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On vehicle routing problems with stochastic demands -- Scenario-optimal recourse policies

Matheus J. Ota, Ricardo Fukasawa

Abstract

Two-Stage Vehicle Routing Problems with Stochastic Demands (VRPSDs) form a class of stochastic combinatorial optimization problems where routes are planned in advance, demands are revealed upon vehicle arrival, and recourse actions are triggered whenever capacity is exceeded. Following recent works, we consider VRPSDs where demands are given by an empirical probability distribution of scenarios. Existing approaches rely on integer L-shaped (ILS) cuts, whose coefficients are tailored for specific recourse policies. In contrast, we propose a framework that casts recourse policies as solutions of a higher-dimensional mixed-integer program, and we characterize its convex hull in the original lower-dimensional space via a new class of inequalities called scenario recourse inequalities (SRIs). We show that SRIs are valid for any recourse policy satisfying mild assumptions and are sufficient for formulating the VRPSD under a scenario-optimal recourse policy, where the recourse actions are chosen optimally for each scenario. Under this latter policy, we also show that SRIs dominate several ILS cuts. We conduct computational experiments on the VRPSD with scenarios under both the classical and the scenario-optimal recourse policies. By using the SRIs, our algorithm solves 329 more instances to optimality than the previous state-of-the-art ILS algorithm.

On vehicle routing problems with stochastic demands -- Scenario-optimal recourse policies

Abstract

Two-Stage Vehicle Routing Problems with Stochastic Demands (VRPSDs) form a class of stochastic combinatorial optimization problems where routes are planned in advance, demands are revealed upon vehicle arrival, and recourse actions are triggered whenever capacity is exceeded. Following recent works, we consider VRPSDs where demands are given by an empirical probability distribution of scenarios. Existing approaches rely on integer L-shaped (ILS) cuts, whose coefficients are tailored for specific recourse policies. In contrast, we propose a framework that casts recourse policies as solutions of a higher-dimensional mixed-integer program, and we characterize its convex hull in the original lower-dimensional space via a new class of inequalities called scenario recourse inequalities (SRIs). We show that SRIs are valid for any recourse policy satisfying mild assumptions and are sufficient for formulating the VRPSD under a scenario-optimal recourse policy, where the recourse actions are chosen optimally for each scenario. Under this latter policy, we also show that SRIs dominate several ILS cuts. We conduct computational experiments on the VRPSD with scenarios under both the classical and the scenario-optimal recourse policies. By using the SRIs, our algorithm solves 329 more instances to optimality than the previous state-of-the-art ILS algorithm.

Paper Structure

This paper contains 31 sections, 20 theorems, 56 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Notation.
  3. Problem description and ILS formulations
  4. Input data
  5. Problem description
  6. Recourse disaggregations and feasible regions
  7. ILS cuts
  8. Partial route cuts.
  9. Set cuts.
  10. Recourse policies
  11. Definition via network flows
  12. The convex hull of recourse policies
  13. Scenario-optimal recourse policies and scenario-recourse inequalities
  14. Assigning weights to recourse policies
  15. Formulations under a scenario-optimal recourse policy
  16. ...and 16 more sections

Key Result

Lemma 1

Let $R$ be a route and $\xi \in [N]$. For any vector $y^\xi \in \mathbb{Z}^{V_+}$, we have that $y^\xi \in \mathcal{Y}^\xi(\vec{R})$ if and only if there exist $f^\xi \in \mathbb{R}^{A}_+$ and $g^\xi \in \mathbb{R}^{V_+}_+$ such that In particular, this implies that $\mathcal{Y}^\xi(\vec{R}) = \mathcal{Y}^\xi( { )$.

Figures (3)

  • Figure 1: Illustration for a route with 4 customers and $C = 10$. We show two recourse actions for a given scenario $\xi \in [N]$ and a directed route $\vec{R} = (v_1, v_2, v_3, v_4)$. The black numbers refer to the demands in scenario $\xi$, the blue numbers (solid arcs) indicate the values of $f^\xi$ and the red numbers (dashed arcs) indicate the values of $g^\xi$.
  • Figure 2: Graphical representation of $(\bar{x}, \bar{y}^\xi, \bar{f}^\xi, \bar{g}^\xi)$. Here we have $C = 10$, $w_{v_3} = 6$ and $w_{v_1} = w_{v_2} = w_{v_4} = 2$. The black numbers next to each vertex indicate the scenario demand vector $d^\xi$. The blue and solid arcs represent the vector $\bar{f}^\xi$, while the red and dashed arcs represent the vector $\bar{g}^\xi$.
  • Figure 3: Empirical cumulative distribution of the execution times. The legend (ils+) sri refers to algorithm sri when $\mathcal{Q} = \ref{['definition:scen_opt_policy']}$, and to algorithm ils+sri when $\mathcal{Q} = \ref{['eq:formula_dror']}$.

Theorems & Definitions (35)

  • Example 1
  • Definition 1
  • Example 2
  • Lemma 1
  • Theorem 1: Theorem 3.18 of cook2011combinatorial
  • Lemma 2
  • proof
  • Theorem 2
  • proof
  • Corollary 1
  • ...and 25 more