘 ∶ 𝐼𝑓 𝑥 1 𝑖𝑛 𝐴1𝑘 ,· · · , 𝑥 𝑚 𝑖𝑛 𝐴 𝑚𝑘 𝑇ℎ𝑒𝑛
(5)
𝑦 1 𝑖𝑛 𝐵1𝑘 ,· · · , 𝑦 𝑠 𝑖𝑛 𝐵𝑠𝑘
𝑓𝑎𝑐𝑡 ∶ 𝐼𝑓 𝑥 1 = 𝑥1 ,· · · , 𝑥𝑚 = 𝑥𝑚
Trapezoidal fuzzy numbers are special cases of
𝑐𝑜𝑛𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒 ∶ 𝑦 1 𝑖𝑛 𝐵1 ,· · · , 𝑦 𝑠 𝑖𝑛 𝐵𝑠
fuzzy numbers which are denoted by 𝐹 𝑇 (𝑅)
959
(8)
M. Adabitabar firozja, et al /IJDEA Vol.4, No.2, (2016).957-965
The firing levels of the rules, denoted by
units. Without loss of generality argument
𝛼𝑗 : 𝑗 = 1, … , 𝑘 are computed by
suppose that 𝑥 𝑖1 ≤ 𝑥 𝑖2 ≤ . . . ≤ 𝑥 𝑖𝑘 for 𝑖−th
𝛼 𝑗 = 𝐴1𝑗 𝑥1 ∧ … ∧ 𝐴𝑚𝑗 𝑥𝑚 ;
input and 𝑦 𝑟1 ≤ 𝑦 𝑟2 ≤ . . . ≤ 𝑦 𝑟𝑘 for 𝑟−th
𝑗 = 1, … , 𝑘
9
output.
The individual rule outputs are obtained by
Now we partition the interval [𝑥𝑖1 , 𝑥𝑖𝑘 ] for
𝐶𝑟𝑗 𝑤𝑟 = 𝐵𝑟𝑗 𝑤𝑟 ∧ 𝛼𝑗 ; 𝑗 = 1, … , 𝑘,
𝑖 = 1, . . . , 𝑚 with triangular fuzzy numbers
𝑟 = 1, … , 𝑠
10
are as follows:
𝑥𝑖2 − 𝑥
𝐴𝑖1 𝑥 = 𝑥𝑖2 − 𝑥𝑖1
0
𝐶𝑟 𝑤𝑟 = 𝐶𝑟1 𝑤𝑟 ∨ … ∨ 𝐶𝑟𝑘 𝑤𝑟 ;
𝑟 = 1, . . . , 𝑠
(11)
𝑥𝑖1 ≤ 𝑥 ≤ 𝑥𝑖2
13
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Finally, to obtain a deterministic control
𝐴𝑖𝑗 𝑥
action, we employ any defuzzification strategy
𝑥 − 𝑥𝑖 𝑗 −1
𝑥𝑖𝑗 − 𝑥𝑖 𝑗 −1
𝑥𝑖 𝑗 +1 − 𝑥
=
𝑥𝑖 𝑗 +1 − 𝑥𝑖𝑗
0
[10].
2.2.3 Middle-of-Maxima defuzzification
If 𝐶 is not discrete then defuzzified value of a
fuzzy set 𝐶 is defined as [10]
𝑍0 =
𝐺
𝑧𝑑𝑧
=
where 𝐺 denotes the set of maximizing
element of 𝐶.
𝐷𝑀𝑈𝑠 ∶ 𝐷𝑀𝑈1 , 𝐷𝑀𝑈2 , … , 𝐷𝑀𝑈𝑛 .
Let
and
also,
𝐵𝑟1 𝑦
𝑦𝑟2 − 𝑦
= 𝑦𝑟2 − 𝑦𝑟1
0
the
input and output data for 𝐷𝑀𝑈𝑗 be 𝑋 𝑗 =
𝑥 1𝑗 , … , 𝑥 𝑚𝑗
𝑌 𝑗 = 𝑦 1𝑗 , … , 𝑦𝑠𝑗 ,
respectively.
𝐵𝑟𝑗 𝑦
We want to create a new similar production
𝑦 − 𝑦𝑟 𝑗 −1
𝑦𝑟𝑗 − 𝑦𝑟 𝑗 −1
= 𝑦𝑟 𝑗 +1 − 𝑦
𝑦𝑟 𝑗 +1 − 𝑦𝑟𝑗
0
unit
with
the
specified
input
(14)
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
we
(15)
partition
the
𝑟 = 1, . . . , 𝑠 with
triangular fuzzy numbers are as follows:
𝑛
are
; 𝑗 = 2, … , 𝑘 − 1
𝑗 +1
𝑥𝑖(𝑘−1) ≤ 𝑥 ≤ 𝑥𝑖𝑘
interval [𝑦𝑟1 , 𝑦𝑟𝑘 ] for
with fuzzy Inference mechanism
there
≤ 𝑥 ≤ 𝑥𝑖𝑗
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
𝑥 − 𝑥𝑖(𝑘−1)
𝑥𝑖𝑘 − 𝑥𝑖(𝑘−1)
0
And
3 Estimate output of a production units
Suppose
𝑥𝑖𝑗 ≤ 𝑥 ≤ 𝑥𝑖
𝑗 −1
𝐴𝑖𝑗 𝑥
12
𝑑𝑧
𝐺
𝑥𝑖
𝑋 =
(𝑥1 , . . . , 𝑥𝑚 ) were new input should be in the
convex hull of the units input observed. For
this purpose, first we identify efficient units by
𝑦𝑟1 ≤ 𝑦 ≤ 𝑦𝑟2
16
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
𝑦𝑟
𝑦𝑟𝑗 ≤ 𝑦 ≤ 𝑦𝑟
𝑗 −1
𝑗 +1
≤ 𝑦 ≤ 𝑦𝑟𝑗
; 𝑗 = 2, … , 𝑘 − 1
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
17
𝐵𝑟𝑗 𝑦
existing methods such as 𝐶𝐶𝑅 or 𝐵𝐶𝐶, etc.
𝑦 − 𝑦𝑟(𝑘−1)
= 𝑦𝑟𝑘 − 𝑦𝑟(𝑘−1)
0
Let, 𝑘 𝐷𝑀𝑈𝑠 be effective. Then, we arranged
increasing order inputs and outputs of effecient
960
𝑦𝑟(𝑘−1) ≤ 𝑦 ≤ 𝑦𝑟𝑘
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
(18)
M. Adabitabar firozja, et al /IJDEA Vol.4, No.2, (2016). 957-965
𝑗 = 1, . . . , 𝑘, 𝑟 = 1, . . . , 𝑠
We consider each efficient unit observed as a
𝑗 =
fuzzy control rules as follows for
𝐶𝑟 𝑤𝑟 = 𝐶𝑟1 𝑤𝑟 ∨ … ∨ 𝐶𝑟𝑘 𝑤𝑟 ;
1, . . . , 𝑘:
𝑟 = 1, . . . , 𝑠
𝑅𝑢𝑙𝑒1 ∶ 𝐼𝑓 𝑥 1 𝑖𝑛 𝐴11 ,· · · , 𝑥 𝑚 𝑖𝑛 𝐴 𝑚1 𝑇ℎ𝑒𝑛
(22)
Finally, to obtain a deterministic control action
𝑦 1 𝑖𝑛 𝐵11 ,· · · , 𝑦 𝑠 𝑖𝑛 𝐵𝑠1
the outputs, we employ Middle-of-Maxima
defuzzification method.
⋮
𝑅𝑢𝑙𝑒 𝑘 ∶ 𝐼𝑓 𝑥 1 𝑖𝑛 𝐴1𝑘 ,· · · , 𝑥 𝑚 𝑖𝑛 𝐴 𝑚𝑘 𝑇ℎ𝑒𝑛
𝑦 1 𝑖𝑛 𝐵1𝑘 ,· · · , 𝑦 𝑠 𝑖𝑛 𝐵𝑠𝑘
(21)
if 𝑥 1 = 𝑥1𝑡 ,· · · , 𝑥 𝑚 = 𝑥𝑚𝑡 ,
Now
for
𝑡 = 1, . . . , 𝑘.
(19)
With Eq. (20)
Where 𝐴𝑖𝑗 corresponding to 𝑖−th input of 𝑗−th
𝛼𝑗 =
unit. And also, 𝐵𝑟𝑗 corresponding to 𝑟−th
1
0
𝑗=𝑡
𝑗≠𝑡
(23)
With Eq. (21) for 𝑟 = 1, . . . , 𝑠
output of 𝑗−th unit.
𝐵𝑟𝑡 (𝑦𝑟𝑡 )
0
𝑗=𝑡
𝑗≠𝑡
Now suppose the input the new unit is as:
𝐶𝑟𝑗 (𝑤𝑟 ) =
𝑥 1 = 𝑥1,· · · , 𝑥 𝑚 = 𝑥𝑚 ,
And with Eq. (22) for 𝑟 = 1, . . . , 𝑠
we obtain with
(24)
Mamdanis max min method and Middle-of-
𝐶𝑟 𝑤𝑟 = 𝐶𝑟1 𝑤𝑟 ∨ … ∨ 𝐶𝑟𝑘 𝑤𝑟 = 𝐵𝑟𝑡 𝑦𝑟𝑡 ;
Maxima defuzzification method the outputs
𝑟 = 1, . . . , 𝑠
of 𝑦 1 = 𝑦1 ,· · · , 𝑦 𝑠 = 𝑦𝑠 .
Now, with Middle-of-Maxima defuzzification
(25)
method 𝑦 𝑟 = 𝑦𝑟𝑡 for 𝑟 = 1, . . . , 𝑚.
Proposition
1.
In
provided
inference
method, interpolation is maintains. In other
3.1 Examples
words, if input is: 𝑥 1 = 𝑥1𝑡 ,· · · , 𝑥 𝑚 = 𝑥𝑚𝑡 ,
In this section, we consider 19 𝐷𝑀𝑈𝑠 with
then with Mamdanis max min method and
two inputs and two outputs.(Table 1). First,
Middle-of-Maxima defuzzification method
we identify efficient units by CCR model
the outputs is as y1 = 𝑦1𝑡 ,· · · , 𝑦 𝑠 = 𝑦𝑠𝑡 for
where has come on table 2 (6 DMUs are
𝑡 = 1, . . . , 𝑘.
efficient). To implement the example we use
MATLAB software. We partition the interval
Proof. With rules (19) and fuzzy partition of
[0, 216] (range of input 1) and [0, 203] (range
input interval and fuzzy partition of output
of input 2) with triangular fuzzy numbers
interval, Mamdanis max min method is as
regarding to Eqs (13)-(15) are as figures 1
following
and 2. And also, we partition the interval
𝛼 𝑗 = 𝐴1𝑗 𝑥1 ∧ … ∧ 𝐴𝑚𝑗 𝑥𝑚𝑗 ;
[0, 5368] (range of output 1) and [0, 639]
𝑗 = 1, . . . , 𝑘
(20)
(range of output 2) with triangular fuzzy
numbers regarding to Eqs (16) − (18) are as
The individual rule outputs are obtained by
𝐶𝑟𝑗 𝑤𝑟 = 𝐵𝑟𝑗 𝑤𝑟 ∧ 𝛼 𝑗 ;
figures 3 and 4.
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M. Adabitabar firozja, et al /IJDEA Vol.4, No.2, (2016).957-965
Table 1. 19 DMUs with two inputs and two outputs.
DMU
input 1
input 2
output 1
output 21
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
81
85
56.7
91
216
58
112.2
293.2
186.6
143.4
108.7
105.7
235
146.3
57
118.7
58
146
0
87.6
12.8
55.2
78.8
72
25.6
8.8
52
0
105.2
127
134.4
236.8
124
203
48.2
47.4
50.8
91.3
5191
3629
3302
3379
5368
1674
2350
6315
2865
7689
2165
3963
6643
4611
4869
3313
1853
4578
0
205
0
0
8
639
0
0
414
0
66
266
315
236
128
540
16
230
217
508
Table 2. DMUs are CCR efficient.
DMU
input 1
input 2
output 1
output 21
1
2
5
9
15
19
81
85
216
186.6
57
0
87.6
12.8
72
0
203
91.3
5191
3629
5368
2865
4869
0
205
0
639
0
540
508
Figure 1: Partition of interval [0, 216] for input 1 with triangular fuzzy numbers.
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M. Adabitabar firozja, et al /IJDEA Vol.4, No.2, (2016). 957-965
Figure 2: Partition of interval [0, 203] for input 2 with triangular fuzzy numbers.
Figure 3: Partition of interval [0, 5368] for output 1 with triangular fuzzy numbers.
Figure 4: Partition of interval [0, 639] for output 2 with triangular fuzzy numbers.
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We consider each efficient unit observed as a
For example, Approximation output to several
fuzzy control rule as figure 5 with partitions of
units with specific inputs is shown in Table 3.
inputs and outputs.
Figure 5: Fuzzy control rules
Table 3. Approximation output with known input.
Input 1
Input 2
Outnput 1
Outnput 2
𝒙𝟏 =90
𝒙𝟐 =20
𝑦1 ≈ 3.65𝑒 + 003
𝑦2 ≈ 9.58
𝒙𝟏 =200
𝒙𝟐 =80
𝑦1 ≈ 5.34𝑒 + 003
𝑦2 ≈ 613
𝒙𝟏 =118
𝒙𝟐 =42.8
𝑦1 ≈ 3.76𝑒 + 003
𝑦2 ≈ 51.1
𝒙𝟏 =69
𝒙𝟐 =180
𝑦1 ≈ 4.62𝑒 + 003
𝑦2 ≈ 559
Figure 6: Approximation outputs with 𝑥1 = 200 and 𝑥2 = 80.
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4 Conclusions
[5] R.D. Banker, Estimating most productive
In this paper, we consider the production
scale size using data envelopment analysis,
possibility set with n production units such
European Journal of Operational Research 17
that the following four principles that govern:
(1984)35-44.
inclusion
conceivability,
[6] Chang SL, Zadeh LA, On fuzzy mapping
immensity and convexity. We proposed a
and control. IEEE Trans Sys Man Cybernet
method for estimate the output of a same and
1972; 2:304.
new production unit with existing production
[7] A. Charnes, W.W. Cooper, E. Rhodes,
possibility and amount of input is specified.
Measuring efficiency of decision making units,
We in Proposition 1. Prove that in this
European Operational Research; (1978), 2,
inference method, interpolation is maintains.
429-444.
In the end, we present an example of
[8] W. W. Cooper, L. M. Sieford and K. Tone,
implementation.
Data envelopment analysis: a comprehensive
observations,
text with models, applications, References and
References
DEA Solver Software, Kluwer Academic
[1] S. Abbasbandy, M. Adabitabar Firozja,
Publishers, 2000.
Fuzzy linguistic model for interpolation,
[9]
Chaos, Solitons and Fractals 34 (2007) 551-
Grzegorzewski, M. Adabitabar Firozja, T.
556.
Houlari, Piecewise Linear Approximation of
[2] M. Adabitabar Firozja, G.H. Fath-Tabar, Z.
Fuzzy Numbers Preserving the Support and
Eslampia,
of
Core, In: Laurent A. et al. (Eds.), Information
generalized fuzzy numbers based on interval
Processing and Management of Uncertainty in
distance, Applied Mathematics Letters 25
Knowledge-Based Systems, Part II, (CCIS
(2012) 1528-1534.
443), Springer, 2014, pp. 244-254.
[3] T. Allahviranloo, F. Hosseinzadeh lotfi and
[10] R. Fullier, Neural Fuzzy Systems, ISBN
M. Adabitabar firozja, Efficiency in fuzzy
951-650-624-0, ISSN 0358-5654.
production possibility set, Iranian Journal of
[11] S. Martin, B. Michal, S. Lenk, Fuzzy Rule
Fuzzy Systems Vol. 9, No. 4, (2012) pp. 17-
Base Ensemble Generated from Data by
30.
Linguistic Associations Mining, Fuzzy Sets
[4] R.D. Banker, A. Charnes, W.W. Cooper,
and System, Accepted date: 22 April 2015.
Some models for estimating technical and
[12] L.A. Zadeh, Fuzzy sets, Inform. Control
scale
8(1965)338-353.
The
inefficiency
similarity
in
data
measure
envelopment
analysis, Manage. Sci 30(1984), pp. 10781092.
965
L.
Coroianu,
M.
Gagolewski,
P.

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