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An approximation algorithm for fuzzy polynomial interpolation with Artificial Bee Colony algorithm
Applied Soft Computing 13 (2013) 1997–2002
Contents lists available at SciVerse ScienceDirect
Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc
An approximation algorithm for fuzzy polynomial interpolation with Artificial Bee Colony algorithm P. Mansouri a,b,∗ , B. Asady a , N. Gupta b a b
Department of Mathematics, Arak Branch, Islamic Azad University, Arak, Iran Department of Computer Science, Delhi University, Delhi, India
a r t i c l e
i n f o
Article history: Received 28 February 2012 Received in revised form 5 November 2012 Accepted 19 November 2012 Available online 13 December 2012 Keywords: Fuzzy polynomial interpolation Artificial Bee Colony algorithm
a b s t r a c t In this paper, a novel approximation algorithm for fuzzy polynomial interpolation using Artificial Bee Colony algorithm to interpolate fuzzy data is discussed. However, we use our modified ABC (MABC; Mansouri et al. [13]) to perform the required task. Some examples (including the benchmark functions Griewank and Rastrigin) illustrate the rationality of the method and the validity of the solution. We compare our results with other methods including Genetic Algorithm (GA), Particle Swarm Algorithm (PSO). The results show that proposed method outperforms the other algorithms. © 2012 Elsevier B.V. All rights reserved.
1. Introduction The problem of interpolating fuzzy values was initially stated by Bellman and Zadeh [3]. Fuzzy polynomial interpolation (FPI) is an extension of polynomial interpolation (PI), in such a way we look at its applications in a wider way. Many years ago interval extensions of PI have been utilized in CAD/CAM applications, especially in the reconstruction (by approximation) of curves and surfaces [17]. We focused our attention on FPI as they could be very interesting for applications in several research areas such as computer graphics, image analysis and processing, engineering design and statistical studies. A new optimal watermarking scheme based on lifting wavelet transform (LWT) and singular value decomposition (SVD) using multi-objective and colony optimization (MOACO) was introduced in 2010 [12]. A refactoring method for cache-efficient swarm intelligence algorithms was presented in 2012 [4]. They focus on two schemes: one was the memory hierarchy, and the other was the algorithm design. Both the cache properties and the cache-aware development were investigated. The ABC-DE algorithm presented in 2011 [20] that combines the ABC and DE approaches and evaluates the results of proposed algorithm on well-known test beds. Approximately, for all of test beds, they show algorithm work better than original ABC algorithm.
∗ Corresponding author at: Department of Mathematics, Arak Branch, Islamic Azad University, Arak, Iran. E-mail addresses: [email protected], [email protected] (P. Mansouri). 1568-4946/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.asoc.2012.11.040
Alternatively, a fuzzy least-squares approach directly uses information included in the input-output data set and considers the measure of best fitting based on distance under fuzzy consideration. Fuzzy least-squares are fuzzy extensions of ordinary least-squares. The least square method is defined as finding a polynomial equation with degree n(n = 1, 2, . . .), so that for some given points of objective function, the total points’ distance from this space becomes minimal. In this work, for finding a fuzzy polynomial interpolation with degree n, we must find n fuzzy coefficients. So that, for finding this coefficients, we must solve 2n × 2n equations that will be very complex for large values of n and sometimes it does not have a fuzzy solution, for more illustrate see [1,6,7]. In order to, we use Artificial Bee Colony optimization algorithm. Because, it is a relatively new meta-heuristic designed to deal with hard combinatorial optimization problems. And it is a biologically inspired method that explores collective intelligence applied by the honey bees during nectar collecting process. Finally, we use modified ABC (MABC [13]) to perform the required task. In Section 2, the ABC algorithm, fuzzy numbers and Fuzzy least square method are described. In Section 3, the approximation fuzzy polynomial interpolation will be suggested and in Section 4 experiments and results are presented. We discuss about our method in Section 5.
2. Preliminaries 2.1. Artificial Bee Colony algorithm Artificial Bee Colony (ABC) algorithm is an algorithm based on the intelligent foraging behavior of honey bee swarm, purposed by
1998
P. Mansouri et al. / Applied Soft Computing 13 (2013) 1997–2002
1. Initial food sources are produced for all employed. 2. Repeat UNTIL (requirements are met) (a) Each employed bee goes to a food source in her memory and determines a neighbor source, then evaluates its nectar amount and dances in the hive. (b) Each onlooker watches the dance of employed bees and chooses one of their sources depending on the dances, and then goes to that source. After choosing a neighbor around that, she evaluates its nectar amount. (c) Abandoned food sources are determined and are replaced with the new food sources discovered by scouts. item The best food source found so far is registered. Fig. 1. Artificial Bee Colony algorithm.
Karaboga [8]. In ABC model, the colony consists of three groups of bees: employed bees, onlookers and scouts. It is assumed that there is only one artificial employed bee for each food source. In other words, the number of employed bees in the colony is equal to the number of food sources around the hive. Employed bees go to their food source and come back to hive and dance on this area. The employed bee whose food source has been abandoned becomes a scout and starts to search for a new food source. Onlookers watch the dances of employed bees and choose food sources depending on dances. The pseudo-code of the ABC algorithm is given in Fig. 1. In ABC which is a population based algorithm, the position of a food source represents a possible solution to the optimization problem and the nectar amount of a food source corresponds to the quality (fitness) of the associated solution. The number of the employed bees is equal to the number of solutions in the population. At the first step, a randomly distributed initial population (food source positions) is generated. After initialization, the population is subjected to repeat the cycles of the search processes of the employed, onlooker, and scout bees, respectively. An employed bee produces a modification on the source position in her memory and discovers a new food source position. Provided that the nectar amount of the new one is higher than that of the previous source, the bee memorizes the new source position and forgets the old one. Otherwise she keeps the position of the one in her memory. After all employed bees complete the search process, they share the position information of the sources with the onlookers on the dance area. Each onlooker evaluates the nectar information taken from all employed bees and then chooses a food source depending on the nectar amounts of sources. As in the case of the employed bee, she produces a modification on the source position in her memory and checks its nectar amount. Providing that its nectar is higher than that of the previous one, the bee memorizes the new position and forgets the old one. The sources abandoned are determined and new sources are randomly produced to be replaced with the abandoned ones by artificial scouts.
2.2. The modified version of ABC algorithm (MABC) In this section the MABC framework and the principal algorithms are presented. We studied modify version of ABC algorithm by adding some conditions to ABC algorithm to increase accuracy and decrease computing time of ABC algorithm in same number of iteration [13]. In MABC model, the colony consists of three groups of bees same as ABC model: employed bees, onlookers and scouts. It is assumed that there is only one artificial employed bee for each food source. In other words, the number of employed bees in the colony is equal to the number of food sources around the hive. Employed bees go to their food source and come back to hive and dance on this area. The employed bee whose food source has been abandoned
becomes a scout and starts to search for a new food source. Onlookers watch the dances of employed bees and choose food sources depending on dances. The pseudo-codes of the MABC algorithm is given in Fig. 2. 2.3. Fuzzy arithmetic Fuzzy sets theory, as it is known, is a powerful mathematical tool for tackling real-world problems, particularly for its capability in treating uncertainty, ambiguity, complex systems, natural language and for the wide range of its applications. Definition 1. Let X be a nonempty set. A fuzzy set A in X is characterized by its membership function A : X → [0, 1] and A (x) is interpreted as the degree of membership of element x in fuzzy set A for each x ∈ X. It is clear that A is completely determined by the set of tuples A = {(x, A (x))|x ∈ X}. It should be noted that the terms membership function and fuzzy subset get used interchangeably and frequently we will write simply A(x) instead of A (x). The family of all fuzzy subsets in X is denoted by F(X). Definition 2. A fuzzy number A in parametric form is a pair (A, A) of functions A(r), A(r), 0 ≤ r ≤ 1, which satisfies the following requirements: 1. A(r) is a bounded increasing continuous function, 2. A(r) is a bounded decreasing continuous function, 3. A(r) ≤ A(r), 0 ≤ r ≤ 1. A popular fuzzy number is the trapezoidal fuzzy number A(x0 , y0 , , ˇ)(L,R) with two defuzzifiers x0 , y0 and left fuzziness and right fuzziness ˇ where the membership function is
⎧ x − x0 + L ⎪ x0 − ≤ x ≤ x0 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨1 [x0 , y0 ], A(x) = R ⎪ y0 − x + ˇ ⎪ ⎪ y0 ≤ x ≤ y0 + ˇ, ⎪ ⎪ ˇ ⎪ ⎪ ⎪ ⎩ 0
otherwise.
where L, R > 0, is denoted by A = (x0 , y0 , , ˇ)(L,R) and if L = R by A = (x0 , y0 , , ˇ)L and if L = R = 1 with A = (x0 , y0 , , ˇ). Its parametric form is A(r) = x0 − + r 1/L ,
A(r) = y0 + ˇ − ˇr 1/R .
Set support function is defined as follows: supp(A) = {x|A(x) > 0}
P. Mansouri et al. / Applied Soft Computing 13 (2013) 1997–2002
1999
1. Initial food sources are produced for all employed. 2. Repeat UNTIL (requirements are met) (a) Each employed bee goes to a food source in her memory and determines a neighbor source, then evaluates its nectar amount and dances in the hive. (b) i = 0(number of cycles) (c) Repeat Until (I = M axCycles) i. N = 0 ii. Repeat until (N = EmployedBee) A. Each onlooker watches the dance of employed bees and chooses on of their sources depending on the dances, and then goes to that source. After choosing a neighbor around that, she evaluates its nectar amount(find best feasible onlooker, replace with best solution) B. Abandoned food sources are determined and are replaced with the new food sources discovered by scouts. C. if f itness(bestf easibleonlooker) < f itness(bestsolution)) D. N = N + 1 iii. The best food source(GlobalParams) found so far is registered. iv. For each bee employed,improve initial lower(lb) and upper(ub) bounds intervals after some arbitrary cycles as bellow: if ((ub(bee) > GlobalP arams(bee)) >= lb(bee))) ub(bee) = (GlobalP arams(bee) + ub(bee))/2, lb(bee) = (GlobalP arams(bee) + lb(bee))/2, v. Check distance of this source and (GlobalMins(GlobalParams)) with arbitrary accuracy. if((Cycle >= k) and |(GlobalM ins(i) − GlobalM ins(i − 1)| < ε, 1 ≤ k ≤ M axCycles and ((|(GlobalM ins(Cycle − 2)− GlobalM ins(Cycle − 1)| < ε))) GlobalM in = GlobalM ins(i), ε = 1.e−t vi. else i = i + 1 Fig. 2. Modified Artificial Bee Colony algorithm.
which that {x|A(x) > 0} is closure of set {x|A(x) > 0}. The addition and scalar multiplication of fuzzy numbers are defined by the extension principle and can be equivalently represented as follows: For arbitrary A = (A, A),B = (B, B) and k > 0 we define addition (A + B) and multiplication by scaler k as (A + B)(r) = A(r) + B(r), (kA)(r) = kA(r),
(A + B)(r) = A(r) + B(r),
(1)
(kA)(r) = kA(r).
(2)
A fuzzy polynomial of degree
n at imost n is a function Pn (x) from R to F(R) such that Pn (x) = a x in which that x is crisp and ai , i=1 i i = 1, 2, .
. ., n are fuzzy. Let ˘ be the set of all fuzzy polynomials n a xi of degree at most n. A fuzzy polynomial of degree Pn (x) = i=1 i at most n from R to F(R), can be put in the following parametric form: P n (x, r) = P n (x, r) =
xi ≥0
ai (r)xi + ai (r)xi +
xi <0
xi <0
ai (r)xi
xi ≥0
1 2 (A(r) − B(r)) dr +
D(A, B) = 0
3. Fuzzy polynomial interpolation (FPI) Let = {x1 , . . ., xn }, be a set of m distinct points of R, and F = {y1 , . . . , yn }, be the value of a triangular fuzzy function f, at the point xi , i = 1, . . ., n. Also, it is easy to fit any polynomial of m degree m
Pm (x) =
j=0
aj xj =
m
(aj (r), aj (r))xj ,
j=0
in which that ai is a trapezoidal fuzzy number with parametric form (aj (r), aj (r)) for j = 0, 1, . . ., m. To experimental data (x1 , y1 ), (x2 , y2 ), . . . (xn , yn ) and xi ∈ and yi ∈ F (provided that n ≥ m + 1) so that the sum of squared residuals S is minimized:
ai (r)xi
Definition 3. For arbitrary fuzzy numbers A = (A, A) and B = (B, B) the quantity
is the distance between A and B, [15,2,14]. The function D(A, B) is a metric in F(R) and (F(R), D) is a complete metric space.
0
1
2
A(r) − B(r)
S=
,
(3)
D(yi , Pm (xi ))2 =
i=1
1/2 dr
n
n i=1
1
(yi (r) − P m (xi , r))2
+ 0
0
1
(yi (r) − P m (xi , r))2
2000
P. Mansouri et al. / Applied Soft Computing 13 (2013) 1997–2002
n
=
0
i=1
⎛ 1
+ 0
forall
⎛ 1
⎛
⎞ ⎞2
In order to, by using of Eq. (4) and define the trapezoidal fuzzy number, we can writing
⎜ ⎜ j j ⎟⎟ aj (r)xi ⎠⎠ ⎝yi (r) − ⎝ aj (r)xi + j i
j i
x ≥0
⎛
x <0
n
⎞ ⎞2
S=
⎜ ⎜ j j ⎟⎟ aj (r)xi ⎠⎠ , ⎝yi (r) − ⎝ aj (r)xi + j i
x <0
1
+
(4)
0
⎞ ⎞2
⎜ ⎜ j j ⎟⎟ aj (r)xi ⎠⎠ ⎝yi (r) − ⎝ aj (r)xi + j i
j i
x ≥0
⎛
x <0
⎞ ⎞2 ⎜ ⎜ j j ⎟⎟ aj (r)xi ⎠⎠ ⎝yi (r) − ⎝ aj (r)xi + j i
j i
x ≥0
x <0
⎛ ⎞2 n 1 ⎜ j j⎟ = aj0 + j(r − 1)xi − bj0 + ˇj (1 − r)xi ⎠ ⎝yi (r) −
with respect to the fuzzy coefficients aj (r) and aj (r) for j = 0, . . ., m . Therefore, we have to find partial derivatives and then solve following fuzzy linear systems:
i=1
.
⎛
⎛
j i
i = 1, 2, . . . , n.
⎧ ∂S ⎪ ⎪ ⎪ ∂a0 (r) ⎪ ⎪ ⎪ ⎪ ∂S ⎪ ⎪ ⎪ ⎪ ⎪ ∂ a 0 (r) ⎪ ⎪ ⎪ ⎨ ..
0
i=1
x ≥0
⎛ 1
1
+ 0
= 0,
j i
j i
x ≥0
⎛
= 0,
0
⎜ ⎝yi (r) −
x <0
j
aj0 + j (r − 1)xi −
j x <0 i
⎞2 j
⎟
bj0 + ˇj (1 − r)xi ⎠
(6)
j x ≥0 i
that parametric form of each fuzzy coefficients are as follow:
= 0,
(5)
⎪ ⎪ ∂S ⎪ = 0, ⎪ ⎪ ⎪ ∂ a m (r) ⎪ ⎪ ⎪ ⎪ ∂S ⎪ ⎪ = 0. ⎪ ⎪ ∂am (r) ⎪ ⎩
aj = (a(j), a(j)) = (aj0 + j(r − 1), bj0 + ˇj (1 − r)), j, ˇj ∈ R, j ≥ 0, ˇj ≥ 0
(7)
Also, we have minimize following function min
But unfortunately, sometimes system 4 has not a fuzzy solution i.e. we can not to have a fuzzy solution (for more illustrate see [1,6,7]). Therefore, we propose a method that, out-put always is a fuzzy polynomial without using of Eq. (5).
S
s.t. aj0 ≤ bj0
for
j = 0, 1, 2, . . . , m,
j and ˇj ≥ 0, aj0 and bj0
for
are free for
j = 0, 1, 2, . . . , m, j = 0, 1, 2, . . . , m.
1. Initial food sources are produced for all employed bees(generate randomly values of coefficients). 2. Repeat UNTIL requirements are met(we obtain minimum optimal solution of fuzzy distance function ) (a) Each employed bee goes to a food source in her memory and determines a neighbor source( nearest values to initial values of coefficients),then evaluates its nectar amount(determined minimum distance of FPI-ABC function and fuzzy optimization problem) and dances in the hive. (b) Each onlooker watches the dance of employed bees and chooses one of their sources depending on the dances, and then goes to that source. After choosing a neighbor around that, she evaluates its nectar amount(find best feasible onlooker, replace with best solution). (c) i = 0(number of cycles) (d) Repeat Until (I = M axCycles) i. N = 0 ii. Repeat until (N = EmployedBee) A. Abandoned food sources are determined( some generated coefficient are not suitable and give us bad result for distance function then, we leave them)from and are replaced with the new food sources discovered by scouts. B. The best food source found so far is registered(best values for coefficients). Fig. 3. The fuzzy polynomial interpolation.
(8)
P. Mansouri et al. / Applied Soft Computing 13 (2013) 1997–2002 Table 1 Fuzzy data of Example 1. x y
Table 3 Data (Grinwank).
0.1 0.5 + 0.2(r − 1), 2.45 + 0.7(1 − r)
0.4 0.8 + 0.3(r − 1), 3 + 0.5(1 − r)
with respect to crisp coefficients aj0 , bj0 , j and ˇj for j = 0, 1, . . ., m. In which that, S is same as follows (see Eq. (6)) S =
n
1
(
0
i=1
+ j(r
(yi (r) −
aj0
j x ≥0 i
j − 1)xi
−
j bj0 + ˇj (1 − r)xi )2
j i
x <0
−
j
aj0 + j (r − 1)xi −
j i
+
1
j
bj0 + ˇj (1 − r)xi )2 )
Example 1. Consider the following fuzzy data (x, y), y = y(r) (see Table 1). where r ∈ [0, 1]. We determine a fuzzy linear approximation P(x) = a1 x + a0 , in which that, aj = (aj0 , bj0 , j , ˇj ), j = 0, 1 are the trapezoidal fuzzy numbers. That, parametric forms of they are same as follows aj = (aj0 + j (r − 1), bj0 + ˇj (1 − r)) for ∀r ∈ [0, 1], j = 0, 1. As, we obtain the fuzzy coefficients a0 and a1 by solve following problem.
j +ˇj (1 − r)xi )2
−
j i
(yi (r) −
(
0
+
j
aj0 + j(r − 1)xi −
j i
1
(yi (r) −
S
(A, B)
Cycle
0.0140 0.0093 0.0116 0.0093 0.4124
(0.4368,−0.0533) (0.5113, −0.0995) (0.5167,−0.1170) (0.5103 −0.0983) (0.381,0.0439)
63 63 63 63 –
And finally, the fuzzy linear approximation is P(x) = (0, 0.0100, 0.1563, 0.4718)x + (0.1602, 0.2301, 0.7372, 0.0100). Our algorithm was coded in Matlab and Table 5 shows that result of proposed method is better than other algorithms. Also, each algorithm was run for 50 iterations (see Table 2). Example 2. Next, we consider the benchmark Grinwank function defined as follows: f (x) =
S =
bj0
2
+
j
for
1
((a1 )xi + a0 − yi )2 dr
0 2 1
2
1
((a1 )xi + a0 − yi (r)
dr
0
or equality
aj0 + j (r − 1)xi
j x <0 i
(9)
10
S=2
(a1 xi + a0 − yi (r))2 ,
xi ∈ R,
yi = yi (x).
i=1
Because parametric forms of ai and yi can define same as follows:
for j = 0, 1,
aj0 and bj0
1 x2 − cos x + 1 4000
The following table provides the input data for the example (see Table 3): We determine real coefficients of linear interpolation P(x) = a1 x + a0 , where a0 , a1 ∈ R, for given crisp data (x, y), y = f(x) . By using of algorithm 3, we have minimize the following distance function
x ≥0
j and ˇj ≥ 0,
1.0 0.4599
ABC MABC GA PSO LSQ
x <0
j bj0 + ˇj (1 − r)xi )2 ))
s.t. aj0 ≤ bj0
0.9 0.3786
Method
j i
x ≥0
0
0.8 0.3035
Table 4 Comparison of ABC, MABC, GA, PSO and LSQ algorithms for Example 2.
1 1
... ...
and a1 = (0, 0.0100, 0.1563, 0.4718).
In this section, we illustrate our algorithm with some examples and compare the results with the classical Least Square Algorithm and other evolutionary optimization algorithms like GA and PSO.
0.3 0.0447
a0 = (0.1602, 0.2301, 0.7372, 0.0100)
4. Numerical examples
2
0.2 0.0199
By using of proposed algorithm, optimal solution is same as follows
Finally, for solving above hard (complex) fuzzy optimization problem, we used MABC method [13] that, the pseudo-codes of the propose method is given in Fig. 3 and all commutations are in Euclidian space, at least we find the best values of fuzzy coefficients.
i=1
0.1 0.0050
(yi (r)
j i
S = min(
x y
0
x ≥0
x <0
min
2001
j = 0, 1,
arefreefor j = 0, 1.
aj = (a(j), a(j)) = [aj , aj ]
(10)
yi = (y(i), y(i)) = [yi , yi ]
(11)
Table 2 Comparison of ABC, MABC, GA and PSO algorithms for Example 1. Algorithm
S
a0
a1
Cycle
ABC MABC GA PSO
0.0297 0.0265 23.9363 0.0463
(0.1602, 0.2301, 0.7372, 0.0100) (0.2172, 0.1921, 0.6951, 0.1464) (1.9287,0.9471, 1.4589, 0.0332) (0.2651, 0.1693, 0.7810,0.0000)
(0, 0.0100, 0.1563,0.4718) (0,0.0172, 0.1344,0.0247) (0.5852,0.4057, 2.1433,5.9380) (0.0000,0.0000, 0.1511,0.0000)
50 50 50 50
2002
P. Mansouri et al. / Applied Soft Computing 13 (2013) 1997–2002
Table 5 Data (Rastrigin). x y
0.1 1.9198
0.2 6.9498
0.3 13.1802
... ...
0.8 7.5498
0.9 2.7198
1.0 1.0
Colony algorithm. With the help of examples including the benchmark functions like Griewank and Rastrigin, we illustrate that our method outperforms most of the existing algorithms like the classical LSQ, GA, PSO and ABC as well most of the times. Acknowledgements
Table 6 Comparison of ABC, MABC, GA, PSO and LSQ algorithms for Example 3. Method
S (Rastrigin)
(a3 , a2 , a1 , a0 )
Cycle
ABC MABC GA PSO LSQ
26.3303 23.722 339.06826 78.7857 106.6611
(18.0731,20.0000, 20.0000, −0.0140) (20.0000, 24.7056, 29.8661, −0.2812) (9.2885, 9.8383, −4.8347, −9.2889) (18.5378, 19.4732, 19.5071, 1.7475) (−52.8101,17.4945,28.4236, 4.4452)
100 100 100 100 –
So that, Table 4 shows the results of proposed method and PSO algorithm is same as and they are better than GA, LSQ and ABC methods (see following table). Example 3. In this example, consider the benchmark Rastrigin function defined as follows: f (x) = x2 − 10 cos(2x) + 10, x ∈ [−15, 15], that the input data for it is same as follows In this part, we want to determine a third degree polynomial P(x) = a3 x3 + a2 x2 + a1 x + a0 , so that, results of proposed method and other methods are same as Table 6. As results of examples show, proposed method outperforms all the other algorithms. 5. Conclusion In this paper, we presented a novel approximation method to fuzzy polynomial interpolation by using iterative Artificial Bee
The authors are very grateful to the anonymous referees and the editor for their comments and suggestions, which have been very helpful in improving the presentation of this paper. References [1] B. Asady, S. Abbasbandy, M. Alavi, Fuzzy general linear systems, Applied Mathematics and Computation 169 (2005) 34–40. [2] S. Abbasbandy, B. Asady, A note on “A new approach for defuzzification”, Fuzzy Sets and Systems 128 (2002) 131–132. [3] R. Bellman, L.A. Zadeh, Decision making in a fuzzy environment, Management Science 17 (1970) 141–164. [4] F.C. Chang, H.C. Huang, A refactoring method for cache-efficient swarm intelligence algorithms, Information Sciences 192 (June (1)) (2012) 39–49. [6] M. Friedman, M. Ming, A. Kandel, Fuzzy linear systems, Fuzzy Sets and Systems 96 (1998) 201–209. [7] M. Friedman, M. Ming, A. Kandel, Duality in fuzzy linear systems, Fuzzy Sets and Systems 109 (2000) 55–58. [8] D. Karaboga, An Idea Based on Honey Bee Swarm for Numerical Optimization, Technical report-TR06, Erciyes University, Engineering Faculty, Computer Engineering Department, 2005. [12] K. Loukhaoukha, J.Y. Chouinard, M.H. Taieb, Optimal image watermarking algorithm based on LWT–SVD via multi-objective ant colony optimization, Journal of Information Hiding and Multimedia Signal Processing 2 (October (4)) (2011) 303–319. [13] P. Mansouri, B. Asady, N. Gupta, The modify version of artificial bee colony algorithm, International Journal Electrical and Computer Engineering 2 (4) (2012). [14] M. Ma, A. Kandel, M. Friedman, Correction to “A new approach for defuzzification”, Fuzzy Sets and Systems 128 (2002) 133–134. [15] M. Ma, A. Kandel, M. Friedman, A new approach for defuzzification, Fuzzy Sets and Systems 111 (2000) 351–356. [17] T.W. Sederberg, R.T. Farouki, Approximation by interval Bezier curves, Institute of Electrical and Electronics Engineers Computer Graphics and Applications 12 (1992) 87–95. [20] A. Alizadegan, M.R. Meybodi, B. Asady, A novel hybrid Artificial Bee Colony algorithm and differential evolution for unconstrained optimization problem, Advances in Computer Science and Engineering 8 (February (1)) (2012) 45–56.
Contents lists available at SciVerse ScienceDirect
Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc
An approximation algorithm for fuzzy polynomial interpolation with Artificial Bee Colony algorithm P. Mansouri a,b,∗ , B. Asady a , N. Gupta b a b
Department of Mathematics, Arak Branch, Islamic Azad University, Arak, Iran Department of Computer Science, Delhi University, Delhi, India
a r t i c l e
i n f o
Article history: Received 28 February 2012 Received in revised form 5 November 2012 Accepted 19 November 2012 Available online 13 December 2012 Keywords: Fuzzy polynomial interpolation Artificial Bee Colony algorithm
a b s t r a c t In this paper, a novel approximation algorithm for fuzzy polynomial interpolation using Artificial Bee Colony algorithm to interpolate fuzzy data is discussed. However, we use our modified ABC (MABC; Mansouri et al. [13]) to perform the required task. Some examples (including the benchmark functions Griewank and Rastrigin) illustrate the rationality of the method and the validity of the solution. We compare our results with other methods including Genetic Algorithm (GA), Particle Swarm Algorithm (PSO). The results show that proposed method outperforms the other algorithms. © 2012 Elsevier B.V. All rights reserved.
1. Introduction The problem of interpolating fuzzy values was initially stated by Bellman and Zadeh [3]. Fuzzy polynomial interpolation (FPI) is an extension of polynomial interpolation (PI), in such a way we look at its applications in a wider way. Many years ago interval extensions of PI have been utilized in CAD/CAM applications, especially in the reconstruction (by approximation) of curves and surfaces [17]. We focused our attention on FPI as they could be very interesting for applications in several research areas such as computer graphics, image analysis and processing, engineering design and statistical studies. A new optimal watermarking scheme based on lifting wavelet transform (LWT) and singular value decomposition (SVD) using multi-objective and colony optimization (MOACO) was introduced in 2010 [12]. A refactoring method for cache-efficient swarm intelligence algorithms was presented in 2012 [4]. They focus on two schemes: one was the memory hierarchy, and the other was the algorithm design. Both the cache properties and the cache-aware development were investigated. The ABC-DE algorithm presented in 2011 [20] that combines the ABC and DE approaches and evaluates the results of proposed algorithm on well-known test beds. Approximately, for all of test beds, they show algorithm work better than original ABC algorithm.
∗ Corresponding author at: Department of Mathematics, Arak Branch, Islamic Azad University, Arak, Iran. E-mail addresses: [email protected], [email protected] (P. Mansouri). 1568-4946/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.asoc.2012.11.040
Alternatively, a fuzzy least-squares approach directly uses information included in the input-output data set and considers the measure of best fitting based on distance under fuzzy consideration. Fuzzy least-squares are fuzzy extensions of ordinary least-squares. The least square method is defined as finding a polynomial equation with degree n(n = 1, 2, . . .), so that for some given points of objective function, the total points’ distance from this space becomes minimal. In this work, for finding a fuzzy polynomial interpolation with degree n, we must find n fuzzy coefficients. So that, for finding this coefficients, we must solve 2n × 2n equations that will be very complex for large values of n and sometimes it does not have a fuzzy solution, for more illustrate see [1,6,7]. In order to, we use Artificial Bee Colony optimization algorithm. Because, it is a relatively new meta-heuristic designed to deal with hard combinatorial optimization problems. And it is a biologically inspired method that explores collective intelligence applied by the honey bees during nectar collecting process. Finally, we use modified ABC (MABC [13]) to perform the required task. In Section 2, the ABC algorithm, fuzzy numbers and Fuzzy least square method are described. In Section 3, the approximation fuzzy polynomial interpolation will be suggested and in Section 4 experiments and results are presented. We discuss about our method in Section 5.
2. Preliminaries 2.1. Artificial Bee Colony algorithm Artificial Bee Colony (ABC) algorithm is an algorithm based on the intelligent foraging behavior of honey bee swarm, purposed by
1998
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1. Initial food sources are produced for all employed. 2. Repeat UNTIL (requirements are met) (a) Each employed bee goes to a food source in her memory and determines a neighbor source, then evaluates its nectar amount and dances in the hive. (b) Each onlooker watches the dance of employed bees and chooses one of their sources depending on the dances, and then goes to that source. After choosing a neighbor around that, she evaluates its nectar amount. (c) Abandoned food sources are determined and are replaced with the new food sources discovered by scouts. item The best food source found so far is registered. Fig. 1. Artificial Bee Colony algorithm.
Karaboga [8]. In ABC model, the colony consists of three groups of bees: employed bees, onlookers and scouts. It is assumed that there is only one artificial employed bee for each food source. In other words, the number of employed bees in the colony is equal to the number of food sources around the hive. Employed bees go to their food source and come back to hive and dance on this area. The employed bee whose food source has been abandoned becomes a scout and starts to search for a new food source. Onlookers watch the dances of employed bees and choose food sources depending on dances. The pseudo-code of the ABC algorithm is given in Fig. 1. In ABC which is a population based algorithm, the position of a food source represents a possible solution to the optimization problem and the nectar amount of a food source corresponds to the quality (fitness) of the associated solution. The number of the employed bees is equal to the number of solutions in the population. At the first step, a randomly distributed initial population (food source positions) is generated. After initialization, the population is subjected to repeat the cycles of the search processes of the employed, onlooker, and scout bees, respectively. An employed bee produces a modification on the source position in her memory and discovers a new food source position. Provided that the nectar amount of the new one is higher than that of the previous source, the bee memorizes the new source position and forgets the old one. Otherwise she keeps the position of the one in her memory. After all employed bees complete the search process, they share the position information of the sources with the onlookers on the dance area. Each onlooker evaluates the nectar information taken from all employed bees and then chooses a food source depending on the nectar amounts of sources. As in the case of the employed bee, she produces a modification on the source position in her memory and checks its nectar amount. Providing that its nectar is higher than that of the previous one, the bee memorizes the new position and forgets the old one. The sources abandoned are determined and new sources are randomly produced to be replaced with the abandoned ones by artificial scouts.
2.2. The modified version of ABC algorithm (MABC) In this section the MABC framework and the principal algorithms are presented. We studied modify version of ABC algorithm by adding some conditions to ABC algorithm to increase accuracy and decrease computing time of ABC algorithm in same number of iteration [13]. In MABC model, the colony consists of three groups of bees same as ABC model: employed bees, onlookers and scouts. It is assumed that there is only one artificial employed bee for each food source. In other words, the number of employed bees in the colony is equal to the number of food sources around the hive. Employed bees go to their food source and come back to hive and dance on this area. The employed bee whose food source has been abandoned
becomes a scout and starts to search for a new food source. Onlookers watch the dances of employed bees and choose food sources depending on dances. The pseudo-codes of the MABC algorithm is given in Fig. 2. 2.3. Fuzzy arithmetic Fuzzy sets theory, as it is known, is a powerful mathematical tool for tackling real-world problems, particularly for its capability in treating uncertainty, ambiguity, complex systems, natural language and for the wide range of its applications. Definition 1. Let X be a nonempty set. A fuzzy set A in X is characterized by its membership function A : X → [0, 1] and A (x) is interpreted as the degree of membership of element x in fuzzy set A for each x ∈ X. It is clear that A is completely determined by the set of tuples A = {(x, A (x))|x ∈ X}. It should be noted that the terms membership function and fuzzy subset get used interchangeably and frequently we will write simply A(x) instead of A (x). The family of all fuzzy subsets in X is denoted by F(X). Definition 2. A fuzzy number A in parametric form is a pair (A, A) of functions A(r), A(r), 0 ≤ r ≤ 1, which satisfies the following requirements: 1. A(r) is a bounded increasing continuous function, 2. A(r) is a bounded decreasing continuous function, 3. A(r) ≤ A(r), 0 ≤ r ≤ 1. A popular fuzzy number is the trapezoidal fuzzy number A(x0 , y0 , , ˇ)(L,R) with two defuzzifiers x0 , y0 and left fuzziness and right fuzziness ˇ where the membership function is
⎧ x − x0 + L ⎪ x0 − ≤ x ≤ x0 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨1 [x0 , y0 ], A(x) = R ⎪ y0 − x + ˇ ⎪ ⎪ y0 ≤ x ≤ y0 + ˇ, ⎪ ⎪ ˇ ⎪ ⎪ ⎪ ⎩ 0
otherwise.
where L, R > 0, is denoted by A = (x0 , y0 , , ˇ)(L,R) and if L = R by A = (x0 , y0 , , ˇ)L and if L = R = 1 with A = (x0 , y0 , , ˇ). Its parametric form is A(r) = x0 − + r 1/L ,
A(r) = y0 + ˇ − ˇr 1/R .
Set support function is defined as follows: supp(A) = {x|A(x) > 0}
P. Mansouri et al. / Applied Soft Computing 13 (2013) 1997–2002
1999
1. Initial food sources are produced for all employed. 2. Repeat UNTIL (requirements are met) (a) Each employed bee goes to a food source in her memory and determines a neighbor source, then evaluates its nectar amount and dances in the hive. (b) i = 0(number of cycles) (c) Repeat Until (I = M axCycles) i. N = 0 ii. Repeat until (N = EmployedBee) A. Each onlooker watches the dance of employed bees and chooses on of their sources depending on the dances, and then goes to that source. After choosing a neighbor around that, she evaluates its nectar amount(find best feasible onlooker, replace with best solution) B. Abandoned food sources are determined and are replaced with the new food sources discovered by scouts. C. if f itness(bestf easibleonlooker) < f itness(bestsolution)) D. N = N + 1 iii. The best food source(GlobalParams) found so far is registered. iv. For each bee employed,improve initial lower(lb) and upper(ub) bounds intervals after some arbitrary cycles as bellow: if ((ub(bee) > GlobalP arams(bee)) >= lb(bee))) ub(bee) = (GlobalP arams(bee) + ub(bee))/2, lb(bee) = (GlobalP arams(bee) + lb(bee))/2, v. Check distance of this source and (GlobalMins(GlobalParams)) with arbitrary accuracy. if((Cycle >= k) and |(GlobalM ins(i) − GlobalM ins(i − 1)| < ε, 1 ≤ k ≤ M axCycles and ((|(GlobalM ins(Cycle − 2)− GlobalM ins(Cycle − 1)| < ε))) GlobalM in = GlobalM ins(i), ε = 1.e−t vi. else i = i + 1 Fig. 2. Modified Artificial Bee Colony algorithm.
which that {x|A(x) > 0} is closure of set {x|A(x) > 0}. The addition and scalar multiplication of fuzzy numbers are defined by the extension principle and can be equivalently represented as follows: For arbitrary A = (A, A),B = (B, B) and k > 0 we define addition (A + B) and multiplication by scaler k as (A + B)(r) = A(r) + B(r), (kA)(r) = kA(r),
(A + B)(r) = A(r) + B(r),
(1)
(kA)(r) = kA(r).
(2)
A fuzzy polynomial of degree
n at imost n is a function Pn (x) from R to F(R) such that Pn (x) = a x in which that x is crisp and ai , i=1 i i = 1, 2, .
. ., n are fuzzy. Let ˘ be the set of all fuzzy polynomials n a xi of degree at most n. A fuzzy polynomial of degree Pn (x) = i=1 i at most n from R to F(R), can be put in the following parametric form: P n (x, r) = P n (x, r) =
xi ≥0
ai (r)xi + ai (r)xi +
xi <0
xi <0
ai (r)xi
xi ≥0
1 2 (A(r) − B(r)) dr +
D(A, B) = 0
3. Fuzzy polynomial interpolation (FPI) Let = {x1 , . . ., xn }, be a set of m distinct points of R, and F = {y1 , . . . , yn }, be the value of a triangular fuzzy function f, at the point xi , i = 1, . . ., n. Also, it is easy to fit any polynomial of m degree m
Pm (x) =
j=0
aj xj =
m
(aj (r), aj (r))xj ,
j=0
in which that ai is a trapezoidal fuzzy number with parametric form (aj (r), aj (r)) for j = 0, 1, . . ., m. To experimental data (x1 , y1 ), (x2 , y2 ), . . . (xn , yn ) and xi ∈ and yi ∈ F (provided that n ≥ m + 1) so that the sum of squared residuals S is minimized:
ai (r)xi
Definition 3. For arbitrary fuzzy numbers A = (A, A) and B = (B, B) the quantity
is the distance between A and B, [15,2,14]. The function D(A, B) is a metric in F(R) and (F(R), D) is a complete metric space.
0
1
2
A(r) − B(r)
S=
,
(3)
D(yi , Pm (xi ))2 =
i=1
1/2 dr
n
n i=1
1
(yi (r) − P m (xi , r))2
+ 0
0
1
(yi (r) − P m (xi , r))2
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P. Mansouri et al. / Applied Soft Computing 13 (2013) 1997–2002
n
=
0
i=1
⎛ 1
+ 0
forall
⎛ 1
⎛
⎞ ⎞2
In order to, by using of Eq. (4) and define the trapezoidal fuzzy number, we can writing
⎜ ⎜ j j ⎟⎟ aj (r)xi ⎠⎠ ⎝yi (r) − ⎝ aj (r)xi + j i
j i
x ≥0
⎛
x <0
n
⎞ ⎞2
S=
⎜ ⎜ j j ⎟⎟ aj (r)xi ⎠⎠ , ⎝yi (r) − ⎝ aj (r)xi + j i
x <0
1
+
(4)
0
⎞ ⎞2
⎜ ⎜ j j ⎟⎟ aj (r)xi ⎠⎠ ⎝yi (r) − ⎝ aj (r)xi + j i
j i
x ≥0
⎛
x <0
⎞ ⎞2 ⎜ ⎜ j j ⎟⎟ aj (r)xi ⎠⎠ ⎝yi (r) − ⎝ aj (r)xi + j i
j i
x ≥0
x <0
⎛ ⎞2 n 1 ⎜ j j⎟ = aj0 + j(r − 1)xi − bj0 + ˇj (1 − r)xi ⎠ ⎝yi (r) −
with respect to the fuzzy coefficients aj (r) and aj (r) for j = 0, . . ., m . Therefore, we have to find partial derivatives and then solve following fuzzy linear systems:
i=1
.
⎛
⎛
j i
i = 1, 2, . . . , n.
⎧ ∂S ⎪ ⎪ ⎪ ∂a0 (r) ⎪ ⎪ ⎪ ⎪ ∂S ⎪ ⎪ ⎪ ⎪ ⎪ ∂ a 0 (r) ⎪ ⎪ ⎪ ⎨ ..
0
i=1
x ≥0
⎛ 1
1
+ 0
= 0,
j i
j i
x ≥0
⎛
= 0,
0
⎜ ⎝yi (r) −
x <0
j
aj0 + j (r − 1)xi −
j x <0 i
⎞2 j
⎟
bj0 + ˇj (1 − r)xi ⎠
(6)
j x ≥0 i
that parametric form of each fuzzy coefficients are as follow:
= 0,
(5)
⎪ ⎪ ∂S ⎪ = 0, ⎪ ⎪ ⎪ ∂ a m (r) ⎪ ⎪ ⎪ ⎪ ∂S ⎪ ⎪ = 0. ⎪ ⎪ ∂am (r) ⎪ ⎩
aj = (a(j), a(j)) = (aj0 + j(r − 1), bj0 + ˇj (1 − r)), j, ˇj ∈ R, j ≥ 0, ˇj ≥ 0
(7)
Also, we have minimize following function min
But unfortunately, sometimes system 4 has not a fuzzy solution i.e. we can not to have a fuzzy solution (for more illustrate see [1,6,7]). Therefore, we propose a method that, out-put always is a fuzzy polynomial without using of Eq. (5).
S
s.t. aj0 ≤ bj0
for
j = 0, 1, 2, . . . , m,
j and ˇj ≥ 0, aj0 and bj0
for
are free for
j = 0, 1, 2, . . . , m, j = 0, 1, 2, . . . , m.
1. Initial food sources are produced for all employed bees(generate randomly values of coefficients). 2. Repeat UNTIL requirements are met(we obtain minimum optimal solution of fuzzy distance function ) (a) Each employed bee goes to a food source in her memory and determines a neighbor source( nearest values to initial values of coefficients),then evaluates its nectar amount(determined minimum distance of FPI-ABC function and fuzzy optimization problem) and dances in the hive. (b) Each onlooker watches the dance of employed bees and chooses one of their sources depending on the dances, and then goes to that source. After choosing a neighbor around that, she evaluates its nectar amount(find best feasible onlooker, replace with best solution). (c) i = 0(number of cycles) (d) Repeat Until (I = M axCycles) i. N = 0 ii. Repeat until (N = EmployedBee) A. Abandoned food sources are determined( some generated coefficient are not suitable and give us bad result for distance function then, we leave them)from and are replaced with the new food sources discovered by scouts. B. The best food source found so far is registered(best values for coefficients). Fig. 3. The fuzzy polynomial interpolation.
(8)
P. Mansouri et al. / Applied Soft Computing 13 (2013) 1997–2002 Table 1 Fuzzy data of Example 1. x y
Table 3 Data (Grinwank).
0.1 0.5 + 0.2(r − 1), 2.45 + 0.7(1 − r)
0.4 0.8 + 0.3(r − 1), 3 + 0.5(1 − r)
with respect to crisp coefficients aj0 , bj0 , j and ˇj for j = 0, 1, . . ., m. In which that, S is same as follows (see Eq. (6)) S =
n
1
(
0
i=1
+ j(r
(yi (r) −
aj0
j x ≥0 i
j − 1)xi
−
j bj0 + ˇj (1 − r)xi )2
j i
x <0
−
j
aj0 + j (r − 1)xi −
j i
+
1
j
bj0 + ˇj (1 − r)xi )2 )
Example 1. Consider the following fuzzy data (x, y), y = y(r) (see Table 1). where r ∈ [0, 1]. We determine a fuzzy linear approximation P(x) = a1 x + a0 , in which that, aj = (aj0 , bj0 , j , ˇj ), j = 0, 1 are the trapezoidal fuzzy numbers. That, parametric forms of they are same as follows aj = (aj0 + j (r − 1), bj0 + ˇj (1 − r)) for ∀r ∈ [0, 1], j = 0, 1. As, we obtain the fuzzy coefficients a0 and a1 by solve following problem.
j +ˇj (1 − r)xi )2
−
j i
(yi (r) −
(
0
+
j
aj0 + j(r − 1)xi −
j i
1
(yi (r) −
S
(A, B)
Cycle
0.0140 0.0093 0.0116 0.0093 0.4124
(0.4368,−0.0533) (0.5113, −0.0995) (0.5167,−0.1170) (0.5103 −0.0983) (0.381,0.0439)
63 63 63 63 –
And finally, the fuzzy linear approximation is P(x) = (0, 0.0100, 0.1563, 0.4718)x + (0.1602, 0.2301, 0.7372, 0.0100). Our algorithm was coded in Matlab and Table 5 shows that result of proposed method is better than other algorithms. Also, each algorithm was run for 50 iterations (see Table 2). Example 2. Next, we consider the benchmark Grinwank function defined as follows: f (x) =
S =
bj0
2
+
j
for
1
((a1 )xi + a0 − yi )2 dr
0 2 1
2
1
((a1 )xi + a0 − yi (r)
dr
0
or equality
aj0 + j (r − 1)xi
j x <0 i
(9)
10
S=2
(a1 xi + a0 − yi (r))2 ,
xi ∈ R,
yi = yi (x).
i=1
Because parametric forms of ai and yi can define same as follows:
for j = 0, 1,
aj0 and bj0
1 x2 − cos x + 1 4000
The following table provides the input data for the example (see Table 3): We determine real coefficients of linear interpolation P(x) = a1 x + a0 , where a0 , a1 ∈ R, for given crisp data (x, y), y = f(x) . By using of algorithm 3, we have minimize the following distance function
x ≥0
j and ˇj ≥ 0,
1.0 0.4599
ABC MABC GA PSO LSQ
x <0
j bj0 + ˇj (1 − r)xi )2 ))
s.t. aj0 ≤ bj0
0.9 0.3786
Method
j i
x ≥0
0
0.8 0.3035
Table 4 Comparison of ABC, MABC, GA, PSO and LSQ algorithms for Example 2.
1 1
... ...
and a1 = (0, 0.0100, 0.1563, 0.4718).
In this section, we illustrate our algorithm with some examples and compare the results with the classical Least Square Algorithm and other evolutionary optimization algorithms like GA and PSO.
0.3 0.0447
a0 = (0.1602, 0.2301, 0.7372, 0.0100)
4. Numerical examples
2
0.2 0.0199
By using of proposed algorithm, optimal solution is same as follows
Finally, for solving above hard (complex) fuzzy optimization problem, we used MABC method [13] that, the pseudo-codes of the propose method is given in Fig. 3 and all commutations are in Euclidian space, at least we find the best values of fuzzy coefficients.
i=1
0.1 0.0050
(yi (r)
j i
S = min(
x y
0
x ≥0
x <0
min
2001
j = 0, 1,
arefreefor j = 0, 1.
aj = (a(j), a(j)) = [aj , aj ]
(10)
yi = (y(i), y(i)) = [yi , yi ]
(11)
Table 2 Comparison of ABC, MABC, GA and PSO algorithms for Example 1. Algorithm
S
a0
a1
Cycle
ABC MABC GA PSO
0.0297 0.0265 23.9363 0.0463
(0.1602, 0.2301, 0.7372, 0.0100) (0.2172, 0.1921, 0.6951, 0.1464) (1.9287,0.9471, 1.4589, 0.0332) (0.2651, 0.1693, 0.7810,0.0000)
(0, 0.0100, 0.1563,0.4718) (0,0.0172, 0.1344,0.0247) (0.5852,0.4057, 2.1433,5.9380) (0.0000,0.0000, 0.1511,0.0000)
50 50 50 50
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P. Mansouri et al. / Applied Soft Computing 13 (2013) 1997–2002
Table 5 Data (Rastrigin). x y
0.1 1.9198
0.2 6.9498
0.3 13.1802
... ...
0.8 7.5498
0.9 2.7198
1.0 1.0
Colony algorithm. With the help of examples including the benchmark functions like Griewank and Rastrigin, we illustrate that our method outperforms most of the existing algorithms like the classical LSQ, GA, PSO and ABC as well most of the times. Acknowledgements
Table 6 Comparison of ABC, MABC, GA, PSO and LSQ algorithms for Example 3. Method
S (Rastrigin)
(a3 , a2 , a1 , a0 )
Cycle
ABC MABC GA PSO LSQ
26.3303 23.722 339.06826 78.7857 106.6611
(18.0731,20.0000, 20.0000, −0.0140) (20.0000, 24.7056, 29.8661, −0.2812) (9.2885, 9.8383, −4.8347, −9.2889) (18.5378, 19.4732, 19.5071, 1.7475) (−52.8101,17.4945,28.4236, 4.4452)
100 100 100 100 –
So that, Table 4 shows the results of proposed method and PSO algorithm is same as and they are better than GA, LSQ and ABC methods (see following table). Example 3. In this example, consider the benchmark Rastrigin function defined as follows: f (x) = x2 − 10 cos(2x) + 10, x ∈ [−15, 15], that the input data for it is same as follows In this part, we want to determine a third degree polynomial P(x) = a3 x3 + a2 x2 + a1 x + a0 , so that, results of proposed method and other methods are same as Table 6. As results of examples show, proposed method outperforms all the other algorithms. 5. Conclusion In this paper, we presented a novel approximation method to fuzzy polynomial interpolation by using iterative Artificial Bee
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