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An adaptive large neighborhood search algorithm for Vehicle Routing Problem with Multiple Time Windows constraints

• * Corresponding author: Lixin Wei
• The Vehicle Routing Problem with Multiple Time Windows (VRPMTW) is a generalization of problems in real life logistics distribution, which has a wide range of applications and research values. Several neighborhood search based methods have been used to solve this kind of problem, but it still has drawbacks of generating numbers of infeasible solutions and falling into local optimum easily. In order to solve the problem of arbitrary selection for neighborhoods, a series of neighborhoods are designed and an adaptive strategy is used to select the neighborhood, which constitute the Adaptive Large Neighborhood Search(ALNS) algorithm framework. For escaping from the local optimum effectively in the search process, a local search based on destroy and repair operators is applied to shake the solution by adjusting the number of customers. The proposed method allows infeasible solutions to participate in the iterative process to expand the search space. At the same time, an archive is set to save the high-quality feasible solutions during the search process, and the infeasible solutions are periodically replaced. Computational experimental results on VRPMTW benchmark instances show that the proposed algorithm is effective and has obtained better solutions.

Mathematics Subject Classification: Primary: 90B06, 90B40; Secondary: 90C11.

 Citation:

• Figure 1.  Removal List(RL), Unserved Customers List(UCL) and current solution

Figure 2.  Number of times each destroy and repair method was used to produce a new best solution

Table 1.  Definition of the parameters and variables

 Name Description $x_{ij}^{k}$ binary variable, equal to 1 if and only if arc$(i,j)$ is traversed by vehicle $k$ $y_{ij}^{k}$ real variable, equals to the flow carried on arc $(i,j)$ $r^{k}$ binary variable, equal to 1 if and only if vehicle $k$ is used $v_{i}^{p}$ binary variable, equal to 1 if and only if customer $i$ is served within its time window $p$ $z_{i}^{k}$ binary variable, equals to 1 if and only if customer $i$ is assigned to vehicle $k$ $w_{i}^{k}$ real variable, waiting time of vehicle $k$ at customer $i$ $d_{k}$ route duration of vehicle $k$ $a_{i}^{k}$ arrival time of vehicle $k$ at customer $i$ $b_{i}^{k}$ departure time of vehicle $k$ from customer $i$ $t_{ij}$ travel time associated with the arc $(i,j)$ $s_{i}$ service time at customer $i$ $q_{i}$ demand of customer $i$ $l_{i}^{p}$ lower bound of time window $p$ at customer $i$ $u_{i}^{p}$ upper bound of time window $p$ at customer $i$ $Q_{k}$ capacity of vehicle $k$ $D_{k}$ maximum duration of the route of vehicle $k$ $F^{k}$ fixed cost in time units of using vehicle $k$ $M$ an arbitrary large constant

Table 2.  Parameters of ALNS

 Parameter nsegs nters $\sigma 4$ $\sigma 3$ $\sigma 2$ $\sigma 1$ $\zeta$ $T_{0}$ $c$ $\beta 1$ $\beta 2$ Value 1000 100 10 5 3 1 0.7 0.4 0.9 100 100

Table 3.  Parameters of RemovalList in LocalSearch procedure

 Parameters Average in 3 groups ($\psi^{LS}$, $\psi^{LS}_{max}$, $noi_{max}$) CM RCM RM (12, 20,200) 13574.6 4201.8 4109.6 (12, 20,500) 13341.7 4197.2 4042.6 (12, 20,800) 12947.1 4141.0 4023.6 (15, 22,200) 12927.3 4094.3 3815.3 (15, 22,500) 12724.2 4055.7 3737.7 (15, 22,800) 12729.0 4056.1 3736.4 (18, 25,200) 12845.1 4092.3 3746.5 (18, 25,500) 12813.0 4084.5 3741.0 (18, 25,800) 12748.9 4088.3 3739.2

Table 4.  ALNS results on VRPMTW instances(Group CM)

 Instance m HVNTS HGVNS EAVNS ALNS gap1 gap2 gap3 CM101 10 12320 12319.1 12345.4 12151.0 1.37 1.36 1.57 CM102 11 12492.1 12410.7 12482.3 12467.8 0.19 -0.46 0.12 CM103 11 12641.2 12632.4 12592.2 12450.5 1.51 1.44 1.13 CM104 13 13087.8 13098.0 12927.8 12912.2 1.34 1.42 0.12 CM105 10 12083.4 12027.0 12066.3 12083.4 0.00 -0.47 -0.14 CM106 10 12073.9 12059.0 12066.4 11995.4 0.65 0.53 0.59 CM107 10 12324.2 12318.0 12108.4 12092.2 1.88 1.83 0.13 CM108 10 11990.4 11986.0 11985.9 11970.3 0.17 0.13 0.13 Average 10.6 12376.6 12356.3 12321.8 12265.4 0.89 0.72 0.46 CM201 5 13520.1 13498.8 13468.4 13418.3 0.75 0.60 0.37 CM202 6 14027.3 14025.1 14020.2 14010.4 0.12 0.10 0.07 CM203 5 13497.2 13465.8 13486.5 13464.6 0.24 0.01 0.16 CM204 5 13359.8 13344.0 13356.9 13344.0 0.12 0.00 0.10 CM205 4 12884.1 12827.8 12896.8 12829.5 0.42 -0.01 0.52 CM206 4 12767.7 12713.2 12733.4 12704.6 0.49 0.07 0.23 CM207 4 13009.7 12963.7 12963.7 12933.4 0.59 0.23 0.23 CM208 4 12788.1 12749.7 12756.8 12746.7 0.32 0.02 0.08 Average 4.6 13231.8 13198.5 13210.3 13181.4 0.38 0.13 0.22

Table 5.  ALNS results on VRPMTW instances(Group RCM)

 Instance m HVNTS HGVNS EAVNS ALNS gap1 gap2 gap3 RCM101 10 4098.9 4081.2 4080.6 4076.1 0.55 0.12 0.11 RCM102 10 4222.6 4188.3 4184.3 4122.7 2.73 1.57 1.47 RCM103 10 4174.3 4150.4 4148.3 4140.5 0.81 0.24 0.19 RCM104 10 4156.3 4144.0 4141.2 4128.3 0.67 0.38 0.31 RCM105 10 4216.7 4207.0 4208.2 4190.4 0.62 0.40 0.42 RCM106 10 4219.9 4187.7 4191.8 4173.1 1.11 0.35 0.45 RCM107 11 4542.4 4521.5 4516.5 4511.4 0.68 0.22 0.11 RCM108 11 4614.5 4565.2 4566.2 4532.5 1.78 0.72 0.74 Average 10.3 4280.7 4254.9 4254.6 4234.4 1.07 0.50 0.48 RCM201 2 3783.6 3783.2 3800.1 3726.5 1.51 1.50 1.94 RCM202 2 3847.1 3779.4 3822.9 3756.7 2.35 0.60 1.73 RCM203 2 3721.9 3722.0 3771.7 3719.2 0.07 0.07 1.39 RCM204 2 3726.5 3708.5 3716.0 3701.6 0.67 0.19 0.39 RCM205 2 3754.5 3754.5 3756.0 3730.3 0.64 0.64 0.68 RCM206 2 3812.7 3803.3 3725.0 3725.9 2.28 2.04 -0.02 RCM207 3 4764.2 4761.5 4757.1 4771.2 -0.15 -0.20 -0.30 RCM208 2 3791.4 3742.7 3735.1 3735.9 1.46 0.18 -0.02 Average 2.1 3900.2 3881.9 3885.5 3858.4 1.10 0.63 0.72

Table 6.  ALNS results on VRPMTW instances(Group RM)

 Instance m HVNTS HGVNS EAVNS ALNS gap1 gap2 gap3 RM101 10 4041.9 4027.1 4026.1 4009.4 0.81 0.44 0.42 RM102 9 3765.1 3751.2 3774.8 3730.0 0.93 0.56 1.19 RM103 9 3708.5 3703.0 3700.6 3704.0 0.12 -0.33 -0.09 RM104 9 3718.0 3701.2 3707.1 3697.6 0.55 0.10 0.26 RM105 9 3688.8 3687.2 3690.5 3689.7 -0.02 -0.07 0.02 RM106 9 3692.9 3708.4 3714.8 3713.5 -0.56 -0.14 0.03 RM107 9 3701.4 3692.8 3700.4 3686.0 0.42 0.18 0.39 RM108 9 3792.1 3722.6 3738.1 3727.6 1.70 -0.14 0.28 Average 9.1 3755.7 3749.2 3756.6 3744.7 0.49 0.11 0.31 RM201 2 4808.2 4805.4 3888.9 3804.4 20.88 20.83 2.17 RM202 2 3739.0 3706.8 3721.9 3706.8 0.86 0.00 0.41 RM203 2 3710.3 3696.9 3693.2 3691.7 0.50 0.14 0.04 RM204 2 3691.9 3674.5 3671.7 3674.5 0.47 0.00 -0.08 RM205 2 3689.9 3668.1 3668.4 3671.0 0.51 -0.08 -0.07 RM206 2 3703.4 3684.9 3672.6 3673.5 0.81 0.31 -0.02 RM207 2 3701.7 3664.3 3662.4 3664.3 1.01 0.00 -0.05 RM208 2 3682.8 3664.3 3663.6 3664.3 0.50 0.00 -0.02 Average 2 3840.9 3820.7 3705.3 3693.8 3.19 2.65 0.30

Table 7.  Evaluation of contribution of each operator

 Operator Average degradation(%) Maximum degradation(%) Worst Removal 0.26 0.86 Basic Related Removal 0.20 0.91 Improved Related Removal 0.25 0.99 Route Removal 0.22 1.15 Random Removal 0.17 0.84 Greedy Insertion 0.27 1.82 Regret Insertion 0.35 2.31 Random Insertion 0.17 0.73

Table 8.  Execution counts of the operators leading to the discovery of a new solution

 Operator Best solution Current solution Simulated annealing Worst Removal 922 73627 1007 Basic Related Removal 2341 88112 756 Improved Related Removal 4267 93581 543 Route Removal 45 45990 4352 Random Removal 64 41686 6235 Greedy Insertion 2511 38656 457 Regret Insertion 7539 95684 365 Random Insertion 582 9083 2120
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