MineLib: a library of open pit mining problems

Similar to the mixed-integer programming library (MIPLIB), we present a library of publicly available test problem instances for three classical types of open pit mining problems: the ultimate pit limit problem and two variants of open pit production scheduling problems. The ultimate pit limit problem determines a set of notional three-dimensional blocks containing ore and/or waste material to extract to maximize value subject to geospatial precedence constraints. Open pit production scheduling problems seek to determine when, if ever, a block is extracted from an open pit mine. A typical objective is to maximize the net present value of the extracted ore; constraints include precedence and upper bounds on operational resource usage. Extensions of this problem can include (i) lower bounds on operational resource usage, (ii) the determination of whether a block is sent to a waste dump, i.e., discarded, or to a processing plant, i.e., to a facility that derives salable mineral from the block, (iii) average grade constraints at the processing plant, and (iv) inventories of extracted but unprocessed material. Although open pit mining problems have appeared in academic literature dating back to the 1960s, no standard representations exist, and there are no commonly available corresponding data sets. We describe some representative open pit mining problems, briefly mention related literature, and provide a library consisting of mathematical models and sets of instances, available on the Internet. We conclude with directions for use of this newly established mining library. The library serves not only as a suggestion of standard expressions of and available data for open pit mining problems, but also as encouragement for the development of increasingly sophisticated algorithms.     

带学习效应的两台平行机时间表长问题

研究机器带学习效应, 目标函数为时间表长的两台平行机排序问题, 问题是NP-难的. 首先建立了求解该问题最优解的整数规划模型. 其次, 基于模拟退火算法给出了该问题的近似算法SA, 并证明了该算法依概率1 全局收敛到最优解. 最后, 通过数值模拟对所提出的算法进行了性能分析. 数值模拟结果表明, 近似算法SA可以达到最优值的99%, 准确度高, 算法较有效.     

Earliness and tardiness single machine scheduling with proportional weights

This paper presents the first efficient method to solve an earliness-tardiness single machine n job scheduling model with idle times permitted. In this model the earliness and tardiness penalties are proportional to the processing times of the jobs. A two stage decomposition process builds the optimal job arrangement as a sequence of single and multijob blocks. The arrangement of the jobs within each multijob block is unspecified since it depends on the start time of this block. This process is followed by a procedure that drastically reduces the number of candidate optimal n job sequences. Finally an available optimal timing algorithm is recommended and implemented to select the best schedule among those sequences. The solution procedure tested on a PC on 400 examples for n=40 and 50 proves to be very fast.     

Implementing a parametric maximum flow algorithm for optimal open pit mine design under uncertain supply and demand

Conventional open pit mine optimization models for designing mining phases and ultimate pit limit do not consider expected variations and uncertainty in metal content available in a mineral deposit (supply) and commodity prices (market demand). Unlike the conventional approach, a stochastic framework relies on multiple realizations of the input data so as to account for uncertainty in metal content and financial parameters, reflecting potential supply and demand. This paper presents a new method that jointly considers uncertainty in metal content and commodity prices, and incorporates time-dependent discounted values of mining blocks when designing optimal production phases and ultimate pit limit, while honouring production capacity constraints. The structure of a graph representing the stochastic framework is proposed, and it is solved with a parametric maximum flow algorithm. Lagragnian relaxation and the subgradient method are integrated in the proposed approach to facilitate producing practical designs. An application at a copper deposit in Canada demonstrates the practical aspects of the approach and quality of solutions over conventional methods, as well as the effectiveness of the proposed stochastic approach in solving mine planning and design problems.     

Comparative analysis of three metaheuristics for short-term open pit block sequencing

This paper presents the application of simulated annealing (SA), Tabu search (TS) and hybrid TS–SA to solve a real-world mining optimisation problem called open pit block sequencing (OPBS). The OPBS seeks the optimum extraction sequences under a variety of geological and technical constraints over short-term horizons. As industry-scale OPBS instances are intractable for standard mixed integer programming (MIP) solvers, SA, TS and hybrid TS–SA are developed to solve the OPBS problem. MIP exact solution is also combined with the proposed metaheuristics to polish solutions in feasible neighbourhood moves. Extensive sensitivity analysis is conducted to analyse the characteristics and determine the optimum sets of values of the proposed metaheuristics algorithms’ parameters. Computational experiments show that the proposed algorithms are satisfactory for solving the OPBS problem. Additionally, this comparative study shows that the hybrid TS–SA is superior to SA or TS in solving the OPBS problem in several aspects.     

The new Fundamental Tree Algorithm for production scheduling of open pit mines

The problem of annual production scheduling in surface mining consists of determining an optimal sequence of extracting the mineralized material from the ground. The main objective of the optimization process is usually to maximize the total Net Present Value of the operation. Production scheduling is typically a mixed integer programming (MIP) type problem. However, the large number of integer variables required in formulating the problem makes it impossible to solve. To overcome this obstacle, a new algorithm termed “Fundamental Tree Algorithm” is developed based on linear programming to aggregate blocks of material and decrease the number of integer variables and the number of constraints required within the MIP formulation. This paper proposes the new Fundamental Tree Algorithm in optimizing production scheduling in surface mining. A case study on a large copper deposit summarized in the paper shows substantial economic benefit of the proposed algorithm compared to existing methods.     

An Application of Branch and Cut to Open Pit Mine Scheduling

The economic viability of the modern day mine is highly dependent upon careful planning and management. Declining trends in average ore grades, increasing mining costs and environmental considerations will ensure that this situation will remain in the foreseeable future. The operation and management of a large open pit mine having a life of several years is an enormous and complex task. Though a number of optimization techniques have been successfully applied to resolve some important problems, the problem of determining an optimal production schedule over the life of the deposit is still very much unresolved. In this paper we will critically examine the techniques that are being used in the mining industry for production scheduling indicating their limitations. In addition, we present a mixed integer linear programming model for the scheduling problems along with a Branch and Cut solution strategy. Computational results for practical sized problems are discussed.     

Block rearranging elements within matrix columns to minimize the variability of the row sums

Several problems in operations research, such as the assembly line crew scheduling problem and the k-partitioning problem can be cast as the problem of finding the intra-column rearrangement (permutation) of a matrix such that the row sums show minimum variability. A necessary condition for optimality of the rearranged matrix is that for every block containing one or more columns it must hold that its row sums are oppositely ordered to the row sums of the remaining columns. We propose the block rearrangement algorithm with variance equalization (BRAVE) as a suitable method to achieve this situation. It uses a carefully motivated heuristic—based on an idea of variance equalization—to find optimal blocks of columns and rearranges them. When applied to the number partitioning problem, we show that BRAVE outperforms the well-known greedy algorithm and the Karmarkar–Karp differencing algorithm.     

A branch and bound algorithm for scheduling trains in a railway network

The paper studies a train scheduling problem faced by railway infrastructure managers during real-time traffic control. When train operations are perturbed, a new conflict-free timetable of feasible arrival and departure times needs to be re-computed, such that the deviation from the original one is minimized. The problem can be viewed as a huge job shop scheduling problem with no-store constraints. We make use of a careful estimation of time separation among trains, and model the scheduling problem with an alternative graph formulation. We develop a branch and bound algorithm which includes implication rules enabling to speed up the computation. An experimental study, based on a bottleneck area of the Dutch rail network, shows that a truncated version of the algorithm provides proven optimal or near optimal solutions within short time limits.     

基于改进模拟退火算法的推动式生产-配送协调优化

为减小物资生产与配送不协调造成的成本及生产资源浪费,建立了考虑推动式生产调度的物资配送优化模型,并针对标准模拟退火算法受随机因素影响易陷入局部最优的缺点,设计带有回火与缓冷操作的改进模拟退火算法对模型求解,确定了优化的车辆配送路线以及物资生产计划。对比实验结果表明：相对于单纯的物资配送优化模型,考虑推动式生产调度的配送优化模型,能够有效减小物资滞留时间以及配送延误成本;相较于标准模拟退火算法,改进算法搜索到了更优解,且计算结果的标准差减小了93.42%,稳定性更好;同时,改进模拟退火算法具有较低的偏差率,在中小规模算例中求解质量较高,平均偏差率在0.5%以内。     

Modeling demand management strategies for evacuations

Evacuations are massive operations that create heavy travel demand on road networks some of which are experiencing major congestions even with regular traffic demand. Congestion in traffic networks during evacuations, can be eased either by supply or demand management actions. This study focuses on modeling demand management strategies of optimal departure time, optimal destination choice and optimal zone evacuation scheduling (also known as staggered evacuation) under a given fixed evacuation time assumption. The analytical models are developed for a system optimal dynamic traffic assignment problem, so that their characteristics can be studied to produce insights to be used for large-scale solution algorithms. While the first two strategies were represented in a linear programming (LP) model, evacuation zone scheduling problem inevitable included integers and resulted in a mixed integer LP (MILP) one. The dual of the LP produced an optimal assignment principle, and the nature of the MILP formulations revealed clues about more efficient heuristics. The discussed properties of the models are also supported via numerical results from a hypothetical network example.     

A sliding time window heuristic for open pit mine block sequencing

The open pit mine block sequencing problem (OPBS) seeks a discretetime production schedule that maximizes the net present value of the orebody extracted from an open-pit mine. This integer program (IP) discretizes the mine’s volume into blocks, imposes precedence constraints between blocks, and limits resource consumption in each time period. We develop a “sliding time window heuristic” to solve this IP approximately. The heuristic recursively defines, solves and partially fixes an approximating model having: (i) fixed variables in early time periods, (ii) an exact submodel defined over a “window” of middle time periods, and (iii) a relaxed submodel in later time periods. The heuristic produces near-optimal solutions (typically within 2% of optimality) for model instances that standard optimization software fails to solve. Furthermore, it produces these solutions quickly, even though our OPBS model enforces standard upper-bounding constraints on resource consumption along with less standard, but important, lower-bounding constraints.     

IPM based sparse LP solver on a heterogeneous processor

We present the parallelization of a linear programming solver using a primal-dual interior point method on one of the heterogeneous processors, namely the Cell BE processor. Focus is given to Cholesky factorization as it is the most computationally expensive kernel in interior point methods. To make it easier to develop and port to other heterogeneous systems, we propose a two-phase implementation procedure where we first develop a shared-memory multithreaded application that executes only on the main processor, and then offload the compute-intensive tasks to execute on the synergistic processors (Cell accelerator cores). We used parent–child supernode amalgamation to increase sizes of the blocks, but we noticed that the existence of many small blocks cause significant performance degradation. To reduce the overhead of small blocks, we extend the block fan-out algorithm such that small blocks are aggregated into large blocks without adding extra zeros. We also use another type of amalgamation that can merge any two consecutive supernodes and use it to avoid having very small blocks in a composed block. The suggested block aggregation method is able to speedup the whole LP solver of up to 2.5 when compared to using parent–child supernode amalgamation alone.     

Some new results on simulated annealing applied to the job shop scheduling problem

We present two results about heuristic solutions to the job shop scheduling problem (JSP). First, we show that the well-known analytical results on convergence of simulated annealing (SA) do not hold in the application to the JSP. We give a simple counterexample where the SA process converges against a suboptimal schedule. To overcome this problem at least heuristically, we present a new approach that uses a small population of SA runs in a genetic algorithm (GA) framework. The novel features are an adaptive temperature control that allows `reheating' of the SA and a new type of time-oriented crossover of schedules. Though the procedure uses only standard properties of the JSP it yields excellent results on the classical test examples.     

A mathematical programming viewpoint for solving the ultimate pit problem

In 1965 Helmut Lerchs and Ingo Grossmann presented to the mining community an algorithm to find the optimum design for an open pit mine. In their words, “the objective is to design the contour of a pit so as to maximize the difference between total mine value of the ore extracted and the total extraction cost of ore and waste”. They modeled the problem in graph theoretic terms and showed that an optimal solution of the ultimate pit problem is equivalent to finding the maximum closure of their graph based model. In this paper, we develop a network flow algorithm based on the dual to solve the same problem. We show how this algorithm is closely related to Lerchs and Grossmann's and how the steps in their algorithm can be viewed in mathematical programming terms. This analysis adds insight to the algorithm of Lerchs and Grossmann and shows where it can be made more efficient. As in the case Lerchs and Grossmann, our algorithm allows us to use very efficient data structures based on graphs that represent the data and constraints.. 1782 1528 V 3     

Learning heuristics for basic block instruction scheduling

Instruction scheduling is an important step for improving the performance of object code produced by a compiler. A fundamental problem that arises in instruction scheduling is to find a minimum length schedule for a basic block—a straight-line sequence of code with a single entry point and a single exit point—subject to precedence, latency, and resource constraints. Solving the problem exactly is known to be difficult, and most compilers use a greedy list scheduling algorithm coupled with a heuristic. The heuristic is usually hand-crafted, a potentially time-consuming process. In contrast, we present a study on automatically learning good heuristics using techniques from machine learning. In our study, a recently proposed optimal basic block scheduler was used to generate the machine learning training data. A decision tree learning algorithm was then used to induce a simple heuristic from the training data. The automatically constructed decision tree heuristic was compared against a popular critical-path heuristic on the SPEC 2000 benchmarks. On this benchmark suite, the decision tree heuristic reduced the number of basic blocks that were not optimally scheduled by up to 55% compared to the critical-path heuristic, and gave improved performance guarantees in terms of the worst-case factor from optimality.     

Solving the discrete-continuous project scheduling problem via its discretization

In this paper a discrete-continuous project scheduling problem is considered. In this problem activities simultaneously require discrete and continuous resources. The processing rate of each activity depends on the amount of the continuous resource allotted to this activity at a time. All the resources are renewable ones. The activities are nonpreemtable and the objective is to minimize the makespan. Discretization of this problem leading to a classical (i.e. discrete) project scheduling problem in the multi-mode version is presented. A simulated annealing (SA) approach to solving this problem is described and tested computationally in two versions: with and without finding an optimal continuous resource allocation for the final schedule. In the former case a nonlinear solver is used for solving a corresponding convex programming problem. The results are compared with the results obtained using SA for the discrete-continuous project scheduling problem where the nonlinear solver is used for exact solving the continuous part in each iteration. The results of a computational experiment are analyzed and some conclusions are included.     

Rescaled Simulated Annealing—Accelerating Convergence of Simulated Annealing by Rescaling the States Energies

This paper presents a new metaheuristic, called rescaled simulated annealing (RSA) which is particularly adapted to combinatorial problems where the available computational effort to solve it is limited. Asymptotic convergence on optimal solutions is established and the results are favorably compared to the famous ones due to Mitra, Romeo, and Sangiovanni-Vincentelli (Mitra, Romeo, and Sangiovanni-Vincentelli. (1986). Adv. Appl. Prob. 18, 747–771.) for simulated annealing (SA). It is based on a generalization of the Metropolis procedure used by the SA algorithm. This generalization consists in rescaling the energies of the states candidate for a transition, before applying the Metropolis criterion. The direct consequence is an acceleration of convergence, by avoiding dives and escapes from high energy local minima. Thus, practically speaking, less transitions need to be tested with RSA to obtain a good quality solution. As a corollary, within a limited computational effort, RSA provides better quality solutions than SA and the gain of performance of RSA versus SA is all the more important since the available computational effort is reduced. An illustrative example is detailed on an instance of the Traveling Salesman Problem.     

An approach to optimize block surgical schedules

We provide an approach to optimize a block surgical schedule (BSS) that adheres to the block scheduling policy, using a new type of newsvendor-based model. We assume that strategic decisions assign a specialty to each Operating Room (OR) day and deal with BSS decisions that assign sub-specialties to time blocks, determining block duration as well as sequence in each OR each day with the objective of minimizing the sum of expected lateness and earliness costs. Our newsvendor approach prescribes the optimal duration of each block and the best permutation, obtained by solving the sequential newsvendor problem, determines the optimal block sequence. We obtain closed-form solutions for the case in which surgery durations follow the normal distribution. Furthermore, we give a closed-form solution for optimal block duration with no-shows.     

具有学习-遗忘效应的半导体批调度问题研究

半导体生产制造系统具有大规模、工艺繁杂、随机性大、可重入等显著特点。以半导体最终测试阶段批处理调度为基础,把学习-遗忘效应应用到典型半导体批调度问题中,构建基于学习-遗忘效应的批调度模型。分别结合调度问题和调度模型对双层算法(粒子群算法&萤火虫算法)进行设计,通过仿真实验检验了双层算法在求解具有学习遗忘效应的批调度模型方面的可行性和有效性,并对比分析以最大完工时间为优化目标的实验结果,探讨学习遗忘效应对半导体批调度问题的影响程度,对实际半导体生产具有重要指导意义。