A Bender’s Algorithm of Decomposition Used for the Parallel Machine Problem of Robotic Cell
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
- B.
- The list of Parameters utilized
- C.
- BM = a great many deal, the historical “Big M”.
- D.
- Variables used to make decisions
- E.
- Model of mathematics
4. Solving Methodology
4.1. Logic-Based Benders’ Decomposition (LBBD)
4.2. Application of LBBD on the Problem
4.2.1. Master Problem
4.2.2. Slave Problem
- (1)
- Solve considering , obtain solution ,
- (2)
- Update
- (3)
- Solve the and obtain the solution
- (4)
- Update
- (5)
- Generate the as a logic-based cut
- (6)
- Update the set of cuts:
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. |
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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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
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 StyleKomari 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
APA StyleKomari Alaei, M. R., Soysal, M., Elmi, A., Banaitis, A., Banaitiene, N., Rostamzadeh, R., & Javanmard, S. (2021). A Bender’s Algorithm of Decomposition Used for the Parallel Machine Problem of Robotic Cell. Mathematics, 9(15), 1730. https://doi.org/10.3390/math9151730