Alexandra M. Newman

A Review of Operations Research in Mine Planning

Applications of operations research to mine planning date back to the 1960s. Since that time, optimization and simulation, in particular, have been applied to both surface and underground mine planning problems, including mine design, long-and short-term production scheduling, equipment selection, and dispatching, inter alia. In this paper, we review several decades of such literature with a particular emphasis on more recent work, suggestions for emerging areas, and highlights of successful industry applications.

Scheduling Direct and Indirect Trains and Containers in an Intermodal Setting

The focus of our research is on rail transportation of intermodal containers. We address the problem of determining day-of-week schedules for both direct and indirect (via a hub) trains and allocating containers to these trains for the rail (linehaul) portion of the intermodal trip. The goal is to minimize operating costs, including a fixed charge for each train, variable transportation and handling costs for each container and yard storage costs, while meeting on-time delivery requirements. We formulate the problem as an integer program and develop a novel decomposition procedure to find near-optimal solutions. We also develop a method to provide relatively tight bounds on the objective function values. Finally, we compare our solutions to those obtained with heuristics designed to mimic current operations, and show that a savings of between 5 and 20% can be gained from using our solution procedure.

Using aggregation to optimize long-term production planning at an underground mine

Motivated by an underground mining operation at Kiruna, Sweden, we formulate a mixed integer program to schedule iron ore production over multiple time periods. Our optimization model determines an operationally feasible ore extraction sequence that minimizes deviations from planned production quantities. The number of binary decision variables in our model is large enough that directly solving the full, detailed problem for a three year time horizon requires hours, or even days. We therefore design a heuristic based on solving a smaller, more tractable, model in which we aggregate time periods, and then solving the original model using information gained from the aggregated model. We compute a bound on the worst case performance of this heuristic and demonstrate empirically that this procedure produces good quality solutions while substantially reducing computation time for problem instances from the Kiruna mine.

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.

Implementing a Production Schedule at LKAB's Kiruna Mine

LKABs Kiruna Mine, located in northern Sweden, produces about 24 million tons of iron ore yearly using an underground mining method known as sublevel caving. To efficiently run the mills that process the iron ore, the mine must deliver planned quantities of three ore types. We used mixed-integer programming to schedule Kirunas operations, specifically, which production blocks to mine and when to mine them to minimize deviations from monthly planned production quantities while adhering to operational restrictions. These production schedules save costs compared to schedules produced manually by meeting desired production quantities more closely and reducing employee time spent on preparing schedules.

Operations research in the natural resource industry

Operations research is becoming increasingly prevalent in the natural resource sector, specifically in agriculture, fisheries, forestry and mining. While there are similar research questions in these areas, e.g., how to harvest and/or extract the resources and how to account for environmental impacts, there are also differences, e.g., the length of time associated with a growth and harvesting or extraction cycle, and whether or not the resource is renewable. Research in all four areas is at different levels of advancement in terms of the methodology currently developed and the acceptance of implementable plans and policies. In this paper, we review the most recent and seminal work in all four areas, considering modeling, algorithmic developments and application.

Scheduling Trains and Containers with Due Dates and Dynamic Arrivals

We consider the problem of scheduling trains and containers (or trucks and pallets) between a depot and a destination. Goods arrive at the depot dynamically over time and have distinct due dates at the destination. There is a fixed-charge transportation cost for each vehicle, and each vehicle has the same capacity. The cost of holding goods may differ between the depot and the destination. The goal is to minimize the sum of transportation and holding costs.For the case in which all goods have the same holding costs, we consider two variations: one in which the holding cost at the destination is less than that at the origin, and one in which the relationship is reversed. For the first variation, we derive properties of the optimal solution which provide the basis for anO( T2) solution procedure. For the second variation, we introduce a new definition of a regeneration state, derive strong characterizations of the shipment schedule within a regeneration interval, and develop anO( T4) procedure.We also analyze two multi-item scenarios. In the first, for each item, the holding cost at the origin is less than that at the destination; in the second, the relationship is reversed for all items. We generalize several of the structural results for the single-item problem to the corresponding multi-item case. We also show that the optimal vehicle schedule can be obtained by solving a related single-item problem in which the item demands are aggregated in a particular way. The optimal assignment of customer orders to vehicles can then be found by solving a linear program.

Centralized and Decentralized Train Scheduling for Intermodal Operations

Abstract We investigate a spectrum of decision-making approaches, from centralized to decentralized, within the context of scheduling direct and indirect (via a hub) trains and assigning containers to trains for the rail (linehaul) portion of the intermodal trip. The goal is to minimize operating costs, including a fixed charge for each train, variable transportation and handling costs for each container and yard storage costs, while meeting on-time delivery requirements. If shipping requirements are known, a centralized solution provides for better coordination, thereby reducing costs. However, information may not be available to support centralized decision-making. We present several methods for obtaining good solutions, and show that carefully-designed decentralized approaches may perform as well as centralized approaches for our problem.

Tailored Lagrangian Relaxation for the open pit block sequencing problem

A common strategic and tactical decision in open pit mining is to determine the sequence of extraction for notional three-dimensional production blocks so as to maximize the net present value of the extracted orebody while adhering to precedence and resource constraints. This problem is commonly formulated as an integer program with a binary variable representing if and when a block is extracted. In practical applications, the number of blocks can be large and the time horizon can be long, and therefore, instances of this NP-hard precedence-constrained knapsack problem can be difficult to solve using an exact approach. The problem is even more challenging to solve when it includes explicit minimum resource constraints. We employ three methodologies to reduce solution times: (i) we eliminate variables which must assume a value of 0 in the optimal solution; (ii) we use heuristics to generate an initial integer feasible solution for use by the branch-and-bound algorithm; and (iii) we employ Lagrangian relaxation, using information obtained while generating the initial solution to select a dualization scheme for the resource constraints. The combination of these techniques allows us to determine near-optimal solutions more quickly than solving the monolith, i.e., the original problem. We demonstrate our techniques to solve instances containing 25,000 blocks and 10 time periods, and 10,000 blocks and 15 time periods, to near-optimality.

Optimizing Military Capital Planning

Planning United States military procurement commits a significant portion of our nations wealth and determines our ability to defend ourselves, our allies, and our principles over the long term. Our military pioneered and has long used mathematical optimization to unravel the distinguishing complexities of military capital planning. The succession of mathematical optimization models we present exhibits increasingly detailed features; such embellishments are always needed for real-world, long-term procurement decision models. Two case studies illustrate practical modeling tricks that are useful in helping decision makers decide how to spend about a trillion dollars.