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

Melis Alpaslan Takan; Refail Kasimbeyli

doi:10.3934/jimo.2020059

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

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

Melis Alpaslan Takan and Refail Kasimbeyli ^,

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

^* Corresponding author: Refail Kasimbeyli

Received January 2019 Revised December 2019 Published 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.

Keywords: Vehicle routing problem, multiobjective optimization, tabu search algorithm.

Mathematics Subject Classification: Primary: 90C08, 90C11; Secondary: 90C29.

Citation: Melis Alpaslan Takan, Refail Kasimbeyli. Multiobjective mathematical models and solution approaches for heterogeneous fixed fleet vehicle routing problems. Journal of Industrial & 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[7]	M. Dror and P. Trudeau, Savings by split delivery routing, Transportation Science, 23 (1989), 141-145. doi: 10.1287/trsc.23.2.141. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[30]	L. Schrage, Formulation and structure of more complex/realistic routing and scheduling problems, Networks, 11 (1981), 229-232. doi: 10.1002/net.3230110212. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[7]	M. Dror and P. Trudeau, Savings by split delivery routing, Transportation Science, 23 (1989), 141-145. doi: 10.1287/trsc.23.2.141. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[30]	L. Schrage, Formulation and structure of more complex/realistic routing and scheduling problems, Networks, 11 (1981), 229-232. doi: 10.1002/net.3230110212. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

Table 7. 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

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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

Table 1. 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

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$ 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

Table 2. 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

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$ 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

Table 3. 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

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$ 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

Table 4. 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

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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

Table 5. 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

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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

Table 6. 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

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$ 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

Table 8. 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
Pr. no	n	tabu size	tabu tenure	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

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Pr. no	n	tabu size	tabu tenure	Taillard's data		TK data
Pr. no	n	tabu size	tabu tenure	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

Table 9. 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
Pr. no	n	tabu size	tabu tenure	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

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Pr. no	n	tabu size	tabu tenure	Taillard's data		TK data
Pr. no	n	tabu size	tabu tenure	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

Table 10. 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
Pr. no	n	tabu size	tabu tenure	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

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Pr. no	n	tabu size	tabu tenure	Taillard's data		TK data
Pr. no	n	tabu size	tabu tenure	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

Table 11. 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
Pr. no	n	tabu size	tabu tenure	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

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Pr. no	n	tabu size	tabu tenure	Taillard's data		TK data
Pr. no	n	tabu size	tabu tenure	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

Table 12. 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
Pr. no	n	tabu size	tabu tenure	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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Download as excel

Pr. no	n	tabu size	tabu tenure	Taillard's data		TK data
Pr. no	n	tabu size	tabu tenure	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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2019 Impact Factor: 1.366

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2019 Impact Factor: 1.366

American Institute of Mathematical Sciences

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

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

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