In practice, the parameters of the vehicle routing problem are uncertain, which is called the uncertain vehicle routing problem (UVRP). Therefore, a data-driven robust optimization approach to solve the heterogeneous UVRP is studied. The uncertain parameters of customer demand are introduced, and the uncertain model is established. The uncertain model is transformed into a robust model with adjustable parameters. At the same time, we use a least-squares data-driven method combined with historical data samples to design a function of robust adjustable parameters related to the maximum demand, demand range, and given vehicle capacity to optimize the robust model. We improve the deep Q-learning-based reinforcement learning algorithm for the fleet size and mix vehicle routing problem to solve the robust model. Through test experiments, it is proved that the robust optimization model can effectively reduce the number of customers affected by the uncertainty, greatly improve customer satisfaction, and effectively reduce total cost and demonstrate that the improved algorithm also exhibits good performance.

To effectively allocate logistics resources and reduce transportation costs, the vehicle routing problem (VRP) has been a key topic in the field of logistics scheduling. The VRP was first introduced by Dantzig and Ramser in 1959 [

The VRP in which a customer’s demand is unknown before reaching the customer point is called the vehicle routing problem with uncertain demand (VRPUD). Bertsimas assumed the probability distribution function of demand and through analysis of the distribution function, the expectation of the minimum route length was obtained in advance [

At present, research methods addressing the UVRP mostly focus on stochastic programming and fuzzy programming, but stochastic programming requires uncertain parameters to satisfy a certain probability model, while fuzzy programming requires corresponding fuzzy membership. For the actual situation, the solution is relatively difficult to obtain and cannot perfectly match the actual situation. In recent years, there have been increasingly more studies on the application of robust optimization of uncertain parameters. Enlarging the value of uncertain parameters is more suitable for actual situations with uncertainties.

At the same time, with the development of data-driven methods, a large amount of data can be collected and analyzed, which can be used for analysis and prediction. Therefore, the application of robust optimization and data-driven methods to VRP is increasingly studied. Sun combined the predecessor’s robust optimization framework, took the demand and travel time in the VRP as uncertain parameters, deduced the final model as a mixed integer programming problem, and then used branch-and-bound and genetic algorithms to analyze the robust optimization model of the VRP in emergency management [

Based on the VRP, in this study, the heterogeneous vehicle routing problem with uncertain demand (HVRPUD) is studied considering the two constraints of the capacity of multiple vehicles and the uncertainty of customer demand. An uncertain model by introducing uncertain customer parameters is established. The uncertain model is transformed into a robust model with adjustable parameters, and, at the same time, functions of robustly adjustable parameters related to the maximum demand, the demand range, and the capacity of the given vehicle are designed by using a least-squares data-driven method combined with historical data samples. Then, those functions are used to optimize the robust model. Finally, a deep Q-network-based reinforcement learning algorithm is used to solve the robust model. Through test experiments, it is proved that the robust optimization model designed for this problem can effectively reduce the number of customers affected by uncertainty, greatly improve customer satisfaction, and effectively reduce total costs. The improved algorithm also exhibits better performance.

HVRPUD is a variant of CVRP considering the capacity of multiple vehicles and the uncertainty of customer demand [

HVRPUD is defined on a complete directed graph

The mathematical formulation is as follows:

Parameters and symbols.

Parameter | Implication |
---|---|

^{k} | The standard capacity of the |

_{ij} | The length of the edge ( |

Represents the penalty factor | |

Represents the amount of unmet customer needs | |

_{ik} | This is a 0-1 variable. When the value is 1, it means that customers |

_{ijk} | This is a 0-1 variable. When the value is 1, it means that the vehicle |

This is a 0-1 variable. When the value is 1, it means that | |

_{k} | Fixed cost of vehicle type |

Constraints (

We refer to Hu et al. [_{i} be the control variable for the uncertainty of the uncertain parameter; _{i} is set to an absolute value of ≤1. When _{i} = 1, then _{i} takes the maximum value of the set; i.e., _{i} is the boundary of the set. When _{i} = 0, _{i} = _{i} is a deterministic parameter with a fixed value; _{i} is uncertain and has a range of values of

The general uncertainty value is related to the degree of uncertainty, but the actual uncertainty value, such as the uncertain parameter of the VRP, is related not only to the customer demand but also to the vehicle capacity and the number of customer points of the uncertain demand served by the vehicle. The HVRPUD requires more consideration of the selection of the vehicle capacity in the robust model. In designing robust models related to the HVRPUD,

At the time of route planning, because of the uncertainty of demand at customer points, it is not possible to determine exactly which vehicle capacity is the best choice for delivery. If a vehicle with an extremely large capacity is chosen, the uncertainty of demand can be greatly magnified. Conversely, it is necessary to control the uncertainty, and larger uncertainty can lead to irrational vehicle planning; i.e., vehicles with larger capacity selected for planning will lead to larger empty loads being actually transported and a larger number of vehicles being required, finally leading to significant waste of costs. The _{max}, _{min}, _{average},, and _{max}, _{min}, and _{average} reflect the approximate distribution of the different vehicles’ capacity in the HVRP, as the average value alone does not provide a good representation of the distribution when the capacity varies greatly. When the mean value of the uncertain demand to meet the customer’s historical data is much less than the mean value of the available vehicle payload, if _{k} is large and the difference is >_{average}, a vehicle capacity one level greater than _{mid} is chosen. Conversely, if _{mid}. If _{average}, then _{mid} directly. _{mid} represents the median value of the order of vehicle capacity or the lower value if the number of vehicles is even. Using _{max} for judgment, we have

In the case of the VRPUD, the more points of uncertain customers served by the same vehicle with the same capacity, the lower the probability that it will be able to meet all the uncertain customers’ demand, and the only solution is to select the maximum value for each uncertain customer demand, but this will increase the degree of conservatism of the robustness and entail cost waste. Therefore, parameter

In the above constraints,

That is, the problem is transformed into finding the maximum value of _{ik} takes the value of 1 or 0 and the value of _{ik} is determined by whether this customer point is served by the

The above formula (

Because the uncertain variable

According to the duality theory,

Therefore, by finding the minimum value of the set _{i} can be obtained.

Historical data are mainly used to make statistical predictions on some uncertain customers’ demand. Because robust optimization is an estimation of uncertain values and has the property of amplifying or reducing, the least-squares estimation method in statistics can be used to fit a robust level functional formula for uncertain customer demand to optimize data for uncertain robust models and improve model accuracy.

According to the above, the formula _{i} varies widely, and to cover the larger value boundary, it is necessary to expand the value of _{i}, which also depends on ^{k} served by the selected vehicles,

Because _{max} has already been used to estimate the actual vehicle load (i.e., the actual vehicle capacity must be >_{max}), it must be assumed that the value of _{i} takes the maximum value).

The analysis of the HVRPUD shows that when the difference between the uncertain customer demand and the vehicle capacity is greater, the fluctuation of the customer demand will have less impact on the subsequent customers in HVRPUD, and the correspondingly higher robustness level can be adopted. According to stochastically generated uncertain customer demand, formulas (

Uncertain customer demand related values.

Point | |||||
---|---|---|---|---|---|

12.4 | 17 | 12 | 35 | 20.5 | 1.208333 |

12.5 | 20 | 15 | 40 | 25 | 1 |

12.8 | 22 | 18 | 60 | 29 | 1.722222 |

12.10 | 19 | 14 | 40 | 23.5 | 1.178571 |

20.1 | 17 | 10 | 40 | 18.5 | 2.15 |

20.4 | 10 | 7 | 40 | 12 | 4 |

20.6 | 17 | 13 | 40 | 21.5 | 1.423077 |

20.11 | 22 | 17 | 60 | 28 | 1.882353 |

20.14 | 25 | 20 | 60 | 32.5 | 1.375 |

20.15 | 12 | 8 | 40 | 14 | 3.25 |

30.1 | 28 | 24 | 100 | 38 | 2.583333 |

30.2 | 40 | 31 | 100 | 51 | 1.580645 |

30.3 | 13 | 10 | 50 | 16.5 | 3.35 |

30.8 | 34 | 24 | 100 | 41 | 2.458333 |

30.12 | 29 | 23 | 100 | 37.5 | 2.717391 |

30.16 | 12 | 9 | 50 | 15 | 3.888889 |

30.24 | 11 | 8 | 50 | 13.5 | 4.5625 |

30.27 | 46 | 35 | 140 | 58 | 2.342857 |

30.29 | 106 | 79 | 200 | 132 | 0.860759 |

50.1 | 22 | 17 | 70 | 28 | 2.470588 |

50.2 | 30 | 22 | 70 | 37 | 1.5 |

50.7 | 17 | 14 | 40 | 22.5 | 1.25 |

50.11 | 41 | 36 | 120 | 56.5 | 1.763889 |

50.16 | 22 | 17 | 70 | 28 | 2.470588 |

50.17 | 23 | 19 | 70 | 30.5 | 2.078947 |

50.20 | 26 | 21 | 70 | 34 | 1.714286 |

50.26 | 20 | 16 | 70 | 26 | 2.75 |

50.29 | 15 | 12 | 70 | 19.5 | 4.208333 |

50.33 | 31 | 25 | 70 | 40.5 | 1.18 |

50.35 | 12 | 9 | 40 | 15 | 2.777778 |

50.38 | 29 | 22 | 70 | 36.5 | 1.522727 |

50.43 | 21 | 16 | 70 | 26.5 | 2.71875 |

50.44 | 20 | 16 | 70 | 26 | 2.75 |

50.45 | 25 | 19 | 70 | 31.5 | 2.026316 |

75.12 | 19 | 15 | 100 | 24.5 | 5.033333 |

75.14 | 37 | 27 | 120 | 45.5 | 2.759259 |

75.17 | 24 | 18 | 100 | 30 | 3.888889 |

75.20 | 26 | 20 | 100 | 33 | 3.35 |

75.22 | 14 | 11 | 50 | 18 | 2.909091 |

75.24 | 32 | 25 | 120 | 41 | 3.16 |

75.25 | 16 | 12 | 50 | 20 | 2.5 |

75.27 | 20 | 15 | 100 | 25 | 5 |

75.29 | 15 | 12 | 50 | 19.5 | 2.541667 |

75.30 | 25 | 19 | 100 | 31.5 | 3.605263 |

75.33 | 32 | 25 | 120 | 41 | 3.16 |

75.34 | 22 | 17 | 100 | 28 | 4.235294 |

75.40 | 39 | 30 | 120 | 49.5 | 2.35 |

75.43 | 21 | 16 | 100 | 26.5 | 4.59375 |

75.52 | 22 | 16 | 100 | 27 | 4.5625 |

75.54 | 19 | 15 | 100 | 24.5 | 5.033333 |

75.58 | 24 | 20 | 120 | 32 | 4.4 |

75.59 | 28 | 21 | 120 | 35 | 4.047619 |

75.64 | 32 | 26 | 120 | 42 | 3 |

75.66 | 43 | 33 | 150 | 54.5 | 2.893939 |

75.68 | 12 | 9 | 50 | 15 | 3.888889 |

75.72 | 4 | 3 | 50 | 5 | 15 |

75.74 | 12 | 9 | 50 | 15 | 3.888889 |

100.8 | 10 | 8 | 60 | 13 | 5.875 |

100.9 | 19 | 15 | 100 | 24.5 | 5.033333 |

100.10 | 19 | 15 | 100 | 24.5 | 5.033333 |

100.17 | 8 | 5 | 60 | 9 | 10.2 |

100.18 | 14 | 10 | 100 | 17 | 8.3 |

100.21 | 12 | 10 | 100 | 16 | 8.4 |

100.22 | 22 | 16 | 100 | 27 | 4.5625 |

100.24 | 14 | 8 | 60 | 15 | 5.625 |

100.27 | 18 | 14 | 100 | 23 | 5.5 |

100.29 | 10 | 8 | 60 | 13 | 5.875 |

100.37 | 10 | 7 | 60 | 12 | 6.857143 |

100.38 | 19 | 14 | 100 | 23.5 | 5.464286 |

100.45 | 19 | 15 | 100 | 24.5 | 5.033333 |

100.46 | 4 | 3 | 60 | 5 | 18.33333 |

100.53 | 17 | 12 | 100 | 20.5 | 6.625 |

100.54 | 22 | 17 | 100 | 28 | 4.235294 |

100.55 | 9 | 5 | 60 | 9.5 | 10.1 |

100.58 | 21 | 16 | 100 | 26.5 | 4.59375 |

100.61 | 15 | 12 | 100 | 19.5 | 6.708333 |

100.62 | 22 | 16 | 100 | 27 | 4.5625 |

100.66 | 29 | 22 | 100 | 36.5 | 2.886364 |

100.74 | 9 | 7 | 60 | 11.5 | 6.928571 |

100.77 | 17 | 12 | 100 | 20.5 | 6.625 |

100.78 | 14 | 8 | 60 | 15 | 5.625 |

100.84 | 16 | 11 | 100 | 19 | 7.363636 |

100.89 | 17 | 13 | 100 | 21.5 | 6.038462 |

100.92 | 9 | 6 | 60 | 10.5 | 8.25 |

100.93 | 26 | 19 | 100 | 32 | 3.578947 |

100.96 | 13 | 10 | 100 | 16.5 | 8.35 |

100.99 | 10 | 8 | 60 | 13 | 5.875 |

In the following, some common standard instances of the HVRP will be selected. In each instance, a fixed number of customer points will be chosen as uncertain demand customer points to generate stochastic demands, which will be fitted by a least-squares function. The standard calculation instances are selected from the 20 HVRPs mentioned in Golden et al. [

The standard formula fitted by the least-squares method [_{i}) represents the fitted function, and _{i} and _{i} represent the variables in the sample. To ensure that the fitted formula can satisfy the stochastic distribution of demand, 30% of the corresponding customer points are taken as customer points with uncertain demand, 1.5 times the original customer demand is chosen as the maximum possible value, 30% of the customer demand is the minimum value, and the uncertain customer demand is stochastically generated 100 times. Ensuring that the fitted formula can meet the stochastic distribution of the demand of the uncertain customer points makes it is easier to meet the regular demand distribution of the customer points. The generated historical data is used to fit the data. The historical demand for each uncertain customer point is analyzed, and _{min}, _{max}, _{average}, and _{average}/_{max} and _{1} and _{2} at the same time, and the corresponding

Using the fitting method, the final values are approximately determined as given in Table

Coefficient values.

Coefficient | Value (≈) |
---|---|

_{0} | 0.67 |

_{1} | −2.87 |

_{2} | 3.16 |

_{3} | 2.60 |

_{4} | −2.19 |

The formula is

The mean squared error is ∼0.0344. Figure

Fitting case comparison.

It can be seen from Figure

Because _{1} and _{2} correspond to _{average}/_{max} and

By substituting this equation into formula (

Similarly, substitute _{1} into the above formula:

Of course, because

It can be seen that the above formula is more complicated. Consequently, formula (

Because all three variables here need to be obtained based on historical data, the historical demand of different customers and the corresponding vehicle capacity are also different, and the vehicle capacity also varies for different problems, it is impossible to determine the exact range of variables. In the formula, the two variables _{1} and _{2} always satisfy

Graphs of the two formulas are shown in Figure

The value of _{1} will be analyzed. _{1} is _{average}/_{max}, where

Because customer demand is always >0 (i.e., there is always customer demand), and the customer point with uncertain demand has been determined before the vehicle service,

Function image.

The latter part of

We refer to the evolutionary hyperheuristic algorithm proposed in article [

The population is generated for the first time, and a vehicle with a capacity of ^{k} is stochastically selected from the existing models, where ^{1}. Customer points are selected according to the algorithm described above until ^{k}, the last customer point is eliminated and another vehicle is stochastically selected with a vehicle capacity of ^{2}; otherwise, the last customer point is retained. The above steps are repeated until the desired population size is generated.

The population _{i} is stochastically selected for subsequent calculations. The optimal solution _{B} = Ind and the optimal adaptation _{B} = fit for the CVRP of this vehicle capacity combination are finally obtained.

Rewards and punishments are scored for the vehicle capacity combination sequence. For the first scoring,

Vehicle capacity combinations are selected. The auxiliary selection value is set as

The flowchart of the algorithm is given in Figure

Flowchart of improved algorithm.

Because the algorithm solves the CVRP with a single population, several cycles are required. Setting the

To ensure that

When

The above two formulas form a loop of the

Figure

The presented DQN-based hyperheuristic algorithm is coded in MATLAB and runs on a computer with an Intel (R) core-i5-3230M and 12 GB of RAM. After repeated testing, the parameters used in the algorithm are set to the following: discount rate ^{6}, empirical pool ^{E} = 800, and the number of samples selected for learning ^{S} = 600. In the experiment, the G-1 to G-20 standard calculation instances proposed by Golden et al. [

Through the method described above, a stochastic demand based on the demand of the original customer point was generated, and 1000 historical samples were generated for each customer point (the demand of the customer point with a certain demand being always unchanged), and the results were generated. Table

HVRPUD test results.

Instance | Type | Cost | Increase (%) | Total | |||
---|---|---|---|---|---|---|---|

E1 (%) | E2 (%) | E3 (%) | |||||

G-1 | Certain | 12 | 602 | — | 43.30 | 56.70 | 0 |

Robust | 12 | 664 | 10.299 | 100 | 0 | 0 | |

G-2 | Certain | 12 | 722 | — | 34.30 | 63.40 | 2.30 |

Robust | 12 | 798 | 10.526 | 100 | 0 | 0 | |

G-3 | Certain | 20 | 961.03 | — | 35.90 | 61.70 | 2.40 |

Robust | 20 | 1067.64 | 11.093 | 91.50 | 8.50 | 0 | |

G-4 | Certain | 20 | 6437.33 | — | 22.00 | 67.90 | 2.00 |

Robust | 20 | 7470.42 | 16.048 | 99.00 | 1.00 | 0 | |

G65 | Certain | 20 | 1007.05 | — | 35.80 | 61.70 | 2.40 |

Robust | 20 | 1089.54 | 8.191 | 90.70 | 9.30 | 0 | |

G-6 | Certain | 20 | 6516.47 | — | 20.20 | 70.0 | 9.80 |

Robust | 20 | 7563.96 | 16.074 | 92.90 | 7.10 | 0 | |

G-7 | Certain | 30 | 6291 | — | 7.40 | 65.10 | 27.50 |

Robust | 30 | 7246 | 15.180 | 85.80 | 14.20 | 0 | |

G-8 | Certain | 30 | 2005 | — | 41.80 | 56.50 | 1.70 |

Robust | 30 | 2333 | 16.359 | 99.20 | 0.80 | 0 | |

G-9 | Certain | 30 | 1937 | — | 32.20 | 65.90 | 1.90 |

Robust | 30 | 2114 | 9.137 | 98.00 | 2.00 | 0 | |

G-10 | Certain | 30 | 2049 | — | 25.90 | 68.50 | 5.60 |

Robust | 30 | 2331 | 13.762 | 99.90 | 0.10 | 0 | |

G-11 | Certain | 30 | 4109 | — | 2.50 | 57.50 | 40.00 |

Robust | 30 | 4729 | 15.088 | 99.90 | 0.10 | 0 | |

G-12 | Certain | 30 | 3493 | — | 14.30 | 66.80 | 18.90 |

Robust | 30 | 4019 | 15.058 | 65.30 | 34.70 | 0 | |

G-13 | Certain | 50 | 2406.36 | — | 7.50 | 50.70 | 41.80 |

Robust | 50 | 2726.68 | 13.311 | 86.20 | 13.80 | 0 | |

G-14 | Certain | 50 | 9119.03 | — | 9.50 | 58.60 | 31.90 |

Robust | 50 | 10665.35 | 16.957 | 100 | 0 | 0 | |

G-15 | Certain | 50 | 2586.37 | — | 0 | 0 | 100 |

Robust | 50 | 3083.23 | 19.210 | 0 | 0 | 100 | |

G-16 | Certain | 50 | 2720.43 | — | 0 | 0 | 100 |

Robust | 50 | 3238.66 | 19.049 | 0 | 0 | 100 | |

G-17 | Certain1 | 75 | 1734.53 | — | 15.70 | 65.30 | 19.00 |

Certain2 | 75 | 1753.12 | 11.390 | — | — | — | |

Robust | 75 | 1952.81 | 12.584 | 91.10 | 8.90 | 0 |

From the Cost column given in Table

Cost overall results graphs. (a) Cost value comparison graph. (b) Cost increase percentage graph.

The four instances, G-17–G-20, in Table

G-17–G-20 cost results graphs. (a) G-17–G-20 cost value comparison graph. (b) G-17–G-20 cost increase percentage graph.

In these four calculation instances, the smaller the difference between the result obtained by the algorithm and the deterministic result is, the smaller the difference between the result obtained by the robust optimization and the deterministic result is.

Figure

Route results charts. (a) G-3 certain and robust route comparison chart. (b) G-19 certain and robust route comparison chart.

From the Total column in Table

Further analysis of these two calculation instances shows that, according to Table _{max} is 8. The maximum changeable difference is only 50% of the change range. Therefore, it is extremely easy to not meet the demand for this customer point, resulting in the vehicle returning to the distribution center, ultimately affecting the demand for at least one remaining customer point and even affecting the demand for two or more customer points if the remaining five customer points have a demand of <4 or contain other customer points with uncertain demand (as the route contains a total of four customer points with uncertain demand). It will even affect two or even more customer points. Therefore, the points can be appropriately amplified using the abovedescribed formula (

Special demand of G-15 and G16.

Instance | Point | _{average} | _{max} | |||
---|---|---|---|---|---|---|

G-15 | 10 | 23 | 39 | 50 | 0.471 | 31 |

21 | 25 | 42 | 50 | 0.332 | 31 | |

32 | 25 | 42 | 50 | 0.332 | 31 | |

46 | 24 | 40 | 50 | 0.428 | 31 | |

G-16 | 11 | 33 | 55 | 80 | 0.657 | 47 |

34 | 17 | 28 | 40 | 0.636 | 24 | |

45 | 19 | 31 | 40 | 0.487 | 25 |

G-15 route containing customer point 10 results.

Table

HVRPUD increase cost result.

Instance | Type | Customer number | Max cost | Min cost |
---|---|---|---|---|

G-1 | Certain | 2 | 118 | 46 |

Robust | 0 | 0 | 0 | |

G-2 | Certain | 3 | 124 | 28 |

Robust | 0 | 0 | 0 | |

G-3 | Certain | 3 | 110.769 | 27.784 |

Robust | 2 | 95.650 | 46.173 | |

G-4 | Certain | 4 | 165.048 | 78.454 |

Robust | 1 | 42.047 | 42.047 | |

G-5 | Certain | 3 | 199.856 | 82.024 |

Robust | 2 | 106.753 | 62.032 | |

G-6 | Certain | 4 | 137.435 | 60.827 |

Robust | 1 | 15.231 | 15.231 | |

G-7 | Certain | 4 | 522 | 322 |

Robust | 2 | 296 | 186 | |

G-8 | Certain | 3 | 470 | 152 |

Robust | 1 | 144 | 144 | |

G-9 | Certain | 3 | 420 | 160 |

Robust | 1 | 152 | 152 | |

G-10 | Certain | 4 | 620 | 296 |

Robust | 1 | 164 | 164 | |

G-11 | Certain | 5 | 702 | 432 |

Robust | 1 | 110 | 110 | |

G-12 | Certain | 4 | 604 | 458 |

Robust | 2 | 264 | 104 | |

G-13 | Certain | 7 | 235.412 | 130.811 |

Robust | 2 | 94.365 | 22.360 | |

G-14 | Certain | 6 | 174.547 | 50.644 |

Robust | 0 | 0 | 0 | |

G-15 | Certain | 10 | 485.850 | 372.207 |

Robust | 7 | 290.154 | 216.502 | |

G-16 | Certain | 9 | 446.153 | 433.462 |

Robust | 8 | 352.608 | 273.292 | |

G-17 | Certain | 5 | 113.840 | 61.145 |

Robust | 1 | 22.090 | 18.439 | |

G-18 | Certain | 9 | 340.513 | 123.505 |

Robust | 3 | 126.12 | 18.439 | |

G-19 | Certain | 9 | 324.969 | 144.223 |

Robust | 1 | 60.926 | 60.926 | |

G-20 | Certain | 13 | 601.571 | 336.205 |

Robust | 1 | 59.665 | 40.249 |

HVRPUD increase cost results charts. (a) Customer num result chart. (b) Max num result chart. (c) Min cost result chart.

The route obtained after the robust optimization is much more reasonable than that obtained by the deterministic model in terms of the number of customer points impacted by returning customer points and the cost of increased distance. In particular, the G-19 and 20 calculation instances, which have been reduced from a maximum impact of 9 customer points and 13 customer points, respectively, to only 1 customer point, indicate significant improvement in terms of improving customer satisfaction. As can be seen from the Max Cost and Min Cost plots, the difference between the Max Cost values is large in the two series, especially in cases G-11 and G-20, with differences of 598 and 541.906, respectively. The value of the Robust series accounts for the Certain series. The value of Min Cost is ∼10%, except for the difference of 344 and 295.956 between the calculation examples G-12 and G-20. Moreover, it can be seen that the larger the number of customer points, the more the number of customer points with uncertain demand will increase, so the route obtained after robust optimization has a greater effect on saving the cost of distance, reflecting the positive effect of robust optimization on the HVRPUD. From the above table and charts, it can be concluded that the route obtained after the above robust optimization can indeed better cope with uncertain customer demand and greatly improve customer satisfaction. In terms of cost, it can also greatly reduce costs.

In real life, logistics companies are faced with the uncertainty of customer demand and the problems of different fleet size and mix vehicle. The HVRPUD model established above can be used. It is an uncertain model, and how to transform it into a deterministic model is a problem that needs to be solved in the application process. First, the dual theory is applied to transform the uncertain model into a robust model with adjustable parameters; second, the historical data of customer demand is collected, and the data-driven method is used to predict the uncertainty value of the demand, thereby transforming the robust model into a computable deterministic model.

In this paper, the heterogeneous vehicle routing problem with uncertain demand is studied. Based on the HVRPUD, stochastic customer points and corresponding stochastic demand were generated. A least-squares data-driven method in combination with stochastically generated samples is used to design the uncertainty degree formula related to the maximum demand, the range of demand, and the capacity of the given vehicles, and this formula is used to optimize customer demand and obtain a data-driven robust model, thus optimizing the uncertainty robust model. An improved deep-Q-network-based hyperheuristic algorithm was used to solve the problem experimentally. The results prove that the proposed data-driven robust optimization method based on the optimized model can significantly adapt to situations in which customer demand changes within a certain range, effectively reducing the number of subsequent customer points affected by the uncertain customers’ demand and greatly improving the satisfaction level of customers. At the same time, the distance cost after robust optimization is small and can effectively reduce the increased distance cost of vehicles returning to the distribution center, and the improved algorithm also exhibits better performance. In the future, the study of the UVRP will continue, using robust optimization and data-driven methods to explore a more realistic solution model.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

The study was supported by the National Natural Science Foundation, China (no. 61402409), and the Natural Science Foundation of Zhejiang Province, China (no. LY19F030017).