Interval numbers multiple attribute decision-making (MADM) is an important branch of uncertainty decision theory, and the decision result largely depends on the selection of the aggregation operator. In this paper, we analyze the ordered weighted average (OWA) operator, which is an averaging aggregation operator. The OWA operator provides an aggregation method between the minimum and maximum operators. Moreover, we further analyze some of extensions about OWA operator, and pay special attention to the Bonferroni means and OWA (BON-OWA) operator. Note that the BON-OWA operator only aggregate the input arguments which are exact numbers. Under normal circumstances, decision makers is difficult to provide a clear evaluation value for attribute and most of them are described by vague information. When the decision information is an interval numbers, the BON-OWA operator cannot describe decision result accurately. Under these environments, we proposed the interval numbers BON-OWA (IB^r,q-OWA) operator to deal with the vague decision information in this paper. Then we consider their main properties, such as idempotence, monotonicity, and boundedness and prove them. Besides, a wide range of special aggregation operators are found in changing parameter values, such as the square mean and max operator, and so on. We also compare the ranking method of the interval numbers based on Boolean matrix. As a result, the combination of IB^r,q-OWA operator makes the decision result more scientific. Finally, a new approach for decision-making problem is developed based on the IB^r,q-OWA operator, which shows the effectiveness in practical examples.

中文翻译：

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区间数BONr,q-OWA算子及其在多属性决策中的应用

区间数多属性决策（MADM）是不确定性决策理论的一个重要分支，决策结果很大程度上取决于聚合算子的选择。在本文中，我们分析了有序加权平均（OWA）算子，它是一个平均聚合算子。OWA 运算符提供了最小值和最大值运算符之间的聚合方法。此外，我们进一步分析了OWA算子的一些扩展，特别关注Bonferroni均值和OWA（BON-OWA）算子。请注意，BON-OWA 运算符仅聚合精确数字的输入参数。通常情况下，决策者很难为属性提供一个明确的评价值，而且大多是用模糊的信息来描述的。当决策信息为区间数时，BON-OWA 算子无法准确描述决策结果。在这些环境下，我们提出了区间数 BON-OWA (IB^r,q -OWA) 算子来处理本文中的模糊决策信息。然后我们考虑它们的主要性质，例如幂等性、单调性和有界性并证明它们。此外，在改变参数值时还发现了大量特殊的聚合运算符，例如均方根和最大值运算符等。我们还比较了基于布尔矩阵的区间数排序方法。因此，IB ^r,q -OWA 算子的组合使得决策结果更加科学。最后，基于IB ^r,q -OWA算子开发了一种新的决策问题方法，并在实际例子中显示了有效性。