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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.

We study a mobile facility routing and scheduling problem with stochastic demand. The probability distribution of demand is assumed ambiguous, and only the mean and range are known. Therefore, we define a distributionally robust mobile facility routing and scheduling (DMFRS) problem that seeks optimal routing and scheduling decisions for a fleet of mobile facilities to minimize the fixed operation costs and worstcase expected cost generated during the planning horizon. We take the worst-case over an ambiguity set characterized through the known mean and range of random demand. We propose a decomposition-based algorithm to solve DMFRS and derive lower bound and symmetry breaking inequalities to strengthen the master problem and speed up the convergence of the algorithm. Our computational results demonstrate a superior computational and operational performance of our DR approach over the stochastic programming approach.

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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.

We study elective surgery planning in flexible operating rooms where emergency patients are accommodated in the existing elective surgery schedule. Probability distributions of surgeries durations are unknown, and only a small set of historical realizations is available. To address distributional ambiguity, we first construct an ambiguity set that encompasses all possible distributions of surgery duration within a Wasserstein distance from the empirical distribution. We then define a data-driven distributionally robust surgery assignment (DSA) problem, which seeks to determine optimal elective surgery assigning decisions to available surgical blocks in multiple ORs to minimize the sum of patient-related costs and the expectation of OR overtime and idle time costs over all distributions residing in the ambiguity set. Using DSA structural properties, we derive an equivalent mixed-integer linear programming (MILP) reformulation of the min-max DSA model, that can be solved efficiently using off-the-shelf optimization software. Using real-world surgery data, we conduct extensive numerical experiments comparing the operational and computational performance of our approach with two state-of-the-art approaches.