The Vehicle Fleet Sizing, Positioning and Routing Problem with Stochastic Customers (VFSPRP-SC) consists on pairing strategic decisions of depot positioning and fleet sizing with operational vehicle routing decisions while taking into account the inherent uncertainty of demand. We successfully solve the VFSPRP-SC with a methodology comprised of two main blocks: i) a scenario generation phase and ii) a two-stage stochastic program. For the first block, a set of scenarios is selected with a simulation-based approach that captures the behavior of the demand and allows us to come up with different solutions that could match different risk profiles. The second block is comprised of a facility location and allocation model and a Multi Depot Vehicle Routing Problem (MDVRP) assembled under a two-stage stochastic program. We propose several novel ideas within our methodology: problem specific cuts that serve as an approximation of the expected second stage costs as a function of first stage decisions; an activation paradigm that guides our main optimization procedure; and, a way of mapping feasible routes from one second-stage problem data into another; among others. We performed experiments for two cases: the first case considers the expected value of the demand, and the second case considers the right tail of the demand distribution, seeking a conservative solution. By using acceleration techniques we obtain solutions within 1 to 6 hours, reasonable times considering the strategic nature of the decision. For the ex-post evaluation, we solve 75% of the instances in less than 3 minutes, meaning that the methodology used to solve the MDVRP is well suited for daily operation.
Crowd-shipping is a promising shared mobility service that involves the delivery of goods using non-professional shippers. This service is mainly intended to reduce congestion and pollution in city centers but, as some authors observe, in most crowd-shipping initiatives the crowd rely on private motorized vehicles and hence the environmental benefits could be small, if not negative. Conversely, a crowd-shipping service relying on public transport should maximize the environmental benefits. Motivated by this observation, in this study we assess the potentials of crowd-shipping based on metro commuters in the city of Brescia, Italy. Our contribution is twofold. First, we analyze the results of a survey conducted among metro users to assess their willingness to act as crowd-shippers. The main result is that most young commuters and retirees are willing to be crowd-shippers even for a null reward. Second, we assess the potential economic impact of using metro-based crowd-shipping coupled with a traditional home delivery service. To this end, we formulate a variant of the VRP model where the customers closest to the metro stations may be served either by a conventional vehicle or by a crowd-shipper. The model is implemented using Python with Gurobi solver. A computational study based on the Brescia case is performed to get insights on the economic advantages that a metro-based crowd delivery option may have for a retailing company.
The popular $mathcal{AB}$/push-pull method for distributed optimization problem may unify much of the existing decentralized first-order methods based on gradient tracking technique. More recently, the stochastic gradient variant of $mathcal{AB}$/Push-Pull method ($mathcal{S}$-$mathcal{AB}$) has been proposed, which achieves the linear rate of converging to a neighborhood of the global minimizer when the step-size is constant. This paper is devoted to the asymptotic properties of $mathcal{S}$-$mathcal{AB}$ with diminishing stepsize. Specifically, under the condition that each local objective is smooth and the global objective is strongly-convex, we first present the boundedness of the iterates of $mathcal{S}$-$mathcal{AB}$ and then show that the iterates converge to the global minimizer with the rate $mathcal{O}left(1/sqrt{k}right)$. Furthermore, the asymptotic normality of Polyak-Ruppert averaged $mathcal{S}$-$mathcal{AB}$ is obtained and applications on statistical inference are discussed. Finally, numerical tests are conducted to demonstrate the theoretic results.
In this paper we consider the problem of finding a Nash equilibrium (NE) via zeroth-order feedback information in games with merely monotone pseudogradient mapping. Based on hybrid system theory, we propose a novel extremum seeking algorithm which converges to the set of Nash equilibria in a semi-global practical sense. Finally, we present two simulation examples. The first shows that the standard extremum seeking algorithm fails, while ours succeeds in reaching NE. In the second, we simulate an allocation problem with fixed demand.
In this paper, we develop a distributionally robust chance-constrained formulation of the Optimal Power Flow problem (OPF) whereby the system operator can leverage contextual information. For this purpose, we exploit an ambiguity set based on probability trimmings and optimal transport through which the dispatch solution is protected against the incomplete knowledge of the relationship between the OPF uncertainties and the context that is conveyed by a sample of their joint probability distribution. We provide an exact reformulation of the proposed distributionally robust chance-constrained OPF problem under the popular conditional-value-at-risk approximation. By way of numerical experiments run on a modified IEEE-118 bus network with wind uncertainty, we show how the power system can substantially benefit from taking into account the well-known statistical dependence between the point forecast of wind power outputs and its associated prediction error. Furthermore, the experiments conducted also reveal that the distributional robustness conferred on the OPF solution by our probability-trimmings-based approach is superior to that bestowed by alternative approaches in terms of expected cost and system reliability.
We consider the null controllability problem for the wave equation, and analyse a stabilized finite element method formulated on a global, unstructured spacetime mesh. We prove error estimates for the approximate control given by the computational method. The proofs are based on the regularity properties of the control given by the Hilbert Uniqueness Method, together with the stability properties of the numerical scheme. Numerical experiments illustrate the results.
The branched transport problem, a popular recent variant of optimal transport, is a non-convex and non-smooth variational problem on Radon measures. The so-called urban planning problem, on the contrary, is a shape optimization problem that seeks the optimal geometry of a street or pipe network. We show that the branched transport problem with concave cost function is equivalent to a generalized version of the urban planning problem. Apart from unifying these two different models used in the literature, another advantage of the urban planning formulation for branched transport is that it provides a more transparent interpretation of the overall cost by separation into a transport (Wasserstein-1-distance) and a network maintenance term, and it splits the problem into the actual transportation task and a geometry optimization.
This paper is concerned with a linear quadratic optimal control for a class of singular Volterra integral equations. Under proper convexity conditions, optimal control uniquely exists, and it could be characterized via Frechet derivative of the quadratic functional in a Hilbert space or via maximum principle type necessary conditions. However, these (equivalent) characterizations have a shortcoming that the current value of the optimal control depends on the future values of the optimal state. Practically, this is not feasible. The main purpose of this paper is to obtain a causal state feedback representation of the optimal control.
In this paper, we investigate a sparse optimal control of continuous-time stochastic systems. We adopt the dynamic programming approach and analyze the optimal control via the value function. Due to the non-smoothness of the $L^0$ cost functional, in general, the value function is not differentiable in the domain. Then, we characterize the value function as a viscosity solution to the associated Hamilton-Jacobi-Bellman (HJB) equation. Based on the result, we derive a necessary and sufficient condition for the $L^0$ optimality, which immediately gives the optimal feedback map. Especially for control-affine systems, we consider the relationship with $L^1$ optimal control problem and show an equivalence theorem.
Aerodynamic design process of multi - stage axial flow compressor usually uses the way that combines the S2 flow design and three-dimensional CFD numerical simulation analysis. Based on Mr. Wu Zhonghuas Three-dimensional flow theory , aiming at the S2 flow design matching parameters and the three-dimensional CFD numerical simulation data, through autonomous programming, the S2 design parameter distribution and the corresponding CGNS format CFD calculation results are extracted. Then make the comparative analysis of the two and provide the modification suggestion of the design. The examples have been tested by the comparison and correction of the eight-stage axial flow compressor calculation and finally improve the design performance of the compressor design. The design adiabatic efficiency increases by 0.5%. The surge margin increases by more than 5%. The validity and feasibility of the method are verified.
