Distributionally Robust Optimization Approaches for a Stochastic Mobile Facility Fleet Sizing, Routing, and Scheduling Problem

Abstract

We propose two distributionally robust optimization (DRO) models for a mobile facility (MF) fleet-sizing, routing, and scheduling problem (MFRSP) with time-dependent and random demand as well as methodologies for solving these models. Specifically, given a set of MFs, a planning horizon, and a service region, our models aim to find the number of MFs to use (i.e., fleet size) within the planning horizon and a route and time schedule for each MF in the fleet. The objective is to minimize the fixed cost of establishing the MF fleet plus a risk measure (expectation or mean conditional value at risk) of the operational cost over all demand distributions defined by an ambiguity set. In the first model, we use an ambiguity set based on the demand’s mean, support, and mean absolute deviation. In the second model, we use an ambiguity set that incorporates all distributions within a 1-Wasserstein distance from a reference distribution. To solve the proposed DRO models, we propose a decomposition-based algorithm. In addition, we derive valid lower bound inequalities that efficiently strengthen the master problem in the decomposition algorithm, thus improving convergence. We also derive two families of symmetry-breaking constraints that improve the solvability of the proposed models. Finally, we present extensive computational experiments comparing the operational and computational performance of the proposed models and a stochastic programming model, demonstrating when significant performance improvements could be gained, and derive insights into the MFRSP.

A two-stage stochastic programming for the integrated emergency mobility facility allocation and road network design under uncertainty

“…In light of the assumption that stochastic programming necessitates knowledge of the probability distribution of demand and recognizing the inherent conservatism in robust optimization results, recent developments, building upon the groundwork laid by Lei et al (2014Lei et al ( , 2016, have emerged. Shehadeh (2023) introduced a distributionally robust optimization model for addressing the MF allocation problem under stochastic demand. Significantly, Shehadeh (2023) formulated two distinct distributionally robust optimization models, distinguished by the ambiguity sets utilized.…”

“…Shehadeh (2023) introduced a distributionally robust optimization model for addressing the MF allocation problem under stochastic demand. Significantly, Shehadeh (2023) formulated two distinct distributionally robust optimization models, distinguished by the ambiguity sets utilized. One model incorporates the demand's mean, support, and mean absolute deviation, while the other leverages a 1-Wasserstein distance from a reference distribution.…”

“…The dynamics of EMF operations involve travel from one location to another, with nearby disaster-affected individuals traveling to the stationary EMF to receive emergency services. Unlike the simplified assumption of fixed travel times in existing studies on MFs (Halper and Raghavan, 2011;Lei et al, 2014Lei et al, , 2016Li et al, 2021;Shehadeh, 2023), the reality for EMFs involves stochastic travel times influenced by factors such as traffic congestion during peak hours and varying degrees of damage to the road network caused by different disaster sizes. During peak hours, the road network tends to be congested, while off-peak hours may experience less congestion.…”

“…Secondly, the applications of EMFs differ from those of conventional commercial MFs. While commercial MF applications involve determining allocations, including fleet sizing, routing, and scheduling problems (Shehadeh, 2023), EMF applications go beyond mere allocation decisions. In the realm of EMFs, apart from determining the allocation of EMFs, as in conventional MFs, decision-makers can incorporate corresponding traffic management measures to reduce travel time for both EMFs and individuals traveling to EMFs.…”

“…In times of emergencies, such as the recent COVID-19 pandemic, mobile clinics have played pivotal roles by swiftly establishing drive-through testing sites and triage locations. The mobile health clinic market, valued at nearly $2 billion in 2019, is projected to exceed $12 billion by 2028, underscoring the escalating significance of MFs in the healthcare sector (Shehadeh, 2023). Furthermore, in emergency resource allocation scenarios, vans laden with essential supplies, including water, food, and clothing, can mobilize to afflicted areas, offering timely aid to affected populations.…”