In this work, we present a novel approach to topology optimization for robust truss structure design. Trusses are two- or three-dimensional mechanical structures consisting of a collection of nodes connected by bars, typically made of linear elastic, isotropic, and homogeneous materials. These structures are engineered to withstand external loads, which often include both stochastic and deterministic forces applied to the nodes.

Our method considers both deterministic and random loadings, aiming to minimize compliance (a measure of structural energy) while adhering to constraints on volume and failure probability. Managing failure probability is often computationally demanding, but we address it using the Bernstein approximation, resulting in a model that is a linear conic programming problem. Additionally, we present a more efficient problem reformulation with small semidefinite constraints. To demonstrate the practicality of our approach, we provide solutions to several examples of truss topology optimization.

Moment-based distributionally robust optimization (DRO) provides an optimization framework to integrate statistical information with traditional optimization approaches. Under this framework, one assumes that the underlying joint distribution of random parameters runs in a distributional ambiguity set constructed by moment information and makes decisions against the worst-case distribution within the set. Although most moment-based DRO problems can be reformulated as semidefinite programming (SDP) problems that can be solved in polynomial time, solving high-dimensional SDPs is still time-consuming. Unlike existing approximation approaches that first reduce the dimensionality of random parameters and then solve the approximated SDPs, we propose an optimized dimensionality reduction (ODR) approach. We first show that the ranks of the matrices in the SDP reformulations are small, by which we are then motivated to integrate the dimensionality reduction of random parameters with the subsequent optimization problems. Such integration enables two outer and one inner approximations of the original problem, all of which are low-dimensional SDPs that can be solved efficiently, providing two lower bounds and one upper bound correspondingly. More importantly, these approximations can theoretically achieve the optimal value of the original high-dimensional SDPs. As these approximations are nonconvex SDPs, we develop modified Alternating Direction Method of Multipliers (ADMM) algorithms to solve them efficiently. We demonstrate the effectiveness of our proposed ODR approach and algorithm in solving multiproduct newsvendor and production-transportation problems. Numerical results show significant advantages of our approach on the computational time and solution quality over the three best possible benchmark approaches. Our approach can obtain an optimal or near-optimal (mostly within 0.1\%) solution and reduce the computational time by up to three orders of magnitude.

We investigate how crowdsourced delivery platforms with both contracted and ad-hoc couriers can effectively manage their workforce to meet delivery demands amidst uncertainties. Our objective is to minimize the hiring costs of contracted couriers and the crowdsourcing costs of ad-hoc couriers while considering the uncertain availability and behavior of the latter. Due to the complication of calibrating these uncertainties through data-driven approaches, we instead introduce a basic reduced information model to estimate the upper bound of the crowdsourcing cost and a generalized reduced information model to obtain a tighter bound. Subsequently, we formulate a robust satisficing model associated with the generalized reduced information model and show that a binary search algorithm can tackle the model exactly by solving a modest number of convex optimization problems. Our numerical tests using Solomon's data sets show that reduced information models provide decent approximations for practical delivery scenarios. Simulation tests further demonstrate that the robust satisficing model has better out-of-sample performance than the empirical optimization model that minimizes the total cost under historical scenarios. Our framework provides a practical tool for decision-making that enables platforms to effectively manage their workforce resources and balance their cost objectives with service quality requirements.

General upgrading is a strategy by which a firm can upgrade a customer with any higher-end product whenever a lower-end product is out of stock. In this paper, we consider the capacity planning problem to decide the initial capacity for multiple products to maximize the expected total profit when general upgrading is allowed. We formulate the problem as a two-stage distributionally robust optimization (DRO) model under the commonly employed ambiguity set with marginal mean and variance information. To obtain an exact reformulation as a second-order cone program (SOCP) that is directly solvable, one needs to characterize the extreme points of the dual of the second-stage problem. To this end, we first show that the dual of the second-stage problem of general upgrading can be equivalently reformulated as an economic lot-sizing problem with bounded inventory constraints. We then derive a binary extended formulation for the extreme points of the dual polyhedron based on the characterization via a shortest path network, which enables a polynomially solvable SOCP reformulation. Our characterization of the extreme points can also be used when partial correlation information is incorporated, and in this case, it leads to a tractable semidefinite program. Our reformulation of the second-stage problem connects various problems studied separately in the literature, such as appointment scheduling and economic lot-sizing problems. Our extensive numerical studies show that our DRO solution performs best with limited training data or in highly uncertain environments with non-stationary demand distributions. Using a real data set from a cosmetic company, we demonstrate that our DRO model yields a higher mean profit and lower variability than the sample average approximation.