Logic-Based Benders Decomposition for Time Slot Management with Mixed Logit Demand

TL;DR

This paper introduces a logic-based Benders decomposition approach to solve complex, integrated time slot management problems with customer choice modeling and routing, significantly improving solvability of large-scale instances.

Contribution

It develops a novel LBBD framework with problem-specific cuts and strengthening strategies for choice-based, routing-integrated optimization under mixed logit demand.

Findings

Successfully solves instances with up to 10 customers to proven optimality.

Achieves tight optimality gaps for instances with 15-20 customers.

Provides meaningful upper bounds for larger, more complex instances.

Abstract

This paper develops an exact solution framework for the choice-based time slot management problem under mixed logit demand in attended home delivery systems. The problem jointly optimizes delivery slot offerings, price discounts, and routing decisions, with customer choices endogenously modeled through a simulation-based mixed logit formulation embedded via sample average approximation, resulting in a large-scale stochastic mixed-integer program. To address this complexity, we propose a logic-based Benders decomposition (LBBD) that separates strategic assortment and pricing decisions, together with customer choice, from scenario-specific vehicle routing subproblems. We derive problem-specific optimality cuts that exploit the routing structure to provide stronger bounds than generic cuts, and establish their validity. To enhance computational performance, we introduce and systematically evaluate several strengthening strategies, including relaxation-based cut generation and capacity- and flow-based valid inequalities. Computational experiments on benchmark instances show that the proposed framework significantly extends the range of solvable instances compared to direct MILP approaches. The method yields proven optimal solutions for instances with up to 10 customers and consistently tight optimality gaps for instances with 15-20 customers. For larger instances, the approach provides meaningful upper bounds, while remaining computationally challenging for larger problem sizes. Overall, the results highlight the interaction between stochastic choice modeling, routing complexity, and decomposition design, and demonstrate the potential of LBBD for solving integrated choice-based optimization problems.