Distributionally robust facility location with uncertain facility capacity and customer demand

Reliable Facility Location Under Event-Correlated Uncertainty

Problem definition: In this paper, we study the fleet repositioning problem for a free-float vehicle sharing system, aiming to dynamically match the vehicle supply and travel demand at the lowest total cost of repositioning and lost sales. Academic/practical relevance: Besides the analytical results on the optimal repositioning policy, the proposed optimization framework is applicable to practical problems by its computational efficiency as well as the capability to handle temporally dependent demands. Methodology: We first formulate the problem as a stochastic dynamic program. To solve for a multiregion system, we deploy the distributionally robust optimization (DRO) approach that can incorporate demand temporal dependence, motivated by real data. We first propose a “myopic” two-stage DRO model that serves as both an illustration of the DRO framework and a benchmark for the later multistage model. We then develop a computationally efficient multistage DRO model with an enhanced linear decision rule (ELDR). Results: Under a two-region system, we find a simple reposition up-to and down-to policy to be optimal, when the demands are temporally independent. Such a structure is also preserved by our ELDR solution. We also provide new analytical insights by proving the optimality of ELDR in solving the single-period DRO problem. We then show that the numerical performance of the ELDR solution is close to the exact optimal solution from the dynamic program. Managerial implications: In a real-world case study of car2go, we quantify the “value of repositioning” and compare with several benchmarks to demonstrate that the ELDR solutions are computationally scalable and in general result in lower cost with less frequent repositioning. We also explore several managerial implications and extensions from the experiments.

Distributionally robust optimization for collaborative emergency response network design

This paper considers facility location, inventory pre‐positioning and vehicle routing as strategic and operational decisions corresponding to preparedness and response phases in disaster relief logistics planning. For balancing surpluses and shortages, an effective lateral transshipment strategy is proposed to evenly distribute the relief resources between warehouses after the disaster occurs. To handle ambiguity in the probability distribution of demand, we develop a risk‐averse two‐stage distributionally robust optimization (DRO) model for the disaster relief logistics planning problem, which specifies the worst‐case mean‐conditional value‐at‐risk (CVaR) as a risk measure. For computationally tractability, we transform the robust counterpart into its equivalent linear mixed‐integer programming model under the discrepancy‐based ambiguity set centered at the nominal (empirical) distributions on the observed demand from the historical data. We verify the effectiveness of the proposed DRO model and the value of lateral transshipment strategy by an illustrative small‐scale example. The numerical results show that the proposed DRO model has advantage on avoiding over‐conservative solutions compared to the classic robust optimization model. We also illustrate the applicability of the proposed DRO model by a real‐world case study of hurricanes in the southeastern United States. The computational results demonstrate that the proposed DRO model has superior out‐of‐sample performance and can mitigate the adverse effects of Optimizers' Curse compared with the traditional stochastic programming model.

Distributionally robust optimization (DRO) has been introduced for solving stochastic programs in which the distribution of the random variables is unknown and must be estimated by samples from that distribution. A key element of DRO is the construction of the ambiguity set, which is a set of distributions that contains the true distribution with a high probability. Assuming that the true distribution has a probability density function, we propose a class of ambiguity sets based on confidence bands of the true density function. As examples, we consider the shape-restricted confidence bands and the confidence bands constructed with a kernel density estimation technique. The former allows us to incorporate the prior knowledge of the shape of the underlying density function (e.g., unimodality and monotonicity), and the latter enables us to handle multidimensional cases. Furthermore, we establish the convergence of the optimal value of DRO to that of the underlying stochastic program as the sample size increases. The DRO with our ambiguity set involves functional decision variables and infinitely many constraints. To address this challenge, we apply duality theory to reformulate the DRO to a finite-dimensional stochastic program, which is amenable to a stochastic subgradient scheme as a solution method.

Quay cranes are among the most important resources in port terminals, and their efficient use is crucial for port terminals to remain competitive in the market. The problem of determining the sequence of tasks performed by the quay cranes that minimizes the turnaround time of a vessel is an NP‐hard problem known as the quay crane scheduling problem (QCSP).In this paper, we consider the unidirectional QCSP under uncertain processing times, where the cranes are only allowed to move in a specific direction. We start by presenting the first distributionally robust optimization (DRO) model for this problem and an exact decomposition algorithm to solve it. The proposed DRO model makes it possible to derive a stochastic programming model and a robust optimization model for the unidirectional QCSP by an appropriate choice of the risk‐averse parameter of the model. Through extensive numerical results, we compare these three approaches—stochastic programming, robust optimization, and DRO—to investigate whether significant differences among the solutions obtained exist. Finally, we propose a new method for helping practitioners to determine a representative set of different DRO solutions.

TimeGAN based distributionally robust optimization for biomass-photovoltaic-hydrogen scheduling under source-load-market uncertainties

Coordinated operations of electricity and district heating networks offer a potential for mitigating inherent variability of renewable energy sources (RES) in the ongoing transition to smart grids. This paper proposes a two-stage distributionally robust optimization (DRO) approach to determine the optimal day-ahead unit commitment in coordinated electricity and district heating networks with variable RES power output. The proposed formulation is to minimize the worst-case expected total cost over an ambiguity set comprising a family of probability distributions with given support and moments of RES power output. As such, the proposed DRO approach can overcome the limitations of stochastic programming in its inherent dependence of exact probability distributions along with a huge computational burden, but also becomes less conservative than classical robust optimization. The pertinent DRO model is eventually reformulated as a tractable mixed-integer second-order cone (SOC) programming after employing linear decision rules and the SOC duality. Simplified affine policies are utilized to further improve computational tractability and performance. Finally, case studies are conducted based on Barry Island electricity and district heating networks. The numerical results demonstrate the decision-making superiority of the proposed method as compared with deterministic, stochastic programming, and robust optimization approaches. They also validate the computational improvement of the proposed approach by employing simplified affine policies.

Randomized decision making refers to the process of making decisions randomly according to the outcome of an independent randomization device, such as a dice roll or a coin flip. The concept is unconventional, and somehow counterintuitive, in the domain of mathematical programming, in which deterministic decisions are usually sought even when the problem parameters are uncertain. However, it has recently been shown that using a randomized, rather than a deterministic, strategy in nonconvex distributionally robust optimization (DRO) problems can lead to improvements in their objective values. It is still unknown, though, what is the magnitude of improvement that can be attained through randomization or how to numerically find the optimal randomized strategy. In this paper, we study the value of randomization in mixed-integer DRO problems and show that it is bounded by the improvement achievable through its continuous relaxation. Furthermore, we identify conditions under which the bound is tight. We then develop algorithmic procedures, based on column generation, for solving both single- and two-stage linear DRO problems with randomization that can be used with both moment-based and Wasserstein ambiguity sets. Finally, we apply the proposed algorithm to solve three classical discrete DRO problems: the assignment problem, the uncapacitated facility location problem, and the capacitated facility location problem and report numerical results that show the quality of our bounds, the computational efficiency of the proposed solution method, and the magnitude of performance improvement achieved by randomized decisions. Summary of Contribution: In this paper, we present both theoretical results and algorithmic tools to identify optimal randomized strategies for discrete distributionally robust optimization (DRO) problems and evaluate the performance improvements that can be achieved when using them rather than classical deterministic strategies. On the theory side, we provide improvement bounds based on continuous relaxation and identify the conditions under which these bound are tight. On the algorithmic side, we propose a finitely convergent, two-layer, column-generation algorithm that iterates between identifying feasible solutions and finding extreme realizations of the uncertain parameter. The proposed algorithm was implemented to solve distributionally robust stochastic versions of three classical optimization problems and extensive numerical results are reported. The paper extends a previous, purely theoretical work of the first author on the idea of randomized strategies in nonconvex DRO problems by providing useful bounds and algorithms to solve this kind of problems.

A risk-averse distributionally robust optimisation approach for drone-supported relief facility location problem

Decomposition methods have been well studied for solving two-stage and multi-stage stochastic programming problems, see Rockafellar and Wets (Math. Oper. Res. 16:119–147, 1991), Ruszczynski and Shapiro (Stochastic Programming, Handbook in OR & MS, North-Holland Publishing Company, Amsterdam, 2003) and Ruszczynski (Math. Program. 79:333–353, 1997). In this paper, we propose an algorithmic framework based on the fundamental ideas of the methods for solving two-stage minimax distributionally robust optimization (DRO) problems where the underlying random variables take a finite number of distinct values. This is achieved by introducing nonanticipativity constraints for the first stage decision variables, rearranging the minimax problem through Lagrange decomposition and applying the well-known primal-dual hybrid gradient (PDHG) method to the new minimax problem. The algorithmic framework does not depend on specific structure of the ambiguity set. To extend the algorithm to the case that the underlying random variables are continuously distributed, we propose a discretization scheme and quantify the error arising from the discretization in terms of the optimal value and the optimal solutions when the ambiguity set is constructed through generalized prior moment conditions, the Kantorovich ball and $$\phi$$ -divergence centred at an empirical probability distribution. Some preliminary numerical tests show the proposed decomposition algorithm featured with parallel computing performs well.

Distributionally robust optimization for dispatching the integrated operation of rural multi-energy system and waste disposal facilities under uncertainties

With the large-scale displacement of fossil fuel-tired units due to the increasing penetration of renewable generation, the generation expansion planning (GEP) problem has been facing a major challenge to deal with the renewable power uncertainty. To handle this issue, this paper introduces a distributionally robust GEP model which explicitly considering the large-scale displacement of fossil fuel-fired units. In the first stage, the long-period generation expansion planning takes into account the expectation of the worst case; While, in the second stage, the short-term operation is modeled based on the distributionally robust optimization (DRO). The wind power uncertainties are treated as ambiguity sets using the Wasserstein distance technique. Aiming to ease the computational complexity of the proposed model, Benders decomposition with a reformulation technique to deal with the worst-case constraints is introduced. Case studies on a modified IEEE RTS-79 system verify the effectiveness of the proposed approach.

In this article, a novel distributionally robust optimization (DRO) based multistage model predictive control (MPC) framework is proposed to hedge against the uncertainty in control problems. Without a priori knowledge of the exact uncertainty distribution, an ambiguity set, constructed based on principal component analysis, incorporates the first-order moment information instead. A linear performance measure is chosen so that the worst-case expected problem can be exactly dualized by adopting the affine decision rule. By considering input and state constraints robustly with respect to a support set, which specifies the domain of the uncertainty, the DRO-based MPC model is developed as a robust optimization problem. The proposed framework is illustrated on a two-mass-spring system and results show the improved control performance compared to traditional control strategies.

Vehicle routing, with its many variants, is one of the most important and frequently solved problems in transportation engineering. The aim of this paper is to develop a decision-making support tool for addressing the issue of dispatching vehicles in scenarios characterized by uncertain demands within soft time windows. In real-world scenarios, it is not uncommon for customer demands to exhibit flexibility, where certain early arrivals or delays may be deemed acceptable. Therefore, this paper introduces vehicle routing in more realistic contexts, offering potential practical implementations. The methodology for solving the problem is based on a fuzzy logic system whose membership functions are additionally adjusted using a neural network. Such a tool, neuro-fuzzy logic, is suitable for solving a defined routing problem since it can consider all the mentioned uncertainties in the distribution systems. Each user is assigned a performance index that considers travel time, demand, and delivery time windows. Then, the performance index is used as input data in the proposed vehicle routing tool based on the Clarke-Wright algorithm. The described approach has been tested on a concrete example, mimicking a distribution network resembling real world conditions, incorporating estimated travel times between customers. The results demonstrate that the proposed approach can effectively handle customer demands, with an average delay of 5.05 minutes during the 80-minute distribution. In future research, some environmental factors could be included in the proposed model. In addition, one of the directions of future research could be vehicle re-routing using the ideas from this paper.