A Multi-Equipment Task Assignment Model for the Horizontal Stripe Pre-Cut Mining Method

Abstract

This paper proposes a multi-equipment task assignment model for the horizontal stripe pre-cut mining method to address the problem of cooperative scheduling operation of multi-equipment in underground metal mines under complex constraints. The model is constructed with multiple objectives, including operation time, operational efficiency, equipment utilization rate, and ore grade fluctuation by considering the constraints of time, space, equipment, and processes. The NSGA-III algorithm is used to obtain the solution. The effectiveness of the algorithm is tested based on the actual data from the Chambishi Copper Mine. The results show that the average equipment utilization rate is 51.25%, and the average ore output efficiency is 278.71 tons/hour. The NSGA-III algorithm can quickly generate the optimal multi-equipment task assignment solution. The solution reduces the interference of manual experience and theoretically improves the actual operation of the mine.

1. Introduction

With the rapid development of intelligent mining, underground metal mining equipment has begun to shift toward unmanned operation due to this method’s increased safety and efficiency over manned equipment [1,2]. The artificial intelligence for mining equipment is currently based on unmanned operation of only a single piece of equipment at a time, but mining succession and equipment synergy has yet to improve accordingly [3,4,5]. Therefore, in addition to developing intelligent mining equipment, the question of how to improve the overall production efficiency of underground mines using the collaborative operation of multiple pieces of equipment (multi-equipment) has become one of the urgent problems to be solved in the field of intelligent mine construction.

Underground mining involves a complex production system with various environments and processes that are dynamic, discrete, parallel, and time-bound. The primary goal of intelligent mining equipment scheduling is to coordinate the execution equipment of each process efficiently, reasonably arrange the operation site and operation time of the equipment, realize the parallel operation of multiple-sites and multi-equipment, improve the equipment utilization rate, ensure the close connection of each process, and ensure the operational safety.

In this paper, we analyze the horizontal stripe pre-cut mining method for scheduling, which considers (1) the constraints of inter-stripe, inter-block, inter-slice, and inter-process relationships; (2) uncertain operation and movement times; and (3) the effects of blasting and ventilation, ore output fluctuation, and operational continuity. With the multiple-objectives of total time, stripe average time, process interval, block interval, equipment utilization rate, and ore grade fluctuation, we construct a multi-equipment task assignment model and implement NSGA-III [6,7,8] as the solver. Finally, we demonstrate the intelligent scheduling and efficient operation of multi-equipment under complex collaborative operation conditions and demonstrate that our method greatly improves the overall production efficiency of underground metal mines.

The remainder of this paper is organized as follows: In Section 2, we review the underground mine scheduling research in recent years. In Section 3, we analyze the characteristics and processes of the horizontal stripe pre-cut mining method. In Section 4, we construct the multi-equipment task assignment model, and in Section 5, the case study of Chambishi Copper Mine is conducted and the results are discussed. Section 6 concludes.

2. Literature Review

The research methods for production scheduling optimization of underground mines include linear programming, mixed integer programming (MIP), constraint planning, and simulation, etc. [9,10,11,12,13,14]. This research dates back to the 1970s when Williams et al. [15] used linear programming methods to develop underground mine production plans. Trout et al. [16,17,18,19,20,21] proposed to develop underground mine production plans based on MIP.

Campeau et al. [22] proposed an MIP-based optimization model for the short-term planning of underground mines, and Newman et al. [23] focused on the problem of matching short-term plans with long-term plans. Martinez et al. [24] improved this model by introducing a heuristic algorithm to improve the speed of the solution, and Nehring et al. [25] and Little et al. [26] proposed integrated MIP-based optimization models for short-term and medium-term production planning. More recently, Campeau et al. [27] improved an integrated MIP-based optimization model for underground mines’ short-term and medium-term planning.

Regarding research with simulation, Chanda [28] proposed a production scheduling model that combined MIP and simulation, and Nehring et al. [29] proposed a new production scheduling optimization model based on MIP that significantly reduced the model-solving time. Along similar lines, O’Sullivan et al. [30] constructed a MIP production scheduling model and applied a heuristic decomposition method to solve the model, and Foroughi et al. [31] proposed a comprehensive model for integrated optimal mine distribution and production scheduling based on multi-objective integer programming (MOIP), which they solved using the NSGA algorithm. Finally, Lindh et al. [32] proposed a heuristic method based on logical Benders decomposition (LBBD) to solve a large-scale example of the short-term scheduling problem in underground mines.

They were not the first to take novel solution approaches a step further, however. Tsomondo et al. [33] proposed a scheduling model for underground metal mines that integrated short-term production scheduling and equipment assignment and used a fuzzy logic modeling approach to generate shift schedules. Nehring et al. [34] improved this model and proposed a new MIP-based dynamic short-term production scheduling and equipment assignment model, which they solved using the CPLEX tool. Similarly, Song et al. [35] developed a decision support tool for mobile mining equipment in underground mines, proposed algorithms for sorting, grouping, machine set, and machine sharing to assist miners in rapidly rescheduling trackless equipment, and validated the effectiveness of their algorithm with Kittila mine data.

Li et al. [36] constructed a production succession and dynamic scheduling model and solved it using an improved genetic algorithm to generate the optimal scheduling scheme. Improvements to this model were found by Wang et al. [37] who considered the constraints between processes and operation surfaces, the number of pieces of equipment, efficiency, and movement time to construct an underground trackless equipment scheduling model where they also solved the model using a genetic algorithm. The influence of equipment failure was also considered in order to improve the robustness of the model. Hou et al. [38] similarly proposed a dynamic scheduling optimization model for underground mining equipment based on an improved genetic algorithm by considering the uncertainty of equipment operation time wherein the optimal plan was immediately reworked once an unexpected event occurred.

Modeling the concept of real-world constraints explicitly, Åstrand et al. [39] proposed an underground mine equipment scheduling model based on constraint planning that considered production interruptions and delays and generated a scheduling scheme based on actual mine data. Then, they [40] extended the model to consider the effect of equipment movement time and proposed a constraint planning-based fixed-neighborhood-size simple neighborhood search strategy for the short-term scheduling of underground mines. Subsequently, they [41] improved on this and proposed a dynamically adjusted neighborhood size large neighborhood search strategy based on constraint planning for short-term underground mine operation scheduling. Finally, Hammami et al. [42] investigated the underground mine mobile equipment scheduling problem with random operation times for mining equipment.

In summary, much previous research has been conducted on underground mine scheduling. However, the consideration of mining process and spatial constraints in the existing research is still fixed and protocolized. Its application effect will thus be reduced in the face of complex spatial constraints, multi-processes, backfilling, and other real-world conditions. This paper focuses on the horizontal stripe pre-cut mining method based on previous research in order to build a multi-equipment task assignment model. Some complex conditions are considered, such as multiple-slice spatial constraints, complex combined processes, and equipment clustering operations, etc.

3. Analysis of the Horizontal Stripe Pre-Cut Mining Method

3.1. Mining Method Process Analysis

Underground mining is a production activity that consists of discrete or continuous operational processes that crush ore to a suitable size and transport it to the surface. Ore recovery is the most important and complex part of the process and is characterized by a harsh operation environment, complex process links, discrete operation sites, multi-equipment, and complex spatio-temporal restrictions. Therefore, the equipment task assignment under different mining methods is necessarily different.

The horizontal stripe pre-cut mining method is designed for underground metal mines with horizontal orebodies. This method divides the orebody into multiple areas at certain intervals along extending and dip directions. Each area is then divided into multi-stripes by fixed width. The stripes within an area are mined in adjacent groupings of three or more, and the void zone is backfilled to maintain the mechanical stability in the underground. Since these are independent mining units, the stripes are divided into blocks according to a fixed length. As shown in Figure 1, the green color indicates the first-step stripes, while the blue stripes can be further mined only after the completion of backfilling in the first-step stripes.

Each block can be divided into an upper slice and a lower slice. The upper slice is designed to create space for further blasting and is drilled horizontally, which requires scaling and anchor support for the exposed rock. The lower slice is designed to mass-produce ore with vertical deep drilling. There is a binding constraint between the upper and lower slices within the same stripe; the completion of the upper slice in all blocks is required before the start of any lower slice. As shown in Figure 2, the upper slice is mined forward while the lower slice is mined backward, completing the entire stripe block by block.

As the smallest mining units, the upper and lower slices of a block require a series of processes to crush the orebody to a reasonable size and haul it. These processes include drilling, charging, blasting and ventilation, scaling, mucking, bolting, backfilling, and maintenance. The drilling process uses drilling jumbos to drill holes in the orebody for filling with explosives. A horizontal drilling jumbo is used for the upper slice, and a downward drilling jumbo is used for the lower slice. The charging process uses a charging dolly to fill the holes with explosives and to lay detonator fuses and other facilities, and the blasting process is operated by professional blasting personnel. To ensure safety, the operation is usually carried out during a fixed time, and as extraneous operations are interrupted and suspended as extraneous personnel are withdrawn from the faces. Due to the dust and toxic gases that are present after blasting, ventilation is required for a certain period of time, and the ventilation time is related to the amount of explosives used. In addition, the destruction of the orebody caused by blasting usually leaves some unstable rocks on the surface of the stopes that can easily collapse and endanger the personnel.

The scaling process exists to clear the loose rock from the surface using scaling jumbos to ensure the safety of subsequent operations, and the mucking process is necessary in order to load the ore by LHD (Load-Haul-Dump) and haul it to the nearest ore pass. The LHD circulates between the stope and the ore pass until all the ore is delivered. Next, the bolting process installs anchors at the top of the stope so that the goaf can remain stable for a longer period, which provides a safe space for the subsequent block operations. Bolting only ensures the short-term stability of the goaf; backfilling fills the goaf with slurry to bring it into a stable state. This ensures the long-term stability of the rock mechanics while providing the necessary conditions for mining in the surrounding area. Each stripe is usually filled uniformly after the blocks within the stripe are completed, including backfilling and maintenance. Backfilling requires the preparation of a pipeline in order to move the slurry, and this is related to the stripe volume and filling capacity. The maintenance time is usually a fixed value, mainly determined by the solidification of the slurry, and is related to the slurry composition.

The process for the upper slice includes drilling, charging, blasting and ventilation, scaling, mucking, and bolting, and that of the lower slice includes drilling, charging, blasting and ventilation, and mucking. Each stripe includes multi-blocks with the upper and lower slices that are uniformly backfilled and maintained after all recovery. The overall process cycle is shown in Figure 3. Figure 3 integrates the process for the upper and lower slices. The left side of Figure 3 represents the process for the upper slice, and the middle of Figure 3 represents the process for the upper slice. After the upper and lower slices have been completed, backfilling and maintenance will be carried out, as shown on the right side of Figure 3.

3.2. Multi-Equipment Task Assignment Definition

The horizontal stripe pre-cut mining method has complex constraints for areas, stripes, blocks, and slices. The operation sites are discrete, and specific processes require specific mining equipment. Therefore, the multi-equipment task assignment problem for the horizontal stripe pre-cut mining can be transformed into a “multi-sites, multi-processes, multi-equipment” arrangement problem. This problem is essentially a flow shop scheduling problem. However, considerations for specific mining characteristics need to be incorporated to enable the flow shop modeling, detailed below.

Inter-stripe constraint: The stripes within an area are mined in adjacent groups of three or more, and the void zone is backfilled to maintain the mechanical stability under the ground. These stripes should be in an “unmined” or “backfilled and maintained” state before the start of the target stripe.

Inter-block constraint: The upper slice is mined forward, but the lower slice is mined backward, completing the entire stripe block by block. Only when all operations of the previous block are completed can the next block be started.

Inter-slice constraint: Completing the upper slice in all blocks is required before the start of any lower slice.

Inter-process constraint: Each mining unit requires the completion of several different processes, and the sequence of processes within each mining unit is not interchangeable.

Impact of equipment movement: Since the underground stopes are connected by narrow tunnels, the same equipment serving different mining units must take movement time into account, especially the movement of equipment between areas, which takes the most time.

Effects of blasting and ventilation: Blasting and ventilation affect other operations in the surrounding zone. When blasting and ventilation occur in the same area, other operations need to be interrupted and suspended. To improve the regularity of this process, mines stipulate that the operation occurs at the moment of shift changeover (for example, starting at 0:00, 8:00, and 16:00 each day).

The impact of ore output fluctuations: The purpose of mining is to produce ore. Since metal ore has different grades, it is important to both ensure stable ore output per shift and maintain stable ore grade.

Impact of operational continuity: To avoid the risk of rock mechanical instability due to long exposure time in the stope, operational continuity needs to be maximized. The process interval time within the same block should be kept as short as possible, and the block interval time within the same stripe should also be kept as short as possible.

Therefore, the multi-equipment task assignment problem for the horizontal stripe pre-cut mining method is described as follows: there are several areas, several stripes within each area, and several blocks within the stripes. Each block is divided into an upper slice and a lower slice that can together be deemed as an independent mining unit. Each mining unit needs to go through several nontransferable processes. Each process corresponds to several pieces of specific equipment, and this equipment can only work in one block simultaneously. There are different ore volumes, grades, and other parameters. After all mining units within the same stripe have been completed, they must be backfilled and maintained. The core of this problem is when to assign certain equipment to certain mining units to perform certain processes to achieve the expected production results.

4. Modeling and Solution Methodology

We now abstract the multi-equipment task assignment problem for the horizontal stripe pre-cut mining method into a mathematical model and solve it using the NSGA-III algorithm.

4.1. Notations

Table 1. Indices and sets.

Name	Meaning
A	Set of areas, A = {A1, A2…, Ai}, $\mathrm{i}\in \left[1,\mathrm{I}\right]$
Bi	Set of stripes, Bi = {Bi1, Bi2, …, Bik}, $\mathrm{i}\in \left[1,\mathrm{I}\right],\mathrm{k}\in \left[1,\mathrm{K}\right]$
Cik	Set of all blocks, ${\mathrm{C}}_{\mathrm{ik}}=\{\mathrm{C}{\mathrm{ik}}_1, \mathrm{C}{\mathrm{ik}}_2, \dots {\mathrm{C}}_{\mathrm{ikn}}\}, \mathrm{i}\in \left[1,\mathrm{I}\right],\mathrm{k}\in \left[1,\mathrm{K}\right],\mathrm{n}\in \left[1,\mathrm{E}\right](\mathrm{n}\in \left[1,\mathrm{N}\right]$ represents blocks of the upper slice, $\mathrm{n}\in \left[\mathrm{N}+1,\mathrm{E}\right]$ represents blocks of the lower slice)
D	Set of horizontal drilling jumbos, D = {D1, D2, …, Dh}, $\mathrm{h}\in \left[1,\mathrm{H}\right]$
Z	Set of downward drilling jumbos, Z = {Z1, Z2, …, Zr}, $\mathrm{r}\in \left[1,\mathrm{R}\right]$
C	Set of charging dollies, C = {C1, C2, …, Cu}, $\mathrm{u}\in \left[1,\mathrm{U}\right]$
S	Set of scaling jumbos, S = {S1, S2, …, Sx}, $\mathrm{x}\in \left[1,\mathrm{X}\right]$
L	Set of LHDs, L = {L1, L2, …, Ly}, $\mathrm{y}\in \left[1,\mathrm{Y}\right]$
B	Set of bolting jumbos, B = {B1, B2, …, Bw}, $\mathrm{w}\in \left[1,\mathrm{W}\right]$
F	Set of filling pipelines, F = {F1, F2, …, Fv}, $\mathrm{v}\in \left[1,\mathrm{V}\right]$
J	Set of processes, J = {J1, J2, …, Jm}, $\mathrm{m}\in \left[1,8\right]$, J1 = 1 (Drilling), J2 = 2 (Charging), J3 = 3 (Blasting and ventilation), J4 = 4 (Scaling), J5 = 5 (Mucking), J6 = 6 (Bolting), J7 = 7 (Backfilling), J8 = 8 (Maintenance)

,

Table 2. Parameters.

Name	Meaning
Qikn	Ore volume of Cikn, t, $\mathrm{n}\in \left[1,\mathrm{E}\right]$
G+	Upper limit of ore grade, g·t−1
G−	Lower limit of ore grade, g·t−1
Ph	Operational efficiency of Dh, m/h
Pr	Operational efficiency of Zr, m/h
Pu	Operational efficiency of Cu, m/h
Pw	Operational efficiency of Bw, anchors/h
Pv	Filling capacity of Fv, m3/h
Py	Ideal operational efficiency of Ly, t·m/h
${\mathsf{\rho }}_{\mathrm{ikny}}$	Operational efficiency discount factor of Ly in Cikn, %, $\mathrm{n}\in \left[1,\mathrm{E}\right]$
${\mathsf{\phi }}_{\mathrm{ikn}}$	Distance factor from Cikn to the corresponding ore pass, m, $\mathrm{n}\in \left[1,\mathrm{E}\right]$
$\mathsf{\beta }$	Ventilation factor, $\mathrm{n}\in \left[1,\mathrm{E}\right]$
${\mathrm{D}}_{\mathrm{i}}^{\mathrm{i}+1}$	Distance between adjacent areas, m
${\mathrm{D}}_{\mathrm{i},\mathrm{k}}^{\mathrm{i},{\mathrm{k}}^{\prime }}$	Distance between different stripes in Ai, m, ${\mathrm{k}}^{\prime }=\mathrm{k}\pm 4$
Vel	Walking speed of any equipment, m/h
qikn	Avalanche ore volume per meter of Cikn, t/m, $\mathrm{n}\in \left[1,\mathrm{E}\right]$
Vikn	Volume of Cikn, m3, $\mathrm{n}\in \left[1,\mathrm{E}\right]$
Gikn	Grade of Cikn, g·t−1, $\mathrm{n}\in \left[1,\mathrm{E}\right]$
Hikn	Height of Cikn, m, $\mathrm{n}\in \left[1,\mathrm{N}\right]$
Nikn	Number of anchors per square meter of Cikn, anchors/m2, $\mathrm{n}\in \left[1,\mathrm{N}\right]$
Tmt,ik	Maintenance time of Bik, h

and

Table 3. Variables.

Name	Meaning
Lpao,ikn	Total length of the blasting hole for Cikn, m, $\mathrm{n}\in \left[1,\mathrm{E}\right]$
Th,ikn	Operation time of Dh in Cikn, h, $\mathrm{n}\in \left[1,\mathrm{N}\right]$
Tr,ikn	Operation time of Zr in Cikn, h, $\mathrm{n}\in \left[\mathrm{N}+1,\mathrm{E}\right]$
Tu,ikn	Operation time of Cu in Cikn, h, $\mathrm{n}\in \left[1,\mathrm{E}\right]$
Tven,ikn	Blasting and ventilation time in Cikn, h, $\mathrm{n}\in \left[1,\mathrm{E}\right]$
Tx,ikn	Operation time of Sx in Cikn, h, $\mathrm{n}\in \left[1,\mathrm{N}\right]$
Ty,ikn	Operation time of Ly in Cikn, h, $\mathrm{n}\in \left[1,\mathrm{E}\right]$
Tw,ikn	Operation time of Bw in Cikn, h, $\mathrm{n}\in \left[1,\mathrm{N}\right]$
Tv,ik	Backfilling time of Fv in Bik, h
${\mathrm{T}}_{\mathrm{work}}$	Its value is equal to any of {Th,ikn, Tr,ikn, Tu,ikn, Tx,ikn, Ty,ikn, Tw,ikn, Tv,ik, Tmt,ik }
TSiknm	Starting time of jm in Cikn, h, $\mathrm{n}\in \left[1,\mathrm{E}\right]$, $\mathrm{m}\ne 7,8$
TSikm	Starting time of jm in Bik, h, $\mathrm{m}=7,8$
TEiknm	Ending time of jm in Cikn, h, $\mathrm{n}\in \left[1,\mathrm{E}\right]$, $\mathrm{m}\ne 7,8$
TEikm	Ending time of jm in Bik, h, $\mathrm{m}=7,8$
DTikn,m,m+1	The interval time between the ending time of jm and the starting time of jm+1 in Cikn, h, $\left[1,\mathrm{E}\right]$, $m\ne 7,8$
DTik,m,m+1	The interval time between the ending time of jm and the starting time of jm+1 in Bik, h, $\mathrm{m}=7$
${\mathrm{TM}}_{\mathrm{i},\mathrm{i}+1}$	Time of movement between Ai and Ai+1 with any equipment, h
${\mathrm{TM}}_{\mathrm{ik},{\mathrm{k}}^{\prime }}$	Time of movement between Bik and ${\mathrm{B}}_{{\mathrm{ik}}^{\prime }}$ with any equipment, h
${\mathrm{Bl}}_{\mathrm{siknm}}$	Number of blasting and ventilation for jm in Cikn, $\mathrm{n}\in \left[1,\mathrm{E}\right]$

respectively show the indices and sets, parameters, and variables of the model.

4.2. Mathematical Model

4.2.1. Objective Functions

(1) Minimum total time

$$\min \left({\mathrm{maxTE}}_{\mathrm{ikm}}-{\mathrm{minTS}}_{\mathrm{iknm}}\right)$$

(1)

(2) Minimum stripe average time

$$\min \frac{{{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{I}}{{\displaystyle \sum }}_{\mathrm{k}=1}^{\mathrm{K}}{\mathrm{TE}}_{\mathrm{ikm}}}{{{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{I}}{{\displaystyle \sum }}_{\mathrm{k}=1}^{\mathrm{K}}\mathrm{ik}},\mathrm{m}=8$$

(2)

(3) Minimum process interval

$$\min \left({\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{I}}{\displaystyle \sum }_{\mathrm{k}=1}^{\mathrm{K}}({\displaystyle \sum }_{\mathrm{n}=1}^{\mathrm{N}}{\displaystyle \sum }_{\mathrm{m}=1}^5{\mathrm{DT}}_{\mathrm{ikn},\mathrm{m},\mathrm{m}+1}+{\displaystyle \sum }_{\mathrm{n}=\mathrm{N}+1}^{\mathrm{E}}{\displaystyle \sum }_{\mathrm{m}=1}^2{\mathrm{DT}}_{\mathrm{ikn},\mathrm{m},\mathrm{m}+1}+{\displaystyle \sum }_{\mathrm{n}=\mathrm{N}+1}^{\mathrm{E}}{\displaystyle \sum }_{\mathrm{m}=3}^3{\mathrm{DT}}_{\mathrm{ikn},\mathrm{m},\mathrm{m}+2})\right)$$

(3)

(4) Minimum block interval

$$\min \left({\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{I}}{\displaystyle \sum }_{\mathrm{k}=1}^{\mathrm{K}}({\displaystyle \sum }_{\mathrm{n}=1}^{\mathrm{N}}({\displaystyle \sum }_{\mathrm{m}=1}^1{\mathrm{TS}}_{\mathrm{ik},\mathrm{n}+1,\mathrm{m}}-{\displaystyle \sum }_{\mathrm{m}=6}^6{\mathrm{TE}}_{\mathrm{ik},\mathrm{n},\mathrm{m}}\right)+{\displaystyle \sum }_{\mathrm{n}=\mathrm{N}+1}^{\mathrm{E}}({\displaystyle \sum }_{\mathrm{m}=1}^1{\mathrm{TS}}_{\mathrm{ik},\mathrm{n}+1,\mathrm{m}}-{\displaystyle \sum }_{\mathrm{m}=5}^5{\mathrm{TE}}_{\mathrm{ik},\mathrm{n},\mathrm{m}}))$$

(4)

(5) Maximize equipment utilization rate

$$\max \left(\frac{{\sum }_{\mathrm{i}=1}^{\mathrm{I}}{\sum }_{\mathrm{k}=1}^{\mathrm{K}}\left({\sum }_{\mathrm{n}=1}^{\mathrm{E}}({\mathrm{TE}}_{\mathrm{iknm}}-{\mathrm{TS}}_{\mathrm{iknm}}-{\mathrm{Bl}}_{\mathrm{siknm}}\times {\mathrm{T}}_{\mathrm{ven},\mathrm{ikn}}\right)+{\sum }_{\mathrm{i}=1}^{\mathrm{I}}{\mathrm{TM}}_{\mathrm{i},\mathrm{i}+1}+{\sum }_{\mathrm{k}=1}^{\mathrm{K}}{\mathrm{TM}}_{\mathrm{ik},{\mathrm{k}}^{\prime }})}{{\mathrm{maxTE}}_{\mathrm{iknm}}-{\mathrm{minTS}}_{\mathrm{iknm}}-{\sum }_{\mathrm{i}=1}^{\mathrm{I}}{\sum }_{\mathrm{k}=1}^{\mathrm{K}}{\mathrm{Bl}}_{\mathrm{siknm}}\times {\mathrm{T}}_{\mathrm{ven},\mathrm{ikn}}}\right),\mathrm{m}\ne 3,7,8$$

(5)

(6) Minimum ore grade fluctuation

The formula for the ore grade during a certain shift is as follows:

$${\mathrm{G}}_{{\mathrm{t}}_1}^{{\mathrm{t}}_2}=\frac{{\sum }_{\mathrm{i}=1}^{\mathrm{I}}{\sum }_{\mathrm{k}=1}^{\mathrm{K}}{\sum }_{\mathrm{n}=1}^{\mathrm{E}}({\mathrm{T}}_{{\mathrm{t}}_1}^{\mathrm{t}2}/{\mathrm{T}}_{\mathrm{y},\mathrm{ikn}})\times {\mathrm{Q}}_{\mathrm{ikn}}\times {\mathrm{G}}_{\mathrm{ikn}}}{{\sum }_{\mathrm{i}=1}^{\mathrm{I}}{\sum }_{\mathrm{k}=1}^{\mathrm{K}}{\sum }_{\mathrm{n}=1}^{\mathrm{E}}({\mathrm{T}}_{{\mathrm{t}}_1}^{\mathrm{t}2}/{\mathrm{T}}_{\mathrm{y},\mathrm{ikn}})\times {\mathrm{Q}}_{\mathrm{ikn}}}$$

(6)

Note: ${\mathrm{T}}_{{\mathrm{t}}_1}^{\mathrm{t}2}$ means the time of ore output during a certain shift

The formula for the ore grade during the total time is as follows:

$$\mathrm{G}=\frac{{\sum }_{\mathrm{i}=1}^{\mathrm{I}}{\sum }_{\mathrm{k}=1}^{\mathrm{K}}{\sum }_{\mathrm{n}=1}^{\mathrm{E}}{\mathrm{Q}}_{\mathrm{ikn}}\times {\mathrm{G}}_{\mathrm{ikn}}}{{\sum }_{\mathrm{i}=1}^{\mathrm{I}}{\sum }_{\mathrm{k}=1}^{\mathrm{K}}{\sum }_{\mathrm{n}=1}^{\mathrm{E}}{\mathrm{Q}}_{\mathrm{ikn}}}$$

(7)

The formula for shifts is as follows:

$$\mathrm{s}=\frac{{\mathrm{maxTE}}_{\mathrm{ikm}}-{\mathrm{minTS}}_{\mathrm{iknm}}}{8}$$

(8)

Ore grade fluctuation is expressed by variance, and the formula for the minimum ore grade fluctuation is as follows:

$$\min (\frac{{\sum }_{\mathrm{s}=1}^{\mathrm{s}}{({\mathrm{G}}_{{\mathrm{t}}_1}^{{\mathrm{t}}_2}-\mathrm{G})}^2}{\mathrm{s}})$$

(9)

4.2.2. Equivalent Relations

Equivalent Relation 1: The total length of the blasting hole is related to the ore volume and avalanche ore volume per meter.

$${\mathrm{L}}_{\mathrm{pao},\mathrm{ikn}}=\frac{{\mathrm{Q}}_{\mathrm{ikn}}}{{\mathrm{q}}_{\mathrm{ikn}}}$$

(10)

Equivalent Relation 2: The operation time of a horizontal drilling jumbo, downward drilling jumbo, and charging dolly is related to the total length of the blasting holes and the operational efficiency.

$${\mathrm{T}}_{\mathrm{h},\mathrm{ikn}}=\frac{{\mathrm{L}}_{\mathrm{pao},\mathrm{ikn}}}{{\mathrm{P}}_{\mathrm{h}}}=\frac{{\mathrm{Q}}_{\mathrm{ikn}}}{{\mathrm{q}}_{\mathrm{ikn}}\times {\mathrm{P}}_{\mathrm{h}}},\mathrm{n}\in \left[1,\mathrm{N}\right]$$

(11)

$${\mathrm{T}}_{\mathrm{r},\mathrm{ikn}}=\frac{{\mathrm{L}}_{\mathrm{pao},\mathrm{ikn}}}{{\mathrm{P}}_{\mathrm{r}}}=\frac{{\mathrm{Q}}_{\mathrm{ikn}}}{{\mathrm{q}}_{\mathrm{ikn}}\times {\mathrm{P}}_{\mathrm{r}}},\mathrm{n}\in \left[\mathrm{N}+1,\mathrm{E}\right]$$

(12)

$${\mathrm{T}}_{\mathrm{u},\mathrm{ikn}}=\frac{{\mathrm{L}}_{\mathrm{pao},\mathrm{ikn}}}{{\mathrm{P}}_{\mathrm{u}}}=\frac{{\mathrm{Q}}_{\mathrm{ikn}}}{{\mathrm{q}}_{\mathrm{ikn}}\times {\mathrm{P}}_{\mathrm{u}}}$$

(13)

Equivalent Relation 3: The blasting and ventilation time is positively correlated with the total length of the blasting holes and the constant factor β.

$${\mathrm{T}}_{\mathrm{ven},\mathrm{ikn}}=\mathsf{\beta }\times {\mathrm{L}}_{\mathrm{pao},\mathrm{ikn}}=\mathsf{\beta }\times \frac{{\mathrm{Q}}_{\mathrm{ikn}}}{{\mathrm{q}}_{\mathrm{ikn}}}$$

(14)

Equivalent Relation 4: The LHD operation time is related to the ore volume, distance factor, operational efficiency discount factor, and ideal operational efficiency.

$${\mathrm{T}}_{\mathrm{y},\mathrm{ikn}}=\frac{{\mathrm{Q}}_{\mathrm{ikn}}\times {\mathsf{\phi }}_{\mathrm{ikn}}}{{\mathsf{\rho }}_{\mathrm{ikny}}\times {\mathrm{P}}_{\mathrm{y}}}$$

(15)

Equivalent Relation 5: The operation time of the bolting jumbo is related to the volume and height of the block, the number of anchors per square meter, and the operational efficiency.

$${\mathrm{T}}_{\mathrm{w},\mathrm{ikn}}=\frac{{\mathrm{V}}_{\mathrm{ikn}}\times {\mathrm{N}}_{\mathrm{ikn}}}{{\mathrm{H}}_{\mathrm{ik}}\times {\mathrm{P}}_{\mathrm{w}}}$$

(16)

Equivalent Relation 6: The backfilling time of the filling pipeline is related to the total volume of blocks and the filling capacity.

$${\mathrm{T}}_{\mathrm{v},\mathrm{ik}}={\displaystyle \sum }_1^{\mathrm{E}}{\mathrm{V}}_{\mathrm{ikn}}/{\mathrm{P}}_{\mathrm{v}}$$

(17)

Equivalent Relation 7: The process interval is the interval time between the ending time of j_m and the starting time of j_m+1.

$${\mathrm{DT}}_{\mathrm{ikn},\mathrm{m},\mathrm{m}+1}={\mathrm{TS}}_{\mathrm{iknm}+1}-{\mathrm{TE}}_{\mathrm{iknm}}$$

(18)

Equivalent Relation 8: Any equipment movement time is related to distance and walking speed.

$${\mathrm{TM}}_{\mathrm{i},\mathrm{i}+1}=\frac{{\mathrm{D}}_{\mathrm{i}}^{\mathrm{i}+1}}{\mathrm{Vel}}$$

(19)

$${\mathrm{TM}}_{\mathrm{ik},{\mathrm{k}}^{\prime }}=\frac{{\mathrm{D}}_{\mathrm{i},\mathrm{k}}^{\mathrm{i},{\mathrm{k}}^{\prime }}}{\mathrm{Vel}}$$

(20)

4.2.3. Constraints

Constraint 1: Mining sequence constraint, there is a sequential order of mining blocks.

$${\mathrm{TS}}_{\mathrm{ikn}+11}\ge {\mathrm{TE}}_{\mathrm{ikn}6},\mathrm{n}\in \left[1,\mathrm{N}\right]$$

(21)

$${\mathrm{TS}}_{\mathrm{ikn}+11}\ge {\mathrm{TE}}_{\mathrm{ikn}5},\mathrm{n}\in \left[\mathrm{N}+1,\mathrm{E}\right]$$

(22)

Constraint 2: Process constraint, there is a sequence between processes.

$${\mathrm{TS}}_{\mathrm{iknm}+1}\ge {\mathrm{TE}}_{\mathrm{iknm}}$$

(23)

Constraint 3: All other processes are suspended until the blasting and ventilation have been completed.

$${\mathrm{TE}}_{\mathrm{iknm}}={\mathrm{TS}}_{\mathrm{iknm}}+{\mathrm{T}}_{\mathrm{work}}+{\mathrm{Bl}}_{\mathrm{siknm}}\times {\mathrm{T}}_{\mathrm{ven},\mathrm{ikn}},\mathrm{m}\ne 3$$

(24)

Constraint 4: The starting time of other processes lags behind the ending time of blasting and ventilation for the current operation cycle, and does not exceed the starting time of blasting and ventilation for the next operation cycle.

$${\mathrm{TE}}_{\mathrm{ikn},3}\le {\mathrm{TS}}_{\mathrm{iknm}}\le {\mathrm{TS}}_{\mathrm{ikn}+1,3},\mathrm{m}\ne 3$$

(25)

Constraint 5: The equipment movement constraint includes the movement constraint between adjacent areas and the movement constraint between stripes within an area.

$${\mathrm{TS}}_{\mathrm{i}+1\mathrm{knm}}\ge {\mathrm{TE}}_{\mathrm{iknm}}+{\mathrm{TM}}_{\mathrm{i},\mathrm{i}+1}$$

(26)

$${\mathrm{TS}}_{\mathrm{iknm}}\ge {\mathrm{TE}}_{{\mathrm{ik}}^{\prime }\mathrm{nm}}+{\mathrm{TM}}_{\mathrm{ik},{\mathrm{k}}^{\prime }}$$

(27)

Constraint 6: Stripe constraint, the stripes within an area are mined in adjacent groups of three or more.

$${\mathrm{TS}}_{\mathrm{iknm}}\ge {\mathrm{TE}}_{\mathrm{ik}\pm 3\mathrm{nm}}\cup {\mathrm{TE}}_{\mathrm{iknm}}\le {\mathrm{TS}}_{\mathrm{ik}\pm 3\mathrm{nm}}$$

(28)

$${\mathrm{TS}}_{\mathrm{ikm}}\ge {\mathrm{TE}}_{\mathrm{ik}\pm 3\mathrm{m}}\cup {\mathrm{TE}}_{\mathrm{ikm}}\le {\mathrm{TS}}_{\mathrm{ik}\pm 3\mathrm{m}}$$

(29)

$${\mathrm{TS}}_{\mathrm{iknm}}\ge {\mathrm{TE}}_{\mathrm{ik}\pm 2\mathrm{nm}}\cup {\mathrm{TE}}_{\mathrm{iknm}}\le {\mathrm{TS}}_{\mathrm{ik}\pm 2\mathrm{nm}}$$

(30)

$${\mathrm{TS}}_{\mathrm{ikm}}\ge {\mathrm{TE}}_{\mathrm{ik}\pm 2\mathrm{m}}\cup {\mathrm{TE}}_{\mathrm{ikm}}\le {\mathrm{TS}}_{\mathrm{ik}\pm 2\mathrm{m}}$$

(31)

$${\mathrm{TS}}_{\mathrm{iknm}}\ge {\mathrm{TE}}_{\mathrm{ik}\pm 1\mathrm{nm}}\cup {\mathrm{TE}}_{\mathrm{iknm}}\le {\mathrm{TS}}_{\mathrm{ik}\pm 1\mathrm{nm}}$$

(32)

$${\mathrm{TS}}_{\mathrm{ikm}}\ge {\mathrm{TE}}_{\mathrm{ik}\pm 1\mathrm{m}}\cup {\mathrm{TE}}_{\mathrm{ikm}}\le {\mathrm{TS}}_{\mathrm{ik}\pm 1\mathrm{m}}$$

(33)

Constraint 7: Ore grade constraint, there are an upper and lower bound for the ore grade during the total time.

$${\mathrm{G}}^{-}\le \mathrm{G}\le {\mathrm{G}}^{+}$$

(34)

4.3. Solution Method

The proposed model includes six objectives and seven constraints, some of which are nonlinear. Some constraints are also related to time deduction, such as blasting and ventilation windows, which make the problem difficult to solve directly by conventional mathematical programming methods. In this paper, we therefore use the simulation of the mining process based on the time-step method to derive the operational effect of the solution, and the optimization of the solution is carried out by a heuristic algorithm.

The mining process simulation is based on the time-step method. Starting from the system initialization at time 0, the system state is judged at every time t. According to the constraint relationship, the operational process is executed, and the operation state of the operation units and equipment is changed until all the operation units complete the operation.

The optimization of the multi-equipment task assignment solution is then performed iteratively using a genetic algorithm that gradually converges to the optimal solution. The NSGA-III algorithm, which is more suitable for high-dimensional objective optimization, was chosen because of the high objective dimensionality of the model (six objectives). Since the multi-equipment task assignment problem for mining operations has been transformed into a flow shop scheduling problem, we chose to implement it when the MSOS coding method is chosen [43].

5. Case Study

5.1. Datasets

The Chambishi Copper Mine of the China Africa Mining Co., Ltd. is located in the Copper Belt Province of the Republic of Zambia in south-central Africa. The mine has three mining areas. For this paper, we take the southeast mining area as a case study. The southeast mining area is located about 7 km southeast of the main mining area and has a large burial depth, ranging from 469.15 m to 1242.58 m. The boundary of this optimization is shown in Figure 4.

The zone consists of five areas, namely NO.1, NO.2, NO.4, NO.5, and NO.6, which are connected by flat tunnels and ramp roads. The areas are divided into stripes along the inclination with a width of 12.5 m, and each stripe is divided into multiple blocks along the strike with a length of 5 m. The blocks are divided into the upper and lower slices with an upper slice height of 4 m for pre-cutting and a lower slice height that varies with the thickness of the orebody. The main parameters of the stope are shown in

Table 4. Main parameters of the stope.

Name of Areas	Number of Stripes	Number of Ore Blocks	Total Quantities of Ores/Million Tons	Average Grade Cu	Average Grade Co
Area1	16	256	49.59	2.42	0.12
Area2	24	766	86.93	2.18	0.14
Area4	31	984	115.16	2.88	0.08
Area1	24	768	69.78	2.73	0.12
Area6	24	768	70.11	2.74	0.12
Total	119	3798	391.57	2.61	0.11

.

A Sandvik DD422I jumbo is used for the upper slice, and a Sandvik DL421 is used for the lower slice. The emulsion explosive is loaded by charging dolly, and blasting operations are uniformly started at 0:00, 08:00, and 16:00 each day, with ventilation timed according to the charging volume. During the blasts, other operations are suspended in the area. A scaling jumbo is used for scaling, and a Sandvik AutoLH514 LHD is used to load and unload ore in multiple cycles between the stope and the ore pass. A D3411 machine with plus pipe slit anchors is used for bolting, and paste-filling technology is used for backfilling from the surface backfilling station to the stripe void zone via pipelines. After the backfilling is completed, it is left to rest for 28 days for maintenance. The main technical parameters of the mining equipment are shown in

Table 5. Main technical parameters of mining equipment.

Category	Operation Content	Operational Capacity	Average Movement Speed	Number of Equipment
Horizontal drilling jumbo	Drilling horizontal holes	40 m/h	1000 m/h	4
Downward drilling jumbo	Drilling vertical holes	30 m/h	1000 m/h	2
Charging dolly	Drilling holes for charging	90 m/h	1000 m/h	3
Scaling jumbo	Roof scaling	10 m2/h	1000 m/h	1
LHD	Ore mucking	1500 t·m/h	1000 m/h	4
Bolting jumbo	Anchor support	10 anchors/h	1000 m/h	3

.

5.2. Result

To find solve the constrained optimization of the model, the parameters and equipment were as follows. The population of the NSGA III was 800, the evolutionary generation was 400, and the computing platform was an Intel(R) Core(TM) i7-8550U CPU, with 16GB RAM, running Python 3.7 @ Anaconda 3. We took each single objective optimization for comparison. The results are shown in

Table 6. The results of single objective optimization and multi-objective optimization.

Name		Total Time (Days)	Stripe Average Time (Days)	Process Interval (Days)	Block Interval (Days)	Equipment Utilization Rate (%)	Ore Grade Fluctuation (Dimensionless)
Multi-objects		587.67	111.15	2914.54	525.75	51.25	0.06
Single object	Total time	549.43	110.06	3620.48	751.81	49.62	0.34
	Stripe average time	567.91	109.59	3561.94	519.07	50.24	0.11
	Process interval	600.61	111.94	2734.91	571.94	46.19	0.24
	Block interval	626.16	124.09	3381.57	492.44	47.94	0.19
	Equipment utilization rate	591.61	116.80	3519.07	749.51	53.46	0.24
	Ore grade fluctuation	604.58	121.09	3705.06	934.83	47.24	0.04

.

Specifically, we found that the overall completion time of operational progress would be 587.67 d, and the average time for completing each area would be 570.30 d, with Area 1 taking the longest time. The Gantt chart for each area is shown in Figure 5.

The average completion time for each stripe would be 111.15 d. The Gantt chart of operational progress and the continuity of the stripes are shown in Figure 6. From the figure, we can see that the stripes are mined in groups of three or more that are adjacent to each other, and multiple stripes are operating simultaneously in each area.

We also performed further analysis of the block-to-process continuity within the stripe performed. Due to a large amount of data, the Gantt chart is drawn for the first 3000 h of the Area 1 only, as an example, in Figure 7. Here, we can see the inter-process continuity, inter-block continuity, and continuity of the upper and lower slices within the stripe. The average interval time in each stripe is 28.91 d, accounting for 26.01% of the average completion time of the stripe, which ensures a good succession of mining operations.

We also further analyzed the equipment utilization rate statistically, as shown in

Table 7. Equipment utilization rate.

Category	Equipment Utilization Rate
	Average	Maximum	Minimum
Horizontal drilling jumbo	44.27	42.91	45.73
Downward drilling jumbo	48.43	47.95	49.54
Charging dolly	51.43	49.29	53.08
Scaling jumbo	45.05	45.05	45.05
LHD	63.84	65.28	61.34
Bolting jumbo	46.74	47.67	45.33
Average	51.25	65.28	45.05

. The individual equipment is also taken as the vertical axis to draw the Gantt chart for equipment operational succession in Figure 8. Due to the problem of data volume, only a partial diagram is presented. The 3000–5000 h working life of some jumbos was selected as an example for the figure. The figure shows the operation time and movement time of different equipment, where the white part indicates that the equipment is idle. During this period, equipment maintenance and fuel replenishment can be carried out.

Statistical analysis was likewise conducted on the ore output for each time interval to obtain ore output fluctuation, as shown in Figure 9. The average ore output efficiency was 278.71 tons/hour, and the dispersion coefficient of the ore output efficiency fluctuation was 0.33, which is considered a low level.

Finally, we statistical analyzed the ore output grade to obtain ore grade fluctuation, as shown in Figure 10. The average ore grade was 3.19%, and the dispersion coefficient of grade fluctuation was 0.06, which is considered to be a good level that can supply a consistent grade for the concentrator.

5.3. Discussion

In this paper, we built a multi-equipment task assignment model for operation duration, operational efficiency, equipment utilization rate, and grade fluctuation in order to address the multi-equipment task assignment problem for the horizontal stripe pre-cut mining method. Mines usually assign specific equipment to designated areas for cyclic operation until all tasks are completed prior to changing areas. Although this approach is convenient for mine management, with the massive investment in intelligent, unmanned equipment, multi-equipment needs to be suitable for more flexible assigning. Recent papers have recognized this and conducted studies to optimize the combination of multiple areas and multi-equipment. However, the consideration of the mining process and spatial constraints in the existing research is fixed and protocolized, and its applicability is reduced in the face of complex spatial constraints, multiple processes, backfilling, and other conditions that are common in underground metal mines.

This paper makes improvements to the existing literature for these specific mining conditions. By considering the constraints of time, space, equipment, and processes, the operability of the solution is ensured, and conflicts during the execution process that often occur during manual preparation are completely avoided. Through multiple-objective programming, the six most important objectives were selected for integrated optimization, which means that the results not only take into account ore volume, grade fluctuations, and equipment utilization but also ensure operational efficiency, and achievement of comprehensive optimization decisions in multiple objective dimensions. The case study at the Chambishi Copper Mine shows the effectiveness and reliability of our algorithm. We presented the optimization results in Gantt charts with multiple dimensions so as to guide the actual production process visually and effectively. The optimal assigning solution was obtained by computer optimization, and improved production results to a certain extent, reduced the interference of manual experience, and made the solution more consistent with the actual operational process.

However, some limitations of the optimization model must be acknowledged. To ensure a high succession of stripes, blocks, and processes, the equipment utilization rate is sacrificed to some extent in the optimal solution. In the future, the number of various types of equipment can be adjusted, and the priority relationship between multiple objectives can be adjusted to make the optimization results more ideal from a utilization standpoint. In addition, the operating environment of an underground metal mine is complex and harsh, and it is easy to experience emergencies, such as equipment failure, roof collapse, and ore pass blockage. Future improvements can be made by making the tasks dynamic, which is more in line with actual production.

6. Conclusions

We proposed a multi-equipment task assignment model for the horizontal stripe pre-cut mining method in order to solve the problem of collaborative scheduling operations for multi-equipment under complex time-space constraints. By considering the constraints of time, space, equipment, and processes, the conflict problem in the operational process was solved in a way that is more in line with real world mining operations compared to the existing literature. Moreover, multiple objectives, such as operation time, operational efficiency, equipment utilization rate, and ore grade fluctuation, were selected for integrated optimization, which solves the problem of integrated decision-making when facing multiple conflicting objectives. Most importantly, optimization experiments on actual mine data demonstrated the effectiveness of our algorithm. However, the model constructed in this paper still has limitations, as discussed above. We therefore intend to focus our future research efforts on the following aspects: (1) optimally adjusting the priority relationship between multiple objectives to improve the equipment utilization rate; and (2) considering emergency situations such as equipment failure, roof collapse, and ore pass blockage to improve the dynamics of operational tasks.