Heterogeneous fixed fleet vehicle routing considering carbon emission

Transportation Research Part D 23 (2013) 81–89

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Transportation Research Part D journal homepage: www.elsevier.com/locate/trd

Heterogeneous fixed fleet vehicle routing considering carbon emission Yong-Ju Kwon a, Young-Jae Choi a, Dong-Ho Lee b,⇑ a b

Department of Industrial Engineering, Hanyang University, Seoul, Republic of Korea Department of Industrial Engineering, Graduate School of Technology and Innovation Management, Hanyang University, Seoul, Republic of Korea

a r t i c l e

i n f o

Keywords: Heterogeneous vehicle routing Carbon emission and trading Tabu search

a b s t r a c t The paper considers heterogeneous fixed fleet vehicle routing with carbon emission to minimizing the sum of variable operation costs. A cost-benefit assessment of the value of purchasing or selling of carbon emission rights, using a mixed integer-programming model to reflect heterogeneous vehicle routing, is incorporated. Essentially, the use of a carbon market as a means of introducing more flexibility into an environmentally constrained network is considered. Tabu search algorithms are used to obtain solutions within a reasonable amount of computation time. In particular, we show the possibility that the amount of carbon emission can be reduced significantly without sacrificing the cost due to the benefit obtained from carbon trading. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction This study considers carbon emission in logistics systems, focusing on the vehicle routing problem. Most previous studies on vehicle routing have focused on minimizing the distance traveled, the fleet size, and similar traditional parameters. Due to the environmental concerns that have emerged over the last decades, there has been an increasing interest in bring environmentally factors into the routing problem.1 This interest has more recently been compounded as new policies, such as carbon trading, have been introduced for making industry more aware of their environmental cost. In particular, measures such a carbon market can note and stimulate a more environmentally efficient use of transportation, but in some cases, because it allows holders of excessive carbon assets to sell their carbon emission rights, positively affect the financial performance of logistic service providers. The vehicle routing problems can be classified by vehicle types i.e., homogeneous and heterogeneous vehicle routing; we focus on the latter. Given the diversity of the vehicle fleet and variations in carbon emissions, we look specifically at the carbon emission based heterogeneous vehicle routing problem (C-HVRP),2 and further narrow it down to the case of a fixed fleet of vehicles and the problem of determining their optimal routing. To represent the problem mathematically, we adopt a mixed integer-programming model for objective of minimizing the sum of variable operation costs, including a cost-benefit assessment of acquiring carbon rights under a cap-and-trade regime.

⇑ Corresponding author. E-mail address: [email protected] (D.-H. Lee). Ericsson et al. (2006) identifies the impact of traffic disturbances on fuel consumption; Kara et al. (2007) defines an energy-minimizing vehicle routing problem that minimizes the weighted load instead of a distance based objective function; Tavares et al. (2008) look at the effects of road inclination and load; Bektasß and Laporte (2011) develop a model with an objective function that includes the travel distance as well as emission, operational and driver costs; and Xiao et al. (2012) extended the capacitated vehicle routing problem taking into account travel distance and load impacts on fuel costs. 2 Pessoa et al. (2009) and Brandão (2011) provide recent studies of the traditional HVRP. 1

1361-9209/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.trd.2013.04.001

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Fig. 1. Heterogeneous vehicle routes: example.

Table 1 Calculating the amount of carbon emission using the fuel based method. P CO2 Emissions = j(FueljEFj) Fuel consumption (Fuelj)

Represented by fuel sold (TJ)

Activity data

Calorific value Conversion factor

35.4 MJ/‘ 1 TJ = 106 MJ

Coefficient

Emission factor (EFj) Fuel type (j)

74,100 kg CO2/TJ Diesel (road transportation)

2. Problem description Let G = (N, A) be a logistics network, where N = {0, 1, 2, . . . , n} is the set of nodes and A = {(i, j): i, j 2 N, i – j} is the set of arcs. Node 0 denotes the depot, i.e., departure and arrival base of each vehicle, and the remaining nodes 1, 2, . . . , n represent customers. A fleet K of heterogeneous vehicles of varying capacities and carbon emission levels is available at the depot. It is assumed that the fleet size is fixed, and that each vehicle can travel at most one route. Each customer has a nonnegative demand that must be satisfied and a distance between customers i and j is associated with each arc (i, j) 2 A. These distances are symmetric and satisfy the triangle inequality. The problem is to determine vehicle routes, given their capacities and customer demands, that minimize the sum of variable operation costs and carbon emission trading net costs. Here consider a deterministic version of the problem; all data, such as customer demands, vehicle capacities, and carbon emission amounts, are given in advance, and variable operation cost is directly proportional to the distance traveled. The decision variable is the same as that of the ordinary HVRP. More specifically, a solution can be represented as a set of routes R1, R2, . . . , RV, where Rv = (0, iv1, iv2, . . . , 0) and V denotes the number of vehicles, with ivl being an index for the lth node on route v. Fig. 1 shows an example involving three heterogeneous vehicle routes and three truck types; 1-, 2-, and 5-ton vehicles. Here, the cost benefit calculation associated with carbon trading is calculated from the difference between the actual carbon use in the network and the upper limit for the amount of carbon emission. In other words, if carbon emissions are greater than the upper bound, then carbon emission ‘‘cost’’ are incurred because additional carbon allowances have to be purchased. The upper limit for carbon emissions is determined by a short-term cap-and-trade framework where the market has cleared and thus at the margin for any small player the effective carbon price is known in advance. We use the fuel-based method to determine this price in which carbon emissions from all sources of combustion are estimated on the basis of fuel burned and average emission factors. (Eggleston et al., 2006) (Table 1) The C-HVRP deployed involves two main constraints; customer demands must be satisfied and the demand over a vehicle’s route must not exceed its capacity. Other assumptions that remain same as for the ordinary HVRP, i.e., all vehicles start and end at the depot, a set of customers is visited in between, and each customer is visited once. To represent the problem, a mixed integer programming model is used following that of Yaman (2006). The following notations are used.

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Sets and indices i, j k

nodes (customers), i, j 2 N|{0} vehicle types, k = 1, 2, . . . , mk, where mk is the number of type k vehicles

Parameters qj dij Qk ek AE c

demand at node (customer) j distance of between nodes i and j (km) variable operation cost of vehicle k per unit distance ($/km) capacity of type of vehicle k (ton) amount of carbon emitted by vehicle k per unit distance (g/km) upper limit for the amount of carbon emissions (g) net benefit of unit carbon emission trading (¢/g)

Decision variables xijk fij PC

is one if a type k vehicle travels from node i to j, otherwise zero vehicle load from i to j difference between the carbon consumption and the upper limit for carbon emissions

vk

The mixed integer-programming model is given below. The set AK is defined as AK = {(i, j, k): (i, j) 2 A, k 2 Kij}, where Kij is the set of vehicles that can serve both customers i and j without violating capacity constraints

XX ½P Minimize v k  dij  xijk þ c  PC k2K ði;jÞ2A

subject to

XX xijk ¼ 1 for all i

ð1Þ

k2K i j2N

XX xijk ¼ 1 for all j

ð2Þ

k2K i i2N

X

x0jk 6 mk

for all k ¼ 1; 2; . . . ; mk

ð3Þ

j2Nnf0g

X X fji  fij ¼ qi j2N

X

for all i 2 Njf0g

ð4Þ

j2N

qj  xijk 6 fij 6

k2K ij

XX

X

ðQ k  qi Þ  xijk

for all ði; jÞ 2 A

ð5Þ

k2K ij

dij  ek  xijk  AE ¼ PC

ð6Þ

k2K i ði;jÞ2A

xijk 2 f0; 1g for all ði; j; kÞ 2 AK

ð7Þ

fij P 0 for all ði; jÞ 2 A

ð8Þ

The objective function denotes the sum of operation cost and carbon emission trading cost. Constraints (1) and (2) imply that each customer is visited once by a vehicle, and constraint (3) limits the number of vehicles of each type. Constraint (4) ensures that the demand of each customer be satisfied while removing sub-tours, and constraint (5) represents the vehicle capacities, and constraint (6) balances actual and permitted carbon emission. Finally, constraints (7) and (8) represent the conditions of decision variables.3 3. Tabu search algorithms Tabu search (TS) is a meta-heuristic. Starting from an initial solution, the TS algorithm generates a new solution S0 in the neighborhood of the original S, which is called a move. To escape from a local minimum in its search for the global minimum, 3 We see that the C-HVRP is NP-hard because its special case, the capacitated vehicle routing problem (CVRP) with homogeneous vehicles, is proved to be NPhard, and thus the tabu search algorithms offer reasonable solutions within a reasonable amount of computation time.

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Fig. 2. Neighborhood generation methods.

the move can be made to a neighbor solution even though it is worse than the current solution, and to avoid cycling on the search process, the TS defines a set of moves that are tabu (forbidden), and are stored in a set K (the tabu list). The moves in K define all tabu moves that cannot be applied to the current solution. The size of K is bounded by l; the tabu list size or tabu tenure. If |K| = l, before adding a move to K, one must remove a move, usually the oldest in it. A tabu move can be chosen if it creates a solution better than the current best solution: the aspiration criterion to revoke tabu status of some moves (Glover, 1989, 1990). An application of TS is characterized by; solution representation method, initial solution method, neighborhood generation methods, definition of tabu moves with the tabu list size, and termination condition. First, a solution is represented as a set of routes R1, R2, . . . , RV, where Rv = (0, iv1, iv2, . . . , 0). The initial solution is obtained as follows. First, from the set of unassigned nodes, the one with the greatest demand is selected with ties broken arbitrarily. Then, the selected node is added to the route end of the vehicle with the largest capacity. If this capacity is inadequate, it is assigned to the vehicle with the second largest capacity, and so on. Finally, after all nodes are assigned, the solution is improved using the 3-opt heuristic (Lin and Kernighan, 1973). The neighborhood generation structure determines the extent and the quality of the solution space explored. Here we consider three neighborhood generation methods; insertion, swap and hybrid. Unlike the traditional approach to the vehicle routing problem, these three methods evaluate the solutions include criterion involving carbon emissions.4 After the best neighborhood solution is obtained from a neighborhood generation method, it is improved further by applying the 3-opt heuristic to each of the changed routes. The move can be made to the improved neighbor solution, even though it may be worse than the current solution. The detailed descriptions of the insertion and the swap methods are given below, and Fig. 2 offers examples. In the figure, it can be seen that the moves are made to neighbor solutions involving smaller amounts of carbon emission (E) without regard to travel distance (D).  Insertion method: This generates the neighborhood solutions by selecting Rp and Rq and then inserting a node on route Rp to an arbitrary position on Rq.  Swap method: This generates the neighborhood solutions by selecting a node on each Rp and Rq, and interchanging them. 4

See Laporte et al. (2000) and Toth and Vigo (2002) for details of approaches to the classical vehicle routing problems.

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Y.-J. Kwon et al. / Transportation Research Part D 23 (2013) 81–89 Table 2 Modified benchmark test instances. Instances Number Upper Heterogeneous vehicle types of nodes limit (g) A B

T13 T14 T15 T16 T17 T18 T19 T20 a b c d

50 50 50 50 75 75 100 100

510755 398043 435360 453951 666573 653370 850839 874460

C

D

QAa

vAb

nAc eAd

QB

vB

nB e B

QC

vC

nC eC

2 12 5 4 5 2 10 6

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

4 4 4 2 4 4 4 6

3 16 10 8 12 5 20 14

1.1 1.1 1.6 1.6 1.2 1.3 1.4 1.7

2 2 3 4 4 4 3 4

4 30 16 14 20 10 30 20

1.2 1.4 2.0 2.1 1.5 1.9 1.7 2.0

4 1 2 3 2 2 3 3

4.13 25.14 10.44 8.34 10.44 4.13 20.94 12.54

6.23 33.55 20.94 16.74 25.14 10.44 41.95 29.35

QD

E

vD

nD eD

8.34 7 1.7 4 62.96 33.55 29.35 41.95 35 1.8 1 20.94 15 2.4 2 62.96 41.95

QE

F

vE

nE e E

QF

vF

nF e F

14.64 12 2.5 2

25.14 20 3.2 1

41.95

73.47 31.45 25 2.9 1

52.46 40 3.2 1

83.97

Vehicle capacity (unit: ton). Variable transportation cost per unit distance (unit: $/km). Number of available vehicles for each vehicle type. Amount of carbon emission per unit distance (unit: g/km).

 Hybrid method: This generates the neighborhood solutions by selecting the insertion and the swap methods randomly at an iteration and then taking the best as the move at the current iteration after evaluating all possible alternatives for insertion or swap. Regarding tabu moves in the TS algorithms, using the insertion method, if node i on route Rp is moved to route Rq at iteration t, the move of node i from Rp to Rq is declared tabu during t + l. Similarly, in the swap method, the moves between two routes, i.e., moves of node i (j) from Rp (Rq) to Rq (Rp), are declared tabu. The tabu list size l is determined using the simple dynamic tabu term rule of Glover and Laguna (1993); i.e. l is set to an integer uniformly distributed over an interval [lmin, lmax]. The oldest tabu move is removed before adding a new one if the tabu list is full. A tabu move can always be chosen if it creates a solution better than the incumbent solution, i.e., the best objective value obtained so far. Finally, the TS algorithms are terminated if there is no improvement for a certain number LTS after consecutive iterations. 4. Results To test the performance of TS algorithms, computational experiments are applied to modified benchmark instances. We compare TS-insertion, TS-swap and TS-hybrid algorithms, and report the costs, the distances traveled and the amounts of carbon emission for each benchmark instance. Computer central processing unit (CPU) seconds are reported to show their practical applicability.5 We performed sensitivity analyses on the effects the changes in the upper limit for the amount of carbon emitted and the net unit cost of traded carbon. The test uses eight modified benchmark instances with 50, 75 and 100 nodes (Taillard, 1999). The demand (qj) at each node is set to the original value of the benchmarks and the variable operation cost (vk) of each vehicle type was set to the fuel cost per unit distance (km). Because the benchmarks were originally designed for the HVRP, we generate the amount of carbon emission for each vehicle type and the net unit carbon cost (in all cases an actual benefit). Carbon emissions for each vehicle type (ek) are estimated based on Xiao et al. (2012) framework, i.e., Y = 0.0000793X  0.026, where Y and X are fuel the consumption rate (l/kg) and the weight of vehicle (kg). Here, X is replaced by vehicle capacity, Q. We assume that fuel consumption increases linearly with distance traveled. For each vehicle type, carbon emissions are calculated using the data in Table 1. The unit carbon emission cost is set at $25/ton. The upper limit (AE) for carbon emission for each instance is set to the amount obtained from the initial solutions because estimate the exact value cannot be estimated (Table 2). Preliminary calculations are conducted to refine appropriate parameters. First, lmin and lmax for the tabu list are set to 5 and 10, and LTS, the termination condition, to ten times the number of nodes. Finally, each test instance was solved ten times due to the randomness of the TS algorithms, i.e., selecting the neighborhood solutions. The results are summarized in Table 3 where it can be seen that the hybrid neighborhood generation method betters the others. Although this may be expected because the hybrid method includes the insertion and the swap methods, and hence searches a larger solution space, the improvement is significant in terms of both financial cost and the amount of carbon emission, together with the percentage reductions below the upper limit. Finally, although the TS-hybrid algorithm takes more computation time, it can give solutions for the largest situations (with 100 nodes) within a minute. Table 4 summarizes the results of the sensitivity analysis on changes in the net unit cost of carbon trading (which is generally a benefit, but because a cost when additional permits must be purchased), and the upper limit on the carbon emission using the TS-hybrid algorithm. We see that carbon trading increases and hence the cost decreases as unit cost benefits of trading increase. This indicates that the TS-hybrid algorithm can be used to maximize the benefit obtained from carbon 5

The TS algorithms were coded in C, and the test was done on a personal computer with an Intel core i7-2600 processor at 3.40 GHz clock speed.

86

Table 3 Results of the TS algorithms.

a b c d

Nodes

Carbon upper limit

TS-Insertion Cost

Distance

Emission

CPU seconds

Cost

TS-Swap Distance

Emission

CPU seconds

Cost

TS-hybrid Distance

Emission

CPU seconds

T13

50

510,755

2925.06a 2925.06b

1717.91 1717.91

483425 (5.4%)c 483425 (5.4%)d

0.1 0.1

2284.70 2253.97

1253.23 1214.94

368363 (27.9%) 361667 (29.2%)

0.3 0.4

2205.05 1994.76

1228.52 1108.27

357245 (30.1%) 323287 (36.7%)

0.3 0.5

T14

50

398,043

1069.45 994.56

964.31 889.98

379022 (4.8%) 352535 (11.4%)

0.2 0.3

1045.79 1035.38

945.55 936.23

369964 (7.1%) 366139 (8.0%)

0.3 0.3

1087.85 1007.63

982.55 902.91

385531 (3.1%) 357161 (10.3%)

0.2 0.3

T15

50

435,360

1737.26 1588.85

1149.65 1051.51

368619 (15.3%) 338006 (22.4%)

0.2 0.2

1569.60 1401.78

1067.90 952.61

336085 (22.8%) 300447 (31.%)

0.4 0.7

1315.78 1213.04

916.12 856.80

284122 (34.7%) 262869 (39.6%)

0.4 0.7

T16

50

453,951

1808.43 1542.53

1088.44 906.34

344218 (24.2%) 291000 (35.9%)

0.3 0.4

2006.98 1943.50

1220.59 1182.92

383503 (15.5%) 371481 (18.2%)

0.2 0.3

1502.94 1380.58

909.36 829.16

286630 (36.9%) 262577 (42.2%)

0.7 1.3

T17

75

666,573

1944.80 1850.76

1467.31 1391.50

548619 (17.7%) 521368 (21.8%)

0.6 0.6

1870.47 1764.49

1427.63 1344.00

526440 (21.0%) 496826 (25.5%)

1.9 2.5

1735.85 1596.50

1310.29 1210.82

488563 (26.7%) 446790 (33.0%)

1.6 2.6

T18

75

653,370

3668.25 3609.65

1788.03 1742.95

623168 (4.6%) 611991 (6.3%)

0.2 0.2

3484.57 3419.40

1672.01 1628.81

590418 (9.6%) 578619 (11.4%)

0.3 0.4

3268.97 3065.19

1526.99 1434.05

551060 (15.7%) 517619 (20.8%)

0.4 0.9

T19

100

850,839

2221.39 2017.57

1439.63 1304.58

660353 (22.4%) 599019 (29.6%)

4.8 5.4

2202.39 2040.78

1437.51 1344.21

657260 (22.8%) 612086 (28.1%)

13.4 19.7

1825.63 1647.76

1189.22 1070.75

544228 (36.0%) 490621 (42.3%)

12.4 29.2

T20

100

874,460

2945.23 2592.43

1917.89 1674.16

679713 (22.3%) 598976 (31.5%)

1.6 2.2

3300.83 3300.83

2166.38 2166.38

764390 (12.6%) 764390 (12.6%)

0.7 0.9

2304.74 2091.12

1529.19 1405.50

535076 (38.8%) 487362 (44.3%)

3.5 7.2

Average cost. best cost (among 10 repetitions). Average percentage of carbon emission reduction for the upper limit. Best percentage of reduced carbon emission over the CO2 upper limit.

Y.-J. Kwon et al. / Transportation Research Part D 23 (2013) 81–89

Problem ID

Table 4 Results of the sensitivity analysis. Problem ID

c

Carbon upper limit

c = 0.001 (10 $/ton)

c = 0.0025 (25 $/ton)

Cost without trading

Emission

Trading benefit

Total cost

Cost without trading

Emission

c = 0.004 (40 $/ton) Trading benefit

Total cost

Cost without trading

Emission

Trading benefit

Total cost

T13

50

510,755a 459,680b 357,529c

2010.28 2021.52 2018.69

324,193 324,979 327,119

186.56 134.70 30.41

1823.72 1886.82 1988.28

1994.76 2073.72 1994.89

323,287 335,334 324,751

468.67 310.87 81.95

1526.09 1762.86 1912.95

2028.07 2093.73 2125.69

328,848 336,829 341,650

727.63 491.40 63.52

1300.44 1602.33 2062.17

T14

50

398,043 358,239 278,631

1000.87 1033.47 954.46

354,478 366,200 337,532

43.57 7.96 58.90

957.31 1041.43 1013.36

1007.63 1007.63 1083.81

357,161 357,161 383,985

102.21 2.70 263.39

905.43 1004.94 1347.20

911.18 1066.29 1083.81

322,514 377,597 383,985

302.12 77.43 421.42

609.06 1143.72 1505.23

T15

50

435,360 391,824 304,752

1206.61 1282.52 1234.02

258,877 277,421 267,678

176.48 114.40 37.07

1030.13 1170.07 1196.95

1213.04 1246.68 1170.18

262,869 268,416 253,346

431.23 308.52 128.52

781.81 938.16 1041.67

1197.14 1233.55 1206.58

257,960 266,208 262,267

709.60 502.46 169.94

487.54 731.09 1036.64

T16

50

453,951 408,556 317,766

1361.78 1418.19 1397.73

258,657 267,937 266,585

195.29 140.62 51.18

1166.49 1277.57 1347.38

1380.58 1413.56 1343.41

262,577 270,672 256,155

478.44 344.71 154.03

902.15 1068.85 1189.38

1355.96 1438.34 1437.04

255,184 270,978 273,747

795.07 550.31 176.08

560.89 888.03 1260.96

T17

75

666,573 599,916 466,602

1575.41 1600.62 1574.73

442,142 449,494 444,562

224.43 150.42 22.04

1350.98 1450.20 1552.69

1596.50 1627.57 1600.57

446,790 458,030 449,173

549.46 354.72 43.57

1047.04 1272.86 1557.00

1587.35 1624.68 1660.77

445,298 455,790 467,196

885.10 576.50 2.38

702.25 1048.46 1663.15

T18

75

653,370 588,033 457,359

3058.46 3177.43 3007.89

512,982 534,223 507,195

140.39 53.81 49.84

2918.07 3123.62 3057.73

3065.19 3142.13 3047.27

517,619 535,082 515,347

339.38 132.38 144.97

2725.81 3009.75 3192.24

3191.18 3170.47 3171.40

536,965 537,808 533,858

465.62 200.90 306.00

2725.56 2969.57 3477.40

T19

100

850,839 765,756 595,588

1694.48 1621.20 1663.55

505,603 481,557 496,810

345.24 284.20 98.78

1349.24 1337.00 1564.77

1647.76 1704.45 1703.91

490,621 509,788 509,103

900.55 639.92 216.21

747.75 1064.53 1487.70

1689.01 1690.89 1601.59

506,724 502,977 477,479

1376.46 1051.12 472.44

312.55 639.77 1129.15

T20

100

874,460 787,014 612,122

2184.02 2033.66 2090.61

508,537 476,885 486,430

365.92 310.13 125.69

1818.10 1723.53 1964.92

2091.12 1981.51 2073.32

487,362 460,972 482,026

967.75 815.11 325.24

1123.38 1166.41 1748.08

2056.02 2030.69 2006.63

478,622 470,809 466,472

1583.35 1264.82 582.60

472.67 765.87 1424.03

Y.-J. Kwon et al. / Transportation Research Part D 23 (2013) 81–89

a b

Nodes

0% Reduction in the initial upper limit for the amount of carbon emission. 10% Reduction in the initial upper limit for the amount of carbon emission. 30% Reduction in the initial upper limit for the amount of carbon emission.

87

88

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800 700 600 500 400 300 200

c = 0.004

100

c = 0.0025

0

c = 0.001

(0%) (10%) (30%)

(a) T13 (n = 50)

500 400 300 200

c = 0.004

100

c = 0.0025

0 -100 -200

(0%)

c = 0.001 (10%)

(30%)

(b) T18 (n = 75) Fig. 3. Trading benefit or cost for instances T13 and T18.

trading. For example, Fig. 3 shows the trading costs and benefits for T13 and T18. Costs increase as the upper limit on the carbon emission decreases because of the need to purchase the carbon permits at the new lower upper limit for emissions. 5. Concluding remarks This study considered the heterogeneous vehicle routing problem that determines a set of vehicle routes that satisfies customer demands and vehicle capacities. We considered the problem with carbon emission for the objective of minimizing the sum of variable operation costs. To represent the problem mathematically, an integer-programming model was used. Then, due to the problem complexity, tabu search algorithms were deployed together with three neighborhood generation methods. Computational experiments were done on modified benchmark instances, and the test results show that the tabu search algorithm with hybrid neighborhood generation method performs better than the others. In particular, it is shown from an additional test that the amount of carbon emission can be reduced significantly without sacrificing the cost due to the benefit obtained from carbon trading. Acknowledgements The authors would like to appreciate for the valuable comments of the two anonymous referees. References Bektasß, T., Laporte, G., 2011. The pollution-routing problem. Transportation Research Part B: Methodological 45, 1232–1250. Brandão, J., 2011. A tabu search algorithm for the heterogeneous fixed fleet vehicle routing problem. Computers and Operations Research 38, 140–151. Eggleston, H.S., Buendia, L., Miwa, K., Ngara, T., Tanabe, K., 2006. IPCC Guidelines for National Greenhouse Gas Inventories. The National Greenhouse Gas Inventories Programme, 2, Geneva. Ericsson, E., Larsson, H., Brundell-Freij, K., 2006. Optimizing route choice for lowest fuel consumption: potential effects of a new driver support tool. Transportation Research Part C: Emerging Technologies 14, 369–383.

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