# SAIS: A Novel Bio-Inspired Artificial Immune System Based on Symbiotic
Paradigm

Junhao Song  Faculty of EngineeringImperial College LondonLondonUnited Kingdom
<EMAIL_ADDRESS>,  Yingfang Yuan  School of Mathematical and
Computer SciencesHeriot-Watt UniversityEdinburghUnited Kingdom<EMAIL_ADDRESS>and  Wei Pang  School of Mathematical and Computer SciencesHeriot-Watt
UniversityEdinburghUnited Kingdom<EMAIL_ADDRESS>
(2024)

###### Abstract.

We propose a novel type of Artificial Immune System (AIS): Symbiotic
Artificial Immune Systems (SAIS), drawing inspiration from symbiotic
relationships in biology. SAIS parallels the three key stages (i.e.,
mutualism, commensalism and parasitism) of population updating from the
Symbiotic Organisms Search (SOS) algorithm. This parallel approach effectively
addresses the challenges of large population size and enhances population
diversity in AIS, which traditional AIS and SOS struggle to resolve
efficiently. We conducted a series of experiments, which demonstrated that our
SAIS achieved comparable performance to the state-of-the-art approach SOS and
outperformed other popular AIS approaches and evolutionary algorithms across
26 benchmark problems. Furthermore, we investigated the problem of parameter
selection and found that SAIS performs better in handling larger population
sizes while requiring fewer generations. Finally, we believe SAIS, as a novel
bio-inspired and immune-inspired algorithm, paves the way for innovation in
bio-inspired computing with the symbiotic paradigm.

Artificial Immune Systems, Computational Intelligence, Benchmark Problems

††copyright: acmcopyright††journalyear: 2024††doi:
10.1145/1122445.1122456††conference: Genetic and Evolutionary Computation
Conference (GECCO ’24); July 14–18, 2024; Melbourne, Australia††booktitle:
Genetic and Evolutionary Computation Conference (GECCO ’24), July 14–18, 2024,
Melbourne, Australia††price: 00.00††isbn: 978-1-4503-XXXX-X/18/06††ccs:
Computing methodologies Bio-inspired approaches

##  1\. Introduction

With the swift advancement of artificial intelligence, bio-inspired algorithms
have become crucial in tackling complex optimisation challenges (Vikhar, 2016;
Brownlee, 2011; Eiben and Smith, 2015). In this field, Artificial Immune
Systems (AIS) drawing inspiration from biological immune mechanisms, offers a
distinctive approach to addressing optimisation problems. Many AIS algorithms
conceptualize antigens as the objectives or constraints of a problem, while
antibodies are viewed as potential solutions. These AIS systems dynamically
update the antibodies to better align with the antigens (identifying optimal
solutions), mirroring the adaptive mechanisms of the biological immune system
(Brownlee, 2011). Despite the wide-ranging potential applications of AIS
across various fields, contemporary research on AIS has predominantly
concentrated on the refinement of individual antibodies (Brownlee, 2007; Ji
and Dasgupta, 2007; Kurapati and Azarm, 2000; Halavati et al., 2007). As
highlighted in (Halavati et al., 2007), there is a notable lack of emphasis on
the interactions among antibodies within existing AIS algorithms. This limited
scope of research restricts the capabilities of AIS in tackling complex
optimisation challenges. This is particularly evident in scenarios involving
the generation of antibodies at a large scale (Brownlee, 2011; Timmis et al.,
2004), where traditional AIS algorithms struggled to deal with challenges like
loss of population diversity and low computational efficiency (Zheng et al.,
2010; Segel et al., 2001).

This research presents the Symbiotic Artificial Immune System (SAIS), an
innovative AIS algorithm employed to address the aforementioned limitations.
Inspired by the symbiotic relationships in biology, SAIS not only focuses on
interconnections among antibodies, but also enhances their collective problem-
solving capabilities. We parallelized three stages (mutualism, commensalism
and parasitism) of the Symbiotic Organisms Search (SOS) algorithm (Cheng and
Prayogo, 2014) to SAIS. This parallel application significantly boosts the
algorithm’s efficiency and effectiveness in addressing complex optimisation
challenges. Through comparative experiments involving multiple algorithms,
including SAIS and SOS in 26 distinct benchmark problems, we showcase the
results achieved by SAIS in optimising solutions. These findings substantiate
the effectiveness of symbiotic interactions among antibodies, thereby
affirming the validity of our approach.

The rest of this paper is structured as follows. Section 2 introduces the
symbiotic relationship in the biological world and existing research of the
AIS. Section 3 details the algorithmic design and implementation of SAIS. In
Section 4, we evaluate the performance of SAIS through a series of benchmark
tests under varying parameters, and we conduct comparative analysis with other
algorithms. Furthermore, we provide an in-depth discussion of the experimental
results that elucidate the advantages of SAIS. Section 5 evaluates the role of
different population relationships in SAIS by removing different population
relationships (mutualism, commensalism and parasitism) from the symbiotic
relationship in SAIS. Finally, Section 6 summarizes the entire study and
proposes future research directions for SAIS.

##  2\. Related Work

###  2.1. Symbiotic Relationships in Biology

The word ’Symbiotic’ is derived from the Greek, meaning ’to live together’. It
was first used by De Bary in 1878 to describe the act of living together
between different organisms (species) (Sapp, 1994; Cheng and Prayogo, 2014).
Biology has discovered in the last five years that symbiotic relationships are
also influencing the human immune system. For example, probiotics in the human
body can help humans digest and increase immunity (Foundation, 2019; Belkaid
and Hand, 2014). Symbiotic relationship may be positive, negative, or neutral.
There are three basic types of symbiosis: mutualism, commensalism, and
parasitism (Foundation, 2019). Mutualism benefit both species. Commensalism
can be one species benefits and the other species is not affected. However,
parasitism can benefit one species and harm another.

Figure 1. Examples of Mutualism, Commensalism and Parasitism in Biology (CC
BY-ND 2.0 Licenses)

Mutualism between flower and bee, commensalism between shark and Remora fish,
malaria-transmitting mosquitoes infect human

As shown in Figure 1, bees collect nectar as food and help flowers spread
pollen. This mutualism relationship not only promotes their respective
survival and reproduction, but also has a positive impact on the health and
diversity of the entire ecosystem (Foundation, 2019). The relationship between
shark and Remora fish is a typical example of commensalism. The Remora fish
attaches to the shark’s body and uses the shark’s movements to swim and find
food more easily, without the shark itself being significantly affected. In
the rightmost of Figure 1, mosquitoes transmitting malaria to humans is an
example of a parasitic relationship. This mosquito ingests blood by biting
humans, thereby transmitting malaria and posing a serious threat to human
health. Mosquitoes gain benefits, but humans suffer negative consequences
(Foundation, 2019; Rooks and Garrett, 2016).

The study of symbiotic relationships in biology reveals complex interaction
mechanisms. These mechanisms play a key role in maintaining biodiversity and
ecological balance. These findings provide inspiration for simulating
interactions in nature in the field of computational intelligence.

###  2.2. Research on the AIS and Symbiotic AIS Applications

In the field of computational intelligence, Artificial Immune Systems (AIS) is
a collective name for a variety of algorithms inspired by biological immune
systems (Timmis et al., 2004; Brownlee, 2011). Many AIS simulate the dynamic
interactions between antibodies (potential solutions to a problem) and
antigens (the target solutions of the problem), though specific
implementations like the B cell algorithm (Kelsey and Timmis, 2003) and
dendritic cell algorithm (Greensmith and Aickelin, 2008) may not directly use
antibodies. While algorithms for AIS have been extensively studied, most
existing methodologies focus on the generation and variation of individual
antibodies (Brownlee, 2011; Timmis et al., 2004; Vikhar, 2016; Kurapati and
Azarm, 2000). For instance, the Clonal Selection Algorithm (CLONALG)
(Brownlee, 2007, 2011) involves repeated cloning and high-frequency mutation
of the most promising antibodies to enhance solution quality. The Negative
Selection Algorithm (NSA) (Forrest et al., 1994; Brownlee, 2011; Ji and
Dasgupta, 2007) emphasizes the identification and differentiation of normal
and abnormal states to assess solution quality. Most of these AIS algorithms
primarily concentrate on the changes within the antibodies themselves.

Even though AIS algorithms possess many different varieties, most of the AIS
algorithms are based on the core functionality shown in Algorithm 1. SAIS also
grounded in these fundamental AIS principles. Within the main loop of the
algorithm, AIS evaluates the affinity of the antibody population $P$ against
each antigen $atg\in ATG$. Antibodies exhibiting the highest affinity undergo
selection and mutation to generate new antibodies. This concept in AIS bears
resemblance to the elite population strategy commonly employed in evolutionary
algorithms (Brownlee, 2011; Vikhar, 2016). Ultimately, the algorithm yields a
set of memory cells, representing an optimal or near-optimal solution to the
problem. The ’memory’ function of the AIS plays a crucial role similar to the
human immune system, with each iteration enhancing the identification and
improvement of antibody quality.

Algorithm 1 The Core Process of AIS Algorithms

1:Initialise antibody population $P$

2:Define antigens $ATG$

3:while termination condition not met do

4: for each antigen $atg$ in $ATG$ do

5: Evaluate affinity of $P$ to $atg$

6: Select a subset of $P$ with highest affinity to $atg$

7: Generate new antibodies by mutation

8: Replace low affinity antibodies in $P$ with new antibodies

9: end for

10: Update memory cells with high-affinity antibodies

11:end while

12:return memory cells

An innovative approach that amalgamates AIS with the concept of symbiosis was
first explored in (Halavati et al., 2007), which led to the development of the
Symbiotic Artificial Immune Systems with Partially Specified Antibodies
(SymbAIS) algorithm, representing an advanced iteration of CLONALG (Brownlee,
2007). The core methodology of SymbAIS involves employing partially specified
antibodies for the search process, progressively constructing categories of
potential solutions, and iterative refining of these solutions until an
optimal solution is identified. Comparative analyses reveal that while SymbAIS
exhibits a longer duration in identifying the most suitable solution relative
to the conventional clonal selection algorithm, it demonstrates superior
efficacy in addressing problems characterized by a higher complexity of
parameters. Notably, SymbAIS manifests its distinct advantages, particularly
in scenarios where the CLONALG encounters limitations or fails to provide
viable solutions.

Distinct from the SymbAIS algorithm, the Symbiotic Organisms Search (SOS)
algorithm (Cheng and Prayogo, 2014) emerges as a unique meta-heuristic
optimisation algorithm, diverging from the conventional AIS. SOS algorithm
ingeniously leverages the three fundamental symbiotic relationships
(mutualism, commensalism and parasitism) observed in the biological realm. The
SOS algorithm creatively translates these symbiotic relationships into
mathematical formulas (see Section 3.2 for details) that encapsulate the
intricate dynamics of symbiotic relationships between organisms. SOS algorithm
is characterized by its minimal parameter requirements and rapid convergence
in optimisation scenarios. Nonetheless, its efficacy can be influenced by the
specific nature of the problem at hand and intrinsic attributes of the
algorithm itself, such as susceptibility to local optima and the implications
of population size. The empirical findings presented in (Cheng and Prayogo,
2014) indicate that the SOS algorithm demonstrates a broadly effective
performance in processing benchmark functions characterized by multiple local
optima. However, our investigations have identified a limitation of the SOS
algorithm, particularly in scenarios involving large initial population sizes.
In contrast, our SAIS algorithm demonstrates superior performance in
effectively optimising a wide range of mathematical benchmarks under these
conditions. This observation has led us to further explore the SAIS algorithm.
In Section 3, we discuss the symbiotic relationships that are central to the
SAIS algorithm and detail the core principles that guide its operation. Our
approach addresses the question of why not simply reduce the size of the
population by demonstrating that, in certain contexts, a larger population
size can be leveraged for more effective optimisation of complex problems.

##  3\. Symbiotic Artificial Immune Systems

###  3.1. Structure and Design of SAIS

The flow chart of the Symbiotic Artificial Immune Systems algorithm is
depicted in Figure 2. The portion enclosed within the dotted rectangle
delineates the five pivotal steps that constitute the core of the SAIS
algorithm.

Figure 2. SAIS Algorithm Flow Chart

The main flow chart of SAIS Algorithm

In the initial step of the SAIS, the termination conditions for the
algorithm’s iterations are established. This could involve setting an upper
limit on the number of loops or determining whether the conditions for the
target solution have been met. Additionally, during the initialization phase,
SAIS randomly generates an initial population $P$ comprising $N$ antibodies,
based on the known domain of the problem or function.

In the second step, SAIS adopts the characteristics of AIS with memory cells.
SAIS saves the state of the population before the algorithm is executed by
copying/cloning the existing population $P$ to the new memory population
$P_{m}$.

As described by the first three dotted rectangles (three main step) at the end
of the flow chart. SAIS has to not only retain the memory cells in AIS to
judge the superior antibodies but also emulate the three symbiotic
relationships in biology to promote antibody mutation. However, the SOS
algorithm mentioned in the Section 2.2 takes the same population step by step
through the three different symbiotic relationships. SOS performs a mutualism
phase followed by a commensalism phase and finally takes the population
through a parasitism phase in the algorithmic cycle. This indicates that SOS
applies all three symbiotic relationships of organisms to the same species.
The fact is, SOS is not accurate in emulating the symbiotic relationships of
organisms, as every organism in SOS will have all the three symbiotic
relationships. In order to more accurately emulate symbiotic relationships in
the biological area for better optimisation performance, SAIS will randomly
divide the initial population into three sub-populations to perform mutualism,
commensalism and parasitism (the realization of these three symbiotic
relationships will be explained in detail in Section 3.2). If the size of this
initial population is not divisible by three, the remaining individuals will
remain in the initial population. SAIS partitions the population $P$ into
three smaller sub-populations, facilitating concurrent symbiotic operations.
These updated sub-populations are then merged and used to to replace the
original population. The augmented population is subsequently integrated with
the ’memory’ population to produce a new population double the size of the
original population. Thereafter, SAIS selects the top $N$ antibodies (ensuring
that the population size is consistent in each iteration), constituting a new
population via fitness evaluation. This procedure aids SAIS in preserving
populations with lower fitness values for subsequent iterations.

In the field of AIS, the affinity value (fitness value) of an antibody is
indicative of its binding efficacy to the antigen, serving as a measure of the
antibody’s quality (Timmis et al., 2004). Let $atb$ represent an antibody and
$atg$ represent an antigen. The affinity (fitness) function $f$ of the
antibody can be defined as follows:

(1) $f(atb)=\text{function}(atb,atg)$

where the function quantifies the degree of binding between the antibody $ATB$
and the antigen $ATG$. In the experiments in the Section 4, our optimisation
problems focused on locating the minimum value of the benchmark function.
Consequently, the fitness value is expected to decrease progressively with
each iteration until the specified condition is met.

###  3.2. Mutualism, Commensalism and Parasitism

SAIS adopts the mathematical presentation of SOS (Cheng and Prayogo, 2014) for
the process of Mutualism, Commensalism and Parasitism. However, SAIS is
applied to the AIS algorithm in the form of parallel connection as shown in
Figure 2.

In the mutualism phase, $atb_{i}$ is the i-th antibody in the mutualism
population $P0$. Then another antibody $atb_{j}$ is randomly selected from the
mutualism population $P0$ to interact with $atb_{i}$. Both antibodies are in a
mutually beneficial relationship, with the goal of increasing the mutual
survival advantage in the population. The update process can be represented as
follows:

(2) $\displaystyle atb_{i}^{\prime}$
$\displaystyle=atb_{i}+\text{rand}(0,1)\cdot(best\\_atb-\mu\cdot b_{f}),$
$\displaystyle atb_{j}^{\prime}$
$\displaystyle=atb_{j}+\text{rand}(0,1)\cdot(best\\_atb-\mu\cdot b_{f})$

where $atb_{i}^{\prime}$ and $atb_{j}^{\prime}$ are the updated antibodies,
$best\\_atb$ is the best antibody in the mutualism population,
$\mu=\frac{atb_{i}+atb_{j}}{2}$ is the mutual factor, $b_{f}$ is a benefit
factor randomly chosen from $\\{1,2\\}$, and $\text{rand}(0,1)$ generates a
random number between 0 and 1.

In the commensalism phase, for each antibody $atb_{i}$ in the commensalism
population $P1$, another antibody $atb_{j}$ is randomly selected from the same
population. The update process can be represented as follows:

(3) $atb_{i}^{\prime}=atb_{i}+\text{rand}(-1,1)\cdot(best\\_atb-atb_{j})$

where $atb_{i}^{\prime}$ is the updated antibody, $best\\_atb$ is the best
antibody in the commensalism population, $atb_{j}$ is another randomly
selected antibody, and $\text{rand}(-1,1)$ generates a random number between
-1 and 1. This phase aims to improve the fitness of each antibody $atb_{i}$ by
leveraging the position of another antibody $atb_{j}$ in relation to the best-
performing antibody in the population ($atb_{j}$ does not change in this
phase).

In the parasitism phase, each antibody $atb_{i}$ in the parasitism population
$P2$ undergoes a potential replacement. The process can be represented as
follows:

(4) $atb_{i}^{\prime}=\begin{cases}new\\_atb,&\text{if }f(new\\_atb)\text{
better than }f(atb_{i})\\\ atb_{i},&\text{otherwise}\end{cases}$

where $atb_{i}^{\prime}$ is the updated antibody, $new\\_atb$ is a newly
generated antibody, and $f$ is the fitness function. In this phase, a new
antibody (parasite) replaces an existing one if it demonstrates superior
fitness, where ’better’ is defined according to the specific optimisation
criteria of the problem (either maximization or minimisation).

###  3.3. Pseudocode for SAIS

Algorithm 2 SAIS Algorithm

1:Initialise population $P$ of $N$ antibodies

2:Determine globally optimal $global\\_opt$

3:while termination condition not met do

4: Copy $P$ to memory population $P_{m}$

5: Divide $P$ into $P0$, $P1$, $P2$ populations

6: if $len(P)\%3\neq 0$ then

7: Retain remainder in $P$

8: end if

9: Mutualism Phase:

10: for each antibodies $atb_{i}$ in $P0$ do

11: Select another antibody $atb_{j}$ randomly

12: Perform mutualism equation 2 on $atb_{i}$, $atb_{j}$

13: end for

14: Commensalism Phase:

15: for each antibody $atb_{i}$ in $P1$ do

16: Select another antibody $atb_{j}$ randomly

17: Perform commensalism equation 3 on $atb_{i}$ using $atb_{j}$

18: end for

19: Parasitism Phase:

20: for each antibody $atb_{i}$ in $P2$ do

21: Generate a new antibody $new\\_atb$

22: Perform parasitism equation 4 on $atb_{i}$ using $new\\_atb$

23: end for

24: Replace updated $P0$, $P1$, $P2$ into $P$

25: $P\leftarrow P+P_{m}$

26: Sort $P$ by fitness and retain the best half

27: $best\\_fitness\leftarrow$ best fitness value in $P$

28: if $best\\_fitness\text{ better than }global\\_opt$ then

29: break

30: end if

31:end while

32:$best\\_antibody\leftarrow$ Antibody with best fitness in $P$

33:return $best\\_antibody$, $best\\_fitness$

The pseudocode of SAIS is shown in Algorithm 2. A population containing $N$
antibodies is first initialised. During the iterative process, the population
will be randomly divided into three sub-populations of the same population
size. These three sub-populations will undergo mutualism, commensalism and
parasitism phases, respectively. If the population size is not divisible by
three, the remaining antibodies are retained in the original population and no
operation will be performed. In each phase, the antibodies are optimised by
specific update rules. The algorithm continues to iterate until the
termination condition is met, and finally returns the antibody with the best
fitness. The innovative design of the SAIS enhances the AIS’s capability to
simulate symbiotic mechanisms found in biology with greater accuracy. As
depicted in Figure 3, SAIS concurrently processes the three symbiotic
relationships (mutualism, commensalism, and parasitism) corresponding to
population update methods within biology. This parallel processing approach
allows each subpopulation to use a different update method at the same time,
which is different from the sequential step-by-step execution of population
$P$ in SOS. Consequently, SAIS’s concurrent execution, demonstrated by the
simultaneous mutualism, commensalism, and parasitism phases in subpopulations
$P0$, $P1$, and $P2$, not only significantly enhances population diversity but
also reduces the time to completion. Such a parallel structure is particularly
beneficial when handling large-scale and complex tasks, as the capacity to
perform multiple operations concurrently is crucial for optimising
computational resource utilisation.

Figure 3. Comparison of Population Updating Methods in SAIS and SOS

Comparison between SAIS and SOS

##  4\. Experiments

###  4.1. The Analysis of Max Iteration and Population Size

In Section 3, we have presented SAIS in detail, with a specific emphasis on
population size and iteration number as critical parameters. Our preliminary
experiments revealed performance variations in SAIS under different parameter
settings. To explore the selection of suitable values for these parameters, we
initially chose to conduct an analysis based on 26 benchmark functions. The 26
benchmark functions initially employed in (Karaboga and Akay, 2009) to
evaluate the efficacy of algorithms like GA (Brownlee, 2011), DE (Karaboga and
Akay, 2009) and PSO (Kennedy and Eberhart, 1995) have been widely recognized
for their comprehensiveness. Further scrutiny by (Cheng and Lien, 2012; Cheng
and Prayogo, 2014) involved testing SOS and Particle Bee Algorithm (PBA)
(Cheng and Lien, 2012) against these benchmark functions, yielding positive
outcomes. Encompassing a diverse range of problem types including uni-modal,
multi-modal, and multidimensional features, these benchmark functions are
instrumental in delineating the scope of problems an algorithm can adeptly
address, thereby providing robust experiments for SAIS. Table 3 contains the
details of these 26 benchmark functions, including their names and optimal
solutions. Our primary objective was to assess the performance of SAIS and
determine the required number of iterations and population size for achieving
optimal solutions.

The benchmark study conducted in (Cheng and Prayogo, 2014) fixed the
population size ($p$) at $50$ and the maximum iteration number ($i$) at
$500,000$. To ensure a fair comparison among different combinations of $p$ and
$i$, we maintained a constant maximum computational cost ($B$) for each
combination, which was set to $B=p\cdot i=2.5\times 10^{7}$. Consequently, we
established specific experimental settings, as outlined in Table 1. We chose
to increase the population size ($p$) from $50$ to $50,000$ because we
observed that SAIS tends to perform better with larger population sizes.

Table 2 presents the results for four different combinations of $p$ and $i$.
It is worth noting that, for each parameter combination, we conducted $30$
trials of SAIS on a single problem to ensure the robustness and statistical
significance of the results. We calculated the success rate, which serves as
an evaluation metric for SAIS performance. The success rate is defined as the
number of times, out of $30$ trials, that SAIS successfully found the optimal
solutions.

In addition, we employed the iteration mean to measure the efficiency of SAIS
configured with different parameters, but only when SAIS successfully found
the optimal solutions. To clarify, those trials in which SAIS did not find the
optimal solutions after reaching the preset iteration limit were excluded from
this calculation. Over the process of $30$ trials for a single benchmark
problem, SAIS might attain the optimal solution using varying numbers of
iterations. While the four settings inherently correspond to four different
values of $i$, our goal was to examine whether the required number of
iterations decreased as the population size increased. The iteration STD
represents the standard deviation of the required iterations to find the
optimal solutions.

In Table 2, we observed that SAIS performed best with $p=50,000$ and $i=500$,
achieving optimal solutions for $22$ out of $26$ tasks with a $100\%$ success
rate. In comparison, SAIS with $p=5,000$ outperformed the other two
configurations, while SAIS with $p=50$ demonstrated the worst performance.
Moreover, as we increased the value of $p$, we noted a corresponding
improvement in the success rate. For example, on benchmark problems $11$ and
$13$, SAIS achieved a $100\%$ success rate only with $p=50,000$, demonstrating
a clear monotonic increase in success rate from $p=50$ to $p=50,000$.
Therefore, based on these $26$ benchmark problems, it can be concluded that,
overall, SAIS tends to perform better when configured with a larger population
size to achieve promising results. However, there is an exception in the case
of benchmark problem 16, where the success rate increased from $46.67\%$ to
$100\%$ as $p$ increased from $500$ to $5,000$, while SAIS with the other two
settings obtained 0%. This underscores the importance of carefully considering
the choice of $p$ and $i$ as these parameters play a crucial role in SAIS
performance and may vary depending on the specific problem.

In terms of the iteration mean, the required number of iterations to reach
convergence decreases overall as the population size increases. However, it is
worth noting that the actual computational cost increases, even when we assign
an equal value of $B$ to all four situations. For example, in benchmark
problem 1, SAIS with $p=50,000$ takes an average of 23. 23 iterations less
than the other three situations to find optimal solutions, but the actual
computational cost is higher compared to SAIS with $p=500$ and $p=5,000$. It
is advisable to use a large population size when dealing with specific and
unfamiliar problems, as it ensures better performance at the expense of
computational cost. Furthermore, in Table 2, for benchmark problems 20, 24,
25, and 26, SAIS with larger population sizes inversely requires more
iterations to find optimal solutions, possibly due to the characteristics of
these benchmark problems. Therefore, SAIS may exhibit lower efficiency when
configured with a larger population size, but it consistently achieves a
higher success rate and maintains stable performance.

| $p$ | $i$
---|---|---
1 | $50$ | $500,000$
2 | $500$ | $50,000$
3 | $5,000$ | $5,000$
4 | $50,000$ | $500$
Table 1. The summary of the experiment design with an aim to investigate the
performance of SAIS in relation to various selections of population size and
iteration.

| Success rate (%) | Iteration Mean | Iteration STD
---|---|---|---
Benchmark Problem | 50 | 500 | 5000 | 50000 | 50 | 500 | 5000 | 50000 | 50 | 500 | 5000 | 50000
1 | 0.00 | 96.67 | 100.00 | 100.00 | n/a | 33.00 | 27.70 | 23.23 | n/a | 4.38 | 3.05 | 2.21
2 | 13.33 | 100.00 | 100.00 | 100.00 | 10966.50 | 35.77 | 32.13 | 27.50 | 13995.32 | 4.35 | 2.92 | 1.63
3 | 100.00 | 100.00 | 100.00 | 100.00 | 25.27 | 19.90 | 17.73 | 15.43 | 3.34 | 1.67 | 1.05 | 1.28
4 | 100.00 | 100.00 | 100.00 | 100.00 | 28.07 | 24.90 | 23.30 | 21.47 | 3.10 | 1.35 | 1.47 | 1.04
5 | 0.00 | 83.33 | 100.00 | 100.00 | n/a | 37.84 | 29.63 | 27.03 | n/a | 5.82 | 2.89 | 2.28
6 | 30.00 | 100.00 | 100.00 | 100.00 | 132.44 | 11.87 | 8.93 | 6.40 | 200.69 | 2.50 | 2.27 | 1.65
7 | 100.00 | 100.00 | 100.00 | 100.00 | 41.10 | 26.83 | 24.13 | 22.60 | 19.99 | 2.56 | 0.86 | 1.28
8 | 43.33 | 96.67 | 100.00 | 100.00 | 22667.08 | 31.10 | 26.37 | 22.63 | 41911.18 | 4.14 | 2.89 | 2.79
9 | 100.00 | 100.00 | 100.00 | 100.00 | 24.80 | 23.20 | 20.97 | 19.53 | 2.14 | 1.52 | 1.27 | 0.82
10 | 100.00 | 100.00 | 100.00 | 100.00 | 31.60 | 26.37 | 24.43 | 22.63 | 4.97 | 2.11 | 1.45 | 1.22
11 | 36.67 | 80.00 | 93.33 | 100.00 | 136.91 | 27.92 | 21.75 | 12.33 | 179.66 | 22.62 | 9.20 | 4.38
12 | 0.00 | 0.00 | 26.67 | 90.00 | n/a | n/a | 117.25 | 100.41 | n/a |  | 18.41 | 12.93
13 | 0.00 | 16.67 | 76.67 | 100.00 | n/a | 91.00 | 71.52 | 59.70 | n/a | 11.77 | 5.78 | 4.63
14 | 100.00 | 100.00 | 100.00 | 100.00 | 77.40 | 80.40 | 79.93 | 76.77 | 5.55 | 2.14 | 1.11 | 0.68
15 | 0.00 | 0.00 | 3.33 | 6.67 | n/a | n/a | 305.00 | 256.00 | n/a | n/a | n/a | 1.41
16 | 0.00 | 46.67 | 100.00 | 0.00 | n/a | 1098.71 | 668.07 | n/a | n/a | 132.97 | 19.82 | n/a
17 | 100.00 | 100.00 | 100.00 | 100.00 | 89.13 | 105.90 | 105.23 | 100.33 | 3.17 | 1.42 | 0.86 | 0.76
18 | 100.00 | 100.00 | 100.00 | 100.00 | 84.20 | 99.40 | 98.40 | 93.93 | 2.81 | 1.35 | 0.89 | 0.52
19 | 0.00 | 0.00 | 0.00 | 0.00 | n/a | n/a | n/a | n/a | n/a | n/a | n/a | n/a
20 | 100.00 | 100.00 | 100.00 | 100.00 | 140.50 | 177.17 | 180.03 | 172.63 | 3.21 | 1.78 | 1.27 | 0.81
21 | 100.00 | 100.00 | 100.00 | 100.00 | 94.80 | 111.87 | 110.97 | 105.90 | 2.46 | 1.25 | 0.72 | 0.55
22 | 0.00 | 0.00 | 0.00 | 0.00 | n/a | n/a | n/a | n/a | n/a | n/a | n/a | n/a
23 | 0.00 | 0.00 | 0.00 | 0.00 | n/a | n/a | n/a | n/a | n/a | n/a | n/a | n/a
24 | 100.00 | 100.00 | 100.00 | 100.00 | 92.50 | 117.10 | 128.33 | 136.87 | 4.22 | 6.70 | 5.66 | 9.10
25 | 100.00 | 100.00 | 100.00 | 100.00 | 92.47 | 108.47 | 107.90 | 103.73 | 2.70 | 1.61 | 1.65 | 1.96
26 | 100.00 | 100.00 | 100.00 | 100.00 | 145.63 | 174.90 | 174.20 | 166.50 | 3.47 | 1.99 | 0.85 | 0.82

Table 2. The experimental results of investigating the correlation between the
performance of SAIS and parameter selection.

###  4.2. SAIS on 26 Benchmark Unconstrained Mathematical Problems

Based on the analysis results obtained from Section 4.1, we have opted to use
$p=50,000$ and $i=500$ for SAIS for the purpose of comparison with other
approaches discussed in (Cheng and Prayogo, 2014). All the selected approaches
in (Cheng and Prayogo, 2014) are listed in Table 3 and are configured with
$p=50$ and $i=500,000$. Although the settings differ, we have ensured that the
value of $B$ remains consistent to maintain experimental fairness. It’s
important to note that our proposed SAIS typically employs a larger value of
$p$, setting it apart from the other approaches. In our experiments, as shown
in Table 3, each method has been allocated the same maximum computational cost
and run 30 times for each problem.

In Table 3, the “Min” column contains 26 optimal solutions for the
corresponding problems. The “Mean” and ”StdDev” columns respectively represent
the average values that each approach obtained in 30 runs and the standard
deviation for each problem. Overall, SAIS achieved results comparable to SOS,
with both approaches successfully finding the global minimum in 20 and 21 out
of 26 tasks, respectively. It is worth noting that SOS was initially reported
to have solved 22 problems, but upon reproducing the experiments, we
discovered that SOS could only solve 21 problems. In Table 3, we have
highlighted in red the discrepancies observed in our replication experiments.
Specifically, for the SOS applied to benchmark functions 22 and 23, the
results did not align with those reported in (Cheng and Prayogo, 2014). In
terms of computational cost, in theory, SAIS has an advantage over SOS. In
SOS, the population goes through three phases to update solutions, and in one
iteration, each phase requires a round of evaluation on the whole population.
Meanwhile, due to the design of SAIS, the population has been randomly divided
into three groups for evolving the promising solutions in a single iteration,
with each group undergoing different evolutionary operators (Mutualism,
Commensalism and Parasitism). Consequently, the computational cost of SOS is
three times higher than that of SAIS. We believe this difference makes SAIS
stand out, especially when applied to solving deep learning-related problems
(e.g., hyperparameter optimisation), as the evaluation of a deep learning
model is a very expensive task (Yuan et al., 2021b) (Yuan et al., 2021a) (Yuan
et al., 2021c).

Additionally, in the case of Benchmark 23, we observed that SAIS outperformed
other approaches, while most of them remained trapped at the local optimal
point of $0.66667$. Although SAIS did not reach the global optimum, it
demonstrated superior performance compared to the other methods. Furthermore,
regarding Benchmark 12, even though SAIS did not reach the optimum, it came
very close. In our experiments, we set the convergence criterion to require
results to match to at least 12 decimal places. In contrast, for Benchmark 5,
SOS is further from the global minimum.

Functions | Metric | Min | GA | DE | PSO | BA | PBA | SOS | SAIS
---|---|---|---|---|---|---|---|---|---
1\. Beale | Mean | 0 | 0 | 0 | 0 | 1.88E-05 | 0 | 0 | 0
| StdDev |  | 0 | 0 | 0 | 1.94E-05 | 0 | 0 | 0
2\. Easom | Mean | -1 | -1 | -1 | -1 | -0.99994 | -1 | -1 | -1
| StdDev |  | 0 | 0 | 0 | 4.50E-05 | 0 | 0 | 0
3\. Matyas | Mean | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
| StdDev |  | 0 | 0 | 0 | 0 | 0 | 0 | 0
4\. Bohachevsky1 | Mean | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
| StdDev |  | 0 | 0 | 0 | 0 | 0 | 0 | 0
5\. Booth | Mean | 0 | 0 | 0 | 0 | 0.00053 | 0 | 0.03382 | 0
| StdDev |  | 0 | 0 | 0 | 0.00074 | 0 | 0.1287 | 0
6\. Michalewicz2 | Mean | -1.8013 | -1.8013 | -1.8013 | -1.57287 | -1.8013 | -1.8013 | -1.8013 | -1.8013
| StdDev |  | 0 | 0 | 0.11986 | 0 | 0 | 0 | 0
7\. Schaffer | Mean | 0 | 0.00424 | 0 | 0 | 0 | 0 | 0 | 0
| StdDev |  | 0.00476 | 0 | 0 | 0 | 0 | 0 | 0
8\. Six Hump Camel Back | Mean | -1.03163 | -1.03163 | -1.03163 | -1.03163 | -1.03163 | -1.03163 | -1.03163 | -1.03163
| StdDev |  | 0 | 0 | 0 | 0 | 0 | 0 | 0
9\. Boachevsky2 | Mean | 0 | 0.06829 | 0 | 0 | 0 | 0 | 0 | 0
| StdDev |  | 0.07822 | 0 | 0 | 0 | 0 | 0 | 0
10\. Boachevsky3 | Mean | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
| StdDev |  | 0 | 0 | 0 | 0 | 0 | 0 | 0
11\. Shubert | Mean | -186.73 | -186.73 | -186.73 | -186.73 | -186.73 | -186.73 | -186.73 | -186.73
| StdDev |  | 0 | 0 | 0 | 0 | 0 | 0 | 0
12\. Colville | Mean | 0 | 0.01494 | 0.04091 | 0 | 1.1176 | 0 | 0 | 2.79E-08
| StdDev |  | 0.00736 | 0.08198 | 0 | 0.46623 | 0 | 0 | 1.50E-07
13\. Michalewicz5 | Mean | -4.6877 | -4.64483 | -4.68348 | -2.49087 | -4.6877 | -4.6877 | -4.6877 | -4.6877
| StdDev |  | 0.09785 | 0.01253 | 0.25695 | 0 | 0 | 0 | 0
14\. Zakharov | Mean | 0 | 0.01336 | 0 | 0 | 0 | 0 | 0 | 0
| StdDev |  | 0.00453 | 0 | 0 | 0 | 0 | 0 | 0
15\. Michalewicz10 | Mean | -9.6602 | -9.49683 | -9.59115 | -4.00718 | -9.6602 | -9.6602 | -9.65982 | -9.42346
| StdDev |  | 0.14112 | 0.06421 | 0.50263 | 0 | 0 | 0.00125 | 0.25186
16\. Step | Mean | 0 | 1.17E+03 | 0 | 0 | 5.1237 | 0 | 0 | 1.19E-10
| StdDev |  | 76.56145 | 0 | 0 | 0.39209 | 0 | 0 | 5.78E-11
17\. Sphere | Mean | 0 | 1.11E+03 | 0 | 0 | 0 | 0 | 0 | 0
| StdDev |  | 74.21447 | 0 | 0 | 0 | 0 | 0 | 0
18\. SumSquares | Mean | 0 | 1.48E+02 | 0 | 0 | 0 | 0 | 0 | 0
| StdDev |  | 12.40929 | 0 | 0 | 0 | 0 | 0 | 0
19\. Quartic | Mean | 0 | 0.1807 | 0.00136 | 0.00116 | 1.72E-06 | 0.00678 | 9.13E-05 | 8.86E-05
| StdDev |  | 0.02712 | 0.00042 | 0.00028 | 1.85E-06 | 0.00133 | 3.71E-05 | 1.29E-05
20\. Schwefel 2.22 | Mean | 0 | 11.0214 | 0 | 0 | 0 | 7.59E-10 | 0 | 0
| StdDev |  | 1.38686 | 0 | 0 | 0 | 7.10E-10 | 0 | 0
21\. Schwefel 1.2 | Mean | 0 | 7.40E+03 | 0 | 0 | 0 | 0 | 0 | 0
| StdDev |  | 1.14E+03 | 0 | 0 | 0 | 0 | 0 | 0
22\. Rosenbrock | Mean | 0 | 1.96E+05 | 18.20394 | 15.088617 | 28.834 | 4.2831 | 1.04E-07 (0.9335) | 26.40837
| StdDev |  | 3.85E+04 | 5.03619 | 24.170196 | 0.10597 | 5.7877 | 2.95E-07 (1.2681) | 0.16131
23\. Dixon Price | Mean | 0 | 1.22E+03 | 0.66667 | 0.66667 | 0.66667 | 0.66667 | 0 (0.66667) | 0.5
| StdDev |  | 2.66E+02 | E-9 | E-8 | 1.16E-09 | 5.65E-10 | 0 | 0
24\. Rastrigin | Mean | 0 | 52.92259 | 11.71673 | 43.97714 | 0 | 0 | 0 | 0
| StdDev |  | 4.56486 | 2.53817 | 11.72868 | 0 | 0 | 0 | 0
25\. Griewank | Mean | 0 | 10.63346 | 0.00148 | 0.01739 | 0 | 0.00468 | 0 | 0
| StdDev |  | 1.16146 | 0.00296 | 0.02081 | 0 | 0.00672 | 0 | 0
26\. Ackley | Mean | 0 | 14.67178 | 0 | 0.16462 | 0 | 3.12E-08 | 0 | 0
| StdDev |  | 0.17814 | 0 | 0.49387 | 0 | 3.98E-08 | 0 | 0
Count of algorithm found global minimum |  | 9 | 18 | 17 | 18 | 20 | 22 (21) | 20

Table 3. Algorithm experiment results, the definitions of 26 benchmark
functions can be found in (Cheng and Prayogo, 2014). It should be noted that
we have replicated all SOS experiments and have identified differences between
the results reported in the original paper and our experimental findings,
which are highlighted.

###  4.3. Comparison of SAIS with other AIS

Building upon the innovative concept of our proposed SAIS inspired by the
immunology system, we sought to benchmark its efficacy against other AIS
algorithms: CLONALG (Brownlee, 2007) and NSA (Forrest et al., 1994; Ji and
Dasgupta, 2007). We have opted to use $p=50,000$ and $i=500$ for SAIS for the
purpose of comparison with other AIS algorithms. To ensure a fair and rigorous
comparison, we adhered to parameter configurations for competing algorithms as
delineated in (Brownlee, 2011, 2007; Ji and Dasgupta, 2007). The initial
population size of both GA and NSA in this experiment is $50$. GA and NSA
allocated the same maximum number of iterations $50,000$ and ran $30$ times
for each benchmark. This setup allows us to ensure that the value of $B$ is
consistent in each different experiment to maintain the fairness of the
experiments.

Table 4 presents the outcome of our experiments. A cursory glance reveals the
distinct advantage SAIS holds across multiple benchmarks. Notably, in
benchmark functions where traditional AIS algorithms such as CLONALG and NAS
exhibit variability in performance, SAIS consistently achieves lower mean
values, signifying superior optimisation capabilities. Furthermore, the
standard deviation (StdDev) associated with SAIS results is significantly
lower, indicating a higher degree of stability in finding optimal solutions.

In Benchmarks 1, 2, 3 and 4, the mean fitness values of SAIS achieved were in
the order of $10^{-13}$. For instance, in Benchmark 1, SAIS achieved a mean
fitness of $4.07E{-13}$ with a standard deviation of $3.25E{-13}$, and in
Benchmark 2, SAIS achieved a perfect mean fitness of $-1$ with a nearly zero
standard deviation of $2.67E{-13}$.

In Benchmark 22 (Rosenbrock), SAIS significantly outperformed both CLONALG and
NAS, achieving a mean fitness value of $26.4083792$ with a standard deviation
of $0.16131842$. Although the mean fitness indicates a departure from the
near-zero values observed in other benchmarks, the relative comparison with
CLONALG and NAS demonstrates SAIS’s superior handling of Benchmark 22’s
notorious complexity and its propensity for local optima. Benchmark 23 (Dixon-
Price) presented an intriguing scenario where SAIS attained a mean fitness of
$0.5$ with a standard deviation of $0$, indicating perfect convergence to a
solution. Considering Benchmark 23 as a thirty-dimensional benchmark, the
performance of SAIS contrasts with that of the other two AIS algorithms. And
we can find that SAIS presents an absolute advantage in handling all these
thirty-dimensional benchmarks from 16 to 26. This highlights the adaptability
and efficiency of SAIS in optimising high-dimensional problems. Through the
benchmark functions with uni-modal characteristics such as Benchmarks 1, 2, 3,
14, 20, 21, etc., SAIS presents the best adaptation and the most stable
standard deviation among the three algorithms, which illustrates the
characteristics of SAIS itself in terms of its high robustness.

These observations highlight several key advantages of SAIS: the high accuracy
achieved in various benchmarks, the robustness in complex optimisation
environments. In previous comparisons with other non-AIS algorithms and with
other AIS algorithms, we can identify the precision of SAIS in solving complex
optimisation problems. In Section 5 we will delve into the role of different
symbiotic relationships in SAIS and compare it with ablation experiments.

| CLONALG | NAS | SAIS
---|---|---|---
Benchmark | Mean | StdDev | Mean | StdDev | Mean | StdDev
1 | 0.17781805 | 0.32231834 | 1.69E-06 | 1.61E-06 | 4.07E-13 | 3.25E-13
2 | -0.9999981 | 1.45E-06 | -0.9991504 | 0.00080705 | -1 | 2.67E-13
3 | 9.95E-08 | 8.71E-08 | 5.21E-07 | 5.60E-07 | 5.34E-13 | 3.14E-13
4 | 2.53E-05 | 3.02E-05 | 0.01145236 | 0.00782652 | 4.51E-13 | 2.70E-13
5 | 0.33816697 | 0.23910833 | 6.41E-05 | 6.45E-05 | 3.42E-13 | 2.55E-13
6 | -1.8012783 | 2.74E-05 | -1.8012998 | 3.67E-06 | -1.8013023 | 9.24E-07
7 | 7.71E-07 | 6.21E-07 | 0.00056121 | 0.00051364 | 4.34E-13 | 2.90E-13
8 | -1.031624 | 4.57E-06 | -1.0316254 | 2.35E-06 | -1.0316285 | 3.19E-13
9 | 1.23E-06 | 1.23E-06 | 0.00108329 | 0.00105122 | 3.93E-13 | 2.90E-13
10 | 6.24E-06 | 5.93E-06 | 0.00368062 | 0.00348445 | 4.01E-13 | 3.02E-13
11 | -186.72915 | 0.00172978 | -186.73007 | 0.00064008 | -186.73061 | 0.00025162
12 | 0.06594855 | 0.03271116 | 0.98124422 | 0.53639183 | 2.80E-08 | 1.51E-07
13 | -4.2838533 | 0.13369352 | -4.4518724 | 0.08294569 | -4.6876581 | 4.40E-08
14 | 0.45738845 | 0.09352363 | 9.62588572 | 2.02332255 | 7.29E-13 | 1.47E-13
15 | -5.9811174 | 0.26750757 | -6.6448745 | 0.28594838 | -9.4234669 | 0.25186417
16 | 7.66642108 | 0.76361459 | 67.8664793 | 5.45301191 | 1.20E-10 | 5.79E-11
17 | 7.50320437 | 0.62416044 | 25566.1748 | 2195.2511 | 8.32E-13 | 1.07E-13
18 | 95.672352 | 14.3965129 | 2898.3747 | 264.168392 | 7.80E-13 | 1.34E-13
19 | 45.0154933 | 8.05428601 | 15.212667 | 2.25707204 | 8.86E-05 | 1.30E-05
20 | 15.1282702 | 17.4169497 | 81.3986359 | 9.52909104 | 8.99E-13 | 6.02E-14
21 | 101.165018 | 9.19969676 | 295078.108 | 24316.8356 | 8.39E-13 | 1.00E-13
22 | 1382.9522 | 689.132506 | 39026615.6 | 3193770.32 | 26.4083792 | 0.16131842
23 | 340.203437 | 63.1686022 | 217674.762 | 44274.2452 | 0.5 | 0
24 | 203.690271 | 13.5334897 | 268.288934 | 7.90957871 | 7.75E-13 | 1.18E-13
25 | 0.31171311 | 0.02257614 | 229.772105 | 20.2426248 | 7.99E-13 | 1.08E-13
26 | 13.8488845 | 7.78037007 | 18.5381361 | 0.28837621 | 8.92E-13 | 6.17E-14

Table 4. The experimental results of comparing SAIS with CLONALG and NAS.

##  5\. Ablation Study

Ablation study is important for understanding the inner workings of SAIS and
for efficiency studies of the algorithm. We elucidate their role by
systematically removing three different symbiotic operators (Mutualism,
Commensalism and Parasitism) of SAIS and by running $30$ experiments on the
feature benchmark.

We have selected four benchmark functions for our ablation study by analysing
Table 2 respectively. Among them, Benchmark 6 and 20 represent the performance
of SAIS on the basis of a population of $50000$ for the least number of
iterations and the most number of iterations, respectively. Benchmark $2$ is
selected empirically, as we observed some interesting phenomena of SAIS based
on this benchmark question. For a more comprehensive analysis, we searched for
Benchmark 13 as part of our ablation experiments using a calculation of the
median of the average number of iterations for a population of $50,000$ over
the 26 benchmark functions. Figures 4, 5, 6, 7 show the change in average
fitness over $500$ iterations for each of our four models with different
symbiotic operators.

The horizontal coordinates of all four graphs indicate the average number of
iterations for each algorithm. The vertical coordinate represents the average
of the best fitness of the iterative process for each algorithm over $30$
experiments. And a zoomed-in line graph at $50$ iterations is placed in each
image for ease of observation. The four different coloured lines represent the
four different algorithms. SAIS denotes the algorithm with all symbiotic
operators. The remaining three represent algorithms that contain only one
symbiotic operator. It can be seen from these figures that the algorithms that
reach $500$ iterations are those that do not find the global optimal solution
and exceed our preset number of iterations. It is worth noting that each line
represents the average value of the fitness and the point of convergence is
different on each fold. For example, some algorithms that require only $15$
iterations to find the optimal antibody in the first experiment may require
$17$ iterations to find the optimal antibody in the second experiment. Where
the fitness of the first 15 iterations will be the average of the results of
all runs up to 15 iterations included. The next two iterations will be the
average of all runs up to 16 and 17 iterations. The algorithm stops as soon as
it finds the optimal solution for the benchmark function, so except for the
algorithm containing only $Parasitism$, which did not succeed in finding a
solution for any of the four problems, the other three algorithms each
demonstrated strengths for a different benchmark function.

A comparison of these experiments shows that SAIS converges to the optimal
solution of the function on all four benchmark functions. Algorithms
containing $Commensalism$ only were unable to find the optimal antibody on
Benchmark 20. The algorithm containing only $Mutualism$ symbiotic operators is
unable to find the optimal solution on Benchmark 13 and the final average
fitness function only converges to $4.6609$, which clearly shows that the
speed and the number of iterations for finding the optimal antibody in
Benchmark 2, 6, and 13 are not as good as that of SAIS.

Figure 4. The change in the average fitness values of four models over 30
experimental runs on Benchmark Easom Figure 5. The change in the average
fitness values of four models over 30 experimental runs on Benchmark
Michalewicz2 Figure 6. The change in the average fitness values of four models
over 30 experimental runs on Benchmark Michalewicz5 Figure 7. The change in
the average fitness values of four models over 30 experimental runs on
Benchmark Schwefel 2.22

##  6\. Conclusions

In this paper, we propose a new AIS algorithm inspired by symbiotic
relationships in biology. Our experimental results show that SAIS can achieve
excellent results in optimization problems. We experimentally investigated and
found that SAIS tends to perform better with a larger population size.
Additionally, in an ablation study, we discovered that SAIS with three
symbiotic operators simultaneously outperforms the algorithm with only one of
them in handling the benchmark function. In our ablation experiments, SAIS is
the only algorithm that successfully finds the optimal solution for the
selected benchmark function. We hope that SAIS can pave the way for new
developments in bio-inspired and immune-inspired computing.

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