Stochastic Problem Research Articles

Are weaker stationarity concepts of stochastic MPCC problems significant in absence of SMPCC-LICQ?

Are weaker stationarity concepts of stochastic MPCC problems significant in absence of SMPCC-LICQ?

Decomposition methods for multi-horizon stochastic programming

Multi-horizon stochastic programming includes short-term and long-term uncertainty in investment planning problems more efficiently than traditional multi-stage stochastic programming. In this paper, we exploit the block separable structure of multi-horizon stochastic linear programming, and establish that it can be decomposed by Benders decomposition and Lagrangean decomposition. In addition, we propose parallel Lagrangean decomposition with primal reduction that, (1) solves the scenario subproblems in parallel, (2) reduces the primal problem by keeping one copy for each scenario group at each stage, and (3) solves the reduced primal problem in parallel. We apply the parallel Lagrangean decomposition with primal reduction, Lagrangean decomposition and Benders decomposition to solve a stochastic energy system investment planning problem. The computational results show that: (a) the Lagrangean type decomposition algorithms have better convergence at the first iterations to Benders decomposition, and (b) parallel Lagrangean decomposition with primal reduction is very efficient for solving multi-horizon stochastic programming problems. Based on the computational results, the choice of algorithms for multi-horizon stochastic programming is discussed.

Space‐time stochastic Galerkin boundary elements for acoustic scattering problems

SummaryAcoustic emission or scattering problems naturally involve uncertainties about the sound sources or boundary conditions. This article initiates the study of time domain boundary elements for such stochastic boundary problems for the acoustic wave equation. We present a space‐time stochastic Galerkin boundary element method which is applied to sound‐hard, sound‐soft and absorbing scatterers. Uncertainties in both the sources and the boundary conditions are considered using a polynomial chaos expansion. The numerical experiments illustrate the performance and convergence of the proposed method in model problems and present an application to a problem from traffic noise.

ODC and ROC Curves, Comparison Curves and Stochastic Dominance†

SummaryWe discuss two novel approaches to inter‐distributional comparisons in the classical two‐sample problem. Our starting point is properly standardised and combined, very popular in several areas of statistics and data analysis, ordinal dominance and receiver characteristic curves, denoted by ODC and ROC, respectively. The proposed new curves are termed the comparison curves. Their estimates, being weighted rank processes on (0,1), form the basis of inference. These weighted processes are intuitive, well‐suited for visual inspection of data at hand and are also useful for constructing some formal inferential procedures. They can be applied to several variants of two‐sample problem. Their use can help improve some existing procedures both in terms of power and the ability to identify the sources of departures from the postulated model. To simplify interpretation of finite sample results, we restrict attention to values of the processes on a finite grid of points. This results in the so‐called bar plots (B‐plots), which readably summarise the information contained in the data. What is more, we show that B‐plots along with adjusted simultaneous acceptance regions provide principled information about where the model departs from the data. This leads to a framework that facilitates identification of regions with locally significant differences.We show an implementation of the considered techniques to a standard stochastic dominance testing problem. Some min‐type statistics are introduced and investigated. A simulation study compares two tests pertinent to the comparison curves to well‐established tests in the literature and demonstrates the strong and competitive performance of the former in many typical situations. Some real data applications illustrate simplicity and practical usefulness of the proposed approaches. A range of other applications of considered weighted processes is briefly discussed too.

Sample intelligence-based progressive hedging algorithms for the stochastic capacitated reliable facility location problem

Selecting facility locations requires significant investment to anticipate and prepare for disruptive events like earthquakes, floods, or labor strikes. In practice, location choices account for facility capacities, which often cannot change during disruptions. When a facility fails, demand transfers to others only if spare capacity exists. Thus, capacitated reliable facility location problems (CRFLP) under uncertainty are more complex than uncapacitated versions. To manage uncertainty and decide effectively, stochastic programming (SP) methods are often employed. Two commonly used SP methods are approximation methods, i.e., Sample Average Approximation (SAA), and decomposition methods, i.e., Progressive Hedging Algorithm (PHA). SAA needs large sample sizes for performance guarantee and turn into computationally intractable. On the other hand, PHA, as an exact method for convex problems, suffers from the need to iteratively solve numerous sub-problems which are computationally costly. In this paper, we developed two novel algorithms integrating SAA and PHA for solving the CRFLP under uncertainty. The developed methods are innovative in that they blend the complementary aspects of PHA and SAA in terms of exactness and computational efficiency, respectively. Further, the developed methods are practical in that they allow the specialist to adjust the tradeoff between the exactness and speed of attaining a solution. We present the effectiveness of the developed integrated approaches, Sampling Based Progressive Hedging Algorithm (SBPHA) and Discarding SBPHA (d-SBPHA), over the pure strategies (i.e. SAA). The validation of the methods is demonstrated through two-stage stochastic CRFLP. Promising results are attained for CRFLP, and the method has great potential to be generalized for SP problems.

Time-reliability optimization for the stochastic traveling salesman problem

This paper presents a novel approach to addressing the Stochastic Traveling Salesman Problem (STSP), a classical problem in combinatorial optimization, by integrating travel time and reliability factors into the decision-making process. Traditional TSP models primarily focus on minimizing the total travel distance or cost without considering the reliability of each route. In real-world situations, especially in logistics and network design, it's just as important to have reliable routes. A reliable route means there's a good chance it will be completed successfully and on time. Our research extends the conventional STSP framework by incorporating a reliability metric for each route, alongside the standard travel time metric. A tri-objective optimization model is proposed to minimize the mean and standard deviation of travel time and maximize route reliability simultaneously. A new algorithm called Permutation Binary-Addition-Tree (BAT) is proposed to solve the problem more efficiently when there is uncertainty. Our approach marks a significant step towards more realistic and practical solutions for route optimization problems in dynamic and uncertain environments. We also present a complexity analysis of our model against traditional cost-only TSP solutions, demonstrating the efficacy of considering reliability in route planning.

Optimal scheduling of active distribution network based on RBF-stochastic response surface method

As the high proportion of distributed generation (DG) and diversification of loads increase, the distribution network faces increasingly greater impact. To address the high-dimensional stochastic problems arising from the multi-dimensional uncertainty of power source and load, this paper proposes a two-stage optimization scheduling scheme for AC distribution networks based on RBF-stochastic response surface method (RBF-SRSM). This scheme takes into account economy, environmental protection and safety, and consists of day-ahead and intra-day stages. The day-ahead decision scheduling is based on an optimal low-carbon economy operation plan, while the intra-day reactive power optimization is achieved by converting nonlinear integral indicators into discrete linear indicators through probabilistic power flow calculation and the first moment derivation principle inspired by SRSM. The mixed integer second-order cone programming (MISOCP) method is used to solve the dispatching plan, and the results suggest that the proposed method is more efficient than the SRSM.

Adapting reservoir operations for optimal water management under varying climate and demand scenarios using metaheuristic algorithms

The operational rule curve is significant for proper reservoir operations and water resources management of the dam. The optimisation of the release policy is rather a complicated and stochastic problem. Various types of algorithms have been used to develop the release policy for the Klang Gate Dam (KGD), but alas, climate parameters associated with climate change, be it regional or globally, e.g., temperature factors, had not been considered then. In this study, the consideration of maximum and minimum temperature factors is given emphasis, while water demand is evaluated in the optimisation problem using simulation-optimisation approaches. The support vector regression simulation model (SVRS), and the multilayer perceptron simulation model (MLPS) were used to simulate the demand for the scenario cases. SVRS utilises the hyperplane theory to study the relationship between the input and output data, whereas MLPS employs the human neural system to weight and process the inputs via multiple layers of neurons into a targeted output. Four distinct types of simulation-optimisation combination model, including the SVRS-firefly algorithm (SVRS-FA), the SVRS-particle swarm optimisation (SVRS-PSO), the MLPS-firefly algorithm (MLPS-FA), and the MLPS-particle swarm optimisation (MLPS-PSO), were applied and investigated. The ultimate goal of this study was to minimise the water deficit. The PSO optimisation models overall have outperformed the FA optimisation models in all aspects. The periodic reliability of the MLPS-PSO was found to be the highest when the minimum temperature scenario was considered, and is 0.6491, with the least shortage period of 80 months. The MLPS-PSO is also the most resilient model in the same scenario, with a resilience value of 0.6750. Whereas the SVRS-PSO was the least vulnerable model in the minimum temperature scenario with the lowest shortage index of 0.0047 and a volumetric deficiency of about 2.44 MCM in total. The SVRS-PSO model showed excellent performance for high and low inflow conditions, while the MLPS-PSO model was better during medium inflows. In short, each of the models investigated has its particular field of advancement, but since the maximum temperatures sounded critical, additional research for the maximum temperature case could be pursued further in depth.

A disaggregated integer L-shaped method for stochastic vehicle routing problems with monotonic recourse

This paper proposes a new integer L-shaped method for solving two-stage stochastic integer programs whose first-stage solutions can decompose into disjoint components, each one having a monotonic recourse function. In a minimization problem, the monotonicity property stipulates that the recourse cost of a component must always be higher or equal to that of any of its subcomponents. The method exploits new types of optimality cuts and lower bounding functionals that are valid under this property. The stochastic vehicle routing problem is particularly well suited to be solved by this approach, as its solutions can be decomposed into a set of routes. We consider the variant with stochastic demands in which the recourse policy consists of performing a return trip to the depot whenever a vehicle does not have sufficient capacity to accommodate a newly realized customer demand. This work shows that this policy can lead to a non-monotonic recourse function, but that the monotonicity holds when the customer demands are modeled by several commonly used families of probability distributions. We also present new problem-specific lower bounds on the recourse that strengthen the lower bounding functionals and significantly speed up the solving process. Computational experiments on instances from the literature show that the new approach achieves state-of-the-art results.

Matched queues with matching batch pair [formula omitted]

In this paper, we develop the matrix-analytic method to discuss an interesting but challenging bilateral stochastic matching problem: A matched queue with matching batch pair (m,n) and two types of impatient customers, where the two types of customers arrive according to two independent Poisson processes. Once m A-customers and n B-customers are matched as a group, the m+n customers immediately leave the system. We show that this matched queue can be expressed as a novel bidirectional level-dependent quasi-birth-and-death (QBD) process whose analysis has its own interests, and specifically, computing the maximal non-positive inverse matrices of bidirectional infinite sizes by using the RG-factorizations. Based on this, we can provide an effective matrix-analytic method to deal with this matched queue, including the system stability, the average stationary queue lengths, the average sojourn times, and the departure process. We believe that the methodology and results developed in this paper can be applicable to studying more general matched queueing systems, which are widely encountered in many practical areas.

Investments in transmission lines and storage units considering second-order stochastic dominance constraints

Increasing intermittent renewable generation to meet the climate goals entails a deep transformation of current power systems. The transmission system must adapt to ensure a rapid and flexible response to the changes in the energy flows. Flexibility can be provided by reinforcing the interconnection among all the market agents and/or by installing facilities with fast response to the changes. In this work, we study the investment problem of a central planner that seeks to expand the transmission network and install storage units considering decarbonizing measures. We propose a two-stage stochastic problem with uncertainty on the demand growth and including representative days to characterize the hourly demand and renewable power variability. To obtain better expansion strategies according to a limit on the carbon emissions, second-order stochastic dominance constraints are imposed. Numerical analyses based on the power system of the Canary island (Spain) are provided. The results show that to install storage units is key to efficiently integrate renewable generation. The investments in the transmission system are mostly in lower-voltage lines. The formulation with stochastic dominance constraints results in higher second-stage investments, allowing a better adaptation to the demand growth evolution.

The Stochastic Transportation Problem with Imprecise Data using Lomax Distribution

The stochastic transportation problem with imprecise data is a probabilistic chance-constrained programming (CCP) problem in which the objective function is fuzzy and the supply and demand are random. Three models for the STP with ID with mixed-type restrictions that follow the Lomax distribution (LD) are created in this research. Optimizing the transportation cost in FTP under probabilistic mixed constraints is the goal of the research project. To do this, the probabilistic mixed constraints are transformed into deterministic form using the LD, and the cost coefficient of the fuzzy objective function is changed with alpha cut representation. Numerical examples are presented to demonstrate the suggested models.

Nature of the Volcano Transition in the Fully Disordered Kuramoto Model.

Randomly coupled phase oscillators may synchronize into disordered patterns of collective motion. We analyze this transition in a large, fully connected Kuramoto model with symmetric but otherwise independent random interactions. Using the dynamical cavity method, we reduce the dynamics to a stochastic single-oscillator problem with self-consistent correlation and response functions that we study analytically and numerically. We clarify the nature of the volcano transition and elucidate its relation to the existence of an oscillator glass phase.

The dynamic stochastic container drayage problem with truck appointment scheduling

The dynamic stochastic container drayage problem with truck appointment scheduling

A Boundary Control Problem for Stochastic 2D-Navier–Stokes Equations

AbstractWe study a stochastic velocity tracking problem for the 2D-Navier–Stokes equations perturbed by a multiplicative Gaussian noise. From a physical point of view, the control acts through a boundary injection/suction device with uncertainty, modeled by stochastic non-homogeneous Navier-slip boundary conditions. We show the existence and uniqueness of the solution to the state equation, and prove the existence of an optimal solution to the control problem.

Adaptive Lagrangian Policies for a Multiwarehouse, Multistore Inventory System with Lost Sales

Adaptive Lagrangian Policies for Inventory Control Consider the problem of managing inventory in a multiwarehouse, multistore setting where each store can be periodically replenished from potentially a set of nearby warehouses. This is a practically relevant problem, and its optimal policy is not easy to compute because of the curse of dimensionality. Existing works in the literature have proposed a policy based on Lagrangian relaxation of the original stochastic problem. This vanilla policy applies the control parameters derived from the Lagrangian model and has been shown to perform well in some cases. In “Adaptive Lagrangian Policies for a Multiwarehouse, Multistore Inventory System with Lost Sales,” we go one step farther and develop adaptive Lagrangian policies that update (at certain update times) the control parameters of the vanilla policy based on the historical demand realizations. We analytically show that our proposed adjustment significantly improves the performance of the vanilla policy.

A reinforcement learning framework for improving parking decisions in last-mile delivery

This study leverages simulation-optimisation with a Reinforcement Learning (RL) model to analyse the routing behaviour of delivery vehicles (DVs). We conceptualise the system as a stochastic k-armed bandit problem, representing a sequential interaction between a learner (the DV) and its surrounding environment. Each DV is assigned a random number of customers and an initial delivery route. If a loading zone is unavailable, the RL model is used to select a delivery strategy, thereby modifying its route accordingly. The penalty is gauged by the additional trucking and walking time incurred compared to the originally planned route. Our methodology is tested on a simulated network featuring realistic traffic conditions and a fleet of DVs employing four distinct lastmile delivery strategies. The results of our numerical experiments underscore the advantages of providing DVs with an RL-based decision support system for en-route decision-making, yielding benefits to the overall efficiency of the transport network. Highlights Combining simulation and optimisation algorithms with reinforcement learning Model DVs en-route parking decisions with a k-armed bandit algorithm Evaluating the impacts of delivery strategies on traffic congestion and in last-mile delivery efficiency

Power-Efficient Dispatching of Time-Sensitive Requests in the Fog-Enabled Industrial Internet of Things

Recent advancements in service offerings and the emergence of new application areas have reveal the limitations of cloud computing as a solely solution. One of these new areas is the industrial internet of things, where industrial processes are controlled in a network-based and automated way which brings many new challenges. To solve the challenges, fog computing has been introduced as a complementary paradigm to the cloud. Resource management is one of those challenges need further investigation. The industrial internet of things by its own characteristics has led traditional methods to be inefficient within this evolving ecosystem. Therefore, fully utilizing potential of the new system necessitates effective resource management. This manuscript aims to investigate the problem of resource management in the emerging cloud-fog ecosystem of the industrial internet of things. Considering the dynamic nature of users, this study proposes the algorithms for dynamic resource allocation and provisioning based on deadlines. We formulate the problem as a multiple criteria stochastic problem and introduced an online solution. Our proposed solution takes into account the unique characteristics of Fog computing and the dynamic behavior of users. To effectively handle the scheduling of requests, the research suggests utilizing the Lyapunov optimization method and Lyapunov drift theory. The Lyapunov method helps to merge and measure all the related objectives in terms drift. The objective function is optimized, in each time slot, using a drift-plus-penalty weight parameter. By adopting this approach, the system can operate efficiently while meeting the constraints and performance requirements of IoT applications.

Comparative study of robust optimization model of multimaterial emergency logistics in an uncertain environment

AbstractGiven the significance of objective functions in alleviating human suffering within the humanitarian relief supply chain, it is imperative to consider both logistics costs and the cost of deprivation as ideal targets. This study investigates a multimaterial single‐cycle stochastic mixed integer model, integrating uncertainties about post‐disaster demand, supply survival, and road networks. An innovative heuristic approach is used to solve the stochastic facility location problem, which minimizes unfulfilled requests and response times in lexicographic order. This research also compares the cost‐minimization model of the pre‐positioning problem to an alternative model that maximizes demand satisfaction under resource constraints. Empirical validation through numerous case studies and solutions confirms the effectiveness and feasibility of the alternative model. The results of our experiments consistently demonstrate that the best solution provided by the alternative model not only satisfies the minimum cost model but also surpasses the conventional problem, without compromising performance. Consequently, it is crucial to avoid the cost‐minimization approach at the expense of human suffering.

Navigating uncertain distribution problem: a new approach for resolution optimization of transportation with several objectives under uncertainty

Amidst uncertainty, decision-making in manufacturing becomes a central focus due to its complexity. This study explores complex transportation constraints and uses novel ways to guide manufacturers. The Multi-objective Stochastic Linear Fractional Transportation Problem (MOSLFTP) is a crucial tool for managing supply chains, manufacturing operations, energy distribution, emergency routes, healthcare logistics, and other related areas. It adeptly addresses uncertainty, transforming efficiency and effectiveness in several domains. Stochastic programming is the process of converting theoretical probabilities into concrete certainties. The artistic compromise programming technique acts as a proficient mediator, reconciling opposing objectives and enabling equitable decision-making. This novel approach also addresses the Multi-objective Stochastic Linear plus Linear Fractional Transportation Problem (MOSLPLFTP), which involves two interconnected issues. The effectiveness of these principles is clearly shown with the help of the LINGO® 18 optimization solver. This study uses a ranking method to compare the similar methods to solve the current problems. A meticulously designed example acts as a significant achievement, shedding light on our method in a practical setting. It serves as a distinctive instrument, leading manufacturers through the maze of uncertainty and assisting them in determining the most advantageous course of action. This journey involves subtle interactions between complexity and simplicity, uncertainty is overcome by decisiveness, and invention is predominant.