# Hybrid Particle Swarm Algorithm for Products’ Scheduling Problem in Cellular Manufacturing System

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Performance Analysis

- No pre-emption is allowed and setup times are sequence independent.
- Breakdown and re-work is neglected and all parts are available.
- Buffer size is not considered.

_{c}) for the current scenario. Furthermore, after completion of parts for the first cycle, all three parts will be processed completely after a cycle time of 7 min.

_{i}as a total duration or throughput time (from start to end of point). If TT

_{i}is superior to T

_{c}, the operating sequence time is longer than cycle time and must be cut into several cycles. This is achieved by introduction of several parts of this sequence: WIP. It is noteworthy to mention that this number has to be at least equal to an integer superior or equal to TT

_{i}by T

_{c}(see [34]). The waiting time of a part due to resource conflict makes the operating sequence longer which proportionally increases WIP.

_{3}= 6 + 5 = 11 min, while its waiting time is ‘0’, hence, TT

_{3}= 6 + 5 + 0 = 11 min with the cycle time of 7 min. For this, we need an integer which is ≥ 11/7 (= 1.57) = 2 parts. Furthermore, for accumulative WIP, summation of WIP for all parts is carried out (See Equation (1) and objective function in Section 4.1). It should also be noted that in the elaborated example, setup and transport times are inclusive in the processing times, while waiting time is zero here, which ultimately leads to a lower bound or optimal point of a solution. Practically, waiting times are mostly a part of an operating schedule.

## 4. Methodology

#### 4.1. Mathematical Modeling

^{th}part, n = last part, j = j

^{th}machine, and m = last machine.

- ${\mathrm{AT}}_{\mathrm{ij}}$ = Arrival time of part “i” on machine “j”
- ${\mathrm{PT}}_{\mathrm{ij}}$ = Process time of part “i” on machine “j”
- ${\mathrm{t}}_{\mathrm{s}\left(\mathrm{ij}\right)}$ = Setup time of part “i” on machine “j”
- ${\mathrm{ST}}_{\mathrm{ij}}$ = Start time of part “i” on machine “j”
- ${\mathrm{C}}_{\mathrm{ij}}$ = Completion time of part “i” on machine “j”
- ${\mathrm{WQ}}_{\mathrm{i},\mathrm{j}}$ = Waiting time in queue of part “i” on machine “j”

^{th}job is expressed by integer number of the cycle times, essential to complete all operations in order to produce that part. Therefore, WIP of i

^{th}job is stated as follows:

^{th}job.

^{th}machine is expressed by the ratio of total processing time for all parts required on that machine to the total run time of the machine in a pattern schedule. Hence, utilization of machine j (U

_{j}) is termed as follows:

^{th}part on j

^{th}machine and ${\mathrm{RT}}_{\mathrm{j}}$ is total machine run time to process all the parts in a schedule pattern. In other words, ${\mathrm{RT}}_{\mathrm{j}}$ = max. machine stop time (makespan) – min machine start time. For calculating under-utilization:

#### 4.2. Genetic Algorithm

- Evaluation: Determine the aptitude of each individual according to certain criteria or measure, that is, which individuals are the most apt to survive.
- Selection: Select the strongest individuals of a generation that must pass to the next generation or must reproduce. In our case, roulette wheel selection scheme [38] was applied.
- Crossing: Take two individuals to reproduce and cross their genetic information to generate new individuals that will form a new population. Several crossover operators have been proposed for the permutation encoded chromosomes. In the current study, the Similar Block 2—Point Order Crossover (SB2OX) was applied as according to results of [39], it is an efficient technique for a permutation solution space. The crossover probability considered was for two levels (0.2 and 0.3)
- Mutation: Randomly alters the genes of an individual because of errors in the reproduction or deformation of genes. In other words, genetic variability is introduced into the population for diversity enhancement of a population. In addition, early convergence is also avoided to any local optimum. In this study, swap technique was used with two level probabilities, 0.01 and 0.02.
- Replacement: Procedure to create a new generation of individuals.

#### 4.3. NEPSO in Scheduling

#### 4.3.1. PSO Terms to clear:

- Particle: At iteration k, ${\mathrm{X}}_{\mathrm{i}}^{\mathrm{k}}$ i
^{th}particle in swarm is denoted by ${\mathrm{X}}_{\mathrm{i}}^{\mathrm{k}}$ and is represented by number of dimensions as ${\mathrm{X}}_{\mathrm{i}}^{\mathrm{k}}$ = $\left[{\mathrm{x}}_{\mathrm{i},1}^{\mathrm{k}},\dots ,{\mathrm{x}}_{\mathrm{i},\mathrm{n}}^{\mathrm{k}}\right]$; ${\mathrm{x}}_{\mathrm{i},\mathrm{j}}^{\mathrm{k}}$ refers to position value of ith particle w.r.t to dimension j (j = 1,..., n). - Population: The group of m particles in swarm at iteration k, i.e. ${\mathrm{p}}^{\mathrm{k}}$ = $\left[{\mathrm{X}}_{1}^{\mathrm{k}},\dots ,{\mathrm{X}}_{\mathrm{m}}^{\mathrm{k}}\right]$.
- Sequence: ${\mathrm{S}}_{\mathrm{i}}^{\mathrm{k}}$ is a sequence of jobs inferred by the particle ${\mathrm{X}}_{\mathrm{i}}^{\mathrm{k}}$. ${\mathrm{S}}_{\mathrm{i}}^{\mathrm{k}}$ = $\left[{\mathrm{s}}_{\mathrm{i},1}^{\mathrm{k}},\dots ,{\mathrm{s}}_{\mathrm{i},\mathrm{n}}^{\mathrm{k}}\right]$; ${\mathrm{s}}_{\mathrm{i},\mathrm{j}}^{\mathrm{k}}$ is the arrangement of job j in domain of the particle i at iteration k w.r.t. j
^{th}dimension. - Particle velocity: ${\mathrm{V}}_{\mathrm{i}}^{\mathrm{k}}$ is the velocity of i
^{th}particle at k^{th}iteration; ${\mathrm{V}}_{\mathrm{i}}^{\mathrm{k}}$ = $\left[{\mathrm{v}}_{\mathrm{i},1}^{\mathrm{k}},\dots ,{\mathrm{v}}_{\mathrm{i},\mathrm{n}}^{\mathrm{k}}\right]$. - Inertia weight: Parameter used for controlling impact of the previous velocity on the current velocity and denoted by w
^{k}. - Personal best: Position values determining best fitness values for i
^{th}particle so far until iteration k is known as personal best denoted by ${\mathrm{PB}}_{\mathrm{i}}^{\mathrm{k}}$. Updated for every particle in swarm, such as when fitness (${\mathrm{S}}_{\mathrm{i}}^{\mathrm{k}}$) < fitness (${\mathrm{S}}_{\mathrm{i}}^{\mathrm{k}-1}$), personal best is updated else it stays the same. - Global best: Among all particles in the swarm at iteration k, position values with a sequence giving overall best fitness represents global best particle.

#### 4.3.2. Initial Seed Solution

- For each job i, find the total processing time T
_{i}with help of (14).$${\mathrm{T}}_{\mathrm{i}}={\displaystyle \sum}_{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{PT}}_{\mathrm{ij}}$$ - Sort the n jobs in descending order of their total processing times.
- Take the first four jobs from the sorted list and form 4! = 24 partial sequences. The best k out of these 24 partial sequences is selected for further processing. The relative positions of jobs in any partial sequence are not altered in any later (larger) sequence.
- Let x be the position numbers. Set x = 5, because we are already filled with 4 positions, that is, jobs.
- The next job on the sorted list is inserted at each of the x positions in each of the k (x−1)-job partial sequences, resulting in x × k x-job partial sequences.
- The best k out of the x × k sequences is selected for further processing. Here we distinguish and find out two sequences, that is, kth sequence with minimum total flow time and kth sequence with minimum makespan.
- Increment x by 1.
- If x > n, accept the best of the k n-job sequences as the final solution and stop.
- Otherwise go to step 7.

- Determine the initial solution with the aid of proposed heuristic.
- Set iteration k=0 and number of particles—each job represent a dimension. Set values of cognitive and social co-efficient C
_{1}, C_{2}[18]. - Linking to the initial solution, generate position values according to SPV rule for m particles in search space, i.e.{${\mathrm{X}}_{\mathrm{i}}^{0},\mathrm{i}=1,\dots ,\mathrm{m}$} where ${\mathrm{X}}_{\mathrm{i}}^{0}=\left\{{\mathrm{x}}_{\mathrm{i}1}^{0},\dots ,{\mathrm{x}}_{\mathrm{in}}^{0}\right\}$ and n is last job’s dimension.
- Generate initial velocities for the particles in random manner with help of Equation (15), where Rn(0,1) is random number in between 0 and 1,$\{{\mathrm{V}}_{\mathrm{i}}^{0},\mathrm{i}=1\dots \mathrm{m}\}$ where we have ${\mathrm{V}}_{\mathrm{i}}^{0}=[{\mathrm{V}}_{\text{i,1}}^{0}\dots ,{\mathrm{V}}_{\text{i,n}}^{0}\}$$${v}_{ij}^{0}={v}_{min}+\left({v}_{max}-{v}_{min}\right)Rn\left(0,1\right)$$
- Evaluate the sequences by finding out the function values with help of simulation.
- Simulation model assists in finding throughput time for machines on basis of which WIP is calculated.
- Model also determines if component of time machine is idle, calculating average utilization of the machines.

- Set these function values to be personal best for each particle in the swarm at first iteration.
- Select minimum of the personal best values and set it as global best.
- In next step, update velocity for the particles, keeping track of previous personal best value of the particle and global best of the swarm; this helps in convergence to achieve good fitness values.
- Updated velocities change the particles position and new sequence is developed and evaluated.
- Apply smallest position value (SPV) rule on the positions of generated particles, thus, determining its sequence. SPV rule gives the job its sequential number according to its dimensions adjusted by arranging particle position values in ascending order.${\mathrm{S}}_{\mathrm{i}}^{\mathrm{k}}$ = $\left[{\mathrm{s}}_{\mathrm{i},1}^{\mathrm{k}},\dots ,{\mathrm{s}}_{\mathrm{i},\mathrm{n}}^{\mathrm{k}}\right]$, where ${\mathrm{s}}_{\mathrm{i},\mathrm{j}}^{\mathrm{k}}$ is the assignment of job j of the particle i in the processing order at iteration k with respect to the jth dimension.
- Evaluate function values; if the achieved are minimum than previous, their particle positions are updated in library of personal and global best positions.
- If stopping criteria is met, stop the iteration.

- Interchange the two jobs between y
^{th}and z^{th}dimension, y ≠ z (interchange). - Remove the job at the y
^{th}dimension and then insert in z^{th}dimension, y ≠ z (insert).

## 5. Case Problems

#### 5.1. Application of NEPSO on Case Study

#### 5.2. Comparison and Model Analysis

## 6. Results

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Dimension J | PV | Velocity | Ascending Order of PV | Sequence |
---|---|---|---|---|

${\mathbf{x}}_{\mathbf{i}\mathbf{j}}^{0}$ | ${\mathbf{v}}_{\mathbf{i}\mathbf{j}}^{0}$ | |||

1 | 5.160 | 1.073 | 1.476 | 6 |

2 | 9.000 | −0.288 | 2.810 | 5 |

3 | 5.449 | −0.348 | 5.160 | 1 |

4 | 8.916 | 3.184 | 5.449 | 3 |

5 | 2.810 | −3.214 | 8.916 | 4 |

6 | 1.476 | 3.918 | 9.000 | 2 |

7 | 11.844 | 0.371 | 9.156 | 8 |

8 | 9.156 | 2.016 | 11.844 | 7 |

Dimension J | PV | Sequence | ${\mathit{x}}_{\mathit{i}\mathit{j}}^{\mathit{k}}$ | ${\mathit{s}}_{\mathit{i}\mathit{j}}^{\mathit{k}}$ |
---|---|---|---|---|

${\mathit{x}}_{\mathit{i}\mathit{j}}^{\mathit{k}}$ | ${\mathit{s}}_{\mathit{i}\mathit{j}}^{\mathit{k}}$ | |||

1 | 5.16 | 6 | 5.16 | 6 |

2 | 9 | 5 | 9 | 5 |

3 | 5.44 | 1 | 5.44 | 3 |

4 | 8.91 | 3 | 8.91 | 1 |

5 | 2.81 | 4 | 2.81 | 4 |

6 | 1.47 | 2 | 1.47 | 2 |

7 | 11.84 | 8 | 11.84 | 8 |

8 | 9.15 | 7 | 9.15 | 7 |

Dimension J | PV | Sequence | ${\mathit{x}}_{\mathit{i}\mathit{j}}^{\mathit{k}}$ | ${\mathit{s}}_{\mathit{i}\mathit{j}}^{\mathit{k}}$ |
---|---|---|---|---|

${\mathit{x}}_{\mathit{i}\mathit{j}}^{\mathit{k}}$ | ${\mathit{s}}_{\mathit{i}\mathit{j}}^{\mathit{k}}$ | |||

1 | 5.16 | 6 | 5.44 | 6 |

2 | 9 | 5 | 9 | 5 |

3 | 5.44 | 1 | 5.16 | 3 |

4 | 8.91 | 3 | 8.91 | 1 |

5 | 2.81 | 4 | 2.81 | 4 |

6 | 1.47 | 2 | 1.47 | 2 |

7 | 11.84 | 8 | 11.84 | 8 |

8 | 9.15 | 7 | 9.15 | 7 |

Machines. Parts | M1 | M2 | M3 | M4 | M5 | M6 | M7 | M8 | M9 | M10 |
---|---|---|---|---|---|---|---|---|---|---|

P1 | 2.3/1/0 | 2.5/0.15/0.15 | 0/0/0.25 | 2/0.65/0.15 | 0/0/0.15 | 0/0/0.2 | 2.8/0.25/0.25 | 1.75/0.5/0.15 | 0/0/0.2 | 0/0/0.25 |

P2 | 2.5/0.25/0 | 1.6/0.3/0.15 | 0/0/0.2 | 3.3/0.25/0.25 | 1.25/0.75/0.15 | 0/0/0.25 | 0/0/0.2 | 0/0/0.15 | 3.25/0.33/0.3 | 0/0/0.2 |

P3 | 0/0/0 | 2.31/0.25/0.2 | 2.4/0.2/0.2 | 1.5/0.5/0.25 | 1.55/0.25/0.2 | 3.1/0.45/0.3 | 0/0/0.25 | 1.5/1/0.2 | 0/0/0.15 | 2.35/0.15/0.25 |

P4 | 3.2/1.25/0 | 0/0/0.2 | 2.3/0.26/0.25 | 0/0/0.15 | 4/0.55/0.15 | 0/0/0.2 | 2.25/0.35/0.25 | 2.75/0.85/0.15 | 1.6/1.11/0.2 | 1.85/0.45/0.3 |

P5 | 1.2/0.35/0 | 2.25/0.35/0.2 | 2.5/0.15/0.15 | 2/0.75/0.25 | 0/0/0.2 | 3.25/0.75/0.25 | 1.75/0.75/0.3 | 0/0/0.2 | 0/0/0.2 | 3.33/0.75/0.3 |

P6 | 2.25/0.45/0 | 1.51/0.5/0.15 | 1.6/1/0.2 | 2.75/0.35/0.15 | 2.85/0.35/0.35 | 0/0/0.15 | 1.25/0.56/0.2 | 3.5/1/0.3 | 2/1.52/0.15 | 0/0/0.15 |

P7 | 1.35/1.5/0 | 0/0/0.2 | 2.33/0.55/0.15 | 0/0/0.2 | 0/0/0.2 | 2.45/0.35/0.15 | 1.85/0.85/0.2 | 1.75/0.65/0.3 | 0/0/0.2 | 2/1.5/0.15 |

P8 | 1.9/0.5/0 | 1.35/0.52/0.15 | 0/0/0.2 | 2.6/0.25/0.25 | 1.85/0.26/0.15 | 0/0/0.15 | 1.45/0.15/0.25 | 0/0/0.15 | 2.33/0.45/0.2 | 0/0/0.2 |

P9 | 1.75/0.4/0 | 0/0/0.2 | 1.3/0.75/0.25 | 1.85/0.46/0.15 | 0/0/0.2 | 1.75/0.5/0.3 | 3.5/0.45/0.2 | 1.5/0.75/0.25 | 2.25/0.5/0.35 | 0/0/0.2 |

P10 | 1.5/0.75/0 | 0/0/0.15 | 2/1.5/0.3 | 0/0/0.25 | 2/1.1/0.15 | 2.25/0.75/0.15 | 0/0/0.2 | 2.45/0.35/0.35 | 3.5/1.75/0.3 | 3.5/0.25 |

Particles | Iteration | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

1 | 8 | 7 | 9 | 2 | 5 | 1 | 3 | 10 | 4 | 6 |

2 | 8 | 7 | 6 | 2 | 5 | 1 | 3 | 10 | 4 | 9 |

3 | 8 | 7 | 9 | 2 | 5 | 1 | 3 | 10 | 6 | 4 |

4 | 2 | 5 | 1 | 8 | 7 | 9 | 3 | 10 | 4 | 6 |

5 | 10 | 4 | 6 | 2 | 5 | 1 | 3 | 8 | 7 | 9 |

6 | 2 | 10 | 7 | 8 | 4 | 9 | 6 | 5 | 3 | 1 |

7 | 2 | 1 | 7 | 8 | 4 | 9 | 6 | 5 | 3 | 10 |

8 | 2 | 7 | 10 | 8 | 4 | 9 | 6 | 5 | 3 | 1 |

9 | 8 | 4 | 9 | 2 | 10 | 7 | 6 | 5 | 3 | 1 |

10 | 5 | 3 | 1 | 8 | 4 | 9 | 6 | 2 | 10 | 7 |

Criteria | Value |
---|---|

Swarm/Population Size | 10 |

Inertia Weight | 1.2 |

Constriction Factor | 0.97 |

Total Number of Generations | 50 × 3 |

Problem Set | Iterations | ILOG IB CPLEX | Time (sec) | Proposed NEPSO | Time (sec) |
---|---|---|---|---|---|

5 × 10 | 20 | 40.9 | 31.232 | 37.2 | 12.133 |

7 × 7 | 20 | 48.8 | 29.220 | 44.6 | 11.544 |

8 × 10 | 20 | 52.3 | 52.644 | 45.1 | 15.330 |

12 × 5 | 20 | 46.7 | 48.321 | 39.5 | 14.221 |

Problem Set | Iterations | % Gap | |||
---|---|---|---|---|---|

ILOG IB CPLEX | Time (sec) | Proposed MPSO | Time (sec) | ||

5 × 10 | 20 | 0.091 | 0.612 | 0.00 | 0.00 |

7 × 7 | 20 | 0.86 | 0.605 | 0.00 | 0.00 |

8 × 10 | 20 | 0.16 | 0.709 | 0.00 | 0.00 |

12 × 5 | 20 | 0.154 | 0.706 | 0.00 | 0.00 |

S.No. | Cell Size | EDD | NEH | GA | PSO | NEPSO | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Normalized WIP | Normalized UU | C.O.F | Normalized WIP | Normalized UU | C.O.F | Normalized WIP | Normalized UU | C.O.F | Normalized WIP | Normalized UU | C.O.F | Normalized WIP | Normalized UU | C.O.F | % Improvement | ||

1 | 5 × 8 | 12 | 28.3 | 40.3 | 8 | 27.3 | 35.3 | 8 | 25.2 | 33.2 | 8 | 25.3 | 33.3 | 7 | 24.3 | 31.3 | 22.3 |

2 | 5 × 9 | 12 | 38.8 | 50.8 | 9 | 36.8 | 45.8 | 8 | 34.2 | 42.2 | 7 | 33.9 | 40.9 | 7 | 31.4 | 38.4 | 24.4 |

3 | 5 × 10 | 14 | 33.33 | 47.33 | 11 | 34.33 | 45.33 | 9 | 31.5 | 40.5 | 10 | 30.3 | 40.3 | 9 | 28.2 | 37.2 | 21.4 |

4 | 7 × 7 | 17 | 32.6 | 49.6 | 16 | 36.6 | 52.6 | 13 | 31.7 | 44.7 | 13 | 31.6 | 44.6 | 12 | 32.6 | 44.6 | 10.1 |

5 | 7 × 9 | 18 | 33.6 | 51.6 | 18 | 30.3 | 48.3 | 17 | 31.6 | 48.6 | 16 | 27.5 | 43.5 | 16 | 30.5 | 46.5 | 9.9 |

6 | 7 × 12 | 23 | 37.2 | 60.2 | 20 | 35.3 | 55.3 | 19 | 26.3 | 45.3 | 17 | 26.3 | 43.3 | 16 | 25.6 | 41.6 | 30.9 |

7 | 8 × 7 | 21 | 28.2 | 49.2 | 18 | 28.2 | 46.2 | 14 | 27.2 | 41.2 | 13 | 27.2 | 40.2 | 13 | 27.1 | 40.1 | 18.5 |

8 | 8 × 9 | 22 | 21.3 | 43.3 | 18 | 23.6 | 41.6 | 13 | 23.1 | 36.1 | 14 | 23.1 | 37.1 | 12 | 23.1 | 35.1 | 18.9 |

9 | 8 × 10 | 25 | 33.4 | 58.4 | 22 | 34.5 | 56.5 | 18 | 29.8 | 47.8 | 20 | 30.3 | 50.3 | 17 | 28.1 | 45.1 | 22.8 |

10 | 10 × 5 | 22 | 28.3 | 50.3 | 19 | 28.3 | 47.3 | 15 | 26.1 | 41.1 | 15 | 25.3 | 40.3 | 13 | 25.9 | 38.9 | 22.7 |

11 | 12 × 5 | 27 | 29.1 | 56.1 | 24 | 26.8 | 50.8 | 20 | 23.4 | 43.4 | 21 | 24.8 | 45.8 | 17 | 22.5 | 39.5 | 29.6 |

12 | 12 × 6 | 33 | 29 | 62 | 29 | 30.4 | 59.4 | 23 | 26.3 | 49.3 | 23 | 27.3 | 50.3 | 21 | 26.3 | 47.3 | 23.7 |

13 | 12 × 10 | 42 | 32.2 | 74.2 | 38 | 31 | 69 | 34 | 25.5 | 59.5 | 33 | 27.2 | 60.2 | 29 | 25.1 | 54.1 | 27.1 |

14 | 12 × 11 | 44 | 31 | 75 | 39 | 28.3 | 67.3 | 33 | 24.5 | 57.5 | 34 | 24.5 | 58.5 | 27 | 26.5 | 53.5 | 28.7 |

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## Share and Cite

**MDPI and ACS Style**

Khalid, Q.S.; Arshad, M.; Maqsood, S.; Jahanzaib, M.; Babar, A.R.; Khan, I.; Mumtaz, J.; Kim, S.
Hybrid Particle Swarm Algorithm for Products’ Scheduling Problem in Cellular Manufacturing System. *Symmetry* **2019**, *11*, 729.
https://doi.org/10.3390/sym11060729

**AMA Style**

Khalid QS, Arshad M, Maqsood S, Jahanzaib M, Babar AR, Khan I, Mumtaz J, Kim S.
Hybrid Particle Swarm Algorithm for Products’ Scheduling Problem in Cellular Manufacturing System. *Symmetry*. 2019; 11(6):729.
https://doi.org/10.3390/sym11060729

**Chicago/Turabian Style**

Khalid, Qazi Salman, Muhammad Arshad, Shahid Maqsood, Mirza Jahanzaib, Abdur Rehman Babar, Imran Khan, Jabir Mumtaz, and Sunghwan Kim.
2019. "Hybrid Particle Swarm Algorithm for Products’ Scheduling Problem in Cellular Manufacturing System" *Symmetry* 11, no. 6: 729.
https://doi.org/10.3390/sym11060729