Sample Average Approximation Research Articles

A 0/1 Constrained Optimization Solving Sample Average Approximation for Chance Constrained Programming

Sample average approximation (SAA) is a tractable approach for dealing with chance constrained programming, a challenging stochastic optimization problem. The constraint of SAA is characterized by the 0/1 loss function, which results in considerable complexities in devising numerical algorithms. Most existing methods have been devised based on reformulations of SAA, such as binary integer programming or relaxed problems. However, the development of viable methods to directly tackle SAA remains elusive, let alone providing theoretical guarantees. In this paper, we investigate a general 0/1 constrained optimization, providing a new way to address SAA rather than its reformulations. Specifically, starting with deriving the Bouligand tangent and Fréchet normal cones of the 0/1 constraint, we establish several optimality conditions. One of them can be equivalently expressed by a system of equations, enabling the development of a semismooth Newton-type algorithm. The algorithm demonstrates a locally superlinear or quadratic convergence rate under standard assumptions along with nice numerical performance compared with several leading solvers. Funding: This research was supported by the National Key R&D Program of China [Grant 2023YFA1011100], the Fundamental Research Funds for the Central Universities, and the National Natural Science Foundation of China [Grant 12271309].

The impact of transshipment and staggered deliveries on pharmaceutical supply chain performance

ABSTRACT This paper studies the impact of proactive transshipment and staggered deliveries on the pharmaceutical supply chain performance. Therefore, a mathematical model considering transshipment and staggered deliveries is developed to minimise the total inventory management costs of a pharmaceutical supply chain under uncertain demand. To deal with this problem, a two-stage stochastic model is presented and solved using the sample average approximation. The proposed solution shows an important gain of 5% on average in the total inventory management costs when incorporating transshipment of drugs between hospitals and a gain of 32% on average when considering both transshipment and staggered deliveries.

Sample Average Approximation for Stochastic Programming with Equality Constraints

Sample Average Approximation for Stochastic Programming with Equality Constraints

A two-stage stochastic programming model for comprehensive risk response action selection: A case study in Industry 4.0

Effective project risk management is critical in environments where both micro-level and macro-level risks are present. Traditional models often focus on micro-level risks, neglecting broader macroeconomic uncertainties such as geopolitical instability and supply chain disruptions. This research introduces a two-stage stochastic programming model designed to optimize the selection of Risk Response Actions (RRAs) under uncertainty while addressing both types of risk. The model incorporates “here-and-now” decisions at the planning stage and “wait-and-see” decisions as uncertainties unfold, enabling adaptive risk management throughout the project lifecycle. To solve the model efficiently, we employ an evolutionary algorithm combined with Sample Average Approximation (SAA) to handle the computational complexity of multiple scenarios. The model is applied to a real-world case study involving the integration of IoT and ERP systems in a smart factory in Iran, a project characterized by significant macroeconomic and geopolitical risks. Our key contribution lies in providing a comprehensive risk response strategy selection model that simultaneously addresses micro- and macro-level risks while incorporating strategic flexibility through outsourcing decisions. The results demonstrate that our model outperforms traditional deterministic models, offering enhanced resilience against macro-level risks and improved project performance under uncertainty. These findings provide valuable insights for project managers aiming to increase resilience and adaptability in volatile environments. By integrating both internal and external risk factors, our model offers a robust tool for managing complex projects, enhancing decision-making and project outcomes in uncertain conditions

Data-driven inventory control for large product portfolios: A practical application of prescriptive analytics

Motivated by the real-world inventory management problem of a large network of pharmacies, this paper proposes and studies a practically relevant Prescriptive Analytics approach for data-driven dynamic inventory control of large portfolios of interrelated products. We extend existing research on weighted Sample Average Approximation by integrating a ‘global learning’ model that effectively exploits cross-learning opportunities within the product portfolio. The results of an extensive numerical evaluation on real-world data suggest that our approach outperforms relevant benchmarks—in particular, models that rely on ‘local learning’ strategies where weight functions are trained separately for each product. The numerical results also allow us to derive important practical and structural insights regarding the value of contextual information in our global learning framework.

Flexibility-Oriented AC/DC Hybrid Grid Optimization Using Distributionally Robust Chance-Constrained Method

With the increasing integration of stochastic sources and loads, ensuring the flexibility of AC/DC hybrid distribution networks has become a pressing challenge. This paper aims to enhance the operational flexibility of AC/DC hybrid distribution networks by proposing a flexibility-oriented optimization framework that addresses the growing uncertainties. Notably, a comprehensive evaluation method for operational flexibility assessment is first established. Based on this, this paper further proposes a flexibility-oriented operation optimization model using the distributionally robust chance-constrained (DRCC) method. A customized solution method utilizing second-order cone relaxation and sample average approximation (SAA) is also introduced. The results of case studies indicate that the flexibility of AC/DC hybrid distribution networks is enhanced through sharing energy storage among multiple feeders, adaptive reactive power regulation using soft open points (SOPs) and static var compensators (SVCs), and power transfer between feeders via SOPs.

A Sample Average Approximation Approach for Stochastic Optimization of Flight Test Planning with Sorties Uncertainty

In the context of flight test planning, numerous uncertainties exist, encompassing aircraft status, number of flights, and weather conditions, among others. These uncertainties ultimately manifest significantly in the actual number of flight sorties executed, rendering high significance to engineering problems related to the execution of flight test missions. However, there is a dearth of research in this specific aspect. To address this gap, this paper proposes an opportunity-constrained integer programming model tailored to the unique characteristics of the problem. To handle the uncertainties, Sample Average Approximation (SAA) is employed to perform oversampling of the uncertain parameters, followed by the Adaptive Large Neighborhood Search (ALNS) algorithm to solve for the optimal solution and objective function value. Results from numerical experiments conducted at varying scales and validated with diverse sampling distributions demonstrate the effectiveness and robustness of the proposed methodology. By decoding the generated execution sequences, comprehensive mission planning schemes can be derived. This approach yields sequences that exhibit commendable feasibility and robustness for the flight test planning problem with sorties uncertainty (FTPPSU), offering valuable support for the efficient execution of future flight test missions.

A distribution-free-based approach for stochastic food closed-loop supply chain

ABSTRACT Resource scarcity has driven growing interest in circular economy (CE). Closed-loop supply chain (CLSC) with returnable transport items (RTIs) in the food industry is an important component of CE. However, existing works on food CLSC with RTIs have not simultaneously considered the perishability, facility location, and uncertain demand under limited information. Therefore, this work addresses a new food CLSC optimisation problem. We first propose a non-linear chance-constrained programming model. It is then transformed into a mixed-integer linear programming model via using the distribution-free (DF) method and sample average approximation (SAA) method, respectively. An illustrative example reveals that the DF method needs only 10.50% of the computation time of the SAA method. To address large-scale problems, an improved Lagrangian relaxation (LR) method is developed. To address the computational challenge in large-scale problems, an improved Lagrangian relaxation (LR) algorithm is developed. Results show that CPLEX achieves a gap of 75.57%, while the LR surpasses it by finding near-optimal solutions with a gap of 1.22%, using only 31.82% of the computation time required by CPLEX. For this work, the main insights are summarised: (1) extending product shelf life can reduce the total cost; and (2) to alleviate uncertain demand and production risks, production capacity and product inventory capacity can be appropriately expanded, but excessive investment may not improve returns.

Meal Delivery Routing Problem with Stochastic Meal Preparation Times and Customer Locations

AbstractWe investigate the Meal Delivery Routing Problem (MDRP), managing courier assignments between restaurants and customers. Our proposed variant considers uncertainties in meal preparation times and future order numbers with their locations, mirroring real challenges meal delivery providers face. Employing a rolling-horizon framework integrating Sample Average Approximation (SAA) and the Adaptive Large Neighborhood Search (ALNS) algorithm, we analyze modified Grubhub MDRP instances. Considering route planning uncertainties, our approach identifies routes at least 25% more profitable than deterministic methods reliant on expected values. Our study underscores the pivotal role of efficient meal preparation time management, impacting order rejections, customer satisfaction, and operational efficiency.

A choice-based approach to dynamic capacitated multi-item lot sizing with demand uncertainty

With the purpose of planning and implementing pricing decisions on a tactical level as well as production decisions on an operational level, we consider – in an integrated form – the capacitated multi-item lot sizing problem with uncertain item demands and price-dependent discrete choice demand. The model is embedded into an overarching rolling horizon procedure allowing for adaptations to changes in demand and cost parameters. We first formulate the static problem version as a nonlinear mathematical program with underlying multinomial logit demand and subsequently linearize it to make it viable for mathematical programming solvers. Uncertainty of demands is taken into account by Monte Carlo simulation. More specifically, we generate random demand scenarios and utilize them as input data for the sample average approximation problem version. We further endow the problem setting with possibilities to incorporate pricing policy requirements such as restricting the number of price adaptations or defining periods without price adaptations. Overall, the developed approach yields a powerful tool for balancing item demands via pricing in a way favorable for adhering to available production capacities and thereby striking a balance between revenues and costs. Computational results confirm that adapting prices to time-dependent demand and cost parameters is exploited effectively to maintain a deliberately controlled production environment. Moreover, the integrated pricing and production setting allows to study the effect of pricing policy restrictions and demand uncertainties upon attainable profits.

Analysis of a Class of Minimization Problems Lacking Lower Semicontinuity

The minimization of nonlower semicontinuous functions is a difficult topic that has been minimally studied. Among such functions is a Heaviside composite function that is the composition of a Heaviside function with a possibly nonsmooth multivariate function. Unifying a statistical estimation problem with hierarchical selection of variables and a sample average approximation of composite chance constrained stochastic programs, a Heaviside composite optimization problem is one whose objective and constraints are defined by sums of possibly nonlinear multiples of such composite functions. Via a pulled-out formulation, a pseudostationarity concept for a feasible point was introduced in an earlier work as a necessary condition for a local minimizer of a Heaviside composite optimization problem. The present paper extends this previous study in several directions: (a) showing that pseudostationarity is implied by (and thus, weaker than) a sharper subdifferential-based stationarity condition that we term epistationarity; (b) introducing a set-theoretic sufficient condition, which we term a local convexity-like property, under which an epistationary point of a possibly nonlower semicontinuous optimization problem is a local minimizer; (c) providing several classes of Heaviside composite functions satisfying this local convexity-like property; (d) extending the epigraphical formulation of a nonnegative multiple of a Heaviside composite function to a lifted formulation for arbitrarily signed multiples of the Heaviside composite function, based on which we show that an epistationary solution of the given Heaviside composite program with broad classes of B-differentiable component functions can in principle be approximately computed by surrogation methods. Funding: The work of Y. Cui was based on research supported by the National Science Foundation [Grants CCF-2153352, DMS-2309729, and CCF-2416172] and the National Institutes of Health [Grant 1R01CA287413-01]. The work of J.-S. Pang was based on research supported by the Air Force Office of Scientific Research [Grant FA9550-22-1-0045].

Stochastic and robust truck-and-drone routing problems with deadlines: A Benders decomposition approach

Time-definite delivery service requires efficient and punctual delivery within specified deadlines. Enhancing the service level of delivery in last-mile logistics prompted the application of a truck-and-drone cooperative system. However, the inherent uncertainty in travel time contributes to frequent service delays. To address this challenge, our study addresses the stochastic and robust truck-and-drone routing problems with deadlines, aiming to minimize service delay risk under uncertainty. We leverage the sample average approximation (SAA) and robust optimization (RO) approaches to handle uncertainty in scenarios involving both big and small datasets. Benders decomposition (BD) algorithms are developed to solve the SAA and RO truck-and-drone routing problems efficiently. Numerical experiments showcase the algorithms’ effectiveness and reveal interesting insights. These findings suggest the importance of tailoring the approach to data availability for optimizing truck-and-drone delivery under uncertainty. Furthermore, the proposed model formulations and algorithms are generalized to address truck-and-drone routing problems under more complex scenarios, such as that with uncertainty of service time, or with nonlinear objective functions.

Surgery scheduling in flexible operating rooms by using a convex surrogate model of second-stage costs

We study the elective surgery planning problem in a hospital with operating rooms shared by elective and emergency patients. This problem is split in two distinct phases. First, a subset of patients to be operated in the next planning period is selected and the selected patients are assigned to a block and a tentative starting time. Then, in the online phase of the problem, a policy decides how to insert the emergency patients in the schedule and may cancel planned surgeries. The overall goal is to minimize the expectation of a cost function representing the assignment of patient to blocks, case cancellations, overtime, waiting time and idle time. We model the offline problem by a two-stage stochastic program, and show that the optimal second-stage costs can be approximated by a convex piecewise linear surrogate model that can be computed in a preprocessing step. This results in a mixed integer program which can be solved very fast, even for large instances of the problem. We also describe a greedy policy for the online phase of the problem, and analyze the performance of our approach by comparing it to both heuristic methods or approaches relying on sampling average approximation (SAA) on a large set of benchmarking instances. Our simulations indicate that our approach can reduce the expected costs by as much as 30% compared to heuristic methods and it can solve problems with 1000 patients in about one minute, while SAA-approaches fail to obtain good solutions within 30 min on small instances.

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 Model-Free Approach for Solving Choice-Based Competitive Facility Location Problems Using Simulation and Submodularity

This paper considers facility location problems in which a firm entering a market seeks to open facilities on a subset of candidate locations so as to maximize its expected market share, assuming that customers choose the available alternative that maximizes a random utility function. We introduce a deterministic equivalent reformulation of this stochastic problem as a maximum covering location problem with an exponential number of demand points, each of which is covered by a different set of candidate locations. Estimating the prevalence of these preference profiles through simulation generalizes a sample average approximation method from the literature and results in a maximum covering location problem of manageable size. To solve it, we develop a partial Benders reformulation in which the contribution to the objective of the least influential preference profiles is aggregated and bounded by submodular cuts. This set of profiles is selected by a knee detection method that seeks to identify the best tradeoff between the fraction of the demand that is retained in the master problem and the size of the model. We develop a theoretical analysis of our approach and show that the solution quality it provides for the original stochastic problem, its computational performance, and the automatic profile-retention strategy it exploits are directly connected to the entropy of the preference profiles in the population. Computational experiments on existing and new benchmark sets indicate that our approach dominates the classical sample average approximation method on large instances of the competitive facility location problem, can outperform the best heuristic method from the literature under the multinomial logit model, and achieves state-of-the-art results under the mixed multinomial logit model. We characterize a broader class of problems, which includes assortment optimization, to which the solving methodology and the analyses developed in this paper can be extended. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms—Discrete. Funding: This research was supported by Fonds de Recherche du Québec-Nature et Technologies and Institut de Valorisation des Données through scholarships to R. Legault. E. Frejinger was partially supported by the Canada Research Chair program [Grant 950-232244]. Supplemental Material: The online appendices are available at https://doi.org/10.1287/ijoc.2023.0280 .

Project selection and scheduling with multiplicative enhancement effects and delay risk: An application in intelligent manufacturing technologies

In the realm of intelligent manufacturing, driven by the push towards smart factories and Industry 4.0, optimizing technology selection and sequencing is paramount for intelligent transformation. This complex decision-making process resembles a novel project selection and scheduling problem, characterized by Multiplicative Enhancement Effects (MEEs) where the collective benefits of multiple projects can exceed their individual contributions. Additionally, the uncertain duration of projects further adds complexity. Motivated by these practical challenges, a stochastic mixed-integer programming model that incorporates MEEs and uncertain delays is established. A conditional value-at-risk-based risk measure is integrated to control delay risk. The solution approach leverages an efficient branch and bound-based approach with cutting strategies, integrating a sample average approximation framework and a backward labeling algorithm to handle stochastic elements. A real-world technology implementation case study highlights the advantages of considering MEEs, uncovers myopic biases in technology selection, and identifies implementation patterns centred around core Industry 4.0 technologies. This research enhances our understanding of intelligent manufacturing decision-making, offering valuable insights and practical implications for technology-driven transformations.

A distributionally robust optimization approach for the potassium fertilizer product transportation considering transshipment through crossdocks

Potassium fertilizer is an essential input for agricultural productivity, and plays a critical role in various plant processes, influencing water uptake, enzyme activation, and photosynthesis. The efficient delivery of potassium fertilizer products to the end-users, typically farmers and agricultural enterprises, is of utmost importance. Due to the long traveling distance between the supplier and customers, decision makers of supplier normally set up multiple crossdocks to aid in transition and storage of potassium fertilizer products. In this situation, making an optimal transportation plan as well as the inventory level of crossdocks will directly enhance the efficiency and effectiveness of long-distance transportation. In this study, we model this problem as a dynamic transportation problem with transshipment considering the uncertain time-varying customers’ demand. To characterize the demand information, we first construct an ambiguity set of customers’ demand based in limited historical data and then develop a distributionally robust optimization-based (DRO) framework to optimize the transportation plan and related inventory level of crossdocks simultaneously. We also propose a general approach to overcome the computational challenges of DRO by transforming the original DRO into a second-order cone programming based on duality theory. Additionally, we introduce linear decision rules to adjust the optimization strategy based on the new observed demand, thus lead the model to handle the dynamic information flow in real time. In case study, all involved data are collected or derived from a real potassium fertilizer company located at Western China. The results show our model reduces the cost by 18.07% compared to stochastic model (sample average approximation), indicating a significant effectiveness of our model on the improvement of delivery efficiency and cost saving in real dynamic logistic systems. Also, we conclude the managerial insight that decision-makers should develop a comprehensive strategy, including improving communication to ensure order status updates, planning rationally to evenly distribute orders, and proactively allocating resources to meet operational demands.

A stochastic-MILP dispatch optimization model for concentrated solar thermal under uncertainty

Concentrated Solar Thermal (CST) offers a promising solution for large-scale solar energy utilization as Thermal Energy Storage (TES) enables electricity generation independent of daily solar fluctuations, shifting to high-priced electricity intervals. The development of dispatch planning tools is mandatory to account for uncertainties associated with weather and electricity price forecasts. A Stochastic Mixed-Integer Linear Program (SMILP) is proposed to maximize Sample Average Approximation (SAA) of expected profit within a specified scenario space. The SMILP exhibits robust performance, yet its computational time poses a challenge. Three heuristic solutions are developed which run a set of deterministic optimizations on different historical weather profiles to generate candidate Dispatch Plans (DPs). Subsequently, the DP with the best average performance on all profiles is selected. The new methods are applied to a hypothetical 115 MW CST plant in South Australia. When the historical database has a limited set of historical weather profiles, the SMILP achieves 6–9 % higher profit than the closest heuristic when the DPs are applied to novel weather conditions. With a large historical weather dataset, the performance of the SMILP and closet heuristic becomes nearly identical since the SMILP can only utilize a limited number of trajectories for optimization without becoming computationally infeasible. In this case, the heuristic emerges a practical alternative, providing similar average profit in a reasonable time. Taken together, the results illustrate the importance of considering uncertainty in DP optimization and indicate that straightforward heuristics on a large database are a practical method for addressing uncertainty.

Quantitative stability for stochastic tensor variational inequalities

ABSTRACT The goal of this article is to investigate a class of stochastic tensor variational inequalities (denoted by STVI), which is a generalization of the stochastic linear variational inequalities (denoted by SLVI) and a subclass of stochastic variational inequalities (denoted by SVI). The quantitative stability of the expected residual minimization (ERM) problem for the STVI is analysed. Firstly, the existence of solutions of the ERM problem and its perturbed problem are considered. Then, the quantitative stability of the ERM problem is derived under suitable probability metrics. Finally, under some suitable assumptions, the rates of convergence of the sample average approximation (SAA) method for the optimal solution sets of the perturbed problems are discussed.

Stochastic Optimization and Learning for Two-Stage Supplier Problems

The main focus of this paper is radius-based (supplier) clustering in the two-stage stochastic setting with recourse, where the inherent stochasticity of the model comes in the form of a budget constraint. In addition to the standard (homogeneous) setting where all clients must be within a distance \(R\) of the nearest facility, we provide results for the more general problem where the radius demands may be inhomogeneous (i.e., different for each client). We also explore a number of variants where additional constraints are imposed on the first-stage decisions, specifically matroid and multi-knapsack constraints, and provide results for these settings. We derive results for the most general distributional setting, where there is only black-box access to the underlying distribution. To accomplish this, we first develop algorithms for the polynomial scenarios setting; we then employ a novel scenario-discarding variant of the standard Sample Average Approximation (SAA) method, which crucially exploits properties of the restricted-case algorithms. We note that the scenario-discarding modification to the SAA method is necessary in order to optimize over the radius.