Underground mine scheduling of mobile machines using Constraint Programming and Large Neighborhood Search
Introduction
Minerals are increasingly important in today’s modern society. In nature, minerals are found in ore deposits, sometimes located several kilometers under the surface. To extract this ore safely and profitably, mining machines are increasingly automated. In addition, many modern mines are now using mathematical programming to create optimized long-term plans. However, transforming the long-term plan into activities to be performed by these increasingly automated mining machines is still most often a manual process; unfortunately both error-prone and time-consuming.
The missing gap is short-term mine scheduling, the process of answering what to do where and when. In recent years, a few works on optimizing and automating the underground mine scheduling process have been published (e.g. Nehring et al., 2012, O’Sullivan and Newman, 2015, Song et al., 2015, Schulze et al., 2016, Åstrand et al., 2018). The proposed methods are mostly based on Mixed Integer Linear Programming (MILP) or greedy construction procedures. Another efficient method for solving scheduling problems is Constraint Programming (CP) which has proven its worth by closing many open job-shop instances (Vilím et al., 2015). Recently, Laborie (2018) compared a variety of methods (CP, MILP, SAT-modulo theory) on a wide range of scheduling problems. In this comparison, CP outperformed the other approaches thanks to a mix of constraint inference, automatic search and Large Neighborhood Search (LNS). Baptiste et al. (2006) argue that the reason why CP is successful for scheduling problems is that it combines the preciseness of methods from operations research, by heavily exploiting the problem structure, with the focus on generality from research on artificial intelligence.
In this paper, we contribute to the scarce literature on underground mine scheduling, building on Åstrand et al. (2018), where CP was used for the first time to solve an underground fleet scheduling problem. An issue that is often overlooked in previous works is that underground mines can have road networks spanning several hundreds of kilometers. This makes it important to account for travel times of the mobile machines to ensure that a feasible schedule is created. Therefore, we improve on previous work by extending the constraint model in Åstrand et al. (2018) to include travel times of the entire mobile fleet. Additionally, to scale to larger problem instances, we introduce a novel way of modeling the scheduling problem based on solving a related, but modified, scheduling problem and transforming the solution back to the original problem domain. Finally, we introduce a domain-specific neighborhood definition that is used in LNS to improve the quality of the mine schedules. This is a significant contribution over many construction procedures in the literature which typically do not support explicit optimization of performance objectives. Lastly, we evaluate the proposed methods on numerous instances derived from real-life data from an operational mine in Sweden.
The outline of this work is as follows; the underground mining process is introduced together with the particular mine scheduling problem in Section 2. The following section summarizes related research as to position this work. In Section 4 we introduce the central concepts of CP, while in Section 5 and Section 6 the main contributions are presented by the proposed CP and LNS methods together with a numerical evaluation of these. Lastly, conclusions are drawn in Section 7.
Section snippets
Underground mine scheduling
There are two ways of accessing an ore-body in order to retrieve material: either from the surface in an open pit mine, or from beneath the surface in an underground mine. Underground mine operations are planned in different levels of granularity. Fig. 1 depicts the different planning functions of the particular mine we study in this work. The life-of-mine plan has the longest horizon and considers the entire remainder of the estimated time of operation. The life-of-mine plan balances the
Related work
In this section, we will first review short-term underground mine scheduling. Second, we will introduce some related works from the open-pit mining industry. Lastly, we will review previous works on scheduling using CP.
Although rare, there are some works on short-term scheduling in underground mining. Most notably, Schulze et al. (2016) study how to schedule the mobile production fleet in an underground potash mine with the objective of minimizing makespan. This is done by formulating a MILP
Constraint Programming
Constraint Programming (CP) is an exact method for combinatorial optimization that has been successful in diverse areas such as planning, scheduling and vehicle routing (Baptiste et al., 2012). CP solves Constraint Satisfaction Problems (CSPs) defined by a set of n decision variables each having a corresponding domain of allowed values. Further, a set of m constraints defines valid combinations of the values that individual variables can take. A solution to a CSP is an
Modeling
In this section, we will first show how the original CP model proposed in Åstrand et al. (2018) can be adapted to include machine travel times. We will then introduce a new model, in which blast times are compressed and the obtained solutions are post-processed back to the original time domain. Lastly, we will introduce a neighborhood definition that can be used together with Large Neighborhood Search to improve the constructed schedules.
The scheduling problem is to take all unscheduled
Result and discussion
We evaluate the performance of the methods introduced in Section 5 by solving instances generated from real data. To ease presentation, the instances are encoded as e.g. 3F:2C:2M which should be interpreted as an instance with 3 parallel Faces, 2 full production Cycles to be scheduled on each face, and 2 Machines of each type. Since the production cycle contains 11 steps, this means that the instance 3F:2C:2M contains 3 faces 2 cycles per face 11 activities per cycle = 66 activities in
Conclusion
As increasingly advanced methods are used to create long-term plans and the actual extraction is being progressively automated, mapping the long-term plan into activities by short-term scheduling is becoming a limiting segment in the mining optimization chain. In this paper we have contributed to the literature on underground mine scheduling by introducing two new models. One model extends previous work by including travel times of the mobile machines. The other model is a novel approach based
CRediT authorship contribution statement
Max Åstrand: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Software, Validation, Visualization, Writing - original draft, Writing - review & editing. Mikael Johansson: Conceptualization, Methodology, Software, Supervision, Validation, Writing - original draft, Writing - review & editing. Alessandro Zanarini: Methodology, Software, Supervision, Validation, Writing - original draft.
Acknowledgement
This work was partially supported by the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation. We would also like to thank the Swedish mining company Boliden for their collaboration.
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