Applied Soft Computing 13 (2013) 1997–2002 Contents lists available at SciVerse ScienceDirect Applied Soft Computing j ourna l ho mepage: 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 Department of Mathematics, Arak Branch, Islamic Azad University, Arak, Iran b 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 develop- ment 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: pmansouri393@yahoo.com, p-mansouri@iau-arak.ac.ir (P. Mansouri). Alternatively, a fuzzy least-squares approach directly uses infor- mation 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 objec- tive 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 Artifi- cial Bee Colony optimization algorithm. Because, it is a relatively new meta-heuristic designed to deal with hard combinatorial opti- mization 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 exper- iments 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 1568-4946/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.asoc.2012.11.040 Author's Personal Copy