# American Institute of Mathematical Sciences

July  2021, 17(4): 2073-2095. doi: 10.3934/jimo.2020059

## Multiobjective mathematical models and solution approaches for heterogeneous fixed fleet vehicle routing problems

 Department of Industrial Engineering, Eskisehir Technical University, Eskisehir, Turkey

* Corresponding author: Refail Kasimbeyli

Received  January 2019 Revised  December 2019 Published  July 2021 Early access  March 2020

In this paper, we study three types of heterogeneous fixed fleet vehicle routing problems, which are capacitated vehicle routing problem, open vehicle routing problem and split delivery vehicle routing problem. We propose new multiobjective linear binary and mixed integer programming models for these problems, where the first objective is the minimization of a total routing and usage costs for vehicles, and the second one is the vehicle type minimization, respectively. The proposed mathematical models are all illustrated on test problems, which are investigated in two groups: small-sized problems and the large-sized ones. The small-sized test problems are first scalarized by using the weighted sum scalarization method, and then GAMS software is used to compute efficient solutions. The large-sized test problems are solved by utilizing the tabu search algorithm.

Citation: Melis Alpaslan Takan, Refail Kasimbeyli. Multiobjective mathematical models and solution approaches for heterogeneous fixed fleet vehicle routing problems. Journal of Industrial and Management Optimization, 2021, 17 (4) : 2073-2095. doi: 10.3934/jimo.2020059
##### References:
 [1] C. Archetti, M. G. Speranza and A. Hertz, Tabu search algorithm for the split delivery vehicle routing problem, Transportation Science, 40 (2006), 64-73.  doi: 10.1287/trsc.1040.0103. [2] M. C. Bolduc, G. Laporte, J. Renaud and F. F. Boctor, A tabu search heuristic for the split delivery vehicle routing problem with production and demand calendars, European Journal of Operational Research, 202 (2010), 122-130.  doi: 10.1016/j.ejor.2009.05.008. [3] J. Brandão, A tabu search algorithm for the open vehicle routing problem, European Journal of Operational Research, 157 (2004), 552-564.  doi: 10.1016/S0377-2217(03)00238-8. [4] P. Belfiore and H. T. Y. Yoshizaki, Scatter search for a real-life heterogeneous fleet vehicle routing problem with time windows and split deliveries in Brazil, European Journal of Operational Research, 199 (2009), 750-758.  doi: 10.1016/j.ejor.2008.08.003. [5] P. Chen, B. Golden, X. Wang and E. Wasil, A novel approach to solve the split delivery vehicle routing problem, International Transactions in Operational Research, 24 (2017), 27-41.  doi: 10.1111/itor.12250. [6] G. B. Dantzig and J. H. Ramser, The truck dispatching problem, Management Science, 6 (1959/60), 80-91.  doi: 10.1287/mnsc.6.1.80. [7] M. Dror and P. Trudeau, Savings by split delivery routing, Transportation Science, 23 (1989), 141-145.  doi: 10.1287/trsc.23.2.141. [8] M. Dror, G. Laporte and P. Trudeau, Vehicle routing with split deliveries, Discrete Applied Mathematics, 50 (1994), 239-254.  doi: 10.1016/0166-218X(92)00172-I. [9] J. Euchi and H. Chabchoub, A hybrid tabu search to solve the heterogeneous fixed fleet vehicle routing problem, Logistics Research, 2 (2010), 3-11.  doi: 10.1007/s12159-010-0028-3. [10] M. L. Fisher and R. Jaikumar, A Decomposition Algorithm for Large-Scale Vehicle Routing, Wharton School, University of Pennsylvania Department of Decision Sciences, Philadelphia, PA, 1978. [11] K. Fleszar, I. H. Osman and K. S. Hindi, A variable neighbourhood search algorithm for the open vehicle routing problem, European Journal of Operational Research, 195 (2009), 803-809.  doi: 10.1016/j.ejor.2007.06.064. [12] R. N. Gasimov, A. Sipahioglu and T. Saraç, A multi-objective programming approach to 1.5-dimensional assortment problem, European Journal of Operational Research, 179 (2007), 64-79.  doi: 10.1016/j.ejor.2006.03.016. [13] M. Gendreau, G. Laporte, C. Musaraganyi and E. D. Taillard, A tabu search heuristic for the heterogeneous fleet vehicle routing problem, Computers and Operations Research, 26 (1999), 1153-1173.  doi: 10.1016/S0305-0548(98)00100-2. [14] F. Glover, Future paths for integer programming and links to artificial intelligence, Computers and Operations Research, 13 (1986), 533-549.  doi: 10.1016/0305-0548(86)90048-1. [15] B. Golden, A. Assad, L. Levy and F. Gheysens, The fleet size and mix vehicle routing problem, Computers and Operations Research, 11 (1984), 49-66.  doi: 10.1016/0305-0548(84)90007-8. [16] S. C. Ho and D. Haugland, A tabu search heuristic for the vehicle routing problem with time windows and split deliveries, Computers and Operations Research, 31 (2004), 1947-1964.  doi: 10.1016/S0305-0548(03)00155-2. [17] N. A. Ismayilova, M. Saǧir and R. N. Gasimov, A multiobjective faculty-course-time slot assignment problem with preferences, Mathematical and Computer Modelling, 46 (2007), 1017-1029.  doi: 10.1016/j.mcm.2007.03.012. [18] M. Jin, K. Liu and B. Eksioglu, A column generation approach for the split delivery vehicle routing problem, Operations Research Letters, 36 (2008), 265-270.  doi: 10.1016/j.orl.2007.05.012. [19] N. Kasimbeyli, T. Sarac and R. Kasimbeyli, A two-objective mathematical model without cutting patterns for one-dimensional assortment problems, Journal of Computational and Applied Mathematics, 235 (2011), 4663-4674.  doi: 10.1016/j.cam.2010.07.019. [20] R. Kasimbeyli, A nonlinear cone separation theorem and scalarization in nonconvex vector optimization, SIAM Journal on Optimization, 20 (2009), 1591-1619.  doi: 10.1137/070694089. [21] R. Kasimbeyli, A conic scalarization method in multi-objective optimization, Journal of Global Optimization, 56 (2013), 279-297.  doi: 10.1007/s10898-011-9789-8. [22] R. Kasimbeyli, Z. K. Ozturk, N. Kasimbeyli, G. D. Yalcin and B. I. Erdem, Comparison of some scalarization methods in multiobjective optimization: Comparison of scalarization methods, Bulletin of the Malaysian Mathematical Sciences Society, 42 (2019), 1875-1905.  doi: 10.1007/s40840-017-0579-4. [23] F. Li, B. Golden and E. Wasil, A record-to-record travel algorithm for solving the heterogeneous fleet vehicle routing problem, Computers and Operations Research, 34 (2007), 2734-2742.  doi: 10.1016/j.cor.2005.10.015. [24] X. Li, S. C. H. Leung and P. Tian, A multi start adaptive memory-based tabu search algorithm for the heterogeneous fixed fleet open vehicle routing problem, Expert Systems with Applications, 39 (2012), 365-374.  doi: 10.1016/j.eswa.2011.07.025. [25] S. Liu, A hybrid population heuristic for the heterogeneous vehicle routing problems, Transportation Research Part E: Logistics and Transportation Review, 54 (2013), 67-78.  doi: 10.1016/j.tre.2013.03.010. [26] A. J. Pedraza-Martinez and L. N. Van Wassenhove, Transportation and vehicle fleet management in humanitarian logistics: Challenges for future research, EURO Journal on Transportation and Logistics, 1 (2012), 185-196.  doi: 10.1007/s13676-012-0001-1. [27] C. E. Miller, A. W. Tucker and R. A. Zemlin, Integer programming formulations and traveling salesman problems, Journal of the Association for Computing Machinery, 7 (1960), 326-329.  doi: 10.1145/321043.321046. [28] K. Nesbitt and D. Sperling, Fleet purchase behavior: Decision processes and implications for new vehicle technologies and fuels, Transportation Research Part C: Emerging Technologies, 9 (2001), 297-318.  doi: 10.1016/S0968-090X(00)00035-8. [29] J. Renaud and F. F. Boctor, A sweep-based algorithm for the fleet size and mix vehicle routing problem, European Journal of Operations Research, 140 (2002), 618-628.  doi: 10.1016/S0377-2217(01)00237-5. [30] L. Schrage, Formulation and structure of more complex/realistic routing and scheduling problems, Networks, 11 (1981), 229-232.  doi: 10.1002/net.3230110212. [31] E. Taillard, A heuristic column generation method for the heterogeneous fleet VRP, RAIRO - Operations Research, 33 (1999), 1-14.  doi: 10.1051/ro:1999101. [32] C. D. Tarantilis, C. T. Kiranoudis and V. S. Vassiliadis, A threshold accepting metaheuristic for the heterogeneous fixed fleet vehicle routing problem, European Journal of Operational Research, 152 (2004), 148-158.  doi: 10.1016/S0377-2217(02)00669-0. [33] O. Ustun and R. Kasimbeyli, Combined forecasts in portfolio optimization: A generalized approach, Computers & Operations Research, 39 (2012), 805–819. doi: 10.1016/j.cor.2010.09.008. [34] M. Yousefikhoshbakht, F. Didehvar and F. Rahmati, Solving the heterogeneous fixed fleet open vehicle routing problem by a combined metaheuristic algorithm, International Journal of Production Research, 52 (2014), 2565-2575.  doi: 10.1080/00207543.2013.855337. [35] S. Yu, C. Ding and K. Zhu, A hybrid GA-TS algorithm for open vehicle routing optimization of coal mines material, Expert Systems with Applications, 38 (2011), 10568-10573.  doi: 10.1016/j.eswa.2011.02.108.

show all references

##### References:
 [1] C. Archetti, M. G. Speranza and A. Hertz, Tabu search algorithm for the split delivery vehicle routing problem, Transportation Science, 40 (2006), 64-73.  doi: 10.1287/trsc.1040.0103. [2] M. C. Bolduc, G. Laporte, J. Renaud and F. F. Boctor, A tabu search heuristic for the split delivery vehicle routing problem with production and demand calendars, European Journal of Operational Research, 202 (2010), 122-130.  doi: 10.1016/j.ejor.2009.05.008. [3] J. Brandão, A tabu search algorithm for the open vehicle routing problem, European Journal of Operational Research, 157 (2004), 552-564.  doi: 10.1016/S0377-2217(03)00238-8. [4] P. Belfiore and H. T. Y. Yoshizaki, Scatter search for a real-life heterogeneous fleet vehicle routing problem with time windows and split deliveries in Brazil, European Journal of Operational Research, 199 (2009), 750-758.  doi: 10.1016/j.ejor.2008.08.003. [5] P. Chen, B. Golden, X. Wang and E. Wasil, A novel approach to solve the split delivery vehicle routing problem, International Transactions in Operational Research, 24 (2017), 27-41.  doi: 10.1111/itor.12250. [6] G. B. Dantzig and J. H. Ramser, The truck dispatching problem, Management Science, 6 (1959/60), 80-91.  doi: 10.1287/mnsc.6.1.80. [7] M. Dror and P. Trudeau, Savings by split delivery routing, Transportation Science, 23 (1989), 141-145.  doi: 10.1287/trsc.23.2.141. [8] M. Dror, G. Laporte and P. Trudeau, Vehicle routing with split deliveries, Discrete Applied Mathematics, 50 (1994), 239-254.  doi: 10.1016/0166-218X(92)00172-I. [9] J. Euchi and H. Chabchoub, A hybrid tabu search to solve the heterogeneous fixed fleet vehicle routing problem, Logistics Research, 2 (2010), 3-11.  doi: 10.1007/s12159-010-0028-3. [10] M. L. Fisher and R. Jaikumar, A Decomposition Algorithm for Large-Scale Vehicle Routing, Wharton School, University of Pennsylvania Department of Decision Sciences, Philadelphia, PA, 1978. [11] K. Fleszar, I. H. Osman and K. S. Hindi, A variable neighbourhood search algorithm for the open vehicle routing problem, European Journal of Operational Research, 195 (2009), 803-809.  doi: 10.1016/j.ejor.2007.06.064. [12] R. N. Gasimov, A. Sipahioglu and T. Saraç, A multi-objective programming approach to 1.5-dimensional assortment problem, European Journal of Operational Research, 179 (2007), 64-79.  doi: 10.1016/j.ejor.2006.03.016. [13] M. Gendreau, G. Laporte, C. Musaraganyi and E. D. Taillard, A tabu search heuristic for the heterogeneous fleet vehicle routing problem, Computers and Operations Research, 26 (1999), 1153-1173.  doi: 10.1016/S0305-0548(98)00100-2. [14] F. Glover, Future paths for integer programming and links to artificial intelligence, Computers and Operations Research, 13 (1986), 533-549.  doi: 10.1016/0305-0548(86)90048-1. [15] B. Golden, A. Assad, L. Levy and F. Gheysens, The fleet size and mix vehicle routing problem, Computers and Operations Research, 11 (1984), 49-66.  doi: 10.1016/0305-0548(84)90007-8. [16] S. C. Ho and D. Haugland, A tabu search heuristic for the vehicle routing problem with time windows and split deliveries, Computers and Operations Research, 31 (2004), 1947-1964.  doi: 10.1016/S0305-0548(03)00155-2. [17] N. A. Ismayilova, M. Saǧir and R. N. Gasimov, A multiobjective faculty-course-time slot assignment problem with preferences, Mathematical and Computer Modelling, 46 (2007), 1017-1029.  doi: 10.1016/j.mcm.2007.03.012. [18] M. Jin, K. Liu and B. Eksioglu, A column generation approach for the split delivery vehicle routing problem, Operations Research Letters, 36 (2008), 265-270.  doi: 10.1016/j.orl.2007.05.012. [19] N. Kasimbeyli, T. Sarac and R. Kasimbeyli, A two-objective mathematical model without cutting patterns for one-dimensional assortment problems, Journal of Computational and Applied Mathematics, 235 (2011), 4663-4674.  doi: 10.1016/j.cam.2010.07.019. [20] R. Kasimbeyli, A nonlinear cone separation theorem and scalarization in nonconvex vector optimization, SIAM Journal on Optimization, 20 (2009), 1591-1619.  doi: 10.1137/070694089. [21] R. Kasimbeyli, A conic scalarization method in multi-objective optimization, Journal of Global Optimization, 56 (2013), 279-297.  doi: 10.1007/s10898-011-9789-8. [22] R. Kasimbeyli, Z. K. Ozturk, N. Kasimbeyli, G. D. Yalcin and B. I. Erdem, Comparison of some scalarization methods in multiobjective optimization: Comparison of scalarization methods, Bulletin of the Malaysian Mathematical Sciences Society, 42 (2019), 1875-1905.  doi: 10.1007/s40840-017-0579-4. [23] F. Li, B. Golden and E. Wasil, A record-to-record travel algorithm for solving the heterogeneous fleet vehicle routing problem, Computers and Operations Research, 34 (2007), 2734-2742.  doi: 10.1016/j.cor.2005.10.015. [24] X. Li, S. C. H. Leung and P. Tian, A multi start adaptive memory-based tabu search algorithm for the heterogeneous fixed fleet open vehicle routing problem, Expert Systems with Applications, 39 (2012), 365-374.  doi: 10.1016/j.eswa.2011.07.025. [25] S. Liu, A hybrid population heuristic for the heterogeneous vehicle routing problems, Transportation Research Part E: Logistics and Transportation Review, 54 (2013), 67-78.  doi: 10.1016/j.tre.2013.03.010. [26] A. J. Pedraza-Martinez and L. N. Van Wassenhove, Transportation and vehicle fleet management in humanitarian logistics: Challenges for future research, EURO Journal on Transportation and Logistics, 1 (2012), 185-196.  doi: 10.1007/s13676-012-0001-1. [27] C. E. Miller, A. W. Tucker and R. A. Zemlin, Integer programming formulations and traveling salesman problems, Journal of the Association for Computing Machinery, 7 (1960), 326-329.  doi: 10.1145/321043.321046. [28] K. Nesbitt and D. Sperling, Fleet purchase behavior: Decision processes and implications for new vehicle technologies and fuels, Transportation Research Part C: Emerging Technologies, 9 (2001), 297-318.  doi: 10.1016/S0968-090X(00)00035-8. [29] J. Renaud and F. F. Boctor, A sweep-based algorithm for the fleet size and mix vehicle routing problem, European Journal of Operations Research, 140 (2002), 618-628.  doi: 10.1016/S0377-2217(01)00237-5. [30] L. Schrage, Formulation and structure of more complex/realistic routing and scheduling problems, Networks, 11 (1981), 229-232.  doi: 10.1002/net.3230110212. [31] E. Taillard, A heuristic column generation method for the heterogeneous fleet VRP, RAIRO - Operations Research, 33 (1999), 1-14.  doi: 10.1051/ro:1999101. [32] C. D. Tarantilis, C. T. Kiranoudis and V. S. Vassiliadis, A threshold accepting metaheuristic for the heterogeneous fixed fleet vehicle routing problem, European Journal of Operational Research, 152 (2004), 148-158.  doi: 10.1016/S0377-2217(02)00669-0. [33] O. Ustun and R. Kasimbeyli, Combined forecasts in portfolio optimization: A generalized approach, Computers & Operations Research, 39 (2012), 805–819. doi: 10.1016/j.cor.2010.09.008. [34] M. Yousefikhoshbakht, F. Didehvar and F. Rahmati, Solving the heterogeneous fixed fleet open vehicle routing problem by a combined metaheuristic algorithm, International Journal of Production Research, 52 (2014), 2565-2575.  doi: 10.1080/00207543.2013.855337. [35] S. Yu, C. Ding and K. Zhu, A hybrid GA-TS algorithm for open vehicle routing optimization of coal mines material, Expert Systems with Applications, 38 (2011), 10568-10573.  doi: 10.1016/j.eswa.2011.02.108.
Vehicle type data for tabu search algorithm
 Problem no vehicle type Taillard's original vehicle number data TK data vehicle type cost (penalty cost) 13 1 4 10 10 2 2 10 15 3 4 10 20 4 4 10 35 5 2 10 60 6 1 10 100 14 1 4 10 100 2 2 10 150 3 1 10 290 15 1 4 10 10 2 3 10 25 3 2 10 45 16 1 2 10 10 2 4 10 20 3 3 10 40 17 1 4 10 10 2 4 10 22 3 2 10 40 4 1 10 70 18 1 4 10 10 2 4 10 25 3 2 10 50 4 2 10 75 5 1 10 125 6 1 10 200 19 1 4 10 10 2 3 10 20 3 3 10 30 20 1 6 10 10 2 4 10 23 3 3 10 35
 Problem no vehicle type Taillard's original vehicle number data TK data vehicle type cost (penalty cost) 13 1 4 10 10 2 2 10 15 3 4 10 20 4 4 10 35 5 2 10 60 6 1 10 100 14 1 4 10 100 2 2 10 150 3 1 10 290 15 1 4 10 10 2 3 10 25 3 2 10 45 16 1 2 10 10 2 4 10 20 3 3 10 40 17 1 4 10 10 2 4 10 22 3 2 10 40 4 1 10 70 18 1 4 10 10 2 4 10 25 3 2 10 50 4 2 10 75 5 1 10 125 6 1 10 200 19 1 4 10 10 2 3 10 20 3 3 10 30 20 1 6 10 10 2 4 10 23 3 3 10 35
GAMS results for the multiobjective heterogeneous fixed fleet capacitated vehicle routing problem
 $w_1$ $w_2$ $z_1$ $z_2$ Routes Vehicles 1 9 523.81 1 1-2-10-6-1 vehicle 1 of type 3 1-4-9-1 vehicle 2 of type 3 1-5-3-1 vehicle 3 of type 3 1-7-8-1 vehicle 4 of type 3 2 8 523.81 1 1-3-5-1 vehicle 1 of type 3 1-6-10-2-1 vehicle 2 of type 3 1-8-7-1 vehicle 3 of type 3 1-9-4-1 vehicle 4 of type 3 3 7 523.81 1 1-3-5-1 vehicle 1 of type 3 1-6-10-2-1 vehicle 2 of type 3 1-8-7-1 vehicle 3 of type 3 1-4-9-1 vehicle 4 of type 3 4 6 468.91 2 1-4-3-6-1 vehicle 2 of type 4 1-5-10-1 vehicle 1 of type 1 1-7-8-9-2-1 vehicle 1 of type 4 5 5 468.91 2 1-5-10-1 vehicle 1 of type 1 1-6-3-4-1 vehicle 1 of type 4 1-7-8-9-2-1 vehicle 2 of type 4 6 4 468.91 2 1-2-9-8-7-1 vehicle 1 of type 4 1-5-10-1 vehicle 1 of type 1 1-6-3-4-1 vehicle 2 of type 4 7 3 468.91 2 1-2-9-8-7-1 vehicle 1 of type 4 1-6-3-4-1 vehicle 2 of type 4 1-10-5-1 vehicle 1 of type 1 8 2 468.91 2 1-5-10-1 vehicle 1 of type 1 1-6-3-4-1 vehicle 1 of type 4 1-7-8-9-2-1 vehicle 2 of type 4 9 1 468.91 2 1-2-9-8-7-1 vehicle 1 of type 4 1-4-3-6-1 vehicle 2 of type 4 1-5-10-1 vehicle 1 of type 1
 $w_1$ $w_2$ $z_1$ $z_2$ Routes Vehicles 1 9 523.81 1 1-2-10-6-1 vehicle 1 of type 3 1-4-9-1 vehicle 2 of type 3 1-5-3-1 vehicle 3 of type 3 1-7-8-1 vehicle 4 of type 3 2 8 523.81 1 1-3-5-1 vehicle 1 of type 3 1-6-10-2-1 vehicle 2 of type 3 1-8-7-1 vehicle 3 of type 3 1-9-4-1 vehicle 4 of type 3 3 7 523.81 1 1-3-5-1 vehicle 1 of type 3 1-6-10-2-1 vehicle 2 of type 3 1-8-7-1 vehicle 3 of type 3 1-4-9-1 vehicle 4 of type 3 4 6 468.91 2 1-4-3-6-1 vehicle 2 of type 4 1-5-10-1 vehicle 1 of type 1 1-7-8-9-2-1 vehicle 1 of type 4 5 5 468.91 2 1-5-10-1 vehicle 1 of type 1 1-6-3-4-1 vehicle 1 of type 4 1-7-8-9-2-1 vehicle 2 of type 4 6 4 468.91 2 1-2-9-8-7-1 vehicle 1 of type 4 1-5-10-1 vehicle 1 of type 1 1-6-3-4-1 vehicle 2 of type 4 7 3 468.91 2 1-2-9-8-7-1 vehicle 1 of type 4 1-6-3-4-1 vehicle 2 of type 4 1-10-5-1 vehicle 1 of type 1 8 2 468.91 2 1-5-10-1 vehicle 1 of type 1 1-6-3-4-1 vehicle 1 of type 4 1-7-8-9-2-1 vehicle 2 of type 4 9 1 468.91 2 1-2-9-8-7-1 vehicle 1 of type 4 1-4-3-6-1 vehicle 2 of type 4 1-5-10-1 vehicle 1 of type 1
GAMS results for the single-objective heterogeneous fixed fleet capacitated vehicle routing problem
 $z_2$ $z_1$ Routes Vehicles $z_2\leq 1$ 523.81 1-2-10-6-1 vehicle 1 of type 3 1-4-9-1 vehicle 2 of type 3 1-5-3-1 vehicle 3 of type 3 1-7-8-1 vehicle 4 of type 3 $z_2\leq 2$ 505.78 1-7-9-4-3-10-6-1 vehicle 1 of type 5 1-2-8-5-1 vehicle 1 of type 3 $z_2\leq 3$ 494.54 1-6-5-1 vehicle 1 of type 2 1-7-8-9-4-3-2-1 vehicle 1 of type 5 1-10-1 vehicle 1 of type 1 $z_2\leq 4$ 480.63 1-2-3-10-6-1 vehicle 1 of type 4 1-4-9-1 vehicle 1 of type 3 1-7-5-1 vehicle 1 of type 2 1-8-1 vehicle 1 of type 1 $z_2\leq 5$ 480.63 1-2-3-10-6-1 vehicle 1 of type 4 1-4-9-1 vehicle 1 of type 3 1-7-5-1 vehicle 1 of type 2 1-8-1 vehicle 1 of type 1
 $z_2$ $z_1$ Routes Vehicles $z_2\leq 1$ 523.81 1-2-10-6-1 vehicle 1 of type 3 1-4-9-1 vehicle 2 of type 3 1-5-3-1 vehicle 3 of type 3 1-7-8-1 vehicle 4 of type 3 $z_2\leq 2$ 505.78 1-7-9-4-3-10-6-1 vehicle 1 of type 5 1-2-8-5-1 vehicle 1 of type 3 $z_2\leq 3$ 494.54 1-6-5-1 vehicle 1 of type 2 1-7-8-9-4-3-2-1 vehicle 1 of type 5 1-10-1 vehicle 1 of type 1 $z_2\leq 4$ 480.63 1-2-3-10-6-1 vehicle 1 of type 4 1-4-9-1 vehicle 1 of type 3 1-7-5-1 vehicle 1 of type 2 1-8-1 vehicle 1 of type 1 $z_2\leq 5$ 480.63 1-2-3-10-6-1 vehicle 1 of type 4 1-4-9-1 vehicle 1 of type 3 1-7-5-1 vehicle 1 of type 2 1-8-1 vehicle 1 of type 1
GAMS results for the multiobjective heterogeneous fixed fleet open vehicle routing problem
 $w_1$ $w_2$ $z_1$ $z_2$ Routes Vehicles 1 9 408.43 1 1-2-6-10 vehicle 1 of type 3 1-5-3 vehicle 2 of type 3 1-7-8 vehicle 3 of type 3 1-9-4 vehicle 4 of type 3 2 8 408.43 1 1-7-8 vehicle 1 of type 3 1-2-6-10 vehicle 2 of type 3 1-5-3 vehicle 3 of type 3 1-9-4 vehicle 4 of type 3 3 7 408.43 1 1-2-6-10 vehicle 1 of type 3 1-5-3 vehicle 2 of type 3 1-7-8 vehicle 3 of type 3 1-9-4 vehicle 4 of type 3 4 6 352.76 2 1-2-9-8 vehicle 1 of type 4 1-3-4 vehicle 2 of type 4 1-5 vehicle 1 of type 1 1-6 vehicle 3 of type 4 1-7 vehicle 4 of type 4 1-10 vehicle 2 of type 1 5 5 352.76 2 1-2-9-8 vehicle 1 of type 4 1-5 vehicle 1 of type 1 1-6-3-4 vehicle 2 of type 4 1-7 vehicle 3 of type 4 1-10 vehicle 2 of type 1 6 4 352.76 2 1-2-9-8 vehicle 1 of type 4 1-5 vehicle 1 of type 1 1-6-3-4 vehicle 2 of type 4 1-7 vehicle 3 of type 4 1-10 vehicle 2 of type 1 7 3 352.76 2 1-2-9-8 vehicle 1 of type 4 1-5 vehicle 1 of type 1 1-6-3-4 vehicle 2 of type 4 1-7 vehicle 3 of type 4 1-10 vehicle 2 of type 1 8 2 352.76 2 1-2-9-8 vehicle 1 of type 4 1-5 vehicle 1 of type 1 1-6-3-4 vehicle 2 of type 4 1-7 vehicle 3 of type 4 1-10 vehicle 2 of type 1 9 1 347.01 3 1-2-9 vehicle 1 of type 2 1-5 vehicle 1 of type 1 1-6-3-4 vehicle 1 of type 4 1-7 vehicle 2 of type 1 1-8 vehicle 3 of type 1 1-10 vehicle 4 of type 1
 $w_1$ $w_2$ $z_1$ $z_2$ Routes Vehicles 1 9 408.43 1 1-2-6-10 vehicle 1 of type 3 1-5-3 vehicle 2 of type 3 1-7-8 vehicle 3 of type 3 1-9-4 vehicle 4 of type 3 2 8 408.43 1 1-7-8 vehicle 1 of type 3 1-2-6-10 vehicle 2 of type 3 1-5-3 vehicle 3 of type 3 1-9-4 vehicle 4 of type 3 3 7 408.43 1 1-2-6-10 vehicle 1 of type 3 1-5-3 vehicle 2 of type 3 1-7-8 vehicle 3 of type 3 1-9-4 vehicle 4 of type 3 4 6 352.76 2 1-2-9-8 vehicle 1 of type 4 1-3-4 vehicle 2 of type 4 1-5 vehicle 1 of type 1 1-6 vehicle 3 of type 4 1-7 vehicle 4 of type 4 1-10 vehicle 2 of type 1 5 5 352.76 2 1-2-9-8 vehicle 1 of type 4 1-5 vehicle 1 of type 1 1-6-3-4 vehicle 2 of type 4 1-7 vehicle 3 of type 4 1-10 vehicle 2 of type 1 6 4 352.76 2 1-2-9-8 vehicle 1 of type 4 1-5 vehicle 1 of type 1 1-6-3-4 vehicle 2 of type 4 1-7 vehicle 3 of type 4 1-10 vehicle 2 of type 1 7 3 352.76 2 1-2-9-8 vehicle 1 of type 4 1-5 vehicle 1 of type 1 1-6-3-4 vehicle 2 of type 4 1-7 vehicle 3 of type 4 1-10 vehicle 2 of type 1 8 2 352.76 2 1-2-9-8 vehicle 1 of type 4 1-5 vehicle 1 of type 1 1-6-3-4 vehicle 2 of type 4 1-7 vehicle 3 of type 4 1-10 vehicle 2 of type 1 9 1 347.01 3 1-2-9 vehicle 1 of type 2 1-5 vehicle 1 of type 1 1-6-3-4 vehicle 1 of type 4 1-7 vehicle 2 of type 1 1-8 vehicle 3 of type 1 1-10 vehicle 4 of type 1
Data related to customers for the multiobjective heterogeneous fixed fleet split delivery vehicle routing problem
 customer x coordinate y coordinate demand 1 145 215 0 2 151 264 10 3 159 261 8 4 130 254 12 5 128 252 14
 customer x coordinate y coordinate demand 1 145 215 0 2 151 264 10 3 159 261 8 4 130 254 12 5 128 252 14
Data related to vehicles for the multiobjective heterogeneous fixed fleet split delivery vehicle routing problem
 Vehicle type Vehicle capacity Number of vehicles Routing Cost Usage Cost Type cost 1 10 1 1 20 100 2 15 2 1.1 35 150 3 20 3 1.2 50 200
 Vehicle type Vehicle capacity Number of vehicles Routing Cost Usage Cost Type cost 1 10 1 1 20 100 2 15 2 1.1 35 150 3 20 3 1.2 50 200
GAMS results for the multiobjective heterogeneous fixed fleet split delivery vehicle routing problem
 $w_1$ $w_2$ $z_1$ $z_2$ Routes Vehicles 1 9 926.40 1 1-4-1 vehicle 1 of type 3 1-2-3-1 vehicle 2 of type 3 1-5-1 vehicle 3 of type 3 2 8 926.40 1 1-2-3-1 vehicle 1 of type 3 1-4-1 vehicle 2 of type 3 1-5-1 vehicle 3 of type 3 3 7 926.40 1 1-4-1 vehicle 1 of type 3 1-5-1 vehicle 2 of type 3 1-2-3-1 vehicle 3 of type 3 4 6 926.40 1 1-4-1 vehicle 1 of type 3 1-2-3-1 vehicle 2 of type 3 1-5-1 vehicle 3 of type 3 5 5 812.40 2 1-5(10)-1 vehicle 1 of type 1 1-2-3-1 vehicle 1 of type 3 1-5(4)-4-1 vehicle 2 of type 3 6 4 812.40 2 1-5(10)-1 vehicle 1 of type 1 1-2-3-1 vehicle 1 of type 3 1-5(4)-4-1 vehicle 2 of type 3 7 3 812.40 2 1-5(10)-1 vehicle 1 of type 1 1-5(4)-4-1 vehicle 1 of type 3 1-3-2-1 vehicle 1 of type 3 8 2 770.60 3 1-4(2)-1 vehicle 1 of type 1 1-5-1 vehicle 1 of type 2 1-3-2-4(10)-1 vehicle 1 of type 3 9 1 770.60 3 1-4(10)-1 vehicle 1 of type 1 1-5-1 vehicle 1 of type 2 1-4(2)-2-3-1 vehicle 1 of type 3
 $w_1$ $w_2$ $z_1$ $z_2$ Routes Vehicles 1 9 926.40 1 1-4-1 vehicle 1 of type 3 1-2-3-1 vehicle 2 of type 3 1-5-1 vehicle 3 of type 3 2 8 926.40 1 1-2-3-1 vehicle 1 of type 3 1-4-1 vehicle 2 of type 3 1-5-1 vehicle 3 of type 3 3 7 926.40 1 1-4-1 vehicle 1 of type 3 1-5-1 vehicle 2 of type 3 1-2-3-1 vehicle 3 of type 3 4 6 926.40 1 1-4-1 vehicle 1 of type 3 1-2-3-1 vehicle 2 of type 3 1-5-1 vehicle 3 of type 3 5 5 812.40 2 1-5(10)-1 vehicle 1 of type 1 1-2-3-1 vehicle 1 of type 3 1-5(4)-4-1 vehicle 2 of type 3 6 4 812.40 2 1-5(10)-1 vehicle 1 of type 1 1-2-3-1 vehicle 1 of type 3 1-5(4)-4-1 vehicle 2 of type 3 7 3 812.40 2 1-5(10)-1 vehicle 1 of type 1 1-5(4)-4-1 vehicle 1 of type 3 1-3-2-1 vehicle 1 of type 3 8 2 770.60 3 1-4(2)-1 vehicle 1 of type 1 1-5-1 vehicle 1 of type 2 1-3-2-4(10)-1 vehicle 1 of type 3 9 1 770.60 3 1-4(10)-1 vehicle 1 of type 1 1-5-1 vehicle 1 of type 2 1-4(2)-2-3-1 vehicle 1 of type 3
Computational results for heterogeneous fixed fleet capacitated VRP, with initial solutions obtained using the NNH algorithm
 Pr. no n tabu size tabu tenure Taillard's data TK data obj. value number of vehicles obj. value n. of vehicles vehicle types used 13 50 7 5 4962.84 17 5596.35 6 type 6 13 50 7 10 4962.84 17 5596.35 6 type 6 13 50 10 5 4962.84 17 5596.35 6 type 6 13 50 10 10 4962.84 17 5596.35 6 type 6 14 50 7 5 11717.23 7 15277.12 4 type 3 14 50 7 10 11717.23 7 15277.12 4 type 3 14 50 10 5 11717.23 7 15277.12 4 type 3 14 50 10 10 11717.23 7 15277.12 4 type 3 15 50 7 5 4109.82 9 4411.77 9 type 2 15 50 7 10 4109.82 9 4411.77 9 type 2 15 50 10 5 4109.82 9 4411.77 9 type 2 15 50 10 10 4109.82 9 4411.77 9 type 2 16 50 7 5 4967.63 9 5014.15 6 type 3 16 50 7 10 4967.63 9 5014.15 6 type 3 16 50 10 5 4967.63 9 5014.15 6 type 3 16 50 10 10 4967.63 9 5014.15 6 type 3 17 75 7 5 3581.81 11 3819.1 8 type 3 17 75 7 10 3581.81 11 3819.1 8 type 3 17 75 10 5 3581.81 11 3819.1 8 type 3 17 75 10 10 3581.81 11 3819.1 8 type 3 18 75 7 5 8251.13 14 7535.78 10 type 4 18 75 7 10 8251.13 14 7535.78 10 type 4 18 75 10 5 8251.13 14 7535.78 10 type 4 18 75 10 10 8251.13 14 7535.78 10 type 4 19 100 7 5 12635.68 9 12712.45 8 type 2 19 100 7 10 12635.68 9 12712.45 8 type 2 19 100 10 5 12635.68 9 12712.45 8 type 2 19 100 10 10 12635.68 9 12712.45 8 type 2 20 100 7 5 6982.42 13 8497.79 8 type 3 20 100 7 10 6982.42 13 8497.79 8 type 3 20 100 10 5 6982.42 13 8497.79 8 type 3 20 100 10 10 6982.42 13 8497.79 8 type 3
 Pr. no n tabu size tabu tenure Taillard's data TK data obj. value number of vehicles obj. value n. of vehicles vehicle types used 13 50 7 5 4962.84 17 5596.35 6 type 6 13 50 7 10 4962.84 17 5596.35 6 type 6 13 50 10 5 4962.84 17 5596.35 6 type 6 13 50 10 10 4962.84 17 5596.35 6 type 6 14 50 7 5 11717.23 7 15277.12 4 type 3 14 50 7 10 11717.23 7 15277.12 4 type 3 14 50 10 5 11717.23 7 15277.12 4 type 3 14 50 10 10 11717.23 7 15277.12 4 type 3 15 50 7 5 4109.82 9 4411.77 9 type 2 15 50 7 10 4109.82 9 4411.77 9 type 2 15 50 10 5 4109.82 9 4411.77 9 type 2 15 50 10 10 4109.82 9 4411.77 9 type 2 16 50 7 5 4967.63 9 5014.15 6 type 3 16 50 7 10 4967.63 9 5014.15 6 type 3 16 50 10 5 4967.63 9 5014.15 6 type 3 16 50 10 10 4967.63 9 5014.15 6 type 3 17 75 7 5 3581.81 11 3819.1 8 type 3 17 75 7 10 3581.81 11 3819.1 8 type 3 17 75 10 5 3581.81 11 3819.1 8 type 3 17 75 10 10 3581.81 11 3819.1 8 type 3 18 75 7 5 8251.13 14 7535.78 10 type 4 18 75 7 10 8251.13 14 7535.78 10 type 4 18 75 10 5 8251.13 14 7535.78 10 type 4 18 75 10 10 8251.13 14 7535.78 10 type 4 19 100 7 5 12635.68 9 12712.45 8 type 2 19 100 7 10 12635.68 9 12712.45 8 type 2 19 100 10 5 12635.68 9 12712.45 8 type 2 19 100 10 10 12635.68 9 12712.45 8 type 2 20 100 7 5 6982.42 13 8497.79 8 type 3 20 100 7 10 6982.42 13 8497.79 8 type 3 20 100 10 5 6982.42 13 8497.79 8 type 3 20 100 10 10 6982.42 13 8497.79 8 type 3
Computational results for heterogeneous fixed fleet capacitated VRP, with initial solutions obtained using the RNH algorithm
 Pr. no n tabu size tabu tenure Taillard's data TK data obj. value n. of vehicles obj. value n. of vehicles vehicle types used 13 50 7 5 5226.86 17 5390.14 6 type 6 13 50 7 10 5438.5 17 6127.04 6 type 6 13 50 10 5 5322.39 17 6085.09 6 type 6 13 50 10 10 4979.07 17 5859.03 6 type 6 14 50 7 5 11655.46 7 15133.31 4 type 3 14 50 7 10 11747.49 7 15168.62 4 type 3 14 50 10 5 11780.2 7 15392.41 4 type 3 14 50 10 10 11755.49 7 15125.57 4 type 3 15 50 7 5 4108.74 9 4141.99 9 type 2 15 50 7 10 4374.32 9 4149.24 9 type 2 15 50 10 5 3945.91 9 4242.02 9 type 2 15 50 10 10 4175.87 9 4224.99 9 type 2 16 50 7 5 5042.91 9 5146.22 6 type 3 16 50 7 10 4651.12 9 5028.29 6 type 3 16 50 10 5 4471.94 9 5521.87 6 type 3 16 50 10 10 4887.57 9 4845.65 6 type 3 17 75 7 5 3668.52 11 4018.59 8 type 3 17 75 7 10 3170.15 11 3566.41 8 type 3 17 75 10 5 3722.46 11 3694.85 8 type 3 17 75 10 10 3749.49 11 3772.36 8 type 3 18 75 7 5 7012.94 14 6661.09 10 type 4 18 75 7 10 8142.29 14 6977.01 10 type 4 18 75 10 5 7304.11 14 6080.45 10 type 4 18 75 10 10 7370.3 14 6405.93 10 type 4 19 100 7 5 14452.39 9 13337.45 8 type 2 19 100 7 10 13523.82 9 12986.54 8 type 2 19 100 10 5 14516.23 9 15678.2 8 type 2 19 100 10 10 15692.56 9 16994.23 8 type 2 20 100 7 5 7140.26 13 8527.07 8 type 3 20 100 7 10 7267.18 13 8396.23 8 type 3 20 100 10 5 7816.72 13 7918.24 8 type 3 20 100 10 10 7329.74 13 8531.93 8 type 3
 Pr. no n tabu size tabu tenure Taillard's data TK data obj. value n. of vehicles obj. value n. of vehicles vehicle types used 13 50 7 5 5226.86 17 5390.14 6 type 6 13 50 7 10 5438.5 17 6127.04 6 type 6 13 50 10 5 5322.39 17 6085.09 6 type 6 13 50 10 10 4979.07 17 5859.03 6 type 6 14 50 7 5 11655.46 7 15133.31 4 type 3 14 50 7 10 11747.49 7 15168.62 4 type 3 14 50 10 5 11780.2 7 15392.41 4 type 3 14 50 10 10 11755.49 7 15125.57 4 type 3 15 50 7 5 4108.74 9 4141.99 9 type 2 15 50 7 10 4374.32 9 4149.24 9 type 2 15 50 10 5 3945.91 9 4242.02 9 type 2 15 50 10 10 4175.87 9 4224.99 9 type 2 16 50 7 5 5042.91 9 5146.22 6 type 3 16 50 7 10 4651.12 9 5028.29 6 type 3 16 50 10 5 4471.94 9 5521.87 6 type 3 16 50 10 10 4887.57 9 4845.65 6 type 3 17 75 7 5 3668.52 11 4018.59 8 type 3 17 75 7 10 3170.15 11 3566.41 8 type 3 17 75 10 5 3722.46 11 3694.85 8 type 3 17 75 10 10 3749.49 11 3772.36 8 type 3 18 75 7 5 7012.94 14 6661.09 10 type 4 18 75 7 10 8142.29 14 6977.01 10 type 4 18 75 10 5 7304.11 14 6080.45 10 type 4 18 75 10 10 7370.3 14 6405.93 10 type 4 19 100 7 5 14452.39 9 13337.45 8 type 2 19 100 7 10 13523.82 9 12986.54 8 type 2 19 100 10 5 14516.23 9 15678.2 8 type 2 19 100 10 10 15692.56 9 16994.23 8 type 2 20 100 7 5 7140.26 13 8527.07 8 type 3 20 100 7 10 7267.18 13 8396.23 8 type 3 20 100 10 5 7816.72 13 7918.24 8 type 3 20 100 10 10 7329.74 13 8531.93 8 type 3
Computational results for heterogeneous fixed fleet open VRP, with initial solutions obtained using the NNH algorithm
 Pr. no n tabu size tabu tenure Taillard's data TK data obj. value n. of vehicles obj. value n. of vehicles vehicle types used 13 50 7 5 4381.93 14 6376.82 6 type 6 13 50 7 10 4386.98 14 6376.82 6 type 6 13 50 10 5 4381.93 14 6376.82 6 type 6 13 50 10 10 4386.98 14 6376.82 6 type 6 14 50 7 5 12609.85 7 10864.35 9 type 1 14 50 7 10 12609.85 7 10864.35 9 type 1 14 50 10 5 12609.85 7 10864.35 9 type 1 14 50 10 10 12609.85 7 10864.35 9 type 1 15 50 7 5 3878.93 9 10054.3 9 type 1 15 50 7 10 3878.93 9 10054.3 9 type1 15 50 10 5 3878.93 9 10054.3 9 type1 15 50 10 10 3878.93 9 10054.3 9 type 1 16 50 7 5 4473.47 9 4845.49 6 type 3 16 50 7 10 4473.47 9 4845.49 6 type 3 16 50 10 5 4473.47 9 4845.49 6 type 3 16 50 10 10 4473.47 9 4845.49 6 type 3 17 75 7 5 3125.69 11 3652.77 8 type 3 17 75 7 10 3125.69 11 3652.77 8 type 3 17 75 10 5 3125.69 11 3652.77 8 type 3 17 75 10 10 3125.69 11 3652.77 8 type 3 18 75 7 5 7803.44 14 6062.63 10 type 4 18 75 7 10 7803.44 14 6062.63 10 type4 18 75 10 5 7803.44 14 6062.63 10 type 4 18 75 10 10 7803.44 14 6062.63 10 type 4 19 100 7 5 12274.38 10 12405.72 8 type 2 19 100 7 10 12274.38 10 12405.72 8 type 2 19 100 10 5 12274.38 10 12405.72 8 type 2 19 100 10 10 12274.38 10 12405.72 8 type 2 20 100 7 5 6782.57 13 8497.79 8 type 3 20 100 7 10 6782.57 13 8497.79 8 type 3 20 100 10 5 6782.57 13 8497.79 8 type 3 20 100 10 10 6782.57 13 8497.79 8 type 3
 Pr. no n tabu size tabu tenure Taillard's data TK data obj. value n. of vehicles obj. value n. of vehicles vehicle types used 13 50 7 5 4381.93 14 6376.82 6 type 6 13 50 7 10 4386.98 14 6376.82 6 type 6 13 50 10 5 4381.93 14 6376.82 6 type 6 13 50 10 10 4386.98 14 6376.82 6 type 6 14 50 7 5 12609.85 7 10864.35 9 type 1 14 50 7 10 12609.85 7 10864.35 9 type 1 14 50 10 5 12609.85 7 10864.35 9 type 1 14 50 10 10 12609.85 7 10864.35 9 type 1 15 50 7 5 3878.93 9 10054.3 9 type 1 15 50 7 10 3878.93 9 10054.3 9 type1 15 50 10 5 3878.93 9 10054.3 9 type1 15 50 10 10 3878.93 9 10054.3 9 type 1 16 50 7 5 4473.47 9 4845.49 6 type 3 16 50 7 10 4473.47 9 4845.49 6 type 3 16 50 10 5 4473.47 9 4845.49 6 type 3 16 50 10 10 4473.47 9 4845.49 6 type 3 17 75 7 5 3125.69 11 3652.77 8 type 3 17 75 7 10 3125.69 11 3652.77 8 type 3 17 75 10 5 3125.69 11 3652.77 8 type 3 17 75 10 10 3125.69 11 3652.77 8 type 3 18 75 7 5 7803.44 14 6062.63 10 type 4 18 75 7 10 7803.44 14 6062.63 10 type4 18 75 10 5 7803.44 14 6062.63 10 type 4 18 75 10 10 7803.44 14 6062.63 10 type 4 19 100 7 5 12274.38 10 12405.72 8 type 2 19 100 7 10 12274.38 10 12405.72 8 type 2 19 100 10 5 12274.38 10 12405.72 8 type 2 19 100 10 10 12274.38 10 12405.72 8 type 2 20 100 7 5 6782.57 13 8497.79 8 type 3 20 100 7 10 6782.57 13 8497.79 8 type 3 20 100 10 5 6782.57 13 8497.79 8 type 3 20 100 10 10 6782.57 13 8497.79 8 type 3
Computational results for heterogeneous fixed fleet open VRP, with initial solutions obtained using the RNH algorithm
 Pr. no n tabu size tabu tenure Taillard's data TK data obj. value n. of vehicles obj. value n. of vehicles vehicle types used 13 50 7 5 4100.25 14 6211.45 6 type 6 13 50 7 10 4215.89 14 6544.47 6 type 6 13 50 10 5 4390.47 14 6632.15 6 type 6 13 50 10 10 4412.69 14 6542.12 6 type 6 14 50 7 5 15463.88 7 16250.75 4 type 3 14 50 7 10 16213.24 7 16350.2 4 type 3 14 50 10 5 15478.23 7 16272.54 4 type 3 14 50 10 10 16952.3 7 16897.56 4 type 3 15 50 7 5 3988.45 9 9956.47 9 type 1 15 50 7 10 3995.64 9 10056.23 9 type 1 15 50 10 5 3654.21 9 9854.12 9 type 1 15 50 10 10 3875.46 9 9932.41 9 type 1 16 50 7 5 4852.77 9 4912.55 6 type 3 16 50 7 10 4744.46 9 4753.21 6 type 3 16 50 10 5 4715.23 9 4655.12 6 type 3 16 50 10 10 4879.56 9 4899.52 6 type 3 17 75 7 5 3478.56 11 3678.99 8 type 3 17 75 7 10 3654.61 11 3245.61 8 type 3 17 75 10 5 3541.72 11 3755.17 8 type 3 17 75 10 10 3655.77 11 3547.89 8 type 3 18 75 7 5 7653.45 14 7456.33 10 type 4 18 75 7 10 7664.52 14 7895.41 10 type 4 18 75 10 5 7569.44 14 7754.13 10 type 4 18 75 10 10 8004.56 14 7965.52 10 type 4 19 100 7 5 16542.33 10 17841.22 8 type 2 19 100 7 10 15478.99 10 16984.53 8 type 2 19 100 10 5 14563.01 10 16547.99 8 type 2 19 100 10 10 15879.22 10 15642.33 8 type 2 20 100 7 5 6961.08 13 7918.54 8 type 3 20 100 7 10 6742.13 13 7654.13 8 type 3 20 100 10 5 6830.15 13 7326.58 8 type 3 20 100 10 10 6955.41 13 7456.23 8 type 3
 Pr. no n tabu size tabu tenure Taillard's data TK data obj. value n. of vehicles obj. value n. of vehicles vehicle types used 13 50 7 5 4100.25 14 6211.45 6 type 6 13 50 7 10 4215.89 14 6544.47 6 type 6 13 50 10 5 4390.47 14 6632.15 6 type 6 13 50 10 10 4412.69 14 6542.12 6 type 6 14 50 7 5 15463.88 7 16250.75 4 type 3 14 50 7 10 16213.24 7 16350.2 4 type 3 14 50 10 5 15478.23 7 16272.54 4 type 3 14 50 10 10 16952.3 7 16897.56 4 type 3 15 50 7 5 3988.45 9 9956.47 9 type 1 15 50 7 10 3995.64 9 10056.23 9 type 1 15 50 10 5 3654.21 9 9854.12 9 type 1 15 50 10 10 3875.46 9 9932.41 9 type 1 16 50 7 5 4852.77 9 4912.55 6 type 3 16 50 7 10 4744.46 9 4753.21 6 type 3 16 50 10 5 4715.23 9 4655.12 6 type 3 16 50 10 10 4879.56 9 4899.52 6 type 3 17 75 7 5 3478.56 11 3678.99 8 type 3 17 75 7 10 3654.61 11 3245.61 8 type 3 17 75 10 5 3541.72 11 3755.17 8 type 3 17 75 10 10 3655.77 11 3547.89 8 type 3 18 75 7 5 7653.45 14 7456.33 10 type 4 18 75 7 10 7664.52 14 7895.41 10 type 4 18 75 10 5 7569.44 14 7754.13 10 type 4 18 75 10 10 8004.56 14 7965.52 10 type 4 19 100 7 5 16542.33 10 17841.22 8 type 2 19 100 7 10 15478.99 10 16984.53 8 type 2 19 100 10 5 14563.01 10 16547.99 8 type 2 19 100 10 10 15879.22 10 15642.33 8 type 2 20 100 7 5 6961.08 13 7918.54 8 type 3 20 100 7 10 6742.13 13 7654.13 8 type 3 20 100 10 5 6830.15 13 7326.58 8 type 3 20 100 10 10 6955.41 13 7456.23 8 type 3
Computational results for heterogeneous fixed fleet split delivery VRP, with initial solutions obtained using the NNH algorithm
 Pr. no n tabu size tabu tenure Taillard's data TK data obj. value n. of vehicles obj. value n. of vehicles vehicle types used 13 50 7 5 4248.33 17 4539.33 9 type 5 13 50 7 10 4248.33 17 4539.33 9 type 5 13 50 10 5 4236.18 17 4514.64 9 type 5 13 50 10 10 4236.18 17 4514.64 9 type 5 14 50 7 5 12264.91 7 9879.73 9 type 1 14 50 7 10 12264.91 7 9879.73 9 type 1 14 50 10 5 12255.03 7 9869.85 9 type 1 14 50 10 10 12255.03 7 9869.85 9 type 1 15 50 7 5 3750.95 9 3536.62 8 type 2 15 50 7 10 3750.95 9 3536.62 8 type 2 15 50 10 5 3750.95 9 3538.62 8 type 2 15 50 10 10 3750.95 9 3538.62 8 type 2 16 50 7 5 4047.47 9 4317.46 5 type 3 16 50 7 10 4047.47 9 4317.46 5 type 3 16 50 10 5 3988.54 9 4305.2 5 type 3 16 50 10 10 3988.54 9 4305.2 5 type 3 17 75 7 5 2850.35 11 2635.93 7 type 3 17 75 7 10 2850.35 11 2635.93 7 type 3 17 75 10 5 2839.2 11 2612.02 7 type 3 17 75 10 10 2839.2 11 2612.02 7 type 3 18 75 7 5 5121.54 14 5053.58 10 type 4 18 75 7 10 5121.54 14 5053.58 10 type 4 18 75 10 5 5094.1 14 4986.7 10 type 4 18 75 10 10 5094.1 14 4986.7 10 type 4 19 100 7 5 11492 10 11405.7 8 type 2 19 100 7 10 11492 10 11405.7 8 type 2 19 100 10 5 11484.76 10 11411 8 type 2 19 100 10 10 11484.76 10 11411 8 type 2 20 100 7 5 5606.26 13 6630.8 8 type 3 20 100 7 10 5606.26 13 6630.8 8 type 3 20 100 10 5 5541.51 13 6652.7 8 type 3 20 100 10 10 5541.51 13 6652.7 8 type 3
 Pr. no n tabu size tabu tenure Taillard's data TK data obj. value n. of vehicles obj. value n. of vehicles vehicle types used 13 50 7 5 4248.33 17 4539.33 9 type 5 13 50 7 10 4248.33 17 4539.33 9 type 5 13 50 10 5 4236.18 17 4514.64 9 type 5 13 50 10 10 4236.18 17 4514.64 9 type 5 14 50 7 5 12264.91 7 9879.73 9 type 1 14 50 7 10 12264.91 7 9879.73 9 type 1 14 50 10 5 12255.03 7 9869.85 9 type 1 14 50 10 10 12255.03 7 9869.85 9 type 1 15 50 7 5 3750.95 9 3536.62 8 type 2 15 50 7 10 3750.95 9 3536.62 8 type 2 15 50 10 5 3750.95 9 3538.62 8 type 2 15 50 10 10 3750.95 9 3538.62 8 type 2 16 50 7 5 4047.47 9 4317.46 5 type 3 16 50 7 10 4047.47 9 4317.46 5 type 3 16 50 10 5 3988.54 9 4305.2 5 type 3 16 50 10 10 3988.54 9 4305.2 5 type 3 17 75 7 5 2850.35 11 2635.93 7 type 3 17 75 7 10 2850.35 11 2635.93 7 type 3 17 75 10 5 2839.2 11 2612.02 7 type 3 17 75 10 10 2839.2 11 2612.02 7 type 3 18 75 7 5 5121.54 14 5053.58 10 type 4 18 75 7 10 5121.54 14 5053.58 10 type 4 18 75 10 5 5094.1 14 4986.7 10 type 4 18 75 10 10 5094.1 14 4986.7 10 type 4 19 100 7 5 11492 10 11405.7 8 type 2 19 100 7 10 11492 10 11405.7 8 type 2 19 100 10 5 11484.76 10 11411 8 type 2 19 100 10 10 11484.76 10 11411 8 type 2 20 100 7 5 5606.26 13 6630.8 8 type 3 20 100 7 10 5606.26 13 6630.8 8 type 3 20 100 10 5 5541.51 13 6652.7 8 type 3 20 100 10 10 5541.51 13 6652.7 8 type 3
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