Sample Average Approximation Method Research Articles

A Revisit to Sunk Cost Fallacy for Two-Stage Stochastic Binary Decision Making

This paper undertakes a revisit of the sunk cost fallacy, which refers to the tendency of people to persist investing resources into something, even if it is destined to have no good outcome. We emphasize that the utilities associated with different alternatives are not static for decision makers, which is exactly opposite to the traditional perspective. This paper argues that the utility of an option may change due to the choice of another option, suggesting that decisions considered irrational by the traditional analytical method, i.e., sunk cost fallacy, may be rational. We propose a novel analytical method for decision making with sunk cost when considering the utility change and validate the effectiveness of this method through mathematical modeling and computational experiments. This paper mathematically describes such decision-making problems, analyzing the impact of changes in the utilities across different alternatives on decision making with a real-world example. Furthermore, we develop a two-stage stochastic optimization model for such decision-making problems and employ the sample average approximation (SAA) method to solve them. The results from computational experiments indicate that some decisions traditionally considered irrational are, in fact, rational when the utility of an option changes as a result of choosing another option. This paper, therefore, highlights the significance of incorporating utility changes into the decision-making process and stands as a valuable addition to the literature, offering a refreshed and effective decision-making method for improved decision making.

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.

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.

AI-enhanced robust method for integrated healthcare resource pre-positioning and patient scheduling

This paper aims to optimise healthcare resource pre-positioning, patient scheduling, and patient transferring under uncertain demands and stochastic resource consumption. We propose a two-stage stochastic programming model that formulates the patient scheduling problem as a Markov decision process. To address complexities and uncertainties, we use Artificial Intelligence (AI) techniques to improve both model formulation and algorithmic performance. To tackle the limited data challenge, we introduce a Wasserstein distance-based ambiguity set and propose a two-stage distributionally robust optimisation (DRO) model, which derives a deterministic equivalent using the Lagrangian dual of non-anticipativity constraints. The solution process is accelerated with a scenario decomposition approach and the K-means clustering method. Both theoretical and numerical results demonstrate the consistency of the two-stage DRO model with the sample average approximation (SAA) method. The potential of AI to improve the model's performance is evident through the significant reduction in computation time achieved with the K-means clustering approach without compromising solution quality. Compared to the SAA method, the DRO model exhibits a considerable reduction in the waiting penalty cost for out-of-sample cases, ranging from 43.06 % to 81.23 % . Numerical results show that the proposed algorithm outperforms SAA methods and several benchmark policies in terms of computational efficiency and solutions.

Multi-period time window assignment for attended home delivery

We study a multi-period stochastic variant of the Time Window Assignment Vehicle Routing Problem, where customers’ demands, locations, and service times are uncertain. Customers are partitioned into geographical zones, each of which has to be visited a predetermined number of times over a planning period of several days. Whenever a zone is visited, a time window is assigned. Time windows are decided before knowing customers and their demands. A fleet of homogeneous vehicles is available to serve customers each day. At a tactical level, the problem looks for a static time window assignment that minimizes the expected traveling costs plus the expected penalty costs for unserved customers. We propose a two-stage formulation and a solution approach, which relies on the Sample Average Approximation Method, while encompassing a perturbation method to assign time windows in the first stage and an Adaptive Large Neighborhood Search to optimize routes in the second stage. We experimentally evaluate three instance sets, including real ones from a Canadian company, comparing our results to lower bounds from the exact solution of a deterministic equivalent formulation over a finite number of scenarios. Our method outperforms the manual approach used by the company.

Fourth-party logistics network design with demand surge: A greedy scenario-reduction and scenario-price based decomposition algorithm

The sharing economy has offered new viable approaches for firms to tackle demand surges. In this paper, we study a novel network design problem for the fourth-party logistics network (4PLN) to cope with unpredictable demand surges, in which temporary renting of logistics resources from a sharing market is available to supplement network capacity. A chance-constrained stochastic programming model is established to minimize the overall cost for 4PLN under service level targets. To deal with the difficulties brought by the evaluation of service level, we reformulate a mixed-integer linear programming (MILP) model, that can be solved straightly, based on the sample average approximation method. Then, to address the enormous challenge posed by the coupling of the basic NP-hard network design problem and the large number of demand scenarios in the MILP version, the Scenario-Price based Decomposition Algorithm (P-DA) is designed based on the key idea of decomposing the above-coupled factors. Further, to mitigate the performance deterioration brought on by large system scale and/or sample size, we expand our base algorithm to the Greedy Scenario-Reduction and Scenario-Price based Decomposition Algorithm (GR&P-DA) through the fast processing of chance constraints by introducing a greedy method. Through computational experiments, we demonstrate the effectiveness of our proposed model and algorithms, and we analyze the impact of various model parameters, including demand level, surge magnitude, surge probability, and rental price of third-party logistics resources, on 4PLN design. Furthermore, our comparative analysis reveals that renting logistics resources to address demand surges can significantly reduce costs and discovers that deploying resources in advance does not always provide an advantage, a temporary “case-by-case” planning approach for renting will give a more cost-saving scheme when surge probability is at a low level as it avoids idle resources by not being stuck with a conservative strategy.

Design of micromachines under uncertainty with the sample-average approximation method

Design of micromachines under uncertainty with the sample-average approximation method

First order asymptotics of the sample average approximation method to solve risk averse stochastic programs

AbstractWe investigate statistical properties of the optimal value of the Sample Average Approximation of stochastic programs, continuing the study (Krätschmer in Nonasymptotic upper estimates for errors of the sample average approximation method to solve risk averse stochastic programs, 2023. Forthcoming in SIAM J. Optim.). Central Limit Theorem type results are derived for the optimal value. As a crucial point the investigations are based on a new type of conditions from the theory of empirical processes which do not rely on pathwise analytical properties of the goal functions. In particular, continuity or convexity in the parameter is not imposed in advance as usual in the literature on the Sample Average Approximation method. It is also shown that the new condition is satisfied if the paths of the goal functions are Hölder continuous so that the main results carry over in this case. Moreover, the main results are applied to goal functions whose paths are piecewise Hölder continuous as e.g. in two stage mixed-integer programs. The main results are shown for classical risk neutral stochastic programs, but we also demonstrate how to apply them to the Sample Average Approximation of risk averse stochastic programs. In this respect we consider stochastic programs expressed in terms of absolute semideviations and divergence risk measures.

A two-stage stochastic model for an industrial symbiosis network under uncertain demand

Industrial Symbiosis (IS) networks are structures built by volunteer companies with the aim of exchanging unused or residual resources, benefiting all participating companies. The profitability of these volunteer companies is critical as it affects the sustainability of these networks. Fluctuations in a participating company's production level can potentially disrupt its network by altering the quantity and availability of wastes and by-products. In light of these considerations, we analyse the impact of fluctuations in demand for final products of the companies on company profitability, and waste and by-product usage. For this purpose, we formulated a two-stage stochastic programming model and solved it using the Sample Average Approximation (SAA) method. We tested our model on a theoretical IS network comprising companies in the forest products industry. The results demonstrate that companies in the network keep exchanging by-products and remain profitable despite uncertainties in demand. Consequently, we conclude that the established network exhibits resilience to demand fluctuations, which is an important aspect of its sustainability.

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.

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.

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.

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.

A new and general stochastic parallel machine ScheLoc problem with limited location capacity and customer credit risk

Scheduling-Location (ScheLoc) problem considering machine location and job scheduling simultaneously is a relatively new and hot topic. The existing works assume that only one machine can be placed at a location, which may not be suitable for some practical applications. Besides, the customer credit risk which largely impacts the manufacturer’s profit has not been addressed in the ScheLoc problem. Therefore, in this work, we study a new and general stochastic parallel machine ScheLoc problem with limited location capacity and customer credit risk. The problem consists of determining the machine-to-location assignment, job acceptance, job-to-machine assignment, and scheduling of accepted jobs on each machine. The objective is to maximize the worst-case probability of manufacturer’s profit being greater than or equal to a given profit (referred to as the profit likelihood). For the problem, a distributionally robust chance-constrained (DRCC) programming model is proposed. Then, we develop two model-based approaches: (1) a sample average approximation (SAA) method; (2) a model-based constructive heuristic. Numerical results of 300 instances adapted from the literature show the average profit likelihood proposed by the constructive heuristic is 9.43% higher than that provided by the SAA, while the average computation time of the constructive heuristic is only 4.24% of that needed by the SAA.

Stochastic planning for low-carbon building integrated energy system considering electric-heat-V2G coupling

The low-carbon building has been proposed to mitigate the climate change caused by environmental problems and realize carbon neutrality in urban areas. In addition, the integrated energy system (IES) has been developed to reduce renewable energy curtailment in the power distribution system and improve energy efficiency due to the independent operation of traditional energy systems. In this paper, we propose a stochastic planning method for low-carbon building IES, in which the Vehicle to Grid (V2G) is also considered to further increase the flexibility of low-carbon buildings. The proposed planning method optimizes the investment and operation costs, and CO2 emission of the building IES, to achieve the maximum benefit of the low-carbon building and help realize carbon neutrality. By considering the uncertainty of distributed renewable energy, multi-energy load fluctuation and the random behavior of EV users, a two-stage stochastic programming model is formulated with chance constraints, in which the heuristic moment matching scenario generation (HMMSG) and sample average approximation (SAA) methods are applied to deal with the uncertainties. In the case study, a real IES office building in Shanghai, where photovoltaic (PV), energy storage system (ESS), fuel cell (FC), EV, etc. are included as planning options, is used as the test system to verify the effectiveness of the proposed planning method, and the functions of the ESS and EV in IES are analyzed in detail in different operation scenarios. The case study results show that the proposed planning scheme can reduce the total cost and carbon emission significantly.© 2017 Elsevier Inc. All rights reserved.

Solving the stochastic team orienteering problem: comparing simheuristics with the sample average approximation method

AbstractThe team orienteering problem (TOP) is an NP‐hard optimization problem with an increasing number of potential applications in smart cities, humanitarian logistics, wildfire surveillance, etc. In the TOP, a fixed fleet of vehicles is employed to obtain rewards by visiting nodes in a network. All vehicles share common origin and destination locations. Since each vehicle has a limitation in time or traveling distance, not all nodes in the network can be visited. Hence, the goal is focused on the maximization of the collected reward, taking into account the aforementioned constraints. Most of the existing literature on the TOP focuses on its deterministic version, where rewards and travel times are assumed to be predefined values. This paper focuses on a more realistic TOP version, where travel times are modeled as random variables, which introduces reliability issues in the solutions due to the route‐length constraint. In order to deal with these complexities, we propose a simheuristic algorithm that hybridizes biased‐randomized heuristics with a variable neighborhood search and MCS. To test the quality of the solutions generated by the proposed simheuristic approach, we employ the well‐known sample average approximation (SAA) method, as well as a combination model that hybridizes the metaheuristic used in the simheuristic approach with the SAA algorithm. The results show that our proposed simheuristic outperforms the SAA and the hybrid model both on the objective function values and computational time.

Stochastic green profit-maximizing hub location problem

This article proposes a two-stage stochastic profit-maximizing hub location problem (HLP) with uncertain demand. Additionally, the model incorporates several carbon regulations, such as carbon tax policy (CTP), carbon cap-and-trade policy (CCTP), carbon cap policy (CCP), and carbon offset policy (COP). In the proposed models, an enhanced sample average approximation (ESAA) method was used to obtain a suitable number of scenarios. To cluster similar samples, k-means clustering and self-organizing map (SOM) clustering algorithms were embedded in the ESAA. The L-shaped algorithm was employed to solve the model inside the ESAA method more efficiently. The proposed models were analyzed using the well-known Australian Post (AP) data set. Computational experiments showed that all of the carbon regulations could reduce overall carbon emissions. Among carbon policies, CCTP could achieve better economic results for the transportation sector. The results also demonstrated that the SOM clustering algorithm within the ESAA method was superior to both k-means inside ESAA and classical SAA algorithms according to the %gap and standard deviation measures. In addition, the results showed that the L-shaped algorithm performed better than the commercial solver in large-scale instances.

Mixed-Integer Conic Formulation of Unit Commitment with Stochastic Wind Power

Due to the high randomness and volatility of renewable energy sources such as wind energy, the traditional thermal unit commitment (UC) model is no longer applicable. In this paper, in order to reduce the possible negative effects of an inaccurate wind energy forecast, the chance-constrained programming (CCP) method is used to study the UC problem with uncertainty wind power generation, and chance constraints such as power balance and spinning reserve are satisfied with a predetermined probability. In order to effectively solve the CCP problem, first, we used the sample average approximation (SAA) method to transform the chance constraints into deterministic constraints and to obtain a mixed-integer quadratic programming (MIQP) model. Then, the quadratic terms were incorporated into the constraints by introducing some auxiliary variables, and some second-order cone constraints were formed by combining them with the output characteristics of thermal unit; therefore, a tighter mixed-integer second-order cone programming (MISOCP) formulation was obtained. Finally, we applied this method to some systems including 10 to 100 thermal units and 1 to 2 wind units, and we invoked MOSEK in MATLAB to solve the MISOCP formulation. The numerical results obtained within 24 h confirm that not only is the MISOCP formulation a successful reformulation that can achieve better suboptimal solutions, but it is also a suitable method for solving the large-scale uncertain UC problem. In addition, for systems of up to 40 units within 24 h that do not consider wind power and pollution emissions, the numerical results were compared with those of previously published methods, showing that the MISOCP formulation is very promising, given its excellent performance.

A multidiscipline collaborative optimization approach with acceleration strategies for generator maintenance scheduling in large hydrothermal power systems considering uncertainty of natural inflows

Generator maintenance scheduling (GMS) is an important task in the dispatch and operation of hydrothermal power systems. On a long-term timescale, the uncertainty of natural inflows makes GMS essentially a stochastic optimization problem. The sample average approximation (SAA) method is used to convert the stochastic hydro and thermal generator maintenance model into a large-scale deterministic optimization problem, but it is difficult to solve. This paper proposes a multidisciplinary collaborative optimization approach with acceleration strategies (MCO-AS) for GMS, where the deterministic optimization problem is decomposed into a system-level optimization problem and discipline-level optimization subproblems through decoupling of the nonanticipative constraints and the maintenance variable consistency constraints. Three acceleration strategies, replacement and linearization, information pooling, and parallel calculation, are proposed and incorporated. Finally, computational experiments on a real large hydrothermal power system are conducted, and the results illustrate the effectiveness of the approach and acceleration strategies proposed in this paper compared to the traditional SAA method as a benchmark. • Stochastic maintenance scheduling is considered in large hydrothermal power systems. • A multidisciplinary collaborative approach is designed to decompose the problem. • Three strategies are proposed to further accelerate the solution. • Results on a real-world power system show the effectiveness of the proposed method.

A stochastic optimization approach for staff scheduling decisions at inpatient units

AbstractThis paper describes a solution approach for stochastic multi‐resource multi‐patient staff scheduling problems in inpatient units. Our solution approach has four steps. First, we classify patients into a number of groups with similar care‐provider requirements. Second, a predictive Markov model captures patients' flow in the inpatient unit and provides a prediction of the number of patients of each group in the future. This predictive model allows us to generate a potentially large set of possible system utilization scenarios over the planning horizon. Third, a mixed‐integer programming model with an expected value objective function seeks to minimize the expected over‐staffing and under‐staffing costs across all possible scenarios. Lastly, we use simulation to sample system utilization scenarios and the sample average approximation method to find a reliable and generalizable solution to the model across all possible scenarios. We evaluate the performance of the proposed solution using real data from the Children's Hospital of Eastern Ontario's inpatient mental health unit. The results show that the proposed approach significantly decreases the expected cost of the schedules in comparison to the traditional approaches.