Average Approximation Research Articles

Determinism versus uncertainty: Examining the worst-case expected performance of data-driven policies

This paper explores binary decision making, a critical domain in areas such as finance and supply chain management, where decision makers must often choose between a deterministic-cost option and an uncertain-cost option. Given the limited historical data on the uncertain cost and its unknown probability distribution, this research aims to ascertain how decision makers can optimize their decisions. To this end, we evaluate the worst-case expected performance of all possible data-driven policies, including the sample average approximation policy, across four scenarios differentiated by the extent of knowledge regarding the lower and upper bounds of the first moment of the uncertain cost distribution. Our analysis, using worst-case expected absolute regret and worst-case expected relative regret metrics, consistently shows that no data-driven policy outperforms the straightforward strategy of choosing either a deterministic-cost or uncertain-cost option in these scenarios. Notably, the optimal choice between these two options depends on the specific lower and upper bounds of the first moment. Our research contributes to the literature by revealing the minimal worst-case expected performance of all possible data-driven policies for binary decision-making problems.

AN-SPS: adaptive sample size nonmonotone line search spectral projected subgradient method for convex constrained optimization problems

We consider convex optimization problems with a possibly nonsmooth objective function in the form of a mathematical expectation. The proposed framework (AN-SPS) employs Sample Average Approximations (SAA) to approximate the objective function, which is either unavailable or too costly to compute. The sample size is chosen in an adaptive manner, which eventually pushes the SAA error to zero almost surely (a.s.). The search direction is based on a scaled subgradient and a spectral coefficient, both related to the SAA function. The step size is obtained via a nonmonotone line search over a predefined interval, which yields a theoretically sound and practically efficient algorithm. The method retains feasibility by projecting the resulting points onto a feasible set. The a.s. convergence of AN-SPS method is proved without the assumption of a bounded feasible set or bounded iterates. Preliminary numerical results on Hinge loss problems reveal the advantages of the proposed adaptive scheme. In addition, a study of different nonmonotone line search strategies in combination with different spectral coefficients within AN-SPS framework is also conducted, yielding some hints for future work.

A logistics provider’s profit maximization facility location problem with random utility maximizing followers

We introduce a strategic decision-making problem faced by logistics providers (LPs) seeking facility location decisions that lead to profitable operations. The profitability depends on the revenue generated through agreements with shippers, and the costs arising when satisfying these agreements. The latter depends, in turn, on the service levels and characteristics of the shippers’ customers. However, at a strategic level, the LP has imperfect information thereof.We propose a stochastic bilevel formulation where a given LP (leader) anticipates the decisions of shippers (followers) arising from a random utility maximization model. Using a sample average approximation and properties of the associated optimal solutions, we propose a heuristic that can compute high-quality solutions. It is based on a reduced single-level mixed integer linear programming formulation that can be solved by a general-purpose solver after a preprocessing phase. We can quickly identify situations that lead to zero expected profit for the LP. Experimental results show that expected profit highly depends on shippers’ price sensitivities. Underestimating price sensitivities can lead to an overestimation of the expected profit.

An enhanced stochastic optimization for more flexibility on integrated energy system with flexible loads and a high penetration level of renewables

Full utilization of the potential regulation ability of flexible loads in an integrated energy system (IES) and expanding its structure for more flexibility via integrating auxiliary devices is the key to realizing its high-efficiency operation and high-proportion renewable consumption. The present paper builds the flexible regulation models of thermal and electrical loads and introduces them to the structure expansion planning of IES. A two-stage stochastic probability optimization method integrating the uncertain operation of introducing flexible loads is then proposed to balance the additional costs of device integration and the benefits of performance promotion. Herein, an enhanced sample average approximation based on a stochastic hierarchy scenario generation method is developed to solve the optimization. The regulation mechanisms of flexible loads and their influence on expansion planning are then analyzed by comparing the optimized results. The results show that the path to decreasing the cost via introducing flexible thermal and electrical loads into optimization is increasing renewable consumption and coordinating an increment in renewable consumption and average electrical efficiency, respectively. The regulation between them has synergism on cost reduction and saturation on increasing renewable consumption. The synergism can reduce the cost by 0.62 %. Besides, there is a synergism between electric boiler integration and flexible loads, which further reduces the total cost of the expanded IES by 24.86 %.

A stochastic programming approach to the antibiotics time machine problem

Antibiotics Time Machine is an important problem to understand antibiotic resistance and how it can be reversed. Mathematically, it can be modeled as follows: Consider a set of genotypes, each of which contain a set of mutated and unmutated genes. Suppose that a set of growth rate measurements of each genotype under a set of antibiotics is given. The transition probabilities of a ‘realization’ of a Markov chain associated with each arc under each antibiotic are computable via a predefined function given the growth rate realizations. The aim is to maximize the expected probability of reaching to the genotype with all unmutated genes given the initial genotype in a predetermined number of transitions, considering the following two sources of uncertainties: (i) the randomness in growth rates, (ii) the randomness in transition probabilities, which are functions of growth rates. We develop stochastic mixed-integer linear programming and dynamic programming approaches to solve static and dynamic versions of the Antibiotics Time Machine Problem under the aforementioned uncertainties. We adapt a Sample Average Approximation approach that exploits the special structure of the problem and provide accurate solutions that perform very well in an out-of-sample analysis.

Resource allocation and route planning under the collection price uncertainty for the biomass supply chain

The escalating severity of global warming has drawn worldwide attention to ecological problems. Nevertheless, with the introduction of environmental policies, biomass resources have been effectively developed as a renewable energy source. This paper investigates an advanced biomass supply chain design wherein biomass resources are initially converted into bio-oil through widely distributed fast pyrolysis facilities, and are subsequently transported to a centralised biorefinery for further refining into biofuels. This novel biomass supply chain addressed three key issues: (1) The number of fast pyrolysis facilities, (2) The allocation of resources, and (3) The routes of resources transport. In respect of these problems, a two-stage stochastic mixed integer programming model is established to minimise the total cost of the biomass supply chain considering the uncertainty collection price of fast pyrolysis facilities. A hybrid simulated annealing algorithm which incorporates the sample average approximation method is proposed to solve the stochastic model and is effectiveness for large-scale examples. Finally, a sensitivity analysis is performed using the proposed algorithm and the results show that the proposed stochastic model outperforms the deterministic model under uncertain collection price. The model allows optimising the biomass supply chain economic performances and minimise financial risk on investment by determining the fast pyrolysis facility locations, reasonable resource allocation and optimal transport routes under uncertain collection price.

Asymptotics of Average Case Approximation Complexity for Tensor Products of Euler Integrated Processes

Asymptotics of Average Case Approximation Complexity for Tensor Products of Euler Integrated Processes

A sample average approximation-based heuristic for the stochastic production routing problem

AbstractThe Production Routing Problem under demand uncertainty is an integrated problem containing production, inventory, and distribution decisions. At the planning level, the aim is to meet retailers demand, when only the demand distribution is known in advance, while minimizing the corresponding costs. In this study, a two-stage formulation is presented in which the routing can be adjusted at short notice. In the first stage, only production decisions are made, while delivery and inventory quantities and retailer visit schedules are determined in the second stage. To handle a large number of scenarios, two solution methods based on Sample Average Approximation are introduced. Furthermore, the impact of the routing quality is explored by applying a simple heuristic and an effective metaheuristic on the routing part. It is shown that, on average, the simple heuristic within an adjustable Sample Average Approximation approach provides better objective function values than the metaheuristic within a non-adjustable approach. Also all solution approaches outperform an expected value based approach in terms of runtime and objective function value.

Learning Fair Policies for Multi-Stage Selection Problems from Observational Data

We consider the problem of learning fair policies for multi-stage selection problems from observational data. This problem arises in several high-stakes domains such as company hiring, loan approval, or bail decisions where outcomes (e.g., career success, loan repayment, recidivism) are only observed for those selected. We propose a multi-stage framework that can be augmented with various fairness constraints, such as demographic parity or equal opportunity. This problem is a highly intractable infinite chance-constrained program involving the unknown joint distribution of covariates and outcomes. Motivated by the potential impact of selection decisions on people’s lives and livelihoods, we propose to focus on interpretable linear selection rules. Leveraging tools from causal inference and sample average approximation, we obtain an asymptotically consistent solution to this selection problem by solving a mixed binary conic optimization problem, which can be solved using standard off-the-shelf solvers. We conduct extensive computational experiments on a variety of datasets adapted from the UCI repository on which we show that our proposed approaches can achieve an 11.6% improvement in precision and a 38% reduction in the measure of unfairness compared to the existing selection policy.

A data-driven newsvendor model for elective-emergency admission control under uncertain inpatient bed capacity.

Elective-emergency admission control referred to allocating available inpatient bed capacity between elective and emergency hospitalization demand. Existing approaches for admission control often excluded several complex factors when making decisions, such as uncertain bed capacity and unknown true probability distributions of patient arrivals and departures. We aimed to create a data-driven newsvendor framework to study the elective-emergency admission control problem to achieve bed operational efficiency and effectiveness. We developed a data-driven approach that utilized the newsvendor framework to formulate the admission control problem. We also created approximation algorithms to generate a pool of candidate admission control solutions. Past observations and relevant emergency demand and bed capacity features were modeled in a newsvendor framework. Using approximation algorithmic approaches (sample average approximation, separated estimation and optimization, linear programing-LP, and distribution-free model) allowed us to derive computationally efficient data-driven solutions with tight bounds on the expected in-sample and out-of-sample cost guaranteed. Tight generalization bounds on the expected out-of-sample cost of the feature-based model were derived with respect to the LP and quadratic programing (QP) algorithms, respectively. Results showed that the optimal feature-based model outperformed the optimal observation-based model with respect to the expected cost. In a setting where the unit overscheduled cost was higher than the unit under-scheduled cost, scheduling fewer elective patients would replace the benefit of incorporating related features in the model. The tighter the available bed capacity for elective patients, the bigger the difference of the schedule cost between the feature-based model and the observation-based model. The study provides a reference for the theoretical study on bed capacity allocation between elective and emergency patients under the condition of the unknown true probability distribution of bed capacity and emergency demand, and it also proves that the approximate optimal policy has good performance.

A chance-constrained optimization approach integrating project scheduling and material ordering to manage the uncertain material supply

To deal with the impact of uncertain material supply on the implementation of projects, we investigate a problem of concurrently finding a robust baseline schedule and a material ordering plan under different levels of uncertainty. Assuming the material warehouses are restricted, a chance-constrained model of the integrated resource-constrained project scheduling and material ordering problem with a confidence level (1−θ) is thus formulated. To tackle the difficult joint chance constraints, we firstly transform the chance-constrained model into a scenario-based integer programming model by the sample average approximation approach. Based on dominance relationship among scenarios, we then derive an exact branch-and-bound algorithm and a sampling-based genetic algorithm (SGA) to solve the model. For the branch-and-bound algorithm, a tailor-made branching scheme and four pruning rules are developed to accelerate the search process. For the SGA, two sampling methods and a local search procedure are presented to boost solution quality. Furthermore, extensive numerical experiments are carried out to test the performance of the proposed algorithms. Specifically, 4608 instances with four different confidence levels and six different sample sizes are generated. Then the Taguchi method is employed to calibrate the parameters of the SGA. Besides, the numerical results demonstrate that the proposed branch-and-bound algorithm can solve more instances with shorter CPU times than the CPLEX solver, and the solution quality of the proposed SGA is far better than that of two existing meta-heuristics and two variants of SGA. Finally, a sensitivity analysis for the key parameters of the model is conducted. By considering the uncertain material supply, limited warehouse capacity and various confidence levels, our research contributes to the effective decision on project scheduling and material ordering in real-world contexts.

Stochastic Modelling Frameworks for Dragon Fruit Supply Chains in Vietnam under Uncertain Factors

Managing uncertainties and risks is always a difficult but fascinating task in fresh fruit supply chains, especially when dealing with the strategy for the production and conveyance of fresh fruit in Vietnam. Following the COVID-19 outbreak, the confluence of economic recession and persistent adverse weather conditions has exacerbated challenges faced by dragon fruit cultivators. This research investigates a two-stage stochastic programming (TSSP) approach which is developed and served as a valuable tool for analyzing uncertainties, optimizing operations, and managing risks in the fresh fruit industry, ultimately contributing to the sustainability and resilience of supply chains in the agricultural sector. A prototype is provided to illustrate the complex and dynamic nature of dragon fruit cultivation and consumption in Vietnam. Data on the selling prices of dragon fruit were collected from several sources between 2013 and 2022 in Binh Thuan Province, Vietnam. The results were obtained from the model by using three different approaches in order of their versatility and efficacy: (1) Scenario tree generation; (2) Sample average approximation; (3) Chance-constrained programming.

Stochastic optimal investment strategy for net‐zero energy houses

AbstractIn this research, Net‐Zero Energy Houses (ZEHs) is investigated that harness regionally produced electricity from photovoltaic (PV) panels and fuel cells, integrating them into a local power system in pursuit of achieving carbon neutrality. This article examines the impact of electricity sharing among users who are working towards attaining ZEH status through the integration of PV panels and battery storage devices. Two potential scenarios are proposed: the first assumes that all users individually invest in storage devices, hence minimizing their costs on a local level without energy sharing; the second envisions cost minimization through the collective use of a shared storage device, managed by a central manager. These two scenarios are formulated as a stochastic convex optimization and a cooperative game, respectively. To tackle the stochastic challenges posed by multiple random variables, the Monte Carlo sample average approximation (SAA) is applied to obtain solvable deterministic convex problems. Last, to demonstrate the practical applicability of these models, the proposed scenarios are implemented in the Jono neighborhood in Kitakyushu, Japan.

Stochastic ambulance dispatching and routing in mass casualty incident under road vulnerability

This paper investigates the stochastic ambulance dispatching and routing (SADR) problem with multiple casualty collection points in the mass casualty incidents (MCI) under uncertainty of road vulnerability and traffic congestion caused by an earthquake. A two-stage stochastic mixed-integer nonlinear programming model is formulated to derive a reliable and high-quality ambulance scheduling, dispatching and routing solution for maximizing expected number of survivors of all casualties. An integrated simulation optimization approach combining a data-driven travel time scenario generation algorithm, sample average approximation, and a column-generation-based heuristic method is developed to efficiently solve this complicated two-stage SADR model with a nonlinear objective function and continuous travel time distributions. Collaborating with the National Science and Technology Center for Disaster Reduction (NCDR), an empirical study using a potential real-world earthquake scenario occurring in Taiwan is conducted to demonstrate the usefulness and effectiveness of our proposed model and solution approach compared to the deterministic model and two current-practice heuristics.

Computational and experimental approaches into molecular structure mechanism of ZQV coal and the COx gas releases during pyrolysis

Understanding the structural features of coal and its reactive behaviors in thermal processing is the fundamental to achieve high-efficiency utilization of coal. However, the molecular structural features and the evolution characteristics at atomic scales in thermal processing remain unclear in the microscopic views, particularly for the low-rank coals, which usually exhibits high oxygen content with a variety of oxygen-containing organic functional groups and gets lots of CO/CO2 release during pyrolysis. This research provides a comprehensive analysis of the ZaoQuan vitrinite (ZQV) bituminous coal's structural attributes and its pyrolysis-induced reactivity, which is crucial for optimizing the thermal conversion of low-rank coals. The structural parameters characteristics and molecular structures of ZQV have been revealed based on series of materials characterizations and molecular modeling calculations. The single molecular description of ZQV was conceptualized to be C209H132O41N2 under the average molecular approximation, and the particle models with size about 10 nm was also been obtained. Based on the built molecular models, reactive force field molecular dynamics (ReaxFF MD) simulations then provided insights into the dynamic behavior of molecular structures under pyrolysis conditions. Experimental validation corroborates the computational predictions, revealing a strong coherence between observed gas products and those derived from simulation data. Crucially, this study vividly captures the dynamic migration patterns of oxygen-containing groups during coal's thermal decomposition, thus contributing to the understanding of gasification processes. The release mechanisms for CO and CO2 are systematically dissected, proposing three and two distinct pathways respectively. These findings not only consolidate the molecular narrative of coal's structure and pyrolysis reactivity but also serve as a reliable foundation for interpreting the COx evolution of low-rank coals. Our integrated approach propels the scientific comprehension of coal pyrolysis and is anticipated to inform the development of more effective coal utilization pathways and thermal treatment technologies.

Theory-guided materials design of multiphase alloys with superior stiffness at finite temperatures

Aiming at designing Mg alloys with a significantly increased stiffness at elevated temperatures, we propose a high-precision multi-scale methodology to compute temperature-dependent elastic modulus of homogeneous multiphase alloys. The proposed method combines (i) first-principles calculations of temperature- and solid solubility-dependent elastic properties in individual phases with (ii) a two-level multiphase homogenization at the continuum level and (iii) a phenomenological thermodynamic modeling by the CALPHAD approach. According to the post-average approximation during the calculation, the stresses and strains of individual phases in the alloy are superimposed before undergoing the average approximation. This is different from the traditional method which consider forces as acting axially assumptions of isostrain and isostress model and followed by numerical averaging. We applied the proposed method in light-weight Mg-alloys, in particular, a three-phase alloy with the composition Mg0.878Al0.083Si0.039, and verified it through experimental measurements. The present work identifies that MgxAlySi1-x-y with 0 ≤ x ≤ 0.87 and 0 ≤ y ≤ 0.87-x are candidate compositions with Young's modulus exceeding 60 GPa at 623 K.

Global multi-sourcing network design with inventory planning under uncertainty

ABSTRACT In the current era of turbulence, demand uncertainty and significant supply delays have become recurrent challenges for every global supply chain, which are accentuated by geopolitical issues, regional conflicts, natural disasters, etc. Companies are keen to decide when and where nearshoring and offshoring their supply locations and to adapt their inventory control policies to cope with significant perturbations. To deal with these challenges, we investigate the integration of inventory planning at the strategic level through a novel setting of the location-inventory problem under uncertainty. This problem is characterised by inbound multi-sourcing, order splitting among multiple suppliers, and optimised inventory decisions from sourcing to customers under uncertain demand and lead time. A two-stage stochastic optimisation model is proposed and solved using a Benders decomposition algorithm, supported by a sample average approximation approach. The results show that both lead time and demand uncertainties impact strategic and tactical decisions of the global supply network. Specifically, multi-sourcing and inbound order splitting policies have proven beneficial when the network faces a high lead time uncertainty. Based on an illustrative case of a global automobile spare parts supply network under different supply-side scenarios, managerial insights are gained on the benefit of onshore, nearshore, and offshore decisions. When facing a high lead time uncertainty, as in the pandemic case or any geopolitical episode, our findings suggest the necessity to rely on nearshoring for the supply side.

Distributionally robust planning method for expressway hydrogen refueling station powered by a wind-PV system

Hydrogen fuel has received widespread attention as one of the critical solutions to global energy issues. The popularization of hydrogen fuel cell vehicles (HFCV) requires the deep integration of transportation and energy fields. Considering the complementarity of wind and PV power in remote areas, this paper proposes a planning model of an expressway hydrogen refueling station, including a hydrogen production system (HPS) powered by a wind-PV system and several hydrogen refueling systems (HRSs). The interaction and operation constraints of hydrogen production, storage, delivery, and demand are employed to complete hydrogen supply chain integration in the deterministic planning model. The spatiotemporal distribution of refueling demand at each HRS is obtained by Monte Carlo Simulation. Furthermore, we consider the impact of uncertainty related to renewable energy generation and refueling demand on planning and depict them by ambiguity sets based on Kullback Leibler divergence. The planning problem is transformed into a data-driven distributionally robust model and solved by Value-at-Risk and sample average approximation method. Results indicate that 1) The proposed model can simultaneously collaborate with the segments of the hydrogen supply chain. 2) The proposed method balances the planning reliability and economic efficiency and enhances solution robustness.

Multi-period reverse logistics network design for water resource management in hydraulic fracturing

This paper addresses the effective management of water resources during the hydraulic fracturing process used for natural gas extraction. With increasing demand and scrutiny on shale gas production due to economic, environmental, and social factors, effective water management strategies are needed to address the challenges of flow-back water usage during hydraulic fracturing. This research contributes to the existing literature from a multi-period reverse logistics network design perspective with realistic modeling approaches. A mathematical formulation is developed for optimizing the design of a multi-period reverse logistics network for effective water resource management that incorporates production scheduling of a hydraulic fracturing process. This mixed integer programming model determines the optimal locations and capacities of impoundments, treatment facilities, and disposal sites while minimizing fixed and operational costs under constraints such as freshwater withdrawal limits, production schedules, inventory balance, and treatment and disposal options. A two-stage stochastic programming formulation is further developed as a means to address the uncertainty associated with water flows during the fracturing operation. The strategic location and capacity decisions are given in the first stage of this formulation, whereas operational decisions such as water flows are handled in the second stage under independently sampled scenarios. A sample average approximation algorithm is proposed to solve the stochastic formulation. The model is tested on a case study from the United States that explores potential impacts on water resource management due to cost variations, changes in operational parameters, and uncertainty in demands and supplies. The base case setup delivers a 12.8% reduction in freshwater usage. The benefit of multi-period reverse logistics network design is further demonstrated through variations in water demand and equipment degradation. The managerial insights derived through sensitivity analysis highlight the values of the proposed multi-period formulation, capacity expansions, and parameter estimations.

Adaptive Sequential Procedures for Quantile Selection with Importance Sampling

In this paper, we consider the problem of identifying the simulated system configuration with optimal quantile performance among a finite set of alternative systems. We first propose adaptive sequential selection procedures via direct simulation, where the point and variance estimators for quantiles are sequentially updated. For the sake of extreme quantiles, we develop adaptive quantile selection procedures using importance sampling. Our procedures apply sample average approximation and stochastic approximation for iteratively computing the optimal importance-sampling parameter and the importance-sampling quantile estimators. The statistical validity of the proposed selection procedures is proven in the asymptotic regime. We particularly investigate the consistency and the central limit convergence using both sample average and stochastic approximation for quantile estimation during the adaptive procedures. Empirical results and an illustrative example show the efficiency of the developed procedures.