Water pollutants can be classified into three categories, each of which includes several classifications of substances. In this paper, we present a methodology based on bankruptcy models to determine the emission limits of polluting substances belonging to more than one category. We model the problem as a multi-issue allocation problem with crossed claims and introduce the constrained proportional awards rule to obtain the emission limits. This rule is based on the concept of proportionality and extends the proportional rule for bankruptcy problems. We also provide an axiomatic characterization of this rule. Moreover, this allocation rule is illustrated by means of a numerical example based on real-world data. Finally, managerial and policy implications of this approach for water pollution control are given.

水污染物可分为三类，每一类又包括几类物质。在本文中，我们提出了一种基于破产模型的方法来确定属于多个类别的污染物质的排放限值。我们将问题建模为具有交叉声明的多问题分配问题，并引入约束比例奖励规则来获得排放限制。该规则以比例原则为基础，并扩展了破产问题的比例规则。我们还提供了这条规则的公理特征。此外，通过基于现实世界数据的数值示例来说明该分配规则。最后，给出了这种水污染控制方法的管理和政策影响。