The F/DR-D-10 Algorithm: A Novel Heuristic Strategy to Solve the Minimum Span Frequency Assignment Problem Embedded in Mobile Applications

Abstract

Wireless communication supports various real-world applications, such as aeronautical navigation, satellite and TV broadcasting, wireless LANs, and mobile communications. The inherent characteristics of the electromagnetic spectrum impose constraints on telecommunication channels and their frequency bandwidths within mobile networks. A persistent challenge in these applications is providing high-demand services to mobile users, where frequency assignment problems, also known as channel assignment problems, assume significance. Researchers have developed several modeling approaches to address different facets of this problem, including the management of interfering radio signals, the assessment of available frequencies, and optimization criteria. In this paper, we present improved algorithms for solving the Minimum Span Frequency Assignment Problem in mobile communication systems using the greedy optimization approach known as F/DR. We solved and evaluated twenty well-known benchmark cases to assess the efficacy of our algorithms. Our findings consistently demonstrate that the modified algorithms outperform the F/DR approach with comparable computational complexity. The proposed algorithm notably achieves the following benchmarks: The modified algorithms consistently produce at least one local optimum better than the traditional algorithm in all benchmark tests. In 95% of the benchmarks evaluated, the probability of discovering a local optimum value (calculated by the modified algorithm) that is better than or equal to the one found by the conventional algorithm exceeds 50%.

1. Introduction

1.1. Overview

The electromagnetic spectrum is a finite and non-renewable resource [1]. Consequently, optimizing its utilization becomes imperative in various real-world engineering applications, such as aeronautical navigation systems, satellite communications, TV broadcasting, wireless LANs, and mobile communications. In addition, the inherent characteristics of the electromagnetic spectrum impose constraints on telecommunications channels and their frequency bandwidths in mobile networks, highlighting the critical necessity for efficient utilization. To address the high demand from mobile users, it is crucial to efficiently allocate and reuse channels to optimize specific system objectives. This challenge is commonly referred to as the Channel Assignment Problem (CAP) or Frequency Assignment Problem (FAP) [2,3]. The majority of CAP-related optimization problems seek to enhance network traffic, improve service quality, reduce the number of frequencies allocated to a given service, or minimize overall network interference [2,3,4,5]. Two main reasons justify the modeling and resolution of this issue in wireless networks:

• Allocation problems concern not only physical channels, but also the allocation of network resources to be reused, such as time-slots, sub-carries, sub-channels, channel coding, or transmission rate for a given QoS, which are required in a variety of modern technologies based on Frequency Division Multiple Access (FDMA), Time Division Multiple Access (TDMA), Code Division Multiple Access (CDMA), Space Division Multiple Access (SDMA), Orthogonal Frequency Division Multiple Access (OFDMA), and Non-orthogonal Multiple Access (NOMA) [6,7].
• The physical channels are closely associated with the transmitted physical frequencies within the system. These frequencies undergo various propagation phenomena, including attenuation, selective fading, and the Doppler effect, which impact the system’s coexistence due to their influence on electromagnetic interference conditions from neighboring users.

1.2. Channel Assignment Framework

In [8], Hale formalized this optimization problem using graph theory, establishing its equivalence to generalized graph coloring problems. Within this context, the resultant optimization problem is fundamentally combinatorial and falls into the category of NP-hardness problems [8]. The CAP problem in wireless networks is commonly addressed through four standard formulations, as outlined in [2]:

• The Maximum Service and Minimum Blocking Frequency Assignment Problem (referred to as Max-FAP) aims to assign the largest number of channels required by a cell in a wireless network while ensuring interference restrictions are met. This approach is essential for determining the maximum capacity of the network under quality of service conditions.
• The Minimum Order Frequency Assignment Problem (MO-FAP) seeks to reduce the number of different channels used in the wireless network while adhering to interference restrictions. This formulation is crucial for achieving a cost-effective network for the utilization of wireless frequencies.
• The Minimum Span Frequency Assignment Problem (MS-FAP) focuses on minimizing the bandwidth of the networks, ensuring that channel demand and interference restrictions are met. This approach plays a critical role in determining the channels required for the network’s service grade.
• The Minimum Interference Frequency Assignment Problem (MI-FAP) aims to decrease the overall interference created by the wireless network to fulfill channel demand. This approach is relevant in scenarios where the network is extensive, and a solution without interference is not feasible.

1.3. State-of-the-Art

Multiple researchers have attempted to efficiently approximate the CAP solution using various optimization techniques [9], such as Linear Programming, Graph Theory, Heuristics, and Bio-inspired Evolutionary Algorithms. In linear programming, the goal is to simplify the formulation into binary or integer programming, relying on linear constraints [10]. In graph theory, each cell is represented as a vertex, and the edges symbolize connections between devices. This approach facilitates the formulation and resolution of the graph coloring problem, which involves finding the minimum assignment of distinct colors within the graph created by its nodes and edges [11,12]. Due to the potentially significant computational costs associated with the two aforementioned approaches, alternative optimization strategies have been developed for identifying local optima in a wide range of real-world problems.

On one hand, heuristics represent straightforward strategies that guide the discovery of local optima. These strategies rely on logical or procedural rules closely aligned with the problem’s nature. Noteworthy among the successful heuristics employed in channel assignment are Taboo Search [13,14], Simulated Annealing [15,16], and Greedy Algorithms [17,18,19,20]. On the other hand, a wide range of modern Bio-inspired Evolutionary Algorithms [21] have been developed for more difficult optimization problems. As examples, we find Genetic Algorithms (GA) [22,23,24,25], Neural Networks (NN) [26,27], Particle Swarm Optimization (PSO) [28,29], Ant Colony Algorithms (ACO) [30], Grey Wolf algorithms (GWO) [31,32], Artificial Bee Colony Algorithms (ABC) [33,34], Firefly Algorithms (FA) [35,36], Whale Algorithms (WOA) [37], Quantum-based Avian Navigation Algorithms (QANA) [38], Zebra Optimization Algorithms (ZOA) [39], and the more general Swarm Intelligence (SI) [40].

1.4. Directly Related Works

In the context of the channel assignment problem, heuristics are of significant importance for two key reasons. Firstly, heuristics enable the rapid identification of feasible solutions, making them well-suited for real-time applications with satisfactory outcomes. Secondly, and equally vital, they facilitate the generation of a well-distributed initial population (i.e., the first generation) for all Bio-inspired Evolutionary Algorithms.

A popular greedy algorithm for solving the MS-FAP is the heuristic algorithm based on the frequency allocation strategy with node degree reordering (F/DR). This algorithm was originally proposed in [41,42] and further refined in [43]. It incorporates a cell prioritization strategy and an exhaustive frequency allocation algorithm for channel assignment [43]. This strategy is highly valuable in CAP because it enables the finding of solutions that, in many cases, nearly approximate the global optimum while requiring little computing complexity, execution time, and implementation cost [44].

For these reasons, the F/DR algorithm has been modified and applied in several problems, including Max-FAP with non-uniform cells, channel assignment incorporating adjacent channel information, and power allocation problems [45,46,47]. It has also been integrated as part of other meta-heuristics, as a local optimization algorithm to tune each generation [48,49,50]. Despite the algorithm’s simplicity, low computational cost and effectiveness, we identified limited literature focused on enhancing this basic heuristic for MS-FAP [43,51,52,53]. This could possibly be attributed to its integration with other meta-heuristics, where the algorithm assumes a secondary role. The F/DR algorithm can be improved because its cell prioritization strategy employs an ordering that favors cells with a higher level of difficulty related to the criteria used to create the sorted list. For instance, employing this heuristic may result in poor channel allocation in scenarios with high-demand cells that do not electromagnetically interfere with their neighboring cells.

1.5. Paper Outline

This paper proposes two modified algorithms based on F/DR heuristic, which were evaluated using several international benchmarks. Our contributions can be summarized as follows:

• The primary contribution of this research is the development of algorithms that increase the likelihood of discovering better solutions when compared to the existing F/DR algorithm. Our proposed methods achieve this without introducing additional computational complexity or prolonging execution times.
• A second significant contribution of our work lies in the statistical nature of the proposed algorithms. These algorithms are designed with the versatility to generate well-distributed initial populations. This feature has broader implications, as it facilitates the efficient use of these algorithms as initialization strategies for various complex meta-heuristics.

The remaining parts of this article are organized as follows: the problem statement will be presented in Section 2, where the formulation of MS-FAP and the associated F/DR algorithm are introduced. In Section 3, the two modified algorithms will be presented. The evaluation methodology of these algorithms and the benchmarks used are summarized in Section 4. In addition, we have selected benchmarks that are traditionally used to evaluate these heuristics. We made this choice due to their inherent complexity and the presence of well-established lower bounds. The evaluation of the proposed algorithms will be presented comparatively in Section 5. Finally, the last section presents the conclusions of this research.

2. CAP in Cellular Mobile Networks

2.1. The Minimum Span Frequency Assignment Problem (MS-FAP)

The Minimum Span Frequency Assignment Problem (MS-FAP) is a CAP model in which the resulting channel assignment satisfies the demand of each cell, adheres to the channel spacing constraints between cells, and minimizes the system’s bandwidth usage. This involves minimizing the gap between the maximum and minimum channel assignments in the entire system [54,55].

The conventional variables and nomenclature for the MS-FAP problem are given by [56], where:

• The network capacity is determined by M different non-negative channels, all of which are accessible to N different cells.
• The demand vector is given by $\mathbf{D}={[D_1,\cdots ,D_i,\cdots ,D_N]}_{1\times N}$, where it contains the number of channels requested by the i-th cell.
• The electromagnetic compatibility matrix $\mathbf{C}={\left[C_{ij}\right]}_{N\times N}$ contains the channel spacing constraints between the i-th and j-th cells. These constraints can be classified and interpreted as co-channel (CCC), adjacent (ACC), or co-site (CSC) limits [56].
• The channel assignment matrix $\mathbf{F}={\left[F_{ik}\right]}_{N\times M}$ represents the assigned channel number $F_{ik}$ to the i-th cell. These channels are chosen from a list of positive integers denoted by $\{1,2,\cdots ,M\}$ or assigned as zero if no channel was allocated.

Using the variables and nomenclature previously shown, the MS-FAP is formulated, and the resulting optimization problem is given by:

$$min\left\{\underset{\forall i,k}{max}\left\{F_{ik}\right\}\right\}$$

(1)

$$\begin{array}{cc}& \forall i,j\in \{1,\cdots ,N\}\\ \mathrm{s}.\mathrm{t}.\left|F_{ik}-F_{jl}\right|\ge C_{ij},& \forall k,l\in \{1,\cdots ,M\}\\ & \forall i\ne j,\forall k\ne l,\forall F_{ik,}{F}_{jl}\in \mathbb{N}\end{array}$$

(2)

$$\mathrm{s}.\mathrm{t}.\sum _{k=1}^M\{1-\delta \left[F_{ik}\right]\}=D_i,\forall i\in \{1,\cdots ,N\}$$

(3)

where $\delta \left[x\right]$ denotes the Delta–Kronecker function, which returns 1 if $x=0$ and 0 if $x\ne 0$. In this formulation, the goal of reducing network bandwidth utilization is represented by (1), compliance with electromagnetic limitations is represented by (2), and fulfillment of network demand is represented by (3).

To clarify the nomenclature, we observe two cells being solved using the MS-FAP in Figure 1, where the electromagnetic compatibility matrix is given by

$$\mathbf{C}=\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right],$$

(4)

the demand vector is $\mathbf{D}=[3,2]$, and the non-zero frequencies are $F_1=\{F_{11},F_{12},F_{13}\}$ and $F_2=\{F_{21},F_{22}\}$ for cells one and two, respectively. For example, the assigned channels $F_1=\{2,4,6\}$ and $F_2=\{1,3\}$ are feasible solutions considering the electromagnetic compatibility and demand constraints. However, the MS-FAP aims to minimize the overall span (i.e., the network’s bandwidth). Therefore, the only valid solution in this example is $F_1=\{1,3,5\}$ and $F_2=\{2,4\}$, where $min\left\{{max}_{\forall i,k}\left\{F_{ik}\right\}\right\}=5$ represents the minimum of the maximum frequencies that satisfy all constraints. As a result, the network’s bandwidth decreases due to the utilization of only five channels to fulfill all service requirements.

2.2. F/DR Algorithm

The F/DR algorithm, based on a frequency allocation strategy with node degree reordering, is a well-known greedy algorithm used to solve FAP problems [55]. To tackle the FAP problem, the conventional F/DR algorithm splits it into two sub-problems. The first deals with the optimal cell ordering, while the second focuses on the optimal channel assignment to these ordered cells [41,42,43]. However, these two problems are closely related [43]. For instance, using two ordered cell lists (based on two different criteria) with the same channel allocation algorithm (in the second sub-problem) yields different results. Furthermore, if the correct ordering of the list or the correct prioritization criteria were previously known, the problem would ideally be solved. In the conventional F/DR algorithm, a cell organization criterion based on the highest level of assignment difficulty is used. Consequently, cells (or groups of cells) with the same difficulty level, which may be equally relevant, could experience inefficient channel allocation. This could be the case for a high-demand cell whose immediate relationship with other cells is irrelevant.

To address optimal cell ordering, the F/DR algorithm employs a strategy known as node-degree sorting (DR), which generates an ordered list, L, from the highest to the lowest cell degree value, eliminating cells with no demand ($D_i=0$). The sorting criterion is as follows:

$$P_i=\left\{\begin{array}{cc}\sum _{j=1}^N{D}_j{C}_{ij},& \mathrm{Si}{D}_i\ne 0\\ 0,& \mathrm{Si}{D}_i=0\end{array}\right\},\forall i\in \{1,\cdots ,N\},$$

(5)

Here, $P_i$ represents the degree of node i. According to this definition, the prioritization criterion is based on the demand of neighboring cells as well as the number of separation channels required between them. This criterion was compared with some non-traditional ones in [43], and its performance was found to be the worst case. On the other hand, the F/DR algorithm employs an exhaustive algorithm in frequency (F) to address optimal channel allocation. This method identifies the first frequency that can be assigned to the first cell in list L. Subsequently, the demand for that cell is reduced, and the list is rearranged. This sequence is repeated until the demand in all cells is zero. Algorithm 1 presents the pseudo-code for this heuristic algorithm.

Algorithm 1 Exhaustive frequency strategy (F)

Inputs: $\mathbf{C}={\left[C_{ij}\right]}_{N\times N}$ and $\mathbf{D}={[D_1,\cdots ,D_i,\cdots ,D_N]}_{1\times N}$ Output: $\mathbf{F}={\left[F_{ik}\right]}_{N\times M}$ 1: While ($\mathbf{D}\ne {[0,\cdots ,0]}_{1\times N}$) 2: Generate an ordered list L using the DR criteria (or replace by DR-k, DR-D-k or PR criteria) 3: $f=1$ 4: Identify the cell located in the first list position, denoted by $L\left\{1\right\}$ 5: While ($f\ne 0$) 6: Check the constraints for the i-th cell and the frequency f 7: If assignment possible, assign f to i-th cell in $F_{ik}$, update $D_i=D_i-1$, $f=0$ 8: If assignment is not possible, do $f=f+1$ 9: End While 10: End While 11: The answer is $\mathbf{F}$ and exit

3. Modified F/DR Algorithms

Two novel ordering strategies were proposed to modify the traditional F/DR algorithm. These strategies were designed to introduce statistical diversity to the list, L, created under DR sorting. Therefore, the modified F/DR algorithms have a pseudo-code that is very similar to the conventional one, as summarized in Algorithm 1. However, they generate the ordered list, L, using a modified DR criteria, as follow:

3.1. DR-k Algorithm

To introduce statistical diversity into the list generated by the DR method (i.e., L), the DR-k algorithm proposes to swap the cells in the list following the algorithm:

• Generate an ordering list following the DR criteria.
• Exchange the cell $L\left\{i\right\}$ at the end of the list with a probability of $\frac{k}{100}$, where k is a number between 0 and 100.

3.2. DR-D-k Algorithm

In this strategy, cells with the highest demand are less likely to be exchanged in the sorting list. This approach probabilistically avoids the creation of a local minimum that results from the DR priority scheme. In particular, the DR-D-k algorithm proposes to swap the cells in the list following the algorithm:

• Generate an ordering list following the DR criteria.
• Exchange the cell $L\left\{i\right\}$ at the end of the list with a probability of $\frac{k}{100}\left\{1-\frac{D_{L\left\{i\right\}}}{max\mathbf{D}}\right\}$, where k is a number between 0 and 100, $D_{L\left\{i\right\}}$ is the demand of the cell $L\left\{i\right\}$, and $max\mathbf{D}$ is the maximum value of the demand vector D.

4. Materials and Methods

International benchmark problems with well-known theoretical lower limits (calculated using graph theory) were selected to assess the proposed modified F/DR algorithms. A summary of these scenarios can be found in

Table 1. Benchmark description, where the compatibility matrices ${\mathbf{C}}_1,\cdots ,{\mathbf{C}}_{10}$ and ${\mathbf{C}}_{11}$ are described in [43,56,57] and [58,59], respectively.

Problem	Compatibility	Demand Vector
	Matrix
P1	${\mathbf{C}}_1$	${\mathbf{D}}_1:=[20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20]$
P2		${\mathbf{D}}_2:=[5,5,5,8,12,25,30,25,30,40,40,45,20,30,25,15,15,30,20,20,25]$
P3		${\mathbf{D}}_3:=[8,25,8,8,8,15,18,52,77,28,13,15,31,15,36,57,28,8,10,13,8]$
P4		${\mathbf{D}}_4:=2\times {\mathbf{D}}_3$
P5		${\mathbf{D}}_5:=4\times {\mathbf{D}}_3$
P6	${\mathbf{C}}_2$	${\mathbf{D}}_1$
P7		${\mathbf{D}}_2$
P8		${\mathbf{D}}_3$
P9	${\mathbf{C}}_3$	${\mathbf{D}}_3$
P10	${\mathbf{C}}_4$	${\mathbf{D}}_3$
P11	${\mathbf{C}}_5$	${\mathbf{D}}_3$
P12	${\mathbf{C}}_6$	${\mathbf{D}}_3$
P13	${\mathbf{C}}_7$	${\mathbf{D}}_2$
P14		${\mathbf{D}}_3$
P15	${\mathbf{C}}_8$	${\mathbf{D}}_2$
P16		${\mathbf{D}}_3$
P17	${\mathbf{C}}_9$	${\mathbf{D}}_2$
P18		${\mathbf{D}}_3$
P19	${\mathbf{C}}_{10}$	${\mathbf{D}}_2$
P20	${\mathbf{C}}_{11}$	${\mathbf{D}}_6:=[10,11,9,5,9,4,5,7,4,8,8,9,10,7,7,6,4,5,5,7,6,4,5,7,5]$

. The problems from P1 to P19 are described in [2,43,56,57] and correspond to the Philadelphia problem, which formulates a cellular network comprising 21 cells. The final problem (i.e., P20) corresponds to the requirements of an operational cellular network consisting of 25 cells in the city of Helsinki, Republic of Finland, as described in [58,59].

All benchmarks were implemented and solved in MATLAB^® V.9.11. The conventional F/DR algorithm and the two modified algorithms (as described in the previous section) were employed to solve these scenarios. Additionally, to assess the impact of both the prioritization algorithm and channel assignment strategy in the selected scenarios, all benchmarks were solved by generating a list, L, with a random permutation. That algorithm will be referred to as F/PER in this paper. This ordering strategy is equivalent to not having a specific prioritization algorithm before applying the allocation algorithm F.

Each heuristic strategy (i.e., F/PER, F/DR-k, and F/DR-D-k) was simulated 100 times, whereas the standard F/DR algorithm was simulated only once due to its deterministic nature. The percentages of better, worse, and equal optimizations were used to assess the merit of the proposed modified algorithms. These percentages were calculated by dividing the number of times the random scenario found a local optimum that was lower, higher, or equal (to the one found by the deterministic conventional F/DR algorithm) by the total number of calculated solutions (i.e., 100).

Furthermore, the performance assessment of the algorithms was conducted through a rigorous comparison between the local optimum value obtained by the conventional algorithm and the best local optimum value achieved by each novel strategy. To quantify this evaluation, we introduced two essential metrics. Firstly, we defined the Probability of One Better Result (PoOBR), representing the percentage of instances in which at least one local optimum value, as computed by the modified algorithm, exhibited superior performance when compared to the conventional algorithm. Secondly, we introduced the Probability of One Better or Equal Result (PoOBE), signifying the percentage of instances whose worsening probability is less than 50%.

5. Results and Discussion

Following the methodology described in the previous section, we calculated and summarized the results listed in

Table 2. Comparison between F/DR and F/PER.

Conventional Approach			Random Permutation of the DR List
Problem	Theoretical Lower Limits $\mathbf{min}\left\{\mathbf{max}\left\{{\mathbf{F}}_{\mathbf{ik}}\right\}\right\}$	F/DR $\mathbf{min}\left\{\mathbf{max}\left\{{\mathbf{F}}_{\mathbf{ik}}\right\}\right\}$	The Best F/PER $\mathbf{min}\left\{\mathbf{max}\left\{{\mathbf{F}}_{\mathbf{ik}}\right\}\right\}$	Improve Probability {F/DR}	Worsening Probability {F/DR}	Average Execution Time per Test [s]
P1	239	259	262	0%	100%	1.0
P2	257	292	315	0%	100%	1.4
P3	426	448	536	0%	100%	1.9
P4	855	895	1082	0%	100%	9.4
P5	1713	1801	2137	0%	100%	62.4
P6	179	240	222	58%	42%	0.8
P7	252	269	294	0%	100%	1.1
P8	426	476	520	0%	100%	1.6
P9	524	593	612	0%	100%	2.7
P10	426	467	493	0%	100%	2.1
P11	426	472	513	0%	100%	2.2
P12	532	533	538	0%	100%	1.8
P13	308	310	309	27%	56%	0.8
P14	532	533	534	0%	100%	1.2
P15	308	315	319	0%	100%	1.1
P16	532	533	562	0%	100%	1.6
P17	220	222	225	0%	100%	0.8
P18	380	381	392	0%	100%	1.1
P19	528	533	529	46%	44%	1.4
P20	72	74	73	4%	90%	0.1

,

Table 3. Comparison between F/DR and F/DR-k.

Conventional Approach				F/DR-10			F/DR-50			F/DR-90
Problem	Theoretical Lower Limits $\mathbf{min}\left\{\mathbf{max}\left\{{\mathbf{F}}_{\mathbf{ik}}\right\}\right\}$	F/DR $\mathbf{min}\left\{\mathbf{max}\left\{{\mathbf{F}}_{\mathbf{ik}}\right\}\right\}$		The Best F/DR-10 $\mathbf{min}\left\{\mathbf{max}\left\{{\mathbf{F}}_{\mathbf{ik}}\right\}\right\}$	Improve Probability {F/DR}	Worsening Probability {F/DR}	The Best F/DR-50 $\mathbf{min}\left\{\mathbf{max}\left\{{\mathbf{F}}_{\mathbf{ik}}\right\}\right\}$	Improve Probability {F/DR}	Worsening Probability {F/DR}	The Best F/DR-90 $\mathbf{min}\left\{\mathbf{max}\left\{{\mathbf{F}}_{\mathbf{ik}}\right\}\right\}$	Improve Probability {F/DR}	Worsening Probability {F/DR}	Average Execution Time per Test [s]
P1	239	259		251	69%	21%	251	65%	23%	254	11%	81%	1.2
P2	257	292		282	85%	8%	283	92%	6%	285	11%	86%	1.5
P3	426	448		438	92%	2%	445	3%	94%	452	0%	100%	1.9
P4	855	895		881	93%	4%	904	0%	100%	907	0%	100%	8.9
P5	1713	1801		1765	100%	0%	1822	0%	100%	1816	0%	100%	58.6
P6	179	240		231	65%	12%	230	57%	36%	219	88%	10%	0.9
P7	252	269		266	16%	47%	268	1%	92%	269	0%	99%	1.1
P8	426	476		451	94%	2%	449	100%	0%	456	87%	8%	1.7
P9	524	593		592	2%	94%	599	0%	100%	566	96%	2%	3.1
P10	426	467		463	45%	39%	461	15%	81%	453	40%	49%	2.4
P11	426	472		464	58%	27%	460	67%	29%	464	18%	76%	2.3
P12	532	533		533	0%	80%	533	0%	53%	533	0%	81%	1.8
P13	308	310		309	5%	5%	309	26%	38%	309	37%	40%	0.8
P14	532	533		533	0%	12%	533	0%	51%	533	0%	80%	1.2
P15	308	315		310	10%	0%	309	67%	0%	309	32%	55%	1.1
P16	532	533		533	0%	10%	533	0%	46%	533	0%	87%	1.6
P17	220	222		221	7%	7%	221	19%	33%	221	3%	89%	0.8
P18	380	381		381	0%	10%	381	0%	42%	381	0%	84%	1.1
P19	528	533		530	19%	1%	529	63%	20%	529	71%	21%	1.4
P20	72	74		73	16%	8%	73	84%	7%	73	20%	54%	0.2

and

Table 4. Comparison between F/DR and F/DR-D-k.

Conventional Approach				F/DR-D-10			F/DR-D-50			F/DR-D-90
Problem	Theoretical Lower Limits $\mathbf{min}\left\{\mathbf{max}\left\{{\mathbf{F}}_{\mathbf{ik}}\right\}\right\}$	F/DR $\mathbf{min}\left\{\mathbf{max}\left\{{\mathbf{F}}_{\mathbf{ik}}\right\}\right\}$		The Best F/DR-D-10 $\mathbf{min}\left\{\mathbf{max}\left\{{\mathbf{F}}_{\mathbf{ik}}\right\}\right\}$	Improve Probability {F/DR}	Worsening Probability {F/DR}	The Best F/DR-D-50 $\mathbf{min}\left\{\mathbf{max}\left\{{\mathbf{F}}_{\mathbf{ik}}\right\}\right\}$	Improve Probability {F/DR}	Worsening Probability {F/DR}	The Best F/DR-D-90 $\mathbf{min}\left\{\mathbf{max}\left\{{\mathbf{F}}_{\mathbf{ik}}\right\}\right\}$	Improve Probability {F/DR}	Worsening Probability {F/DR}	Average Execution Time per Test [s]
P1	239	259		252	69%	14%	251	74%	17%	251	69%	22%	1.2
P2	257	292		284	86%	8%	282	96%	3%	283	97%	3%	1.6
P3	426	448		438	66%	8%	437	93%	5%	438	75%	17%	2.0
P4	855	895		878	87%	4%	882	82%	15%	882	24%	71%	9.3
P5	1713	1801		1761	100%	0%	1771	100%	0%	1786	29%	67%	60.6
P6	179	240		231	52%	6%	229	70%	13%	231	82%	13%	1.0
P7	252	269		267	21%	25%	265	20%	58%	266	9%	81%	1.1
P8	426	476		461	72%	5%	450	98%	1%	451	100%	0%	1.7
P9	524	593		589	15%	77%	592	1%	99%	592	1%	99%	3.6
P10	426	467		464	44%	20%	462	28%	58%	463	13%	84%	2.8
P11	426	472		468	33%	26%	463	84%	8%	459	86%	11%	2.6
P12	532	533		533	0%	0%	533	0%	0%	533	0%	0%	1.9
P13	308	310		310	0%	0%	309	1%	14%	309	8%	18%	0.9
P14	532	533		533	0%	0%	533	0%	0%	533	0%	0%	1.1
P15	308	315		312	1%	0%	310	11%	0%	310	24%	0%	1.1
P16	532	533		533	0%	0%	533	0%	0%	533	0%	0%	1.7
P17	220	222		221	1%	1%	221	2%	9%	221	7%	20%	0.9
P18	380	381		381	0%	0%	381	0%	0%	381	0%	0%	1.2
P19	528	533		530	3%	0%	530	8%	2%	530	16%	13%	1.4
P20	72	74		73	2%	0%	73	40%	1%	73	89%	1%	0.2

. Each table compares the conventional algorithm, the new modified algorithms, and the theoretical bounds of each problem using two metrics. The first metric is the associated probabilities of improving, equaling, or worsening the local optimum result of each MS-FAP problem. For example, the probability of improving the result was calculated by dividing the number of times that the new algorithm discovered a better solution than the conventional algorithms by the total number of solutions (i.e., 100).

To simplify the tables, the probability of matching the conventional algorithm’s result was not displayed, but it can be calculated using the reported probabilities. The second metric compares the MS-FAP problem bounds to the best algorithm result, which is the lowest value among the 100 optimum values found by the modified algorithms. It is essential to emphasize that the theoretical bound column does not imply that the minimum result is that value but rather that no algorithm can find a result lower than that value as a theoretical limit. Furthermore, to aid in data comparison, results that are equal to or better than the F/DR algorithm are highlighted in bold, with an underline, while results that are very close to the theoretical limits are highlighted with a gray background.

The comparison between the traditional F/DR algorithm and the F/PER algorithm is presented in Table 2. With the exception of four cases, the results demonstrate the relevance of the F/DR strategy, as random list organization does not yield better results than the conventional algorithm. Problems P6, P13, and P19 are so complex that the F/DR algorithm cannot properly solve them because even a random organization of the list offers a 58%, 27%, and 46% probability of finding a better result, respectively. Problem P20 is excluded from this classification because, although F/PER produces a better result, it occurs in only 4% of the simulations. These findings indicate that the F/DR algorithm does not consistently produce global or near-global optimal results, despite its efficiency in local list prioritization. Analyzing the specificity of problems P6, P13, and P19 in terms of their electromagnetic compatibility matrix or demand vector does not provide a simple hypothesis that allows us to infer the fundamental challenge or restriction causing the F/DR algorithm to fail. However, in MS-FAP problems with high cell demand (e.g., P6), the F/PER algorithm outperforms the conventional one, possibly because the F strategy favors cells with a non-uniform distribution of channels [43].

As evident from Table 3 and Table 4, the proposed heuristics (i.e., F/DR-k and F-DR-D-k) exhibit a high likelihood of outperforming F/DR due to the introduction of statistical diversity into priority list L. In particular, the results obtained using the proposed F/DR-10 algorithm outperform those achieved with the conventional F/DR algorithm. Moreover, all assessed F/DR-D-k algorithms also surpass the performance of the conventional F/DR algorithm. When focusing solely on high-demand scenarios (i.e., P4 and P5), both F/DR-10 and F/DR-D-k strategies exhibit a high likelihood of improvement or equaling the F/DR algorithm’s result. In

Table 5. Some results of MS-FAP using main heuristic approaches in literature.

Problem	Optimal Value	F/DR [43]	Best Greedy Algorithms from [60]	This Work (Best Result)	Tabu Search from [61]	Simulated Annealing from [61]
P1	239	259	250	251	240	239
P2	257	292	284	282	269	260
P3	426	448	447	437	428	428
P4	855	895	894	878	858	858
P5	1713	1801	1800	1761	1724	1724

, we compare results for instances P1–P5 from the literature, including the optimal value, the well-established greedy heuristic algorithm developed in [60], Tabu Search and Simulated Annealing as discussed in [61], the F/DR algorithm [43], and our modified algorithms. We refrain from comparing our algorithm to Bio-inspired algorithms due to their distinct characteristics, namely, differences in implementation complexity and simulation time. In fact, Tabu Search and Simulated Annealing, for instance, enhance the performance of greedy algorithms, yielding results closely approaching the optimal value. However, it is worth noting that greedy algorithms exhibit significantly faster execution time. For instance, in the case of the largest network evaluated (i.e., P5), the F/DR-D and F/DR-D-k algorithms require only 1 minute of simulation time and they achieve an error rate of less than $3\%$ for the MS-FAP solution.

Based on Table 3 and Table 4, the PoOBR and the PoOBE were calculated and are summarized in

Table 6. Summary of the percentage of cases with improved performance.

	F/PER	F/DR-10	F/DR-D-10	F/DR-50	F/DR-D-50	F/DR-90	F/DR-D-90
PoOBR	20%	100%	100%	85%	100%	85%	100%
PoOBE	10%	90%	95%	60%	85%	30%	75%

. Therefore, F/DR-D-10 is characterized by the best results, which are as follows: (1) In 100% of the benchmarks used, at least one local optimum value (calculated by the modified algorithm) outperforms the conventional algorithm’s result, (2) in 95% of the benchmarks used, the probability of discovering a local optimum value (calculated by the modified algorithm) that is better than or equal to the one discovered by the conventional algorithm is greater than 50%.

6. Conclusions

This paper introduces two modifications to the F/DR algorithm for channel assignment problems in telecommunication systems, named as the F/DR-k and F/DR-D-k algorithms. These heuristics either match or improve upon the original algorithm’s results while maintaining the same level of implementation complexity and execution time. Notably, F/DR-D-10 stands out with the following achievements: (1) In 100% of the benchmarks used, at least one local optimum value (calculated by the modified algorithm) outperforms the conventional algorithm’s result, (2) In 95% of instances used to asses the proposed algorithm, the worsening probability was less than 50%. These novel heuristics can be seamlessly integrated into more complex optimization algorithms to enhance the optimization process’s outcomes.