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.     

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.     

Robust Planning for an Open-Pit Mining Problem under Ore-Grade Uncertainty

Open-pit mining production planning is a risky problem: operation costs are considerable, many parameters are inherently subject to uncertainty and, moreover, the mining operation can only be done once. In this work we address uncertainty in the ore-grade, where we only assume the availability of an i.i.d. sample of the joint distribution of ore-grade in the blocks of the mining site. We consider an open-pit mining problem involving extraction and processing decisions under capacity constraints. We apply and compare the risk-hedging performance of three approaches for optimization under uncertainty: Value-at-Risk, Conditional Value-at-Risk and a proposed robust optimization approach. The latter is shown to have desirable risk-averse properties. Computational results on one small size vein-type mine are shown.     

A diversified Tabu search approach for the open-pit mine production scheduling problem with metal uncertainty

This paper presents a metaheuristic solution approach based on Tabu search for the open-pit mine production scheduling problem with metal uncertainty. To search the feasible domain more extensively, two different diversification strategies are used to generate several initial solutions to be optimized by the Tabu search procedure. The first diversification strategy exploits a long-term memory of the search history. The second one relies on the variable neighborhood search method. Numerical results on realistic large-scale instances are provided to indicate the efficiency of the solution approach to produce very good solutions in relatively short computational times.     

CAPACITY AND PRICE OF NONRENEWABLE RESOURCE DEPOSITS

This paper develops a model to determine the planned rate of production from a nonrenewable resource deposit and applies the model to the U.S. copper industry to estimate changes in aggregate planned copper production as a function of copper price expectations. Standard investment theory is applied in the model to examine the choice of the planned rate of extraction from limited volume resource deposits. A setting is analyzed in which the total production volume from any site is fixed by the size of the resource deposit at that site. Within this setting, the planned rate of resource extraction is a fully endogenous solution of the model. Properties of the model are investigated to determine changes in the planned rate of production caused by variation in operating cost, investment cost and output price parameters. The planned rate of production from any deposit is found to increase at a decreasing rate, given incremental increases in the output price, ceteris paribus. The finding is illustrated with data describing mined and yet-to-be mined copper deposits in the United States. A close match is observed between the calculated production rate for mined copper deposits and the mine capacity (rate of production) chosen by the owners of those deposits, given plausible assumptions about mine owner price expectations. The planned rate of production is aggregated across all deposits and plotted as a function of output price. An aggregate medium run copper supply curve is obtained, indicating the elasticity of planned production to changes in the expected long run output price of copper. This empirical example suggests that doubling the output price from $1 to $2 per pound copper, would triple aggregate planned copper production in the United States.     

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.     

Incorporating geo-metallurgical information into mine production scheduling

Economic characterization of mining parcels depends upon geo-metallurgical properties, which vary throughout orebody. Mine production scheduling should aim to obtain maximum utility from orebody in such a way as to ensure mine–mill reconciliation. As heterogeneity of geo-metallurgical variables increases, the scheduling will be a very complicated task. Geo-metallurgical and financial data used in the mine production scheduling are based on simulation and/or estimation generated from sparse drilling and unknown future events. Therefore, the scheduling process involves a significant degree of uncertainty. In order to deal with the uncertainty stemmed from geo-metallurgical and financial variables, two approaches are recommended in this paper. Firstly, mine production scheduling is formulated as a problem of stochastic programming with recourse. The extraction periods of mining blocks are treated as the first-stage variables and the block destinations represents a recourse vector. It is observed that the solution is implicitly robust. Secondly, the scheduling is expressed as a maximin problem to extract more uniform metal quantity in periods to coincide with mill requirements instead of maximization of net present value because the blending constraint in the traditional approach forces more uniform production. In the case where there is correlation between grade and geo-metallurgical variables, this model generates reasonably good results.     

Agricultural planning of annual plants under demand,maturation, harvest,and yield risk

In this study we present a planning methodology for a firm whose objective is to match the random supply of annual premium fruits and vegetables from a number of contracted farms and the random demand from the retailers during the planning period. The supply uncertainty is due to the uncertainty of the maturation time, harvest time, and yield. The demand uncertainty is the uncertainty of weekly demand from the retailers. We provide a planning methodology to determine the farm areas and the seeding times for annual plants that survive for only one growing season in such a way that the expected total profit is maximized. Both the single period and the multi period cases are analyzed depending on the type of the plant. The performance of the solution methodology is evaluated by using numerical experiments. These experiments show that the proposed methodology matches random supply and random demand in a very effective way and improves the expected profit substantially compared to the planning approaches where the uncertainties are not taken into consideration.     

Design and planning of supply chains with integration of reverse logistics activities under demand uncertainty

In this paper, a mixed integer linear programming (MILP) formulation is developed for the design and planning of supply chains with reverse flows while considering simultaneously production, distribution and reverse logistics activities. It is also considered products’ demand uncertainty using a scenario tree approach. As main goal the model defines the maximization of the expected net present value and the results provide details on sizing and location of plants, warehouses and retailers, definition of processes to install, establishment of forward and reverse flows and inventory levels to attain. The model is applied to a representative European supply chain case study and its applicability is demonstrated.     

Hybrid evolutionary optimization of the operation of pipeless plants

Pipeless plants are a new production concept in chemical engineering in which automated guided vehicles (AGVs) transport the substances in mobile vessels between processing stations. In the operation of such plants, decisions have to be made on the scheduling of the production, the assignment of the equipment and the routing of the AGVs that carry the vessels. The large number of interacting degrees of freedom prohibit the use of exact mathematical algorithms to compute optimal schedules. This paper describes the combination of an evolutionary scheduling algorithm with a simulation based schedule builder. The algorithm is tested on a real-life example and on a benchmark problem from the literature and yields considerably shorter makespans than a heuristic solution.     

The treatment of uncertainty in mineral exploration and exploitation

A sequential method of modeling the increase in precision of expected net revenues for a proposed exploration and exploitation program has been developed. Embedded within a computer simulation model of the exploration process, which incorporates a method of learning about deposit characteristics, is a multi-stage stochastic optimization process model to determine the optimal exploitation pattern of the deposit. This approach stresses the interdependence of the planning of the exploration and exploitation processes.The model can be used to determine the amount of exploration which should be undertaken in an area by more precisely predicting the long-range profitability associated with the amount of exploration. Thus, decision makers are provided a capability which reduces the uncertainty in profitability outcomes over future production periods.Time did not permit us to respond fully to the editorial review process. Accordingly., the authors accept full responsibility for any consequences of this shortcoming.     

On conjugate gradient-like methods for eigen-like problems

Numerical analysts, physicists, and signal processing engineers have proposed algorithms that might be called conjugate gradient for problems associated with the computation of eigenvalues. There are many variations, mostly one eigenvalue at a time though sometimes block algorithms are proposed. Is there a correct “conjugate gradient” algorithm for the eigenvalue problem? How are the algorithms related amongst themselves and with other related algorithms such as Lanczos, the Newton method, and the Rayleigh quotient?     

矿山环境恢复治理保证金制度中公众参与的博弈分析:基于合谋与防范的视角

矿产资源开采行为容易造成矿区生态环境的恶化,国内外目前主要采取矿山环境恢复治理保证金制度以强制矿山企业对矿区开采后生态环境进行一系列修复。在缺少公众监督的情况下,保证金制度容易出现道德风险,即矿山企业和地方政府监管者的合谋行为。目前,我国各地推行的矿山环境恢复治理保证金制度均没有让公众参与进来,这就需要对这一制度缺陷进行弥补与修正。本文基于公众参与理论和博弈论构建了公众、矿山企业和政府监管者三方博弈模型,并利用对其混合纳什均衡解的分析,得出结论为:通过提高公众监督概率和公众监督有效概率、降低公众监督成本来防范和控制合谋行为。最后,根据以上分析提出了相关的政策建议。     

Medium-term hydro-thermal coordination via hydro and thermal subsystem decomposition

Summary This paper addresses the medium-term hydro-thermal coordination problem in an electric energy system. That is, the problem of finding the energy production of every power plant (hydro or thermal) in every subperiod of a given planning period, so that the customer load is supplied at minimum cost. The planning horizon is typically one to two months and the first week of this planning period is modeled in detail. The solution method proposed decomposes the problem in two subproblems corresponding to the hydro and thermal subsystems. These two subproblems are coordinated using a coordinating function for every subperiod. The coordinating function of a given subperiod expresses total production cost in that subperiod as a function of the total hydro production in that subperiod. The decomposition proposed makes it possible to use specialized algorithms to solve the hydro and thermal subproblems. This results in a very efficient computational procedure. From an experimental point of view the coordinating mechanism is robust. A case study is provided. It considers 61 thermal plants, a hydro system including 8 cascaded hydro plants and a 48 subperiods planning period.     

Robust supply chain network design with multi-products for a company in the food sector

The paper aims to solve a problem faced by a company competing in the snacks market in Turkey. In line with the growth in this market, the company needs to make important decisions over the next few years about the timing and location of a new plant, its initial capacity, the timing and amount of additional capacity to be installed at the new and existing plants, the assignment of demand points to plants and the amount of raw materials to be shipped from suppliers to the plants in each period. The objective is to minimize the total cost of various components. The problem is formulated as a multi-period supply chain network design model with multi products. The resulting mixed-integer linear programming model is solved by the commercial solver CPLEX. This model enables us to carry out all analyses requested by the company in an efficient way. After this deterministic model is solved on the basis of a 9% annual increase in demand, it is extended to a minimax regret model to deal with uncertainty in demand quantities. The results suggest that opening the new plant in the city of İzmir is indeed a robust solution that is unaffected in different scenarios that are based on three distinct demand increase rates. Even though the location of the new plant remains unchanged with respect to scenarios, the optimal robust solution differs from the optimal solution of each scenario in terms of the capacity expansion decisions. After all obtained results had been communicated to the company managers and executives, the new plant construction was started in 2016 very close to the city that the mathematical model had determined. The new plant is expected to start operating in 2018.     

Single-machine scheduling with maintenance and repair rate-modifying activities

Most papers in the scheduling field are based on the assumption that machines are always available at constant speed. However, in industry applications, it is very common for a machine to be in subnormal condition after running for a certain period of time. Motivated by a problem commonly found in the surface-mount technology of electronic assembly lines, this paper deals with scheduling problems involving repair and maintenance rate-modifying activities. When a machine is running at less than an efficient speed, a production planner can decide to stop the machine and maintain it or wait and maintain it later. If the choice is made to continue running the machine without fixing it, it is possible that the machine will break down and repair will be required immediately. Both maintenance and repair activities can change the machine speed from a sub-normal production rate to a normal one. Hence, we call them rate-modifying activities. Our purpose here is to simultaneously sequence jobs and schedule maintenance activity to optimize regular performance measures. In this paper, we assume that processing time is deterministic, while machine break down is a random process following certain distributions. We consider two types of processing cases: resumable and nonresumable. We study problems with objective functions such as expected makespan, total expected completion time, maximum expected lateness, and expected maximum lateness, respectively. Several interesting results are obtained, especially for the nonresumable case.     

OPTIMAL R&D FOR NONRENEWABLE RESOURCES: THE CASE OF COST REDUCING TECHNOLOGY

A model is developed to determine optimal R&D spending and completion time when R&D results in lower extraction costs of a nonrenewable resource deposit. Examples of R&D projects for which the model is designed are in-situ leaching for mining and carbon dioxide injection in petroleum. The model is a combined R&D/nonrenewable resource model (CM). Results from the CM are compared to simulations of an R&D model which ignores the nonrenewable resource. The comparison demonstrates the importance of including resource parameters in the R&D spending model. The CM extends the literature by considering R&D spending and exhaustible natural resource production simultaneously. It demonstrates the importance of including the resource deposit when R&D affects the deposit. This is important because more accurate models of R&D will increase the profitability of the R&D projects.     

化工园区应急设施区间规划选址模型研究

应急设施选址受应急物资需求量的影响。为优化应急设施选址布局,提高突发事件应急处置能力,以化工园区突发事件为研究背景,对化工园区突发事故下应急设施选址进行研究。考虑到化工园区突发事件的随机性和复杂性、突发事件应急物资需求的不确定性等特点,以应急设施选址安全性最大、经济性和服务效益最好为目标,基于传统确定性应急设施选址模型,构建了不确定需求条件下化工园区应急设施选址区间规划数学模型。模型中应急物资需求量是一个区间值,通过引入区间规划理论和模糊理论对模型进行求解,不仅避免了不确定参数随机概率分布的波动率,而且也降低了模型求解过程中的不确定性。最后,以园区各企业潜在事故为工程背景进行实例分析,得到园区应急设施的布局方案。结果表明,模型的求解效果较好,可为园区应急设施选址决策提供参考依据。     

On the average performance of the adjustable RO and its use as an offline tool for multi-period production planning under uncertainty

Robust optimization (RO) is a distribution-free worst-case solution methodology designed for uncertain maximization problems via a max-min approach considering a bounded uncertainty set. It yields a feasible solution over this set with a guaranteed worst-case value. As opposed to a previous conception that RO is conservative based on optimal value analysis, we argue that in practice the uncertain parameters rarely take simultaneously the values of the worst-case scenario, and thus introduce a new performance measure based on simulated average values. To this end, we apply the adjustable RO (AARC) to a single new product multi-period production planning problem under an uncertain and bounded demand so as to maximize the total profit. The demand for the product is assumed to follow a typical life-cycle pattern, whose length is typically hard to anticipate. We suggest a novel approach to predict the production plan’s profitable cycle length, already at the outset of the planning horizon. The AARC is an offline method that is employed online and adjusted to past realizations of the demand by a linear decision rule (LDR). We compare it to an alternative offline method, aiming at maximum expected profit, applying the same LDR. Although the AARC maximizes the profit against a worst-case demand scenario, our empirical results show that the average performance of both methods is very similar. Further, AARC consistently guarantees a worst profit over the entire uncertainty set, and its model’s size is considerably smaller and thus exhibit superior performance.     

Behavior prediction of washing system in a paper industry using GA and fuzzy lambda–tau technique

Availability analysis has been an important issue in the design field of any Industrial system as the system structure has become more complicated. Also, the system availability is affected by many factors such as design, manufacturing, installation, etc., and so it may be extremely difficult to model, analyze and predict the failure behavior of the system. The purpose of this paper is to develop a new approach for computing various performance measures, namely reliability, availability, MTBF (mean time between failures), ENOF (expected number of failures), failure rate and repair time, for any industrial system. In the proposed approach, the failure rates and repair times of all constituent components are obtained using genetic algorithms and then various performance measures are computed using fuzzy lambda–tau methodology. Washing system, the major part of paper industry is the subject of study. The interactions among the working components are modeled using Petri nets. Failure and repair rates are represented using triangular fuzzy numbers as they allow expert opinion, linguistic variables, operating conditions, uncertainty and imprecision in reliability information to be incorporated into system model. Based on calculated reliability parameters, a structured framework has been developed that may help the maintenance engineers to analyze and predict the system behavior.