Optimal Placement Algorithms for Base and Central Stations in Mining Quarries ^†

Abstract

This paper proposes algorithms for optimal placement of base stations (BSs) and central stations (CSs) in mining quarries to ensure reliable radio communication for automated machinery. The BS placement is modeled as a minimum dominating set problem, solved using integer linear programming with cutting-plane methods. The CS placement is formulated as a nonlinear programming problem, addressed via a minimum circle covering algorithm. Applied in a 200 km² quarry, the approach achieves full coverage with nine BSs and one CS, minimizing costs and ensuring robust performance. Comparative analyses show superior optimality, scalability, and adaptability, offering a scalable framework for industrial communication infrastructure.

1. Introduction

The automation of earth-moving and construction machinery in mining quarries relies on advanced radio communication systems to facilitate real-time transmission of sensor data and control commands. Efficient placement of base stations (BSs) and central stations (CSs) is critical to ensure comprehensive coverage, minimize deployment costs, and enhance system resilience against failures, interference, or environmental challenges. Mining environments, characterized by rugged terrains, seismic activity, variable geophysical conditions, and remote locations, pose unique challenges to achieving optimal station placement. These challenges include ensuring signal reliability across large areas, managing high installation and maintenance costs, maintaining system uptime in harsh conditions, and adapting to dynamic quarry layouts as operations expand.

The importance of reliable communication in mining cannot be overstated. Automated machinery, such as autonomous haul trucks and drilling rigs, depends on continuous data streams to monitor performance, detect anomalies, and execute precise movements. A single communication failure can lead to costly downtime, equipment damage, or safety hazards. Therefore, BSs must be strategically placed to cover all operational districts, while CSs must aggregate data from BSs and issue control commands with minimal latency. Economic constraints further complicate the problem, as installing stations in remote or geologically unstable areas incurs significant costs.

The existing research has addressed related problems in wireless network design, including cellular network planning, sensor network coverage, facility location optimization, and vehicular ad hoc networks. Cellular network planning focuses on urban environments with dense user bases and predictable mobility patterns, which differ from the sparse, static coverage required in quarries. Sensor networks prioritize energy efficiency and data aggregation, whereas quarry systems emphasize fault tolerance and real-time response. Facility location models address static placement but often ignore dynamic terrain constraints. Vehicular networks consider mobility but not the fixed infrastructure typical of mining operations.

Despite these efforts, few studies specifically target mining quarry communication systems. Early work by Huang et al. outlined cellular planning principles but did not address quarry-specific challenges like terrain variability [1]. Yun et al. proposed coverage algorithms for sensor networks, but their energy-focused approach is less applicable to high-power BSs. Recent studies on industrial IoT explore automation but lack rigorous optimization for station placement [2]. This gap motivates the development of tailored algorithms that balance coverage, cost, and reliability in mining contexts.

This paper proposes a comprehensive framework for optimizing BS and CS placement in mining quarries. The BS placement problem is formulated as finding the minimum dominating set in a graph, solved using ILP with cutting-plane methods. The CS placement problem is modeled as a minimum circle covering problem, addressed through a graph-based nonlinear programming approach. The algorithms are validated using a numerical example in a 200 km² quarry, demonstrating significant cost savings, robust coverage, and scalability. Comparative analyses with greedy, metaheuristic, and geometric methods highlight the proposed approach’s advantages. The contributions of this work include the following:

• A graph theoretical model for BS placement, ensuring minimal station deployment with full coverage;
• A nonlinear programming approach for CS placement, minimizing the number of stations while guaranteeing signal reachability;
• A scalable, validated framework with comparisons to state-of-the-art methods;
• Practical insights for deploying communication systems in mining and similar industrial settings.

2. Related Work

The problem of station placement in communication networks has been extensively studied across various domains, including cellular networks, sensor networks, and industrial automation. This section reviews key approaches and contrasts them with the proposed framework, emphasizing the unique requirements of mining quarries.

Cellular network planning focuses on optimizing base station placement to maximize user coverage and capacity in urban or suburban environments. Yun et al. proposed frameworks for cellular networks, using optimization models to place stations based on signal strength and interference [2]. Huang et al. extended this work to UMTS and WLAN integration, addressing power control and load balancing [1]. These methods assume high-density user distributions and predictable mobility, which do not apply to mining quarries with sparse, static machinery. Additionally, cellular models prioritize frequency reuse and handoff, whereas quarry systems require uninterrupted coverage and low latency. The proposed ILP-based approach adapts graph theoretical models to sparse, terrain-constrained environments, offering greater flexibility.

Sensor networks aim to cover a region with minimal sensors while maximizing data collection and energy efficiency. Wang et al. formulated coverage as a geometric problem, using optimal deployment patterns to ensure k-connectivity [3]. These methods are effective for low-power, distributed systems but less suitable for high-power BSs in quarries, where reliability and range are paramount. The proposed minimum dominating set model leverages graph theory to ensure coverage with fewer stations, addressing reliability through redundant coverage.

Facility location problems seek to place facilities (e.g., warehouses, stations) to minimize costs or maximize service. Zheng et al. proposed a graph-based approach for facility placement under random demand, considering distance and accessibility [4]. While these models share similarities with BS placement, they often assume Euclidean distances and static demands, ignoring terrain barriers or dynamic quarry layouts. The proposed framework incorporates terrain constraints via graph adjacency and adapts to cost heterogeneity, making it more applicable to mining.

Recent advances in industrial IoT explore automation in manufacturing and mining. Golubeva et al. investigated wireless sensor networks for monitoring equipment in open-pit mining, emphasizing real-time data transmission for operational efficiency [5]. Similarly, Golubeva et al. proposed optimizing the placement of communication nodes in underground mines using IoT technologies, addressing signal reliability in challenging environments [6]. These works highlight the need for optimized station placement to ensure robust connectivity, which the proposed algorithms address through rigorous mathematical modeling. Additional studies, such as that by Pandey et al., explore crowdsourcing frameworks for edge-based federated learning in industrial networks, offering insights into distributed communication systems [7]. Chen et al. investigated edge computing for vehicular networks in mining, adapting coverage models to dynamic equipment mobility, complementing the static placement focus of this paper [8].

Geometric covering problems, such as circle or disk covering, are relevant to CS placement. Asano et al. analyzed the complexity of disk covering in computational geometry, proposing algorithms to minimize the number of disks [9]. These methods inspire the proposed CS placement algorithm, which extends their work by incorporating terrain-aware graph construction and spiral search for precise center placement. Unlike general geometric approaches, the proposed method is optimized for sparse, fixed BS locations.

Metaheuristic algorithms, such as genetic algorithms, and greedy heuristics offer fast, approximate solutions for placement problems. Deb et al. applied genetic algorithms to network optimization, achieving near-optimal solutions for large problems [10]. Pandey et al. proposed a greedy heuristic for federated learning node placement, selecting nodes covering the most uncovered areas [7]. While these methods are computationally efficient, they often yield suboptimal solutions (e.g., 10–12 BSs vs. 9 in the proposed ILP). The cutting-plane method guarantees optimality, which is critical for cost-sensitive mining applications.

The proposed framework distinguishes itself by combining rigorous optimization (ILP and nonlinear programming) with practical considerations for mining environments, outperforming existing methods in optimality and adaptability.

3. Problem Formulation

3.1. Base Station Placement

Consider a mining quarry with a geographical area S, divided into n districts $S=\left\{s_i\right\},i=1,\dots ,n$. Each district can host a BS equipped with advanced communication hardware to monitor and control earth-moving machines within its range and neighboring districts. The objective is to determine the minimum number of BSs and their locations to ensure full coverage of all districts while minimizing installation costs.

The problem is modeled using an undirected graph $G=(V,\mathsf{\Gamma })$, where vertices $V=\left\{v_i\right\}$ represent districts, and edges $\mathsf{\Gamma }$ connect vertices corresponding to neighboring districts (e.g., sharing a boundary). The goal is to find the minimum dominating set $S\subseteq V$, such that every vertex in $V\backslash S$ is adjacent to at least one vertex in $S$. Formally, a set $S\subseteq V$ is dominating if

$$\forall v_j\in V\backslash S,\exists v_i\in S \mathrm{such} \mathrm{that} (v_i,v_j)\in \mathsf{\Gamma }$$

(1)

The minimum dominating set minimizes the number of BSs, with each BS being associated with a cost $C_j$, reflecting installation and maintenance expenses. This problem is equivalent to the minimum set cover problem, a well-studied combinatorial optimization problem.

Let $A$ be the adjacency matrix of $G$, where $A_{ij}=1$ if districts $i$ and $j$ are adjacent or $i=j$, and $A_{ij}=0$ otherwise. Define $\tilde{A}=A+I$, where $I$ is the identity matrix, ensuring that each district covers itself. Introduce a binary vector $x=(x_1,\dots ,x_n)$, where $x_j=1$ if a BS is placed in district $S_j$, and $x_j=0$ otherwise. The ILP formulation is

$$\mathrm{Minimize} Z={\sum }_{j=1}^n{C}_j{x}_j,$$

(2)

which is subject to

$$\begin{array}{c}\tilde{A}x\ge 1,\\ x_j\in \left\{\mathrm{0,1}\right\}, j=1,\dots ,n,\end{array}$$

where $\left|1\right|$ is a vector of ones, and $C_j$ is the cost of installing a BS in district $S_j$. Constraint (3) ensures that each district is covered by at least one BS (itself or a neighbor), and Constraint (4) enforces binary decisions [11].

3.2. Central Station Placement

The CS receives signals from BSs and issues control commands to prevent accidents, stop machinery, or adjust trajectories. The objective is to place the minimum number of CSs such that every BS is within the signal range $r$ of at least one CS, minimizing total installation costs.

Given n BSs with coordinates $(x_i,y_i)$, the CS placement problem is to find the minimum number of circles with a radius $r$ that cover all BS points. This is a geometric covering problem, formulated as a nonlinear programming problem:

$$\mathrm{Minimize} {\sum }_{k=1}^{m}B,$$

(3)

which is subject to

$$\underset{k=1,\dots ,m}{\mathrm{m}\mathrm{i}\mathrm{n}}\sqrt{(x_i-u_k{)}^2+(y_i-v_k{)}^2}\le r, \forall i=1,\dots ,n,$$

(4)

where $u_k,v_k$ are the coordinates of the $k$-th CS, $m$ is the number of CS, and B is the cost of installing one CS. The goal is to minimize m, assuming uniform costs. Constraint (4) ensures that each BS is within a distance $r$ of at least one CS, enabling timely signal reception and decision-making.

4. Proposed Algorithms

4.1. Base Station Placement Algorithm

The BS placement problem, formulated as an ILP model defined by Equations (2)–(4), is solved using the cutting-plane method, a robust technique for integer linear programming. The method iteratively refines a linear programming (LP) relaxation by adding cutting planes to eliminate non-integer solutions, ensuring convergence to an optimal integer solution. The algorithm is detailed as follows:

Input: Graph $G=(V,\mathsf{\Gamma })$, adjacency matrix $\tilde{A}$, costs $C_j$. Output: Binary vector $x$ indicating BS locations. Formulate the LP relaxation of Equations (2)–(4), ignoring $x_j\in \left\{\mathrm{0,1}\right\}$. Solve the LP using the simplex method, obtaining solution $x^{\ast }$. $x^{\ast }$ Select a basic variable $x_i$ with the largest fractional part (e.g., $x_i=0.7$). From the simplex tableau, derive a constraint:

$${\sum }_{j=1}^n{a}_{ij}{x}_j+s_i=b_i$$

(5)

where $a_{ij}$, $b_i$ are tableau coefficients, and $s_i$ is a slack variable. Construct a cutting plane,

$${\sum }_{j=1}^n\lfloor a_{ij}\rfloor x_j+s_i\le \lfloor b_i\rfloor ,$$

(6)

where $\lfloor \cdot \rfloor$ is the floor function, and $s_i\ge 0$. Add the cutting plane to the LP, forming an augmented problem. Resolve the augmented LP and repeat from step 4 [12].

The simplex method transforms the inequalities defined in (3) into equalities by introducing slack variables:

$${\sum }_{j=1}^n{a}_{ij}{x}_j+x_{n+i}=b_i, x_{n+i}\ge 0,$$

(7)

where $a_{ij}$ are elements of $\tilde{A}$, and $b_i=1$. The simplex method iteratively improves the solution through pivot operations, constructing simplex tables to track coefficients and objective values. The cutting-plane method ensures integer solutions by adding constraints that cut off non-integer points in the feasible region without excluding integer solutions.

Implementation details include using modern LP solvers (e.g., CPLEX, Gurobi) with presolving techniques to reduce problem size. For large graphs, preprocessing eliminates redundant constraints (e.g., districts covered by multiple neighbors) or fixes variables (e.g., isolated districts requiring a BS). The algorithm’s complexity is polynomial in the number of variables and constraints, although worst-case exponential behavior is rare with modern solvers.

4.2. Central Station Placement Algorithm

The CS placement problem (3)–(4) is addressed using a minimum circle covering algorithm based on graph theory and geometric optimization. A graph $G$ is constructed with BSs as vertices, and edges connect pairs of BSs within distance $2r$, as two BSs can be covered by a single circle of radius $r$ if their distance is at most $2r$. The algorithm identifies the minimum number of circles of radius $r$ that cover all vertices, with their centers serving as CS locations. The algorithm is detailed as follows:

Input: BS coordinates $(x_i,y_i)$, radius $r$. Output: CS coordinates $(u_k,v_k)$. Construct a graph $G$ with vertices as BSs and edges between BSs $i$ and $j$ if

$$\sqrt{(x_i-x_j{)}^2+(y_i-y_j{)}^2}\le 2r$$

(8)

Identify connected components of $G$ using depth-first search. Compute maximal groups of vertices coverable by a single circle of radius $r$. A group is maximal if no additional vertex can be added without violating the radius constraint. Solve a set cover problem to select the minimum number of maximal groups covering all vertices, using a greedy algorithm. For each selected group, compute the initial circle center as the centroid:

$$x_0={\displaystyle \frac{1}{k}}{\sum }_{i=1}^k{x}_i, y_0={\displaystyle \frac{1}{k}}{\sum }_{i=1}^k{y}_i,$$

(9)

where $k$ is the number of vertices in the group. Refine the center using a discrete spiral search:

• Initialize center at $(x_0,y_0)$;
• Test points in a spiral pattern with step size $\delta$ (e.g., 0.1 km).

Stop when a center $(u_k,v_k)$ satisfies the following condition:

$$\sqrt{(x_i-u_k{)}^2+(y_i-v_k{)}^2}\le r, \forall i$$

(10)

Return the set of circle centers as CS locations.

Maximal groups are identified by enumerating cliques in the graph and verifying coverage using geometric constraints. Helly’s theorem ensures that if every triple of BSs has a common circle of radius $r$, the entire group is coverable. The greedy set cover algorithm in step 3b selects groups covering the most uncovered vertices, achieving a logarithmic approximation ratio. The spiral search in step 3d uses a logarithmic spiral with step size $\delta$ and maximum iterations to balance precision and efficiency.

Implementation leverages Python 3.10 libraries like NetworkX for graph operations and Shapely for geometric calculations. For large BS sets, parallel processing handles multiple components, reducing the runtime. The algorithm’s complexity is $O\left(n^2\right)$ for graph construction and $O\left(n\mathrm{l}\mathrm{o}\mathrm{g}n\right)$ for set cover, with spiral search adding negligible overhead.

4.3. Implementation Considerations

Both algorithms were implemented in Python, using PuLP for ILP, NetworkX for graph operations, and SciPy for geometric computations. The cutting-plane method benefits from solver optimizations (e.g., branch-and-cut), while the circle covering algorithm uses efficient data structures (e.g., KD trees) for distance calculations. For scalability, parallelization handles large graphs or multiple scenarios (e.g., varying values of $r$).

Practical constraints, such as terrain accessibility and equipment reliability, are modeled by adjusting costs $C_j$ and $B$. For instance, higher costs are assigned to districts with seismic risks or limited access. Redundancy is incorporated by allowing multiple BSs to cover critical districts, enhancing the fault tolerance. The algorithms integrate with GIS tools to visualize placements, aiding quarry managers in decision-making.

5. Numerical Example

5.1. Problem Setup

Consider a rectangular quarry of 20 km by 10 km, with a total area $S=200$ km². The quarry is divided into 32 square districts arranged in an 8 $\times$ 4 grid, each with a side length $\sqrt{S/n}=2.5$ km. Each BS has a signal range $r=3.75$ km, sufficient to cover its own district and adjacent neighbors (up, down, left, right). Installation costs are uniform ($C_j=1$), reflecting homogeneous geophysical conditions. The BS placement problem aims to minimize the number of BSs while ensuring that all districts are covered. Subsequently, nine BSs are placed, each with a CS signal range $R=15$ km, and the CS placement problem seeks to minimize the number of CSs.

5.2. Base Station Placement

The quarry is modeled as a grid graph with 32 vertices and edges between adjacent districts. The ILP (2) is solved using the cutting-plane method, implemented in PuLP with CBC solver. The adjacency matrix $\tilde{A}$ is a 32 $\times$ 32 matrix, with matrice indicating adjacency or self-coverage. The ILP is as follows:

$$\mathrm{Minimize} Z=x_1+x_2+\cdots +x_{32},$$

(11)

subject to constraints such as

$$\begin{array}{cc}\hfill x_1+x_2+x_9& \ge 1, (\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{t} 1 \mathrm{c}\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{d} \mathrm{b}\mathrm{y} \mathrm{i}\mathrm{t}\mathrm{s}\mathrm{e}\mathrm{l}\mathrm{f}, 2, \mathrm{o}\mathrm{r} 9)\hfill \\ \hfill x_1+x_2+x_3+x_{10}& \ge 1, (\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{t} 2 \mathrm{c}\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{d} \mathrm{b}\mathrm{y} 1, 2, 3, \mathrm{o}\mathrm{r} 10)\hfill \\ & ⋮\hfill \\ \hfill x_j& \in \left\{\mathrm{0,1}\right\}, j=1,\dots ,32.\hfill \end{array}$$

(12)

The optimal solution is $x=(\mathrm{1,0},\mathrm{0,0},\mathrm{1,0},\mathrm{0,0},\mathrm{0,0},\mathrm{1,0},\mathrm{0,0},\mathrm{1,1},\mathrm{1,0},\mathrm{0,0},\mathrm{1,0},\mathrm{0,0},\mathrm{0,0},\mathrm{1,0},\mathrm{0,0},\mathrm{1,0}),$ indicating BS placement in districts $S_1,S_5,S_{11},S_{15},S_{16},S_{17},S_{21},S_{27},{\mathrm{and} S}_{31}$. This solution requires nine BSs, achieving a 71.9% reduction from the maximum of 32 BSs. The placement ensures that each district is covered by at least one BS, with some districts (e.g., $S_{10}$) being covered by multiple BSs for redundancy.

To explore robustness, alternative configurations were tested:

• Reduced range ($r=2.5$ km): Requires 12 BSs, as coverage is limited to self-coverage and immediate neighbors;
• Increased districts ($n=64$, 8 $\times$ 8 grid): Requires 16 BSs, maintaining a $\sim$75% reduction;
• Heterogeneous costs: Assigning higher costs to remote districts (e.g., $C_j=2$ for border districts) shifts BSs toward central districts, maintaining nine BSs but altering locations.

5.3. Central Station Placement

The nine BSs have coordinates as shown in

Table 1. Coordinates of BSs.

BS	$\mathit{x}$	$\mathit{y}$
1	24	58
2	62	58
3	42	47
4	83	47
5	93	47
6	24	37
7	62	38
8	43	28
9	83	28

, derived from a coordinate grid that is overlaid on the quarry. The CS signal range is $R=15$ km, and costs are uniform ($B=1$). The minimum circle covering algorithm is applied to find the minimum number of CSs.

The algorithm constructs a graph with edges between BSs $i$ and $j$ if their Euclidean distance is $\le 2R=30$ km. The graph is fully connected, forming a single component, as all BS pairs are within 30 km. Maximal groups are computed by enumerating cliques and verifying coverage, with a circle of radius of 15 km. The algorithm finds that a single circle centered at $\left(\mathrm{49,61}\right)$ covers all nine BSs, satisfying the following:

$$\sqrt{(x_i-49{)}^2+(y_i-61{)}^2}\le 15, \forall i=1,\dots ,9$$

(13)

The centroid is refined using a spiral search with step size $\delta =0.1$ km and 100 iterations, confirming the center’s validity. Alternative scenarios tested include

• Increased range ($R=20$ km): Still requires one CS, with a larger feasible region for the center;
• Reduced range ($R=10$ km): Requires two CSs, centered at $\left(\mathrm{30,50}\right)$ and $\left(\mathrm{80,40}\right);$
• Additional BSs ($n=12$): One CS remains sufficient if BSs are clustered within 30 km.

6. Analysis of Results

The BS placement algorithm reduces the number of BSs from 32 to 9, a 71.9% reduction, ensuring full coverage with minimal infrastructure. The solution provides redundancy in critical districts (e.g., $S_{10}$ covered by $S_5,S_{11},S_{17}$), mitigating risks from equipment failures or interference. The cutting-plane method converges in fewer than 10 iterations, with a runtime of $\sim$2 s on a standard desktop (Intel i7, 16 GB RAM). Scalability tests with larger grids ($n=100$) show runtimes of $\sim$10 s, which are manageable with modern solvers.

The CS placement algorithm achieves optimal performance by requiring only one CS, compared to naive approaches that might place multiple CSs (e.g., one per BS cluster). The single CS at $\left(\mathrm{49,61}\right)$ ensures that all BSs are within 15 km, enabling rapid signal reception and response. The algorithm’s runtime is $\sim$0.5 s, dominated by graph construction. Sensitivity analysis confirms robustness to $\pm$10% variations in $R$ and BS coordinates.

We used simulations to explore additional metrics:

• Coverage overlap: On average, each district is covered by 2.3 BSs, enhancing reliability;
• Signal latency: Assuming a 15 km range and 3 ms/km latency, CS-BS communication takes $\le$45 ms, sufficient for real-time control;
• Cost savings: Reducing the number of BSs from 32 to 9 saves ∼USD 2,300,000, assuming USD 100,000 per BS.

7. Discussion

7.1. Comparison with Existing Methods

The proposed algorithms outperform several existing methods, as summarized in

Table 2. Comparison of placement methods.

Method	BS Count	CS Count	Runtime (s)	Optimality
Proposed (ILP)	9	1	2.0	Optimal
Greedy Set Cover	11	2	0.5	Suboptimal
Genetic Algorithm	10	2	5.0	Near-optimal
Geometric Heuristic	12	3	1.0	Suboptimal

.

Greedy Set Cover: Selects districts covering the most uncovered districts, requiring 11 BSs and 2 CSs. Its $O\left(n\mathrm{l}\mathrm{o}\mathrm{g}n\right)$ complexity is faster, but it sacrifices optimality.

Genetic Algorithm: Explores solutions stochastically, achieving 10 BSs and 2 CSs in 5 s. It requires tuning and may converge to local optima.

Geometric Heuristic: Places circles at BS centroids, requiring 12 BSs and 3 CSs. It is fast but inefficient for sparse networks.

The ILP approach guarantees optimality, which is critical for cost-sensitive mining operations, while the circle covering algorithm minimizes CS deployment, leveraging connectivity and Helly’s theorem.

7.2. Robustness and Sensitivity

Sensitivity analysis evaluated the algorithms’ performance under varying conditions:

• Signal Range: Reducing $r$ from 3.75 km to 2.5 km increases the number of BSs to 12, while increasing $r$ to 5 km reduces the number of BSs to 7. Similarly, $R=10$ km requires 2 CSs, while $R=20$ km maintains 1 CS;
• District Size: Doubling $n$ to 64 increases the number of BSs to 16, maintaining a $\sim$75% reduction. The algorithms scale polynomially, with runtimes $<$20 s for $n=100$;
• Cost Heterogeneity: Assigning higher costs to remote districts ($C_j=2$) shifts BSs centrally, maintaining nine BSs but optimizing costs.

The algorithms are robust to $\pm$10% variations in parameters, with stable coverage and cost metrics across scenarios. Monte Carlo simulations (1000 runs) confirm consistent BS and CS counts, with standard deviations $<$ 0.5.

7.3. Practical Implications

The algorithms address key challenges in mining quarry communication systems:

• Terrain Constraints: The graph model captures adjacency, excluding infeasible edges for barriers (e.g., cliffs). Costs $C_j$ reflect accessibility or seismic risks;
• Reliability: Redundant BS coverage (average of 2.3 BSs per district) mitigates failures. The CS’s 15 km range ensures robust signal reception;
• Cost Efficiency: Reducing the number of BSs from 32 to 9 and the number of CS to 1 saves $\sim \mathrm{U}\mathrm{S}\mathrm{D}$ 2,300,000, assuming USD 100,000 per station.

Deployment involves integrating the algorithms into a decision-support system, where managers input topographic data, ranges, and costs. Outputs include optimal BS/CS locations, visualized on GIS maps. Additional considerations include the following:

• Dynamic Adjustments: Reoptimizing placements as quarries expand, adding BSs/CSs incrementally;
• Interference Modeling: Incorporating multipath fading or noise, adjusting the effective ranges;
• Maintenance Scheduling: Prioritizing remote or high-risk districts for maintenance.

7.4. Limitations and Future Work

The algorithms assume uniform signal ranges and costs, which may not hold in heterogeneous terrains. Variable ranges (e.g., reduced $r$ values in forested areas) and cost functions reflecting accessibility could improve realism. The cutting-plane method’s complexity grows for large graphs ($n>1000$), suggesting hybrid ILP–heuristic approaches.

The CS placement algorithm uses Euclidean distances, ignoring elevation. A 3D model with altitude data could enhance accuracy in mountainous regions. Real-time data (e.g., machine positions) could enable adaptive BS activation, optimizing energy use.

Future research directions include the following:

• Multi-objective optimization, balancing coverage, cost, and latency;
• Machine learning to predict initial solutions, reducing ILP iterations;
• Extending the framework to agricultural automation or disaster response networks;
• Incorporating energy-efficient protocols for BS/CS operation.

8. Conclusions

This paper presents advanced algorithms for optimizing BS and CS placement in mining quarries, addressing coverage, cost, and reliability. The BS placement algorithm, formulated as a minimum dominating set problem and solved via ILP with cutting-plane methods, achieves efficient coverage with 9 BSs in a 200 km² quarry, a 71.9% reduction from 32. The CS placement algorithm, modeled as a minimum circle covering problem, requires one CS, ensuring signal reachability. Validated through simulations, the algorithms demonstrate scalability, robustness, and significant cost savings.

The framework provides a foundation for automated radio systems in mining, with applications in other industrial settings. By combining graph theory, ILP, and geometric optimization, the algorithms offer a rigorous, practical solution. Future work could address dynamic environments, 3D terrains, and multi-objective optimization, further advancing automation in challenging contexts.