Short-term planning of a work shift for open-pit mines: A case study

Abstract

This work deals with the short-term planning problem of a work shift for open-pit mines. The problem involves ore and waste fronts, shovels, heterogeneous truck fleets, and discharge points. The allocation of trucks is dynamic to allow multiple routes to be assigned to each truck. The problem consists of deciding which fronts must be mined and establishing the number of trucks, their routes, and the amount of material transported by them to each discharge point, satisfying a stripping ratio at the desired level. The objectives are to minimize the deviations from the targets for production, chemical grade, and particle size range of each control parameter at each plant and reduce the number of trucks needed for the process. To solve the problem, we developed a mixed-integer linear goal programming model and tested it using real data from an iron ore mine. The results showed that the proposed approach supports decision-makers in the sizing and allocation of truck fleets and in meeting the production and control parameter targets required by the ore processing plants according to the daily scenario, such as low availability of shovels and trucks, flexibility in ore quality, and need for increased production.

Keywords:

• Short-term planning
• open-pit mining
• blending
• truck allocation
• mixed-integer linear programming
• goal programming

1. Introduction

After the last commodities supercycle of iron ore, mining companies started to have as their main objective the fulfillment of ore quality goals and reduction of the variability of certain control parameters (such as the grade of iron, silica, manganese, and particle size range) without detriment to the production volume. Due to the large supply of iron ore, the market began to demand that more and more companies offer a product that adhered to customers’ specifications.

A product with greater added value can be obtained by generating a material within the specifications. Quality is essential in production since volume alone is not a guarantee of profit and cash generation for companies.

On the other hand, failure to meet the required quality, in some cases, directly impacts the reduction in production volume, with a consequent increase in operating costs. Reducing operating costs, with the maintenance of productivity and efficiency of production processes, becomes a challenge for the mining industries.

The literature addresses three types of problems on the short-term horizon. The first deals with the production scheduling of an annual or monthly plan broken into weeks or days (Blom et al., 2019; Eivazy & Askari-Nasab, 2012; Kozan & Liu, 2018). The second deals with shift production planning, also known as the upper stage of the multi-stage method of the dispatch problem (Martins & Souza, 2023; Souza et al., 2010). In this second type, the planner must decide on the amount of ore and waste extracted from the mining fronts before the beginning of each shift, according to the available resources. The third type refers to the instantaneous decisions that must be made to choose the route each truck should take after each unloading or loading activity, known as the lower stage of the multi-stage method of the dispatch problem.

This paper deals with short-term planning for open-pit mines involving the last two types of short-term problems. The objective is to perform a more adherent planning to the allocation and route decisions made in the dispatch problem. In this problem, it is necessary to meet several indicators in order to have a quality schedule, for example: $\mathit{i}$) Meeting the targets for production, chemical grade, and particle size range of each control parameter at each plant; $\mathit{ii}$) Reduction in the number of trucks in operation; $\mathit{iii}$) Considering the cycle time of the trucks, that is, the time taken for a truck to travel between the loading and discharge points, and $\mathit{iv}$) Compliance with the stripping ratio so that this indicator adheres to strategic planning. The use only of the operator’s experience in searching for a mining schedule that is efficient, agile, and capable of meeting all these indicators has proven to be not enough.

This article presents a goal programming formulation capable of generating good solutions to solve this problem considering a work shift. The main contributions of this work are as follows:

(1) The characterization of the problem under study with multiple loading and discharge points, a heterogeneous fleet of trucks, dynamic truck allocation, ore blending, determination of the number of trucks and their routes, and compliance with the stripping ratio;

(2) A new mixed-integer linear goal programming formulation for solving it;

(3) Approach validation using a case study involving two iron ore mines;

(4) Analysis of different production scenarios regarding ore’s control parameters, production, and the number of trucks.

The rest of this work is organized as follows. In Section 2, the problem is characterized. In Section 3, a literature review is provided. Section 4 presents the goal programming formulation proposed to solve it. Section 5 reports the results of the computational experiments. Section 6 concludes this work and presents perspectives for future work.

2. Problem statement

The problem under study deals with the operational planning problem of an open-pit mine of a work shift. The planning consists of selecting the mining fronts used to extract material and determining the number of trips for each truck to its destination. The objective is to build a short-term plan that meets the targets for chemical grades, particle size ranges, and production required by the processing plants and minimizes the number of trucks in operation.

Figure 1 illustrates the trips of a truck in a scenario. As can be seen in Figure 1, it consists of a set of mining fronts, loading equipment, and trucks. Each truck, loaded by a shovel, transports the material extracted from the mining front to its unloading point. The truck’s allocation is dynamic, i.g. in the same figure, the truck leaves Front 1 towards Ore discharge point 1. Then, it goes to Front 2. Finally, it concludes its last trip in the Ore discharge point 3.

Figure 1. Illustration of the operational planning problem.

The mining fronts differ from each other by the ore’s control parameters, such as the percentage of certain elements, %Fe and %SiO, for example. Thus, to meet the composition required in the monthly mining plan, it is necessary to select the mining fronts that will be used.

The monthly mining plan consists of a geometry that considers several operational parameters for its preparation. This plan makes available the mining fronts that will compose the production shift schedule. The purpose of the mining plan is to direct the operational teams (mine operation, topography, quality control) to the fronts that can be mined. Currently, the weekly and daily schedule is prepared by a mining technician, who informs which fronts are available in the period and verifies the quality of ore present and the historical productivity of the existing equipment. To evaluate the execution of the monthly plan, the topography team performs the mine topographic measurements at the end of the last day of the planned month. With the results of these measurements, adherence and reconciliation indicators are obtained. The first indicator reports the proportion between the planned mass mined and the total mass extracted. This second indicator compares the predicted and actual masses and qualities. The technician responsible for quality control performs a visual assessment of the fronts and calculates the amount of material each shovel must extract based on his experience in the mine and through trial and error. However, given the complexity of the problem, the production goal, the control parameter targets, and the minimum stripping ratio (WOR, Waste/Ore Ratio) are not always met.

For the mining industry, WOR is considered one important indicator of the mine. The extraction of waste rock aims to guarantee the opening of new ore mining fronts and, thus, to advance the mine’s development. Failure to comply with the WOR may impact the supply of ore quickly with the loss of volume, quality, and, consequently, loss of market due to non-compliance with product specifications.

On the other hand, the mining company that operates in this mine uses shovels to exploit ore from the mining fronts. Each shovel is associated with minimum and maximum production rates. The maximum production rate is determined by the equipment’s maximum production capacity, while the minimum production rate is imposed to justify the shovel’s use economically feasible. The maximum production rate of each shovel is estimated based on the monthly mining plan, which considers the available mining fronts, the time spent with displacements between the mining fronts, stoppages for fuel supply, and work shift change of shovel operators, among others.

To determine each truck’s number of trips to each front, it is necessary to consider which shovel is assigned to each mining front, the truck availability, cycle times, and the ore production goal. The shovel assigned to the front must be operationally compatible with the trucks designed for it since there may be trucks and shovels of different sizes, preventing, for example, a small shovel from carrying a large truck.

This work considers the existence of a heterogeneous fleet of trucks, i.e., they have different payloads. Besides, the dynamic allocation method is adopted, which means that the same truck can be allocated to different mining fronts after each material discharge. This technique decreases the queuing time and increases the utilization rate of the trucks.

3. Literature review

This section presents a bibliographic review of the short-term planning for open-pit mining problems and associated problems.

To supply ore of uniform quality to the processing plants, it is necessary to blend ore of different qualities from several mining fronts. The objective is to ensure the uniformity of feed as changes are usually accompanied by an increase in the total cost of the operation (Alarie & Gamache, 2002). According to these authors, material transport is one of the essential activities in the operation of open-pit mines due to its high cost. Thus, minimizing the transport cost from the mine to the plant is essential.

In Souza et al. (2010), the authors treated a short-term open-pit mining problem considering only a single point of loading and discharge. They proposed a mixed-integer linear programming formulation and a General Variable Neighborhood Search-based heuristic algorithm. The model generates one-hour production plans, updated when a front is exhausted or when the mine’s operating conditions change. One of the model’s limitations is the short planning time horizon of only one hour.

Subtil et al. (2011) addressed the problem through a multi-stage approach. This approach is the same as that implemented in the MineOperate ® commercial package (formerly known as SmartMine ®), currently marketed by Hexagon Mining ${ }^{\mathit{TM}}$. The first stage defines the ideal number of trucks that maximizes production through a linear programming model, which considers the mine’s operational restrictions. The second stage uses a dynamic truck dispatch system through computer simulation techniques combined with heuristics and multicriteria optimization. The authors validated this approach using a computer simulation model of discrete events. The results show that it is possible to increase the load and transport equipment’s productivity through this approach. However, their study does not clarify how to address the metal grades of the processing plants.

Zhang and Xia (2015) formulated the problem through an integer programming model to achieve the production goal with a minimum operating cost but without considering the targets for chemical grades. Still, the authors determined the ideal size of the homogeneous fleet of vehicles by taking advantage of the proposed model’s result. Through the experiments, the authors’ approach was capable of saving 15.65% of the truck operational cost concerning the studied iron mine’s current truck allocation strategy. However, in this model is not possible to measure a trade-off between the reduction of these costs and the revenue gain from producing a better quality ore.

Patterson et al. (2017) proposed a mixed-integer linear programming model that seeks to minimize the truck and shovel energy consumption required to meet production targets. At any given instant, only one truck at a time is allocated to the loading or unloading points. A Tabu Search-based algorithm is applied to a mine in South East Queensland, Australia, to solve the problem. The authors performed a sensitivity analysis of the model and found significant potential for improvement. They also discussed various ways of using the model as a decision support tool, including examples of how it can be used in the short, medium, and long-term planning process.

Kozan and Liu (2016) focused on short-term drilling, blasting and excavating operations problems for an open-pit iron mine in Australia. They considered different types of equipment and their capacities, times and operation velocities in a multi-resource multi-stage mine production timetabling model that synchronizes all activities. The results showed that it is possible to increase the productivity and use of the equipment through the proposed model.

Blom et al. (2017) dealt with the short-term schedule of an open-pit mine that produces a single ore product. The authors adapted the rolling-horizon-based algorithm proposed by Blom et al. (2016) to determine the sequencing of material extraction and its destination. This adaptation consists of a procedure that discretizes the time into two grouped periods. At each iteration, the algorithm consists of solving integer programming models and updating the state of the mine. A different goal is considered for each model resolution, which determines the sequencing. The results were obtained from two datasets from a mine. According to the authors, while the available tools need days, the proposed algorithm can generate multiple plans in a few minutes.

Liu and Kozan (2019) integrated the problems of mine design planning, mine block sequencing, and mine production scheduling in an open-pit mine. Mine design planning involves determining the pit limits and which blocks should be extracted. Mine block sequencing determines the sequence of extracting blocks within a period. The objective function of the first two problems is to maximize the net present value. In turn, mine production scheduling consists of assigning the trucks and shovels to the blocks that must be mined so that the total weighted tardiness is minimal. The models were integrated into a system that solves the models individually and uses the solution of one model as an input parameter for the next. From a case study, the authors concluded that the proposed system significantly improves the overall efficiency of the mining process.

Upadhyay and Askari-Nasab (2019) dealt with integrating short, medium, and long-term mine planning. A multi-objective model, based on the Souza et al. (2010) model, was proposed to maximize production and minimize the deviation from the feed rate of processing plants and crushers, as well as the total travel time of the shovel. They tested the model in a case study of an iron ore mine.

Upadhyay and Askari-Nasab (2018) presented an optimization and simulation framework tool to address uncertainties in short-term mine planning. This framework uses a simulation model interacting with a goal programming-based optimization tool. The authors used a case study of an iron ore mine to evaluate the proposed approach. According to the authors, this framework allows the planner to make proactive decisions to achieve operational objectives and provide realistic and practical short-term schedules.

Upadhyay et al. (2021) modeled the short-term production scheduling using continuous-time frames for shovel allocation. This implementation only allows allocating a shovel in one front after the shovel extracts all material from a previously allocated front. The model was tested with four instances, each with a different quantity of fronts in an iron ore mine. The results showed that the number of binary variables did not increase considerably with the increase of the fronts, which would not occur if discrete variables were considered for allocating shovels.

Afrapoli et al. (2019) approached a short-term planning problem through a multi-objective model. The proposed model seeks to minimize shovel idle times, truck waiting times, and deviations from production targets. The results were evaluated using a simulation model through a case study of an iron ore mine. The authors observed that it is possible to reduce the size of the current truck fleet through the multi-objective model.

Samavati et al. (2020) used conveyor belts instead of trucks to transport material. This replacement introduces a series of additional sequencing constraints that are not required in systems that use trucks. The authors presented an integer programming model and an algorithm based on its previous study (Samavati et al., 2017) to treat instances extracted from the literature. According to the authors, the results of the experiments were satisfactory.

Bakhtavar and Mahmoudi (2020) took a scenario-based robust optimization (SBRO) approach to solve a short-term planning problem. The problem was formulated in two phases through the four concepts: the SBRO approach, the minimum path problem, network analysis, and binary programming. The authors assumed the uncertainties regarding shovel production and the crusher capacity in the first phase. In the second phase, they considered the number of trucks available and the different routes between loading and unloading points. They applied the proposal to a copper mine and showed that it outperformed the company’s current strategy, increasing production and reducing operating costs.

Kumar et al. (2020) addressed the short-term planning for a copper mining complex. This complex contains two mines and seven material destination points. The objective is to determine the material flow (ore/waste), mainly its destination. The authors used the continuous updating framework that combines policy gradient reinforcement learning with an extended ensemble Kalman filter (EnKF; Evensen, 1994). They assumed the uncertainties of the grades of the blocks and the operation of the equipment involved: trucks, shovels, and processing. Production, blending, transportation costs, and equipment production capacity are also considered. This approach made it possible to increase cash flows and ore production compared to the company’s current strategy.

Both and Dimitrakopoulos (2020) addressed short-term planning in an open-pit mining complex. The authors presented a new stochastic model that simultaneously addresses the sequencing of blocks, the relocation of shovels, scheduling a heterogeneous fleet of trucks, and the extracted material’s destination. The results reported for a gold mining complex showed the method’s efficiency compared to the company’s current methodology.

Mohtasham et al. (2021a) dealt with the problem of truck allocation in open-pit mines under uncertainty on the shovel’s production capacity. They used a model based on random constrained goal programming (CCGP; Charnes et al., 1958) to address the stochastic nature of this problem. This model is an extension of Mohtasham et al. (2017) to address uncertainties and has four different objectives, namely: minimizing the deviations of the ore or waste rock production; the total cost of operating the trucks; the grade and tonnage deviations from the plant targets. It was tested with eleven schedule scenarios, with four confidence levels for the operation of the shovels in the Sungun copper mine, Iran. The results showed that the model achieved production and chemical grade targets.

Mohtasham et al. (2021b) addressed the short-term problem by analyzing the impact of changing the priority order of objectives for mine equipment fleets. The proposed mixed-integer linear goal programming model is solved hierarchically, considering the order of objectives according to the operational conditions at each shift. A drawback of this work is the not consideration of the dynamic allocation of each truck in the mining fronts.

Flores-Fonseca et al. (2022) addressed the mine planning problem by dividing it into two stages. The first is determining when each block will be extracted and its destination. The second stage consists of defining the blocks’ optimal extraction sequence, considering the shovel’s operation. The objective function of the first subproblem is to maximize the net present value (NPV), and the second is to maximize the efficiency of the shovel’s operation. They developed a mixed-integer linear programming model for each subproblem. These subproblems were tested using CPLEX and Gurobi solvers in real and random instances. The results showed that including the destination option in the mine planning increases the operational NPV.

Martins and Souza (2023) treated the short-term planning problem lexicographically. An optimization model with four objectives solved hierarchically was proposed. The authors generated 24 scenarios from a Brazilian iron mine to analyze the impact of varying the number of shovels and the tolerance of the plant targets on other objectives, mainly the ore particle size ranges. The scenarios with fewer shovels and lower values of the plant’s grade target tolerances generated higher deviations from the targets.

A comprehensive review of papers on the topic can be found in Blom et al. (2019), Afrapoli and Askari-Nasab (2019), and Franco-Sepúlveda et al. (2019).

This work differs from others for simultaneously considering the following features in a unique model: 1) the equipment’s travel time is dependent on the mining front, the truck type, the type of material transported (ore or waste), and the destination location; 2) there are multiple loading and discharge points; 3) the control parameter bounds and targets of the ore should be respected; 4) there may be an incompatibility between a shovel and a truck type, preventing the truck from loading through the shovel; 5) the stripping ratio must be obeyed; 6) the number of trucks and their trips must be determined; 7) there is a heterogeneous fleet of trucks, and 8) the allocation of trucks is dynamic; that is, the same truck can unload the material transported at one discharge point, return to be loaded at another mining front, and unload at another. Therefore, each truck’s route during a work shift can be determined.

Table 1 shows which features of this work differ from other studies in the literature dealing with short-term planning.

Table 1. Comparison of the features present in each study

	Features
References	1	2	3	4	5	6	7	8
Afrapoli et al. (2019)		✓	✓			✓
Bakhtavar and Mahmoudi (2020)		✓	✓			✓	✓
Blom et al. (2017)		✓	✓
Both and Dimitrakopoulos (2020)		✓	✓				✓
Flores-Fonseca et al. (2022)		✓	✓
Kozan and Liu (2016)		✓				✓
Kumar et al. (2020)		✓	✓
Liu and Kozan (2019)		✓				✓
Martins and Souza (2023)	✓	✓	✓			✓	✓
Mohtasham et al. (2021b)		✓	✓	✓		✓	✓
Mohtasham et al. (2021a)		✓	✓			✓	✓
Patterson et al. (2017)		✓		✓		✓
Samavati et al. (2020)		✓	✓
Souza et al. (2010)			✓	✓	✓	✓	✓
Subtil et al. (2011)		✓	✓		✓
Upadhyay and Askari-Nasab (2019)		✓	✓			✓	✓
Upadhyay and Askari-Nasab (2018)		✓	✓		✓	✓
Upadhyay et al. (2021)		✓	✓
Zhang and Xia (2015)		✓	✓			✓	✓
Our proposal	✓	✓	✓	✓	✓	✓	✓	✓

As can be seen, our proposal has the advantage of being more comprehensive and handling several operational features in a single model.

4. Proposed mathematical model

This section presents a mixed-integer linear goal programming model for the short-term mining planning problem of a work shift.

Let the following input parameters:

(1) Sets:

$\mathcal{O}$: Set of ore fronts;

$\mathcal{W}$: Set of waste fronts;

$\mathcal{F}$: $\mathcal{O}\cup \mathcal{W}$;

$\mathcal{Q}$: Set of ore control parameters regarding chemical grades and particle size ranges;

$\mathcal{S}$: Set of shovels;

$\mathcal{V}$: Set of trucks;

$\mathcal{P}$: Set of processing plants;

$\mathcal{U}$: Set of the source nodes $s_l$ of each truck $\mathit{l}\in \mathcal{V}$;

$\mathcal{N}$: $\mathcal{F}\cup \mathcal{P}\cup \mathcal{U}\cup \{\mathit{w}\}$, where $\mathit{w}$ is the waste dumping point.

(2) Indices:

$\mathit{p}$: Index for an element of the set $\mathcal{P}\cup \{\mathit{w}\}$;

$\mathit{i}$: Index for an element of the set $\mathcal{F}$;

$\mathit{j}$: Index for an element of the set; $$

$\mathit{k}$: Index for an element of the set $\mathcal{S}$;

$\mathit{l}$: Index for an element of the set $\mathcal{V}$;

$(\mathit{a},\mathit{d})$: Edge where $\mathit{a},\mathit{d}\in \mathcal{N}$.

(3) Production:

$\mathit{TP}{P}_p$: Production rate target for the plant $\mathit{p}$ $\in \mathcal{P}$ (t/h)

$\mathit{LP}{P}_p$: Minimum production rate for the plant $\mathit{p}\in \mathcal{P}$ (t/h);

$\mathit{UP}{P}_p$: Maximum production rate for the plant $\mathit{p}\in \mathcal{P}$ (t/h);

${\alpha }_p$: Penalty for production deviation in each plant $\mathit{p}\in \mathcal{P}$;

$\bar{\alpha }$: Global weight penalty for production deviation;

$\mathit{UP}{F}_i$: Ore production rate in the front $\mathit{i}\in \mathcal{O}$ (t/h);

$\mathit{WOR}$: Stripping ratio target;

$\mathit{PH}$: Planning horizon (minutes).

(4) Control parameters:

${\beta }_{\mathit{jp}}$: Penalty for deviations from the target established for the control parameter $\mathit{j}\in \mathcal{Q}$ in the blending of the plant $\mathit{p}\in \mathcal{P}$;

$\bar{\beta }$: Global weight penalty for the deviations from the control parameter targets;

${\delta }_{\mathit{jp}}$: Penalty due to control parameter $\mathit{j}\in \mathcal{Q}$ fall below or exceed its lower and upper bounds in the blending of plant $\mathit{p}\in \mathcal{P}$;

$\bar{\delta }$: Global weight penalty for which control parameters fall below or exceed their lower and upper bounds;

$C_{\mathit{ij}}$: Grade or particle size range of the control parameter $\mathit{j}\in \mathcal{Q}$ in front $\mathit{i}\in \mathcal{O}$ (%);

$\mathit{T}{P}_{\mathit{jp}}$: Target for the grade or particle size range of the control parameter $\mathit{j}\in \mathcal{Q}$ in the blending of plant $\mathit{p}\in \mathcal{P}$ (%);

$\mathit{L}{P}_{\mathit{jp}}$: Lower bound for the grade or particle size range of the control parameter $\mathit{j}\in \mathcal{Q}$ in the blending of plant $\mathit{p}\in \mathcal{P}$ (%);

$\mathit{U}{P}_{\mathit{jp}}$: Upper bound for the grade or particle size range of the control parameter $\mathit{j}\in \mathcal{Q}$ in the blending of plant $\mathit{p}\in \mathcal{P}$ (%);

(5) Shovels and trucks:

$\mathit{LT}$: Lower bound for the number of trucks;

$\mathit{\omega }$-: Penalty for using more than $\mathit{LT}$ trucks;

$\mathit{S}{W}_{\mathit{ik}}$: $\left\{\begin{array}{cc}1& \mathrm{If}\mathrm{shovel}\mathit{k}\in \mathcal{S}\mathrm{can}\mathrm{be}\mathrm{used}\mathrm{at}\mathrm{front}\mathit{i}\in \mathcal{O};\\ 0& \mathrm{Otherwise}.\end{array}\right.$

$\mathit{LO}{S}_k$: Minimum ore production rate of the shovel $\mathit{k}\in \mathcal{S}$ (t/h);

$\mathit{UO}{S}_k$: Maximum ore production rate of the shovel $\mathit{k}\in \mathcal{S}$ (t/h);

$\mathit{LW}{S}_k$: Minimum waste production rate of the shovel $\mathit{k}\in \mathcal{S}$ (t/h);

$\mathit{UW}{S}_k$: Maximum waste production rate of the shovel $\mathit{k}\in \mathcal{S}$ (t/h);

$\mathit{URT}$: Maximum utilization rate of each truck;

$\mathit{OC}{T}_l$: Ore load capacity of truck $\mathit{l}\in \mathcal{V}$ (t);

$\mathit{WC}{T}_l$: Waste load capacity of truck $\mathit{l}\in \mathcal{V}$ (t);

$\mathit{T}{T}_{\mathit{lad}}$: Travel time of truck $\mathit{l}\in \mathcal{V}$ from the source node $\mathit{a}$ to the sink node $\mathit{d}$ added to the loading and unloading time (minutes);

$\mathit{T}{O}_l$: Total operating time of truck $\mathit{l}\in \mathcal{V}$ to transport ore during a work shift (minutes);

$\mathit{T}{W}_l$: Total operating time of truck $\mathit{l}\in \mathcal{V}$ to transport waste during a work shift (minutes);

$\mathit{COM}{P}_{\mathit{kl}}$: $\left\{\begin{array}{cc}1& \mathrm{If}\mathrm{shovel}\mathit{k}\in \mathcal{S}\mathrm{is}\mathrm{compatible}\mathrm{with}\mathrm{truck}\mathit{l}\in \mathcal{V};\\ 0& \mathrm{Otherwise}.\end{array}\right.$

The decision variables of the model are as follows:

$x_{\mathit{ip}}$: Production rate of front $\mathit{i}\in \mathcal{F}$ to the discharge $\mathit{p}\in \mathit{P}\cup \{\mathit{w}\}$ (t/h);

$\mathit{to}{t}_l$: Total operating time of truck $\mathit{l}\in \mathcal{V}$ during a work shift (minutes);

$\mathit{d}{t}_{jp}^{+}$: Positive deviation from the target for control parameter $\mathit{j}\in \mathcal{Q}$ in the blending of plant $\mathit{p}\in \mathcal{P}$ (t/h);

$\mathit{d}{t}_{jp}^{-}$: Negative deviation from the target for control parameter $\mathit{j}\in \mathcal{Q}$ in the blending of plant $\mathit{p}\in \mathcal{P}$ (t/h);

$\mathit{d}{b}_{jp}^{-}$: Negative deviation from the lower bound for control parameter $\mathit{j}\in \mathcal{Q}$ in the blending of plant $\mathit{p}\in \mathcal{P}$ (t/h);

$\mathit{d}{b}_{jp}^{+}$: Positive deviation from the upper bound for control parameter $\mathit{j}\in \mathcal{Q}$ in the blending of plant $\mathit{p}\in \mathcal{P}$ (t/h);

$\mathit{d}{p}_p^{-}$: Negative deviation from the target rate of ore production in plant $\mathit{p}\in \mathcal{P}$ (t/h);

$\mathit{d}{p}_p^{+}$: Positive deviation from the target rate of ore production in plant $\mathit{p}\in \mathcal{P}$ (t/h);

$u_l$: $\left\{\begin{array}{cc}1& \mathrm{If}\mathrm{truck}\mathit{l}\in \mathcal{V}\mathrm{is}\mathrm{used};\\ 0& \mathrm{Otherwise}.\end{array}\right.$

$z_{\mathit{lad}}$: Number of times that truck $\mathit{l}\in \mathcal{V}$ traverses the edge ($\mathit{a},\mathit{d}$), with $\mathit{a},\mathit{d}\in \mathcal{N}$;

$n_{\mathit{il}}$: Number of trips in which truck $\mathit{l}\in \mathcal{V}$ leaves front $\mathit{i}\in \mathcal{F}$.

The mixed-integer linear goal programming formulation for the problem is presented through Equations (1)(1) $\begin{array}{rl}& min\mathit{z}=\bar{\alpha }\sum _{\mathit{p}\in \mathcal{P}}{\alpha }_p\frac{d{p}_p^{-}+\mathit{d}{p}_p^{+}}{max\{\mathit{TP}{P}_p-\mathit{LP}{P}_p,\mathit{UP}{P}_p-\mathit{TP}{P}_p\}}\\ & +\bar{\beta }\sum _{\mathit{p}\in \mathcal{P}}\sum _{\mathit{j}\in \mathcal{Q}}{\beta }_{\mathit{jp}}\frac{d{t}_{jp}^{-}+\mathit{d}{t}_{jp}^{+}}{max\{\mathit{T}{P}_{\mathit{jp}}\mathit{TP}{P}_p-\mathit{L}{P}_{\mathit{jp}}\mathit{LP}{P}_p,\mathit{U}{P}_{\mathit{jp}}\mathit{UP}{P}_p-\mathit{T}{P}_{\mathit{jp}}\mathit{TP}{P}_p\}}\\ & +\bar{\delta }\sum _{\mathit{p}\in \mathcal{P}}\sum _{\mathit{j}\in \mathcal{Q}}{\delta }_{\mathit{jp}}\frac{d{b}_{jp}^{-}+\mathit{d}{b}_{jp}^{+}}{\mathit{U}{P}_{\mathit{jp}}\mathit{UP}{P}_p-\mathit{L}{P}_{\mathit{jp}}\mathit{LP}{P}_p}+\mathit{\omega }\frac{\left(\sum _{\mathit{l}\in \mathcal{V}}{u}_l-\mathit{LT}\right)}{|\mathcal{V}|}\end{array}$(1) , (6(6) $\sum _{\mathit{i}\in \mathcal{O}}(C_{\mathit{ij}}-\mathit{U}{P}_{\mathit{jp}})x_{\mathit{ip}}-\mathit{d}{b}_{jp}^{+}\le 0\mathrm{\forall j}\in \mathcal{Q},\mathit{p}\in \mathcal{P}$(6) )–(38(38) $\mathit{to}{t}_l\ge 0\mathrm{\forall l}\in \mathcal{V}$(38) ).

(1) Objective function:

Equation (1)(1) $\begin{array}{rl}& min\mathit{z}=\bar{\alpha }\sum _{\mathit{p}\in \mathcal{P}}{\alpha }_p\frac{d{p}_p^{-}+\mathit{d}{p}_p^{+}}{max\{\mathit{TP}{P}_p-\mathit{LP}{P}_p,\mathit{UP}{P}_p-\mathit{TP}{P}_p\}}\\ & +\bar{\beta }\sum _{\mathit{p}\in \mathcal{P}}\sum _{\mathit{j}\in \mathcal{Q}}{\beta }_{\mathit{jp}}\frac{d{t}_{jp}^{-}+\mathit{d}{t}_{jp}^{+}}{max\{\mathit{T}{P}_{\mathit{jp}}\mathit{TP}{P}_p-\mathit{L}{P}_{\mathit{jp}}\mathit{LP}{P}_p,\mathit{U}{P}_{\mathit{jp}}\mathit{UP}{P}_p-\mathit{T}{P}_{\mathit{jp}}\mathit{TP}{P}_p\}}\\ & +\bar{\delta }\sum _{\mathit{p}\in \mathcal{P}}\sum _{\mathit{j}\in \mathcal{Q}}{\delta }_{\mathit{jp}}\frac{d{b}_{jp}^{-}+\mathit{d}{b}_{jp}^{+}}{\mathit{U}{P}_{\mathit{jp}}\mathit{UP}{P}_p-\mathit{L}{P}_{\mathit{jp}}\mathit{LP}{P}_p}+\mathit{\omega }\frac{\left(\sum _{\mathit{l}\in \mathcal{V}}{u}_l-\mathit{LT}\right)}{|\mathcal{V}|}\end{array}$(1) represents the weighted sum of four objectives to be minimized. The first seeks to minimize deviation from the production rate target. The second objective aims to minimize deviations from the targets for the control parameters. The third objective seeks to minimize the extent to which control parameters fall below or exceed their lower and upper bounds. Finally, the last objective aims to minimize the number of trucks used above $\mathit{LT}$.

(1) $\begin{array}{rl}& min\mathit{z}=\bar{\alpha }\sum _{\mathit{p}\in \mathcal{P}}{\alpha }_p\frac{d{p}_p^{-}+\mathit{d}{p}_p^{+}}{max\{\mathit{TP}{P}_p-\mathit{LP}{P}_p,\mathit{UP}{P}_p-\mathit{TP}{P}_p\}}\\ & +\bar{\beta }\sum _{\mathit{p}\in \mathcal{P}}\sum _{\mathit{j}\in \mathcal{Q}}{\beta }_{\mathit{jp}}\frac{d{t}_{jp}^{-}+\mathit{d}{t}_{jp}^{+}}{max\{\mathit{T}{P}_{\mathit{jp}}\mathit{TP}{P}_p-\mathit{L}{P}_{\mathit{jp}}\mathit{LP}{P}_p,\mathit{U}{P}_{\mathit{jp}}\mathit{UP}{P}_p-\mathit{T}{P}_{\mathit{jp}}\mathit{TP}{P}_p\}}\\ & +\bar{\delta }\sum _{\mathit{p}\in \mathcal{P}}\sum _{\mathit{j}\in \mathcal{Q}}{\delta }_{\mathit{jp}}\frac{d{b}_{jp}^{-}+\mathit{d}{b}_{jp}^{+}}{\mathit{U}{P}_{\mathit{jp}}\mathit{UP}{P}_p-\mathit{L}{P}_{\mathit{jp}}\mathit{LP}{P}_p}+\mathit{\omega }\frac{\left(\sum _{\mathit{l}\in \mathcal{V}}{u}_l-\mathit{LT}\right)}{|\mathcal{V}|}\end{array}$(1)

In Eq. (1(1) $\begin{array}{rl}& min\mathit{z}=\bar{\alpha }\sum _{\mathit{p}\in \mathcal{P}}{\alpha }_p\frac{d{p}_p^{-}+\mathit{d}{p}_p^{+}}{max\{\mathit{TP}{P}_p-\mathit{LP}{P}_p,\mathit{UP}{P}_p-\mathit{TP}{P}_p\}}\\ & +\bar{\beta }\sum _{\mathit{p}\in \mathcal{P}}\sum _{\mathit{j}\in \mathcal{Q}}{\beta }_{\mathit{jp}}\frac{d{t}_{jp}^{-}+\mathit{d}{t}_{jp}^{+}}{max\{\mathit{T}{P}_{\mathit{jp}}\mathit{TP}{P}_p-\mathit{L}{P}_{\mathit{jp}}\mathit{LP}{P}_p,\mathit{U}{P}_{\mathit{jp}}\mathit{UP}{P}_p-\mathit{T}{P}_{\mathit{jp}}\mathit{TP}{P}_p\}}\\ & +\bar{\delta }\sum _{\mathit{p}\in \mathcal{P}}\sum _{\mathit{j}\in \mathcal{Q}}{\delta }_{\mathit{jp}}\frac{d{b}_{jp}^{-}+\mathit{d}{b}_{jp}^{+}}{\mathit{U}{P}_{\mathit{jp}}\mathit{UP}{P}_p-\mathit{L}{P}_{\mathit{jp}}\mathit{LP}{P}_p}+\mathit{\omega }\frac{\left(\sum _{\mathit{l}\in \mathcal{V}}{u}_l-\mathit{LT}\right)}{|\mathcal{V}|}\end{array}$(1) ), the penalty weights $\bar{\alpha }$, ${\alpha }_p$, $\bar{\beta }$, ${\beta }_{\mathit{jp}}$, $\bar{\delta }$, ${\delta }_{\mathit{jp}}$, and $\mathit{\omega }$ must satisfy the following equations:

(2) $\bar{\alpha }+\bar{\beta }+\bar{\delta }+\mathit{\omega }=1$(2)

(3) $\sum _{\mathit{p}\in \mathcal{P}}{\alpha }_p=1$(3)

(4) $\sum _{\mathit{p}\in \mathcal{P}}\sum _{\mathit{j}\in \mathcal{Q}}{\beta }_{\mathit{jp}}=1$(4)

(5) $\sum _{\mathit{p}\in \mathcal{P}}\sum _{\mathit{j}\in \mathcal{Q}}{\delta }_{\mathit{jp}}=1$(5)

Note that each objective has a different measurement. Therefore, for an adequate comparison between these measures, each of them was normalized. Thus, excluding multiplication by weight, each parcel assumes a value between 0 and 1 after normalization.

In the first objective, the deviation from the production target is divided by the maximum value between two parcels in each plant. The first parcel is the difference between the target and minimum production rate. The second parcel is the difference between the maximum and target production rate.

In the second objective, the deviation from the target for each control parameter in each plant is divided by the maximum value between two parcels. The first parcel is the difference between the target for the control parameter and its respective lower bound. The second parcel is the difference between the upper bound for the control parameter and its target.

In the third objective, the deviation from the bounds for each control parameter in each plant is divided by the difference between its upper and lower bounds.

Finally, in the last objective, the number of trucks used above $\mathit{LT}$ is divided by the total number of trucks.

(2) Control parameter constraints:

Constraints (6) and (7) define the quantity of each control parameter outside of its upper and lower bounds at each ore discharge point, respectively.

Constraints (8) define that the quantity of each control parameter deviates from its target at each ore discharge point.

Constraints (8) define the quantity of each control parameter $\mathit{j}$ that can deviate from its target at each ore discharge point $\mathit{p}$.

(6) $\sum _{\mathit{i}\in \mathcal{O}}(C_{\mathit{ij}}-\mathit{U}{P}_{\mathit{jp}})x_{\mathit{ip}}-\mathit{d}{b}_{jp}^{+}\le 0\mathrm{\forall j}\in \mathcal{Q},\mathit{p}\in \mathcal{P}$(6)

(7) $\sum _{\mathit{i}\in \mathcal{O}}(C_{\mathit{ij}}-\mathit{L}{P}_{\mathit{jp}})x_{\mathit{ic}}+\mathit{d}{b}_{jp}^{-}\ge 0\mathrm{\forall j}\in \mathcal{Q},\mathit{p}\in \mathcal{P}$(7)

(8) $\sum _{\mathit{i}\in \mathcal{O}}(C_{\mathit{ij}}-\mathit{T}{P}_{\mathit{jp}})x_{\mathit{ip}}+\mathit{d}{t}_{jp}^{-}-\mathit{d}{t}_{jp}^{+}=0\mathrm{\forall j}\in \mathcal{Q},\mathit{p}\in \mathcal{P}$(8)

(3) Production constraints:

Constraints (9) and (10) prevent the mining rate from exceeding the minimum and maximum production bounds at the processing plants, respectively. Constraints (11) define the deviation from the production rate target of a solution. Constraints (12) prevent more ore from being mined than the mass $\mathit{UP}{F}_i$ available in each front $\mathit{i}$. Constraints (13) ensure that the minimum WOR is met. Constraints (14) and (15) determine the mining rate in each ore and waste front, respectively. Constraints (16) and (17) determine the number of trips of each truck to each ore and waste front, respectively.

(9) $\sum _{\mathit{i}\in \mathcal{O}}{x}_{\mathit{ip}}-\mathit{LP}{P}_p\ge 0\mathrm{\forall p}\in \mathcal{P}$(9)

(10) $\sum _{\mathit{i}\in \mathcal{O}}{x}_{\mathit{ip}}-\mathit{UP}{P}_p\le 0\mathrm{\forall p}\in \mathcal{P}$(10)

(11) $\sum _{\mathit{i}\in \mathcal{O}}{x}_{\mathit{ip}}-\mathit{TP}{P}_p+\mathit{d}{p}_p^{-}-\mathit{d}{p}_p^{+}=0\mathrm{\forall p}\in \mathcal{P}$(11)

(12) $x_{\mathit{ip}}-\mathit{UP}{F}_i\le 0\mathrm{\forall i}\in \mathcal{F},\mathit{p}\in \mathcal{P}$(12)

(13) $\sum _{\mathit{i}\in \mathcal{W}}{x}_{\mathit{iw}}-\mathit{WOR}\sum _{\mathit{p}\in \mathcal{P}}\sum _{\mathit{i}\in \mathcal{O}}{x}_{\mathit{ip}}\ge 0$(13)

(14) $\sum _{\mathit{p}\in \mathcal{P}}{x}_{\mathit{ip}}-\sum _{\mathit{l}\in \mathcal{V}}\sum _{\mathit{p}\in \mathcal{P}}{z}_{\mathit{lip}}\mathit{OC}{T}_l=0\mathrm{\forall i}\in \mathcal{O}$(14)

(15) $x_{\mathit{iw}}-\sum _{\mathit{l}\in \mathcal{V}}{z}_{\mathit{liw}}\mathit{WC}{T}_l=0\mathrm{\forall i}\in \mathcal{W}$(15)

(16) $n_{\mathit{il}}-\sum _{\mathit{p}\in \mathcal{P}}{z}_{\mathit{lip}}=0\mathrm{\forall i}\in \mathcal{O},\mathit{l}\in \mathcal{V}$(16)

(17) $n_{\mathit{il}}-z_{\mathit{liw}}=0\mathrm{\forall i}\in \mathcal{W},\mathit{l}\in \mathcal{V}$(17)

(4) Shovel capacity constraints:

Constraints (18) and (19) ensure that the hourly capacity of the shovels is respected when loading ore and waste, respectively. Constraints (20) and (21) ensure that each shovel operates above the minimum hourly production bound when loading ore and waste, respectively.

(18) $\sum _{\mathit{p}\in \mathcal{P}}{x}_{\mathit{ip}}-\sum _{\mathit{k}\in \mathcal{S}}\mathit{UO}{S}_k\mathit{S}{W}_{\mathit{ik}}\le 0\mathrm{\forall i}\in \mathcal{O}$(18)

(19) $x_{\mathit{iw}}-\sum _{\mathit{k}\in \mathcal{S}}\mathit{UW}{S}_k\mathit{WS}{ }_i\mathit{k}\le 0\mathrm{\forall i}\in \mathcal{W}$(19)

(20) $\sum _{\mathit{p}\in \mathcal{P}}{x}_{\mathit{ip}}-\sum _{\mathit{k}\in \mathcal{S}}\mathit{LO}{S}_k\mathit{S}{W}_{\mathit{ik}}\ge 0\mathrm{\forall i}\in \mathcal{O}$(20)

(21) $x_{\mathit{iw}}-\sum _{\mathit{k}\in \mathcal{S}}\mathit{LW}{S}_k\mathit{S}{W}_{\mathit{ik}}\ge 0\mathrm{\forall i}\in \mathcal{W}$(21)

(5) Truck allocation constraints:

Constraints (22), (23) and (24) consider a six-hour shift planning horizon, that is, $\mathit{PH}$ = 360 minutes. Constraints (22) ensure that the total operation time of each truck does not surpass the work shift and only a compatible shovel can load it. Constraints (23) prevent the maximum utilization rate of each truck from being violated during its work shift. Constraints (24) return a unit value for the variable $u_l$ whenever a truck $\mathit{l}$ is used.

(22) $\mathit{to}{t}_l-\mathit{PH}\sum _{\mathit{k}\in \mathcal{S},COM{\mathit{P}}_{\mathit{lk}}=1}\mathit{S}{W}_{\mathit{ik}}\le 0\mathrm{\forall l}\in \mathcal{V},\mathit{i}\in \mathcal{F}$(22)

(23) $\frac{to{t}_l}{\mathit{PH}}\le \mathit{URT}\mathrm{\forall l}\in \mathcal{V}$(23)

(24) $\frac{to{t}_l}{\mathit{PH}}\le u_l\mathrm{\forall l}\in \mathcal{V}$(24)

The total operating time ($\mathit{to}{t}_l$) of the truck $\mathit{l}$ for all trips allocated to it, in minutes, is calculated through Eq. (25)(25) $\mathit{to}{t}_l=\mathit{T}{O}_l+\mathit{T}{W}_l\mathrm{\forall l}\in \mathcal{V}$(25) :

(25) $\mathit{to}{t}_l=\mathit{T}{O}_l+\mathit{T}{W}_l\mathrm{\forall l}\in \mathcal{V}$(25)

where $\mathit{T}{O}_l$ and $\mathit{T}{W}_l$ are calculated according to Equations (26)(26) $\mathit{T}{O}_l=\sum _{\mathit{i}\in \mathcal{O}}{z}_{l{s}_l\mathit{i}}\mathit{T}{T}_{l{s}_l\mathit{i}}+\sum _{\mathit{i}\in \mathcal{O}}\sum _{\mathit{p}\in \mathcal{P}}(z_{\mathit{lip}}\mathit{T}{T}_{\mathit{lip}}+z_{\mathit{lpi}}\mathit{T}{T}_{\mathit{lpi}}+z_{\mathit{lwi}}\mathit{T}{T}_{\mathit{lwi}})+\sum _{\mathit{p}\in \mathcal{P}}{z}_{lp{s}_l}\mathit{T}{T}_{lp{s}_l}\mathrm{\forall l}\in \mathcal{V}$(26) and (27(27) $\mathit{T}{W}_l=\sum _{\mathit{i}\in \mathcal{W}}{z}_{l{s}_l\mathit{i}}\mathit{T}{T}_{l{s}_l\mathit{i}}+\sum _{\mathit{i}\in \mathcal{W}}\sum _{\mathit{p}\in \mathcal{P}}(z_{\mathit{liw}}\mathit{T}{T}_{\mathit{liw}}+z_{\mathit{lwi}}\mathit{T}{T}_{\mathit{lwi}}+z_{\mathit{lpi}}\mathit{T}{T}_{\mathit{lpi}})+z_{lw{s}_l}\mathit{T}{T}_{lw{s}_l}\mathrm{\forall l}\in \mathcal{V}$(27) ), respectively:

(26) $\mathit{T}{O}_l=\sum _{\mathit{i}\in \mathcal{O}}{z}_{l{s}_l\mathit{i}}\mathit{T}{T}_{l{s}_l\mathit{i}}+\sum _{\mathit{i}\in \mathcal{O}}\sum _{\mathit{p}\in \mathcal{P}}(z_{\mathit{lip}}\mathit{T}{T}_{\mathit{lip}}+z_{\mathit{lpi}}\mathit{T}{T}_{\mathit{lpi}}+z_{\mathit{lwi}}\mathit{T}{T}_{\mathit{lwi}})+\sum _{\mathit{p}\in \mathcal{P}}{z}_{lp{s}_l}\mathit{T}{T}_{lp{s}_l}\mathrm{\forall l}\in \mathcal{V}$(26)

(27) $\mathit{T}{W}_l=\sum _{\mathit{i}\in \mathcal{W}}{z}_{l{s}_l\mathit{i}}\mathit{T}{T}_{l{s}_l\mathit{i}}+\sum _{\mathit{i}\in \mathcal{W}}\sum _{\mathit{p}\in \mathcal{P}}(z_{\mathit{liw}}\mathit{T}{T}_{\mathit{liw}}+z_{\mathit{lwi}}\mathit{T}{T}_{\mathit{lwi}}+z_{\mathit{lpi}}\mathit{T}{T}_{\mathit{lpi}})+z_{lw{s}_l}\mathit{T}{T}_{lw{s}_l}\mathrm{\forall l}\in \mathcal{V}$(27)

(6) Material flow constraints:

Constraints (28) ensure that, for each truck $\mathit{l}$ and each ore front $\mathit{i}$, the number of trips leaving the origin node of this truck ($s_l$) is equal to the number of trips arriving at the processing plants. Likewise, constraints (29) determine that for each truck $\mathit{l}$ and each waste front $\mathit{i}\in \mathcal{W}$, the number of trips of the truck leaving its source node ($s_l$) is equal to the number of trips arriving at the waste dumping point. Constraints (30) determine that the number of trips of each truck departing from each ore discharge point towards the source node ($s_l$) of the truck $\mathit{l}$ is equal to the number of trips of this truck for all ore fronts. Similarly, Constraints (31) define the flow for trucks’ trips carrying waste.

(28) $z_{l{s}_l\mathit{i}}-\sum _{\mathit{p}\in \mathcal{P}}{z}_{\mathit{lip}}=0\mathrm{\forall l}\in \mathcal{V},\mathit{i}\in \mathcal{O}$(28)

(29) $z_{l{s}_l\mathit{i}}-z_{\mathit{liw}}=0\mathrm{\forall l}\in \mathcal{V},\mathit{i}\in \mathcal{W}$(29)

(30) $z_{lp{s}_l}-\sum _{\mathit{i}\in \mathcal{O}}{z}_{\mathit{lip}}=0\mathrm{\forall l}\in \mathcal{V},\mathit{p}\in \mathcal{P}$(30)

(31) $z_{lw{s}_l}-\sum _{\mathit{i}\in \mathcal{W}}{z}_{\mathit{liw}}=0\mathrm{\forall l}\in \mathcal{V}$(31)

(7) Variable domain constraints:

Constraints (32), (33), (34), (35), (36), (37), and (38) define the domain of the decision variables.

(32) $u_l\in \{0,1\}\mathrm{\forall }\mathit{l}\in \mathcal{V}$(32)

(33) $n_{\mathit{il}}\in Z^{+}\mathrm{\forall i}\in \mathcal{F},\mathit{l}\in \mathcal{V}$(33)

(34) $z_{\mathit{lad}}\in Z^{+}\mathrm{\forall l}\in \mathcal{V},\mathit{a}\in \mathcal{N},\mathit{d}\in \mathcal{N}$(34)

(35) $x_{\mathit{ip}}\ge 0\mathrm{\forall p}\in \mathit{C}\cup \{\mathit{w}\}$(35)

(36) $\mathit{d}{t}_{jp}^{+},\mathit{d}{t}_{jp}^{-},\mathit{d}{b}_{jp}^{+},\mathit{d}{b}_{jp}^{-}\ge 0\mathrm{\forall j}\in \mathcal{Q},\mathit{p}\in \mathcal{P}$(36)

(37) $\mathit{d}{p}_p^{+},\mathit{d}{p}_p^{-}\ge 0\mathrm{\forall p}\in \mathcal{P}$(37)

(38) $\mathit{to}{t}_l\ge 0\mathrm{\forall l}\in \mathcal{V}$(38)

5. Computational experiments

The proposed mathematical model was implemented in the Gurobi solver, version 8.1.1, interfacing with a Microsoft Excel spreadsheet, and the mipgap was set to 0.5%. The computer used in the computational experiments was a Dell Inspiron 7572 with an Intel i7-8550 U @ 1.80 GHz $\times$ 4, 16 GB of RAM, running under the Windows 10 operational system.

This study used data from a mining complex located in the central region of Minas Gerais state, Brazil. This mining complex has the Capão Xavier and Mar Azul mines of Vale S.A. The ore from these mines feeds four plants with three different discharge points, and all waste is dumped in a unique waste dump. Figure 2 illustrates the haulage routes in this mining complex for transporting ore and waste between the two mines and the possible points for discharging mined material.

Figure 2. Possible ore and waste routes for trucks. Image from Google Earth—Coordinates: Lat = ${20}^{\circ }{3.0}^{\prime }$ S, Lon = ${43}^{\circ }{58.6}^{\prime }$ O.

5.1. Generation of scenarios

To estimate $\mathit{LT}$ in Equation (1)(1) $\begin{array}{rl}& min\mathit{z}=\bar{\alpha }\sum _{\mathit{p}\in \mathcal{P}}{\alpha }_p\frac{d{p}_p^{-}+\mathit{d}{p}_p^{+}}{max\{\mathit{TP}{P}_p-\mathit{LP}{P}_p,\mathit{UP}{P}_p-\mathit{TP}{P}_p\}}\\ & +\bar{\beta }\sum _{\mathit{p}\in \mathcal{P}}\sum _{\mathit{j}\in \mathcal{Q}}{\beta }_{\mathit{jp}}\frac{d{t}_{jp}^{-}+\mathit{d}{t}_{jp}^{+}}{max\{\mathit{T}{P}_{\mathit{jp}}\mathit{TP}{P}_p-\mathit{L}{P}_{\mathit{jp}}\mathit{LP}{P}_p,\mathit{U}{P}_{\mathit{jp}}\mathit{UP}{P}_p-\mathit{T}{P}_{\mathit{jp}}\mathit{TP}{P}_p\}}\\ & +\bar{\delta }\sum _{\mathit{p}\in \mathcal{P}}\sum _{\mathit{j}\in \mathcal{Q}}{\delta }_{\mathit{jp}}\frac{d{b}_{jp}^{-}+\mathit{d}{b}_{jp}^{+}}{\mathit{U}{P}_{\mathit{jp}}\mathit{UP}{P}_p-\mathit{L}{P}_{\mathit{jp}}\mathit{LP}{P}_p}+\mathit{\omega }\frac{\left(\sum _{\mathit{l}\in \mathcal{V}}{u}_l-\mathit{LT}\right)}{|\mathcal{V}|}\end{array}$(1) , that is, the lower limit for the number of trucks needed for the production, we proceed as follows: 1) only Equation (39)(39) $min\sum _{\mathit{l}\in \mathcal{V}}{u}_l$(39) was used as the objective function instead Equation (1)(1) $\begin{array}{rl}& min\mathit{z}=\bar{\alpha }\sum _{\mathit{p}\in \mathcal{P}}{\alpha }_p\frac{d{p}_p^{-}+\mathit{d}{p}_p^{+}}{max\{\mathit{TP}{P}_p-\mathit{LP}{P}_p,\mathit{UP}{P}_p-\mathit{TP}{P}_p\}}\\ & +\bar{\beta }\sum _{\mathit{p}\in \mathcal{P}}\sum _{\mathit{j}\in \mathcal{Q}}{\beta }_{\mathit{jp}}\frac{d{t}_{jp}^{-}+\mathit{d}{t}_{jp}^{+}}{max\{\mathit{T}{P}_{\mathit{jp}}\mathit{TP}{P}_p-\mathit{L}{P}_{\mathit{jp}}\mathit{LP}{P}_p,\mathit{U}{P}_{\mathit{jp}}\mathit{UP}{P}_p-\mathit{T}{P}_{\mathit{jp}}\mathit{TP}{P}_p\}}\\ & +\bar{\delta }\sum _{\mathit{p}\in \mathcal{P}}\sum _{\mathit{j}\in \mathcal{Q}}{\delta }_{\mathit{jp}}\frac{d{b}_{jp}^{-}+\mathit{d}{b}_{jp}^{+}}{\mathit{U}{P}_{\mathit{jp}}\mathit{UP}{P}_p-\mathit{L}{P}_{\mathit{jp}}\mathit{LP}{P}_p}+\mathit{\omega }\frac{\left(\sum _{\mathit{l}\in \mathcal{V}}{u}_l-\mathit{LT}\right)}{|\mathcal{V}|}\end{array}$(1) , that is, here the objective is to minimize only the number of trucks; 2) the original constraints of the problem was maintained; 3) the minimum production and the lower bounds for the control parameters (that is, Equations (7)(7) $\sum _{\mathit{i}\in \mathcal{O}}(C_{\mathit{ij}}-\mathit{L}{P}_{\mathit{jp}})x_{\mathit{ic}}+\mathit{d}{b}_{jp}^{-}\ge 0\mathrm{\forall j}\in \mathcal{Q},\mathit{p}\in \mathcal{P}$(7) and (9(9) $\sum _{\mathit{i}\in \mathcal{O}}{x}_{\mathit{ip}}-\mathit{LP}{P}_p\ge 0\mathrm{\forall p}\in \mathcal{P}$(9) )) were redefined to take their target values. Thus, production and control parameter values below their targets are not allowed.

(39) $min\sum _{\mathit{l}\in \mathcal{V}}{u}_l$(39)

In the mine under study, there are 15 mining ore fronts, 5 waste fronts, and 13 control parameters involving the chemical grades and particle size range of the ore. The available fronts have high variability in the grade and the particle size range of the ore. For example, the iron grade varies from 39.75% to 66.79%, and the manganese grade from 0.0108% to 1.3308%.

Table 2 shows the percentage of each control parameter in each mining front. The control parameters analyzed are: $\mathit{i}$) Phosphorus grade (P); $\mathit{ii}$) Iron grade (Fe); $\mathit{iii}$) Silica grade (Si); $\mathit{iv}$) Alumina grade (Al); $\mathit{v}$) Loss On Ignition (LOI) percentage; and $\mathit{vi}$) Manganese grade (Mn). There are two variations for all these control parameters: percentage of material with a size range greater than 8 mm (represented by +8 mm) and “Global”, which is the global percentage of the control parameter, regardless of its size range. Finally, the control parameter +8 mm indicates the overall percentage of material with a size range above 8 mm.

Table 2. Grade and particle size range of the control parameters in the ore fronts in %

Control parameter		Front
		1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
Global	Fe	46.8	46.3	63.8	46.0	47.5	66.8	43.5	39.8	65.2	42.5	51.3	63.8	59.6	54.5	64.3
	Si	31.3	31.8	2.3	32.6	28.5	1.7	36.2	38.5	1.3	37.6	22.6	3.4	9.5	12.8	2.1
	Al (${10}^{-2}$)	58.0	60.0	126.0	40.0	52.0	68.0	82.0	165.0	106.0	76.0	59.0	187.0	211.0	443.0	150.0
	P (${10}^{-3}$)	30.0	28.0	72.0	23.0	63.0	40.0	18.0	22.0	79.0	18.0	86.0	20.0	20.0	98.0	83.0
	Mn (${10}^{-2}$)	7.0	3.0	4.0	1.0	2.0	3.0	2.0	18.0	6.0	2.0	2.0	10.0	18.0	18.0	13.0
	LOI	0.8	1.1	4.6	1.0	2.6	1.9	0.9	2.2	4.0	0.9	3.0	3.4	2.4	4.3	3.8
+ 8 mm	Size	14.3	11.4	38.0	12.1	62.6	31.2	8.2	11.0	75.2	10.1	80.1	21.5	18.0	25.4	32.7
	Fe	7.4	6.2	24.2	6.5	32.1	20.6	4.5	6.0	49.2	5.4	42.4	13.9	11.8	15.3	21.1
	Si	3.3	2.0	0.6	2.4	14.2	0.3	1.3	2.0	0.6	2.0	16.1	0.2	0.3	1.4	0.3
	Al (${10}^{-2}$)	4.4	5.5	31.9	3.9	20.6	17.1	7.8	5.6	60.1	7.2	34.4	46.0	28.9	89.1	30.0
	P (${10}^{-3}$)	6.0	6.0	34.5	4.9	48.2	20.9	2.7	1.5	63.1	3.1	75.2	4.5	3.6	34.5	35.3
	Mn (${10}^{-2}$)	0.8	0.4	1.5	0.1	1.2	0.9	0.1	0.1	3.0	0.2	0.8	1.5	1.0	2.0	3.2
	LOI	0.1	0.2	2.1	0.2	2.0	1.0	0.1	0.2	3.1	0.1	2.5	0.8	0.4	0.9	1.6
−8 mm	Size	85.7	88.6	62.0	87.9	37.4	68.8	91.8	89.0	24.8	89.9	19.9	78.5	82.0	74.6	67.3
	Fe	39.4	40.1	39.6	39.5	15.4	46.2	39.0	33.8	16.0	37.1	8.9	49.9	47.8	39.2	43.2
	Si	28.0	29.8	1.7	30.2	14.3	1.4	34.9	36.5	0.7	35.6	6.5	3.2	9.2	11.4	1.8
	Al (${10}^{-2}$)	53.6	54.5	94.1	36.1	31.4	50.9	74.2	159.4	45.9	68.8	24.6	141.0	182.1	353.9	120.0
	P (${10}^{-3}$)	24.0	22.0	37.5	18.1	14.8	19.1	15.3	20.5	15.9	14.9	10.8	15.5	16.4	63.5	47.7
	Mn (${10}^{-2}$)	6.2	2.6	2.5	0.9	0.8	2.1	1.9	17.9	3.0	1.8	1.2	8.5	17.0	16.0	9.8
	LOI	0.7	0.9	2.5	0.8	0.6	0.9	0.8	2.0	0.9	0.8	0.5	2.6	2.0	3.4	2.2

Table 3 details the main characteristics of the mine.

Table 3. Main characteristics of the mine

Characteristic	Values
Number of fronts	20 (15 ore and 5 waste fronts)
Stripping ratio target		0.3
Shift duration (h)		6
Shovels:
Quantity		9
Capacities in t/h		800
Number of truck fleet		2
Trucks	Fleet 1	Fleet 2
Quantity:	20	24
Ore payload (t):	90	64
Waste Payload (t):	88	61
Number of plants	3
Capacity in t/h:	Plant 1	Plant 2	Plant 3
Minimum	450	550	1,550
Target	550	650	1,650
Maximum	600	700	1,700

In mining, each control parameter is classified according to its priority, given the ore that will be produced. Table 4 shows this classification and the target values for the control parameters. Its columns show the control parameters, the lower and upper bounds, the target, the assigned weight, and the priority description, respectively.

Table 4. The lower and upper bounds, targets, and weights for the control parameters

Control parameter	Lower bound	Upper bound	Target	Weight	Priority description
	(%)	(%)	(%)
+8 mm	27.0	32.0	31.0	1	Low
P Global	1.0	3.5	2.0
Al Global	0.9	1.6	1.3
Si Global	11.0	14.0	13.0
P +8 mm	1.5	4.0	3.0
Al +8 mm	0.5	1.0	0.9
Si +8 mm	7.0	10.0	9.0
LOI Global	1.0	3.5	2.0	5	Medium
LOI +8 mm	1.5	4.0	3.0
Mn Global	0.045	0.200	0.100	10	High
Mn +8 mm	0.030	0.200	0.100
Fe Global	56.0	58.5	58.0	100	Critical
Fe +8 mm	58.0	61.0	60.0

The mine studied is responsible for producing iron ore, so the iron grade is the critical control parameter since it has the highest priority. In addition, this ore produced requires more significant control over its manganese grade since it is used to adjust the manganese grade of ore from other mines during the ore blending at the port.

Below are described four scenarios representing real mine cases to evaluate the proposed mathematical model. These scenarios consider that the source node of each truck is located near Plant 1 and all trucks depart from the same location at the beginning of the work shift.

• Scenario 1: The standard operating condition of the mine, in which the highest priority is to achieve the production target; the second one is to reduce the deviations outside the control parameter bounds. The third one is to minimize the deviations from the control parameter targets. The lowest priority is the minimization of the number of trucks.

• Scenario 2: The operating condition of the mine in which there are specific customer demands to be attained, and they require high rigor in checking the quality of the ore produced. In this scenario, the highest priority is to reduce the deviations outside the control parameter bounds. The second priority is to minimize the deviations from the control parameter targets, the third is to achieve the production target, and the lowest priority consists of minimizing the number of trucks.

• Scenario 3: The mine has a high demand for ore, and all of it is sold. In this scenario, the quality control of the ore has low relevance. Therefore, the highest priority is to meet the production target. As greater operational efficiency in using trucks is also sought, the second priority is minimizing the number of trucks. In turn, the deviations from the control parameter bounds and targets have the lowest priority.

• Scenario 4: This scenario reproduces the goals of Scenario 1 but without considering the minimization of the number of trucks used. So, it seeks to strictly meet the production goal and reduce deviations from the control parameter bounds and targets, disregarding the operational costs related to the use of trucks. Therefore, the highest priority is to meet the production goal. The second priority is to reduce deviations outside the control parameter bounds. Finally, the third one is to reach the control parameter targets.

Table 5 lists the weights assigned to each component of the weighted objective function (1) for each of the four scenarios. These weights are only suggestions for the decision-makers. It is up to them to choose the most appropriate weights according to the mining scenario.

Table 5. The weights adopted for each objective of the weighted objective function in the four scenarios

Scenario	Objective
	Production	Control Parameter		Number of Trucks
		Bounds	Targets
1	0.50	0.25	0.15	0.10
2	0.10	0.40	0.40	0.10
3	0.50	0.05	0.05	0.40
4	0.60	0.25	0.15	0.00

5.2. Results

This subsection reports the model’s results in three different situations for each scenario. For each situation, here called an instance, the number of shovels is varied (5, 7, and 9), resulting in 12 instances to be analyzed. This variation directly impacts meeting the production goal and the goals established for the ore control parameters, as only one shovel can be allocated to each mining front.

Table 6 shows the results of the 12 instances. The first to the fifth column reports the instance index, the scenario index, the number of available shovels, the run time, and the gap result, respectively. The sixth column presents the plants’ production deviations from the targets. The next twelve columns show the control parameters’ deviation from the plants’ targets. Finally, the last column reports the number of trucks used in each instance. In bold are the results of the control parameters with high and critical priority.

Table 6. Instances’ results ${ }^{\star }$

Inst.	Scen.	Shovels	Run	Gap	Deviations in tonnes													Trucks
			time		Prod.	Global						+8 mm
		(#)	(sec.)	(%)		Fe	Si	Al	P	Mn	LOI	Fe	Si	Al	P	Mn	LOI	(#)
1	1	5	1.0	0.17	1102	63.67	48.73	9.26	0.589	1.05	111.96	0.30	1.09	0.24	0.025	0.05	0.52	25
2	2	5	0.4	0.46	1100	62.17	46.16	9.13	0.584	1.03	111.88	0.29	1.06	0.24	0.025	0.05	0.52	24
3	3	5	0.5	0.44	1106	277.58	339.93	12.49	0.486	1.75	100.47	0.08	0.08	0.26	0.012	0.03	0.34	24
4	4	5	0.6	0.15	1100	68.42	55.94	9.74	0.611	1.06	111.47	0.34	1.15	0.24	0.026	0.05	0.51	45
5	1	7	1.8	0.38	2	0.69	72.10	0.52	0.286	0.04	132.29	1.40	1.24	0.26	0.021	0.05	0.57	27
6	2	7	0.6	0.26	0	0.03	28.95	3.82	0.419	0.02	97.05	1.43	1.09	0.22	0.018	0.05	0.45	26
7	3	7	934.2	0.45	12	685.02	932.85	13.41	1.271	2.80	85.18	1.33	3.50	0.24	0.000	0.03	0.12	23
8	4	7	0.5	0.46	0	11.15	47.33	8.16	0.003	0.56	113.20	0.89	0.48	0.38	0.019	0.05	0.41	45
9	1	9	5.4	0.29	0	0.34	52.40	5.55	0.908	0.02	98.05	2.87	2.33	0.26	0.001	0.08	0.31	34
10	2	9	11.7	0.49	0	0.17	52.92	1.89	0.000	0.01	116.82	1.39	1.15	0.24	0.020	0.05	0.53	27
11	3	9	161.2	0.42	0	8.27	43.07	2.36	0.006	0.15	117.27	1.39	1.16	0.24	0.019	0.05	0.54	26
12	4	9	0.3	0.49	0	3.75	49.29	1.48	0.032	0.01	117.98	1.35	1.32	0.23	0.019	0.05	0.53	45

${ }^{\star }$ Control parameters with critical or high priorities are highlighted in bold.

5.3. Discussion of results

Figures 3, 4 , and 5 illustrate the results. In these figures, the first vertical axis indicates the sum of the production deviations, in tonnes, concerning the target of the three ore plants. The following four axes report the deviations, in tonnes, concerning Fe Global, Mn Global, Mn +8 mm, and Fe +8 mm parameters from their target values. Finally, the last axis shows the number of trucks used.

Figure 3. Results of the scenarios with five shovels available.

Figure 4. Results of the scenarios with seven shovels available.

Figure 5. Results of the scenarios with nine shovels available.

All instances have production deviations equal to or near zero, except those with five shovels. In the latter, the deviations were large since only one shovel was assigned to a waste front. That is, the maximum amount of waste material allowed for mining was equal to the shovel’s capacity (4,800 t per shift) in the instances with five shovels. Consequently, according to Constraint (13), the maximum tonnes of mined ore depends on the waste tonnes necessary to meet the WOR (i.e., 4,800 t/WOR = 4,800/0.3 = 16,000 t of ore), that is, 1,100 t below the capacity of the plants (17,100 t per shift).

Regarding the grade and particle size range of the critical and high priority control parameters (Fe and Mn, respectively), it was not possible to reach the target in all scenarios with 5, 7, and 9 shovels.

However, in the instances with more available resources (7 and 9 shovels), the deviations were smaller in Scenario 2, in which the sum of the deviations from the control parameter bounds and targets has the highest weight (0.80).

Regarding the number of trucks allocated, in all instances (with 5, 7, and 9 shovels), Scenario 3 required the smallest number of trucks, and Scenario 4 required the largest number of trucks. These facts occurred because, in Scenario 3, the highest weight in the weighted objective function is to minimize the number of trucks used. In contrast, this parcel of the objective function has a weight equal to zero in Scenario 4.

Figure 6 illustrates in blue and red colors the travel flow of two trucks in Scenario 1. In this scenario, the two trucks performed trips to fronts 1, 6, 9, 12, 13, and 15. Solid lines represent trips from a mining front to a discharge point, while dashed lines indicate trips to the fronts. The starting position of each truck is the one in which it finished the previous work shift. This position is referred as the origin.

Figure 6. Travel flow of two trucks in Scenario 1 involving nine shovels.

As can be seen in Figure 6, the truck that makes the trips highlighted in blue leaves the origin towards Front 1. On its first trip, it leaves Front 1, goes to Plant 2, and returns to Front 1 on its second trip. Then, on the third trip, it goes to Plant 3 and then to Front 6 on its fourth trip. On its fifth trip, it goes to Plant 2 and then to Front 9 on its sixth trip. On its seventh trip, it travels to Plant 1 and returns to the same front on its eighth trip. On the ninth trip, it goes to Plant 2, and after unloading, it goes to Front 12 on its tenth trip. Then, it makes five trips, where the 11th, 13th, and 15th are to Plant 1, and the 12th and 14th are return trips. On its sixteenth trip, it exits Plant 1 towards Front 15. Then, it executes two trips (17th and 19th) to the waste dumping and one return trip to Front 15 (18th trip). Finally, this truck returns to the origin, completing its scheduled trips for the work shift. This sequence of trips made by this truck is shown in Table 7.

Table 7. Sequence of travels made by the truck that makes the trips highlighted in blue in Figure 6

Trip order	Origin	Destination	Trip order	Origin	Destination
1 ${ }^{\mathrm{st}}$	Front 1	Plant 2	11 ${ }^{\mathrm{th}}$	Front 12	Plant 1
2 ${ }^{\mathrm{nd}}$	Plant 2	Front 1	12 ${ }^{\mathrm{th}}$	Plant 1	Front 12
3 ${ }^{\mathrm{rd}}$	Front 1	Plant 3	13 ${ }^{\mathrm{th}}$	Front 12	Plant 1
4 ${ }^{\mathrm{th}}$	Plant 3	Front 6	14 ${ }^{\mathrm{th}}$	Plant 1	Front 12
5 ${ }^{\mathrm{th}}$	Front 6	Plant 2	15 ${ }^{\mathrm{th}}$	Front 12	Plant 1
6 ${ }^{\mathrm{th}}$	Plant 2	Front 9	16 ${ }^{\mathrm{th}}$	Plant 1	Front 15
7 ${ }^{\mathrm{th}}$	Front 9	Plant 1	17 ${ }^{\mathrm{th}}$	Front 15	Waste dumping
8 ${ }^{\mathrm{th}}$	Plant 1	Front 9	18 ${ }^{\mathrm{th}}$	Waste dumping	Front 15
9 ${ }^{\mathrm{th}}$	Front 9	Plant 2	19 ${ }^{\mathrm{th}}$	Front 15	Waste dumping
10 ${ }^{\mathrm{th}}$	Plant 2	Front 12

The proposed mathematical model found high-quality solutions (with gap up to 0.49%), in all instances, in suitable computational time. In nine of 12 instances, the model needed less than 3 seconds to solve the problem, and the maximum run time was equal to 15.5 minutes. So, the results validate the model as a tool to support decision-making.

6. Conclusions

This work dealt with the short-term planning problem of a work shift for open-pit mines. The objective is to minimize the deviations from the production goal, the bounds and targets of the control parameters, and the number of trucks needed.

We proposed a new mixed-integer linear goal programming formulation to solve it. This approach was applied to a mining complex with two mines that supply material to three mining plants and one waste dumping. Four scenarios simulating some operating conditions of the mines and the iron ore market were analyzed. These scenarios differ concerning the weights given to each objective of the weighted objective function.

The results showed that the proposed approach can support decision-makers in the sizing and allocation of truck fleets and in determining the amount of ore extracted from each mining front. Thus, the decision-makers can meet the production and control parameter targets required by the ore processing plants according to the daily mining scenario, as the need for increased production, low equipment availability, and ore quality flexibility. Furthermore, for the scenarios analyzed, only about 3 minutes are needed to plan a work shift.

In future work, we suggest to do a sensitivity analysis of the weights adopted in the weighted objective function. We also intend to evaluate the solutions generated by the optimizer through a simulator. Thus, the simulator could consider stochastic aspects of the process, such as travel time and truck queue time, to validate the results provided by the optimizer.

Acknowledgements

The authors are grateful for the support provided by the Vale S.A., Instituto Tecnológico Vale, Universidade Federal de Ouro Preto, and by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brazil (CAPES) - Finance Code 001, CNPq (grant 303266/2019-8), and FAPEMIG (grant PPM CEX 676/17).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This manuscript is the result of the research project funded by the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq, grant 303266/2019-8), the Fundação de Amaparo à Pesquisa de Minas Gerais (FAPEMIG, grant 00676-17), and the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES, 001).