The robust vehicle routing problem with time windows

TLDR: This paper addresses the robust vehicle routing problem with time windows by proposing two new formulations for the robust problem, each based on a different robust approach, and develops a new cutting plane technique for robust combinatorial optimization problems with complicated constraints.

About: This article is published in Computers & Operations Research.The article was published on 2013-03-01 and is currently open access. It has received 173 citations till now. The article focuses on the topics: Robust optimization & Vehicle routing problem.

Figures

Table 1: Average numbers of extreme points generated by (T -RI).

Table 2: Average numbers of cuts generated by (T -PI).

Table 3: Average solution times in seconds for the three formulations for instances with 20 nodes.

Figure 2: Polytope T and its projections T k1 and T k2 .

Figure 1: The set of scenarios {t, t̂} dominates t̃.

Table 4: Average solution times in seconds for (T -RI) and (T -PI) for larger instances.

Frequently asked questions

Q1. What are the contributions mentioned in the paper "The robust vehicle routing problem with time windows" ?

This paper addresses the robust vehicle routing problem with time windows. The authors propose two new formulations for the robust problem, each based on a different robust approach. The authors propose two techniques, which, using the structure of the problem, allow to reduce significantly the number of extreme points of the uncertainty polytope. In particular, efficient separation procedures are discussed. 

Q2. What is the classical approach to linear robust programming?

The classical approach in linear robust programming under polyhedral uncertainty to handle the infinite set of constraints (12) [5], relies on dualizing constraints (12). 

Q3. What is the classical approach for robust programming?

The classical approach for robust programming relies on static models where the variables of the problem are not allowed to vary to account for the different values taken by the uncertain parameters. 

Q4. What is the benefit of the addition in complexity?

The benefit of the addition in complexity is that the model from [1] is more flexible than the one from [25] and leads to less conservative robust solutions. 

Q5. How can the authors extend a model to handle uncertain polytopes?

Model (RI) can be naturally extended to handle uncertain polytope T : x becomes the set of first-stage variables, while y becomes y(t), a function of t ∈ T . 

Q6. How can the authors check whether x satisfies the robust inequalities?

checking whether x satisfies the robust inequalities (27) can be done in polynomial time, more specifically, by applying a sorting algorithm |K| times. 

Q7. How did Ben-Tal et al. prove that adjustable robust optimization is computationally?

Ben-Tal et al. [4] proved that adjustable robust optimization is computationally intractable so that they introduced an approximation scheme that relied on affine decision rules. 

Q8. How can one extend (T R-P) to the case of uncertain cost c1?

Notice that (T R-P ) can be extended to the case of uncertain cost c1 by replacing the objective function with min z andadding the restrictions z ≥ (c1)Tx1 to the set of uncertain constraints. 

Q9. What is the way to withdraw from ext(T k)?

the authors can withdraw from ext(T kΓ), and therefore from diag(T Γ), all vectors where the delay occurs on two arcs that enter or leave the same node. 

Q10. How is the necessity checked for the extreme points?

In addition, their necessity is checked only after an optimal integer solution has been found for the previous set of extreme points.