Second-stage Problem Research Articles

Efficient optimization algorithms for surgical scheduling under uncertainty

In this paper, we develop a stochastic optimization model for a surgical scheduling problem considering a single operating room. We arrange a set of elective surgeries into appropriate time blocks, and determine their planned start time and specific sequence. Due to the complexity of the original formulation, we reformulate our model as a two-stage mixed-integer problem. We consider the planning decision in the first stage and the sequencing decision in the second stage (based on the first one). The goal of this paper is to obtain a nearly optimal schedule in reasonable computational time. The term “optimal” is defined as the lowest surgically related cost while achieving the given threshold with respect to some specific deterministic or stochastic performance measures.The optimization model involves expected and probabilistic formulations that are analytically intractable. This implies that traditional mathematical programming techniques cannot be used directly. Therefore, we propose adapted rapid-screening and stochastic-approximation algorithms to deal with the first-stage and the second-stage problems, respectively. In both algorithms, we can apply either the Laplace transform or simulation methods to either evaluate or estimate the desired performance measures. The experimental results demonstrate that the proposed algorithms are more favorable compared to existing approaches.

Two-Stage Robust Optimization for the Orienteering Problem with Stochastic Weights

In this paper, the two-stage orienteering problem with stochastic weights is studied, where the first-stage problem is to plan a path under the uncertain environment and the second-stage problem is a recourse action to make sure that the length constraint is satisfied after the uncertainty is realized. First, we explain the recourse model proposed by Evers et al. (2014) and point out that this model is very complex. Then, we introduce a new recourse model which is much simpler with less variables and less constraints. Based on these two recourse models, we introduce two different two-stage robust models for the orienteering problem with stochastic weights. We theoretically prove that the two-stage robust models are equivalent to their corresponding static robust models under the box uncertainty set, which indicates that the two-stage robust models can be solved by using common mathematical programming solvers (e.g., IBM CPLEX optimizer). Furthermore, we prove that the two two-stage robust models are equivalent to each other even though they are based on different recourse models, which indicates that we can use a much simpler model instead of a complex model for practical use. A case study is presented by comparing the two-stage robust models with a one-stage robust model for the orienteering problem with stochastic weights. The numerical results of the comparative studies show the effectiveness and superiority of the proposed two-stage robust models for dealing with the two-stage orienteering problem with stochastic weights.

Study on optimal design of hatch cover via a three-stage optimization method involving material selection, size, and plate layout arrangement

The rising prices of materials for ship construction is one of the primary problems faced by global shipbuilding industries. A potential solution is to reduce the material cost through a structure optimization technique. In this study, the plate material and stiffener types, plate thickness, and plate layout are optimized in order to minimize the cost of materials. The genetic algorithm (GA) is a popular optimization technique for determining design variables. However, the GA is not suitable for dealing with continuously changing design variables, such as plate thickness. Hence, this study proposes a three-stage optimization method that consists of the optimization of plate material and stiffener types, plate thickness, and plate layout. The first and second stages are part of an integrated process called the hybrid GA, which starts with the plate material and stiffener-type selections, and then performs plate thickness optimization. The hybrid GA is effective in simultaneously determining design variables with different characteristics, such as the first - and second-stage problems. The last stage is layout optimization in which the plate material and stiffener types handled in the previous stage are considered to have been decided, thereby resulting in minimal effort during this stage.

A Stochastic Integer Programming Approach to Air Traffic Scheduling and Operations

Air traffic management measures comprise tactical operating procedures to minimize delay costs and strategic scheduling interventions to control overcapacity scheduling. Although interdependent, these problems have been treated in isolation. This paper proposes an integrated model of scheduling and operations in airport networks that jointly optimizes scheduling interventions and ground-holding operations across airports networks under operating uncertainty. It is formulated as a two-stage stochastic program with integer recourse. To solve it, we develop an original decomposition algorithm with provable solution quality guarantees. The algorithm relies on new optimality cuts—dual integer cuts—that leverage the reduced costs of the dual linear programming relaxation of the second-stage problem. The algorithm also incorporates neighborhood constraints, which shift from exploration to exploitation at later stages. We also use a scenario generation approach to construct representative scenarios from historical records of operations—using integer programming. Computational experiments show that our algorithm yields near-optimal solutions for the entire U.S. National Airspace System network. Ultimately, the proposed approach enhances airport demand management models through scale integration (by capturing network-wide interdependencies) and scope integration (by capturing interdependencies between scheduling and operations).

A Stochastic Optimization Framework for Channel Bonding in Wireless LANs Under Demand Uncertainty

Channel bonding is one promising approach to cope with rising WLAN data demand, given scarce spectrum resources. An access point (AP) can aggregate multiple contiguous channels to satisfy demand. We discuss how to optimally utilize available frequency bands under uncertainty in AP demands using two stochastic optimization frameworks: a static scheme which minimizes the total occupied bandwidth while satisfying the demand of each AP with probability at least β, and an adaptive scheme that allows adaptability of the bandwidth allocation in response to the AP demand variations. Given its complexity, we propose a novel framework to solve the adaptive stochastic optimization problem efficiently. The proposed framework exploits the special structure of the problem through decomposition into two subproblems. A particle swarm optimization (PSO)-based algorithm is tailored to the first-stage problem in order to obtain good solutions. The second-stage problem is further decomposed into several subproblems that can be solved independently in parallel. Our numerical results (i) demonstrate the advantages of stochastic compared to deterministic allocation, (ii) illustrate that the proposed framework reaches the optimal solution for the two-stage problem in few iterations, and (iii) explain the bandwidth-user satisfaction trade-off provided by the adaptive allocation approach.

Optimal sizing of wind/ concentrated solar plant/ electric heater hybrid renewable energy system based on two-stage stochastic programming

Component capacity and energy management strategy are two key issues for the optimal sizing of a hybrid renewable energy system. In this study, a two-stage stochastic programming problem is proposed for the optimal sizing of a hybrid renewable energy system consisted of wind turbine, concentrated solar plant, and electric heater. In the problem, component capacity optimization is the first-stage problem to minimize the levelized cost of energy while satisfying the reliability constraint, and energy management strategy optimization is the second-stage problem to minimize the loss of power supply probability while satisfying the operation constraints. The problem is solved by combining of an improved differential evolution algorithm, namely JADE, and Cplex, and the superiority of JADE is validated by algorithm comparisons with several popular intelligent optimization algorithms. Furthermore, economic benefits of the electric heater in the hybrid system are investigated by techno-economic comparisons with a reference wind turbine/concentrated solar plant hybrid energy system without electric heater under their optimal capacity. The results show the electric heater is beneficial for a lower levelized cost of energy, reducing by 0.004 ($/kWh) and 0.009 ($/kWh) respectively when the loss of power supply probability is 2% and 5%.

Data-driven stochastic programming for energy storage system planning in high PV-penetrated distribution network

Energy storage systems (ESSs) facilitate the reliable and economic operation of distribution systems with high PV penetration. Establishing uncertainty models is the key to the optimal planning and operation of ESSs in distribution systems. Widely used parametric models cannot describe the variability of uncertainties thoroughly. In this paper, a data-driven method is designed for uncertainty modeling, and a distributionally robust optimization (DRO) model is developed to determine the optimal ESS planning strategy in distribution systems. First, a deterministic optimization model is established considering both ESS planning and distribution system operation. Then, the Wasserstein-metric-based ambiguity set is designed for the probability distributions of random variables. To hedge against the distributional ambiguity, the optimization problem is expanded into a two-stage DRO problem, of which the second-stage problem minimizes the expected operating cost under the worst-case probability distribution. Finally, the two-stage DRO model is reformulated as a mixed-integer second-order cone programming (MISOCP) problem, which is solved by optimization solvers. The proposed method is tested on modified 33-bus and 123-bus distribution systems with actual solar irradiance and load data. The influence of different strategies on ESS planning and operation is discussed. The ESS planning results obtained by DRO are compared with those of conventional stochastic optimization and deterministic optimization to verify the superiority of the proposed method.

On a multistage discrete stochastic optimization problem with stochastic constraints and nested sampling

We consider a multistage stochastic discrete program in which constraints on any stage might involve expectations that cannot be computed easily and are approximated by simulation. We study a sample average approximation (SAA) approach that uses nested sampling, in which at each stage, a number of scenarios are examined and a number of simulation replications are performed for each scenario to estimate the next-stage constraints. This approach provides an approximate solution to the multistage problem. To establish the consistency of the SAA approach, we first consider a two-stage problem and show that in the second-stage problem, given a scenario, the optimal values and solutions of the SAA converge to those of the true problem with probability one when the sample sizes go to infinity. These convergence results do not hold uniformly over all possible scenarios for the second stage problem. We are nevertheless able to prove that the optimal values and solutions of the SAA converge to the true ones with probability one when the sample sizes at both stages increase to infinity. We also prove exponential convergence of the probability of a large deviation for the optimal value of the SAA, the true value of an optimal solution of the SAA, and the probability that any optimal solution to the SAA is an optimal solution of the true problem. All of these results can be extended to a multistage setting and we explain how to do it. Our framework and SAA results cover a large variety of resource allocation problems for which at each stage after the first one, new information becomes available and the allocation can be readjusted, under constraints that involve expectations estimated by Monte Carlo. As an illustration, we apply this SAA method to a staffing problem in a call center, in which the goal is to optimize the numbers of agents of each type under some constraints on the quality of service (QoS). The staffing allocation has to be decided under an uncertain arrival rate with a prior distribution in the first stage, and can be adjusted at some additional cost when better information on the arrival rate becomes available in later stages.

Data-Driven Wind Generation Admissibility Assessment of Integrated Electric-Heat Systems: A Dynamic Convex Hull-Based Approach

This paper proposes a data-driven approach to assess the wind generation accommodation capability of the integrated electric-heat system. The overall assessment model is constructed based on the two-stage robust decision-making framework, where the uncertainty of wind generation is described by a convex hull formed by historical data. To take a balance between modeling accuracy and computation tractability, the convex hull is firstly approximated by the intersection of a set of low-dimensional convex hulls, named the approximated convex hull, and then the approximated convex hull based uncertainty set is further simplified by the vertex coefficient discretization treatment. A uniform-compression-expansion scheme for the identified worst-case data samples is derived to realize the evolution of the dynamic convex hull as well as to bridge the gap between the first- and the second-stage problems. A modified column-and-constraint generation algorithm is devised to solve the overall model. Simulation results on two test systems verify the effectiveness and the scalability of the proposed method.

Risk-Loss Coordinated Admissibility Assessment of Wind Generation for Integrated Electric-Gas Systems

Due to the widely deployment of gas-fired units and the emerging power-to-gas (P2G) technologies, the interdependencies between power systems and gas systems have been enhanced, yet the impacts on the ability of accommodating the uncertain and variable wind generation are unclear. On one hand, the regulation capability of gas-fired units might be restricted due to pipeline congestion or gas supply shortage; on the other hand, the P2G facilities could operate as interruptible electrical loads to enhance the operational flexibility of the power network. This paper aims to revisit the wind generation admissibility assessment problem in view of integrated electric-gas systems. The overall model is established according to the two-stage robust optimization framework. By introducing the stepwise penalty coefficients for power shortage and surplus, the risk preference of the operator can be reflected. The mixed integer linear programming-based approximation methods is adopted to tackle the computational challenging Weymouth equation in the gas network, which requires an additional column-and-constraint (C&CG) loop to solve the second-stage max-min problem, other than the common one to coordinate the first- and second-stage problems, resulting in a nested C&CG algorithmic structure. The effectiveness of the proposed model and solution methods are validated by the simulation results.

Robust Day-Ahead Dispatch for Integrated Power-Heat-Gas Microgrid considering Wind Power Uncertainty

Wind power generation has been widely deployed in the modern power system due to the issues of energy crisis and environment pollution. Meanwhile, the microgrid is gradually regarded as a feasible way to connect and accommodate the distributed wind power generations. Recently, more research studies also focus on incorporating various energy systems, for example, heat and gas into the microgrid in terms of satisfying different types of load demands. However, the uncertainty of wind power significantly impacts the economy of the integrated power-heat-gas microgrid. To deal with this issue, this paper presents a two-stage robust model to achieve the optimal day-ahead economic dispatch strategy considering the worst-case wind power scenarios. The first stage makes the initial day-ahead dispatch decision before the observation of uncertain wind power. The additional adjustment action is made in the second stage once the wind power uncertainty is observed. Based on the duality theory and Big-M approach, the original second-stage problem can be dualized and linearized. Therefore, the column-and-constraint generation algorithm can be further implemented to achieve the optimal day-ahead economic dispatch strategy for the integrated power-heat-gas microgrid. The experimental results indicate the effectiveness of the presented approach for achieving operation cost reduction and promoting wind power utilization. The robustness and the economy of the two-stage robust model can be balanced, of which the performances significantly outperform those of the single-stage robust model and the deterministic model.

Mixed Integer Programming models for planning maintenance at offshore wind farms under uncertainty

We introduce the Stochastic Maintenance Fleet Transportation Problem for Offshore wind farms (SMFTPO), in which a maintenance provider determines an optimal, medium-term planning for maintaining multiple wind farms while controlling for uncertainty in the maintenance tasks and weather conditions. Since the maintenance provider is typically not the owner of a wind farm, it needs to adhere minimum service requirements that specify the required service. We consider three of such settings: (1) perform all maintenance tasks, (2) allow for a fraction of unscheduled tasks, and (3) incentivize to perform maintenance rather quickly. We provide a two-stage stochastic mixed integer programming model for the three SMFTPO settings, and solve it by means of Sample Average Approximation. In addition, we provide an overview of the, what we discovered, non-aligned modeling assumptions in the literature regarding operational decisions. By providing a series of special cases of the second-stage problem resembling the different modeling assumptions, we aim to establish a common consensus regarding the key modeling decisions to be taken in maintenance planning problems for offshore wind farms. We provide newly constructed, and publicly available, benchmark sets. We extensively compare the different SMFTPO settings and its special cases on those benchmark sets, and we show that the special case reformulations are very effective for solving the second-stage problems. In addition, we find that for particular cases, established modeling techniques result in overestimations and increased running times.

A Co-Scheduling Problem of Ship Lift and Ship Lock at the Three Gorges Dam

Due to the unprecedentedly increasing demands of waterway transportation on the Yangtze River during the latest years, traffic congestion at the Three Gorges Dam (TGD) has becoming a serious problem. In busiest seasons, a vessel may need to wait up to 3 days for passing the TGD via the ship lift or the five-stage ship lock. In order to improve the navigation efficiency and the utilization of passing facilities, a co-scheduling problem of ship lift and ship lock at the TGD is modeled in this paper. Besides, the improvement on effectiveness, safety and fairness of the navigation scheduling by introducing the ship lift is taken into account. The mathematical model proposed is, by nature, a two-stage problem, where the first-stage variables determine the assignments of passing facilities and the second-stage variables schedule the vessels. In order to solve the proposed mathematical model, a hybrid metaheuristic algorithm is used at the first stage to determine the allocation of vessels to different passing facilities and to different lockage through the maximization of facility utilization rate. The second-stage problem is then solved by CPLEX. The proposed mathematical model and algorithm are validated through a set of numerical experiments and a case study. The results show both the navigation efficiency and the utilization of passing facilities can be improved by using the proposed methods.

Two-stage stochastic variational inequalities for Cournot-Nash equilibrium with risk-averse players under uncertainty

<p style='text-indent:20px;'>A convex two-stage non-cooperative game with risk-averse players under uncertainty is formulated as a two-stage stochastic variational inequality (SVI) for point-to-set operators. Due to the indifferentiability of function <inline-formula><tex-math id="M1">\begin{document}$ (\cdot)_+ $\end{document}</tex-math></inline-formula> and the discontinuity of solution mapping of the second-stage problem, under standard assumptions, we propose a smoothing and regularization method to approximate it as a two-stage SVI in point-to-point case with continuous second stage solution functions. The corresponding convergence analysis is also given.

Box-Based Temporal Decomposition of Multi-Period Economic Dispatch for Two-Stage Robust Unit Commitment

To ensure the feasibility of a unit commitment (UC) schedule under uncertainty, most existing two-stage robust UC methods formulate their second-stage problems as a multi-period economic dispatch (ED) problem with dynamic constraints, which does not properly model real-time ED where dynamic constraints relevant to the future operation are not certifiable. This paper proposes a two-stage robust UC method where a UC schedule and a box in a feasible operation set of each power source as its new feasible operation set are determined. As the multi-period ED problem with the refined feasible set at the second stage has no dynamic constraint, it is identical to a series of single-period ED problems. By modeling real-time ED as the single-period ED problems whose solutions depend only on uncertainties at the corresponding timeslots, the proposed method presents a practical framework of non-anticipative robust UC. Simulation results with a 118-bus system demonstrate the performance of the proposed method.

Robust Two-Stage Regional-District Scheduling of Multi-carrier Energy Systems With a Large Penetration of Wind Power

This paper proposes a robust day-ahead scheduling method for a multi-carrier energy system (MES), which would enhance the flexibility of power systems with a large sum of variable wind power. We build an MES model and propose an optimal MES schedule which helps MES reduce wind power curtailment in power systems. At first, electricity and natural gas networks are coordinated at the transmission (regional) level for accommodating the large penetration of wind power in regional MES. The distribution (district) level MES coordinates energy conversion and storage to jointly supply the electricity, natural gas, and heat loads. The transmission level MES is modeled using detailed network equations while the distribution level MES is modeled as a device with multiple input/output ports using the linear branch-flow-based energy hub model. A two-stage robust model is established to consider the variability of wind power at the two MES levels. The proposed problem is solved by a nested column-and-constraint (C&CG) generation method. The first-stage problem which schedules the hourly unit commitment is solved in the outer loop, while the inner loop solves the second-stage problem to realize the worst scenario. Several acceleration strategies are utilized to enhance the computational performance of the nested C&CG. Numerical results offered for a 6-bus 3-node system and a modified IEEE 118-bus 10-node system show the effectiveness of the proposed MES model and solution technique for enhancing the power system flexibility.

A robust two-stage transit-based evacuation model for large-scale disaster response

After a natural or man-made large-scale disaster occurs, it is a great danger to the residents who are living in the affected area. Evacuees in the (potential) impacted area need to be assembled at pick-up points and evacuated within the specified time by using vehicles that transport them to the safe shelters, potentially multiple times. It is necessary to consider this transit-based evacuation problem right after the occurrence of a large-scale disaster with different time windows caused by different radius to the disaster center point. As the pick-up points of assembling evacuees can greatly influence the evacuation process, it is crucial to identify the critical pick-up point locations to assemble evacuees. We decompose the problem into two stages: determination of pick-up point locations, vehicle routing and scheduling. In the first stage, the goal is to determine a set of pick-up points to assemble evacuees while minimizing the total walking time of evacuees from their locations to pick-up points. In the second stage, the aim is to allocate vehicles to safe shelters to evacuate evacuees from pick-up points to safer shelters to minimize the total transit-based evacuation time. The first-stage problem is formulated as an integer nonlinear programming model and the second-stage problem is modeled as a mixed-integer programming model. To better recognize the locations of pick-up points, a hybrid genetic algorithm (HGA) is developed. An interval/roundtrip-based routing and scheduling heuristic (IRRSH) algorithm is proposed to route and schedule the vehicles under time-window constraint. Finally, computational results are provided to demonstrate the validity and robustness of the proposed model.

Robust Transmission Network Expansion Planning Under Correlated Uncertainty

This paper addresses the transmission network expansion planning problem under uncertain demand and generation capacity. A two-stage adaptive robust optimization framework is adopted whereby the worst-case operating cost is accounted for under a given user-defined uncertainty set. This work differs from previously reported robust solutions in two respects. First, the typically disregarded correlation of uncertainty sources is explicitly considered through an ellipsoidal uncertainty set relying on their variance-covariance matrix. In addition, we describe the analogy between the corresponding second-stage problem and a certain class of mathematical programs arising in structural reliability. This analogy gives rise to a relevant probabilistic interpretation of the second stage, thereby revealing an undisclosed feature of the worst-case setting characterizing robust optimization with ellipsoidal uncertainty sets. More importantly, a novel nested decomposition approach based on results from structural reliability is devised to solve the proposed robust counterpart, which is cast as an instance of mixed-integer trilevel programming. Numerical results from several case studies demonstrate that the effect of correlated uncertainty can be captured by the proposed robust approach.

A simulation-optimization approach for the stochastic discrete cost multicommodity flow problem

ABSTRACTThis article addresses a variant of the Discrete Cost Multicommodity Flow (DCMF) problem with random demands, where a penalty is incurred for each unrouted demand. The problem requires finding a network topology that minimizes the sum of the fixed installation facility costs and the expected penalties of unmet multicommodity demands. A two-stage stochastic programming with recourse model is proposed. A simulation-optimization approach is developed to solve this challenging problem approximately. To be precise, the first-stage problem requires solving a specific multi-facility network design problem using an exact enhanced cut-generation procedure coupled with a column generation algorithm. The second-stage problem aims at computing the expected penalty using a Monte Carlo simulation procedure together with a hedging strategy. To assess the empirical performance of the proposed approach, a Sample Average Approximation (SAA) procedure is developed to derive valid lower bounds. Results of extensive computational experiments attest to the efficacy of the proposed approach.

Product quality, consumption externalities, and the role of National Treatment

Article III of the WTO (National Treatment) limits domestic policy in an effort to curtail protectionist discrimination against foreign products. In this paper, we examine the role of National Treatment and prominent Article XX exceptions when goods are vertically differentiated and generate consumption externalities. We study the domestic policy choices of two countries (first-stage problem) and quantity and price equilibria in two product markets (second-stage problem) under a strict application of Article III and compare them to the policy choices and product market equilibria prevailing when countries can misrepresent external damages and discriminate. While one may expect that high externalities would lead to a greater need for exceptions to equal treatment, we show that this is often not the case and, furthermore, that non-discrimination can lead to a cleaner world environment.