Sample Average Approximation Research Articles

Stochastic approximation based confidence regions for stochastic variational inequalities

The sample average approximation (SAA) and the stochastic approximation (SA) are two popular schemes for solving the stochastic variational inequalities problem (SVIP). In the past decades, theories on the consistency of the SAA solutions and SA solutions have been well studied. More recently, the asymptotic confidence regions of the true solution to SVIP have been constructed when the SAA scheme is implemented. It is of fundamental interest to develop confidence regions of the true solution to the SVIP when the SA scheme is employed. In this paper, we discuss the framework of constructing asymptotic confidence regions for the true solution of SVIP with a focus on stochastic dual average method. We first establish the asymptotic normality of the SA solutions both in ergodic sense and non-ergodic sense. Then the online methods of estimating the covariance matrices in the normal distributions are studied. Finally, practical procedures of building the asymptotic confidence regions of solutions to SVIP with numerical simulations are presented.

Sample average approximation method for a class of stochastic vector variational inequalities

In the present paper, the expected-value (EV) reformulation of a class of stochastic vector variational inequalities (SVVI) is investigated. By using the regularized gap function, the EV reformulation of SVVI is transformed into a constrained optimization problem. Then a sample average approximation (SAA) method is proposed for solving the constrained optimization problem. Under suitable assumptions, the limiting behaviors of the optimal values and optimal solutions of the approximation problem are investigated. Finally, the rates of convergence in the different senses of optimal solutions for sample average approximation problem are discussed under the error bound condition.

Resilient Supply Chain Optimization Considering Alternative Supplier Selection and Temporary Distribution Center Location

The global supply chain is facing huge uncertainties due to potential emergencies, and the disruption of any link may threaten the security of the supply chain. This paper considers a disruption scenario in which supply disruption and distribution center failure occur simultaneously from the point of view of the manufacturer. A resilient supply chain optimization model is developed based on a combination of proactive and reactive defense strategies, including manufacturer’s raw material mitigation inventory, preference for temporary distribution center locations, and product design changes, with the objective of obtaining maximum expected profit. The proposed stochastic planning model with demand uncertainty is approximated as a mixed integer linear programming model using Latin hypercube sampling (LHS), sample average approximation (SAA), and scenario reduction (SR) methods. In addition, an improved genetic algorithm (GA) is also developed to determine the approximate optimal solution. The algorithm ensures the feasibility of the solution and improves the solving efficiency through specific heuristic repair strategies. Numerical experiments are conducted to verify the application and advantages of the proposed disruption recovery model and approach. The experimental results show that the proposed resilient supply chain optimization model can effectively reduce the recovery cost of manufacturers after disruption, and the proposed approach performs well in dealing with related problems.

Two-stage travel itinerary recommendation optimization model considering stochastic traffic time

In an increasingly competitive market, most online tour operators have launched personalized travel itinerary recommendation services. Different from previous studies, this research considers the travel itinerary planning problem under total time limit and uncertain traffic time. This problem requires two stages of decision-making: firstly, selecting the tourist attractions to visit from the candidate attractions based on maximizing the popularity utility of tourists; secondly, planning the tourist attraction visit sequence under stochastic traffic time to maximize the tourist activity utility. Therefore, a two-stage stochastic optimization model with chance constraint is constructed and then solved by the sample average approximation (SAA) method. In order to verify the effectiveness of our model, we introduced two benchmark models for comparative analysis. The results show that in the worst case, our model increases the reliability level of travel itineraries by almost 40% compared with the two benchmark models. In addition, case studies of two large cities, i.e., Nanjing and Beijing, indicate that a tour operator can optimize travel itinerary recommendations by improving tourists utility as well as their own profitability without much loss of reliability.

Robust optimization of control parameters for WEC arrays using stochastic methods

This work presents a new computational optimization framework for the robust control of parks of Wave Energy Converters (WEC) in irregular waves. The power of WEC parks is maximized with respect to the individual control damping and stiffness coefficients of each device. The results are robust with respect to the incident wave direction, which is treated as a random variable. Hydrodynamic properties are computed using the linear potential model, and the dynamics of the system is computed in the frequency domain. A slamming constraint is enforced to ensure that the results are physically realistic. We show that the stochastic optimization problem is well posed. Two optimization approaches for dealing with stochasticity are then considered: stochastic approximation and sample average approximation. The outcomes of the above mentioned methods in terms of accuracy and computational time are presented. The results of the optimization for complex and realistic array configurations of possible engineering interest are then discussed. Results of extensive numerical experiments demonstrate the efficiency of the proposed computational framework.

Data-driven inventory policy: Learning from sequentially observed non-stationary data

This paper aims to find dynamic inventory policies for retailers that have limited knowledge about future demand and sequentially observe the unprecedented demand data. We assume the demand is non-stationary; it follows different distributions for different time periods, and the data distributions and the transition behavior are unknown. Two solution approaches are presented to tackle this problem. Integrated-Bayesian (IB) approach is a parametric approach and is introduced for the case when an uncertainty set of possible demand distributions is available. A non-parametric approach, separate-lasso (SL), is proposed for the case that the uncertainty set possible demand distributions is not known. Both methods are theoretically analyzed and empirically benchmarked against several state-of-the-art heuristics. The theoretical analyses provide easy-to-implement algorithms for both approaches, while performance guarantees are derived for the separate-lasso approach. Computational studies show that the proposed methods outperform state-of-the-art heuristics–namely, sample average approximation, rolling horizon, and exponential smoothing–in nine different data environments. The optimal dynamic policy is not obtainable in this dynamic setting as reliable demand forecasts are not available. Therefore, we derive an approximated optimal policy, OPT, assuming the complete knowledge of the demand data in advance. The empirical results reveal that the cost of the proposed approaches is only 12% higher than that of OPT on average. Furthermore, we show that the proposed methods capture the hidden patterns inside the highly non-stationary real-world demand data of one of the largest e-commerce websites.

Robustness in optimal design of Eco-Industrial Parks under the lens of two-stage stochastic optimization

An eco-industrial park (EIP) is a community of businesses located together on a common property, that seek to reduce environmental and economical impact of their operation by collaborating and sharing materials and wastes. In practice, operations within an EIP have daily variations, and therefore are constantly facing uncertainty. In this work, a methodology to design efficient EIPs that are also robust to daily (uncertain) variations of the nominal operation of the enterprises is proposed. The attention is mainly focused in water exchange networks. Probability functions are used to measure robustness and propose a Sample Average Approximation technique to solve the associated optimization problem. Simulations based on literature examples are performed to illustrate the approach.

Distributed appointment assignment and scheduling under uncertainty

We investigate a stochastic distributed appointment assignment and scheduling problem, which consists of assigning appointments to distributed service units and determining service sequences at each service unit. In particular, the service time duration and release time uncertainties are well-considered. The solution to this generic problem finds interesting applications in distributed production systems, healthcare systems, and post-disaster operations. We formulate the problem as a two-stage stochastic program to minimise the total transportation cost and expected makespan, idle time or overtime, and apply the sample average approximation method to make the problem tractable. We then develop a stochastic logic-based Benders decomposition method, decomposing the problem into a master problem and a subproblem. The master problem determines the appointment assignment variables, and the subproblem handles the sequence and service start time variables. Benders optimality cuts are generated from the subproblem's solution and added to the master problem. The developed stochastic logic-based method is advantageous since it can manage many scenarios in parallel. We further consider each appointment's due date, minimise the weighted earliness and tardiness, and adjust the developed method to solve this variant. Experiments on random instances demonstrate the excellent performance of the proposed model and methods.

The framework of parametric and nonparametric operational data analytics

This paper introduces the general philosophy of the Operational Data Analytics (ODA) framework for data‐based decision modeling. The fundamental development of this framework lies in establishing the direct mapping from data to decision by identifying the appropriate class of operational statistics. The efficient decision making relies on a careful balance between data integration and decision validation. Through a canonical decision making problem under uncertainty, we show that the existing approaches (including statistical estimation and then optimization, retrospective optimization, sample average approximation, regularization, robust optimization, and robust satisficing) can all be unified through the lens of the ODA formulation. To make the key concepts accessible, we demonstrate, using a simple running example, how some of the existing approaches may become equivalent under the ODA framework, and how the ODA solution can improve the decision efficiency, especially in the small sample regime.

Explainable subgradient tree boosting for prescriptive analytics in operations management

Motivated by the success of gradient boosting approaches in machine learning and driven by the need for explainable prescriptive analytics approaches in operations management (OM), we propose subgradient tree boosting (STB) as an explainable prescriptive analytics approach to solving convex stochastic optimization problems that frequently arise in OM. The STB approach combines the well-known method of subgradient descent in function space with sample average approximation, and prescribes decisions from a problem-specific loss function, historical demand observations, and prescriptive features. The approach provides a decision-maker with detailed explanations for the prescribed decisions, such as a breakdown of individual features’ impact. These explanations are particularly valuable in practice when the decision-maker has the discretion to adjust the recommendations made by a decision support system. We show how subgradients can be derived for common single-stage and two-stage stochastic optimization problems; demonstrate the STB approach’s applicability to two real-world, complex capacity-planning problems in the service industry; benchmark the STB approach’s performance against those of two prescriptive approaches—weighted sample average approximation (wSAA) and kernelized empirical risk minimization (kERM); and show how the STB approach’s prescriptions can be explained by estimating the impact of individual features. The results suggest that the quality of the STB approach’s prescriptions is comparable to that of wSAA’s and kERM’s prescriptions while also providing explanations.

Technical Note—A Data-Driven Approach to Beating SAA Out of Sample

A Little Pessimism Goes a Long Way Data-driven optimization is concerned with finding a decision, using data and perhaps a model, that performs well when it is applied on a new unseen data point. Data-driven optimization is challenging because data are limited or the model is wrong or the environment in which the decision is being applied is different from the one in which the training data were collected. Distributionally robust optimization (DRO), a worst case optimization method for finding decisions that are insensitive to model error, can sometimes but not always deliver a decision that has a larger out-of-sample expected reward than the sample average approximation (SAA). “A Data Driven Approach to Beating SAA out of Sample” by Jun-ya Gotoh, Michael Kim, and Andrew Lim shows that if worst case (DRO) solutions fail at this task, then the solution of a best case distributionally optimistic optimization problem will do the job. As good as this sounds, there is a catch: whereas an optimistic decision might beat SAA, the improvement is very modest and comes at the cost of being much more sensitive to model misspecification than both the SAA and the DRO decisions. Moreover, it is easy to make a mistake: it can be difficult to determine with a modestly sized data set whether the best or worst case solution will have the higher expected reward than SAA. In summary, data driven optimization is a trade-off between maximizing the expected reward and controlling the sensitivity of this expectation to model misspecification. When both are considered, a little bit of pessimism goes a long way.

Service-oriented container slot allocation policy under stochastic demand

The liner shipping industry plays a pivotal role in global cargo transportation, catering to both contract and spot shippers. Proper capacity allocation between these shippers is vital for maintaining service quality and improving revenue. This research investigates the service-oriented container slot allocation problem under stochastic demand, aiming to maximize total freight revenue while providing adequate service levels to contract shippers for sustaining their market loyalty. We use the fill rate (i.e., the proportion of satisfied demand) as a metric for service level and formulate the research problem as a stochastic linear programming model. To solve this model, we convert it into a multi-objective attainability problem by setting a target for total revenue, and apply Blackwell’s Approachability Theorem to theoretically determine the feasibility of any given revenue and service level requirements. Leveraging these insights, we devise near-optimal policies to guide the slot allocation decision under each demand scenario. Numerical experiments demonstrate that our approach outperforms the benchmark policies in the literature. Furthermore, it can also achieve near-optimal performance closely resembling the sampling average approximation (SAA) solution while significantly reducing the computational time.

Tight and Compact Sample Average Approximation for Joint Chance-Constrained Problems with Applications to Optimal Power Flow

In this paper, we tackle the resolution of chance-constrained problems reformulated via sample average approximation. The resulting data-driven deterministic reformulation takes the form of a large-scale mixed-integer program (MIP) cursed with Big-Ms. We introduce an exact resolution method for the MIP that combines the addition of a set of valid inequalities to tighten the linear relaxation bound with coefficient strengthening and constraint screening algorithms to improve its Big-Ms and considerably reduce its size. The proposed valid inequalities are based on the notion of k-envelopes and can be computed off-line using polynomial-time algorithms and added to the MIP program all at once. Furthermore, they are equally useful to boost the strengthening of the Big-Ms and the screening rate of superfluous constraints. We apply our procedures to a probabilistically constrained version of the DC optimal power flow problem with uncertain demand. The chance constraint requires that the probability of violating any of the power system’s constraints be lower than some positive threshold. In a series of numerical experiments that involve five power systems of different size, we show the efficiency of the proposed methodology and compare it with some of the best performing convex inner approximations currently available in the literature. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms – Discrete. Funding: This work was supported in part by the European Research Council under the EU Horizon 2020 research and innovation program [Grant 755705], in part by the Spanish Ministry of Science and Innovation [Grant AEI/10.13039/501100011033] through project PID2020-115460GB-I00, and in part by the Junta de Andalucía and the European Regional Development Fund through the research project P20_00153. Á. Porras is also financially supported by the Spanish Ministry of Science, Innovation and Universities through the University Teacher Training Program with fellowship number FPU19/03053. Supplemental Material: The online supplement is available at https://doi.org/10.1287/ijoc.2022.0302 .

Deterministic Bi-Criteria Model for Solving Stochastic Mixed Vector Variational Inequality Problems

In this paper, we consider stochastic mixed vector variational inequality problems. Firstly, we present an equivalent form for the stochastic mixed vector variational inequality problems. Secondly, we present a deterministic bi-criteria model for giving the reasonable resolution of the stochastic mixed vector variational inequality problems and further propose the approximation problem for solving the given deterministic model by employing the smoothing technique and the sample average approximation method. Thirdly, we obtain the convergence analysis for the proposed approximation problem while the sample space is compact. Finally, we propose a compact approximation method when the sample space is not a compact set and provide the corresponding convergence results.

Combining optimisation and simulation using logic-based Benders decomposition

Operations Research practitioners often want to model complicated functions that are difficult to encode in their underlying optimisation framework. A common approach is to solve an approximate model, and then use a simulation to evaluate the true objective value of one or more solutions. We propose a new approach to integrating simulation into the optimisation model itself. The idea is to run the simulation at each incumbent solution to a master problem. The simulation results are then used to guide the trajectory of the optimisation model itself using logic-based Benders cuts. We test the approach on a class of stochastic resource allocation problems with monotonic performance measures. We derive strong novel Benders cuts that are provably valid for all problems of the given form. We consider two concrete examples: a nursing home shift scheduling problem, and an airport check in counter allocation problem. While previous papers on these applications could only approximately solve realistic instances, we are able to solve them exactly within a reasonable amount of time. Moreover, while those papers account for the inherent variance of the problem by including estimates of the underlying random variables as model parameters, we are able to compute sample-average approximations to optimality with up to 100 scenarios.

On the unconstrained optimization reformulations for a class of stochastic vector variational inequality problems

In this paper, a class of stochastic vector variational inequality (SVVI) problems are considered. By employing the idea of a D-gap function, the SVVI problem is reformulated as a deterministic model, which is an unconstrained expected residual minimization (UERM) problem, while it is reformulated as a constrained expected residual minimization problem in the work of Zhao et al. Then, the properties of the objective function are investigated and a sample average approximation approach is proposed for solving the UERM problem. Convergence of the proposed approach for global optimal solutions and stationary points is analyzed. Moreover, we consider another deterministic formulation, i.e., the expected value (EV) formulation for an SVVI problem, and the global error bound of a D-gap function based on the EV formulation is given.

Managing interruptions in appointment schedules via patient notification

Interruptions in appointment schedules are common in practice, yet many clinics use generic appointment scheduling templates and sub-optimal interruption management strategies that fail to address dynamics aspects of daily healthcare operations. In this study, we consider patient no-shows, patient punctuality, service time variability, random walk-in patient arrivals and other common appointment interruptions to determine optimal single-server, intra-day appointment schedules. Using a modified depth first search (DFS) algorithm that relies on a sample average approximation (SAA) technique for objective function evaluation, we minimize a weighted sum of expected waiting times for scheduled and walk-in patients, physician idle time, and physician overtime. Extensive experimentation shows the proposed algorithm solves realistic problems in reasonable time. Furthermore, we propose a sequential notification procedure (SNP) that serves as a dynamic correcting mechanism to address infrequent and unexpected interruptions with a relatively long duration. Considering possible patient responses to notification, SNP notifies scheduled patients in advance to prevent excessive patient waiting and physician overtime. Furthermore, we propose an easy-to-use patient notification heuristic (NH) that can be adopted by clinics using traditional appointment systems. Experimentation shows that both SNP and NH provide significant value, improving operational outcomes. We demonstrate the value of patient notification increases as interruption levels increase and when patients are responsive to notifications.

On the value of operational flexibility in the trailer shipment and assignment problem: Data-driven approaches and reinforcement learning

This paper addresses the trailer shipment problem—the task of managing the optimal weight of products in a trailer, taking into consideration the uncertain weight of tractors provided by Third-Party Logistics (3PL) providers, and abiding by the gross weight regulation. We propose a series of data-analytics methodologies, including Sample Average Approximation (SAA), Empirical Risk Minimization (ERM), and a dynamic trailer assignment methodology using Reinforcement Learning (RL), to optimize the trailer shipment process. The introduction of operational flexibility and the dynamic utilization of tractor weight information upon arrival are pivotal to the effectiveness of the RL-based methodology. To validate our approaches, we apply them to transaction-level shipping data from a real company. The results demonstrate significant cost reductions in the logistics process, driven by the dynamic assignment methodology which allows efficient selection of trailers to suit varying tractor weights. This research proposes an innovative approach to the prevalent trailer shipment problem, applicable to a wide range of industries using 3PL outsourcing. Through this work, we demonstrate the transformative potential of data-analytics methodologies to enhance efficiency and profitability in logistics operations.

Distributionally Robust Chance-Constrained p-Hub Center Problem

The p-hub center problem is a fundamental model for the strategic design of hub location. It aims at constructing p fully interconnected hubs and links from nodes to hubs so that the longest path between any two nodes is minimized. Existing literature on the p-hub center problem under uncertainty often assumes a joint distribution of travel times, which is difficult (if not impossible) to elicit precisely. In this paper, we bridge the gap by investigating two distributionally robust chance-constrained models that cover, respectively, an existing stochastic one under independent normal distribution and one that is based on the sample average approximation approach as a special case. We derive deterministic reformulations as a mixed-integer program wherein a large number of constraints can be dynamically added via a constraint-generation approach to accelerate computation. Counterparts of our models in the emerging robust satisficing framework are also discussed. Extensive numerical experiments demonstrate the encouraging out-of-sample performance of our proposed models as well as the effectiveness of the constraint-generation approach. History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods & Analysis. Funding: This work is partially supported by the National Natural Science Foundation of China [Grants 72101187 and 72021002] and Early Career Scheme from the Hong Kong Research Grants Council General Research Fund [Grant 9043424] and NSFC/RGC Joint Research Scheme N_CityU105/21. Y. Zhao is supported by the Ministry of Education, Singapore, under its 2019 Academic Research Fund Tier 3 grant call [Grant MOE-2019-T3-1-010]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0113 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0113 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

A Deficiency of the Weighted Sample Average Approximation (wSAA) Framework: Unveiling the Gap between Data-Driven Policies and Oracles

This paper critically examines the weighted sample average approximation (wSAA) framework, a widely used approach in prescriptive analytics for managing uncertain optimization problems featuring non-linear objectives. Our research pinpoints a key deficiency of the wSAA framework: when data samples are limited, the minimum relative regret—the discrepancy between the expected optimal profit realized by an oracle aware of the genuine distribution, and the maximum expected out-of-sample profit garnered by the data-driven policy, normalized by the former profit—can approach towards one. To validate this assertion, we scrutinize two distinct contextual stochastic optimization problems—the production decision-making problem and the ship maintenance optimization problem—within the wSAA framework. Our study exposes a potential deficiency of the wSAA framework: its decision performance markedly deviates from the full-information optimal solution under limited data samples. This finding offers valuable insights to both researchers and practitioners employing the wSAA framework.