运筹学论文中英文翻译

A maximum flow formulation of a multi-period open-pitmining problem多期露天采矿的最大流公式Henry Amankwah •Torbjo¨rn Larsson •Bjo¨rn TextoriusAbstract 摘要We consider the problem of finding an optimal mining sequence for anopen pit during a number of time periods subject to only spatial and temporal precedence constraints. This problem is of interest because such constraints are generic to any open-pit scheduling problem and, in particular, because it arises as a Lagrangean relaxation of an open-pit scheduling problem. We show that this multi-period open-pit mining problem can be solved as a maximum flow problem in a time-expanded mine graph. Further, the minimum cut in this graph will define an optimal sequence of pits. This result extends a well-known result of J.-C. Picard from 1976 for the open-pit mine design problem, that is, the single-period case, to the case of multiple time periods.我们认为在若干期间，找到一个最佳的露天采矿的开采顺序，只会受到空间和时间的优先约束。

这个问题很有趣，因为这些约束通用于任何露天矿调度问题，特别是因为它是作为露天矿调度的拉格朗日松弛算法而出现的。

我们可以将这种多期露天开采的问题作为一个时间扩张矿图的最大流问题来解决。

此外，本图中的最小割集将定义的矿坑的最佳顺序。

这个结果是J. C. Picard著名理论的延伸，J. C. Picard从1976年就研究露天矿设计问题，即，单周期的情况下，对多个时间段的研究。

1 IntroductionOpen-pit mining is a surface mining operation whereby ore, or waste, is excavated from the surface of the land, and in so doing a deeper and deeper pit is formed.Before the mining begins, the volume of the ore deposit is usually partitioned into blocks and the value of the ore in each block is estimated by using geological information from drill holes. The cost of mining and processing each block is also estimated. A profit can thus be assigned to each block of the mine model, as illustrated in Fig.1 引言露天开采是一个表面采矿作业，从地面挖掘出矿石或废物，因此形成一个越来越深的坑。

在开始采矿前，通常把矿床划分成块，通过钻孔的地质信息估计每块矿床的价值。

每块矿床的开采和加工成本也能估计。

利润可以分配给每个矿山模型，如图所示。

A fundamental problem in open-pit mine planning is to decide which blocks to mine. This is known as the problem of finding an open-pit mine design, or an ultimate contour for the pit. The only restrictions are spatial precedence relationships, stating that in order to extract any given block, so must all blocks immediately above and within a required wall slope angle. Lerchs and Grossmann (1965) showed that the design problem can be stated as the problem of finding a maximal closure in a mine graph which represents the blocks and the precedencerestrictions, as shown in Fig. 1 (for a safe slope angle of 45 ). Their algorithm for finding a maximal closure in the mine graph has over the years been commonly used by the mining industry for the design of open pits.在露天矿山规划中，一个基本问题是决定开采哪块矿。

这被称为寻找露天矿山设计的问题，或矿井的最终轮廓问题。

唯一的限制是空间优先的关系，指出为了提取任何给定的矿块，所有矿块必须在上面和在要求的壁面收敛角范围内。

勒奇斯和格罗斯曼（1965）表明，设计问题可以表述为在矿图中寻找最大闭合的问题，这幅矿图能表现矿块和优先级限制，如图1所示（45°的安全坡度）。

多年来，他们用来在矿图中寻找最大闭合的公式，在采矿业的露天矿设计中也被普遍使用。

The practical significance of the open-pit mine design problem makes it an important instance of the maximal closure problem (Picard and Queyranne, 1982). As shown by Picard (1976), the problem of finding a maximal closure in a mine graph can be solved as a maximum flow problem in a network derived from the mine graph, and where a minimum cut determines an optimal pit contour. Later, Hochbaum and Chen (2000) and Hochbaum (2001) developed efficient maximum flow algorithms for the open-pit mining problem.露天矿设计问题的实际意义是使其成为最大闭合问题中的一个重要实例（皮卡德和凯拉纳，1982）。

皮卡德（1976）表明，在矿图中寻找最大闭合的问题可以作为矿图中网状图的最大流问题来解决，矿图中，最小割集确定最佳矿井轮廓。

后来，陈（2000）和霍赫鲍姆（2001）研究出高效的露天开采的最大流算法。

In reality, the profit of a block depends on when it is mined, for example due to discounting. This fact leads to another crucial issue in open-pit mine planning, namely scheduling. This is the process of deciding how and when to mine the blocks so as to maximize profit (typically the net present value), while obeying the wall slope and precedence constraints, as well as various mining capacity restrictions. Contributions within open-pit mine scheduling, from the view of mathematical optimization, have been given by Gershon (1983), Dagdelen and Johnson (1986), Caccetta and Hill (2003), Ramazan (2007), Rafiee and Asghari (2008), Bley et al. (2010), and Cullenbine et al. (2011), among others.在现实中，一个矿块的利润取决于开采时，例如由于打折。

这个事实导致了露天矿山规划的另一个关键问题，即调度。

这是决定何时开采、如何开采并使利润最大化（通常是净现值）的过程，同时遵守墙坡和优先约束，并受到各种开采能力的限制。

Gershon (1983),Dagdelen 和Johnson (1986), Caccetta和Hill (2003), Ramazan (2007), Rafiee和Asghari (2008), Bley et al. Cullenbineetal (2011)等人已经从数学优化的角度给出了露天矿山调度的贡献。

We consider a multi-period open-pit mining problem with only spatial and temporal precedence constraints. The latter simply state that once a block has been mined, it shall remain mined. The spatial and temporal precedence constraints are generic to open-pit mine scheduling and the multi-period problem arises as a Lagrangean relaxed open-pit scheduling problem, when capacity restrictions are Lagrangean dualized.我们只考虑有空间和时间优先约束的多周期的露天开采问题。