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A Set Cover Mapping Heuristic for Demand-Robust Fleet Size Vehicle Routing Problem with Time Windows and Compatibility Constraints

Jordan Makansi

TL;DR

The paper tackles the demand-robust fleet size problem with time windows and compatibility constraints (DRFSP) by formulating a two-stage decision framework that minimizes the worst-case cost across finite scenarios. It introduces a three-phase Set Cover Mapping (SCM) heuristic: Generate Routes via an insertion-based Fleet Size with Time Windows solver, Construct Route Sets through greedy coverage, and solve a Demand-Robust Weighted Set Cover to select routes under a max-scenario objective. The authors prove an exponential gap in worst-case time between the heuristic and a direct MILP solved by branch-and-bound, and demonstrate empirically that SCM achieves a sub-2.0 approximation ratio on Solomon benchmark instances while significantly reducing computation time. These results suggest a practical, scalable method for large-scale fleet sizing under uncertain routing constraints, with potential applicability to other demand-robust routing problems. The work also frames avenues for future research, including approximation guarantees and comparisons with constraint programming approaches.

Abstract

We study the demand-robust fleet size vehicle routing problem with time windows and compatibility constraints. Unlike traditional robust optimization, which considers uncertainty in the data, demand-robust optimization considers uncertainty in which constraints must be satisfied. This paper is the first to solve a practical demand-robust optimization problem at large scale. We present an MILP formulation and also propose a heuristic that maps the problem to set cover in polynomial time. We show that under modest assumptions the relative difference in time complexity from a standard branch-and-bound algorithm to the proposed heuristic scales exponentially with the size of the problem. We evaluate our heuristic using a simulation case study on the Solomon benchmark instances for a variety of practical problem sizes, and compare with Gurobi. The empirical approximation ratio remains below 2.0.

A Set Cover Mapping Heuristic for Demand-Robust Fleet Size Vehicle Routing Problem with Time Windows and Compatibility Constraints

TL;DR

The paper tackles the demand-robust fleet size problem with time windows and compatibility constraints (DRFSP) by formulating a two-stage decision framework that minimizes the worst-case cost across finite scenarios. It introduces a three-phase Set Cover Mapping (SCM) heuristic: Generate Routes via an insertion-based Fleet Size with Time Windows solver, Construct Route Sets through greedy coverage, and solve a Demand-Robust Weighted Set Cover to select routes under a max-scenario objective. The authors prove an exponential gap in worst-case time between the heuristic and a direct MILP solved by branch-and-bound, and demonstrate empirically that SCM achieves a sub-2.0 approximation ratio on Solomon benchmark instances while significantly reducing computation time. These results suggest a practical, scalable method for large-scale fleet sizing under uncertain routing constraints, with potential applicability to other demand-robust routing problems. The work also frames avenues for future research, including approximation guarantees and comparisons with constraint programming approaches.

Abstract

We study the demand-robust fleet size vehicle routing problem with time windows and compatibility constraints. Unlike traditional robust optimization, which considers uncertainty in the data, demand-robust optimization considers uncertainty in which constraints must be satisfied. This paper is the first to solve a practical demand-robust optimization problem at large scale. We present an MILP formulation and also propose a heuristic that maps the problem to set cover in polynomial time. We show that under modest assumptions the relative difference in time complexity from a standard branch-and-bound algorithm to the proposed heuristic scales exponentially with the size of the problem. We evaluate our heuristic using a simulation case study on the Solomon benchmark instances for a variety of practical problem sizes, and compare with Gurobi. The empirical approximation ratio remains below 2.0.
Paper Structure (9 sections, 2 theorems, 3 equations, 6 tables, 2 algorithms)

This paper contains 9 sections, 2 theorems, 3 equations, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. 3-Phase Set Cover Mapping (SCM) Heuristic Algorithm
  4. Generate Routes
  5. Construct Route Sets
  6. Demand-Robust Weighted Set Cover
  7. Time Complexity Analysis
  8. Computational Results
  9. Conclusions and Future Work

Key Result

lemma 1

The asymptotic time complexities of SCM and MIP are $O(W^3 T m + WT + 2^{WT(m+1)})$, and $O(2^{2mTPW^2 + TP})$, respectively.

Theorems & Definitions (10)

  • remark 1
  • definition 1: route
  • definition 2: route set
  • remark 2
  • lemma 1
  • proof
  • lemma 2
  • proof
  • remark 3
  • remark 4