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doi: 10.3934/jimo.2022097
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A directional heuristics pulse algorithm for a two resources constrained shortest path problem with reinitialization

1. 

School of Sciences, Southwest Petroleum University, Chengdu, 610500, China

2. 

State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu, 610500, China

*Corresponding author: Bin Zhou

Received  September 2021 Revised  May 2022 Early access June 2022

In this paper, the path planning problem of UAV flight error correction is studied, and a mixed-integer linear programming (MILP) model is proposed to solve the resource constrained shortest path problem (RCSPP-R) with reinitialization, and its resource accumulation can be reinitialized to zero at the reset node. Since RCSPP-R is a NP-hard problem, Pulse algorithm is used to solve the proposed model. Aiming at the high computational cost of pulse algorithm, three pruning strategies are proposed to improve the effectiveness and efficiency of the algorithm, namely, preprocessing strategy based on direction constraint to reduce the search space, dynamic relaxation strategy to improve the quality of boundary, and heuristic search strategy to optimize the node expansion sequence. Then the time-consuming of the algorithm and the number of dominant solutions are used as evaluation indexes for comparison. Numerical results show that the proposed algorithm can deal with this problem more reliably. Specifically, compared with the Label Correction algorithm, the original Pulse algorithm and the Ant Colony algorithm, there are advantages in the number of superior solutions, and compared with the above three algorithms, the time consumption is reduced by 31.64%, 88.52% and 83.5% respectively.

Citation: Bin Zhou, Xinghao Chen. A directional heuristics pulse algorithm for a two resources constrained shortest path problem with reinitialization. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022097
References:
[1]

F. H. AjeilI. K. IbraheemA. T. Azar and A. J. Humaidi, Grid-based mobile robot path planning using aging-based ant colony optimization algorithm in static and dynamic environments, Sensors, 20 (2020), 1880.  doi: 10.3390/s20071880.

[2]

R. BaldacciA. Mingozzi and R. Roberti, New route relaxation and pricing strategies for the vehicle routing problem, Operations research, 59 (2011), 1269-1283.  doi: 10.1287/opre.1110.0975.

[3]

E. A. Cabral, Wide Area Telecommunication Network Design: Problems and Solution Algorithms with Application to the Alberta SuperNet., Ph.D thesis, University of Alberta (Canada)., 2005.

[4]

E. A. CabralE. ErkutG. Laporte and R. A. Patterson, The network design problem with relays, European Journal of Operational Research, 180 (2007), 834-844.  doi: 10.1016/j.ejor.2006.04.030.

[5]

N. CabreraA. L. MedagliaL. Lozano and D. Duque, An exact bidirectional pulse algorithm for the constrained shortest path, Networks, 76 (2020), 128-146.  doi: 10.1002/net.21960.

[6]

W. M. CarlyleJ. O. Royset and R. Kevin Wood, Lagrangian relaxation and enumeration for solving constrained shortest-path problems, Networks: An International Journal, 52 (2008), 256-270.  doi: 10.1002/net.20247.

[7]

J. Chen, C. Du, Y. Zhang, P. Han and W. Wei, A clustering-based coverage path planning method for autonomous heterogeneous uavs, IEEE Transactions on Intelligent Transportation Systems. doi: 10.1109/TITS.2021.3066240.

[8]

J. Chen, M. Li, Z. Yuan and Q. Gu, An improved a* algorithm for uav path planning problems, in 2020 IEEE 4th Information Technology, Networking, Electronic and Automation Control Conference (ITNEC), 1, IEEE, 2020,958–962. doi: 10.1109/ITNEC48623.2020.9084806.

[9]

J. H. ChoH. S. Kim and H. R. Choi, An intermodal transport network planning algorithm using dynamic programming–a case study: From busan to rotterdam in intermodal freight routing, Applied Intelligence, 36 (2012), 529-541.  doi: 10.1007/s10489-010-0223-6.

[10]

D. G. Corneil, Lexicographic breadth first search–A survey, in International Workshop on Graph-Theoretic Concepts in Computer Science, Springer, 2004, 1–19. doi: 10.1007/978-3-540-30559-0_1.

[11]

R. K. DewanganA. Shukla and W. W. Godfrey, Three dimensional path planning using grey wolf optimizer for uavs, Applied Intelligence, 49 (2019), 2201-2217.  doi: 10.1007/s10489-018-1384-y.

[12]

H. Ding, Motion path planning of soccer training auxiliary robot based on genetic algorithm in fixed-point rotation environment, Journal of Ambient Intelligence and Humanized Computing, 11 (2020), 6261-6270.  doi: 10.1007/s12652-020-01877-4.

[13]

I. Dumitrescu and N. Boland, Improved preprocessing, labeling and scaling algorithms for the weight-constrained shortest path problem, Networks: An International Journal, 42 (2003), 135-153.  doi: 10.1002/net.10090.

[14]

D. FeroneP. Festa and F. Guerriero, An efficient exact approach for the constrained shortest path tour problem, Optimization Methods and Software, 35 (2020), 1-20.  doi: 10.1080/10556788.2018.1548015.

[15]

Y. GaoM. SchmidtL. Yang and Z. Gao, A branch-and-price approach for trip sequence planning of high-speed train units, Omega, 92 (2020), 102150.  doi: 10.1016/j.omega.2019.102150.

[16]

A. Goli and B. Malmir, A covering tour approach for disaster relief locating and routing with fuzzy demand, International Journal of Intelligent Transportation Systems Research, 18 (2020), 140-152.  doi: 10.1007/s13177-019-00185-2.

[17]

W. HeX. Qi and L. Liu, A novel hybrid particle swarm optimization for multi-uav cooperate path planning, Applied Intelligence, 51 (2021), 7350-7364.  doi: 10.1007/s10489-020-02082-8.

[18]

S. Irnich, Resource extension functions: Properties, inversion, and generalization to segments, OR Spectrum, 30 (2008), 113-148.  doi: 10.1007/s00291-007-0083-6.

[19]

Y. Jie, W. Xinmin and X. Rong, Route planning of uav formation based on improved apf, Journal of Northwestern Polytechnical University, 02.

[20]

E. Lambert, R. Romano and D. Watling, Optimal path planning with clothoid curves for passenger comfort, in Proceedings of the 5th International Conference on Vehicle Technology and Intelligent Transport Systems (VEHITS 2019), 1, SciTePress, 2019,609–615. doi: 10.5220/0007801806090615.

[21]

G. Laporte and M. M. Pascoal, Minimum cost path problems with relays, Computers & Operations Research, 38 (2011), 165-173.  doi: 10.1016/j.cor.2010.04.010.

[22]

B. LiJ. ZhangL. DaiK. L. Teo and S. Wang, A hybrid offline optimization method for reconfiguration of multi-uav formations, IEEE Transactions on Aerospace and Electronic Systems, 57 (2020), 506-520.  doi: 10.1109/TAES.2020.3024427.

[23]

L. LozanoD. Duque and A. L. Medaglia, An exact algorithm for the elementary shortest path problem with resource constraints, Transportation Science, 50 (2016), 348-357.  doi: 10.1287/trsc.2014.0582.

[24]

L. Lozano and A. L. Medaglia, On an exact method for the constrained shortest path problem, Computers & Operations Research, 40 (2013), 378-384.  doi: 10.1016/j.cor.2012.07.008.

[25]

D. LyuZ. ChenZ. Cai and S. Piao, Robot path planning by leveraging the graph-encoded floyd algorithm, Future Generation Computer Systems, 122 (2021), 204-208.  doi: 10.1016/j.future.2021.03.007.

[26]

I. MuterJ.-F. Cordeau and G. Laporte, A branch-and-price algorithm for the multidepot vehicle routing problem with interdepot routes, Transportation Science, 48 (2014), 425-441.  doi: 10.1287/trsc.2013.0489.

[27]

N. J. Nilsson, Principles of Artificial Intelligence, Morgan Kaufmann, (2014). 

[28]

L. D. P. PuglieseF. Guerriero and M. Poss, The resource constrained shortest path problem with uncertain data: a robust formulation and optimal solution approach, Computers & Operations Research, 107 (2019), 140-155.  doi: 10.1016/j.cor.2019.03.010.

[29]

M. Reihaneh and A. Ghoniem, A branch-and-price algorithm for a vehicle routing with demand allocation problem, European Journal of Operational Research, 272 (2019), 523-538.  doi: 10.1016/j.ejor.2018.06.049.

[30]

G. Righini and M. Salani, New dynamic programming algorithms for the resource constrained elementary shortest path problem, Networks: An International Journal, 51 (2008), 155-170.  doi: 10.1002/net.20212.

[31]

O. J. SmithN. Boland and H. Waterer, Solving shortest path problems with a weight constraint and replenishment arcs, Computers & Operations Research, 39 (2012), 964-984.  doi: 10.1016/j.cor.2011.07.017.

[32]

B. W. ThomasT. Calogiuri and M. Hewitt, An exact bidirectional a* approach for solving resource-constrained shortest path problems, Networks, 73 (2019), 187-205.  doi: 10.1002/net.21856.

[33]

E. B. TirkolaeeA. GoliP. Ghasemi and F. Goodarzian, Designing a sustainable closed-loop supply chain network of face masks during the covid-19 pandemic: Pareto-based algorithms, Journal of Cleaner Production, 333 (2022), 130056.  doi: 10.1016/j.jclepro.2021.130056.

[34]

B. YıldızO. E. Karaşan and H. Yaman, Branch-and-price approaches for the network design problem with relays, Computers & Operations Research, 92 (2018), 155-169.  doi: 10.1016/j.cor.2018.01.004.

[35]

Y. ZhangS. SongZ.-J. M. Shen and C. Wu, Robust shortest path problem with distributional uncertainty, IEEE Transactions on Intelligent Transportation Systems, 19 (2017), 1080-1090.  doi: 10.1109/TAC.2000.855580.

show all references

References:
[1]

F. H. AjeilI. K. IbraheemA. T. Azar and A. J. Humaidi, Grid-based mobile robot path planning using aging-based ant colony optimization algorithm in static and dynamic environments, Sensors, 20 (2020), 1880.  doi: 10.3390/s20071880.

[2]

R. BaldacciA. Mingozzi and R. Roberti, New route relaxation and pricing strategies for the vehicle routing problem, Operations research, 59 (2011), 1269-1283.  doi: 10.1287/opre.1110.0975.

[3]

E. A. Cabral, Wide Area Telecommunication Network Design: Problems and Solution Algorithms with Application to the Alberta SuperNet., Ph.D thesis, University of Alberta (Canada)., 2005.

[4]

E. A. CabralE. ErkutG. Laporte and R. A. Patterson, The network design problem with relays, European Journal of Operational Research, 180 (2007), 834-844.  doi: 10.1016/j.ejor.2006.04.030.

[5]

N. CabreraA. L. MedagliaL. Lozano and D. Duque, An exact bidirectional pulse algorithm for the constrained shortest path, Networks, 76 (2020), 128-146.  doi: 10.1002/net.21960.

[6]

W. M. CarlyleJ. O. Royset and R. Kevin Wood, Lagrangian relaxation and enumeration for solving constrained shortest-path problems, Networks: An International Journal, 52 (2008), 256-270.  doi: 10.1002/net.20247.

[7]

J. Chen, C. Du, Y. Zhang, P. Han and W. Wei, A clustering-based coverage path planning method for autonomous heterogeneous uavs, IEEE Transactions on Intelligent Transportation Systems. doi: 10.1109/TITS.2021.3066240.

[8]

J. Chen, M. Li, Z. Yuan and Q. Gu, An improved a* algorithm for uav path planning problems, in 2020 IEEE 4th Information Technology, Networking, Electronic and Automation Control Conference (ITNEC), 1, IEEE, 2020,958–962. doi: 10.1109/ITNEC48623.2020.9084806.

[9]

J. H. ChoH. S. Kim and H. R. Choi, An intermodal transport network planning algorithm using dynamic programming–a case study: From busan to rotterdam in intermodal freight routing, Applied Intelligence, 36 (2012), 529-541.  doi: 10.1007/s10489-010-0223-6.

[10]

D. G. Corneil, Lexicographic breadth first search–A survey, in International Workshop on Graph-Theoretic Concepts in Computer Science, Springer, 2004, 1–19. doi: 10.1007/978-3-540-30559-0_1.

[11]

R. K. DewanganA. Shukla and W. W. Godfrey, Three dimensional path planning using grey wolf optimizer for uavs, Applied Intelligence, 49 (2019), 2201-2217.  doi: 10.1007/s10489-018-1384-y.

[12]

H. Ding, Motion path planning of soccer training auxiliary robot based on genetic algorithm in fixed-point rotation environment, Journal of Ambient Intelligence and Humanized Computing, 11 (2020), 6261-6270.  doi: 10.1007/s12652-020-01877-4.

[13]

I. Dumitrescu and N. Boland, Improved preprocessing, labeling and scaling algorithms for the weight-constrained shortest path problem, Networks: An International Journal, 42 (2003), 135-153.  doi: 10.1002/net.10090.

[14]

D. FeroneP. Festa and F. Guerriero, An efficient exact approach for the constrained shortest path tour problem, Optimization Methods and Software, 35 (2020), 1-20.  doi: 10.1080/10556788.2018.1548015.

[15]

Y. GaoM. SchmidtL. Yang and Z. Gao, A branch-and-price approach for trip sequence planning of high-speed train units, Omega, 92 (2020), 102150.  doi: 10.1016/j.omega.2019.102150.

[16]

A. Goli and B. Malmir, A covering tour approach for disaster relief locating and routing with fuzzy demand, International Journal of Intelligent Transportation Systems Research, 18 (2020), 140-152.  doi: 10.1007/s13177-019-00185-2.

[17]

W. HeX. Qi and L. Liu, A novel hybrid particle swarm optimization for multi-uav cooperate path planning, Applied Intelligence, 51 (2021), 7350-7364.  doi: 10.1007/s10489-020-02082-8.

[18]

S. Irnich, Resource extension functions: Properties, inversion, and generalization to segments, OR Spectrum, 30 (2008), 113-148.  doi: 10.1007/s00291-007-0083-6.

[19]

Y. Jie, W. Xinmin and X. Rong, Route planning of uav formation based on improved apf, Journal of Northwestern Polytechnical University, 02.

[20]

E. Lambert, R. Romano and D. Watling, Optimal path planning with clothoid curves for passenger comfort, in Proceedings of the 5th International Conference on Vehicle Technology and Intelligent Transport Systems (VEHITS 2019), 1, SciTePress, 2019,609–615. doi: 10.5220/0007801806090615.

[21]

G. Laporte and M. M. Pascoal, Minimum cost path problems with relays, Computers & Operations Research, 38 (2011), 165-173.  doi: 10.1016/j.cor.2010.04.010.

[22]

B. LiJ. ZhangL. DaiK. L. Teo and S. Wang, A hybrid offline optimization method for reconfiguration of multi-uav formations, IEEE Transactions on Aerospace and Electronic Systems, 57 (2020), 506-520.  doi: 10.1109/TAES.2020.3024427.

[23]

L. LozanoD. Duque and A. L. Medaglia, An exact algorithm for the elementary shortest path problem with resource constraints, Transportation Science, 50 (2016), 348-357.  doi: 10.1287/trsc.2014.0582.

[24]

L. Lozano and A. L. Medaglia, On an exact method for the constrained shortest path problem, Computers & Operations Research, 40 (2013), 378-384.  doi: 10.1016/j.cor.2012.07.008.

[25]

D. LyuZ. ChenZ. Cai and S. Piao, Robot path planning by leveraging the graph-encoded floyd algorithm, Future Generation Computer Systems, 122 (2021), 204-208.  doi: 10.1016/j.future.2021.03.007.

[26]

I. MuterJ.-F. Cordeau and G. Laporte, A branch-and-price algorithm for the multidepot vehicle routing problem with interdepot routes, Transportation Science, 48 (2014), 425-441.  doi: 10.1287/trsc.2013.0489.

[27]

N. J. Nilsson, Principles of Artificial Intelligence, Morgan Kaufmann, (2014). 

[28]

L. D. P. PuglieseF. Guerriero and M. Poss, The resource constrained shortest path problem with uncertain data: a robust formulation and optimal solution approach, Computers & Operations Research, 107 (2019), 140-155.  doi: 10.1016/j.cor.2019.03.010.

[29]

M. Reihaneh and A. Ghoniem, A branch-and-price algorithm for a vehicle routing with demand allocation problem, European Journal of Operational Research, 272 (2019), 523-538.  doi: 10.1016/j.ejor.2018.06.049.

[30]

G. Righini and M. Salani, New dynamic programming algorithms for the resource constrained elementary shortest path problem, Networks: An International Journal, 51 (2008), 155-170.  doi: 10.1002/net.20212.

[31]

O. J. SmithN. Boland and H. Waterer, Solving shortest path problems with a weight constraint and replenishment arcs, Computers & Operations Research, 39 (2012), 964-984.  doi: 10.1016/j.cor.2011.07.017.

[32]

B. W. ThomasT. Calogiuri and M. Hewitt, An exact bidirectional a* approach for solving resource-constrained shortest path problems, Networks, 73 (2019), 187-205.  doi: 10.1002/net.21856.

[33]

E. B. TirkolaeeA. GoliP. Ghasemi and F. Goodarzian, Designing a sustainable closed-loop supply chain network of face masks during the covid-19 pandemic: Pareto-based algorithms, Journal of Cleaner Production, 333 (2022), 130056.  doi: 10.1016/j.jclepro.2021.130056.

[34]

B. YıldızO. E. Karaşan and H. Yaman, Branch-and-price approaches for the network design problem with relays, Computers & Operations Research, 92 (2018), 155-169.  doi: 10.1016/j.cor.2018.01.004.

[35]

Y. ZhangS. SongZ.-J. M. Shen and C. Wu, Robust shortest path problem with distributional uncertainty, IEEE Transactions on Intelligent Transportation Systems, 19 (2017), 1080-1090.  doi: 10.1109/TAC.2000.855580.

Figure 1.  The feasible path from node $ i $ to $ k $
Figure 2.  The feasible solution set of local path $ \{i, j, k \} $
Figure 3.  The frequency distribution of the included angle $ < ij, ik > $ with $ n = 400 $
Figure 4.  The back path
Figure 5.  The difference between forward and backward search
Figure 6.  Node extension for traditional and heuristics
Figure 7.  the precursor node of $ j $
Figure 8.  The number of dominant solutions
Figure 9.  The time consumption
Figure 10.  The number of dominant solutions with relaxation coefficient
Figure 11.  The time of computation with relaxation coefficient
Figure 12.  The number of dominant solutions with angle restriction coefficient
Figure 13.  The time of computation with angle restriction coefficient
Figure 14.  The number of dominant solutions for various algorithms with n = 200
Figure 15.  The number of dominant solutions for various algorithms with n = 400
Figure 16.  The number of dominant solutions for various algorithms with n = 600
Table 1.  Average times for various algorithms in different constraint
$ \rho $ 0.9 0.95
$ \xi_\alpha $ Grid(n) LC PA DA HPA AC DHPA LC PA DA HPA AC DHPA
2.5 200 0.154 1.129 0.035 0.045 0.4459 0.0542 0.150 1.156 0.033 0.042 0.6494 0.0822
400 0.468 2.548 0.051 0.080 1.1706 0.2622 0.574 3.190 0.054 0.083 1.6444 0.2322
600 1.122 5.431 0.102 0.160 4.0981 0.9275 1.308 6.334 0.107 0.171 3.2794 0.6897
AVG 0.581 3.036 0.063 0.095 1.9049 0.4146 0.677 3.560 0.065 0.099 1.8577 0.3347
2.75 200 0.154 1.268 0.038 0.043 0.8006 0.0909 0.161 1.355 0.037 0.046 0.6494 0.0944
400 0.556 3.567 0.054 0.080 1.7941 0.4072 0.611 3.940 0.051 0.078 2.0397 0.2959
600 1.550 9.037 0.113 0.169 6.7900 1.0525 1.743 10.067 0.121 0.184 3.9750 0.9078
AVG 0.753 4.624 0.068 0.097 3.1282 0.5169 0.838 5.121 0.070 0.103 2.2542 0.433
3 200 0.160 1.460 0.038 0.047 1.0484 0.1163 0.158 1.508 0.041 0.042 0.8772 0.0966
400 0.523 3.859 0.054 0.076 4.5391 0.5625 0.611 6.042 0.045 0.073 3.0772 0.4091
600 1.385 7.720 0.098 0.154 13.4325 1.3094 1.386 8.467 0.095 0.152 9.5213 1.2447
AVG 0.689 4.346 0.063 0.092 6.3400 0.6627 0.718 5.339 0.060 0.089 4.4919 0.5835
$ \rho $ 0.9 0.95
$ \xi_\alpha $ Grid(n) LC PA DA HPA AC DHPA LC PA DA HPA AC DHPA
2.5 200 0.154 1.129 0.035 0.045 0.4459 0.0542 0.150 1.156 0.033 0.042 0.6494 0.0822
400 0.468 2.548 0.051 0.080 1.1706 0.2622 0.574 3.190 0.054 0.083 1.6444 0.2322
600 1.122 5.431 0.102 0.160 4.0981 0.9275 1.308 6.334 0.107 0.171 3.2794 0.6897
AVG 0.581 3.036 0.063 0.095 1.9049 0.4146 0.677 3.560 0.065 0.099 1.8577 0.3347
2.75 200 0.154 1.268 0.038 0.043 0.8006 0.0909 0.161 1.355 0.037 0.046 0.6494 0.0944
400 0.556 3.567 0.054 0.080 1.7941 0.4072 0.611 3.940 0.051 0.078 2.0397 0.2959
600 1.550 9.037 0.113 0.169 6.7900 1.0525 1.743 10.067 0.121 0.184 3.9750 0.9078
AVG 0.753 4.624 0.068 0.097 3.1282 0.5169 0.838 5.121 0.070 0.103 2.2542 0.433
3 200 0.160 1.460 0.038 0.047 1.0484 0.1163 0.158 1.508 0.041 0.042 0.8772 0.0966
400 0.523 3.859 0.054 0.076 4.5391 0.5625 0.611 6.042 0.045 0.073 3.0772 0.4091
600 1.385 7.720 0.098 0.154 13.4325 1.3094 1.386 8.467 0.095 0.152 9.5213 1.2447
AVG 0.689 4.346 0.063 0.092 6.3400 0.6627 0.718 5.339 0.060 0.089 4.4919 0.5835
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