Computing the centroid of an interval type-2 fuzzy set (IT2 FS) is an important type-reduction method. The aim of this paper is to develop a new method to calculate the centroid of an IT2 FS when the problems of centroid computation of an IT2 FS are continuous. For the continuous centroid computation problems, the structures of optimal solutions are strictly proven from mathematics for the first time in this paper. Furthermore, we also prove that the structures of the optimal solutions are unique in the sense of almost everywhere equal, i.e., if there are two optimal solutions $f_1(x)$ and $f_2(x)$, the Lebesgue measure of $\{x|f_1(x)\neq f_2(x)\}$ is equal to 0. Subsequently, a combinatorial iterative (CI) method is proposed to solve the roots of the sufficiently differentiable objective functions. It is proven that the convergence of the proposed iterative method is at least sixth order. Based on the proposed iterative method, two algorithms, called CI algorithms, are devised to compute the centroid of an IT2 FS. The efficiencies of CI algorithms are demonstrated by comparing the continuous Karnik–Mendel algorithms and the Hallye's methods with the CI algorithms through three numerical examples.

中文翻译：

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用于计算区间类型 2 模糊集质心的组合迭代算法

计算区间类型 2 模糊集 (IT2 FS) 的质心是一种重要的类型归约方法。本文的目的是开发一种新方法，当 IT2 FS 的质心计算问题是连续的时，计算 IT2 FS 的质心。对于连续质心计算问题，本文首次从数学上严格证明了最优解的结构。此外，我们还证明了最优解的结构在几乎处处相等的意义上是唯一的，即如果有两个最优解$f_1(x)$ 和 $f_2(x)$, Lebesgue 测度 $\{x|f_1(x)\neq f_2(x)\}$等于0。随后，提出了一种组合迭代（CI）方法来求解足够可微的目标函数的根。证明所提出的迭代方法的收敛性至少为六阶。基于所提出的迭代方法，设计了两种称为 CI 算法的算法来计算 IT2 FS 的质心。通过三个数值例子比较连续的 Karnik-Mendel 算法和 Hallye 方法与 CI 算法，证明了 CI 算法的效率。