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The dynamic and stochastic vehicle routing problem (DSVRP) can be modelled as a stochastic program (SP). In a two-stage SP with recourse model, the first stage minimizes the a priori routing plan cost and the second stage minimizes the cost of corrective actions, performed to deal with changes in the inputs. To deal with the problem, approaches based either on stochastic modelling or on sampling can be applied. Sampling-based methods incorporate stochastic knowledge by generating scenarios set on realizations drawn from distributions. In this paper we proposed a robust solution approach for the capacitated DSVRP based on sampling strategies. We formulated the problem as a two-stage stochastic program model with recourse. In the first stage the a priori routing plan cost is minimized, whereas in the second stage the average of higher moments for the recourse cost calculated via a set of scenarios is minimized. The idea is to include higher moments in the second stage aiming to compute a robust a priori routing plan that minimizes transportation costs while permitting small changes in the demands without changing solution structure. Additionally, the approach allows managers to choose between optimality and robustness, that is, transportation costs and reconfiguration. The computational results on a generic dynamic benchmark dataset show that the robust routing plan can cover unmet demand while incurring little extra costs as compared to the preplanning. We observed that the plan of routes is more robust; that is, not only the expected real cost, but also the increment within the planned cost is lower.

The basic task in freight transport is to ship goods from one location to another one, which are typically represented by depots and geographically dispersed points, respectively. Hence, a combinatorial optimization problem arises, which is known as vehicle routing problem (VRP). The VRP aims to determine a set of vehicle routes to perform transportation requests with a given vehicle fleet at minimum cost, that is, to decide which vehicle handles which customer order in which sequence. In this kind of problem, one typically assumes that the values of all inputs are known with certainty and do not change. However, in today’s economy, one issue needs to be integrated: customers desire more flexibility and fast fulfillment of their orders. Besides that, the recent developments in information technology permit a growing amount of available data and both control of a vehicle fleet and management of customer orders in real-time. This context calls for real-time decision support in vehicle routing, motivating a version of the VRP, the so-called dynamic and stochastic vehicle routing problem.

The DSVRP is a generalization of the VRP, where parts or all necessary information regarding inputs is stochastic and the true values become available at runtime only. Usually, the dynamic and stochastic VRP is modelled either as a

In this paper, we formulate the dynamic and stochastic capacitated vehicle routing problem, where the demands are stochastic and dynamic, as a two-stage stochastic program model. Similar to the sampling-based methods, we also make use of scenarios in the proposed robust solution approach. However, different from MSA, the scenarios are generated only once at the beginning of the planning stage and, different from SAA, we do not minimize the average of the second stage cost of a set of sample scenarios. The idea of the robust approach is to address uncertainty using higher moments calculated via scenarios, permitting the solution to be able to adapt to situations when the real demand is greater than expected. Our aim is to develop a solution approach such that the routing plan is robust against small changes in the inputs, that is, allowing to compensate for changes in the input without losing structural properties and optimality. For that, the remainder of the paper is organized as follows. Section

The VRP is a generalization of the Traveling Salesman Problem (TSP). The TSP is a well-studied problem, in which the goal is to minimize the total distance traveled by the salesman while visiting a group of cities and returning to the first visited city. In this problem each city is visited exactly once by the salesman. The vehicle routing problem, in its turn, consists of designing a routing plan to attend to all customers with a given vehicle fleet at minimum cost. Mathematically, the VRP reads as follows.

Consider a set of vehicles

Within Definition

In contrast to the basic definition of the VRP, most real-life applications have to be analyzed with regard to two aspects: evolution and quality of information [

Taxonomy of vehicle routing problems by information evolution and quality.

| Information quality | ||
---|---|---|---|

Deterministic input | Stochastic input | ||

Information evolution | Input known beforehand | | |

Input unknown beforehand | | |

In the

Suppose the setting of the capacitated VRP is to be given with

Traditionally, dynamic and stochastic VRP are formulated as

Several methods based on the formulation described above have been proposed to address the dynamic nature of routing problems. Dynamic methods can be divided into two categories: nonanticipative [

Verweij et al. [

In this work, we formulate the dynamic and stochastic capacitated vehicle routing problem as a two-stage stochastic program with recourse, using a detour to depot as the corrective action. Based on this formulation we propose a robust solution approach. Such approach also tries to optimize the corresponding deterministic expected value cost, like in SAA method. However, our approach uses only one sample of scenario and permits a wider deviation in the cost for the scenarios in the sample, aiming to accommodate changes in the customers’ demands.

Figure

Papers dealing with dynamic vehicle routing problem.

In this paper, we concentrate on the capacitated dynamic and stochastic vehicle routing problem according to Definition

Suppose a set of vehicles

In comparison to (

Exemplary of cost development regarding strategy

The computed solution will typically exhibit higher planned costs than the one of the capacitated vehicle routing problem in Definition

For the problem described before we develop a robust solution approach. The proposed approach includes 4 stages:

In the third stage, a static and deterministic instance of the capacitated DSVRP is set by using equation

Numerical example.

Customer | | | | | |
---|---|---|---|---|---|

1 | 40 | 38 | 43 | 35 | |

2 | 40 | 47 | 41 | 40 | |

3 | 40 | 28 | 35 | 43 | |

In the fourth and last stage we solve the instance defined in the previous stage. For that we use three heuristics: Clarke Wright savings, 2-opt Local Search, and Simulated Annealing. Using Clarke Wright [

In the literature, some authors based their computational experiments on adaptations of the Solomon [

We generated five benchmark test problems

After developing the dynamic benchmark dataset, we applied the proposed solution approach to the dataset using a total of

Note that the first stage of the proposed solution approach is to fit a probability distribution in the customer demand data. To render the approach realistic, we included the fitting for the dynamic benchmark dataset. For that, we assumed that we do not know the PDF used to generate the instances (

The reliability of a plan of routes is defined as the probability that the plan of routes did not suffer a failure. In the context of our Problem

The number of routes in the plan of routes, which suffered a failure, that is,

Extra cost of the robust plan of routes is defined as the additional cost we incur if we apply the robust solution approach from Problem

A solution to the optimization problems from Definition

To assess the solutions, we also computed the difference between the planned cost and the expected real cost:

This performance measure extends the previous one and shows the distance from the planned to the expected real cost:

From our results given in Table

Results for the test problems considering the weight

Test problem | | | Number of routes | | Planned cost | Extra cost | Expected real cost | | | CPU |
---|---|---|---|---|---|---|---|---|---|---|

TP1 | 0 (mean) | 0.87 | 4 | 0.37 | 1471 | - | 1817 | 346 | 1.23 | 8976 |

1 | 0.79 | 5 | 0.30 | 1634 | 1.11 | 1810 | 176 | 1.10 | 8999 | |

2 | 0.77 | 5 | 0.28 | 1634 | 1.11 | 1807 | 173 | 1.10 | 9133 | |

3 | 0.76 | 5 | 0.25 | 1643 | 1.11 | 1720 | 77 | 1.04 | 9567 | |

8 | 0.59 | 5 | 0.19 | 1700 | 1.12 | 1879 | 163 | 1.09 | 9822 | |

10 | 0.52 | 5 | 0.10 | 1772 | 1.20 | 1901 | 129 | 1.07 | 10790 | |

| ||||||||||

TP2 | 0 (mean) | 0.98 | 8 | 0.42 | 2592 | - | 3250 | 658 | 1.25 | 20509 |

1 | 0.98 | 9 | 0.40 | 2747 | 1.05 | 3257 | 510 | 1.18 | 20902 | |

2 | 0.90 | 9 | 0.39 | 2747 | 1.05 | 3221 | 474 | 1.17 | 21112 | |

3 | 0.70 | 9 | 0.35 | 2769 | 1.06 | 3155 | 386 | 1.13 | 21678 | |

8 | 0.68 | 10 | 0.27 | 2807 | 1.08 | 3467 | 660 | 1.23 | 22487 | |

10 | 0.65 | 10 | 0.25 | 2861 | 1.08 | 3441 | 580 | 1.21 | 22756 | |

| ||||||||||

TP3 | 0 (mean) | 0.99 | 12 | 0.44 | 3690 | - | 4969 | 1279 | 1.34 | 38234 |

1 | 0.99 | 14 | 0.42 | 3944 | 1.06 | 4965 | 1021 | 1.25 | 39877 | |

2 | 0.90 | 14 | 0.38 | 3952 | 1.06 | 4960 | 1008 | 1.25 | 43234 | |

3 | 0.89 | 14 | 0.33 | 3941 | 1.06 | 4875 | 934 | 1.23 | 43654 | |

8 | 0.88 | 14 | 0.30 | 3998 | 1.07 | 5023 | 1025 | 1.25 | 44877 | |

10 | 0.80 | 16 | 0.24 | 24184 | 1.32 | 5268 | 18499 | 1.75 | 51001 | |

| ||||||||||

TP4 | 0 (mean) | 1.00 | 16 | 0.46 | 4882 | - | 6757 | 1875 | 1.38 | 51667 |

1 | 1.00 | 18 | 0.46 | 5232 | 1.07 | 6754 | 1522 | 1.29 | 57100 | |

2 | 1.00 | 18 | 0.40 | 5294 | 1.07 | 6753 | 1459 | 1.27 | 57212 | |

3 | 1.00 | 18 | 0.42 | 5294 | 1.07 | 6739 | 1445 | 1.27 | 57302 | |

8 | 1.00 | 18 | 0.35 | 5351 | 1.09 | 6702 | 1351 | 1.25 | 5804 | |

10 | 1.00 | 19 | 0.23 | 5415 | 1.10 | 6675 | 1260 | 1.23 | 62200 | |

| ||||||||||

TP5 | 0 (mean) | 1.00 | 20 | 0.46 | 6057 | - | 8402 | 2345 | 1.38 | 72865 |

1 | 1.00 | 22 | 0.40 | 6278 | 1.03 | 8401 | 2124 | 1.33 | 76965 | |

2 | 1.00 | 22 | 0.40 | 6296 | 1.03 | 8395 | 2099 | 1.33 | 77102 | |

3 | 1.00 | 22 | 0.41 | 6278 | 1.03 | 8390 | 2112 | 1.33 | 77000 | |

8 | 1.00 | 22 | 0.37 | 6329 | 1.04 | 8370 | 2041 | 1.32 | 77590 | |

10 | 1.00 | 23 | 0.33 | 6354 | 1.04 | 8372 | 2018 | 1.31 | 85786 | |

| ||||||||||

TP6 | 0 (mean) | 1.00 | 24 | 0.47 | 7253 | - | 10002 | 2749 | 1.37 | 100856 |

1 | 1.00 | 26 | 0.42 | 7436 | 1.02 | 9987 | 2551 | 1.34 | 114502 | |

2 | 1.00 | 26 | 0.40 | 7543 | 1.03 | 9980 | 2437 | 1.32 | 115121 | |

3 | 1.00 | 26 | 0.40 | 7514 | 1.03 | 9912 | 2398 | 1.31 | 115031 | |

8 | 1.00 | 26 | 0.37 | 7550 | 1.04 | 10020 | 2470 | 1.32 | 115672 | |

10 | 1.00 | 28 | 0.38 | 7585 | 1.05 | 10140 | 2555 | 1.33 | 120876 |

For all test problems the

Plan of routes for Test Problem 2 (

Plan of routes for Test Problem 2 after revealing the real values for the demands (

Comparing CPU time for the same instance, we see that increases on

In this paper we proposed a robust solution approach for the dynamic and stochastic CVRP, where demands are uncertain and dynamic based on sampling strategies. We formulate the problem as a two-stage stochastic program model with recourse. A detour to the depot was defined as corrective action. The two-stage model is a new model, in which the first stage minimizes the a priori routing plan cost whereas in the second stage minimizes the average of higher moments for the recourse cost calculated via a set of scenarios. Different from the other sampling-based methods for the DSCVRP, the proposed solution approach permits deciding between optimality and robustness and computes an a priori robust plan of routes, which allows for small changes in demands without changing solution structure and losing optimality. Using the robust approach, the capacitated dynamic and stochastic VRP is reduced to capacitated static and deterministic VRP, which allows using simple algorithms. The results show that the proposed approach provides significant improvements over the deterministic approach. It is evident that the proposed idea provides a robust plan of routes. That is, for some

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The authors would like to acknowledge the National Council of Technological and Scientific Development (CNPq) for a Doctor’s Degree scholarship granted to the first author.