# An Application of Interval Arithmetic for Solving Fully Fuzzy Linear Systems with Trapezoidal Fuzzy Numbers.

1. IntroductionIn various fields of sciences, in solving real-life problems, where the system of linear equations is noticed, there are situations where the values of the parameters cannot be stated exactly, but their estimation or some bounds on them can be measured. Modeling a lot of these problems leads to a fuzzy linear system because the inexact kind of real numbers can be modeled in fuzzy numbers.

Fuzzy linear systems have been studied by several authors. Friedman et al. [1] proposed a general model for solving an n x n fuzzy linear system whose coefficient matrix is crisp and the right-hand side column is an arbitrary fuzzy vector. Allahviranloo [2] proposed solution of a fuzzy linear system by using an iterative method and later suggested various numerical methods to solve fuzzy linear systems [3]. Some methods for solving these systems can be found in [4-7]. Later, a system, called fully fuzzy linear system (FFLS), is introduced wherein elements in the coefficient matrix, the right-hand side vector, and the vector of unknowns are fuzzy numbers.

This system is solved in [8, 9] using decomposition of the coefficient matrix. Some other methods used iterative methods for solving FFLS [3, 10, 11]. Many researchers studied FFLS and numerical techniques to solve them [1216]. Kochen et al. proposed a method for solving square and nonsquare FFLS with trapezoidal fuzzy numbers in 2016 [17]. In 2017, Edalatpanah reviewed some iterative methods for solving FFLS [18]. Recently, Allahviranloo and Babakordi introduced a new method for solving an extension of FFLS that has two coefficient matrices [19]. A new method for solving FFLS with hexagonal fuzzy numbers is introduced by Malkawi et al. in 2018 [20].

In Section 2, we present some basics about fuzzy numbers and [alpha]-cut arithmetic. Some basics of interval arithmetic, new interval operations and some properties of interval numbers, and also our method for solving FFLS are investigated in Section 3. In Section 4, some examples are solved to show the effectiveness of the proposed method and concluding remarks are contained in Section 5.

2. Preliminaries

In this section we review fuzzy numbers and [alpha]-cut arithmetic on them. [alpha]-cut method is a method for performing interval operations like addition, multiplication, and subtraction on fuzzy numbers. [alpha]-cut fuzzy arithmetic is described in detail in [21].

Let us consider an arbitrary trapezoidal fuzzy number u = ([u.sub.1], [u.sub.2], [u.sub.3], [u.sub.4]), where [u.sub.1], [u.sub.2], [u.sub.3], [u.sub.4] [member of] R and [u.sub.1] [less than or equal to] [u.sub.2] [less than or equal to] [u.sub.3] [less than or equal to] [u.sub.4]. The membership function [[mu].sub.u] of u is defined as follows:

[mathematical expression not reproducible]. (1)

Two trapezoidal fuzzy numbers u and v are said to be equal, if and only if [u.sub.i] = [v.sub.i] for 1 [less than or equal to] i [less than or equal to] 4.

Given a trapezoidal fuzzy number u and membership function [[mu].sub.u] and a real number [alpha] [member of] {0, 1}, then the [alpha]-cut of a fuzzy number u is an interval, which is defined as

[u.sup.[alpha]] = [[[u.sup.[alpha]].bar], [bar.[u.sup.[alpha]]]], (2)

where [[u.sup.[alpha]].bar], [bar.[u.sup.[alpha]]] are

[bar.[u.sup.[alpha]]] = inf {x [member of] R | [[mu].sub.u] (x) [greater than or equal to] [alpha]}, (3)

[bar.[u.sup.[alpha]]] = sup [x [member of] R | [[mu].sub.u] (x) [greater than or equal to] [alpha]}. (4)

Note that

[mathematical expression not reproducible]. (5)

The basic arithmetic of two fuzzy numbers u and v is discussed in [21-23] based on interval arithmetic

[(u + v).sup.[alpha]] = [u.sup.[alpha]] + [v.sup.[alpha]] = [[[u.sup.[alpha]].bar], [bar.[u.sup.[alpha]]]] + [[v.sup.[alpha]].bar], [bar.[v.sup.[alpha]]], (6)

[(u + v).sup.[alpha]] = [u.sup.[alpha]] + [v.sup.[alpha]] = [[[u.sup.[alpha]].bar], [bar.[u.sup.[alpha]]]] - [[v.sup.[alpha]].bar], [bar.[v.sup.[alpha]]], (7)

[(uv).sup.[alpha]] = [u.sup.[alpha]] x [v.sup.[alpha]] = [[[u.sup.[alpha]].bar], [bar.[u.sup.[alpha]]]] x [[v.sup.[alpha]].bar], [bar.[v.sup.[alpha]]]. (8)

Clearly, for a trapezoidal fuzzy number u = ([u.sub.1], [u.sub.2], [u.sub.3], [u.sub.4]), we have [u.sup.0] = [[u.sub.1], [u.sub.4]], [u.sup.1] = [[u.sub.2], [u.sub.3]].

A matrix (vector) A is called a fuzzy matrix (fuzzy vector), if at least one element of matrix (vector) A is a fuzzy number. A matrix A is called a fully fuzzy matrix, whose elements of A are fuzzy numbers. An [alpha]-cut of an n x m matrix A is an interval matrix with the same dimension whose elements are [alpha]-cut of elements of matrix A.

3. Materials and Methods

3.1. Interval Arithmetic. In this section, we briefly explain interval arithmetic and notion. Also, we present a new operation on interval arithmetic.

An interval number [??] is the bounded, closed subset of real numbers, which is indicated by a hat, defined by [??] = [[a.bar], [bar.a]], where [a.bar], [bar.a] [member of] R and [a.bar] [less than or equal to] [bar.a]. Let DR denote the set of such intervals. An interval [??] is a degenerate interval when [a.bar] = [bar.a].

Infimum and supremum of an interval number [??] are [a.bar] and [bar.a], respectively Let a [member of] [??], say a [member of] [??] when [a.bar] [less than or equal to] a [less than or equal to] [bar.a].

Let [omicron] denote a binary operation on real space R. For [mathematical expression not reproducible]

[mathematical expression not reproducible]. (9)

Then, for basic arithmetic operations {+, -, x, /} in DR, we have

[mathematical expression not reproducible] (10)

[mathematical expression not reproducible] (11)

[mathematical expression not reproducible]. (12)

[mathematical expression not reproducible] (13)

[IR.sup.+] is the set of all improper intervals which is defined in [24] as follows:

[IR.sup.+] = {[??] = [[a.bar], [bar.a]] | [a.bar], [bar.a] [member of] R, [a.bar] [greater than or equal to] [bar.a]}. (14)

The set

[IR.sup.*] = {[??] = [[a.bar], [bar.a]] | [a.bar], [bar.a] [member of] R} (15)

contains all interval numbers and improper interval numbers called general interval. The basic arithmetic of general intervals is defined similar to intervals, but the x operation has some differences that are explained in [25].

By an interval matrix, we mean a matrix whose elements are interval numbers. Let [??] be an interval matrix with elements [[??].sub.ij] and let B be a matrix with real elements [B.sub.ij]; we consider B [member of] [??] if [B.sub.ij] [member of] [[??].sub.ij] for all i and j. Basic arithmetic operations on interval matrices are defined similar to real matrices [26].

3.2. New Interval Operations. Let [mathematical expression not reproducible]; we define two new operations [??] and [??] as follows:

[mathematical expression not reproducible], (16)

[mathematical expression not reproducible] (17)

The operation [??] is defined and used in some articles as an operation in extended interval arithmetic [27]. But, we use [??] in a different way. We illustrate that the operation [??] can act as an inverse of + in interval arithmetic.

Theorem 1. Considering [mathematical expression not reproducible], the interval equation [mathematical expression not reproducible] has a unique solution [mathematical expression not reproducible] and [??] [member of] [IR.sup.*].

Proof.

[mathematical expression not reproducible] (18)

[mathematical expression not reproducible] (19)

Corollary 2. Supposing [mathematical expression not reproducible] and [??] [member of] [IR.sup.*] and [??] [not member of] IR, the equation [mathematical expression not reproducible] has no solution in DR but has exactly one solution in [IR.sup.*].

Corollary 3. If [mathematical expression not reproducible] and [mathematical expression not reproducible], then [??] can be computed by [mathematical expression not reproducible] uniquely. On the other hand, the operation [??] is the inverse operation of + in interval arithmetic.

Theorem 4. Considering [mathematical expression not reproducible], the interval equation [mathematical expression not reproducible] has a solution [mathematical expression not reproducible], where [??] [member of] [IR.sup.*].

Remark 5. Solution [??] is not usually unique; e.g., the interval equation [-1,1] x [??] = [-2,2] has solutions [-1,2], [-2, 0], [-2,2]. Actually this equation has unlimited solutions. All intervals [u, 2] and [-2, v] where -2 [less than or equal to] u [less than or equal to] 0 and 0 [less than or equal to] v [less than or equal to] 2 can be solution of this equation.

In cases that [??] is not unique, we consider [??] as the longest interval as possible. In the above example, [??] will be [-2,2].

Proof. Let [mathematical expression not reproducible]; Table 1 shows sign of [??] related to signs of [??] and [??] and also shows formulas of multiplication result.

Now, given [mathematical expression not reproducible] such that [mathematical expression not reproducible] and [b.bar] [less than or equal to] 0 [less than or equal to] [bar.b]. From Table 1, it is deduced that 0 [member of] [??] and [??] = [[bar.a][x.bar], [bar.a][bar.x]]. Therefore

[mathematical expression not reproducible]. (20)

In this case, [??] is calculated uniquely by operation [??]. Other cases, except cases with condition 0 [member of] [??], are provable as above.

In cases 0 [less than or equal to] [a.bar] and 0 [greater than or equal to] [bar.a] among three possible choices, we could choose the appropriate interval according to the signs of [b.bar] and [bar.b], while in case 0 [member of] [??] all three possible choices are similar in terms of signs of [b.bar] and [bar.b]. To conquer this problem in case 0 [member of] [??], we choose the longest interval as the solution from three possibilities 0 [less than or equal to] [x.bar], 0 [member of] [??], and 0 [greater than or equal to] [bar.x] which is 0 [member of] [??].

Now, let [??] be an interval where 0 [member of] [??] and let [??] be the solution of [mathematical expression not reproducible]; then [??] = [min{[bar.a][x.bar], [a.bar][bar.x]}, max[[a.bar][x.bar], [bar.a][bar.x]}]. It follows that

[mathematical expression not reproducible], (21)

[mathematical expression not reproducible]. (22)

Then [x.bar] [greater than or equal to] max{[b.bar]/[bar.a], [bar.b]/[a.bar]}. If [x.bar] [not equal to] max{[b.bar]/[bar.a], [bar.b]/[a.bar]}, then [x.bar] > max{[b.bar]/[bar.a], [bar.b]/[a.bar]}, and [bar.a][x.bar] > max{[b.bar]/, [bar.a][bar.b[/[a.bar]} [greater than or equal to] [b.bar]. This contradicts with [b.bar] = min{[bar.a][x.bar], [a.bar][bar.x]}. Hence

[mathematical expression not reproducible]. (23)

Similarly, [bar.x] is obtained as

[mathematical expression not reproducible]. (24)

It is observed simply that [??] is equal to [mathematical expression not reproducible].

In addition, two special cases should be considered separately:

(i) If [??] = [0,0] and [??] [not equal to] [0,0], there is no [??] such that [mathematical expression not reproducible].

(ii) If [mathematical expression not reproducible] = [0,0], then [??] is free and expression [mathematical expression not reproducible] is consistent for all [??], and [??] is considered as unbounded interval [-[infinity], +[infinity]] that is the longest possible interval satisfying [mathematical expression not reproducible].

Corollary 6. If [mathematical expression not reproducible], and [??] [not member of] IR, then the equation [mathematical expression not reproducible] has no solution in IR but has solution in [IR.sup.*].

Corollary 7. If [mathematical expression not reproducible] and [mathematical expression not reproducible], then [??] can be computed by [mathematical expression not reproducible].

The operation [??] is the inverse operation of x in equation [mathematical expression not reproducible] where 0 [not member of] [??]. In case 0 [member of] [??] the solution [mathematical expression not reproducible] is not unique, but [mathematical expression not reproducible] is the longest interval such that [mathematical expression not reproducible]. In other words, any [??] [member of] IR that satisfies [mathematical expression not reproducible] is a subset of [??].

3.3. Some Properties of Interval Numbers. Interval arithmetic and some of their properties over interval numbers and matrices are discussed in [26]. We note some of them here. For any [mathematical expression not reproducible],

[mathematical expression not reproducible]. (25)

Lemma 8. For all [mathematical expression not reproducible].

Proof.

[mathematical expression not reproducible]. (26)

All these properties are correct for interval matrices as well.

3.4. A Method for Solving a Fully Fuzzy Linear System. Consider an n x n FFLS

[mathematical expression not reproducible], (27)

which is written as

Ax = b (28)

where the coefficient matrix A = [([a.sub.ij]).sub.nxn] is the fuzzy matrix of trapezoidal fuzzy numbers, and x = [([x.sub.i]).sub.nx1], b = [([b.sub.i]).sub.nx1] are column vectors of trapezoidal fuzzy numbers.

Allahviranloo et al. [14] defined the following solution sets for FFLS (27):

(i) United trapezoidal fuzzy solution set:

UTFSS = {x | [A.sup.1][x.sup.1] = [b.sup.1], [A.sup.0][x.sup.0] = [b.sup.0]}, (29)

(ii) Tolerable trapezoidal fuzzy solution set:

TTFSS = {x | [A.sup.1][x.sup.1] = [b.sup.1], [A.sup.0][x.sup.0] [subset or equal to] [b.sup.0]}, (30)

(iii) Controllable trapezoidal fuzzy solution set:

CTFSS = {x | [A.sup.1][x.sup.1] = [b.sup.1], [A.sup.0][x.sup.0] [contains as member of equal to] [b.sup.0]}. (31)

Note that [A.sup.1] = [[[a.sub.ij.sup.1]].sub.nxn] and [A.sup.0] = [[[a.sub.ij.sup.0]].sub.nxn] are interval matrices, and [b.sup.1] = [[[b.sub.i.sup.1]].sub.nx1], [b.sup.0] = [[[b.sub.i.sup.0]].sub.nx1] are known right-hand side interval vectors, and [x.sup.1] = [[[x.sub.i.sup.1]].sub.nx1] and [x.sup.1] = [[[x.sub.i.sup.1]].sub.nx1] and [x.sup.0] = [[[x.sub.i.sup.0]].sub.nx1] are unknown interval vectors.

Now, we aim to propose a practical method to obtain the suitable solution of an FFLS. The suitable solution is defined in [14] as solution x [member of] UTFSS. To find such a solution x [member of] UTFSS, it is sufficient to solve the following two interval linear systems:

[A.sup.0][x.sup.0] = [b.sup.0] (32)

[A.sup.1][x.sup.1] = [b.sup.1]. (33)

Now, consider the n x n linear system as

[mathematical expression not reproducible] (34)

where [mathematical expression not reproducible] for all (1 [less than or equal to] i, j [less than or equal to] n) and [[??].sub.i] are unknown intervals. The interval linear system is represented as

[mathematical expression not reproducible] (35)

Note that two systems (32) and (33) are in the form (34). Using operations [??] and [??], we can propose a new method to decompose an interval matrix [??] to two interval matrices, similar to LU decomposition method [28]. Then, one can solve systems (32) and (33)) using this decomposition method.

Assume that [??] [member of] [IR.sup.nxn], and there exist two interval matrices [mathematical expression not reproducible] such that [mathematical expression not reproducible], where [??] and [??] are lower triangular and upper triangular matrices, respectively. Then using this decomposition method, we can solve the interval linear system (34).

Consider the system

[mathematical expression not reproducible], (36)

where [[??].sub.nxn] is a known interval matrix, [[??].sub.nxn] is an unknown lower triangular interval matrix, and [[??].sub.nxn] is an unknown upper triangular interval matrix. In this system, there exist [n.sup.2] linear interval equations, and there are [n.sup.2] + n unknown variables. This system extends to [n.sup.2] + n linear equations with [n.sup.2] + n unknowns by adding n linear equations [[??].sub.ii] = 1 for 1 [less than or equal to] i [less than or equal to] n.

To solve this linear system, first, we set all diagonal elements of [??] with [1,1]. Then, using reversing operations, all elements of [??] and [??] can be computed by solving linear interval equations [mathematical expression not reproducible]. To solve such linear equations, we need to consider two cases:

Case 1 (i [less than or equal to] j).

[mathematical expression not reproducible] (37)

which results in

[mathematical expression not reproducible]. (38)

Case 2 (i > j).

[mathematical expression not reproducible] (39)

which results in

[mathematical expression not reproducible]. (40)

Using these results, we can compute interval LU decomposition for an interval matrix. The Algorithm ILU_Decomposition computes such interval LU decomposition and is given in Algorithm 1.

If at least one of [[u.bar].sub.ii] or [[bar.u].sub.ii] for 0 [less than or equal to] i [less than or equal to] n - 1 is [0, 0], then a dividing-by-zero error occurs, and decomposition is not possible. Otherwise, there exist [mathematical expression not reproducible] such that [mathematical expression not reproducible].

Example 9. Consider the following 3x3 interval matrix:

[mathematical expression not reproducible]. (41)

Initially, assign [1,1] to [[??].sub.11], [[??].sub.22], [[??].sub.33], and initialize all other elements of [??] and [??] to [0,0]. Then, compute the rows of [??] and columns of [??] by annotating Algorithm 1. Compute the rows and columns index of [??] and [??], following the first loop (lines (4) to (14)).

In case r = 1,

[mathematical expression not reproducible]. (42)

Algorithm 1: Computing the interval LU decomposition. (1) function ILU_Decomposition ([mathematical expression not reproducible], n) (i) Input: [??] [member of] [IR.sup.nxn] (ii) Output: [mathematical expression not reproducible] such that [mathematical expression not reproducible] (2) for i = 1 to n do (3) [[??].sub.ii] [left arrow] [1, 1] (4) for r = 1 to n do (5) i [left arrow] r (6) for j = i to n do (7) sum [mathematical expression not reproducible] (8) [mathematical expression not reproducible] (9) [[??].sub.ij] [left arrow] h (10) j [left arrow] r (11) for i = j + 1 to n do (12) sum [mathematical expression not reproducible] (13) [mathematical expression not reproducible] (14) [mathematical expression not reproducible] return [??] and [??] If L [not member of] [IR.sup.nxn] or U [not member of] [IR.sup.nxn] there is no decomposition in [IR.sup.nxn] but has decomposition in [([IR.sup.*]).sup.nxn].

In case r = 2,

[mathematical expression not reproducible]. (43)

In case r = 3,

[mathematical expression not reproducible]. (44)

It is observed simply that [mathematical expression not reproducible].

Now, to solve [mathematical expression not reproducible] for [??] with [mathematical expression not reproducible], we have

[mathematical expression not reproducible] (45)

[mathematical expression not reproducible] (46)

To solve [mathematical expression not reproducible] we do the following steps:

(1) Decompose [??] into two matrices [??] and [??].

(2) Solve [mathematical expression not reproducible] for [??].

(3) Solve [mathematical expression not reproducible] for [??}.

Step (1) can be performed by Algorithm 1 and step (2) can be performed as follows:

[mathematical expression not reproducible], (47)

and finally, step (3) can be performed as

[mathematical expression not reproducible]. (48)

Algorithm 2: Solving linear system of intervals using interval LU decomposition. (1) function ILU_Solve [mathematical expression not reproducible] (i) Input: [mathematical expression not reproducible] (ii) Output: [mathematical expression not reproducible] (2) call ILU_Decomposition[mathematical expression not reproducible] (3) for i = 1 to n do (4) [mathematical expression not reproducible] (5) for i = n to 1 do (6) [mathematical expression not reproducible] return [??]

Algorithm 2 shows our approach to solve a system of linear interval equation [mathematical expression not reproducible].

Example 10. Considering the interval matrix [??] in Example 9 and the following [??], we aim to solve [mathematical expression not reproducible] for [??].

[mathematical expression not reproducible]. (49)

In Example 9, ILU decomposition of [??] is obtained. So, we need to solve [mathematical expression not reproducible]. By solving [mathematical expression not reproducible] are obtained, respectively, as follows:

[mathematical expression not reproducible], (50)

4. Numerical Results

Example 11. Consider the following FFLS which is solved in [8, 9, 29]:

[mathematical expression not reproducible] (51)

First, we solve 1-cut system which is obtained as follows:

[mathematical expression not reproducible]. (52)

The 1-cut crisp solution will be [(37,62,75).sup.T].

Now, we solve 0-cut system which is obtained as follows:

[mathematical expression not reproducible]. (53)

The left-hand side matrix decomposed into the following two matrices:

[mathematical expression not reproducible] (54)

We solve equations [mathematical expression not reproducible] and the solutions are

[mathematical expression not reproducible]. (55)

Finally, solution of System (53) is obtained as

[mathematical expression not reproducible] (56)

Example 12. Consider a 3x3 FFLS:

Ax = b (57)

where

[mathematical expression not reproducible], (58)

[mathematical expression not reproducible]. (59)

This system is converted to two interval systems [A.sup.0][x.sup.0] = [b.sup.0] and [A.sup.1][x.sup.1] = [b.sup.1] as follows:

[mathematical expression not reproducible]. (60)

[mathematical expression not reproducible]. (61)

Interval matrix [A.sup.0] is decomposed to [mathematical expression not reproducible]:

[mathematical expression not reproducible]. (62)

Algorithm 2 solves [mathematical expression not reproducible] and obtains

[mathematical expression not reproducible]. (63)

Similarly, interval matrix [A.sup.1] is decomposed to [mathematical expression not reproducible] as follows:

[mathematical expression not reproducible]. (64)

Then, [x.sup.1] is obtained from [mathematical expression not reproducible] by Algorithm 2 as

[mathematical expression not reproducible]. (65)

Finally, x is found as

[mathematical expression not reproducible]. (66)

5. Conclusion

In the present paper, we proposed a new method for solving the fully fuzzy linear system with trapezoidal fuzzy numbers. This problem is converted into two interval linear systems; then by introducing a new interval method based on decomposition, we solved these systems. Our method is using two inverse operations [??] and [??] which are presented in this paper for the first time. Numerical results demonstrated the efficiency of our new method.

Data Availability

No data were used to support this study.

https://doi.org/10.1155/2018/2104343

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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[28] A. H. Roger and R. J. Charles, Matrix Analysis, Cambridge University Press, Cambridge, UK, 2nd edition, 2013.

[29] W. Daud, N. Ahmad, and K. Aziz, A Review on Classification Methods for Solving Fully Fuzzy Linear Systems, AIP Publishing LLC, 2015.

Esmaeil Siahlooei (iD) (1) and Seyed Abolfazl Shahzadeh Fazeli (iD) (1,2)

(1) Department of Computer Science, Yazd University, Yazd, Iran

(2) Processing Laboratory, Yazd University, Yazd, Iran

Correspondence should be addressed to Seyed Abolfazl Shahzadeh Fazeli; fazeli@yazd.ac.ir

Received 19 May 2018; Revised 24 June 2018; Accepted 28 June 2018; Published 11 July 2018

Academic Editor: Rustom M. Mamlook

Table 1: Interval multiplication. [mathematical expression 0 [less than or equal to] [a.bar] not reproducible] 0 [less than or equal to] [x.bar] [[a.bar][x.bar], [bar.a][bar.x]] [+, +] 0 [member of] [??] [[bar.a][x.bar], [bar.a][bar.x]] [-, +] 0 [greater than or equal to] [[bar.a][x.bar], [a.bar][x.bar]] [x.bar] [-, -] [mathematical expression 0 [member of] [??] not reproducible] 0 [less than or equal to] [x.bar] [[a.bar][x.bar], [bar.a][bar.x]] [-, +] 0 [member of] [??] [min {[bar.a][x.bar], [a.bar][bar.x]}, max {[a.bar][x.bar], [bar.a][bar.x}}] [-,+] 0 [greater than or equal to] [[bar.a][x.bar], [a.bar][x.bar]] [x.bar] [-, +] [mathematical expression 0 [greater than or equal to] not reproducible] [bar.a] 0 [less than or equal to] [x.bar] [[a.bar][x.bar], [bar.a][x.bar]] [-, -] 0 [member of] [??] [[a.bar][x.bar], [a.bar][x.bar]] [-, +] 0 [greater than or equal to] [[bar.a][bar.x], [a.bar][x.bar]] [x.bar] [+, +]

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Title Annotation: | Research Article |
---|---|

Author: | Siahlooei, Esmaeil; Fazeli, Seyed Abolfazl Shahzadeh |

Publication: | Advances in Fuzzy Systems |

Date: | Jan 1, 2018 |

Words: | 4681 |

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