# An Adaptive Jellyfish Search Algorithm for Packing Items with Conflict

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## Abstract

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## 1. Introduction

- To our knowledge, the BPPCs problem has not been addressed in the literature as one problem. The existing methods pack the conflict items first and then the non-conflict items as a separate problem. This investigation of packing all the conflict and non-conflict items at one stage is the first of its kind.
- To address the BPPCs, we proposed to utilize the first adapted jellyfish algorithm. The proposed work is provided as an open source code on a GitHub repository (https://github.com/Ahmed-Fathalla/An-Adaptive-Jellyfish-Search-Algorithm-for-Packing-Items-with-Conflict, accessed on 1 June 2023).
- Two solution representation methods have been provided at different levels. Subsequently, a comparison was drawn to answer the question as to whether it is better to represent the BPPCs solution at the item level or bin level.
- The proposed methods were evaluated on a standard dataset with different levels of problem difficulties.

## 2. Literature Review

## 3. Problem Formulation

#### 3.1. Description of the Problem

#### 3.2. Mathematical Description of BPPCs

## 4. The Proposed Adaptive Jellyfish Search Algorithm

#### 4.1. The Artificial Jellyfish Search (JS) Optimizer Algorithm

- In order to travel within the swarm, jellyfish can only travel in one of two directions: with or against the ocean current. A time control system is responsible for the switching between both modes of movement.
- Jellyfish are continuously on the move in search of food; they are often more attracted to regions where food is readily available.
- The objective function and location indicate the food quantities discovered.

#### 4.1.1. Moving in the Ocean Current

#### 4.1.2. Moving inside the Swarm

#### 4.2. The Proposed AJS Item-Wise Level (AJS_I)

- Parameters initialization: Population size $ps$, the iteration counter t, the maximum number of iterations ${t}_{max}$, upper $Ub$ and lower $Lb$ number of items, respectively, and bin capacity C.
- A jellyfish representation: Each individual in the population (i.e., JS) represents a solution of the BPPCs, as shown in Figure 2 as an example. This representation can be described by an n-dimensional vector of integer numbers (i.e., $Ub=n$), where the items and bins are identified using indices and values, respectively.Figure 2 represents the packing of five items into two bins. Item 1, 3, and 4 are packed into bin 1, while items 2 and 5 are packed into bin 2 according to the capacity constraint of the bins and the conflict constraint among items.
- Initial population: Each individual ${X}_{i}(i=1,\dots ,ps)$ is generated randomly based on a specific heuristic for achieving a feasible initial solution. Figure 3 illustrates an example of the initial population of three individuals. The example consists of 10 items $(Ub=10)$, the corresponding weight of each item, the bin capacity $(C=20)$, and a conflict set of items.Each population solution ${X}_{i}$, $i=1,\dots ,3$ is generated by packing a collection of items into a collection of bins without violating any constraint. According to ${X}_{1}$, items 3, 4, 5, 1, 10, and 7 are packed into bin 1 with a total weight of 20 and have no conflicts. Items 8, 2, and 9 are packed into bin 2, with a total weight of 14, and item 6 is packed into bin 3, with a total of 6. The initial random-based-heuristic approach guarantees feasible initial solutions to the BPPCs.
- Individual evaluation: Each jellyfish ${X}_{i}$ is evaluated according to the fitness function f at each iteration t, and the one with the smallest fitness value is allocated to the best jellyfish ${X}^{*}$ (i.e., in the case of minimization of the objective function). In contrast, the one with the maximum fitness value is assigned to ${X}^{*}$ (i.e., in the case of the maximization of the objective function). Although minimizing the number of used bins is a primary goal of the BPP, more than one solution with different packing representations and the same total number of bins may exist. Therefore, there should be other criteria to evaluate the solution in terms of bin utility, as in Equations (21) and (22) for minimization and maximization, respectively.$$Minimizef(X)=1-({\displaystyle \sum _{b=1}^{B}}{({U}_{b}\u2044C)}^{k})/B$$$$Maximizef(X)={\displaystyle \sum _{b=1}^{B}}{({U}_{b}\u2044C)}^{k}/B,$$
- Individual update: The location of a JF is updated based on the time control mechanism $c(t)$ to switch the movement between the ocean current and jellyfish swarm.
- (a)
- In the case where a jellyfish follows the ocean current, Equations (13) and (14) can be rewritten to solve the discrete BPPCs problem as follows:$${X}_{i}(t+1)={X}_{i}\u229erand(0,1)\u22a1trend$$$$trend={X}^{*}\boxminus {X}_{random},$$The new operators ⊞, ⊡, and ⊟ can be used instead of the standard operators in the JS algorithm. The operator ⊟ represents the difference between two jellyfish in the current population as a set of swaps S. Each swap s can be defined as a raw vector of three elements $(p,q,r)$, where p represents the $ite{m}_{id}$, q represents the current assigned $bi{n}_{id}$, and r represents the new assigned $bi{n}_{id}$. For example, $s=(7,2,1)$ means item 7 can be packed into bin 1 instead of bin 2. Figure 4 illustrates an example as a difference between two individuals (e.g., ${X}_{1}$ and ${X}_{2}$).The ⊡ operator represents the probability of the chosen random number of swaps from the swap set S. Figure 5 illustrates an illustrative example of the results of applying the ⊡ operator. According to the example, if S has five swaps, the generated random integer number is 0.6; then, the result of ⊡ is a randomly selected set of three swaps from the set S.The ⊞ operator takes a leading role in updating the location of a jellyfish. Therefore, it can be viewed as applying the set of swaps sequentially to the current jellyfish position in the swarm in order to obtain a new position. An illustrative example of the ⊞ operator is shown in Figure 6.From the example in Figure 5, item 5 is currently packed into bin number one and can be swapped and packed into a new bin number three. This reassignment can be done with one condition, which is that the new packing does not violate both the capacity and conflict constraints. As for swap (5, 1, 3), the weight of item 5 should be less than or equal to the remaining capacity of bin 3, and item 5 must have no conflict with all the items in bin 3. For the second swap (6, 3, 2), the reassignment satisfies the capacity constraint but violates the conflict constraint, in which item 6 has a conflict with the items’ list in bin two according to Figure 3. In this case, to guarantee a better exploration of a new solution, we apply an Any-Fit (AF) heuristic [26]. The main idea of an AF heuristic is that, for sequential current nonempty bins (i.e., ${b}_{1}$, ${b}_{2}$, …, ${b}_{j}$), the current item will not be assigned to bin ${b}_{j+1}$ unless it does not fit (i.e., it violates the capacity and conflict constraint) any of the bins in the sequence ${b}_{1}$, ${b}_{2}$, …, ${b}_{j}$. Therefore, item 6 tries to be packed into bin 1 and then tries to be packed into bin 2 (i.e., both bins satisfy the capacity constraint and violate the conflict constraint); then, it remains in bin 3. Item 9 can be reassigned to bin 1 instead of 2, thus satisfying both the capacity and conflict constraint. The remaining capacity and item list for both the current and new bins is updated for any reassignment that is applied.
- (b)
- In the case of a jellyfish moving inside a swarm, the individual can move according to type A (i.e., passive) or type B (i.e., active) movement based on a random number that is generated and compared to $c(t)$:
- Type A update: The updating location of a jellyfish (JF) in a passive way can be achieved by rewriting Equation (15) as follows to solve the discrete problem:$${X}_{i}(t+1)={X}_{i}\u229erand(0,1)\u22a1\gamma (Ub-Lb),$$
- Type B update: This involves updating the current location of an actively based JF on a randomly selected jellyfish ${X}_{j}$ from the current population. In order to cope with the discrete representation of the BPPCs, Equations (17)–(19) can be reformulated as follows:$${X}_{i}(t+1)={X}_{i}\u229erand(0,1)\u22a1direction$$$$direction=\left\{\begin{array}{cc}{X}_{j}\boxminus {X}_{i}\hfill & \mathrm{if}\phantom{\rule{4.pt}{0ex}}f({X}_{i})>=f({X}_{j})\hfill \\ {X}_{i}\boxminus {X}_{j}\hfill & \mathrm{if}\phantom{\rule{4.pt}{0ex}}f({X}_{i})<f({X}_{j}).\hfill \end{array}\right.$$

- Stopping criteria: Steps 4 and 5 are repeated for the new population and update the current best jellyfish ${X}^{*}$ until a maximum number of iterations ${t}_{max}$ is reached or an optimal assignment of items to the BPPCs has been reached.
- Report the best jellyfish, ${X}^{*}$, as the best result so far.

#### 4.3. The Proposed Bin-Wise Level (AJS_B)

Algorithm 1 The proposed AJS algorithm. |

## 5. Results and Discussion

#### 5.1. Dataset

#### 5.2. Setup

#### 5.3. Evaluation Metrics

#### 5.4. Results

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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Paper | BPPC Method | Methodology | Objectives and Results |
---|---|---|---|

[36] | Weight Annealing Methodology | Using the WA concept, develops a straightforward procedure for the one-dimensional BPP. | Covered the majority of relevant Hadoop constraints and achieved comparable performance with FIFO and Fair Schedulers. |

[37] | Multi-Capacity Bin Packing Problems (MCBPP) and Machine Reassignment Problems (MRP) | Multi-Start Iterated Local Search (MS-ILS-PPs) | Maximized machine utilization and cost-efficient task reassignment, improved upper bounds and achieved near-optimal solutions. |

[38] | The Island-Parallel Grouping Genetic Algorithm (IPGGA) | Creates a parallelized version of an evolutionary algorithm for the 1DBPP. Evaluates different model parameters. | Achieved optimal solutions for 23 of 28 instances of the Hard28 dataset, thus outperforming earlier models. Results showed that, for widely distributed computing, dynamic communication topologies are more suitable. |

[43] | One-Dimensional Bin Packing Problem (BPP) | Grouping Genetic Algorithm with Controlled Gene Transmission (GGA-CGT) | The GGA-CGT was created to promote the transmission of the best genes of the chromosomes and to explore the search space while balancing selective pressure and population diversity to prevent premature convergence. With respect to the Hard28 set, the method outperformed state-of-the-art algorithms. |

$d$ | 0 | $0.1$ | $0.2$ | $0.3$ | $0.4$ | $0.5$ | $0.6$ | $0.7$ | $0.8$ | $0.9$ |

$\delta $ | 0 | $0.2$ | $0.8$ | $0.18$ | $0.32$ | $0.5$ | $0.68$ | $0.82$ | $0.92$ | $0.98$ |

Instance No. | Maximum Degree (D) | Optimal No. of Bins (${\mathit{m}}^{*}$) |
---|---|---|

BPPC_1_1_1 | 18 | 48 |

BPPC_1_2_5 | 41 | 50 |

BPPC_1_3_3 | 73 | 46 |

BPPC_1_4_2 | 96 | 53 |

BPPC_1_5_5 | 119 | 57 |

BPPC_1_6_3 | 119 | 75 |

BPPC_1_7_2 | 119 | 91 |

BPPC_1_8_6 | 119 | 99 |

BPPC_1_9_4 | 119 | 110 |

Instance No. | Maximum Degree (D) | Optimal No. of Bins (${\mathit{m}}^{*}$) |
---|---|---|

BPPC_2_1_3 | 40 | 102 |

BPPC_2_2_2 | 99 | 100 |

BPPC_2_3_5 | 149 | 101 |

BPPC_2_4_6 | 199 | 109 |

BPPC_2_5_7 | 249 | 126 |

BPPC_2_6_7 | 249 | 147 |

BPPC_2_7_5 | 249 | 181 |

BPPC_2_8_5 | 249 | 205 |

BPPC_2_9_2 | 249 | 225 |

Instance No. | Maximum Degree (D) | Optimal No. of Bins (${\mathit{m}}^{*}$) |
---|---|---|

BPPC_3_1_3 | 105 | 202 |

BPPC_3_2_1 | 207 | 198 |

BPPC_3_3_9 | 296 | 196 |

BPPC_3_4_5 | 393 | 206 |

BPPC_3_5_8 | 499 | 241 |

BPPC_3_6_4 | 499 | 307 |

BPPC_3_7_5 | 499 | 343 |

BPPC_3_8_5 | 499 | 400 |

BPPC_3_9_2 | 499 | 444 |

Algorithm | Parameters | Definition | Value |
---|---|---|---|

PSO | $c1$ and $c2$ | The acceleration coefficients | 2 |

Jaya | $r1$ and $r2$ | Learning parameters | $[0,1]$ |

AJS | $c(t)$ | Switching parameter | $[0,1]$ |

$\gamma $ | A motion coefficient | $[Ub,Lb]$ |

Instance No. | FF | BF | AJS_I | AJS_B | ||||
---|---|---|---|---|---|---|---|---|

$\mathit{f}\mathit{v}$ | $\mathit{f}\mathit{v}$ | Min. | Max. | Avg. ± Std. | Min. | Max. | Avg. ± Std. | |

BPPC_1_1_1 | 0.132 | 0.107 | 0.068 | 0.071 | 0.070 ± 0.001 | 0.033 | 0.033 | 0.033 ± 0.000 |

BPPC_1_2_5 | 0.109 | 0.109 | 0.066 | 0.072 | 0.069 ± 0.003 | 0.037 | 0.038 | 0.038 ± 0.000 |

BPPC_1_3_3 | 0.223 | 0.198 | 0.094 | 0.097 | 0.095 ± 0.001 | 0.056 | 0.104 | 0.080 ± 0.017 |

BPPC_1_4_2 | 0.266 | 0.267 | 0.191 | 0.207 | 0.197 ± 0.006 | 0.195 | 0.220 | 0.210 ± 0.009 |

BPPC_1_5_5 | 0.297 | 0.268 | 0.225 | 0.250 | 0.239 ± 0.011 | 0.229 | 0.250 | 0.236 ± 0.008 |

BPPC_1_6_3 | 0.586 | 0.583 | 0.561 | 0.572 | 0.568 ± 0.004 | 0.550 | 0.589 | 0.567 ± 0.015 |

BPPC_1_7_2 | 0.677 | 0.665 | 0.655 | 0.664 | 0.661 ± 0.004 | 0.647 | 0.650 | 0.649 ± 0.001 |

BPPC_1_8_6 | 0.734 | 0.730 | 0.714 | 0.718 | 0.716 ± 0.002 | 0.716 | 0.722 | 0.719 ± 0.002 |

BPPC_1_9_4 | 0.775 | 0.776 | 0.770 | 0.772 | 0.771 ± 0.001 | 0.766 | 0.770 | 0.769 ± 0.002 |

Average | 0.422 | 0.411 | 0.372 | 0.359 |

Instance No. | Optimal | FF | BF | AJS_I | AJS_B | ||||
---|---|---|---|---|---|---|---|---|---|

${\mathit{m}}^{*}$ | $\mathit{m}\mathit{b}\mathit{e}\mathit{s}\mathit{t}$ | $\mathit{D}\mathit{e}\mathit{v}$ | $\mathit{m}\mathit{b}\mathit{e}\mathit{s}\mathit{t}$ | $\mathit{D}\mathit{e}\mathit{v}$ | $\mathit{m}\mathit{b}\mathit{e}\mathit{s}\mathit{t}$ | $\mathit{D}\mathit{e}\mathit{v}$ | $\mathit{m}\mathit{b}\mathit{e}\mathit{s}\mathit{t}$ | $\mathit{D}\mathit{e}\mathit{v}$ | |

BPPC_1_1_1 | 48 | 51 | 0.059 | 50 | 0.040 | 49 | 0.020 | 48 | 0.000 |

BPPC_1_2_5 | 50 | 52 | 0.038 | 52 | 0.038 | 51 | 0.020 | 50 | 0.000 |

BPPC_1_3_3 | 46 | 53 | 0.132 | 52 | 0.115 | 48 | 0.042 | 47 | 0.021 |

BPPC_1_4_2 | 53 | 58 | 0.086 | 59 | 0.102 | 55 | 0.036 | 56 | 0.054 |

BPPC_1_5_5 | 57 | 60 | 0.050 | 60 | 0.050 | 58 | 0.017 | 59 | 0.034 |

BPPC_1_6_3 | 75 | 79 | 0.051 | 80 | 0.063 | 77 | 0.026 | 77 | 0.026 |

BPPC_1_7_2 | 91 | 93 | 0.022 | 93 | 0.022 | 91 | 0.000 | 91 | 0.000 |

BPPC_1_8_6 | 99 | 102 | 0.029 | 102 | 0.029 | 99 | 0.000 | 100 | 0.010 |

BPPC_1_9_4 | 110 | 111 | 0.009 | 111 | 0.009 | 110 | 0.000 | 110 | 0.000 |

Average | 0.053 | 0.052 | 0.018 | 0.016 |

instance_id | AJS | Jaya | PSO | |
---|---|---|---|---|

BPPC_1_1_1 | Min. | 0.033 | 0.062 | 0.432 |

Max. | 0.033 | 0.071 | 0.469 | |

Avg. ± Std. | 0.033 ± 0.000 | 0.066 ± 0.001 | 0.455 ± 0.014 | |

BPPC_1_2_5 | Min. | 0.037 | 0.065 | 0.474 |

Max. | 0.038 | 0.068 | 0.485 | |

Avg. ± Std. | 0.038 ± 0.000 | 0.066 ± 0.017 | 0.480 ± 0.005 | |

BPPC_1_3_3 | Min. | 0.056 | 0.098 | 0.522 |

Max. | 0.104 | 0.144 | 0.555 | |

Avg. ± Std. | 0.080 ± 0.017 | 0.121 ± 0.011 | 0.543 ± 0.014 | |

BPPC_1_4_2 | Min. | 0.191 | 0.210 | 0.578 |

Max. | 0.207 | 0.240 | 0.595 | |

Avg. ± Std. | 0.197 ± 0.006 | 0.229 ± 0.014 | 0.589 ± 0.007 | |

BPPC_1_5_5 | Min. | 0.225 | 0.247 | 0.583 |

Max. | 0.250 | 0.281 | 0.628 | |

Avg. ± Std. | 0.239 ± 0.004 | 0.270 ± 0.003 | 0.610 ± 0.019 | |

BPPC_1_6_3 | Min. | 0.561 | 0.566 | 0.726 |

Max. | 0.572 | 0.572 | 0.739 | |

Avg. ± Std. | 0.568 ± 0.004 | 0.569 ± 0.000 | 0.734 ± 0.006 | |

BPPC_1_7_2 | Min. | 0.655 | 0.661 | 0.749 |

Max. | 0.664 | 0.661 | 0.757 | |

Avg. ± Std. | 0.661 ± 0.004 | 0.661 ± 0.002 | 0.753 ± 0.004 | |

BPPC_1_8_6 | Min. | 0.714 | 0.716 | 0.785 |

Max. | 0.718 | 0.721 | 0.791 | |

Avg. ± Std. | 0.716 ± 0.002 | 0.719 ± 0.001 | 0.787 ± 0.003 | |

BPPC_1_9_4 | Min. | 0.770 | 0.770 | 0.795 |

Max. | 0.772 | 0.772 | 0.798 | |

Avg. ± Std. | 0.771 ± 0.001 | 0.771 ± 0.004 | 0.797 ± 0.001 | |

Average | Min. | 0.360 | 0.377 | 0.627 |

Instance No. | Optimal | AJS | Jaya | PSO | |||
---|---|---|---|---|---|---|---|

${\mathit{m}}^{*}$ | $\mathit{m}\mathit{b}\mathit{e}\mathit{s}\mathit{t}$ | $\mathit{D}\mathit{e}\mathit{v}$ | $\mathit{m}\mathit{b}\mathit{e}\mathit{s}\mathit{t}$ | $\mathit{D}\mathit{e}\mathit{v}$ | $\mathit{m}\mathit{b}\mathit{e}\mathit{s}\mathit{t}$ | $\mathit{D}\mathit{e}\mathit{v}$ | |

BPPC_1_1_1 | 48 | 48 | 0.000 | 49 | 0.020 | 65 | 0.262 |

BPPC_1_2_5 | 50 | 50 | 0.000 | 51 | 0.020 | 71 | 0.296 |

BPPC_1_3_3 | 46 | 48 | 0.042 | 48 | 0.042 | 70 | 0.343 |

BPPC_1_4_2 | 53 | 55 | 0.036 | 56 | 0.054 | 79 | 0.329 |

BPPC_1_5_5 | 57 | 58 | 0.017 | 59 | 0.034 | 82 | 0.305 |

BPPC_1_6_3 | 75 | 77 | 0.026 | 78 | 0.038 | 96 | 0.219 |

BPPC_1_7_2 | 91 | 91 | 0.000 | 91 | 0.000 | 104 | 0.125 |

BPPC_1_8_2 | 99 | 99 | 0.000 | 100 | 0.010 | 112 | 0.116 |

BPPC_1_9_4 | 110 | 110 | 0.000 | 110 | 0.000 | 115 | 0.043 |

Average | 0.013 | 0.024 | 0.226 |

Algorithm | Average | Class No. | |
---|---|---|---|

Class 2 | Class 3 | ||

FF | $fv$ | 0.402 | 0.371 |

BF | $fv$ | 0.403 | 0.370 |

AJS | Min. | 0.343 | 0.326 |

Max. | 0.350 | 0.331 | |

Avg. ± Std. | 0.346 ± 0.003 | 0.329 ± 0.002 | |

Jaya | Min. | 0.365 | 0.355 |

Max. | 0.374 | 0.360 | |

Avg. ± Std. | 0.371 ± 0.003 | 0.358 ± 0.002 | |

PSO | Min. | 0.636 | 0.641 |

Max. | 0.651 | 0.653 | |

Avg. ± Std. | 0.644 ± 0.006 | 0.647 ± 0.005 |

Instance No. | FF | BF | AJS | Jaya | PSO | |||||
---|---|---|---|---|---|---|---|---|---|---|

$\mathit{m}\mathit{b}\mathit{e}\mathit{s}\mathit{t}$ | $\mathit{D}\mathit{e}\mathit{v}$ | $\mathit{m}\mathit{b}\mathit{e}\mathit{s}\mathit{t}$ | $\mathit{D}\mathit{e}\mathit{v}$ | $\mathit{m}\mathit{b}\mathit{e}\mathit{s}\mathit{t}$ | $\mathit{D}\mathit{e}\mathit{v}$ | $\mathit{m}\mathit{b}\mathit{e}\mathit{s}\mathit{t}$ | $\mathit{D}\mathit{e}\mathit{v}$ | $\mathit{m}\mathit{b}\mathit{e}\mathit{s}\mathit{t}$ | $\mathit{D}\mathit{e}\mathit{v}$ | |

BPPC_2_1_3 | 109 | 0.064 | 110 | 0.073 | 102 | 0.000 | 104 | 0.019 | 144 | 0.292 |

BPPC_2_2_2 | 108 | 0.074 | 108 | 0.074 | 100 | 0.000 | 102 | 0.020 | 144 | 0.306 |

BPPC_2_3_5 | 110 | 0.082 | 109 | 0.073 | 102 | 0.010 | 105 | 0.038 | 152 | 0.336 |

BPPC_2_4_6 | 117 | 0.068 | 118 | 0.076 | 111 | 0.018 | 116 | 0.060 | 170 | 0.359 |

BPPC_2_5_7 | 132 | 0.045 | 136 | 0.074 | 129 | 0.023 | 132 | 0.045 | 181 | 0.304 |

BPPC_2_6_7 | 156 | 0.058 | 158 | 0.070 | 152 | 0.033 | 155 | 0.052 | 200 | 0.265 |

BPPC_2_7_5 | 186 | 0.027 | 188 | 0.037 | 182 | 0.005 | 185 | 0.022 | 219 | 0.174 |

BPPC_2_8_5 | 209 | 0.019 | 210 | 0.024 | 207 | 0.010 | 208 | 0.014 | 234 | 0.124 |

BPPC_2_9_2 | 228 | 0.013 | 229 | 0.017 | 226 | 0.004 | 226 | 0.004 | 242 | 0.070 |

BPPC_3_1_3 | 212 | 0.047 | 211 | 0.043 | 203 | 0.005 | 209 | 0.033 | 296 | 0.318 |

BPPC_3_2_1 | 211 | 0.062 | 210 | 0.057 | 199 | 0.005 | 205 | 0.034 | 298 | 0.336 |

BPPC_3_3_9 | 209 | 0.062 | 209 | 0.062 | 197 | 0.005 | 205 | 0.044 | 307 | 0.362 |

BPPC_3_4_5 | 226 | 0.088 | 231 | 0.108 | 216 | 0.046 | 223 | 0.076 | 336 | 0.387 |

BPPC_3_5_8 | 253 | 0.047 | 260 | 0.073 | 251 | 0.040 | 259 | 0.069 | 364 | 0.338 |

BPPC_3_6_4 | 312 | 0.016 | 318 | 0.035 | 310 | 0.010 | 316 | 0.028 | 405 | 0.242 |

BPPC_3_7_5 | 360 | 0.047 | 364 | 0.058 | 351 | 0.023 | 355 | 0.034 | 436 | 0.213 |

BPPC_3_8_3 | 410 | 0.024 | 412 | 0.029 | 405 | 0.012 | 410 | 0.024 | 468 | 0.145 |

BPPC_3_9_5 | 448 | 0.009 | 451 | 0.016 | 446 | 0.004 | 449 | 0.011 | 484 | 0.083 |

Average | 0.047 | 0.055 | 0.014 | 0.035 | 0.258 |

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© 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**

El-Ashmawi, W.H.; Salah, A.; Bekhit, M.; Xiao, G.; Al Ruqeishi, K.; Fathalla, A.
An Adaptive Jellyfish Search Algorithm for Packing Items with Conflict. *Mathematics* **2023**, *11*, 3219.
https://doi.org/10.3390/math11143219

**AMA Style**

El-Ashmawi WH, Salah A, Bekhit M, Xiao G, Al Ruqeishi K, Fathalla A.
An Adaptive Jellyfish Search Algorithm for Packing Items with Conflict. *Mathematics*. 2023; 11(14):3219.
https://doi.org/10.3390/math11143219

**Chicago/Turabian Style**

El-Ashmawi, Walaa H., Ahmad Salah, Mahmoud Bekhit, Guoqing Xiao, Khalil Al Ruqeishi, and Ahmed Fathalla.
2023. "An Adaptive Jellyfish Search Algorithm for Packing Items with Conflict" *Mathematics* 11, no. 14: 3219.
https://doi.org/10.3390/math11143219