# Multi-Objective ABC-NM Algorithm for Multi-Dimensional Combinatorial Optimization Problem

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

## Abstract

**:**

## 1. Introduction

- Diminish all the objective functions;
- Increase all the objective functions;
- Diminish a few objectives and increase other objective functions.

- The solution $u$ dominates solution $v$, denoted as ${u}_{i}\prec {v}_{i}$;
- The solution $u$ is dominated by solution $v$, denoted as ${v}_{i}\prec {u}_{i}$;
- Both the solutions $u$ and $v$ are not dominated by each other, and they are said to be non-dominated. It is denoted as $\neg \left({u}_{i}\prec {v}_{i}\right)\wedge \neg ({v}_{i}\prec {u}_{i})$.

- A novel algorithm, MBABC-NM, is proposed to improve the exploitation of the artificial bee colony (ABC) technique. The algorithm incorporates a modified non-dominated sorting and fitness-sharing approach to handle multi-dimensional problems efficiently.
- The proposed MBABC-NM algorithm is tested on two different real-time datasets: the knapsack problem and the nurse scheduling problem.
- The algorithm’s performance is compared with other state-of-the-art algorithms, like genetic algorithm, cyber swarm optimization, and particle swarm optimization.
- The results of the experiments demonstrate that MBABC-NM outperforms the compared algorithms significantly. This result suggests that the proposed algorithm can effectively solve real-world optimization problems.

## 2. Methodology

#### 2.1. Modified Non-Dominated Sorting

Algorithm 1: Non-Dominated Sort ($Z$) |

Input: $Z$For each individual $a\in Z$ doIndividuals dominated by $a$ ${P}_{a}\leftarrow \varnothing $ ${P}_{b}\leftarrow \varnothing $ Solutions which dominate $a$ ${C}_{a}\leftarrow 0$ For each solution $b\in Z$ doif $\left(a\prec b\right)$ thenAdd the individuals $b$ to the set of solutions dominated by $a$ ${P}_{a}\leftarrow {P}_{a}\cup \left\{b\right\}$ else if $\left(b\prec a\right)$ thenIncrement the domination counter $a$ ${C}_{a}\leftarrow {C}_{a}+1$ End ifend forif ${C}_{a}=0$ thenAssign non-dominance rank as 1 for individual $a$ ${a}_{rank}\leftarrow 1$ ${L}_{1}\leftarrow {L}_{1}\cup \left\{a\right\}$ End ifend forInitialize front counter $u\leftarrow 1$ While ${L}_{u}\ne \varnothing $ doMembers of next front $K$ $K\leftarrow \varnothing $ For each solution $a\in {L}_{u}$ doFor each solution $b\in {P}_{a}$ doDecrement the dominant counter of $b$ ${C}_{b}\leftarrow {C}_{b}-1$ if ${C}_{b}=0$ thenAssign rank for the individual $b$ ${b}_{rank}\leftarrow u+1$ $K\leftarrow K\cup \left\{b\right\}$ End ifend forend for$u\leftarrow u+1$ ${L}_{u}\leftarrow K$ The dominant solution of ${L}_{u}$ are stored in ${\stackrel{\xb4}{L}}_{u}$ |

#### 2.2. Fitness Sharing

Algorithm 2: Fitness Sharing (${L}_{u}$) |

Number of solutions in Front counter $L$ $g\leftarrow \left|{L}_{u}\right|$ For $k\leftarrow 1$ to $g$ do${L}_{u}\left({Share}_{k}\right)\leftarrow 0$ For each objective $m$ doSort population with respect to all objectives ${L}_{u}\leftarrow sort({L}_{u},m)$ ${L}_{u}\left[1\right]\leftarrow \infty $ ${L}_{u}\left[g\right]\leftarrow \infty $ For $k\leftarrow 2$ to $g-1$ doCalculate Shared fitness of the ${k}^{th}$ solution with ${fit}_{k}$ ${L}_{u}\left({Share}_{k}\right)\leftarrow \frac{{fit}_{k}}{{n}_{k}}$ Niche count can be measured by ${n}_{k}\leftarrow \sum _{j=1}^{\left|L\right|}{\phi (d}_{kj})$ The sharing function between two population elements can be measured using |

${\phi (d}_{kj})\leftarrow \left\{\begin{array}{l}1-{\left(\frac{{d}_{kj}}{{\theta}_{r}}\right)}^{\rho},d<{\theta}_{r}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}0,otherwise\end{array}\right.$End forEnd for |

End for |

## 3. Multi-Objective BABC-NM for a Multi-Dimensional Combinatorial Problem

Algorithm 3: MBABC-NM |

InputFS: Number of Food Sources MI: Maximum iteration Limit: number of predefined trials Iter = 0 Prepare the population For i = 1 to FS do |

For j = 1 to S do |

Produce ${x}_{i,j}$ solution |

${x}_{i,j}\leftarrow {x}_{min,j}\pm rand\left(0,1\right)\ast ({x}_{max,j}-{x}_{min,j})$ |

Where ${x}_{min,j}$ and ${x}_{max,j}$ are the min and max bound of the dimension j ${\widehat{x}}_{i,j}\leftarrow $ BinaryConv(${x}_{i,j}$) using Algorithm 5 For $h=1$ to $M$ doEvaluate the fitness of the population for a $M$ number of Objectives ${f}_{h}\leftarrow {f}_{h}\left({\widehat{x}}_{i,j}\right)$ End for$trial\left(i\right)\leftarrow 0$ |

End forEnd for |

$iter\leftarrow 1$ |

Repeat |

{ //*Employed Bee process*// For each food source $i$ doCreate new individual ${v}_{i}$ using ${v}_{i,j}\leftarrow {x}_{i,j}+{\varnothing}_{i,j}({x}_{i,j}-{x}_{k,j})$ ${\widehat{v}}_{i,j}\leftarrow $ BinaryConv(${v}_{i,j}$) using Algorithm 5 Evaluate $f\left({\widehat{v}}_{i}\right)$ Select between $f\left({\widehat{v}}_{i}\right)$ and $f\left({\widehat{x}}_{i}\right)$ using greedy method If $f\left({\widehat{v}}_{i}\right)$ < $f\left({\widehat{x}}_{i}\right)$${x}_{i}\leftarrow {v}_{i}$ $f\left({\widehat{x}}_{i}\right)\leftarrow f\left({\widehat{v}}_{i}\right)$ $trial\left(i\right)\leftarrow 0$ Else$trial\left(i\right)\leftarrow trial\left(i\right)+1$ End ifEnd For |

//*Onlooker Bee Phase*//If iter = 1Set $r=0,i=1;$ While (r ≤ FS)Calculate Probabilities for onlooker bees using Algorithm 4 If rand (0, 1) < ${Pro}_{i}$ $r\leftarrow r+1$ For each food source, $i$ doGenerate new individual ${v}_{i}$ using Algorithm 6 NM method (${v}_{i}$) ${\widehat{v}}_{i,j}\leftarrow $ BinaryConv (${v}_{i,j}$) using Algorithm 5 Evaluate $f\left({\widehat{v}}_{i}\right)$ Select between $f\left({\widehat{v}}_{i}\right)$ and $f\left({\widehat{x}}_{i}\right)$ using greedy method If $f\left({\widehat{v}}_{i}\right)$ < $f\left({\widehat{x}}_{i}\right)$${x}_{i}\leftarrow {v}_{i}$ $f\left({\widehat{x}}_{i}\right)\leftarrow f\left({\widehat{v}}_{i}\right)$ $trail\left(i\right)\leftarrow 0$ Else$trial\left(i\right)\leftarrow trial\left(i\right)+1$ End ifEnd ForEnd if$i\leftarrow \left(i+1\right)modFS$ End whileElse For each food source, $i$ doGenerate new individual ${v}_{i}$ using Algorithm 6 NM method (${L}_{u}$) $u\in {L}_{u}$ Divide $\left\{{L}_{u}\right\}$ into $\left|{L}_{u}\right|$ equal chunks ${S}_{u}\leftarrow \frac{\left\{{L}_{u}\right\}}{\left|{L}_{u}\right|}$ $\forall {{L}_{u}}_{i},i\in \mathrm{1,2},\dots ,\left|{L}_{u}\right|$ ${{T}_{x}}_{i}\leftarrow $Rank (${{L}_{u}}_{i},{{S}_{u}}_{i}$) ${{T}_{x}}_{i}\leftarrow $ Delete least rank individual (${{T}_{x}}_{i}$) ${v}_{i}\leftarrow $ celltomat $\left\{{{T}_{x}}_{i}\right\}$ End ForEnd if |

//*Scout Bee Phase*// $q=\left\{i:trial\left(i\right)=\mathrm{max}\left(trial\right)\right\}$ If $trial\left(q\right)>limit$Abandon the food source ${x}_{i}$ ${x}_{q,j}\leftarrow {x}_{min,j}\pm rand\left(0,1\right)\ast ({x}_{max,j}-{x}_{min,j})$ ${\widehat{x}}_{q,j}\leftarrow $ BinaryConv(${x}_{q,j}$) using Algorithm 5 For $h=1$ to $M$ doEvaluate the fitness of the population for a $M$ number of Objectives ${f}_{h}\leftarrow {f}_{h}\left({\widehat{x}}_{q}\right)$ End for $trial\left(q\right)\leftarrow 0$ End ifAdd the new solution obtained to ${Z}_{i}$ Non-Dominated Sort (${Z}_{i})$ using Algorithm 1 $L{\leftarrow Z}_{i}$ Fitness Sharing ($L$) using Algorithm 2//density estimation where $L$ denotes dense population around the individual $i$ |

Memorize the best solution obtained so far |

$iter\leftarrow iter+1$ |

} |

Until $iter=MI$Output: Optimal value of the objective function |

Algorithm 4: Probability Computation |

For i = 1 to FS, do |

Compute the probability ${P}_{ij}$ for the individual ${v}_{i,j}$ |

${Pro}_{i}\leftarrow \frac{{fit}_{i}}{\sum _{j=1}^{FS}{fit}_{j}}$ ${fit}_{i}\leftarrow \left\{\begin{array}{l}\frac{1}{1+{f}_{i}},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{f}_{i}\ge 0\\ 1+abs\left({f}_{i}\right),\hspace{1em}{f}_{i}<0\end{array}\right.$ |

End for |

Algorithm 5: BinaryConv(${x}_{i,j}$) |

For i = 1 to FS do |

For j = 1 to S do |

$bit\left({x}_{i,j}\right)=\mathrm{sin}(2\pi {x}_{i,j}\mathrm{cos}(2\pi {x}_{i,j}\left)\right)$ ${\widehat{x}}_{i,j}=\left\{\begin{array}{l}1,\hspace{1em}\hspace{1em}ifbit\left({x}_{i,j}\right)0\\ 0,\hspace{1em}\hspace{1em}otherwise\end{array}\right.$ |

End forEnd for |

Algorithm 6: NM method (${v}_{i}$) |

Generate new food source ${v}_{i}$ using modified NM technique |

Let ${v}_{i}$ denotes list of vertices ɽ, μ, λ and ζ are the coefficients of reflection, expansion, contraction, and shrinkage ƒ is the objective function to be minimized |

For i = 1, 2,…, n + 1 vertices, do |

Arrange the values from lowest fit value ƒ(${v}_{1}$) to highest fit value ƒ(${v}_{n+1}$) ƒ(${v}_{1}$) $\le $ ƒ(${v}_{2}$) $\le \dots \le $ƒ(${v}_{n+1}$) |

Compute mean for best two summits ${v}_{m}\leftarrow \sum \frac{{v}_{i}}{n}$, where i = 1, 2,…, n |

//*Likeness point ${v}_{r}$*//${v}_{r}\leftarrow {v}_{m}+\mathrm{\u027d}({v}_{m}-{v}_{n+1})$ If ƒ(${v}_{1}$) $\le $ ƒ(${v}_{r}$) $\le $ ƒ(${v}_{n}$) then${v}_{n}$←${v}_{r}$ and go to end condition End if |

//*Enlargement point ${v}_{e}$*//If ƒ(${v}_{r}$) $\le $ ƒ(${v}_{1}$) then${v}_{e}\leftarrow {v}_{r}+\mathsf{\mu}\left({v}_{r}-{v}_{m}\right)$ End ifIf ƒ(${v}_{e}$) < ƒ(${v}_{r}$) then${v}_{n}$←${v}_{e}$ and go to end condition Else${v}_{n}$←${v}_{r}$ and go to end condition End if |

//*Reduction point ${v}_{c}$*//If ƒ(${v}_{n}$) $\le $ ƒ(${v}_{r}$) $\le $ ƒ(${v}_{n+1}$) thenCompute outside reduction ${v}_{c}\leftarrow \mathsf{\lambda}{v}_{r}+(1-\mathsf{\lambda}){v}_{m}$ End ifIf ƒ(${v}_{r}$) $\ge $ ƒ(${v}_{n+1}$) thenCompute inside reduction ${v}_{c}\leftarrow \mathsf{\lambda}{v}_{n+1}+(1-\mathsf{\lambda}){v}_{m}$. End ifIf ƒ(${v}_{r}$) $\ge $ ƒ(${v}_{n}$) thenContraction is done among ${v}_{m}$ and the best vertex among ${v}_{r}$ and ${v}_{n+1}$. End ifIf ƒ(${v}_{c}$) < ƒ(${v}_{r}$) then${v}_{n}$←${v}_{c}$ and go to end condition Else go to Shrinkage partEnd ifIf ƒ(${v}_{c}$) $\ge $ ƒ(${v}_{n+1}$) then${v}_{n+1}$←${v}_{c}$ and go to end condition Else go to Shrinkage partEnd if |

//* Shrinkage part *// Shrink towards the best solution with new vertices ${v}_{i}\leftarrow {\mathsf{\zeta}v}_{i}+{v}_{1}(1-\mathsf{\zeta})$, where $i=2,\dots ,n+1$ |

End condition Arrange and rename the newly constructed simplex’s summits according to their fit values, then carry on with the reflection phase. |

## 4. Experimental and Environment Setup

#### 4.1. Experimental Setup

#### 4.2. Standard 0-1 Knapsack Problem Dataset

## 5. Experimental Result Analysis and Discussion

#### 5.1. Standard NSP Dataset

Case | Nurse | Instance | MBABC-NM | M1 | M2 | M3 | M4 |
---|---|---|---|---|---|---|---|

C-1 | N25 | 1 | 1.1 × 10^{−4} | 3.9 × 10^{−4} | 2.0 × 10^{−4} | 2.1 × 10^{−4} | 4.2 × 10^{−5} |

C-1 | N25 | 7 | 1.1 × 10^{−5} | 8.8 × 10^{−4} | 2.4 × 10^{−4} | 9.2 × 10^{−5} | 3.1 × 10^{−4} |

C-1 | N25 | 12 | 1.1 × 10^{−4} | 9.8 × 10^{−4} | 2.2 × 10^{−4} | 9.4 × 10^{−5} | 8.4 × 10^{−5} |

C-1 | N25 | 19 | 1.9 × 10^{−4} | 7.0 × 10^{−4} | 1.1 × 10^{−4} | 8.1 × 10^{−6} | 1.3 × 10^{−5} |

C-1 | N25 | 25 | 1.1 × 10^{−4} | 2.6 × 10^{−4} | 1.0 × 10^{−4} | 5.6 × 10^{−4} | 3.3 × 10^{−4} |

C-2 | N25 | 2 | 1.8 × 10^{−4} | 3.1 × 10^{−4} | 4.7 × 10^{−4} | 5.5 × 10^{−4} | 2.3 × 10^{−4} |

C-2 | N25 | 5 | 9.7 × 10^{−5} | 6.1 × 10^{−4} | 3.4 × 10^{−4} | 1.1 × 10^{−4} | 2.9 × 10^{−5} |

C-2 | N25 | 9 | 7.4 × 10^{−5} | 8.2 × 10^{−4} | 4.0 × 10^{−4} | 3.4 × 10^{−4} | 9.5 × 10^{−6} |

C-2 | N25 | 15 | 6.6 × 10^{−5} | 9.0 × 10^{−5} | 3.5 × 10^{−4} | 2.5 × 10^{−4} | 3.3 × 10^{−4} |

C-2 | N25 | 27 | 3.7 × 10^{−5} | 4.3 × 10^{−5} | 2.7 × 10^{−4} | 1.6 × 10^{−4} | 1.0 × 10^{−4} |

C-3 | N25 | 1 | 7.2 × 10^{−5} | 7.6 × 10^{−4} | 4.4 × 10^{−4} | 7.4 × 10^{−5} | 1.0 × 10^{−4} |

C-3 | N25 | 3 | 1.4 × 10^{−4} | 9.0 × 10^{−4} | 1.0 × 10^{−4} | 4.7 × 10^{−4} | 3.5 × 10^{−4} |

C-3 | N25 | 16 | 1.5 × 10^{−4} | 3.2 × 10^{−4} | 2.0 × 10^{−4} | 1.7 × 10^{−4} | 1.4 × 10^{−4} |

C-3 | N25 | 27 | 6.6 × 10^{−5} | 2.3 × 10^{−4} | 3.5 × 10^{−4} | 2.3 × 10^{−4} | 2.7 × 10^{−4} |

C-3 | N25 | 35 | 4.5 × 10^{−5} | 5.6 × 10^{−4} | 3.8 × 10^{−4} | 1.9 × 10^{−4} | 2.7 × 10^{−4} |

C-4 | N25 | 5 | 1.9 × 10^{−4} | 5.1 × 10^{−4} | 7.0 × 10^{−5} | 5.6 × 10^{−4} | 1.9 × 10^{−5} |

C-4 | N25 | 10 | 9.4 × 10^{−5} | 9.1 × 10^{−4} | 8.4 × 10^{−5} | 5.0 × 10^{−4} | 1.0 × 10^{−4} |

C-4 | N25 | 25 | 1.6 × 10^{−4} | 6.9 × 10^{−4} | 6.3 × 10^{−5} | 5.4 × 10^{−4} | 2.0 × 10^{−4} |

C-4 | N25 | 38 | 3.2 × 10^{−5} | 4.6 × 10^{−4} | 1.0 × 10^{−4} | 7.8 × 10^{−5} | 8.1 × 10^{−5} |

C-4 | N25 | 41 | 1.4 × 10^{−4} | 5.3 × 10^{−4} | 1.9 × 10^{−4} | 7.2 × 10^{−5} | 1.3 × 10^{−4} |

C-5 | N25 | 7 | 1.8 × 10^{−4} | 3.8 × 10^{−5} | 2.4 × 10^{−4} | 9.9 × 10^{−5} | 1.2 × 10^{−4} |

C-5 | N25 | 11 | 3.5 × 10^{−5} | 4.3 × 10^{−4} | 4.2 × 10^{−4} | 2.4 × 10^{−4} | 1.2 × 10^{−4} |

C-5 | N25 | 30 | 1.2 × 10^{−4} | 2.3 × 10^{−4} | 4.1 × 10^{−4} | 4.9 × 10^{−4} | 2.0 × 10^{−4} |

C-5 | N25 | 42 | 2.1 × 10^{−5} | 2.5 × 10^{−4} | 4.6 × 10^{−4} | 1.0 × 10^{−4} | 2.2 × 10^{−5} |

C-5 | N25 | 47 | 1.8 × 10^{−4} | 2.9 × 10^{−4} | 6.2 × 10^{−5} | 1.4 × 10^{−4} | 1.0 × 10^{−4} |

C-6 | N50 | 1 | 1.2 × 10^{−4} | 4.4 × 10^{−4} | 1.7 × 10^{−4} | 5.3 × 10^{−4} | 2.3 × 10^{−4} |

C-6 | N50 | 4 | 1.9 × 10^{−4} | 6.8 × 10^{−4} | 6.3 × 10^{−5} | 2.5 × 10^{−5} | 3.2 × 10^{−4} |

C-6 | N50 | 12 | 1.4 × 10^{−4} | 3.2 × 10^{−5} | 2.5 × 10^{−5} | 2.5 × 10^{−4} | 1.5 × 10^{−4} |

C-6 | N50 | 26 | 9.4 × 10^{−5} | 8.4 × 10^{−4} | 1.1 × 10^{−4} | 2.6 × 10^{−4} | 3.3 × 10^{−4} |

C-6 | N50 | 29 | 3.4 × 10^{−5} | 2.2 × 10^{−4} | 1.8 × 10^{−4} | 2.9 × 10^{−4} | 3.4 × 10^{−4} |

C-7 | N50 | 3 | 9.2 × 10^{−5} | 3.5 × 10^{−4} | 1.9 × 10^{−4} | 2.4 × 10^{−4} | 3.5 × 10^{−4} |

C-7 | N50 | 6 | 1.3 × 10^{−4} | 4.5 × 10^{−4} | 8.8 × 10^{−5} | 2.5 × 10^{−4} | 3.5 × 10^{−4} |

C-7 | N50 | 12 | 3.1 × 10^{−5} | 4.0 × 10^{−4} | 2.2 × 10^{−4} | 8.0 × 10^{−5} | 9.3 × 10^{−5} |

C-7 | N50 | 26 | 3.4 × 10^{−5} | 1.8 × 10^{−4} | 1.3 × 10^{−4} | 4.3 × 10^{−4} | 4.1 × 10^{−5} |

C-7 | N50 | 36 | 1.2 × 10^{−4} | 2.1 × 10^{−4} | 2.7 × 10^{−4} | 3.7 × 10^{−4} | 9.1 × 10^{−5} |

C-8 | N50 | 4 | 9.2 × 10^{−5} | 1.3 × 10^{−4} | 4.9 × 10^{−4} | 3.5 × 10^{−4} | 2.7 × 10^{−4} |

C-8 | N50 | 9 | 1.9 × 10^{−4} | 1.2 × 10^{−5} | 4.0 × 10^{−5} | 8.6 × 10^{−5} | 3.7 × 10^{−5} |

C-8 | N50 | 15 | 1.8 × 10^{−4} | 1.2 × 10^{−4} | 9.3 × 10^{−5} | 4.3 × 10^{−4} | 1.6 × 10^{−4} |

C-8 | N50 | 40 | 9.1 × 10^{−5} | 5.7 × 10^{−4} | 3.1 × 10^{−4} | 2.1 × 10^{−4} | 1.2 × 10^{−4} |

C-8 | N50 | 47 | 2.0 × 10^{−4} | 8.6 × 10^{−4} | 2.1 × 10^{−4} | 2.3 × 10^{−5} | 3.1 × 10^{−4} |

C-9 | N60 | 5 | 1.0 × 10^{−4} | 9.6 × 10^{−4} | 2.0 × 10^{−4} | 3.5 × 10^{−5} | 2.8 × 10^{−4} |

C-9 | N60 | 10 | 4.4 × 10^{−5} | 6.2 × 10^{−4} | 1.3 × 10^{−4} | 4.5 × 10^{−4} | 1.3 × 10^{−4} |

C-9 | N60 | 23 | 1.6 × 10^{−4} | 5.5 × 10^{−4} | 3.2 × 10^{−4} | 3.6 × 10^{−4} | 1.3 × 10^{−5} |

C-9 | N60 | 29 | 9.8 × 10^{−5} | 6.6 × 10^{−4} | 6.9 × 10^{−5} | 2.3 × 10^{−4} | 6.7 × 10^{−5} |

C-9 | N60 | 40 | 1.2 × 10^{−4} | 2.0 × 10^{−4} | 7.8 × 10^{−5} | 2.6 × 10^{−4} | 3.0 × 10^{−4} |

C-10 | N60 | 6 | 3.9 × 10^{−5} | 2.0 × 10^{−5} | 4.1 × 10^{−4} | 5.4 × 10^{−5} | 3.3 × 10^{−4} |

C-10 | N60 | 14 | 7.2 × 10^{−5} | 2.3 × 10^{−4} | 1.9 × 10^{−4} | 1.4 × 10^{−4} | 1.4 × 10^{−4} |

C-10 | N60 | 20 | 1.5 × 10^{−4} | 8.8 × 10^{−4} | 4.2 × 10^{−5} | 2.1 × 10^{−4} | 2.4 × 10^{−4} |

C-10 | N60 | 32 | 1.5 × 10^{−4} | 8.4 × 10^{−4} | 3.0 × 10^{−4} | 7.0 × 10^{−5} | 1.4 × 10^{−4} |

C-10 | N60 | 41 | 3.6 × 10^{−5} | 7.0 × 10^{−4} | 9.8 × 10^{−5} | 2.4 × 10^{−4} | 2.8 × 10^{−4} |

C-11 | N60 | 2 | 4.0 × 10^{−5} | 9.4 × 10^{−4} | 4.9 × 10^{−4} | 4.1 × 10^{−4} | 7.3 × 10^{−5} |

C-11 | N60 | 8 | 1.4 × 10^{−5} | 4.5 × 10^{−4} | 1.8 × 10^{−4} | 3.8 × 10^{−4} | 1.7 × 10^{−4} |

C-11 | N60 | 14 | 8.7 × 10^{−5} | 2.1 × 10^{−4} | 3.5 × 10^{−4} | 2.7 × 10^{−4} | 2.0 × 10^{−4} |

C-11 | N60 | 20 | 1.9 × 10^{−4} | 4.3 × 10^{−4} | 4.8 × 10^{−4} | 2.3 × 10^{−4} | 4.7 × 10^{−5} |

C-11 | N60 | 32 | 5.2 × 10^{−5} | 2.3 × 10^{−4} | 8.0 × 10^{−5} | 5.4 × 10^{−4} | 3.1 × 10^{−4} |

C-12 | N60 | 3 | 5.8 × 10^{−5} | 1.1 × 10^{−4} | 3.7 × 10^{−4} | 4.5 × 10^{−5} | 2.9 × 10^{−4} |

C-12 | N60 | 12 | 1.9 × 10^{−4} | 9.1 × 10^{−4} | 2.8 × 10^{−4} | 2.0 × 10^{−4} | 1.5 × 10^{−4} |

C-12 | N60 | 19 | 3.4 × 10^{−5} | 7.5 × 10^{−4} | 1.0 × 10^{−4} | 3.3 × 10^{−4} | 2.6 × 10^{−4} |

C-12 | N60 | 23 | 1.8 × 10^{−4} | 3.4 × 10^{−4} | 2.9 × 10^{−4} | 9.3 × 10^{−5} | 7.2 × 10^{−5} |

C-12 | N60 | 34 | 1.2 × 10^{−4} | 5.2 × 10^{−4} | 4.0 × 10^{−4} | 5.3 × 10^{−5} | 1.7 × 10^{−4} |

C-13 | N60 | 1 | 5.4 × 10^{−5} | 9.8 × 10^{−4} | 3.3 × 10^{−4} | 3.6 × 10^{−4} | 3.5 × 10^{−4} |

C-13 | N60 | 4 | 1.3 × 10^{−4} | 6.7 × 10^{−4} | 2.3 × 10^{−4} | 3.5 × 10^{−4} | 2.0 × 10^{−4} |

C-13 | N60 | 19 | 6.6 × 10^{−6} | 7.5 × 10^{−4} | 1.9 × 10^{−4} | 1.4 × 10^{−5} | 2.0 × 10^{−4} |

C-13 | N60 | 29 | 2.0 × 10^{−4} | 4.9 × 10^{−4} | 1.3 × 10^{−4} | 4.3 × 10^{−4} | 1.5 × 10^{−4} |

C-13 | N60 | 40 | 1.1 × 10^{−4} | 5.8 × 10^{−4} | 1.9 × 10^{−4} | 1.3 × 10^{−4} | 1.3 × 10^{−5} |

C-14 | N60 | 5 | 1.1 × 10^{−4} | 7.5 × 10^{−4} | 2.6 × 10^{−4} | 5.0 × 10^{−4} | 3.3 × 10^{−4} |

C-14 | N60 | 9 | 2.0 × 10^{−4} | 3.9 × 10^{−4} | 1.7 × 10^{−4} | 3.0 × 10^{−4} | 3.0 × 10^{−4} |

C-14 | N60 | 15 | 1.1 × 10^{−4} | 1.9 × 10^{−4} | 8.2 × 10^{−5} | 1.3 × 10^{−4} | 3.2 × 10^{−4} |

C-14 | N60 | 30 | 2.3 × 10^{−4} | 6.5 × 10^{−4} | 2.1 × 10^{−4} | 1.1 × 10^{−4} | 3.3 × 10^{−4} |

C-14 | N60 | 43 | 1.5 × 10^{−4} | 8.1 × 10^{−4} | 4.4 × 10^{−4} | 5.4 × 10^{−4} | 1.9 × 10^{−4} |

C-15 | N60 | 6 | 2.1 × 10^{−6} | 5.7 × 10^{−5} | 2.7 × 10^{−4} | 5.5 × 10^{−4} | 3.2 × 10^{−4} |

C-15 | N60 | 15 | 1.5 × 10^{−4} | 3.5 × 10^{−4} | 3.4 × 10^{−4} | 1.4 × 10^{−5} | 1.2 × 10^{−4} |

C-15 | N60 | 26 | 1.2 × 10^{−4} | 7.7 × 10^{−4} | 3.8 × 10^{−4} | 1.6 × 10^{−5} | 2.6 × 10^{−4} |

C-15 | N60 | 35 | 2.2 × 10^{−5} | 1.3 × 10^{−4} | 2.2 × 10^{−4} | 8.1 × 10^{−5} | 1.0 × 10^{−4} |

C-15 | N60 | 44 | 9.6 × 10^{−5} | 8.4 × 10^{−5} | 1.5 × 10^{−4} | 1.6 × 10^{−4} | 9.7 × 10^{−7} |

#### 5.2. Standard 0-1 Knapsack Problem

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Goldberg, D.E.; Korb, B.; Deb, K. Messy genetic algorithms: Motivation, analysis, and first results. Complex Syst.
**1989**, 3, 493–530. [Google Scholar] - Tharwat, A.; Houssein, E.H.; Ahmed, M.M.; Hassanien, A.E.; Gabel, T. MOGOA algorithm for constrained and unconstrained multi-objective optimization problems. Appl. Intell.
**2018**, 48, 2268–2283. [Google Scholar] [CrossRef] - Zitzler, E.; Deb, K.; Thiele, L. Comparison of multi-objective evolutionary algorithms: Empirical results. Evol. Comput.
**2000**, 8, 173–195. [Google Scholar] [CrossRef] [PubMed] - Von Lücken, C.; Barán, B.; Brizuela, C. A survey on multi-objective evolutionary algorithms for many-objective problems. Comput. Optim. Appl.
**2014**, 58, 707–756. [Google Scholar] [CrossRef] - Manzoor, A.; Javaid, N.; Ullah, I.; Abdul, W.; Almogren, A.; Alamri, A. An intelligent hybrid heuristic scheme for smart metering-based demand side management in smart homes. Energies
**2017**, 10, 1258. [Google Scholar] [CrossRef] - Mirjalili, S.Z.; Mirjalili, S.; Saremi, S.; Faris, H.; Aljarah, I. Grasshopper optimization algorithm for multi-objective optimization problems. Appl. Intell.
**2018**, 48, 805–820. [Google Scholar] [CrossRef] - Tamaki, H.; Kita, H.; Kobayashi, S. Multi-objective optimization by genetic algorithms: A review. In Proceedings of the IEEE International Conference on Evolutionary Computation, Nagoya, Japan, 20–22 May 1996; IEEE: Piscataway, NJ, USA; pp. 517–522. [Google Scholar]
- Zhang, Y.; Gong, D.-W.; Gao, X.-Z.; Tian, T.; Sun, X.-Y. Binary differential evolution with self-learning for multi-objective feature selection. Inf. Sci.
**2020**, 507, 67–85. [Google Scholar] [CrossRef] - Wang, Y.; Yang, Y. Particle swarm optimization with preference order ranking for multi-objective optimization. Inf. Sci.
**2009**, 179, 1944–1959. [Google Scholar] [CrossRef] - Mirjalili, S.; Saremi, S.; Mirjalili, S.M.; Coelho, L.D.S. Multi-objective grey wolf optimizer: A novel algorithm for multi-criterion optimization. Expert Syst. Appl.
**2016**, 47, 106–119. [Google Scholar] [CrossRef] - Lv, L.; Zhao, J.; Wang, J.; Fan, T. Multi-objective firefly algorithm based on compensation factor and elite learning. Future Gener. Comput. Syst.
**2019**, 91, 37–47. [Google Scholar] [CrossRef] - Wang, C.-N.; Yang, F.-C.; Nguyen, V.T.T.; Vo Nhut, T.M. CFD analysis and optimum design for a centrifugal pump using an effectively artificial intelligent algorithm. Micromachines
**2022**, 13, 1208. [Google Scholar] [CrossRef] [PubMed] - Huynh, N.T.; Nguyen, T.V.T.; Nguyen, Q.M. Optimum Design for the Magnification Mechanisms Employing Fuzzy Logic-ANFIS. CMC-Comput. Mater. Contin.
**2022**, 73, 5961–5983. [Google Scholar] - Huynh, N.-T.; Nguyen, T.V.T.; Tam, N.T.; Nguyen, Q.-M. Optimizing Magnification Ratio for the Flexible Hinge Displacement Amplifier Mechanism Design. In Proceedings of the 2nd Annual International Conference on Material, Machines and Methods for Sustainable Development (MMMS2020); Springer International Publishing: Berlin/Heidelberg, Germany, 2021; pp. 769–778. [Google Scholar]
- Ramalingam, R.; Saleena, B.; Basheer, S.; Balasubramanian, P.; Rashid, M.; Jayaraman, G. EECHS-ARO: Energy-efficient cluster head selection mechanism for livestock industry using artificial rabbits optimization and wireless sensor networks. Electron. Res. Arch.
**2023**, 31, 3123–3144. [Google Scholar] [CrossRef] - Ramalingam, R.; Karunanidy, D.; Alshamrani, S.S.; Rashid, M.; Mathumohan, S.; Dumka, A. Oppositional Pigeon-Inspired Optimizer for Solving the Non-Convex Economic Load Dispatch Problem in Power Systems. Mathematics
**2022**, 10, 3315. [Google Scholar] [CrossRef] - Kuppusamy, P.; Kumari, N.M.J.; Alghamdi, W.Y.; Alyami, H.; Ramalingam, R.; Javed, A.R.; Rashid, M. Job scheduling problem in fog-cloud-based environment using reinforced social spider optimization. J. Cloud Comput.
**2022**, 11, 99. [Google Scholar] [CrossRef] - Thirugnanasambandam, K.; Ramalingam, R.; Mohan, D.; Rashid, M.; Juneja, K.; Alshamrani, S.S. Patron–Prophet Artificial Bee Colony Approach for Solving Numerical Continuous Optimization Problems. Axioms
**2022**, 11, 523. [Google Scholar] [CrossRef] - Bao, C.; Xu, L.; Goodman, E.D.; Cao, L. A novel non-dominated sorting algorithm for evolutionary multi-objective optimization. J. Comput. Sci.
**2017**, 23, 31–43. [Google Scholar] [CrossRef] - Ye, T.; Si, L.; Zhang, X.; Cheng, R.; He, C.; Tan, K.C.; Jin, Y. Evolutionary large-scale multi-objective optimization: A survey. ACM Comput. Surv.
**2021**, 54, 1–34. [Google Scholar] - Luo, J.; Liu, Q.; Yang, Y.; Li, X.; Chen, M.-R.; Cao, W. An artificial bee colony algorithm for multi-objective optimization. Appl. Soft Comput.
**2017**, 50, 235–251. [Google Scholar] [CrossRef] - Salazar-Lechuga, M.; Rowe, J.E. Particle swarm optimization and fitness sharing to solve multi-objective optimization problems. In Proceedings of the 2005 IEEE Congress on Evolutionary Computation, Edinburgh, UK, 2–5 September 2005; IEEE: Piscataway, NJ, USA; Volume 2, pp. 1204–1211. [Google Scholar]
- Zhang, P.; Qian, Y.; Qian, Q. Multi-objective optimization for materials design with improved NSGA-II. Mater. Today Commun.
**2021**, 28, 102709. [Google Scholar] [CrossRef] - Yin, P.-Y.; Chiang, Y.-T. Cyber swarm algorithms for multi-objective nurse rostering problem. Int. J. Innov. Comput. Inf. Control
**2013**, 9, 2043–2063. [Google Scholar] - Han, F.; Chen, W.-T.; Ling, Q.-H.; Han, H. Multi-objective particle swarm optimization with adaptive strategies for feature selection. Swarm Evol. Comput.
**2021**, 62, 100847. [Google Scholar] [CrossRef] - Li, Y.; Huang, W.; Wu, R.; Guo, K. An improved artificial bee colony algorithm for solving multi-objective low-carbon flexible job shop scheduling problem. Appl. Soft Comput.
**2020**, 95, 106544. [Google Scholar] [CrossRef] - Luo, R.-J.; Ji, S.-F.; Zhu, B.-L. A Pareto evolutionary algorithm based on incremental learning for a kind of multi-objective multi-dimensional knapsack problem. Comput. Ind. Eng.
**2019**, 135, 537–559. [Google Scholar] [CrossRef] - Yuan, J.; Li, Y. Solving binary multi-objective knapsack problems with novel greedy strategy. Memetic Comput.
**2021**, 13, 447–458. [Google Scholar] [CrossRef] - Alharbi, S.T. A hybrid genetic algorithm with tabu search for optimization of the traveling thief problem. Int. J. Adv. Comput. Sci. Appl.
**2018**, 9, 276–287. [Google Scholar] [CrossRef] - Fidanova, S. Hybrid Ant Colony Optimization Algorithm for Multiple Knapsack Problem. In Proceedings of the 2020 5th IEEE International Conference on Recent Advances and Innovations in Engineering (ICRAIE), Jaipur, India, 1–3 December 2020; IEEE: Piscataway, NJ, USA; pp. 1–5. [Google Scholar]
- Beasley, J.E. OR-Library Collection of Test Data Sets for a Variety of OR Problems. World Wide Web. 2005. Available online: http://people.brunel.ac.uk/mastjjb/jeb/orlib/scpinfo.html (accessed on 20 December 2022).

Type | Method | Reference |
---|---|---|

M1 | Multi-objective genetic algorithm: NSGA-II | Zhang et al., 2021 [23] |

M2 | Multi-objective cyber swarm optimization algorithm | Yin et al., 2013 [24] |

M3 | Multi-objective particle swarm optimization | Han et al., 2021 [25] |

M4 | Multi-objective ABC | Li et al., 2015 [26] |

Type | Method |
---|---|

number of bees | 100 |

maximum iterations | 1000 |

initialization technique | binary |

stop criteria | maximum iterations |

run | 20 |

heuristic | Nelder—Mead method |

likeness factor | α > 0 |

enlargement factor | γ > 1 |

reduction factor | 0 > β > 1 |

shrinkage factor | 0 < δ < 1 |

Instance | No. of Objectives | No. of Items |
---|---|---|

kn250_2 | 2 | 250 |

kn250_3 | 3 | 250 |

kn250_4 | 4 | 250 |

kn500_2 | 2 | 500 |

kn500_3 | 3 | 500 |

kn500_4 | 4 | 500 |

kn750_2 | 2 | 750 |

kn750_3 | 3 | 750 |

kn750_4 | 4 | 750 |

Type | Method | Reference |
---|---|---|

M1 | Pareto evolutionary algorithm | Luo et al., 2019 [27] |

M2 | GRASP | Yuan et al., 2021 [28] |

M3 | Genetic Tabu search for MKP | Alharbi et al., 2018 [29] |

M4 | ACO for MKP | Fidanova et al., 2020 [30] |

Case | Type | Instance | MBABC-NM | M1 | M2 | M3 | M4 |
---|---|---|---|---|---|---|---|

C-1 | N25 | 1 | 135 | 121 | 92 | 104 | 53 |

C-1 | N25 | 7 | 132 | 125 | 86 | 106 | 63 |

C-1 | N25 | 12 | 131 | 119 | 91 | 115 | 51 |

C-1 | N25 | 19 | 128 | 119 | 88 | 101 | 50 |

C-1 | N25 | 25 | 128 | 122 | 83 | 104 | 56 |

C-2 | N25 | 2 | 143 | 118 | 86 | 99 | 67 |

C-2 | N25 | 5 | 142 | 121 | 80 | 113 | 69 |

C-2 | N25 | 9 | 136 | 124 | 85 | 116 | 69 |

C-2 | N25 | 15 | 149 | 124 | 78 | 115 | 60 |

C-2 | N25 | 27 | 146 | 124 | 79 | 99 | 63 |

C-3 | N25 | 1 | 145 | 123 | 77 | 97 | 61 |

C-3 | N25 | 3 | 150 | 125 | 82 | 97 | 65 |

C-3 | N25 | 16 | 151 | 125 | 77 | 99 | 71 |

C-3 | N25 | 27 | 146 | 121 | 91 | 113 | 73 |

C-3 | N25 | 35 | 151 | 117 | 93 | 107 | 70 |

C-4 | N25 | 5 | 139 | 122 | 92 | 98 | 63 |

C-4 | N25 | 10 | 136 | 117 | 88 | 112 | 71 |

C-4 | N25 | 25 | 150 | 120 | 78 | 111 | 65 |

C-4 | N25 | 38 | 151 | 121 | 97 | 110 | 59 |

C-4 | N25 | 41 | 135 | 122 | 79 | 99 | 72 |

C-5 | N25 | 7 | 150 | 122 | 78 | 97 | 59 |

C-5 | N25 | 11 | 127 | 118 | 92 | 107 | 70 |

C-5 | N25 | 30 | 135 | 120 | 80 | 114 | 61 |

C-5 | N25 | 42 | 135 | 121 | 91 | 104 | 71 |

C-5 | N25 | 47 | 148 | 118 | 83 | 100 | 64 |

C-6 | N50 | 1 | 192 | 40 | 90 | 109 | 73 |

C-6 | N50 | 4 | 229 | 47 | 91 | 107 | 60 |

C-6 | N50 | 12 | 222 | 35 | 87 | 125 | 73 |

C-6 | N50 | 26 | 244 | 47 | 96 | 114 | 66 |

C-6 | N50 | 29 | 223 | 41 | 87 | 126 | 76 |

C-7 | N50 | 3 | 242 | 36 | 96 | 65 | 57 |

C-7 | N50 | 6 | 248 | 42 | 90 | 60 | 60 |

C-7 | N50 | 12 | 246 | 34 | 87 | 67 | 66 |

C-7 | N50 | 26 | 233 | 36 | 88 | 62 | 65 |

C-7 | N50 | 36 | 214 | 39 | 89 | 72 | 61 |

C-8 | N50 | 4 | 251 | 43 | 95 | 55 | 71 |

C-8 | N50 | 9 | 255 | 48 | 98 | 74 | 55 |

C-8 | N50 | 15 | 249 | 34 | 97 | 65 | 58 |

C-8 | N50 | 40 | 196 | 37 | 87 | 57 | 57 |

C-8 | N50 | 47 | 228 | 47 | 88 | 57 | 73 |

C-9 | N60 | 5 | 225 | 36 | 94 | 63 | 61 |

C-9 | N60 | 10 | 210 | 49 | 89 | 60 | 58 |

C-9 | N60 | 23 | 207 | 33 | 99 | 73 | 63 |

C-9 | N60 | 29 | 203 | 41 | 91 | 65 | 72 |

C-9 | N60 | 40 | 183 | 37 | 100 | 73 | 67 |

C-10 | N60 | 6 | 196 | 49 | 94 | 76 | 58 |

C-10 | N60 | 14 | 180 | 47 | 90 | 65 | 66 |

C-10 | N60 | 20 | 208 | 49 | 95 | 54 | 64 |

C-10 | N60 | 32 | 184 | 42 | 91 | 64 | 60 |

C-10 | N60 | 41 | 218 | 39 | 92 | 69 | 63 |

C-11 | N60 | 2 | 349 | 82 | 137 | 123 | 129 |

C-11 | N60 | 8 | 374 | 98 | 151 | 126 | 121 |

C-11 | N60 | 14 | 316 | 83 | 144 | 113 | 111 |

C-11 | N60 | 20 | 364 | 96 | 145 | 118 | 118 |

C-11 | N60 | 32 | 292 | 96 | 139 | 112 | 134 |

C-12 | N60 | 3 | 327 | 98 | 140 | 121 | 115 |

C-12 | N60 | 12 | 335 | 94 | 151 | 121 | 125 |

C-12 | N60 | 19 | 351 | 98 | 145 | 120 | 124 |

C-12 | N60 | 23 | 384 | 78 | 144 | 111 | 118 |

C-12 | N60 | 34 | 289 | 98 | 140 | 121 | 107 |

C-13 | N60 | 1 | 450 | 97 | 138 | 126 | 108 |

C-13 | N60 | 4 | 438 | 87 | 141 | 118 | 109 |

C-13 | N60 | 19 | 446 | 99 | 149 | 122 | 133 |

C-13 | N60 | 29 | 347 | 81 | 152 | 126 | 109 |

C-13 | N60 | 40 | 464 | 88 | 152 | 120 | 121 |

C-14 | N60 | 5 | 335 | 100 | 141 | 108 | 121 |

C-14 | N60 | 9 | 420 | 96 | 139 | 124 | 107 |

C-14 | N60 | 15 | 400 | 90 | 146 | 116 | 115 |

C-14 | N60 | 30 | 398 | 99 | 144 | 108 | 108 |

C-14 | N60 | 43 | 483 | 94 | 140 | 117 | 110 |

C-15 | N60 | 6 | 380 | 87 | 150 | 119 | 136 |

C-15 | N60 | 15 | 433 | 87 | 141 | 123 | 125 |

C-15 | N60 | 26 | 481 | 88 | 151 | 125 | 108 |

C-15 | N60 | 35 | 477 | 90 | 151 | 124 | 136 |

C-15 | N60 | 44 | 469 | 99 | 149 | 123 | 115 |

Instance | NRS | TNS | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

MBABC-NM | M1 | M2 | M3 | M4 | MBABC-NM | M1 | M2 | M3 | M4 | |

kn250_2 | 288.45 | 125.77 | 120.09 | 3.77 | 61.9 | 304.96 | 156.16 | 201.62 | 194.21 | 198 |

kn250_3 | 549.19 | 191.68 | 183.63 | 0.69 | 92.2 | 663.75 | 207.34 | 376.82 | 1472.45 | 925 |

kn250_4 | 742.71 | 215.63 | 207.90 | 1.28 | 104.6 | 791.22 | 254.71 | 617.90 | 3958.62 | 2288 |

kn500_2 | 5019.36 | 1505.53 | 1500.14 | 1222.24 | 1361.2 | 5198.89 | 1543.83 | 1761.34 | 257.31 | 1009 |

kn500_3 | 6751.66 | 2986.63 | 2980.58 | 1827.03 | 2403.8 | 6935.97 | 3032.85 | 3601.36 | 2368.15 | 2985 |

kn500_4 | 17,156.62 | 4282.56 | 4277.53 | 2258.97 | 3268.2 | 17,255.48 | 5455.92 | 4930.68 | 5705.94 | 5318 |

kn750_2 | 18,236.52 | 4247.75 | 4240.96 | 3765.41 | 4003.2 | 20,515.01 | 6087.35 | 4525.30 | 6362.16 | 5444 |

kn750_3 | 33,682.95 | 8035.34 | 8029.33 | 5336.95 | 6683.1 | 34,520.30 | 9102.70 | 8297.50 | 7915.18 | 8106 |

kn750_4 | 58,129.46 | 11,307.37 | 11,299.73 | 6515.64 | 8907.7 | 60,293.70 | 13,065.30 | 11,648.42 | 6976.39 | 9312 |

Instance | |R| | Davg | ||||
---|---|---|---|---|---|---|

MBABC-NM | M1 | M2 | M3 | M4 | ||

kn250_2 | 320 | 2.10 × 10^{−4} | 9.70 × 10^{−3} | 3.20 × 10^{−3} | 1.48 × 10^{−2} | 7.50 × 10^{−3} |

kn250_3 | 564 | 3.50 × 10^{−4} | 1.50 × 10^{−3} | 4.60 × 10^{−3} | 2.02 × 10^{−2} | 1.06 × 10^{−2} |

kn250_4 | 778 | 1.00 × 10^{−4} | 3.14 × 10^{−3} | 3.10 × 10^{−3} | 3.24 × 10^{−2} | 6.32 × 10^{−3} |

kn500_2 | 8844 | 7.20 × 10^{−4} | 1.78 × 10^{−2} | 4.50 × 10^{−3} | 1.61 × 10^{−2} | 2.36 × 10^{−2} |

kn500_3 | 11978 | 6.00 × 10^{−4} | 1.41 × 10^{−2} | 2.20 × 10^{−3} | 3.24 × 10^{−2} | 1.28 × 10^{−2} |

kn500_4 | 33374 | 2.50 × 10^{−3} | 1.01 × 10^{−2} | 3.60 × 10^{−3} | 5.60 × 10^{−2} | 3.00 × 10^{−3} |

kn750_2 | 34890 | 6.40 × 10^{−3} | 2.46 × 10^{−2} | 6.80 × 10^{−3} | 3.17 × 10^{−2} | 2.15 × 10^{−2} |

kn750_3 | 74504 | 9.50 × 10^{−3} | 3.12 × 10^{−2} | 7.80 × 10^{−3} | 2.80 × 10^{−2} | 9.80 × 10^{−2} |

kn750_4 | 105161 | 7.20 × 10^{−3} | 2.93 × 10^{−2} | 1.32 × 10^{−2} | 3.18 × 10^{−2} | 1.01 × 10^{−2} |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Rajeswari, M.; Ramalingam, R.; Basheer, S.; Babu, K.S.; Rashid, M.; Saranya, R.
Multi-Objective ABC-NM Algorithm for Multi-Dimensional Combinatorial Optimization Problem. *Axioms* **2023**, *12*, 395.
https://doi.org/10.3390/axioms12040395

**AMA Style**

Rajeswari M, Ramalingam R, Basheer S, Babu KS, Rashid M, Saranya R.
Multi-Objective ABC-NM Algorithm for Multi-Dimensional Combinatorial Optimization Problem. *Axioms*. 2023; 12(4):395.
https://doi.org/10.3390/axioms12040395

**Chicago/Turabian Style**

Rajeswari, Muniyan, Rajakumar Ramalingam, Shakila Basheer, Keerthi Samhitha Babu, Mamoon Rashid, and Ramar Saranya.
2023. "Multi-Objective ABC-NM Algorithm for Multi-Dimensional Combinatorial Optimization Problem" *Axioms* 12, no. 4: 395.
https://doi.org/10.3390/axioms12040395