Stochastic Constraints Research Articles

Single-vendor single-buyer multi-product economic production quantity problem with stochastic constraints: a modified generalized elimination method

This study designs a single-buyer single-vendor multi-product Economic Production Quantity (EPQ) model to optimize the total cost in an inventory system. To do so, a Non-Linear Programming (NLP) model is developed considering a set of stochastic constraints on space or warehouse capacity, backordering cost, procurement, ordering, and obtainable budget. A modified Generalized Elimination Method (GEM) is then offered to treat the complexity of the model wherein the variables are eliminated by adding two equations. The developed method generates much more high-quality solutions compared with the classic GEM, while the number of iterations slightly increases. Three problem instances in different scales are then taken into account to assess the efficiency of the modified GEM in terms of optimality criteria. The results demonstrate that the modified GEM has an excellent performance with respect to optimal solutions, infeasibility, number of iterations, complementarity, and errors of optimality. Finally, the behavior of the objective function is analyzed against order quantity fluctuations to draw out practical implications.

Probabilistic branch and bound considering stochastic constraints

In this paper, we investigate a simulation optimization problem that poses challenges due to (i) the inability to evaluate the objective and multiple constraint functions analytically; instead, we rely on stochastic simulation to estimate them, and (ii) a discrete and potentially vast solution space. Rather than providing a single optimal solution, our aim is to identify a set of near-optimal solutions within a specific quantile, such as the top 10%. Our investigation covers two different problem settings or frameworks. The first framework is focused solely on a stochastic objective function, disregarding any stochastic constraints. In this context, we propose employing a probabilistic branch-and-bound algorithm to discover a level set of solutions. Alternatively, the second framework involves stochastic constraints. To address such stochastically constrained problems, we utilize a penalty function methodology in conjunction with a probabilistic branch-and-bound algorithm. Furthermore, we establish a convergence analysis of both algorithms to demonstrate their asymptotic validity and highlight their theoretical properties and behavior. Our experimental results provide evidence of the efficiency of our proposed algorithms, showing that they outperform existing approaches in the field of simulation optimization.

Measures of stochastic non-dominance in portfolio optimization

Stochastic dominance rules are well-characterized and widely used. This work aims to describe and better understand the situations when they do not hold by developing measures of stochastic non-dominance. They quantify the error caused by assuming that one random variable dominates another one when it does not. To calculate them, we search for a hypothetical random variable that satisfies the stochastic dominance relationship and is as close to the original one as possible. The Wasserstein distance between the optimal hypothetical random variable and the original one is considered as the measure of stochastic non-dominance. Depending on the conditions imposed on the probability distribution of the hypothetical random variable, we distinguish between general and specific measures of stochastic non-dominance. We derive their exact values for random variables with uniform, normal, and exponential distributions. We present relations to almost first-order stochastic dominance and to tractable almost stochastic dominance. Using monthly returns of twelve assets captured by the German stock index DAX, we solve portfolio optimization problems with the first-order and second-order stochastic dominance constraints. The measures of stochastic non-dominance allow us to compare the optimal portfolios with respect to different orders of stochastic dominance from a new angle. We also defined the closest dominating and closest approximately dominating portfolios. They brought a better understanding of the relationship between the two types of optimal portfolios. Using moving window analysis, the relationship of the in-sample measure of stochastic non-dominance to out-of-sample performance was studied, too.

Low rank approximation method for perturbed linear systems with applications to elliptic type stochastic PDEs

In this paper, we propose a low rank approximation method for efficiently solving stochastic partial differential equations. Specifically, our method utilizes a novel low rank approximation of the stiffness matrices, which can significantly reduce the computational load and storage requirements associated with matrix inversion without losing accuracy. To demonstrate the versatility and applicability of our method, we apply it to address two crucial uncertainty quantification problems: stochastic elliptic equations and optimal control problems governed by stochastic elliptic PDE constraints. Based on varying dimension reduction ratios, our algorithm exhibits the capability to yield a high-precision numerical solution for stochastic partial differential equations, or provides a rough representation of the exact solutions as a pre-processing phase. Meanwhile, our algorithm for solving stochastic optimal control problems allows a diverse range of gradient-based unconstrained optimization methods, rendering it particularly appealing for computationally intensive large-scale problems. Numerical experiments are conducted and the results provide strong validation of the feasibility and effectiveness of our algorithm.

A gradient-descent-based framework for solving a stochastic two-echelon delivery problem with cargo-bikes

In this paper, we examine a stochastic two-echelon vehicle routing problem (2e-VRP) using cargo bikes within hyperconnected networks. The focus is on the integration of both delivery and pickup of reusable containers, incorporating transshipment operations, time windows, and stochastic demand constraints. The model also considers the flow consolidation for empty and full containers at the satellites and allows for load splitting. Moreover, this study introduces an innovative gradient-descent-based optimization framework to handle the combinatorial complexity of the proposed model, opening new avenues in stochastic programming. Furthermore, the performance of this novel method is compared against the sample average approximation method, evaluating both solution quality and computational efficiency. Experimental results demonstrate the model’s advanced integration and flexibility, significantly enhancing urban delivery systems and advancing logistics and transportation optimization research.

A neural network framework for portfolio optimization under second-order stochastic dominance

Traditional portfolio optimization strategies often rely on statistical methods and linear programming tools to achieve a balance between return and risk. Despite their usefulness for portfolio optimization, they cannot efficiently capture the specific differences and complexity of real-world financial markets. Several modern approaches attempt to overcome these limitations by using nonlinear models, machine learning, and advanced risk measures. In this study, we propose a novel strategy for optimizing portfolios that incorporates second-order stochastic dominance constraints and solves them with neural networks. we show that portfolios subject to second-order stochastic dominance constraints outperform their traditional counterparts, especially in tail-risk situations.

Multi-objective shape optimization of TESLA-like cavities: Addressing stochastic Maxwell's eigenproblem constraints

In this paper, we present a robust formulation to address nonlinear equality-constrained and inequality-constrained multi-objective optimization problems for multi-cell accelerating cavities with TESLA-like geometry under consideration of uncertain parameters. We propose to explore a robust approach that employs the polynomial chaos expansion (PCE) to model the uncertainty. For the uncertainty quantification (UQ), the resulting random-dependent eigenvalue problem is solved using the stochastic collocation method (SCM) combined with PCE. As a result, the shape optimization of the studied multi-cell cavity is defined in terms of statistical moments, which serve as the cost functionals in the multi-objective (MO) steepest descent algorithm. Two versions of this method are developed and implemented. Finally, the optimization results and their implications for the operation of a multi-cell cavity are discussed.

Selection of the Best in the Presence of Subjective Stochastic Constraints

We consider the problem of finding a system with the best primary performance measure among a finite number of simulated systems in the presence of subjective stochastic constraints on secondary performance measures. When no feasible system exists, the decision maker may be willing to relax some constraint thresholds. We take multiple threshold values for each constraint as a user’s input and propose indifference-zone procedures that perform the phases of feasibility check and selection-of-the-best sequentially or simultaneously. Given that there is no change in the underlying simulated systems, our procedures recycle simulation observations to conduct feasibility checks across all potential thresholds. We prove that the proposed procedures yield the best system in the most desirable feasible region possible with at least a pre-specified probability. Our experimental results show that our procedures perform well with respect to the number of observations required to make a decision, as compared with straight-forward procedures that repeatedly solve the problem for each set of constraint thresholds, and that our simultaneously-running procedure provides the best overall performance.

Adjustable light robust optimization with second order stochastic dominance constraints

Robust optimization suffers from excessive conservatism and infeasibility due to stringent requirements on all constraints within the uncertainty set, rendering robust solutions impractical. The light robust approach has been proposed to mitigate these challenges. This method aims to identify a solution that minimizes the weighted sum of all constraint violations while adhering to a predetermined limit on the deterioration of expected return. Drawing inspiration from light robustness, our paper introduces the adjustable light robust optimization models with second-order stochastic dominance constraints. We empirically examine their feasibility, out-of-sample performance, and the dynamic trade-off between return and robustness.

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.

A Bi-Objective Inventory Model for Strategic Items in a Stochastic Environment with Partial Shortage: A Base-Stock Approach

This paper suggests an ordering policy for strategic items. Based on the base stock policy, we modeled a bi-objective, multi-product inventory control system with partial back-ordering. The model aims to minimize total costs and maximize the average service level, considering stochastic constraints such as storage space, maximum allowable shortage, and maximum allowable lost sales to match real-world conditions. We used three Multi-Objective Decision-Making methods, namely Individual optimization, Desirability function, and Goal attainment, to solve the problem. TOPSIS method showed the Goal attainment method’s superiority over the others. The sensitivity analysis revealed that the mean of storage space had the most significant impact on objective functions. Also, increasing the back-order percentage can reduce the system’s total costs.

A MODEL OF GROSS CAPITAL FLOWS: RISK SHARING AND FINANCIAL FRICTIONS

AbstractThis article builds a two‐country model of gross capital flows where agents share tradable output risk using two bonds, subject to stochastic collateral constraints. Equilibrium portfolios are short in domestic bonds and long in foreign bonds because the endogenous movements of the real exchange rate provide a hedge against domestic output shocks. Under negative domestic shocks, these external positions transfer wealth from home to abroad. During the Great Recession, the model shows that such wealth transfer from the United States mitigated the consumption drop abroad. Quantitatively, financial frictions account for about half of the collapse in U.S. gross flows in 2008.

An Optimal Computing Allocation Strategy for Swarm-Based Algorithm in Simulation-Based Optimization Problem with Stochastic Constraint

For realizing smart manufacturing, recent studies have been focusing on the capabilities of digital twins (DTs) that offer a virtual portrayal of a process or system and thereby facilitate real-time monitoring, analysis, and optimization. DTs construct a “digital model” of factories by using advanced technologies such as smart sensors, Internet of Things (IoT), artificial intelligence (AI) techniques, and optimization processes for executing critical decisions for enhancing productivity and predicting future events. Therefore, simulation-based optimization techniques play key supporting roles in the decision-making process of the DT shop. Thus, to enhance competitiveness in industrial companies, discourse on properly designing simulation-based optimization techniques begets significant study. This study aims to address a simulation-based optimization problem with stochastic constraint (SO[Formula: see text] using a hybrid particle swarm optimization (PSO). Given the large solution space, the PSO supports design space exploration. However, since all solutions must be verified for feasibility, the optimal computing allocation strategy (OCAS) strategy for SO[Formula: see text] is proposed to allocate the computing budget efficiently. OCAS is an improved version of optimal computing budget allocation for constrained optimization (OCBA-CO), specifically designed for complex systems with an expansive design space that possesses multiple local optimal solutions. During a typical PSO algorithm searching process, OCBA-CO is applied separately for each generation, resulting in a significant number of replications merely distinguishing similar solutions. To overcome this problem, our study proposes the PSO[Formula: see text], which combines PSO with OCAS to avoid wasting simulation runs on similar solutions. Two quantitative experiments are performed to assess the efficacy of PSO[Formula: see text] compared to other competitors. The simulation results indicate that the PSO[Formula: see text] algorithm presents a competitive and superior solution in terms of both quality and searching efficiency.

Gaussian Process-Based Model Predictive Control for Autonomous Underwater Vehicles

Traditional MPC algorithms, assuming constant values, suffer from performance degradation caused by model mismatch. This paper addresses the enhancement of predictability in Autonomous Underwater Vehicles (AUVs) under uncertain disturbances and unknown system dynamics through the design of Model Predictive Controllers (MPCs). We propose a hybrid model that integrates a first-principles nominal model with a learning-based model utilizing Gaussian Processes (GPs). The algorithm addresses the problem of model mismatch in AUV motion control by constructing a precise GP model, which captures the dynamic characteristics of the process through the collection and learning of deviations between the reference model and the controlled system. Additionally, the GP model transforms stochastic constraints into deterministic convex constraints, enhancing safety guarantees in complex and challenging environments. The effectiveness of the proposed algorithm is demonstrated through two simulation examples.

An approach to the integral optimization of investment portfolios

A comprehensive approach to the optimization of financial portfolios is provided by the Integral Analysis Method (IAM), which comprises four steps: The description of the problem is followed by three mathematical steps, namely, cardinal analysis, ordinal analysis and integration analysis, which allow simultaneously involving ordinal and cardinal variables in an optimization problem. The cardinal analysis uses the L_1 risk model, which is enriched by the inclusion of a stochastic constraint on the successful performance forecast attributed to each of the shares that make up the investment portfolio. This bound, however, does not alter the linear nature of the model. The Ordinal analysis includes expert opinions on the reputation of each company as a qualitative criterion. This completes two of the most relevant aspects of the investment portfolio optimization problem: success probability (assessed through the cardinal analysis) and business reputation (assessed through the ordinal analysis). By resorting to Stochastic Multicriteria Acceptability Analysis, the integration analysis allows obtaining indicators that jointly evaluate the variables resulting from the two previous analyses. As a validation process, IAM was applied to simulated data.

Portfolio reshaping under 1st-order stochastic dominance constraints by the exact penalty function methods

The paper studies a financial portfolio selection problem under 1st-order stochastic dominance constraints. These constraints constitute lower bounds on the return profile of the portfolio. In particular, they allow searching for a better portfolio than some reference portfolio by comparing their cumulative distribution functions. Candidate objective functions are the average return, a value at risk, or the average value at risk. The optimization problems obtained are computationally hard because of possibly non-convex constraints and possibly discontinuous objective functions. In the case of a discrete distribution of the return, we develop numerical procedures to solve the problem. The proposed approach uses new exact penalty functions to tackle with the 1st-order stochastic dominance constraints. The resulting penalized objective function is further optimized be the stochastic successive smoothing method as a local optimizer within some branch and bound global optimization scheme. The approach is numerically and graphically illustrated on small portfolio selection problems up to dimension 10.

Construction of an intelligent optimization model for water resource allocation under drought conditions

Abstract In this paper, we first establish the model of optimal allocation of water resources within the region and convert it into a dynamic planning problem. Then, under drought conditions, the models of optimal allocation of intra-regional water resources among industries and different crops are established, respectively, and the water production function model is established using sensitivity coefficients. Under the condition of satisfying the water quantity opportunity constraints and with the objective of ensuring obtaining a higher system return, the water resources optimal allocation model based on interval multi-stage stochastic opportunity constraint planning is established by combining multi-stage planning, interval planning and stochastic opportunity constraint planning. A hybrid intelligent algorithm combining stochastic simulation, multilayer neural networks, and genetic algorithms is used to solve the model. A case study of optimal water resource allocation is used to verify the feasibility and validity of the proposed model and algorithm. Under the traditional water resource allocation scheme, the total water supply and water shortage of Nandan County are 0.97 and 0.58 billion m3, and the water shortage rate is 37.55%. Under the optimized water resources allocation scheme of this paper, the water shortage of Nandan County is 0.05 billion m³, and the water shortage rate is 6.09%. The model of this paper’s allocation scheme can effectively meet the water demand.

Aptenodytes forsteri optimization algorithm for low-carbon logistics network under demand uncertainty.

As China's "double carbon" goal continues to advance, logistics as a key area of carbon emissions and low-carbon logistics center site selection are key links in the process. However, existing studies on logistics center location often ignore the impact of demand uncertainty, which leads to a waste of resources in the planning and construction processes. We take logistics cost and carbon emission as the objectives, and the multi-objective site selection model established based on stochastic programming theory takes demand uncertainty as a stochastic constraint. We transform the stochastic constraint model into a 0-1 mixed integer multi-objective planning model by utilizing the idea of equivalence transformation. The Aptenodytes Forsteri Optimization (AFO) algorithm is combined with the Ideal Point Method to solve the model, and the algorithm is compared with the Particle Swarm Optimization (PSO), Differential Evolutionary (DE), Tabu Search (TS), Sparrow Search (SS) algorithms, and the exact solver Linear Interactive and General Optimizer (LINGO). The examples verify the validity of the models and algorithms, with an average reduction of 6.2% and 3.6% in logistics costs and carbon emissions in the case of demand determination, and at the confidence level of 0.9 under demand uncertainty, both logistics costs and carbon emissions are decreased to varying degrees. This study provides a new research idea for the low-carbon logistics location problem under demand uncertainty, which helps to promote the transformation of the logistics industry to low-carbon and high-efficiency.

A Belief Propagation Based Algorithm for Solving Stochastic Constraint Satisfaction Problem

A Belief Propagation Based Algorithm for Solving Stochastic Constraint Satisfaction Problem

ROSCOM: Robust Safe Reinforcement Learning on Stochastic Constraint Manifolds

ROSCOM: Robust Safe Reinforcement Learning on Stochastic Constraint Manifolds