# American Institute of Mathematical Sciences

## The heterogeneous fleet location routing problem with simultaneous pickup and delivery and overloads

 School of Business Administration, Jiangxi University of Finance and Economics, Nanchang 30013, China

* Corresponding author: Xuefeng Wang

Received  June 2017 Revised  December 2017 Published  November 2018

This paper addresses a new variant of the location routing problem (LRP), namely the heterogeneous fleet LRP with simultaneous pickup and delivery and overloads (HFLRPSPDO) which has not been previously tackled in literatures. In this problem, the heterogeneous fleet is comprised of vehicles with different capacities, and the vehicle overloads up to a specified upper bound is allowed. This paper proposes a polynomial-size mixed integer linear programming formulation for the problem in which a penalty function, allowing capacity violations of vehicles, is integrated into objective function. Furthermore, two heuristic algorithms, respectively based on tabu search and simulated annealing, are proposed to solve HFLRPSPDO. Computational results on simulated instances show the effectiveness of the proposed problem formulation and the efficiency of the proposed heuristic algorithms.

Citation: Xuefeng Wang. The heterogeneous fleet location routing problem with simultaneous pickup and delivery and overloads. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2019079
##### References:
 [1] D. Ambrosinoa, Distribution network design: New problems and related models, European Journal of Operational Research, 165 (2005), 610-624. doi: 10.1016/j.ejor.2003.04.009. [2] R. T. Berger, C. R. Coullard and M. S. Daskin, Location-routing problems with distance constraints, Transportation Science, 41 (2007), 29-43. [3] T. W. Chien, Heuristic procedures for practical-sized uncapacitated location-capacitated routing problems, Decision Sciences, 24 (1993), 995-1021. [4] C. H. Chu and J. Hopscotch, Further discussion for transit system of chicago, Journal of Discrete Mathematical Sciences & Cryptography, 20 (2017), 717-724. doi: 10.1080/09720529.2016.1197600. [5] G. Clarke and J. W. Wright, Scheduling of vehicles from a central depot to a number of delivery points, Operations Research, 12 (1964), 568-581. [6] J. Dethloff, Vehicle routing and reverse logistics: The vehicle routing problem with simultaneous delivery and pick-up, OR-Spektrum, 23 (2001), 79-96. doi: 10.1007/PL00013346. [7] J. Geunes and B. M. Chang, Operations Research Models for Supply Chain Management and Design, vol. 76, Springer US, 1994. [8] B. Golden, A. Assad, L. Levy and F. Gheysens, The fleet size and mix vehicle routing problem, Computers & Operations Research, 11 (1984), 49-66. [9] G. Ioannou, M. Kritikos and G. Prastacos, A greedy look-ahead heuristic for the vehicle routing problem with time windows, Journal of the Operational Research Society, 52 (2001), 523-537. [10] H. Kamankesh and V. G. Agelidis, A sufficient stochastic framework for optimal operation of micro-grids considering high penetration of renewable energy sources and electric vehicles, Journal of Intelligent & Fuzzy Systems, 32 (2017), 373-387. [11] I. Karaoglan, F. Altiparmak, I. Kara and B. Dengiz, A branch and cut algorithm for the location-routing problem with simultaneous pickup and delivery, European Journal of Operational Research, 211 (2011), 318-332. doi: 10.1016/j.ejor.2011.01.003. [12] I. Karaoglan, F. Altiparmak, I. Kara and B. Dengiz, The location-routing problem with simultaneous pickup and delivery: Formulations and a heuristic approach, Omega, 40 (2012), 465-477. [13] M. N. Kritikos and G. Ioannou, The heterogeneous fleet vehicle routing problem with overloads and time windows, International Journal of Production Economics, 144 (2013), 68-75. [14] G. Laporte, Y. Nobert and D. Arpin, An exact algorithm for solving a capacitated location-routing problem, Annals of Operations Research, 6 (1986), 293-310. [15] G. Laporte, Y. Nobert and S. Taillefer, Solving a family of multi-depot vehicle routing and location-routing problems, Transportation Science, 22 (1988), 161-172. doi: 10.1287/trsc.22.3.161. [16] C. K. Y. Lin, C. K. Chow and A. Chen, A location-routing-loading problem for bill delivery services, Computers & Industrial Engineering, 43 (2002), 5-25. [17] C. K. Y. Lin and R. C. W. Kwok, Multi-objective metaheuristics for a location-routing problem with multiple use of vehicles on real data and simulated data, European Journal of Operational Research, 175 (2006), 1833-1849. [18] M. Lundy and A. Mees, Convergence of an annealing algorithm, Mathematical Programming, 34 (1986), 111-124. doi: 10.1007/BF01582166. [19] H. Min, Consolidation terminal location-allocation and consolidated routing problems, Journal of Business Logistics, 17 (1996), 235-263. [20] H. Min, V. Jayaraman and R. Srivastava, Combined Location-Routing Problems: A Synthesis and Future Research Directions, vol. 108, Springer Berlin Heidelberg, 1998. [21] G. Nagy and S. Salhi, Location-routing: Issues, models and methods, European Journal of Operational Research, 177 (2007), 649-672. doi: 10.1016/j.ejor.2006.04.004. [22] G. Nagy and S. Salhi, Nested heuristic methods for the location-routing problem, 47 (1996), 1166-1174. [23] C. Prodhon,, in http://prodhonc.free.fr/homepage, 2016. [24] S. Salhi and M. Fraser, An integrated heuristic approach for the combined location vehicle fleet mix problem, Studies in Locational Analysis, 8 (1996), 3-21. [25] S. Salhi and G. Nagy, A cluster insertion heuristic for single and multiple depot vehicle routing problems with backhauling, Journal of the Operational Research Society, 50 (1999), 1034-1042. [26] M. Schwardt and J. Dethloff, Solving a continuous location-routing problem by use of a self-organizing map, International Journal of Physical Distribution & Logistics Management, 35 (2005), 390-408. [27] R. Srivastava, Alternate solution procedures for the location-routing problem, Omega, 21 (1993), 497-506. [28] D. Tuzun and L. I. Burke, A two-phase tabu search approach to the location routing problem, European Journal of Operational Research, 116 (1999), 87-99. [29] T. H. Wu, C. Low and J. W. Bai, Heuristic solutions to multi-depot location-routing problems, Computers & Operations Research, 29 (2002), 1393-1415. [30] X. Zhang, Z. B. Zhang, H. Broersma and X. Wen, On the complexity of edge-colored subgraph partitioning problems in network optimization, Discrete Mathematics & Theoretical Computer Science Dmtcs, 17 (2016), 227-244.

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##### References:
 [1] D. Ambrosinoa, Distribution network design: New problems and related models, European Journal of Operational Research, 165 (2005), 610-624. doi: 10.1016/j.ejor.2003.04.009. [2] R. T. Berger, C. R. Coullard and M. S. Daskin, Location-routing problems with distance constraints, Transportation Science, 41 (2007), 29-43. [3] T. W. Chien, Heuristic procedures for practical-sized uncapacitated location-capacitated routing problems, Decision Sciences, 24 (1993), 995-1021. [4] C. H. Chu and J. Hopscotch, Further discussion for transit system of chicago, Journal of Discrete Mathematical Sciences & Cryptography, 20 (2017), 717-724. doi: 10.1080/09720529.2016.1197600. [5] G. Clarke and J. W. Wright, Scheduling of vehicles from a central depot to a number of delivery points, Operations Research, 12 (1964), 568-581. [6] J. Dethloff, Vehicle routing and reverse logistics: The vehicle routing problem with simultaneous delivery and pick-up, OR-Spektrum, 23 (2001), 79-96. doi: 10.1007/PL00013346. [7] J. Geunes and B. M. Chang, Operations Research Models for Supply Chain Management and Design, vol. 76, Springer US, 1994. [8] B. Golden, A. Assad, L. Levy and F. Gheysens, The fleet size and mix vehicle routing problem, Computers & Operations Research, 11 (1984), 49-66. [9] G. Ioannou, M. Kritikos and G. Prastacos, A greedy look-ahead heuristic for the vehicle routing problem with time windows, Journal of the Operational Research Society, 52 (2001), 523-537. [10] H. Kamankesh and V. G. Agelidis, A sufficient stochastic framework for optimal operation of micro-grids considering high penetration of renewable energy sources and electric vehicles, Journal of Intelligent & Fuzzy Systems, 32 (2017), 373-387. [11] I. Karaoglan, F. Altiparmak, I. Kara and B. Dengiz, A branch and cut algorithm for the location-routing problem with simultaneous pickup and delivery, European Journal of Operational Research, 211 (2011), 318-332. doi: 10.1016/j.ejor.2011.01.003. [12] I. Karaoglan, F. Altiparmak, I. Kara and B. Dengiz, The location-routing problem with simultaneous pickup and delivery: Formulations and a heuristic approach, Omega, 40 (2012), 465-477. [13] M. N. Kritikos and G. Ioannou, The heterogeneous fleet vehicle routing problem with overloads and time windows, International Journal of Production Economics, 144 (2013), 68-75. [14] G. Laporte, Y. Nobert and D. Arpin, An exact algorithm for solving a capacitated location-routing problem, Annals of Operations Research, 6 (1986), 293-310. [15] G. Laporte, Y. Nobert and S. Taillefer, Solving a family of multi-depot vehicle routing and location-routing problems, Transportation Science, 22 (1988), 161-172. doi: 10.1287/trsc.22.3.161. [16] C. K. Y. Lin, C. K. Chow and A. Chen, A location-routing-loading problem for bill delivery services, Computers & Industrial Engineering, 43 (2002), 5-25. [17] C. K. Y. Lin and R. C. W. Kwok, Multi-objective metaheuristics for a location-routing problem with multiple use of vehicles on real data and simulated data, European Journal of Operational Research, 175 (2006), 1833-1849. [18] M. Lundy and A. Mees, Convergence of an annealing algorithm, Mathematical Programming, 34 (1986), 111-124. doi: 10.1007/BF01582166. [19] H. Min, Consolidation terminal location-allocation and consolidated routing problems, Journal of Business Logistics, 17 (1996), 235-263. [20] H. Min, V. Jayaraman and R. Srivastava, Combined Location-Routing Problems: A Synthesis and Future Research Directions, vol. 108, Springer Berlin Heidelberg, 1998. [21] G. Nagy and S. Salhi, Location-routing: Issues, models and methods, European Journal of Operational Research, 177 (2007), 649-672. doi: 10.1016/j.ejor.2006.04.004. [22] G. Nagy and S. Salhi, Nested heuristic methods for the location-routing problem, 47 (1996), 1166-1174. [23] C. Prodhon,, in http://prodhonc.free.fr/homepage, 2016. [24] S. Salhi and M. Fraser, An integrated heuristic approach for the combined location vehicle fleet mix problem, Studies in Locational Analysis, 8 (1996), 3-21. [25] S. Salhi and G. Nagy, A cluster insertion heuristic for single and multiple depot vehicle routing problems with backhauling, Journal of the Operational Research Society, 50 (1999), 1034-1042. [26] M. Schwardt and J. Dethloff, Solving a continuous location-routing problem by use of a self-organizing map, International Journal of Physical Distribution & Logistics Management, 35 (2005), 390-408. [27] R. Srivastava, Alternate solution procedures for the location-routing problem, Omega, 21 (1993), 497-506. [28] D. Tuzun and L. I. Burke, A two-phase tabu search approach to the location routing problem, European Journal of Operational Research, 116 (1999), 87-99. [29] T. H. Wu, C. Low and J. W. Bai, Heuristic solutions to multi-depot location-routing problems, Computers & Operations Research, 29 (2002), 1393-1415. [30] X. Zhang, Z. B. Zhang, H. Broersma and X. Wen, On the complexity of edge-colored subgraph partitioning problems in network optimization, Discrete Mathematics & Theoretical Computer Science Dmtcs, 17 (2016), 227-244.
The main steps of the TS-heuristics and SA-heuristics
Problem cost parameters for simulated instances
 Parameter Value Depot Capacity ($CD_j )$ Uniformly distributed over the interval [200,400] Fixed cost ($FD_j )$ 3000 unit per depot Vehicle Type Type A, B, C for customers. Fixed cost ($FV_k )$ Cost (100,150,200) for type (A, B, C), respectively. Capacity ($CV_k )$ Cost (100,200,300) for type (A, B, C), respectively Distance cost ratio Unit cost distance (1, 1.5, 2) for type (A, B, C), respectively
 Parameter Value Depot Capacity ($CD_j )$ Uniformly distributed over the interval [200,400] Fixed cost ($FD_j )$ 3000 unit per depot Vehicle Type Type A, B, C for customers. Fixed cost ($FV_k )$ Cost (100,150,200) for type (A, B, C), respectively. Capacity ($CV_k )$ Cost (100,200,300) for type (A, B, C), respectively Distance cost ratio Unit cost distance (1, 1.5, 2) for type (A, B, C), respectively
Parameter settings of TS-heuristics and SA-heuristics
 TS-heuristics SA -heuristics Parameter Value Parameter Value max-add 5 initial temperature (location and routing phases) 50 max-swap 8 cooling rate (location phase) 0.95 max-route 6 cooling rate (routing phase) 0.9 tabu duration(location phase) 8 final temperature (location and routing phase) 0.15 tabu duration(routing phase) 10 no-improvement number of cycles (location and routing phases) 10
 TS-heuristics SA -heuristics Parameter Value Parameter Value max-add 5 initial temperature (location and routing phases) 50 max-swap 8 cooling rate (location phase) 0.95 max-route 6 cooling rate (routing phase) 0.9 tabu duration(location phase) 8 final temperature (location and routing phase) 0.15 tabu duration(routing phase) 10 no-improvement number of cycles (location and routing phases) 10
Computational results for the formulations on small-size test problems
 Original formulation Strong formulation $N_C$ $N_0$ Gap% CPU #OP Gap% CPU #OP 15 3 45.82 86.29 10 5.04 48.02 10 20 3 36.02 1973.64 8 1.47 2074.65 9 20 4 51.80 2391.82 6 0.85 2619.46 7 30 4 26.39 3729.65 5 13.23 3482.73 6 40 4 38.74 5183.59 5 15.78 5827.84 5 30 5 33.90 4734.64 4 21.92 5418.38 5 40 5 46.29 5374.68 2 3.63 6016.73 3 50 5 34.62 6739.52 2 8.51 6473.63 3 40 6 29.43 7200.00 0 2.58 7200.00 0 50 6 37.58 7200.00 0 16.28 7200.00 0 Average 38.06 5279.52 8.93 4636.14
 Original formulation Strong formulation $N_C$ $N_0$ Gap% CPU #OP Gap% CPU #OP 15 3 45.82 86.29 10 5.04 48.02 10 20 3 36.02 1973.64 8 1.47 2074.65 9 20 4 51.80 2391.82 6 0.85 2619.46 7 30 4 26.39 3729.65 5 13.23 3482.73 6 40 4 38.74 5183.59 5 15.78 5827.84 5 30 5 33.90 4734.64 4 21.92 5418.38 5 40 5 46.29 5374.68 2 3.63 6016.73 3 50 5 34.62 6739.52 2 8.51 6473.63 3 40 6 29.43 7200.00 0 2.58 7200.00 0 50 6 37.58 7200.00 0 16.28 7200.00 0 Average 38.06 5279.52 8.93 4636.14
Computational results for relaxations of the formulations
 LP of original formulation LP of strong formulation SLP of strong formulation $N_C$ $N_0$ Gap% CPU #OP Gap% CPU #OP 15 3 45.82 0.03 18.63 0.02 4.75 0.43 20 3 36.02 1.84 1.76 1.59 2.04 15.71 20 4 51.80 2.53 1.09 2.37 1.84 1.68 30 4 26.39 6.85 16.39 5.89 4.73 184.93 40 4 38.74 6.52 2.54 7.41 5.91 347.55 30 5 33.90 12.48 11.64 11.91 2.43 194.69 40 5 46.29 11.53 10.83 10.53 3.74 842.68 50 5 34.62 14.83 18.46 13.96 6.47 1043.79 40 6 29.43 17.35 13.58 15.37 10.25 357.73 50 6 37.58 28.59 17.36 22.54 12.54 1074.51 Average 38.06 10.26 11.23 13.13 5.47 406.37
 LP of original formulation LP of strong formulation SLP of strong formulation $N_C$ $N_0$ Gap% CPU #OP Gap% CPU #OP 15 3 45.82 0.03 18.63 0.02 4.75 0.43 20 3 36.02 1.84 1.76 1.59 2.04 15.71 20 4 51.80 2.53 1.09 2.37 1.84 1.68 30 4 26.39 6.85 16.39 5.89 4.73 184.93 40 4 38.74 6.52 2.54 7.41 5.91 347.55 30 5 33.90 12.48 11.64 11.91 2.43 194.69 40 5 46.29 11.53 10.83 10.53 3.74 842.68 50 5 34.62 14.83 18.46 13.96 6.47 1043.79 40 6 29.43 17.35 13.58 15.37 10.25 357.73 50 6 37.58 28.59 17.36 22.54 12.54 1074.51 Average 38.06 10.26 11.23 13.13 5.47 406.37
Perform comparison of HFLRPSPDO on the effect of capacity violation
 $N_C$ $N_0$ Cost without violation Cost with violation Improvement on cost (%) CPU times (sec) Capacity violation (%) 15 3 29222 24731 15.37 86.29 6.82 20 3 32626 29742 8.84 1973.64 7.38 20 4 30325 27387 9.69 2391.82 9.65 30 4 51172 48373 5.47 3729.65 4.27 40 4 66101 61382 7.14 5183.59 6.49 30 5 51798 46183 10.84 4734.64 3.85 40 5 62976 53284 15.39 5374.68 6.58 50 5 64828 59337 8.47 6739.52 9.83 40 6 59014 55833 5.39 7200.00 5.48 50 6 64188 57821 9.92 7200.00 6.29 Average 51225 46407 9.65 6.66
 $N_C$ $N_0$ Cost without violation Cost with violation Improvement on cost (%) CPU times (sec) Capacity violation (%) 15 3 29222 24731 15.37 86.29 6.82 20 3 32626 29742 8.84 1973.64 7.38 20 4 30325 27387 9.69 2391.82 9.65 30 4 51172 48373 5.47 3729.65 4.27 40 4 66101 61382 7.14 5183.59 6.49 30 5 51798 46183 10.84 4734.64 3.85 40 5 62976 53284 15.39 5374.68 6.58 50 5 64828 59337 8.47 6739.52 9.83 40 6 59014 55833 5.39 7200.00 5.48 50 6 64188 57821 9.92 7200.00 6.29 Average 51225 46407 9.65 6.66
Computational results of the TS and SA on small-size problems
 TS-heuristics SA-heuristics $N_C$ $N_0$ Gap% CPU #OP Gap% CPU #OP 15 3 0.00 24731 38.13 0.00 24731 40.57 20 3 0.00 29742 63.59 0.00 29742 54.13 20 4 0.01 27387 73.82 0.02 27390 80.53 30 4 0.23 48373 102.43 0.28 48397 138.62 40 4 0.91 61382 162.57 0.73 61273 147.76 30 5 0.35 46183 90.37 0.34 46178 128.54 40 5 0.97 53284 194.63 1.14 53307 251.72 50 5 1.24 59337 288.79 1.45 59460 300.05 40 6 0.99 55833 239.40 1.06 55872 207.24 50 6 1.93 57821 247.56 1.82 57759 277.42 Average 0.66 46407 150.13 0.75 46411 162.66
 TS-heuristics SA-heuristics $N_C$ $N_0$ Gap% CPU #OP Gap% CPU #OP 15 3 0.00 24731 38.13 0.00 24731 40.57 20 3 0.00 29742 63.59 0.00 29742 54.13 20 4 0.01 27387 73.82 0.02 27390 80.53 30 4 0.23 48373 102.43 0.28 48397 138.62 40 4 0.91 61382 162.57 0.73 61273 147.76 30 5 0.35 46183 90.37 0.34 46178 128.54 40 5 0.97 53284 194.63 1.14 53307 251.72 50 5 1.24 59337 288.79 1.45 59460 300.05 40 6 0.99 55833 239.40 1.06 55872 207.24 50 6 1.93 57821 247.56 1.82 57759 277.42 Average 0.66 46407 150.13 0.75 46411 162.66
Computational results of the TS and SA on larger-size problems
 TS-heuristics SA-heuristics $N_C$ $N_0$ Gap% CPU #OP Gap% CPU #OP 50 8 2.41 54823 234.65 2.08 54711 208.40 80 8 1.73 113897 383.59 2.36 114602 361.47 100 8 0.96 137254 437.42 1.53 138029 472.43 80 9 1.24 100286 369.38 2.37 101405 390.22 100 9 0.72 130287 482.36 1.24 130960 538.52 120 9 0.92 157239 501.36 0.83 157099 409.25 150 9 1.09 186275 472.17 0.95 186117 463.47 80 10 3.73 983673 302.54 2.54 982388 378.49 100 10 2.27 125362 261.52 2.36 125472 330.52 120 10 1.34 139927 289.55 2.03 140080 375.38 150 10 1.46 173845 573.82 3.41 174186 593.54 200 10 2.03 237419 479.53 3.56 238279 636.39 Average 1.66 211690 398.99 2.11 211944 429.84
 TS-heuristics SA-heuristics $N_C$ $N_0$ Gap% CPU #OP Gap% CPU #OP 50 8 2.41 54823 234.65 2.08 54711 208.40 80 8 1.73 113897 383.59 2.36 114602 361.47 100 8 0.96 137254 437.42 1.53 138029 472.43 80 9 1.24 100286 369.38 2.37 101405 390.22 100 9 0.72 130287 482.36 1.24 130960 538.52 120 9 0.92 157239 501.36 0.83 157099 409.25 150 9 1.09 186275 472.17 0.95 186117 463.47 80 10 3.73 983673 302.54 2.54 982388 378.49 100 10 2.27 125362 261.52 2.36 125472 330.52 120 10 1.34 139927 289.55 2.03 140080 375.38 150 10 1.46 173845 573.82 3.41 174186 593.54 200 10 2.03 237419 479.53 3.56 238279 636.39 Average 1.66 211690 398.99 2.11 211944 429.84
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