Distributionally robust facility location with bimodal random demand

In this paper, we study a facility location problem in which customer demand is bimodal, i.e., display, or belong to, two spatially distinct distributions. We assume that these two distributions are ambiguous (unknown), and only their mean values and ranges are known.Therefore, we propose a distributionally robust facility location (DRFL) problem that seeks to find a subset of locations from a given set of candidate sites to open facilities to minimize the fixed cost of openingfacilities, and worst-case (maximum) expected costs of transportation andunmet demand over a family of distributions characterized through the known means and support of these distributions. We propose adecomposition-based algorithm to solve DRFL, which include valid lowerbound inequalities to accelerate the convergence of the algorithm. In aseries of numerical experiments, we demonstrate the superior computational and operational performance of our approach as compared with the stochastic programming approach and a DR approach that does notconsider bimodality of the demand. Our results draw attention to the needto consider the impact of uncertainty of customer demand when it does notfollow one distinct and known distribution in many strategic real-world problems

Stochastic Optimization Approaches for Outpatient Appointment Scheduling under Uncertainty

Outpatient clinics (OPCs) are quickly growing as a central component of the healthcare system. OPCs offer a variety of medical services, with benefits such as avoiding inpatient hospitalization, improving patient safety, and reducing costs of care. However, they also introduce new challenges for appointment planning and scheduling, primarily due to the heterogeneity and variability in patient characteristics, multiple competing performance criteria, and the need to deliver care within a tight time window. Ignoring uncertainty, especially when designing appointment schedules, may have adverse outcomes such as patient delays and clinic overtime. Conversely, accounting for uncertainty when scheduling has the potential to create more efficient schedules that mitigate these adverse outcomes. However, many challenges arise when attempting to account for uncertainty in appointment scheduling problems. In this dissertation, we propose new stochastic optimization models and approaches to address some of these challenges. Specifically, we study three stochastic outpatient scheduling problems with broader applications within and outside of healthcare and propose models and methods for solving them. We first consider the problem of sequencing a set of outpatient procedures for a single provider (where each procedure has a known type and a random duration that follows a known probability distribution), minimizing a weighted sum of waiting, idle time, and overtime. We elaborate on the challenges of solving this complex stochastic, combinatorial, and multi-criteria optimization problem and propose a new stochastic mixed-integer programming model that overcomes these challenges in contrast to the existing models in the literature. In doing so, we show the art of, and the practical need for, good mathematical formulations in solving real-world scheduling problems. Second, we study a stochastic adaptive outpatient scheduling problem which incorporates the patients’ random arrival and service times. Finding a provably-optimal solution to this problem requires solving a MSMIP, which in turn must optimize a scheduling problem over each random arrival and service time for each stage. Given that this MSMIP is intractable, we present two approximation based on two-stage stochastic mixed-integer models and a Monte Carlo Optimization approach. In a series of numerical experiments, we demonstrate the near-optimality of the appointment order (AO) rescheduling policy, which requires that patients are served in the order of their scheduled appointments, in many parameter settings. We also identify parameter settings under which the AO policy is suboptimal. Accordingly, we propose an alternative swap-based policy that improves the solution of such instances. Finally, we consider the outpatient colonoscopy scheduling problem, recognizing the impact of pre-procedure bowel preparation (prep) quality on the variability of colonoscopy duration. Data from a large OPC indicates that colonoscopy durations are bimodal, i.e., depending on the prep quality they can follow two different probability distributions, one for those with adequate prep and the other for those with inadequate prep. We define a distributionally robust outpatient colonoscopy scheduling (DRCOS) problem that seeks optimal appointment sequence and schedule to minimize the worst-case weighted expected sum of patient waiting, provider idling, and provider overtime, where the worst-case is taken over an ambiguity set characterized through the known mean and support of the prep quality and durations. We derive an equivalent mixed-integer linear programming formulation to solve DRCOS. Finally, we present a case study based on extensive numerical experiments in which we draw several managerial insights into colonoscopy scheduling.PHDIndustrial & Operations EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/151727/1/ksheha_1.pdfDescription of ksheha_1.pdf : Restricted to UM users only

Convex Fairness Measures: Theory and Optimization

We propose a new parameterized class of fairness measures, convex fairness measures, suitable for optimization contexts. This class includes our new proposed order-based fairness measure and several popular measures (e.g., deviation-based measures, Gini deviation). We provide theoretical analyses and derive a dual representation of these measures. Importantly, this dual representation renders a unified mathematical expression and a geometric characterization for convex fairness measures through their dual sets. Moreover, we propose a generic framework for optimization problems with a convex fairness measure objective, including reformulations and solution methods. Finally, we provide a stability analysis on the choice of convex fairness measures in the objective of optimization models

Stochastic Programming and Distributionally Robust Optimization Approaches for Location and Inventory Prepositioning of Disaster Relief Supplies

In this paper, we study the problem of disaster relief inventory prepositioning under uncertainty. Specifically, we aim to determine where to open warehouses and how much relief item inventory to preposition in each, pre-disaster. During the post-disaster phase, prepositioned items are distributed to demand nodes, and additional items are procured and distributed as needed. There is uncertainty in the (1) disaster level, (2) locations of affected areas, (3) demand of relief items, (4) usable fraction of prepositioned items post-disaster, (5) procurement quantity, and (6) arc capacity. We propose and analyze two-stage stochastic programming (SP) and distributionally robust optimization (DRO) models, assuming known and unknown uncertainty distributions, respectively. The first and second stages correspond to pre- and post-disaster phases, respectively. We propose a Monte Carlo Optimization procedure to solve the SP and a decomposition algorithm to solve the DRO model. To illustrate potential applications of our approaches, we conduct extensive experiments using a hurricane season and an earthquake as case studies. Our results demonstrate the (1) the robustness and superior post-disaster operational performance of the DRO decisions under various distributions compared to SP decisions, especially under misspecified distributions and high variability, (2) the trade-off between considering distributional ambiguity and following distributional belief, and (3) computational efficiency of our approaches

Stochastic Optimization Approaches for an Operating Room and Anesthesiologist Scheduling Problem

We propose combined allocation, assignment, sequencing, and scheduling problems under uncertainty involving multiple operation rooms (ORs), anesthesiologists, and surgeries, as well as methodologies for solving such problems. Specifically, given sets of ORs, regular anesthesiologists, on-call anesthesiologists, and surgeries, our methodologies solve the following decision-making problems simultaneously: (1) an allocation problem that decides which ORs to open and which on-call anesthesiologists to call in, (2) an assignment problem that assigns an OR and an anesthesiologist to each surgery, and (3) a sequencing and scheduling problem that determines the order of surgeries and their scheduled start times in each OR. To address uncertainty of each surgery's duration, we propose and analyze stochastic programming (SP) and distributionally robust optimization (DRO) models with both risk-neutral and risk-averse objectives. We obtain near-optimal solutions of our SP models using sample average approximation and propose a computationally efficient column-and-constraint generation method to solve our DRO models. In addition, we derive symmetry-breaking constraints that improve the models' solvability. Using real-world, publicly available surgery data and a case study from a health system in New York, we conduct extensive computational experiments comparing the proposed methodologies empirically and theoretically, demonstrating where significant performance improvements can be gained. Additionally, we derive several managerial insights relevant to practice

Distributionally robust facility location with bimodal random demand

In this paper, we study a facility location problem in which customer demand is bimodal, i.e., display, or belong to, two spatially distinct distributions. We assume that these two distributions are ambiguous (unknown), and only their mean values and ranges are known.Therefore, we propose a distributionally robust facility location (DRFL) problem that seeks to find a subset of locations from a given set of candidate sites to open facilities to minimize the fixed cost of openingfacilities, and worst-case (maximum) expected costs of transportation andunmet demand over a family of distributions characterized through the known means and support of these distributions. We propose adecomposition-based algorithm to solve DRFL, which include valid lowerbound inequalities to accelerate the convergence of the algorithm. In aseries of numerical experiments, we demonstrate the superior computational and operational performance of our approach as compared with the stochastic programming approach and a DR approach that does notconsider bimodality of the demand. Our results draw attention to the needto consider the impact of uncertainty of customer demand when it does notfollow one distinct and known distribution in many strategic real-world problems

Fleet sizing and allocation for on-demand last-mile transportation systems

The last-mile problem refers to the provision of travel service from the nearest public transportation node to home or other destination. Last-Mile Transportation Systems (LMTS), which have recently emerged, provide on-demand shared transportation. In this paper, we investigate the fleet sizing and allocation problem for the on-demand LMTS. Specifically, we consider the perspective of a last-mile service provider who wants to determine the number of servicing vehicles to allocate to multiple last-mile service regions in a particular city. In each service region, passengers demanding last-mile services arrive in batches, and allocated vehicles deliver passengers to their final destinations. The passenger demand (i.e., the size of each batch of passengers) is random and hard to predict in advance, especially with limited data during the planning process. The quality of fleet allocation decisions is a function of vehicle fixed cost plus a weighted sum of passenger's waiting time before boarding a vehicle and in-vehicle riding time. We propose and analyze two models-a stochastic programming model and a distributionally robust optimization model-to solve the problem, assuming a known and unknown distribution of the demand, respectively. We conduct extensive numerical experiments to evaluate the models and discuss insights and implications into the optimal fleet sizing and allocation for the on-demand LMTS under demand uncertainty

Using stochastic programming to solve an outpatient appointment scheduling problem with random service and arrival times

We study a stochastic outpatient appointment scheduling problem (SOASP) in which we need to design a schedule and an adaptive rescheduling (i.e., resequencing or declining) policy for a set of patients. Each patient has a known type and associated probability distributions of random service duration and random arrival time. Finding a provably optimal solution to this problem requires solving a multistage stochastic mixed‐integer program (MSMIP) with a schedule optimization problem solved at each stage, determining the optimal rescheduling policy over the various random service durations and arrival times. In recognition that this MSMIP is intractable, we first consider a two‐stage model (TSM) that relaxes the nonanticipativity constraints of MSMIP and so yields a lower bound. Second, we derive a set of valid inequalities to strengthen and improve the solvability of the TSM formulation. Third, we obtain an upper bound for the MSMIP by solving the TSM under the feasible (and easily implementable) appointment order (AO) policy, which requires that patients are served in the order of their scheduled appointments, independent of their actual arrival times. Fourth, we propose a Monte Carlo approach to evaluate the relative gap between the MSMIP upper and lower bounds. Finally, in a series of numerical experiments, we show that these two bounds are very close in a wide range of SOASP instances, demonstrating the near‐optimality of the AO policy. We also identify parameter settings that result in a large gap in between these two bounds. Accordingly, we propose an alternative policy based on neighbor‐swapping. We demonstrate that this alternative policy leads to a much tighter upper bound and significantly shrinks the gap.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/163877/1/nav21933_am.pdfhttp://deepblue.lib.umich.edu/bitstream/2027.42/163877/2/nav21933.pdfhttp://deepblue.lib.umich.edu/bitstream/2027.42/163877/3/nav21933-sup-0001-supinfo.pd

An inexact column-and-constraint generation method to solve two-stage robust optimization problems

We propose a new inexact column-and-constraint generation (i-C&CG) method to solve two-stage robust optimization problems. The method allows solutions to the master problems to be inexact, which is desirable when solving large-scale and/or challenging problems. It is equipped with a backtracking routine that controls the trade-off between bound improvement and inexactness. Importantly, this routine allows us to derive theoretical finite convergence guarantees for our i-C\CG method. Numerical experiments demonstrate computational advantages of our i-C&CG method over state-of-the-art column-and-constraint generation methods