Article Contents
Article Contents

# An integrated bi-objective optimization model and improved genetic algorithm for vehicle routing problems with temporal and spatial constraints

• * Corresponding author: Zhong Wan
All the three authors are supported by the National Science Foundation of China (Grant No. 71671190)
• Vehicle routing problem (VRP) is a typical and important combinatorial optimization problem, and is often involved with complicated temporal and spatial constraints in practice. In this paper, the VRP is formulated as an optimization model for minimizing the number of vehicles and the total transportation cost subject to constraints on loading plan, service time and weight capacity. The transportation cost consists of the rent charge of vehicles, fuel cost, and carbon tax. Owing to complexity of the built model, it is divided into two subproblems by a two-stage optimization approach: at the first stage, the number of vehicles is minimized, then the routing plan is optimized at the second stage. For solving the sequential subproblems, two correlated genetic algorithms are developed, which share the same initial population to reduce their computational costs. Numerical results indicate that the developed algorithms are efficient, and a number of important managerial insights are revealed from the model.

Mathematics Subject Classification: Primary: 90B06, 62K05; Secondary: 68T37.

 Citation:

• Figure 1.  Example of crossover

Figure 2.  Minimal total costs by Algorithm 5

Figure 3.  Sensitivity analysis on speed of vehicles

Table 1.  Numerical results in case study

 NC MVN $v$ TD TPV WTV VC TFC TCE TC CPU 1 19 6 50 1544.2 5.14 0.33 600 1160.5 7.09 1767.6 68.5 2 39 10 50 2208.7 3.40 0.28 1000 1654.2 10.1 2664.3 79.4 3 59 19 50 4368.9 4.60 0.32 1900 3145.7 19.2 5064.9 92.9 4 79 26 50 6070.6 4.66 0.32 2600 4379.8 26.8 7006.6 125.7 5 99 35 50 8636.1 4.93 0.27 3500 6133.3 37.5 9670.8 163.0 6 119 38 50 9743.4 5.12 0.25 3800 6970.8 42.6 10813.4 219.3 7 139 45 50 11470.4 5.09 0.43 4500 8213.9 50.2 12764.1 244.3 8 159 49 50 13913.2 5.68 0.44 4900 9938.1 60.8 14898.9 259.0 9 179 60 50 14883.3 4.96 0.50 6000 10673.7 65.3 16739.0 274.4 10 199 71 50 17599.0 4.96 0.62 7100 12544.3 76.7 19721.0 309.7

Table 2.  Routes of all the vehicles in the case $n = 19$

 LV LNC DT AT WT LW LL DD TFC TCEt 1 0 9:00 - - - - - - - 6 12:08 11:48 20 0.32 1.12 130.2 86.6 0.53 4 13:32 13:32 0 0.18 0.48 59.0 38.3 0.23 0 - 15:20 - 0 0 1654.2 50.2 0.31 2 0 9:00 - - - - - - - 5 10:19 10:19 0 1.26 0.52 56.4 43.7 0.27 1 11:20 11:20 0 1.05 0.31 40.8 30.6 0.19 10 13:29 12:28 61 0.59 0.05 46.0 32.0 0.20 0 - 14:24 - 0 0 1654.2 22.6 0.14 3 0 9:00 - - - - - - - 9 11:54 11:54 0 2.57 1.62 135.3 125.7 0.77 17 13:11 13:10 0 1.91 1.35 53.7 45.7 0.28 7 14:35 14:35 0 1.00 0.64 60.5 45.0 0.28 0 - 15:13 0 0 0 1654.2 13.5 0.08 4 0 9:00 - - - - - - - 2 10:57 10:37 20 1.14 1.47 71.1 54.1 0.33 13 13:07 12:07 60 0.91 0.97 48.4 35.5 0.22 19 14:22 14:22 0 0.38 0.42 51.8 34.8 0.21 0 - 15:19 - 0 0 37.7 23.6 0.14 5 0 9:00 - - - - - - - 18 11:55 11:15 39 2.31 2.03 120.9 92.4 0.57 3 12:53 12:53 0 1.93 1.81 38.2 32.6 0.20 12 13:44 13:44 0 1.28 1.54 33.2 25.9 0.16 16 14:23 14:23 0 0.85 0.60 22.4 16.3 0.10 0 - 15:11 - 0 0 29.7 18.6 0.11 6 0 9:00 - - - - - - - 15 11:30 11:30 0 1.98 2.58 115.3 99.1 0.61 8 13:03 13:03 0 1.74 1.82 67.2 55.9 0.35 11 14:14 14:14 0 0.92 0.82 49.5 36.4 0.22 14 15:55 15:55 0 0.33 0.55 73.6 49.0 0.30 0 - 16:48 0 0 0 83.8 52.5 0.32

Table 3.  Comparison of the optimal costs by different models

 Case NC TD TFC TCE CPU Test1$^a$ 20 1544.2 1160.5 7.1 68.5 Test1$^b$ 20 1442.2 1214.1 7.5 102.8 Test2$^a$ 40 2208.7 1654.2 10.1 79.4 Test2$^b$ 40 2304.6 1815.2 11.1 111.2 Test3$^a$ 60 4368.9 3145.7 19.2 92.9 Test3$^b$ 60 4262.7 3351.5 20.5 148.6 Test4$^a$ 80 6070.6 4379.8 26.8 125.7 Test4$^b$ 80 6118.4 4811.0 29.4 188.6 Test5$^a$ 100 8636.1 6133.3 37.5 163.0 Test5$^b$ 100 8288.8 6517.7 39.8 252.7 Test6$^a$ 120 9743.4 6970.8 42.6 219.3 Test6$^b$ 120 9758.8 7693.3 46.9 346.5 Test7$^a$ 140 11470.4 8213.9 50.2 244.3 Test7$^b$ 140 11853.7 9320.9 57.0 359.1 Test8$^a$ 160 13913.2 9938.1 60.8 259.0 Test8$^b$ 160 13011.9 10230.9 62.5 383.3 Test9$^a$ 180 14883.3 10673.7 65.3 274.4 Test9$^b$ 180 15488.2 12178.9 74.4 466.5 Test10$^a$ 200 17599.0 12544.3 76.7 309.7 Test10$^b$ 200 17378.1 13664.8 83.5 526.5

Table 4.  Vehicle parameters

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