On vehicle routing problems with stochastic demands -- Generic integer L-shaped formulations

TL;DR

This paper develops a unifying integer L-shaped framework for vehicle routing problems with stochastic demands, introducing a recourse-disaggregation approach that encompasses and extends prior ILS cuts. By fixing a general disaggregation and defining activation functions and recourse lower bounds, the authors show how route-, partial-route-, path-, and set-cut inequalities can be combined in a coherent branch-and-cut algorithm. The framework is then applied to the VRPSD with scenario-based demands under the classical recourse policy, yielding the first exact algorithm that handles correlations among demands; computational results demonstrate substantial improvements over prior methods. Overall, the work provides a flexible, principled toolkit for designing strong ILS-based relaxations across VRPSD variants and highlights the importance of explicit disaggregation in achieving broader applicability and performance gains.

Abstract

We study a broad class of vehicle routing problems in which the cost of a route is allowed to be any nonnegative rational value computable in polynomial time in the input size. To address this class, we introduce a unifying framework that generalizes existing integer L-shaped (ILS) formulations developed for vehicle routing problems with stochastic demands (VRPSDs). This framework and subsequent analysis allow us to generalize previous ILS cuts and pinpoint which assumptions are needed to apply those generalizations to other problems. Using these tools, we develop the first algorithm for the VRPSD in the case where the demands are given by an empirical probability distribution of scenarios - a data-driven variant that tackles a significant challenge identified in the literature: dealing with correlations. Indeed, all previous ILS-based exact algorithms for the VRPSD assume either independence of customer demands or correlations through a single external factor. This shows the potential of this generic unifying framework to be applied to a multitude of different variants of the problem.

Figures (4)

• Figure 1: Illustration of a route $R = (v_2, v_1, v_3, v_4, v_5, v_6)$ that does not adhere to partial route $\ref{['def:partial_route']} = (\{v_1\}, \{v_2, v_3\}, \{v_4\}, \{v_5, v_6\})$. However, note that $R$ does adhere (but not exactly) to partial route $\ref{['def:partial_route']}' = (\{v_4\}, \{v_5, v_6\})$, since the subroute $R' = (v_4, v_5, v_6) \subseteq R$ exactly adheres to $\ref{['def:partial_route']}'$.
• Figure 2: Comparison of the different ILS branch-and-cut algorithms.
• Figure 3: Example of a solution $\bar{x} \in \ref{['assumption:formulation']}$ where $G(\bar{x}) \setminus \{0\}$ is made of a single component whose total flow to the depot is 4. The block–cut tree associated with $G(\bar{x})$ contains a branching vertex corresponding to the block $\{v_5, v_6, v_7, v_8\}$. In this case, our algorithm considers partial route $\ref{['def:partial_route']} = (\{v_1\}, \{v_5\}, \{v_6\}, \{v_2\})$ and we have that $\ref{['eq:activation_function_exact_adheres']} = 0.5$.
• Figure 4: Instance with $V_+ = \{v_1, v_2, v_3, v_4, v_5\}$, $k = 2$ and $C = 10$ that does not satisfy the monotonicity property. The numbers next to the edges correspond to the cost of the edges incident to the depot, while the numbers on top of the vertices refer to the vector of expected demands $\bar{d} \in \mathbb{Q}^{V_+}_{++}$. This instance has $N = 4$ scenarios, each with a realization probability of $\frac{1}{4}$. The table on the right show the scenario demands vectors $d^\xi \in \mathbb{Q}^{V_+}_{+}$. Note that failures may occur only in scenarios $\xi = 2$ and $\xi = 4$.

Theorems & Definitions (66)

• Definition 1
• Definition 2
• Claim 1
• Remark 1
• Definition 3
• Claim 2
• proof
• Definition 4
• Remark 2
• Definition 5