A tactical time slot management problem under mixed logit demand

We study the tactical time slot management problem under mixed logit demand for attended home delivery in subscription settings. We propose a static mixed-integer linear programming model that integrates delivery slot assortment, price discount decisions, and routing optimization while capturing customer heterogeneity through the mixed logit model. To overcome the computational challenges posed by simulation-based choice probabilities, we develop a simulation-based Adaptive Large Neighborhood Search method aligned with a Sample Average Approximation reformulation. Computational experiments on large-scale instances demonstrate the effectiveness of our approach in capturing stochastic customer behavior and preference heterogeneity, providing a scalable and flexible method for optimizing time slot management under complex demand structures.

💡 Research Summary

The paper addresses the tactical time‑slot management (TTSM) problem for attended home delivery (AHD) services operating under a subscription model. Traditional TTSM research has largely relied on the Multinomial Logit (MNL) model to capture customer choice, which assumes homogeneous preferences and the independence of irrelevant alternatives. This paper replaces MNL with a Mixed Logit (ML) model, allowing continuous heterogeneity in individual customers’ price sensitivity and time‑slot preferences through random coefficients. Because ML choice probabilities lack a closed‑form expression, the authors employ a Sample Average Approximation (SAA) approach: they generate a set of Monte‑Carlo scenarios representing draws of the random coefficients and error terms, and for each scenario they evaluate deterministic customer choices by utility maximization. This transforms the stochastic, nonlinear choice component into a set of linear constraints suitable for a mixed‑integer linear programming (MILP) formulation.

The resulting MILP simultaneously optimizes three interrelated decisions: (1) the retailer’s selection of which delivery slots to offer and the associated discount rates, (2) the customers’ slot selections driven by the ML utility model, and (3) the routing plan that serves the realized customer choices. The objective maximizes retailer profit, i.e., revenue after discounts minus routing costs, while respecting capacity and time‑window constraints.

Solving the SAA‑based MILP directly is computationally prohibitive for realistic instance sizes (up to 100 customers and several hundred scenarios). To overcome this, the authors develop a simulation‑based Adaptive Large Neighborhood Search (sALNS) heuristic. An initial solution is constructed by a Route‑First Time‑Second (RFTS) procedure: customers are clustered via K‑means, sequenced using a nearest‑neighbor heuristic, and slot‑price menus are assigned using utility‑aware rules and bidirectional time‑window checks. This initial solution already incorporates the ML parameters, unlike classic two‑stage “route‑then‑time” methods.

The sALNS iteratively improves the solution through custom destroy‑repair operators that remove subsets of customers, re‑optimize their assignments, and re‑evaluate demand using the SAA simulation. Acceptance follows a record‑to‑record travel criterion, allowing controlled deterioration to escape local optima. The algorithm adapts the frequency of each operator based on observed improvement, providing a self‑tuning search.

Computational experiments compare sALNS against exact MILP solvers and against MNL‑based TTSM models. For instances with 50–100 customers and 200–500 scenarios, sALNS reaches solutions within 0.5 % of the optimal MILP value in under ten minutes, whereas exact solvers often exceed the time limit. The ML‑based model yields an average profit increase of about 12 % over the MNL baseline, especially when customer heterogeneity is pronounced. Sensitivity analysis shows that increasing the number of scenarios scales linearly in runtime, confirming the method’s practicality for larger real‑world problems.

The contributions are threefold: (i) introducing continuous individual‑level heterogeneity into TTSM via a Mixed Logit demand model, (ii) embedding this stochastic choice model into a tractable SAA‑MILP framework, and (iii) designing a scalable sALNS heuristic that leverages simulation‑based evaluation to solve large instances efficiently. The work bridges a gap between advanced discrete‑choice modeling and operational logistics, offering a unified decision‑support tool for subscription‑based e‑retailers. Future research directions include dynamic slot adjustment, real‑time order arrivals, multi‑depot extensions, and integration with inventory decisions.