Worst-case Cost Research Articles

Cost-Effectiveness of Intraoperative Electromyography to Determine Adequate Screw Position.

Prospective observational cohort. To examine the cost-effectiveness of IntraOperative ElectroMyeloGraphy (IO-EMG) by evaluating how often an abnormal IO-EMG signal changed the surgeon's surgical plan, or replaced a pedicle screw either intra-operatively or as a second unplanned surgery. Patients undergoing instrumented posterolateral lumbar fusion were monitored with intraoperative triggered EMG's. Pedicle screws were placed freehand from L1 to S1 by attending physicians and fellows. Concern for pedicle breach was a screw stimulation<10mA. There were 145 cases with a total of 725 pedicle screws placed. Mean age was 57.8 ± 14.2 yrs, OR time was 238 ± 95minutes, EBL was 426.8 ± 354.3cc. Mean number of surgical levels fused was 2.7 ± 1.1. 686 (95%) screws stimulated at >10mA and 39 (5%) screws stimulated at <10mA. All 39 screws were removed and pedicles re-examined. Intraoperative screw repositioning was necessary in 8 of 145 cases (6%). No patient required a return to the OR for screw repositioning. As a worst case cost analysis, assuming the 8 patients requiring intraoperative screw positioning would have returned to the OR at a cost of $11,798 per readmission, the per patient cost is $651 which is less than the ION per patient cost of $750. Only 1% of the 725 lumbar pedicle screws placed in 8 of 145 cases required repositioning. Due to the infrequency of pedicle wall breaches and the cost of ION, the utility of this modality in straightforward lumbar fusions should be critically evaluated.

Online Apprenticeship Learning

In Apprenticeship Learning (AL), we are given a Markov Decision Process (MDP) without access to the cost function. Instead, we observe trajectories sampled by an expert that acts according to some policy. The goal is to find a policy that matches the expert's performance on some predefined set of cost functions. We introduce an online variant of AL (Online Apprenticeship Learning; OAL), where the agent is expected to perform comparably to the expert while interacting with the environment. We show that the OAL problem can be effectively solved by combining two mirror descent based no-regret algorithms: one for policy optimization and another for learning the worst case cost. By employing optimistic exploration, we derive a convergent algorithm with O(sqrt(K)) regret, where K is the number of interactions with the MDP, and an additional linear error term that depends on the amount of expert trajectories available. Importantly, our algorithm avoids the need to solve an MDP at each iteration, making it more practical compared to prior AL methods. Finally, we implement a deep variant of our algorithm which shares some similarities to GAIL, but where the discriminator is replaced with the costs learned by OAL. Our simulations suggest that OAL performs well in high dimensional control problems.

Robust microgrid energy trading and scheduling under budgeted uncertainty

The advent of smart grids came with several technological developments including new electricity market rules and regulation mechanisms. Microgrids can trade energy with the main grid to either sell its production surplus (from renewable energy sources) or buy an additional amount to support local consumers’ demand, which includes flexible loads, such as smart appliances and electric vehicles. In this scenario, smart control devices are important elements, executing real-time energy scheduling according to fluctuations in production and consumption. As we might expect, the main grid’s power generation and supply becomes more unscheduled and risky as energy trading quantities oscillate over time. This work studies a flexible energy contract subscription framework, coupled with a real-time command strategy, suited for energy scheduling of microgrids with uncertainty in both production and consumption. Our main contributions are a Robust Optimization model under budgeted uncertainty for contract subscription and a set of heuristic control strategies for the real-time energy scheduling. The robust model is capable of providing solutions for multi-period-ahead trading of energy, while minimizing the worst-case cost. We run an extensive computational case study on a real microgrid instance to confirm the efficacy of our solution approach.

Recoverable robust representatives selection problems with discrete budgeted uncertainty

Recoverable robust optimization is a multi-stage approach, in which it is possible to adjust a first-stage solution after the uncertain cost scenario is revealed. We analyze this approach for a class of selection problems. The aim is to choose a fixed number of items from several disjoint sets, such that the worst-case costs after taking a recovery action are as small as possible. The uncertainty is modeled as a discrete budgeted set, where the adversary can increase the costs of a fixed number of items. While special cases of this problem have been studied before, its complexity has remained open. In this work we make several contributions towards closing this gap. We show that the problem is NP-hard and identify a special case that remains solvable in polynomial time. We provide a compact mixed-integer programming formulation and two additional extended formulations. Finally, computational results are provided that compare the efficiency of different exact solution approaches.

Distributionally robust optimization of a Canadian healthcare supply chain to enhance resilience during the COVID-19 pandemic

This paper presents a multi-period multi-objective distributionally robust optimization framework for enhancing the resilience of personal protective equipment (PPE) supply chains against disruptions caused by pandemics. The research is motivated by and addresses the supply chain challenges encountered by a Canadian provincial healthcare provider during the COVID-19 pandemic. Supply, price, and demand of PPE are the uncertain parameters. The ∊-constraint method is implemented to generate efficient solutions along the trade-off between cost minimization and service level maximization. Decision makers can easily adjust model conservatism through the ambiguity set size parameter. Experiments investigate the effects of model conservatism on optimal procurement decisions such as the portion of the supply base dedicated to long-term fixed contracts. Other types of PPE sources considered by the model are one-time open-market purchases and federal emergency PPE stockpiles. The study recommends that during pandemics health care providers use distributionally robust optimization with the ambiguity set size falling in one of three intervals based on decision makers’ relative preferences for average cost performance, worst-case cost performance, or cost variance. The study also highlights the importance of surveillance and early warning systems to allow supply chain decision makers to trigger contingency plans such as locking contracts, reinforcing logistical capacities and drawing from emergency stockpiles. These emergency stockpiles are shown to play efficient hedging functions in allowing healthcare supply chain decision makers to compensate variations in deliveries from contract and open-market suppliers.

Day-ahead robust optimal dispatch of integrated energy station considering battery exchange service

In order to improve the energy utilization efficiency and decrease the environmental pollution, multiple energy production, conversion and storage devices are integrated as a whole, called an integrated energy station (IES), which can meet various energy demands. In this paper, a day-ahead robust optimal scheduling strategy of IES for electricity, cold energy and heat energy supply is proposed, where the battery exchange service is incorporated by introducing the electric vehicle exchange station (EVES). Distinguished from the prevalent detailed modeling of each battery, an integrated model of EVES based on the state of charge (SOC) interval of batteries is established to reduce the number of variables, by which the charging priority of batteries is embedded into the optimal dispatch of IES. Considering the uncertainties of renewable energy resources and load demands, a two-stage robust optimization model of day-ahead dispatch of IES is formulated to minimize the total operation cost under the worst case, which can be solved efficiently by the column-and-constraint generation (C&CG) algorithm. The effectiveness of the proposed method is demonstrated in the case study. The results show that the total operation cost of IES decreases by 9.2%, indicating the role played by the flexibility of EVES. Moreover, the worst-case cost is reduced compared with the deterministic approach.

Two-stage robust power cost minimization in a natural gas compressor station

Optimal operation of a compressor station is important since it accounts for 25 % to 50 % of a company's total operating budget. In short-term management of a compressor station, handling demand uncertainty is important yet challenging. Previous studies either require precise information about the distribution of uncertain parameters or greatly simplify the compressor model. We build a two-stage robust optimization framework of power cost minimization in a natural gas compressor station with non-identical compressors. In the first stage, decision variables are the ON/OFF state of each compressor and discharge pressure. The worst-case cost of the second stage is incorporated in the first stage. First-stage decision variables feasibility is discussed and proper feasibility cuts are also proposed for the first stage. We employ a piece-wise approximation and propose accelerate methods. Our numerical results highlight two advantages of robust approach when managing uncertainty in practical settings: (1) the feasibility of first-stage decision can be increased by up to 45 % , and (2) the worst-case cost can be reduced by up to 25 % compared with stochastic programming models. Furthermore, our numerical experiments show that the designed accelerate algorithm has time improvements of 1518.9% on average (3785.9% at maximum).

Distribution System Operation With Renewables and Energy Storage: A Linear Programming Based Multistage Robust Feasibility Approach

Distribution systems are operated with an increasing level of uncertainty. Energy storage is playing an important role in shaving the peak load and mitigating uncertainty. This paper proposes a multistage robust optimization model for distribution system operation with energy storage under uncertainty. Unlike the conventional robust optimization paradigm which minimizes the worst-case cost, the proposed formulation optimizes the cost in the nominal scenario. In analogy to dynamic programming, we define dynamic robust feasible regions in a recursive manner. In each period, the dynamic robust feasible region is shown to be polyhedral, and a linear programming based projection algorithm is developed to compute such regions offline. In the online stage, the method is executed following a rolling horizon manner: renewable output is observed at the beginning of each period, and the cost of remaining periods in the forecast scenario is to be minimized subject to operation constraints and dynamic robust feasible regions, giving rise to a linear program. In this way, the dispatch strategy ensures multistage operation security regardless of future realizations of renewable power. In numeric tests on a modified IEEE 33-bus distribution system, the dynamic robust feasible regions are visualized and analyzed, and the proposed method is compared with two prevailing robust optimization methods, verifying its advantages in terms of optimality and robustness.

Selection of countermeasure portfolio for shipping safety with consideration of investment risk aversion

This paper proposes optimization models to select shipping safety countermeasure portfolios with consideration of the investment risk preferences of decision-makers. The formulation is extended from the basic economic safety model, where losses are scenario-specific and are expressed as a function of the countermeasure selection variables based on analysing the chain of events along accident paths. Conditional value-at-risk is introduced to measure the investment risk represented by the worst-case cost. Several single -objective programming models are formulated first to optimize the safety-related costs using either the expectation criterion in risk-neutral environments or the conditional value-at-risk criterion in risk-averse environments. Then a bi-objective model is presented by combining those two criteria. Simulation results indicate that the best countermeasures for the expectation objectives are different from those for the conditional value-at-risk objectives. Such findings provide shipping safety investment planners insights in balancing not only the active investment and the accident loss but the return and the risk of the investment, to achieve a selection of countermeasures that is cost-effective as well as leads to low costs in extreme situations.

RETRACTED: Integration of hydrogen storage system and wind generation in power systems under demand response program: A novel p-robust stochastic programming

Penetration of renewable energy sources (RESs) in power systems increase all over the world to overcome current challenges, most importantly environmental issues. Beside advantages of RESs, their integration into the power systems have imposed various challenges considering uncertain and intermitted power output. To cope with these challenges, utilizing energy storage systems with renewable energy sources alongside the demand response (DR) programs are considered as reliable solutions. On the other hand, in an uncertain environment, minimizing worst-case cost or regret is counted as an important criterion to evaluate operation of any system under uncertain parameters. Therefore, in this paper, optimal operation of power systems is solved under penetration of wind turbines, hydrogen storage system, and DR programs in an uncertain environment. To guarantee robust operation of the system under the worst-case scenario, a novel stochastic p-robust optimization method (SPROM) is proposed which combines both stochastic programming and robust optimization approaches where minimizes the worst-case cost or regret level. The performance of the developed model is evaluated considering a six-bus test system under two cases as stochastic optimization (SO) and SPROM. Obtained results show that the maximum regret level is reduced considerably using the proposed SPROM comparing with pure stochastic method.

Robust Constrained Model Predictive Control for T-S Fuzzy Uncertain System with Data Loss and Data Quantization

This paper addresses the robust constrained model predictive control (MPC) for Takagi-Sugeno (T-S) fuzzy uncertain quantized system with random data loss. To deal with the quantization error and the data loss over the networks, the sector bound approach and the Bernoulli process are introduced, respectively. The fuzzy controller and new conditions for stability, which are written as the form of linear matrix inequality (LMI), are presented based on nonparallel distributed compensation (non-PDC) control law and an extended nonquadratic Lyapunov function, respectively. In addition, slack and collection matrices are provided for reducing the conservativeness. Based on the obtained stability results, a model predictive controller which explicitly considers the input and state constraints is synthesized by minimizing an upper bound of the worst-case infinite horizon quadratic cost function. The developed MPC algorithm can guarantee the recursive feasibility of the optimization problem and the stability of closed-loop system simultaneously. Finally, the simulation example is given to illustrate the effectiveness of the proposed technique.

Optimal deterministic and robust selection of electricity contracts

We address the Electricity Contract Selection Problem (ECSP), of finding best parameters of an electricity contract for a client based on his/her past records of electricity consumption over a fixed time period. The objective is to optimize the electricity bill composed by some fixed cost, the cost of the subscription of the electricity contract and penalties due to overpowering when consumption exceeds subscribed power. The ECSP can be formulated as a convex separable optimization problem subject to total order constraints. Due to this special structure, ECSP is a special case of two well known classes of convex separable optimization problems, namely the minimum network flow under convex separable cost and minimizing convex separable functions under chain constraints. Both classes are well treated in the litterature and can be solved in polynomial time (Ahuja and Orlin in Oper Res 49(5):784–789, 2001; Ahuja et al. in Manag Sci 49(7):950–964, 2003; Best et al. in SIAM J Optim 10:658–672, 2000; Karzanov and McCormick in SIAM J Comput 26(4):1245–1275, 1997; Minoux in Eur J Oper Res 18(3):377–387, 1984; Minoux, in Gallo and Sandi (eds) Netflow at pisa, mathematical programming studies, Springer, Berlin, 1986). In particular, the algorithm in Ahuja and Orlin (2001) achieves the best theoretical time complexity assuming that computing the objective function value at one specific point can be done in constant time. However, when we work on a big amount of historic data as in ECSP, the time required for evaluating the objective function cannot be assumed to be O(1) anymore. In this paper, we propose a new algorithm for ECSP which is specially designed to reduce the computational effort over large scale historical data. We present numerical results showing that our algorithm outperforms the algorithm in Ahuja and Orlin (2001) when applied to consumption data of various types of clients. A robust version of ECSP based on a Seasonal and Trend decomposition approach for modelling consumption uncertainty is also investigated. The resulting worst-case cost minimization problem is shown to be efficiently solvable using the same algorithm as for deterministic ECSP.

K-adaptability in robust container vessel sequencing problem with week-dependent demands of a service route

ABSTRACT This work investigates a robust container vessel sequencing (RCVS) problem in a service route. As weekly demands vary dramatically and cannot be forecasted accurately, shipping companies need to develop a robust sequence of vessels with different capacities to maximally meet demands. As export heavily depends on economy, demands may share the same pattern in adjacent years, which motivates us to study the problem in a cyclic fashion. To refine the literature, we adopt a robust optimisation model to minimise the worst-case total cost, including container tardy and outsourcing cost, due to reliability guarantee. To accommodate human decision-making, we focus on an associated K-adaptability problem, which pre-selects a number of candidate vessel sequences and implements the best one when the uncertain demands have been observed. A branch-and-bound solution approach is explored. Numerical experiments demonstrate the performance of our approach.

Second-Order Conic Programming Approach for Wasserstein Distributionally Robust Two-Stage Linear Programs

This article proposes a second-order conic programming (SOCP) approach to solve distributionally robust two-stage linear programs over 1-Wasserstein balls. We start from the case with distribution uncertainty only in the objective function and then explore the case with distribution uncertainty only in constraints. The former program is exactly reformulated as a tractable SOCP problem, whereas the latter one is proved to be generally NP-hard as it involves a norm maximization problem over a polyhedron. However, it reduces to an SOCP problem if the extreme points of the polyhedron are given as a prior. This motivates the design of a constraint generation algorithm with provable convergence to approximately solve the NP-hard problem. Moreover, the least favorable distribution achieving the worst case cost is given as an &#x201C;empirical&#x201D; distribution by simply perturbing each original sample for both cases. Finally, experiments illustrate the advantages of the proposed model in terms of the out-of-sample performance and computational complexity. <i>Note to Practitioners</i>&#x2014;The two-stage program with distribution uncertainty is an important decision problem in broad applications, e.g., two-stage schedule problems, facility location problems, and recourse allocation problems. To deal with the uncertainty, this work proposes a novel data-driven model over the 1-Wasserstein ball and develops an efficient second-order conic programming (SOCP)-based solution approach, where the sample data set can be easily exploited to reduce the distribution uncertainty. The good out-of-sample performance and computational complexity of the proposed model are validated by the experiments on the two-stage portfolio programs and material order programs.

A Lipschitz Matrix for Parameter Reduction in Computational Science

We introduce the Lipschitz matrix: a generalization of the scalar Lipschitz constant for functions with many inputs. Among the Lipschitz matrices compatible a particular function, we choose the smallest such matrix in the Frobenius norm to encode the structure of this function. The Lipschitz matrix then provides a function-dependent metric on the input space. Altering this metric to reflect a particular function improves the performance of many tasks in computational science. Compared to the Lipschitz constant, the Lipschitz matrix reduces the worst-case cost of approximation, integration, and optimization; if the Lipschitz matrix is low-rank, this cost no longer depends on the dimension of the input, but instead on the rank of the Lipschitz matrix defeating the curse of dimensionality. Both the Lipschitz constant and matrix define uncertainty away from point queries of the function and by using the Lipschitz matrix we can reduce uncertainty. If we build a minimax space-filling design of experiments in the Lipschitz matrix metric, we can further reduce this uncertainty. When the Lipschitz matrix is approximately low-rank, we can perform parameter reduction by constructing a ridge approximation whose active subspace is the span of the dominant eigenvectors of the Lipschitz matrix. In summary, the Lipschitz matrix provides a new tool for analyzing and performing parameter reduction in complex models arising in computational science.

A Multi-Objective Optimization Problem on Evacuating 2 Robots from the Disk in the Face-to-Face Model; Trade-Offs between Worst-Case and Average-Case Analysis

The problem of evacuating two robots from the disk in the face-to-face model was first introduced by Czyzowicz et al. [DISC’2014], and has been extensively studied (along with many variations) ever since with respect to worst-case analysis. We initiate the study of the same problem with respect to average-case analysis, which is also equivalent to designing randomized algorithms for the problem. In particular, we introduce constrained optimization problem 2EvacF2F, in which one is trying to minimize the average-case cost of the evacuation algorithm given that the worst-case cost does not exceed w. The problem is of special interest with respect to practical applications, since a common objective in search-and-rescue operations is to minimize the average completion time, given that a certain worst-case threshold is not exceeded, e.g., for safety or limited energy reasons. Our main contribution is the design and analysis of families of new evacuation parameterized algorithms which can solve 2EvacF2F, for every w for which the problem is feasible. Notably, the worst-case analysis of the problem, since its introduction, has been relying on technical numerical, computer-assisted calculations, following tedious robot trajectory analysis. Part of our contribution is a novel systematic procedure, which given any evacuation algorithm, can derive its worst- and average-case performance in a clean and unified way.

Seismic effectiveness and robustness of tuned mass dampers versus nonlinear energy sinks in a lifecycle cost perspective

Tuned mass dampers (TMDs) and nonlinear energy sinks (NESs) are two viable options for passively absorbing structural vibrations. In seismic applications, a trade-off exists in their performance, because TMDs’ effectiveness varies with the structural stiffness while NESs’ effectiveness varies with the earthquake intensity. To investigate this trade-off systematically, a lifecycle cost- (LCC-) oriented robust analysis and design method is here proposed, in which the effectiveness of a solution is measured by the reduction it entails in the expected cost of future seismic losses. In it, structural stiffness variability is modelled using a worst-case approach with lower and upper bounds, while seismic intensity variability is inherently captured by the incremental dynamic analyses underlying every LCC evaluation. The resulting worst-case lifetime cost provides a rational metric for discussing pros and cons of TMDs and NESs, and becomes the objective function for their robust optimization. The method is applied to the design of TMDs and NESs on a variety of single- and multi-story linear building models, located in a moderate-to-high seismic hazard region. Mass ratios from 1 to 10% and structural stiffness reductions up to 4 times are considered. Results show that TMDs are consistently more effective than NESs even in the presence of large stiffness reductions, provided that structural stiffness uncertainty is considered in design. They also show that a conventional robust H∞ design provides for TMDs a solution which is very close to that obtained by minimizing the proposed LCC metric.

A risk-averse simulation-based approach for a joint optimization of workforce capacity, spare part stocks and scheduling priorities in maintenance planning

We model a maintenance system consisting of one repair facility, where repairables are kept on inventory to serve assets to prevent downtime and increase availability. We seek optimal values of the repairable spare parts stocks and workforce capacity in the repair facility. Further, we simultaneously search for the best repair scheduling rule that minimizes total inventory holding and backorder costs associated with the downtime of assets. The joint optimization problem under study brings about two additional challenges: (i) the difficulty of analyzing such systems due to the lack of analytical (i.e., queuing) models, and (ii) the difficulty in incorporating the decision maker’s risk attitude regarding uncertainties. We develop a risk-averse simulation-based optimization approach, in which the decision maker’s risk attitude is modeled as a trade-off between the expected and the worst-case costs in the objective function. In the developed approach, the repairable spare part supply system is analyzed with a discrete-event simulation (DES) model. The DES model is coupled with an improved reduced variable neighborhood search (IRVNS) meta-heuristic that seeks the optimal values of decision variables. We compare the performance of the proposed risk-averse simulation-based optimization approach with several plausible benchmark methods commonly used in practice and with well-known meta-heuristic algorithms.

Robust model predictive control for Markovian jump systems under Round-Robin protocol

In this paper, a robust model predictive control problem is investigated for a class of Markovian jump systems with polytopic uncertainties. The Round-Robin protocol is utilised to schedule data transmission from the controllers to the actuator in order to reduce the burden of communication and the occupancy of network effects. Moreover, two kinds of transition probability matrices for the Markovian process are considered due to the practical application process, namely the one with polytopic uncertainties and the other with partially unknown entries. Sufficient conditions are established by means of the mixed-mode and token-dependent Lyapunov-like technique such that the closed-loop system is stable in the mean-square sense. In addition, a set of mixed-mode and token-dependent state feedback controllers are designed in terms of minimising a certain upper bound of the worst-case infinite horizon cost function. Finally, some simulation examples are implemented to verify the effectiveness of the proposed design scheme.

Deterministic Treasure Hunt in the Plane with Angular Hints

A mobile agent equipped with a compass and a measure of length has to find an inert treasure in the Euclidean plane. Both the agent and the treasure are modeled as points. In the beginning, the agent is at a distance at most $$D>0$$ from the treasure, but knows neither the distance nor any bound on it. Finding the treasure means getting at distance at most 1 from it. The agent makes a series of moves. Each of them consists in moving straight in a chosen direction at a chosen distance. In the beginning and after each move the agent gets a hint consisting of a positive angle smaller than $$2\pi$$ whose vertex is at the current position of the agent and within which the treasure is contained. We investigate the problem of how these hints permit the agent to lower the cost of finding the treasure, using a deterministic algorithm, where the cost is the worst-case total length of the agent’s trajectory. It is well known that without any hint the optimal (worst case) cost is $$\varTheta (D^2)$$ . We show that if all angles given as hints are at most $$\pi$$ , then the cost can be lowered to O(D), which is the optimal complexity. If all angles are at most $$\beta$$ , where $$\beta <2\pi$$ is a constant unknown to the agent, then the cost is at most $$O(D^{2-\epsilon })$$ , for some $$\epsilon >0$$ . For both these positive results we present deterministic algorithms achieving the above costs. Finally, if angles given as hints can be arbitrary, smaller than $$2\pi$$ , then we show that cost complexity $$\varTheta (D^2)$$ cannot be beaten.