Production scheduling under uncertainty of an open-pit mine using Lagrangian relaxation and branch-and-cut algorithm

ABSTRACT

The life-of-mine optimization of open pit mine production scheduling under geological uncertainty is a computationally intensive process. Production scheduling determines the optimal extraction sequence by maximizing net present value (NPV). In this paper, an algorithm is proposed to schedule an open pit mine under geological uncertainty, where instead of solving the whole problem at once, the production schedule is generated by sequentially solving sub-problems. The sub-gradient method is used to generate the upper bound solution of a Lagrangian relaxed sub-problem. If the upper bound relaxed solution is infeasible, a mixed integer programming is applied to the latter solution. The algorithm is validated by solving six problems and is compared to the linear relaxation of the original production scheduling problem. The results show that the proposed algorithm generates a solution that is very close to optimal, with less than a 3% optimality gap. An application at a copper mine, where geological uncertainty is quantified with geostatistical simulations of the related orebody, shows that all constraints are satisfied and an 11% higher NPV is generated when compared to the corresponding deterministic equivalent of the proposed approach, while a 26% higher NPV is generated compared to a common conventional industry approach.     

Oil sands mine planning and waste management using mixed integer goal programming

Strategic mine planning and waste management are an important aspect of surface mining operations. Recent environmental and regulatory requirements make waste management an integral part of mine planning in the oil sands industry. The research problem here is determining the order of extraction of ore, dyke material and waste to be removed from a predefined ultimate pit limit over the mine life that maximises the net present value of the operation. We have developed, implemented, and tested a proposed mixed integer goal programming theoretical framework for oil sands open pit production scheduling with multiple material types. The formulation uses binary integer variables to control mining precedence and continuous variables to control mining of ore and dyke material. There are also goal deviational variables and penalty costs and priorities that must be set up by the planner. The optimisation model was implemented in TOMLAB/CPLEX environment. The developed model proved to be able to generate a uniform schedule for ore and dyke material. This is in line with recent regulatory requirements by Alberta Energy Resources and Conservation Board (Directive 074) which requires oil sands mining companies to develop life of mine plans which ties in to their in-pit tailings disposal strategy. It also provides a practical mining sequence that is consistent with mining oil sands deposit. Similarly, tradeoffs between achieving a goal and maximising NPV can be made.     

A new approach to constrained open pit pushback design using dynamic cut-off grades

An integral part of open pit optimization is deciding which section of the ultimate pit to mine during a specific period. For a given period there are often operational and marketing constraints that restrict what can be removed or processed. The operational constraints arise from a number of different limitations such as safe slope of internal mining walls, mill and mining capacity. Traditional methods for pushback (phase) design that incorporate these constraints are ad-hoc and can lead to suboptimal solutions. Another important optimization decision that must be made is the cut-off grade to be used for a specific period. In this paper, a new method is presented that generates near maximal expected profit and dynamically defines the optimal cut-off grade for each mining period or pushback over the life-of-mine, thus deciding whether a block is ore or waste during the optimization process. More specifically, a method for converting a fractional linear program solution into an integral solution known as pipage rounding is applied to an integer program formulation of a pushback design optimization problem. The proposed method aims to produce a set of pushbacks in a way that the total discounted profit to be generated through production scheduling is maximized. Two case studies demonstrate the applied aspects of the proposed method.     

Mixed integer linear programming formulations for open pit production scheduling

One of the main obstacles in using mixed integer linear programming (MILP) formulations for large-scale open pit production scheduling is the size of the problem. The objective of this work is to develop, implement, and verify deterministic MILP formulations for long-term large-scale open pit production scheduling problems. The objective of the model is to maximize the net present value, while meeting grade blending, mining and processing capacities, and the precedence of block extraction constraints. We present four MILP formulations; the first two models are modifications of available models; we also propose, test and validate two new MILP formulations. To reduce the number of binary integer variables in the formulation, we aggregate blocks into larger units referred to as mining-cuts. We compare the performances of the proposed models based on net present value generated, practical mining production constraints, size of the mathematical programming formulations, the number of integer variables required in formulation, and the computational time required for convergence. An iron ore mine case study is represented to illustrate the practicality of the models as well.     

世界上最大的铜矿山-智利埃斯科地达铜矿山

智利埃斯科地达(Escondida)铜矿山是世界上目前最大的铜矿山,具有世界上最丰富的铜资源。根据2009年资料,该矿山的铜矿石储量为41.57亿t,铜品位0.76%,其中含铜3156.7万t铜,矿石资源量为46.50亿t。矿山规划中采用的矿石储量为6.62亿t,其铜品位为2.12%。是世界铜矿山的生产成本最低的企业之一。矿山采用常规露天开采方法。平均每天开采24万t/d高品位硫化矿、4万t/d低品位硫化矿和3.5万t/d氧化矿。矿山由两个露天采场、两个选矿厂、一个氧化矿堆浸场、一个低品位硫化矿生物堆浸场和一个溶剂萃取/电极厂组成。矿山设计铜年产量为120万t。硫化矿选矿厂采用磨矿-浮选流程,得到含金和银的铜精矿。精矿铜品位为38%～43%,回收率为84%～86%。氧化矿堆浸采用破碎-制球-堆浸工艺。低品位硫化矿采用破碎-制球-筑堆-生物堆浸工艺。堆浸场得到的贵液送溶剂萃取/电积厂处理,得到阴极铜。堆浸场对氧化矿石堆浸的铜浸出率分别为80%和54%(对可溶性铜),堆浸场对低品硫化矿石堆浸的铜浸出率分别为37%和29%(对全铜)。矿山和选矿厂基本投资为56.4亿美元,而矿山铜的直接操作费用仅为60.8美分/磅铜。     

露天矿矿体计算机模型的建立

露天矿矿体计算机模型的建立是进行矿床开采境界优化及制定矿山生产计划的基础。为实现矿山生产现代化管理，德兴铜矿建立了大型露天矿山矿体计算机模型，并获得成功应用，对类似矿山具有参考意义。     

多金属露天矿多目标生产计划优化问题建模及求解算法

针对多金属露天矿山生产计划优化问题难以建模、求解复杂等问题,从多种金属元素、采掘运输成本以及矿石品位三个角度出发,综合考虑矿石产量、品位波动、矿石资源利用率、开采和处理能力以及回采率等多种影响因素,构建了一个多金属露天矿多目标生产计划模型。受粒子群算法启发,提出一种改进狼群算法（IGWO）对模型进行求解,并引入反向学习策略和非线性收敛策略来提高算法的求解效率。以国内某露天矿的实际生产为例,分别利用粒子群算法（PSO）、灰狼算法（GWO）和IGWO算法对模型进行求解对比。结果表明：该生产计划模型更加符合露天矿多种矿产资源综合开采利用的实际需求,IGWO算法较PSO算法运行速度上提高了71%,在求解精度上提高16%。该生产计划方案对多金属露天矿山矿产资源综合利用及精细化排产具有重要的指导意义,可促进企业可持续发展。     

Long Term Production Scheduling Optimisation for a Surface Mining Operation: An Application of MineMax™ Scheduling Software

Since long term production scheduling deals with movement of ore and waste during the life of a mine, it has a significant effect on the cash flow of a mining operation. Therefore, mine planners often seek to optimise the production schedule with respect to a given criterion. The most commonly used criterion in long term scheduling optimisation is to maximise net present value (NPV). The process involves sequencing of ore blocks or parcels to be mined in each period over the life of the mine subject to precedence and other physical constraints imposed by the mining system. The complexity of mine production scheduling in practice entails a computer solution using mathematical programming as the optimisation technique. Linear and mixed integer programming techniques have been used to optimise long term production schedules but most of the computer programs based on these make overly simplifying assumptions and lack the flexibility to handle practical considerations of mine scheduling. MineMax? scheduling software is applied to long term scheduling for a multielement surface mining operation. MineMax uses mixed integer programming with the branch and bound algorithm as solution strategy. The system allows the planner to generate a mine production schedule in the same manner as manual scheduling, but making use of the interactivity in planning data input, in-built optimisation algorithm and the speed of the personal computer. This study involves investigation into the effects of reblocking and grade intervals on the mining schedule, specifically, the material movement, milling grade and the cash flow over the life of the mine.     

露天采场排放尾矿后下部矿体开采安全性研究

由于新华山铜矿没有尾矿排放地点,因此在正式生产前决定将首采部位上部露天采场作为排尾地点。本文对排尾后下部矿体开采的安全性展开了研究,主要采用了有限元程序对变形和渗流进行了模拟研究。结果表明:采取一定的措施后,排尾后下部矿体可以回采。     

露天矿生产规模优化的经济动态评估方法

为实现经济动态评估、采剥总量均衡的露天矿生产规模优化,将露天境界优化与生产规模相结合,建立矿床地质数据库、矿床三维地质实体模型和块段品位模型,综合考虑矿床品位分布不均、资金时间及矿石价格的可变性等因素对境界优化和生产规模的影响,借助境界动态优化分析软件Whittle的Milawa算法,快速生成高质量批量的露天境界,并计算出不同境界方案下的矿山动态经济指标,选择出最优境界,模拟矿山经济运营情况,从技术经济角度优化出矿山生产规模和采剥顺序;在项目依托矿山包钢白云鄂博东介勒格勒铁矿的应用结果表明：该矿山最优生产规模为130万t/a,但在当前的市场环境下,投资收益率为7.58%,矿山开采经济风险很大。为矿山企业的决策提供了科学依据。     

基于DIMINE软件系统的露天矿境界优化研究

露天境界优化是露天矿设计和生产经营管理的重要环节,而最终境界优化是一项需要综合考虑矿山资源状况、开采技术条件、矿产品销售价格、矿石开采及处理成本等多种因素,实现经济效益最大化的复杂工作。目前,三维可视化矿床模拟技术和露天境界优化方法的完美结合使其工序更为简化,并可获得良好的动态效果。以广西平果铝土矿为研究对象,借助DIMINE软件系统建立了该露天矿山的三维实体模型,更加准确、直观地反映了矿床与工程实体的形态及其空间分布关系。针对所建的矿体模型进行了品位估值、储量计算等工作,并基于Le-rchs-Grossmann优化方法(L-G法)对该铝土矿进行露天境界优化研究,参照采矿手册及工程经验,确定合理的露天境界优化参数,得出了该矿山露天开采的最优经济境界。结果可用来指导该矿山前期的资源开发和投资决策,并为后期安排生产计划和实时调度控制提供可靠的依据。     

某铜矿区露天开采对地下开采的影响探讨

一个矿床内需要采用露天与地下联合开采时,如果不是将它们在空间和时间上作为一个有机整体同时考虑开采设计,则会带来很多的问题。某铜矿区地表氧化矿的露天开采活动就是在完全不考虑对地下开采的影响下进行的。通过对该铜矿区形成的数处露天采坑的调查,分析其对地下开采造成的影响,并提出灾害防治的对策。     

边界品位的动态规划优化模型及算法

在分析边界品位优化现有方法存在问题的基础上,提出边界品位动态规划优化模型以及算法步骤。该模型考虑矿体的开采顺序和矿床不同区段的品位分布,实现了地下开采边界品位在时间和空间上的动态结合。基于建立的数学模型,开发了相应的计算机应用系统,并在非洲某铜矿进行了应用。     

Cutoff grade optimization algorithm with stockpiling option for open pit mining operations of two economic minerals

Lane's theory of cutoff grade optimization maximizes the Net Present Value (NPV) of an open pit mining operation with a declining effect as the deposit moves toward exhaustion. This declining effect of NPV defines dynamic cutoff grades, i.e. higher cutoff grades in the early years of mine life and lower cutoff grades in the later years. This phenomenon allows the creation of stockpiles with material between the lowest (breakeven) and optimum cutoff grades for processing during later years, when it becomes economical. As an extension to Lane's original theory of cutoff grades in deposits of two economic minerals, the management, i.e. supplies of material from the mine to the stockpile and from the stockpile to the processing plant, is addressed through the development of a cutoff grade optimization algorithm with option to stockpile. The benefits of the methodology are elaborated in a hypothetical case study.     

基于免疫算法的三道庄露天矿车辆调度优化

分析露天矿车辆调度问题,建立求解矿山车辆调度问题的数学模型。矿山车辆调度问题实际上属于NP完全问题,建立多目标优化模型来求解,即解决总运量最小问题。最后,通过免疫算法在三道庄露天矿车辆调度优化中的应用实例表明,本算法可以快速求得优化解,是求解车辆调度问题的一种有效算法。     

Stochastic optimisation of long-term block cave scheduling with hang-up and grade uncertainty

Abstract

Mass mining methods provide alternatives in developing deeper and lower-grade mineral deposits. Consequently, block cave mining has been increasingly popular mass mining method, especially for large copper deposits currently being mined by open pit methods. This study adopts similar concepts as in stochastic open pit production planning to the planning of block cave mines, to evaluate their effectiveness in a different approach to mass mining. The main contribution of this study is the incorporation of the uncertainty of delays from hang-ups and grades directly into the production scheduling process of a cave mining operation. Hang-up uncertainty relates to the uncertainty linked to the occurrence of ore that clogs the production draw points. This clogging causes time delays in the production cycle leading to tonnage losses and additional costs. Grade uncertainty is incorporated by means of stochastic orebody simulations. Both uncertainty sources are directly linked to the extraction decisions and influence the optimized schedules. The proposed stochastic integer programming model is applied to the optimization of the long-term schedule of a large-scale, low-grade copper deposit by taking into account hang-up delays in block caving. The results of the optimization maximizing net present value clearly show the capability of the formulation to mitigate the effects of both grade and hang-up uncertainty.     

刚果民主共和国腾克丰古鲁梅铜钴矿山

刚果民主共和国腾克丰古鲁梅(Tenke Fungurume)铜钴矿山是世界上现有的最大的铜钴矿山,它具有世界上最丰富的铜钴资源。现有的矿石储量可能超过1.19亿t,其中含2.64%铜和0.35%钴。矿山采用常规的露天开采方法。平均每天开采4.65万t物料,其中7000t高品位矿石、7500t低品位矿石和32000t废石。选矿厂日处理矿石7000t,年产11.1万t铜和8545t钴(氢氧化钴)。选矿流程包括磨矿、硫酸浸出和对流倾析、中和、溶剂萃取/电积和钴的回收。矿山和选矿厂基本投资为17.50亿美元,矿石的采矿和选矿总成本为85.10美元/t矿石。     

露天矿计划阶段内离散块体物料运距预测算法

矿岩运距是衡量露天矿山卡车运输经济性的重要指标之一,但多年因受运输系统复杂、多变等多种综合因素限制,在进度计划编制期间进行逐条带的运距计算问题一直并未得到有效解决。特别是需要进行离散化的物料规划时,传统逐阶段的运距量测方法无法对离散块体进行逐个推估。因此,为有效解决计划阶段内离散块体物料运距计算问题,根据露天矿外排土场出入口选址及排土运输干线相对固定等空间特征,将非线性预测理论、因子分析法等技术方法应用于实际的矿山优化问题中,研究了排土场计划阶段内块体运距波动变化的主要控制因素;结合外排土场内运距波动变化的影响要素,提出了采用加权最小二乘支持向量机技术(WLS-SVM)拟合非线性运距曲线的预测算法,并对权向量、核参数等主观经验指标进行了动态修正,以实现对计划阶段内(两阶段工程位置间的)离散块体物料过程运距的时变预测;最终,利用拟合出的时变曲线,实现两连续工程位置间逐个物料块体运距的高效计算。实验结果表明:预测算法的误差期望分别为:训练集0. 93 m,测试集0. 84 m,且在实验中表现出模型精度易受样本规模影响的特性,且当样本规模N90时,可控制绝对误差水平稳定在0. 8%～1. 2%。结论表明:试图应用非线性预测理论处理计划阶段内缺乏时变路网条件的运距预测问题是可行的,提出的算法模型对解决计划阶段内的逐块体运距赋值问题具有现实有效性。常规的露天矿山运输优化问题其优化目标和特征参数之间往往存在紧密耦合关系,无法直接应用特定的优化、规划模型进行计算求解,尝试采用全新的解耦思想和求解思路对于有效解决露天矿山运输优化问题是十分有益的,也应该是今后露天矿山系统工程学科研究的热点问题之一。     

大型深凹露天金属矿倾斜条带式短分期开采工艺及应用

某铜镍矿是世界级大型铜镍矿床,设计采用露天开采,采剥总量达7.4亿t,最高边坡达645m。针对大型深凹露天矿,采用短分期开采,组合台阶和倾斜条带式陡帮作业,有效推迟剥离高峰,减少基建剥离量和初期生产剥采比,降低投资及初期开采成本,经济效益显著提高。     

沙溪铜矿开采边界品位动态优化方法

边界品位对矿山而言是一个尤其重要的决策参数。我国的边界品位指标制定于计划经济时期,随着我国市场经济的不断完善,现行的矿产工业指标越来越不能适应采矿工业的发展。沙溪铜矿具有埋藏深、储量大、品位低等特点,如何实现经济合理开采,其中首要解决的便是边界品位的问题。鉴于此,提出以铜品位0.1%为边界圈定矿化域、进行地质统计学估值得到矿化域模型,从而得到矿石储量、平均品位与边界品位的关系的方法,随后提出以最大净现值为目标的边界品位动态优化数学模型和算法。最后,应用所提方法优化了沙溪铜矿边界品位,与原初步设计结果相比,使得矿山服务年限、回收金属量、总现金流量、总净现值都得到了显著提高。该方法对我国大量的储量大、品位低的地下矿床经济开采具有借鉴意义。