# A Bender’s Algorithm of Decomposition Used for the Parallel Machine Problem of Robotic Cell

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Problem Definition

- Data are considered to be deterministic.
- Input buffer always includes parts and there exists constantly a gap at the output buffer.
- There could not be seen any buffer storage within machines.
- Every part can either be on a machine or be controled by the robot.
- The robot and the machines cannot possess more than one part at any predetermined time span.
- The intended robot and the processing machines considered above never encounter breakdown and never need maintenance.
- Times for setup are regarded as something trivial.
- There isn not any preemption authorized in the processing of any operation.

- A.
- Index properties

_{s}= Related machines’ amount at stage s, while input and output stages have a single machine, m ∈ {1, 2… Ms}

- B.
- The list of Parameters utilized

_{j}= Processing time standard regarding j

^{th}part at the processing stage, through which all parts’ processing time at input and output stages (s = 1, 3 respectively) are equal to zero,

- C.
- BM = a great many deal, the historical “Big M”.
- D.
- Variables used to make decisions

_{j,j’}= 1 if j

^{th}part calculated before ${{j}^{\prime}}^{th}$ part regarding processing stage, 0 if not,

_{j,s,f}= 1 if part j on stage s is the f

^{th}operation that would be transmitted by robot to the next stage,

_{j,s,m}= 1 if machine m processes part j at stage s, 0 if not,

_{j, s}= Time of departure of j

^{th}part from stage s,

_{MAX}= When all parts meet the output device.

- E.
- Model of mathematics

## 4. Solving Methodology

#### 4.1. Logic-Based Benders’ Decomposition (LBBD)

_{x}and D

_{y}are the domain amounts of x and y, consecutively. This issue could be divided into limitations only including the x or variables of master problem and limits that integrate the x and y, variables of slave problem. Because only the x variables are considered by the master problem, the set S is relaxed and also be demonstrated as $\overline{\text{}S}$. On the contrary to Benders’ decomposition theory, there are not any structural constraints (e.g., linearity) regarding the various constituents related to decomposition. As a whole, master problem could be identified in Equations (13)–(16):

^{k}with cost z

^{k}in iteration k. The solution obtained, is used afterwards in order to formulate a single or more slave problems, solved one by one while producing functions of bounding (i.e., the Benders cuts) on z. Suppose that k

^{th}solution method of master problem satisfies all bounding functions encountered, gained in iteration 1 to k, the process is known as to be converted to an optimal solution globally (i.e., z

^{k}= f (x

^{k},y

^{k}), where y

^{k}is the slave problem solution). If not, the master problem could be put for solution again, and the iterations go on. Within the conditions’ boundary on the Benders’ cuts, it is known that converges of the process to an optimal solution in a finite amount of iterations is achieved Chu ve Xia [41].

#### 4.2. Application of LBBD on the Problem

#### 4.2.1. Master Problem

#### 4.2.2. Slave Problem

**Initialization**: $\mathit{U}\mathit{B}=\infty $, $\mathit{L}\mathit{B}=0$, $\mathit{C}\mathit{u}\mathit{t}\mathit{s}\mathit{e}\mathit{t}=\varnothing $

**While**$\left(\mathit{L}\mathit{B}\ne \mathit{U}\mathit{B}\right)$

- (
**1**) - Solve $\left[\mathit{M}\mathit{P}\right]$ considering $\mathit{C}\mathit{u}\mathit{t}\mathit{s}\mathit{e}\mathit{t}$, obtain solution $\left({\overline{\mathit{Y}}}_{\mathit{j},\mathit{s},\mathit{m}}\leftarrow {\mathit{Y}}_{\mathit{j},\mathit{s},\mathit{m}}\right)$,
- (
**2**) - Update $\mathit{L}\mathit{B}\leftarrow {\mathit{C}}_{\mathit{M}\mathit{A}\mathit{X}}$
- (
**3**) - Solve the $\left[\mathit{S}\mathit{P}\right]$ and obtain the solution ${\mathit{C}}_{\mathit{m}\mathit{a}\mathit{x}}^{\mathit{S}\mathit{P}}$
- (
**4**) - Update $\mathit{U}\mathit{B}\leftarrow {\mathit{C}}_{\mathit{m}\mathit{a}\mathit{x}}^{\mathit{S}\mathit{P}}$
- (
**5**) - Generate the $\mathit{l}\mathit{b}\mathit{c}\mathit{u}\mathit{t}$ as a logic-based cut
- (
**6**) - Update the set of cuts: $\mathit{C}\mathit{u}\mathit{t}\mathit{s}\mathit{e}\mathit{t}\leftarrow \mathit{C}\mathit{u}\mathit{t}\mathit{s}\mathit{e}\mathit{t}+\mathit{l}\mathit{b}\mathit{c}\mathit{u}\mathit{t}$

**End While**

## 5. Results of the Computations

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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Project | SN | MS | PCri | RMS | PS | Sol-Proc | Helps |
---|---|---|---|---|---|---|---|

Sethi et al. [10] | 2, 3 | 1 | Free | Given | ✓ | - | Utilization of Gilmore and Gomory algorithms. |

Bilge and Ulusoy [11] | S | 1 | Free | ✓ | ✓ | Parallel | Model of MILP and a resolving procedure in iterative mode. |

Chen et al. [25] | 3, S | 1 | Free | Given | ✓ | - | An algorithm called Branch and Bound. |

Hall et al. [26] | 3 | 1 | Free | ✓ | ✓ | Separate | NP-completeness in sequences specific for robot move. |

Sriskandarajah [12] | S | 1 | Free | ✓ | ✓ | Separate | Classification of part sequences within 4 groups. |

Aneja and Kamoun [27] | 2 | 1 | Free | ✓ | ✓ | Separate | O (n log n) algorithm. |

Agnetis [28] | 2, 3 | 1 | Not-wait | ✓ | ✓ | Parallel | O (n log n) algorithms for the 2-machines case. Analysis of optimality in 3-machine cases. |

Hurink and Knust [29] | 3, S | 1 | Free | ✓ | ✓ | Parallel | Analyzing complexity in 2, 3 and m-machine flow shop robotic cells. |

Hurink and Knust [30] | S | 1 | Free | ✓ | ✓ | Parallel | Probe approach regarding job shop robotic cell. |

Soukhal and Martineau [13] | S | 1 | Free | ✓ | ✓ | Parallel | Genetic algorithm along with MILP models. |

Soukhal et al. [14] | 2 | 1 | Free | ✓ | ✓ | - | Analysis of complexity regarding transportation capacity. |

Carlier et al. [3] | S | 1 | Free | ✓ | ✓ | Separate | Genetic algorithm along with Branch and Bound in robot move sequences. |

Kharbeche et al. [15] | S | 1 | Free | ✓ | ✓ | Parallel | Branch and Bound algorithm along with MILP model and a new lower Bound. |

Zahrouni and Kamoun [16] | 3 | 1 | Free | ✓ | ✓ | Parallel | Heuristic Min MPS cycle provided by a 3-machine robotic cell. |

Zarandi et al. [31] | 2 | 1 | Free | ✓ | ✓ | Parallel | Analysis of robotic flow shops extension into sequence-dependent setup time. |

Elmi and Topaloglu [5] | S | M | Free | ✓ | ✓ | Parallel | New MILP models and simulated annealing (SA) algorithms. Machines with different speed at each stage and machine eligibility limits. |

Majumder and Laha [4] | 1,2 | M | Free | ✓ | ✓ | - | A novel location algorithm for two-machine CS considering sequence-dependent setup times. |

Elmi and Topaloglu [1] | S | M | Free | ✓ | ✓ | Parallel | Multi-degree cyclic problem using multiple robots. |

Elmi and Topaloglu [21] | S | M | Free | ✓ | ✓ | Parallel | Cyclic job shops robotic cell scheduling problem: Ant colony optimization. |

Majumder and Laha [4] | S | M | Free | ✓ | ✓ | - | Bacteria foraging optimization algorithm for robotic cell scheduling problem. |

Far [22] | S | M | Free | ✓ | ✓ | - | A problem of cell scheduling having flexibility and automated guided vehicles and robots using an energy-conscious policy. |

Foumani [8] | S | M | Free | ✓ | ✓ | - | Optimization in two-machine flow shop robotic cells using controllable inspection times: theory to practice. |

Xiuli [23] | S | M | Free | ✓ | ✓ | - | An algorithm of multi objective differential evolution to solve robotic CSP using batch-processing machines. |

Fattahi [24] | S | M | Free | ✓ | ✓ | Parallel | A hybrid particle swarm optimization model and parallel variable neighborhood location algorithm to be used with flexible job shop scheduling along with assembly processes |

Current research | 1 | M | Free | ✓ | ✓ | Parallel | An exact method based on benders decomposition. |

**SN**: number of stage;

**MS**: number of machines (at each stage);

**PCri**: criterion for pickup;

**RMS**: sequence in robot move;

**PS**: sequence of part;

**Sol-Proc**: procedure of solution regarding RMS and PS.

Part | Makespan | CPU Time | Total CPU Time | GAP % | Problem | ||||
---|---|---|---|---|---|---|---|---|---|

MILP | LBBD | MILP | LBBD | MP | SP | ||||

10 | 3 | 492 | 492 | 1105.38 | 24.35 | 7.06 | 17.28 | 0 | 1 |

421 | 421 | 1214.15 | 31.61 | 9.79 | 21.81 | 0 | 2 | ||

503 | 503 | 1005.54 | 22.26 | 6.67 | 15.58 | 0 | 3 | ||

516 | 516 | 1116.54 | 26.81 | 7.50 | 19.30 | 0 | 4 | ||

488 | 488 | 1204.64 | 33.10 | 10.92 | 22.17 | 0 | 5 | ||

20 | 3 | 1058 | 1058 | 2166.72 | 154.75 | 46.42 | 108.32 | 0 | 6 |

1114 | 1114 | 2018.33 | 142.29 | 41.26 | 101.02 | 0 | 7 | ||

1104 | 1104 | 1974.87 | 165.67 | 46.38 | 119.28 | 0 | 8 | ||

1046 | 1046 | 2083.77 | 155.15 | 48.09 | 107.05 | 0 | 9 | ||

1083 | 1083 | 2117.33 | 149.31 | 49.27 | 100.03 | 0 | 10 | ||

5 | 751 | 751 | 3124.12 | 138.85 | 40.26 | 98.58 | 0 | 11 | |

793 | 793 | 3101.52 | 130.19 | 36.45 | 93.73 | 0 | 12 | ||

709 | 709 | 3204.96 | 127.73 | 39.59 | 88.13 | 0 | 13 | ||

815 | 815 | 2961.24 | 139.95 | 46.18 | 93.76 | 0 | 14 | ||

762 | 762 | 3087.57 | 149.11 | 44.73 | 104.37 | 0 | 15 | ||

30 | 3 | 1731 | 1552 | 3600 | 293.47 | 88.04 | 205.42 | −10.34 | 16 |

1803 | 1487 | 3600 | 301.91 | 90.57 | 211.33 | −17.52 | 17 | ||

1792 | 1563 | 3600 | 243.55 | 73.06 | 170.48 | −12.77 | 18 | ||

1649 | 1449 | 3600 | 288.47 | 86.54 | 201.92 | −12.12 | 19 | ||

1711 | 1603 | 3600 | 251.79 | 75.53 | 176.25 | −6.31 | 20 | ||

5 | 1293 | 947 | 3600 | 274.61 | 82.38 | 192.22 | −26.75 | 21 | |

1351 | 1031 | 3600 | 237.89 | 71.36 | 166.52 | −23.68 | 22 | ||

1285 | 918 | 3600 | 208.06 | 62.41 | 145.64 | −28.56 | 23 | ||

1291 | 883 | 3600 | 264.44 | 79.33 | 185.10 | −31.60 | 24 | ||

1163 | 949 | 3600 | 225.71 | 67.71 | 157.99 | −18.40 | 25 | ||

40 | 3 | 1933 | 1737 | 3600 | 341.18 | 105.7658 | 235.41 | −10.13 | 26 |

1967 | 1805 | 3600 | 327.75 | 95.0475 | 232.70 | −8.23 | 27 | ||

1849 | 1597 | 3600 | 339.51 | 112.0383 | 227.47 | −13.62 | 28 | ||

2013 | 1897 | 3600 | 349.47 | 104.841 | 244.62 | −5.76 | 29 | ||

1784 | 1617 | 3600 | 350.47 | 98.1316 | 252.33 | −9.36 | 30 | ||

5 | 1352 | 1149 | 3600 | 328.15 | 91.882 | 236.26 | −15.01 | 31 | |

1374 | 1169 | 3600 | 310.59 | 96.2829 | 214.30 | −14.91 | 32 | ||

1289 | 1091 | 3600 | 325.94 | 107.5602 | 218.37 | −15.36 | 33 | ||

1417 | 1209 | 3600 | 330.07 | 95.7203 | 234.34 | −14.67 | 34 | ||

1286 | 1180 | 3600 | 343.95 | 103.185 | 240.76 | −8.24 | 35 | ||

50 | 3 | 3094 | 2507 | 3600 | 851.85 | 255.55 | 596.29 | −18.97 | 36 |

3271 | 2376 | 3600 | 902.56 | 270.76 | 631.79 | −27.36 | 37 | ||

3158 | 2451 | 3600 | 812.93 | 243.87 | 569.05 | −22.38 | 38 | ||

2974 | 2329 | 3600 | 784.40 | 235.32 | 549.08 | −21.68 | 39 | ||

3052 | 2385 | 3600 | 806.95 | 242.08 | 564.86 | −21.85 | 40 | ||

5 | 2842 | 1749 | 3600 | 752.17 | 225.65 | 526.51 | −38.45 | 41 | |

2937 | 1672 | 3600 | 811.49 | 243.44 | 568.04 | −43.07 | 42 | ||

2566 | 1659 | 3600 | 715.62 | 214.68 | 500.93 | −35.34 | 43 | ||

2732 | 1532 | 3600 | 771.91 | 231.57 | 540.33 | −43.92 | 44 | ||

2691 | 1471 | 3600 | 735.41 | 220.62 | 514.78 | −45.33 | 45 |

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## Share and Cite

**MDPI and ACS Style**

Komari Alaei, M.R.; Soysal, M.; Elmi, A.; Banaitis, A.; Banaitiene, N.; Rostamzadeh, R.; Javanmard, S.
A Bender’s Algorithm of Decomposition Used for the Parallel Machine Problem of Robotic Cell. *Mathematics* **2021**, *9*, 1730.
https://doi.org/10.3390/math9151730

**AMA Style**

Komari Alaei MR, Soysal M, Elmi A, Banaitis A, Banaitiene N, Rostamzadeh R, Javanmard S.
A Bender’s Algorithm of Decomposition Used for the Parallel Machine Problem of Robotic Cell. *Mathematics*. 2021; 9(15):1730.
https://doi.org/10.3390/math9151730

**Chicago/Turabian Style**

Komari Alaei, Mohammad Reza, Mehmet Soysal, Atabak Elmi, Audrius Banaitis, Nerija Banaitiene, Reza Rostamzadeh, and Shima Javanmard.
2021. "A Bender’s Algorithm of Decomposition Used for the Parallel Machine Problem of Robotic Cell" *Mathematics* 9, no. 15: 1730.
https://doi.org/10.3390/math9151730