Average Approximation Research Articles

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.

Simulation of natural convection in a solar collector

The process of natural convection in a flat solar collector, which was designed by the authors, was mathematically modeled using experimental data. Stones were used as storage means. The inclined collector is expressed as a two-dimensional plate for modeling. The initial and boundary conditions were formulated. It was assumed that the flow in this region is laminar. To simulate the flow of natural air convection, a system of non-stationary partial differential equations was chosen based on the laws of conservation of mass, momentum and energy in the Boussinesq approximation. The equations were solved in a dimensionless form using the finite difference method and an explicit scheme. The picture of the temperature fields and the appearance of convection was obtained on the basis of experimental temperature data in the inlet and outlet zones of the collector, as well as on the surface of the heat accumulator during the day. To evaluate the effectiveness of the resulting model an average approximation error (approximation) of 7.7% was established. The results obtained in specific values of the Grashof and Prandtl numbers are presented in the form of graphs.

Quantum approximate optimization algorithm parameter prediction using a convolutional neural network

The Quantum approximate optimization algorithm (QAOA) is a quantum-classical hybrid algorithm aiming to produce approximate solutions for combinatorial optimization problems. In the QAOA, the quantum part prepares a quantum parameterized state that encodes the solution, where the parameters are optimized by a classical optimizer. However, it is difficult to find optimal parameters when the quantum circuit becomes deeper. Hence, there is numerous active research on the performance and the optimization cost of QAOA. In this work, we build a convolutional neural network to predict parameters of depth p + 1 QAOA instance by the parameters from the depth p QAOA counterpart. We propose two strategies based on this model. First, we recurrently apply the model to generate a set of initial values for a certain depth QAOA. It successfully initiates depth 10 QAOA instances, whereas each model is only trained with the parameters from depths less than 6. Second, the model is applied repetitively until the maximum expected value is reached. An average approximation ratio of 0.9759 for Max-Cut over 264 Erdős–Rényi graphs is obtained, while the optimizer is only adopted for generating the first input of the model.

Exact and heuristic algorithms for the domination problem

In a simple connected graph G=(V,E), a subset of vertices S⊆V is a dominating set if any vertex v∈V∖S is adjacent to some vertex x from this subset. A number of real-life problems can be modeled using this problem which is known to be among the difficult NP-hard problems in its class. We formulate the problem as an integer liner program (ILP) and compare the performance with the two earlier existing exact state-of-the-art algorithms and exact implicit enumeration and heuristic algorithms that we propose here. Our exact algorithm was able to find optimal solutions much faster than ILP and the above two exact algorithms for middle-dense instances. For graphs with a considerable size, our heuristic algorithm was much faster than both, ILP and our exact algorithm. It found an optimal solution for more than half of the tested instances, whereas it improved the earlier known state-of-the-art solutions for almost all the tested benchmark instances. Among the instances where the optimum was not found, it gave an average approximation error of 1.18.

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.

Bloch–Siegert Effect for Surface Nuclear Magnetic Resonance Sounding Experiments in the Unsaturated Zone

The Bloch–Siegert effect is relevant for NMR experiments where components of the excitation pulse other than the circularly polarized component have an influence on the evolution of the magnetization of the spin system under consideration. For linearly polarized excitation fields this happens at amplitudes higher than roughly one tenth of the magnitude of the static magnetic field. Since surface nuclear magnetic resonance (SNMR) experiments, also called magnetic-resonance-soundings (MRS), are conducted in the relatively low local field of the earth, the Bloch–Siegert effect can quickly become relevant. This is especially the case for SNMR experiments in the unsaturated zone, where due to short relaxation times fast pulses of high intensity must be used. To describe the Bloch–Siegert effect, we use the average Hamiltonian approximation obtained by the Magnus expansion of up to fifth order, as well as the solution of the Bloch equations. The results of these approximations are tested against the Bloch simulations and it is shown that they are only valid for limited ranges of the excitation field amplitude. The influence of the Bloch–Siegert effect on sensitivity kernels is described and verified with experimental data obtained with a small scale SNMR sensor on water containers.

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.

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.

A distributionally robust production planning model for maximizing customer satisfaction with budget and carbon emissions constraints

This paper investigates a new stochastic green production planning problem without full knowledge about the probability distribution on customer demands. The only information regarding the demand is the means and covariance matrix. The goal of the study is to provide an eco-conscious manufacturer with solution methods capable of yielding, within a reasonable amount of computation time, a robust solution that maximizes customer satisfaction while taking into account the constraints of budgets and periodic carbon caps spontaneously set out of environmental awareness. We formulate the problem into a novel distributionally robust chance-constrained optimization model under demand uncertainty set. To effectively solve the model, two heuristics are proposed. The widely used sample average approximation (SAA) scheme is first deployed as a benchmark method. However, the computation time increases rapidly with the problem size. In some instances, even feasible solutions cannot be found within a reasonable amount of computation time. We then develop an approximated mixed-integer reformulation on the basis of second order cone program (SOCP). A real-world case is solved and extensive numerical experiments show that the SOCP heuristic is superior over the SAA heuristic in terms of computation time while the solution quality is only slightly lower in most of instances. Finally, sensitivity analysis on budgets, periodic carbon caps, the maximum budget overrun risk level, initial inventory levels, and the magnitude of demand uncertainty are conducted and useful managerial insights are derived.

Using GAN-generated market simulations to guide genetic algorithms in index tracking optimization

Index tracking is the problem of building a portfolio that replicates the performance of a market index. The recent applications involving deep learning in index tracking are more focused on the learned information rather than on the framework that consumes this information. The problem is that, until now, a way to enable the extension of the index tracking models that adopt machine learning was not yet proposed. Nowadays, the mathematical programming framework is more flexible when considering model extensions to build realistic portfolios. Thus, this study presents ways to combine generative adversarial networks (GANs) within this framework to generate market simulations incorporated into a base index tracking model. It was verified how the simulations generated by GANs can impact the out-of-sample performance of the portfolio and how to deal with their instability, by using real data from the Brazilian market. To achieve this, two evolutionary metaheuristics were proposed to solve the multiple scenario index tracking problem. The proposed evolutionary algorithms minimize the tracking errors of a portfolio in multiple market simulations generated by GANs. The performance of the proposed algorithms was compared against another evolutionary metaheuristic that solves the index tracking problem using historical data. It was possible to observe that the scenario-based dominance genetic algorithm (SDM-SBDGA-GAN) was able to perform better than the real data genetic algorithm (RDM-GA) and the sample average approximation genetic algorithm (SDM-SAAGA-GAN). These results open doors for new applications of synthetic data in the construction of portfolios using more realistic constraints and other tracking objectives. This work also brings discussions about problems related to the application of GANs in this context.

Moderate Deviations and Invariance Principles for Sample Average Approximations

We study moderate deviations and convergence rates for the optimal values and optimal solutions of sample average approximations. Firstly, we give an extension of the Delta method in large deviations. Then under Lipschitz continuity on the objective function, we establish a moderate deviation principle for the optimal value by the Delta method. When the objective function is twice continuously differentiable and the optimal solution of true optimization problem is unique, we obtain a moderate deviation principle for the optimal solution and a Cramér-type moderate deviation for the optimal value. Motivated by the Donsker invariance principle, we consider a functional form of stochastic programming problem and establish a Donsker invariance principle, a functional moderate deviation principle, and a Strassen invariance principle for the optimal value.