In this work, we consider a risk-averse ultimate pit problem where the grade of the mineral is uncertain. We derive conditions under which we can generate a set of nested pits by varying the risk level instead of using revenue factors. We propose two properties that we believe are desirable for the problem: risk nestedness, which means the pits generated for different risk aversion levels should be contained in one another, and additive consistency, which states that preferences in terms of order of extraction should not change if independent sectors of the mine are added as precedences. We show that only an entropic risk measure satisfies these properties and propose a two-stage stochastic programming formulation of the problem, including an efficient approximation scheme to solve it. We illustrate our approach in a small self-constructed example, and apply our approximation scheme to a real-world section of the Andina mine, in Chile.

在这项工作中，我们考虑了矿物等级不确定的规避风险的最终矿井问题。我们通过改变风险级别而不是使用收入因子来推导产生一系列嵌套坑的条件。我们提出了两个我们认为是该问题所希望的属性：风险嵌套，这意味着针对不同风险规避级别生成的凹坑应相互包含，以及相加一致性，表明在提取顺序方面的偏好不应更改如果将矿山的独立部门作为优先级添加。我们表明，只有熵风险度量能够满足这些属性，并提出了该问题的两阶段随机规划公式，包括用于解决该问题的有效近似方案。我们通过一个小小的自我构建的示例来说明我们的方法，