This paper studies a reliable joint inventory-location problem that optimizes facility locations, customer allocations, and inventory management decisions when facilities are subject to disruption risks (e.g., due to natural or man-made hazards). When a facility fails, its customers may be reassigned to other operational facilities in order to avoid the high penalty costs associated with losing service. The authors propose an integer programming model that minimizes the sum of facility construction costs, expected inventory holding costs and expected customer costs under normal and failure scenarios. The authors develop a Lagrangian relaxation solution framework for this problem, including a polynomial-time exact algorithm for the relaxed nonlinear subproblems. Numerical experiment results show that this proposed model is capable of providing a near-optimum solution within a short computation time. Managerial insights on the optimal facility deployment, inventory control strategies, and the corresponding cost constitutions are drawn.

Joint Inventory-Location Problem Under Risk of Probabilistic Facility Disruptions

Disasters such as fires, earthquakes, and floods cause severe casualties and enormous economic losses. One effective method to reduce these losses is to construct a disaster relief network to deliver disaster supplies as quickly as possible. This method requires solutions to the following problems. 1) Given the established distribution centers, which center(s) should be open after a disaster? 2) Given a set of vehicles, how should these vehicles be assigned to each open distribution center? 3) How can the vehicles be routed from the open distribution center(s) to demand points as efficiently as possible? 4) How many supplies can be delivered to each demand point on the condition that the relief allocation plan is made a priority before the actual demands are realized? This study proposes a model for risk-averse optimization of disaster relief facility location and vehicle routing under stochastic demand that solves the four problems simultaneously. The novel contribution of this study is its presentation of a new model that includes conditional value at risk with regret (CVaR-R)—defined as the expected regret of worst-case scenarios—as a risk measure that considers both the reliability and unreliability aspects of demand variability in the disaster relief facility location and vehicle routing problem. Two objectives are proposed: the CVaR-R of the waiting time and the CVaR-R of the system cost. Due to the nonlinear capacity constraints for vehicles and distribution centers, the proposed problem is formulated as a bi-objective mixed-integer nonlinear programming model and is solved with a hybrid genetic algorithm that integrates a genetic algorithm to determine the satisfactory solution for each demand scenario and a non-dominated sorting genetic algorithm II (NSGA-II) to obtain the non-dominated Pareto solution that considers all demand scenarios. Moreover, the Nash bargaining solution is introduced to capture the decision-maker’s interests of the two objectives. Numerical examples demonstrate the trade-off between the waiting time and system cost and the effects of various parameters, including the confidence level and distance parameter, on the solution. It is found that the Pareto solutions are distributed unevenly on the Pareto frontier due to the difference in the number of the distribution centers opened. The Pareto frontier and Nash bargaining solution change along with the confidence level and distance parameter, respectively.

Risk-averse optimization of disaster relief facility location and vehicle routing under stochastic demand

Wasserstein distributionally robust optimization (DRO) aims to find robust and generalizable solutions by hedging against data perturbations in Wasserstein distance. Despite its recent empirical success in operations research and machine learning, existing performance guarantees for generic loss functions are either overly conservative due to the curse of dimensionality, or plausible only in large sample asymptotics. In this paper, we develop a non-asymptotic framework for analyzing the out-of-sample performance for Wasserstein robust learning and the generalization bound for its related Lipschitz and gradient regularization problems. To the best of our knowledge, this gives the first finite-sample guarantee for generic Wasserstein DRO problems without suffering from the curse of dimensionality. Our results highlight the bias-variation trade-off intrinsic in the Wasserstein DRO, which balances between the empirical mean of the loss and the variation of the loss, measured by the Lipschitz norm or the gradient norm of the loss. Our analysis is based on two novel methodological developments that are of independent interest: 1) a new concentration inequality controlling the decay rate of large deviation probabilities by the variation of the loss and, 2) a localized Rademacher complexity theory based on the variation of the loss.

Finite-Sample Guarantees for Wasserstein Distributionally Robust Optimization: Breaking the Curse of Dimensionality.

This paper proposes a scenario-based three-stage hybrid robust and stochastic model that optimally designs the response network and distributes casualties effectively under uncertain combinational scenarios of primary and secondary disasters. Following the stochastic severity of combinational disasters, the robust counterparts are derived against the ambiguous uncertainty of evacuee scales and transportation time, respectively. A customized progressive hedging algorithm based on the augmented Lagrangian relaxation is developed to solve the problem. We decompose the problem based on the scenario and iteratively solve the adaptively penalized sub-problems with decision variables independent of stages. The results of an illustrative example show that incorporating secondary disaster scenarios can contribute to improving relief coverage. The proposed algorithm is competitive with some benchmarks.

A scenario-based hybrid robust and stochastic approach for joint planning of relief logistics and casualty distribution considering secondary disasters

Tractable reformulations of two-stage distributionally robust linear programs over the type-∞ Wasserstein ball

We consider the risk-averse uncapacitated facility location problem under stochastic disruptions. By the Conditional-value-at-risk, we control the risks at each individual customer, while previous works usually control the entire networks. We show that our model provides more reliable solutions than previous ones. The resulting formulation is a mixed-integer nonlinear programming. In response, we develop a multi-dual decomposition algorithm based on the augmented Lagrangian and classic penalty function. A class of decomposed unconstrained subproblems are then solved by an iterative approach not relying on Lagrange multipliers and differentiability. Our experiments show that the algorithm performs well even for some larger problems.

Multi-dual decomposition solution for risk-averse facility location problem

In a bottleneck combinatorial problem, the objective is to minimize the highest cost of elements of a subset selected from the combinatorial solution space. This paper studies data-driven distributionally robust bottleneck combinatorial problems (DRBCP) with stochastic costs, where the probability distribution of the cost vector is contained in a ball of distributions centered at the empirical distribution specified by the Wasserstein distance. We study two distinct versions of DRBCP from different applications: (i) Motivated by the multi-hop wireless network application, we first study the uncertainty quantification of DRBCP (denoted by DRBCP-U), where decision-makers would like to have an accurate estimation of the worst-case value of DRBCP. The difficulty of DRBCP-U is to handle its max–min–max form. Fortunately, similar to the strong duality of linear programming, the alternative forms of the bottleneck combinatorial problems using clutters and blocking systems allow us to derive equivalent deterministic reformulations, which can be computed via mixed-integer programs. In addition, by drawing the connection between DRBCP-U and its sampling average approximation counterpart under empirical distribution, we show that the Wasserstein radius can be chosen in the order of negative square root of sample size, improving the existing known results; and (ii) Next, motivated by the ride-sharing application, decision-makers choose the best service-and-passenger matching that minimizes the unfairness. That is, we study the decision-making DRBCP, denoted by DRBCP-D. For DRBCP-D, we show that its optimal solution is also optimal to its sampling average approximation counterpart, and the Wasserstein radius can be chosen in a similar order as DRBCP-U. When the sample size is small, we propose to use the optimal value of DRBCP-D to construct an indifferent solution space and propose an alternative decision-robust model, which finds the best indifferent solution to minimize the empirical variance. We further show that the decision robust model can be recast as a mixed-integer conic program. Finally, we extend the proposed models and solution approaches to the distributionally robust $$\varGamma $$
 —sum bottleneck combinatorial problem (
 $$\hbox {DR}\varGamma \hbox {BCP}$$
 ), where decision-makers are interested in minimizing the worst-case sum of $$\varGamma $$
 highest costs of elements.

Distributionally robust bottleneck combinatorial problems: uncertainty quantification and robust decision making

Robust multi-product newsvendor model with uncertain demand and substitution

This paper studies a multiproduct newsvendor problem with customer-driven demand substitution, where each product, once run out of stock, can be proportionally substituted by the others. This probl...

Multiproduct Newsvendor Problem with Customer-Driven Demand Substitution: A Stochastic Integer Program Perspective

This paper studies data-driven distributionally robust bottleneck combinatorial problems (DRBCP) with stochastic costs, where the probability distribution of the cost vector is contained in a ball of distributions centered at the empirical distribution specified by the Wasserstein distance. We study two distinct versions of DRBCP from different applications: (i) Motivated by the multi-hop wireless network application, we first study the uncertainty quantification of DRBCP (denoted by DRBCP-U), where decision-makers would like to have an accurate estimation of the worst-case value of DRBCP. The difficulty of DRBCP-U is to handle its max-min-max form. Fortunately, the alternative forms of the bottleneck combinatorial problems from their blockers allow us to derive equivalent deterministic reformulations, which can be computed via mixed-integer programs. In addition, by drawing the connection between DRBCP-U and its sampling average approximation counterpart under empirical distribution, we show that the Wasserstein radius can be chosen in the order of negative square root of sample size, improving the existing known results; and (ii) Next, motivated by the ride-sharing application, decision-makers choose the best service-and-passenger matching that minimizes the unfairness. This gives rise to the decision-making DRBCP (denoted by DRBCP-D). For DRBCP-D, we show that its optimal solution is also optimal to its sampling average approximation counterpart, and the Wasserstein radius can be chosen in a similar order as DRBCP-U. When the sample size is small, we propose to use the optimal value of DRBCP-D to construct an indifferent solution space and propose an alternative decision-robust model, which finds the best indifferent solution to minimize the empirical variance. We further show that the decision robust model can be recast as a mixed-integer program.

Jie Zhang

Papers

Multi-dual decomposition solution for risk-averse facility location problem

Distributionally robust bottleneck combinatorial problems: uncertainty quantification and robust decision making

Robust multi-product newsvendor model with uncertain demand and substitution

Multiproduct Newsvendor Problem with Customer-Driven Demand Substitution: A Stochastic Integer Program Perspective

Distributionally Robust Bottleneck Combinatorial Problems: Uncertainty Quantification and Robust Decision Making