Abstract In this study, we introduce and analyze the concepts of a fixed ordering structure and a variable ordering structure on intervals. The fixed ordering structures on intervals are defined with the help of a pointed convex cone of intervals. A variable ordering is defined by a set-valued map whose values are convex cones of intervals. In the sequel, a few properties of a cone of intervals are derived. It is shown that a binary relation, defined by a convex cone of intervals, is a partial order relation on intervals; further, the relation is antisymmetric if the convex cone of intervals is pointed. Several results under which a variable ordering map of intervals satisfies the conditions of a partial ordering relation of intervals are provided. The introduced fixed and variable ordering of intervals are applied to define and characterize optimal elements of an optimization problem with interval-valued functions. Finally, we propose a numerical technique and present its algorithmic implementation to obtain the set of optimal elements of an interval optimization problem. We also provide illustrative examples to support the study.

中文翻译：

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区间的变量和固定顺序及其在区间值函数优化中的应用

摘要 在本研究中，我们介绍并分析了区间上的固定排序结构和可变排序结构的概念。区间上的固定排序结构是在区间的尖凸锥体的帮助下定义的。变量排序由一个集合值映射定义，其值是间隔的凸锥。在后续中，导出了区间锥的一些属性。证明了由区间凸锥定义的二元关系是区间上的偏序关系；此外，如果区间的凸锥是尖的，则该关系是反对称的。给出了区间可变排序图满足区间偏序关系条件的几个结果。引入的区间的固定和可变排序用于定义和表征具有区间值函数的优化问题的最优元素。最后，我们提出了一种数值技术并展示了其算法实现，以获得区间优化问题的最佳元素集。我们还提供了说明性的例子来支持这项研究。