# Minimizing the Makespan in Flowshop Scheduling for Sustainable Rubber Circular Manufacturing

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## Abstract

**:**

## 1. Introduction

- As our study scheduling context of two-line circular manufacturing has not been contemplated before, a precise mathematical formulation that clearly defines the objectives and the constraints of the problem has to be presented.
- Previous evolutionary algorithms for job-sequencing problems are not directly applicable to the addressed CRMP problem. Therefore, the developed solution method should take the CRMP properties into account and be customized to the problem context to enhance the effectiveness and efficiency.
- The CRMP problem is inspired by the real circular rubber production context, so the experimental design should include the empirical results with the dataset collected from the real production lines of the research subject. Synthetic datasets should also be tested to validate the robustness of the proposed approach.
- The proposed approach needs to be compared to modified versions of the traditional scheduling heuristics and demonstrate its advantages, such that the need for developing a new method customized to CRMP is justified.

## 2. Problem Statements and Proposed Methods

#### 2.1. Notations

Decision variables | |

${x}_{l,{i}_{1},{i}_{2}}=\{\begin{array}{cc}1,& \mathrm{if}\mathrm{job}{i}_{1}\mathrm{operation}\mathrm{precedes}\mathrm{job}{i}_{2}\mathrm{operation}\mathrm{on}\mathrm{line}l,l\in \left\{\mathrm{A},\mathrm{B}\right\}\\ 0,& \mathrm{otherwise}.\end{array}$ | |

${y}_{{i}_{1},{j}_{1},{i}_{2},{j}_{2}}=\{\begin{array}{cc}1,& \begin{array}{c}\mathrm{if}\mathrm{job}{i}_{1}\mathrm{operation}\mathrm{on}\mathrm{machine}{j}_{1}\mathrm{of}\mathrm{Line}\mathrm{A}\mathrm{is}\mathrm{completed}\mathrm{before}\\ \mathrm{the}\mathrm{starting}\mathrm{of}\mathrm{job}{i}_{2}\mathrm{operation}\mathrm{on}\mathrm{machine}{j}_{2}\mathrm{of}\mathrm{Line}\mathrm{B}\end{array}\\ 0,& \mathrm{otherwise}.\end{array}$ | |

${z}_{{i}_{1},{j}_{1},{i}_{2},{j}_{2}}=\{\begin{array}{cc}1,& \begin{array}{c}\mathrm{if}\mathrm{job}{i}_{1}\mathrm{operation}\mathrm{on}\mathrm{machine}{j}_{1}\mathrm{of}\mathrm{Line}\mathrm{B}\mathrm{starts}\mathrm{before}\\ \mathrm{the}\mathrm{starting}\mathrm{of}\mathrm{job}{i}_{2}\mathrm{operation}\mathrm{on}\mathrm{machine}{j}_{2}\mathrm{of}\mathrm{Line}\mathrm{B}\end{array}\\ 0,& \mathrm{otherwise}.\end{array}$ | |

Parameters | |

C_{max} | Makespan of the entire production consisting of Line A and Line B |

C_{l,i,j} | Completion time of job i operation on machine j of Line l, l ∈ {A, B} |

${p}_{l,i,j}$ | Processing time of job i operation on machine j of Line l, l ∈ {A, B} |

${r}_{i,j,k}$ | Yielded quantity of type k circular materials after completing job i operation on machine j in Line A |

${d}_{i,j,k}$ | Required quantity of type k circular materials before starting job i operation on machine j in Line B |

#### 2.2. Problem Statements and Formulation

_{1}jobs to be processed on line A, and all jobs on this line go through the same m

_{1}operations, which should be performed in a specific order. All of the first operations of these jobs are processed on the first machine, all of the second operations on the second, etc. Each job is processed by only one operation at a time, and no preemption is allowed. The flow control implies that the processed job sequence on each machine is the same. Similar flow control is applied to line B with n

_{2}jobs and m

_{2}operations. Each job operation upon finishing on line A will yield two types of circular materials (scrap rubber granulates and strips), which can be reused or mixed with other materials in the job operations on line B. The start of job operation on line B should wait until all required circular materials are received from the yielding of line A. The aim of our circular rubber manufacturing problem (CRMP) is to determine the optimal flowshop schedules for the two lines such that the makespan of all jobs is a minimum. The CRMP is NP-Hard given that each individual line of CRMP is itself a flowshop scheduling problem, which has been known as NP-Hard when there are more than two machines [16].

_{max}

_{max}> C

_{l,i,j}∀i, ∀j, ∀l;

_{2}can only start after its operation on any preceding machines has completed, so the completion time of job i on machine j

_{2}should be larger than the sum of its processing time and the completion time of any preceding machine j

_{1}in the same production line. Constraint (4) entails that the operation of any two distinct jobs on the same machine (job i

_{1}operation precedes job i

_{2}operation on machine j) cannot be preempted. Equations (5) and (6) guarantee the job flow constraints. Constraint (7) calls for the satisfaction of the circular material demand for processing job i

_{2}on machine j

_{2}in Line B by requiring the demand is less than or equal to all the materials already yielded in Line A minus all the consumed materials in Line B. Equations (8)–(11) are used to validate the decision variables ${y}_{{i}_{1},{j}_{1},{i}_{2},{j}_{2}}$ and ${z}_{{i}_{1},{j}_{1},{i}_{2},{j}_{2}}$, while Equations (12)–(14) claim that all decision variables are binary ones. The integer programming model of small problem sizes can be solved to optimality by using commercial software such as lingo. However, for medium- and large-sized problems, quality solutions may be obtained by applying evolutionary algorithms such as GAs.

#### 2.3. Proposed Methods

_{1}jobs $\left\{{J}_{1}^{\mathrm{A}},{J}_{2}^{\mathrm{A}},\dots ,{J}_{{n}_{1}}^{\mathrm{A}}\right\}$ to be processed by line A and n

_{2}jobs $\left\{{J}_{1}^{\mathrm{B}},{J}_{2}^{\mathrm{B}},\dots ,{J}_{{n}_{2}}^{\mathrm{B}}\right\}$ by line B. A feasible job processing schedule to the CRMP problem can be represented by a chromosome integer string as X = ‘${x}_{1},{x}_{2},\dots ,{x}_{{n}_{1}+{n}_{2}}$’ where $1\le {x}_{i}\le {n}_{1}+{n}_{2}$ and ${x}_{i}\ne {x}_{j}$ if $i\ne j$. In other words, X is an integer permutation where ${x}_{i}$ indicates ${J}_{i}^{\mathrm{A}}$ if $1\le {x}_{i}\le {n}_{1}$ and ${x}_{i}$ indicates ${J}_{i-{n}_{1}}^{\mathrm{B}}$ if ${n}_{1}+1\le {x}_{i}\le {n}_{1}+{n}_{2}$. For example, let there be 4 and 3 jobs to be processed by line A and line B, respectively. A chromosome X = ‘3614725’ indicates the job order $\left\{{J}_{3}^{\mathrm{A}},{J}_{1}^{\mathrm{A}},{J}_{4}^{\mathrm{A}},{J}_{2}^{\mathrm{A}}\right\}$ for line A and $\left\{{J}_{2}^{\mathrm{B}},{J}_{3}^{\mathrm{B}},{J}_{1}^{\mathrm{B}}\right\}$ for line B. The fitness value of the chromosome is evaluated by calculating the makespan with the job orders contained in the chromosome. In this context, the lower the fitness value (makespan) is, the better the chromosome.

**P**of N chromosomes is generated at random to start the evolution. The fitness value for each chromosome in

**P**is evaluated. Natural selection is a process that mimics the survival of the fitter chromosomes. We adopt the tournament selection scheme to generate the next population

**Q**from the current population

**P**. The selection process is as follows. For each empty slot in the next population

**Q**, two chromosomes are randomly selected from

**P**to compete for survival. The winner of the competition is determined by reference to the fitness values, and the winner is cloned to fill the empty slot. Both competitors still remain in

**P**and are eligible for selection in the next round. The selection process is repeated until all the empty slots in the next population

**Q**are filled up. The tournament selection is superior to earlier selection schemes like roulette-wheel selection, which has a high risk of generating large stochastic errors.

_{c}. In other words, N × P

_{c}chromosomes will be manipulated by crossover. As our chromosome structure is represented as a permutation string, the partially matched crossover (PMX) that can reorder the permutation is applied. The PMX crossover resembles the two-point crossover but respects a look-up table to avoid repetitions of gene alleles. The PMX crossover consists of three steps: look-up table establishment, gene exchange, and gene correction. An example is illustrated in Figure 3 where two parental chromosomes X and Y are shown with the PMX crossover to produce two offspring chromosomes X’ and Y’. Firstly, two cutting sites are generated at random, say at 3 and 6. The gene segments between the two cutting sites are used to establish a look-up table, i.e., the list (2, 3, 10) is mapped to (5, 6, 7). Secondly, the gene segments between the two cutting sites are exchanged. Finally, the repetitive genes in the remaining segments due to the exchange are corrected by referring to the look-up table (the corresponding genes before and after correction are printed in red as shown in Figure 3).

_{m}. Mutation is an imperative operation to reserve the gene diversity in the population in order to avoid premature convergence. The ordinary flipping mutation, which mutates the gene by replacing its allele with an alternative one, is not applicable to our permutation-based chromosomes because there is risk of generating repetitive gene alleles in a chromosome. Here, we employ the 2-swap mutation that randomly selects two genes and exchanges them. The 2-swap operation is a common heuristic that has been embedded in sophisticated methods for combinatorial optimization problems.

## 3. Results

#### 3.1. Experiment with Real Dataset

#### 3.2. Experiment with Synthetic Dataset

#### 3.3. Sensitivity Analysis

_{1}= 6, m

_{2}= 3) and varied the number of jobs with different values (j

_{1}= 4, j

_{2}= 3; j

_{1}= 8, j

_{2}= 6; j

_{1}= 16, j

_{2}= 12; j

_{1}= 32, j

_{2}= 24). Secondly, we fixed the value of the number of jobs in the two lines (j

_{1}= 8, j

_{2}= 6) and varied the number of machines with different values (m

_{1}= 4, m

_{2}= 2; m

_{1}= 6, m

_{2}= 3; m

_{1}= 8, m

_{2}= 4; m

_{1}= 10, m

_{2}= 5). The statistics over 30 independent runs of GA for each problem instance are listed in Table 7. To realize the performance of GA, the integer programming proposed in Section 2.2 was solved to optimality or the best feasible bound obtained at a maximal CPU time set to 24 h (86,400 s). The computed result of the integer programming on the same dataset is shown in Table 8.

_{1}and j

_{2}) was a more influential parameter than the number of machines (m

_{1}and m

_{2}). When the number of jobs increased, the best and average makespan from multiple runs of GA may differ, and the consumed CPU time increased due to a longer length (j

_{1}+ j

_{2}) needed for encoding the chromosome. Similarly, the increasing complexity makes the CRMP intractable. The exact solution can only be obtained for small problems (j

_{1}= 4, j

_{2}= 3; j

_{1}= 8, j

_{2}= 6). For larger-sized problems (j

_{1}= 16, j

_{2}= 12; j

_{1}= 32, j

_{2}= 24), no exact solution can be obtained within 86,400 s, and only the best feasible bounds can be returned. For small problems, GA obtained the global optimal solutions as the exact ones. For larger-sized problems, the solutions evolved by GA were significantly better than the best feasible bounds reported by the exact method at the maximal CPU time, validating the robustness of our GA against the number of jobs.

_{1}= 6, m

_{2}= 3; j

_{1}= 8, j

_{2}= 6). The consumed CPU times were also very near to each other because the length (j

_{1}+ j

_{2}) required for encoding the chromosome was the same. For the exact method, the required computation time to obtain the exact solution was very close to the maximal CPU time. When the exact solution cannot be resolved within 86,400 s, the reported best feasible bound is likely the exact solution, which was also the best solution obtained by the GA. Again, this phenomenon validates the robustness of the proposed GA against the variations on the number of machines.

## 4. Discussions

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The typical operations for the production in the rubber manufacturer and exemplar rubber products. (

**a**) Compounding; (

**b**) Coating; (

**c**) Molding; (

**d**) Vulcanization; (

**e**) Demolding; (

**f**) Deflashing; (

**g**) Exemplar products of the first line; (

**h**) Exemplar circular products of the second line; (

**i**) Stone-like wall rubber.

**Figure 3.**Two parental chromosomes X and Y are shown with the PMX crossover to produce two offspring chromosomes X’ and Y’.

**Figure 5.**The Gantt chart of the best schedule of both lines and the availability of the two types of circular materials: (

**a**) FCFS; (

**b**) Campbell-Dudek; (

**c**) GA.

${\mathit{J}}_{1}^{\mathbf{A}}$ | ${\mathit{J}}_{2}^{\mathbf{A}}$ | ${\mathit{J}}_{3}^{\mathbf{A}}$ | |
---|---|---|---|

${M}_{1}^{\mathrm{A}}$ | 4, 3, 1 | 5, 1, 3 | 3, 2, 2 |

${M}_{2}^{\mathrm{A}}$ | 5, 3, 2 | 2, 2, 2 | 6, 1, 1 |

**Table 2.**The processing time and required circular materials of each job on each operation of line B.

${\mathit{J}}_{1}^{\mathbf{A}}$ | ${\mathit{J}}_{2}^{\mathbf{A}}$ | ${\mathit{J}}_{3}^{\mathbf{A}}$ | |
---|---|---|---|

${M}_{1}^{\mathrm{A}}$ | 2, 0, 1 | 3, 2, 2 | 4, 7, 1 |

${M}_{2}^{\mathrm{A}}$ | 1, 1, 0 | 2, 1, 1 | 4, 1, 5 |

${\mathit{J}}_{1}^{\mathbf{A}}$ | ${\mathit{J}}_{2}^{\mathbf{A}}$ | ${\mathit{J}}_{3}^{\mathbf{A}}$ | ${\mathit{J}}_{4}^{\mathbf{A}}$ | ${\mathit{J}}_{5}^{\mathbf{A}}$ | ${\mathit{J}}_{6}^{\mathbf{A}}$ | ${\mathit{J}}_{7}^{\mathbf{A}}$ | ${\mathit{J}}_{8}^{\mathbf{A}}$ | |
---|---|---|---|---|---|---|---|---|

${M}_{1}^{\mathrm{A}}$ | 115, 63, 15 | 77, 74, 35 | 107, 96, 5 | 93, 140, 54 | 91, 74, 49 | 62, 12, 28 | 77, 28, 38 | 72, 46, 40 |

${M}_{2}^{\mathrm{A}}$ | 21, 20, 13 | 5, 4, 1 | 26, 33, 5 | 23, 32, 13 | 15, 6, 4 | 10, 11, 6 | 17, 19, 5 | 25, 22, 3 |

${M}_{3}^{\mathrm{A}}$ | 10, 15, 5 | 14, 17, 7 | 14, 23, 3 | 11, 14, 2 | 10, 7, 4 | 14, 2, 5 | 11, 5, 5 | 14, 12, 8 |

${M}_{4}^{\mathrm{A}}$ | 173, 147, 37 | 113, 122, 66 | 132, 57, 59 | 169, 141, 76 | 92, 29, 29 | 145, 140, 27 | 165, 107, 8 | 114, 150, 63 |

${M}_{5}^{\mathrm{A}}$ | 12, 11, 6 | 7, 9, 2 | 3, 1, 1 | 14, 22, 4 | 8, 6, 2 | 4, 2, 2 | 5, 6, 2 | 11, 4, 6 |

${M}_{6}^{\mathrm{A}}$ | 52, 39, 20 | 111, 33, 68 | 36, 28, 3 | 107, 91, 64 | 53, 37, 8 | 68, 67, 43 | 50, 68, 15 | 66, 107, 11 |

${\mathit{J}}_{1}^{\mathbf{B}}$ | ${\mathit{J}}_{2}^{\mathbf{B}}$ | ${\mathit{J}}_{3}^{\mathbf{B}}$ | ${\mathit{J}}_{4}^{\mathbf{B}}$ | ${\mathit{J}}_{5}^{\mathbf{B}}$ | ${\mathit{J}}_{6}^{\mathbf{B}}$ | |
---|---|---|---|---|---|---|

${M}_{1}^{\mathrm{B}}$ | 51, 134, 42 | 54, 101, 82 | 37, 88, 45 | 71, 75, 37 | 32, 127, 30 | 78, 218, 105 |

${M}_{2}^{\mathrm{B}}$ | 21, 76, 18 | 43, 40, 40 | 40, 114, 21 | 19, 71, 24 | 31, 72, 25 | 26, 65, 41 |

${M}_{3}^{\mathrm{B}}$ | 84, 98, 103 | 75, 114, 44 | 110, 116, 96 | 85, 288, 55 | 96, 196, 50 | 112, 189, 111 |

Best | Average | Std | CPU Time | |
---|---|---|---|---|

FCFS | 1457 | 1457 | 0 | 6.7 × 10^{−6} |

Campbell-Dudek | 1340 | 1361 | 17.46 | 0.0223 |

GA | 1307 | 1315 | 8.05 | 1.2764 |

FCFS | Campbell-Dudek | GA | |||||
---|---|---|---|---|---|---|---|

Instance No. | Best | Best | Average | Std | Best | Average | Std |

1 | 1440 | 1271 | 1323.60 | 28.89 | 1254 | 1258.83 | 9.80 |

2 | 1398 | 1238 | 1257.50 | 17.26 | 1211 | 1211.60 | 2.24 |

3 | 1434 | 1282 | 1282.60 | 1.20 | 1244 | 1249.07 | 7.00 |

4 | 1431 | 1291 | 1319.40 | 31.71 | 1231 | 1233.57 | 3.03 |

5 | 1440 | 1295 | 1321.20 | 20.29 | 1284 | 1285.67 | 5.61 |

6 | 1417 | 1290 | 1301.40 | 9.37 | 1259 | 1266.67 | 5.14 |

7 | 1474 | 1347 | 1366.00 | 15.72 | 1308 | 1316.00 | 7.36 |

8 | 1465 | 1323 | 1348.60 | 25.09 | 1302 | 1305.43 | 5.79 |

9 | 1364 | 1299 | 1317.00 | 14.93 | 1270 | 1273.53 | 3.96 |

10 | 1407 | 1281 | 1299.30 | 11.98 | 1272 | 1278.07 | 6.30 |

Average | 1427.00 | 1291.70 | 1313.66 | 17.64 | 1263.50 | 1267.84 | 5.62 |

Gap ratio | − | 9.5% | 7.9% | 11.5% | 11.2% |

m_{1}-m_{2}-j_{1}-j_{2} | Best | Average | Std | CPU Time |
---|---|---|---|---|

6-3-4-3 | 687 | 687 | 0 | 0.3742 |

6-3-8-6 | 1307 | 1315 | 8.05 | 1.2764 |

6-3-16-12 | 2370 | 2370.40 | 2.15 | 4.6592 |

6-3-32-24 | 4497 | 4501.27 | 4.84 | 16.9457 |

4-2-8-6 | 1277 | 1277 | 0 | 0.9479 |

6-3-8-6 | 1307 | 1315 | 8.05 | 1.2764 |

8-4-8-6 | 1201 | 1201 | 0 | 1.5196 |

10-5-8-6 | 1408 | 1408 | 0 | 2.0288 |

m_{1}-m_{2}-j_{1}-j_{2} | Exact Solution | Feasible Bound | CPU Time |
---|---|---|---|

6-3-4-3 | 687 | 0.5924 | |

6-3-8-6 | 1307 | 66,667.15 | |

6-3-16-12 | 2427 | 86,400 | |

6-3-32-24 | 4619 | 86,400 | |

4-2-8-6 | 1277 | 86,400 | |

6-3-8-6 | 1307 | 66,667.15 | |

8-4-8-6 | 1201 | 74,804.99 | |

10-5-8-6 | 1408 | 86,400 |

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**MDPI and ACS Style**

Yin, P.-Y.; Chen, H.-M.; Cheng, Y.-L.; Wei, Y.-C.; Huang, Y.-L.; Day, R.-F.
Minimizing the Makespan in Flowshop Scheduling for Sustainable Rubber Circular Manufacturing. *Sustainability* **2021**, *13*, 2576.
https://doi.org/10.3390/su13052576

**AMA Style**

Yin P-Y, Chen H-M, Cheng Y-L, Wei Y-C, Huang Y-L, Day R-F.
Minimizing the Makespan in Flowshop Scheduling for Sustainable Rubber Circular Manufacturing. *Sustainability*. 2021; 13(5):2576.
https://doi.org/10.3390/su13052576

**Chicago/Turabian Style**

Yin, Peng-Yeng, Hsin-Min Chen, Yi-Lung Cheng, Ying-Chieh Wei, Ya-Lin Huang, and Rong-Fuh Day.
2021. "Minimizing the Makespan in Flowshop Scheduling for Sustainable Rubber Circular Manufacturing" *Sustainability* 13, no. 5: 2576.
https://doi.org/10.3390/su13052576