Risk-averse Solutions Research Articles

A risk-averse solution for the prescribed burning problem

Hazard reduction is a complex task involving important efforts to prevent and mitigate the consequences of disasters. Many countries around the world have experienced devastating wildfires in recent decades and risk reduction strategies are now more important than ever. Reducing contiguous areas of high fuel load through prescribed burning is a fuel management strategy for reducing wildfire hazard. Unfortunately, this has an impact on the habitat of fauna and thus constrains a prescribed burning schedule which is also subject to uncertainty. To address this problem a mathematical programming model is proposed for scheduling prescribed burns on treatment units on a landscape over a planning horizon. The model takes into account the uncertainty related to the conditions for performing the scheduled prescribed burns as well as several criteria related to the safety and quality of the habitat. This multiobjective stochastic problem is modelled from a risk-averse perspective whose aim is to minimize the worst achievement of the criteria on the different scenarios considered. This model is applied to a real case study in Andalusia (Spain) comparing the solutions achieved with the risk-neutral solution provided by the simple weighted aggregated average. The results obtained show that our proposed approach outperforms the risk-neutral solution in worst cases without a significant loss of quality in the global set of scenarios.

Risk-Averse Co-Decision for Lower-Carbon Product Family Configuration and Resilient Supplier Selection

Due to global pandemics, political unrest and natural disasters, the stability of the supply chain is facing the challenge of more uncertain events. Although many scholars have conducted research on improving the resilience of the supply chain, the research on integrating product family configuration and supplier selection (PCSS) under disruption risks is limited. In this paper, the centralized supply chain network, which contains only one major manufacturer and several suppliers, is considered, and one resilience strategy (i.e., the fortified supplier) is used to enhance the resilience level of the selected supply base. Then, an improved stochastic bi-objective mixed integer programming model is proposed to support co-decision for PCSS under disruption risks. Furthermore, considering the above risk-neutral model as a benchmark, a risk-averse mixed integer program with Conditional Value-at-Risk (CVaR) is formulated to achieve maximum potential worst-case profit and minimum expected total greenhouse gases (GHG) emissions. Then, NSGA-II is applied to solve the proposed stochastic bi-objective mixed integer programming model. Taking the electronic dictionary as a case study, the risk-neutral solutions and risk-averse solutions that optimize, respectively, average and worst-case objectives of co-decision are also compared under two different ranges of disruption probability. The sensitivity analysis on the confidence level indicates that fortifying suppliers and controlling market share in co-decision for PCSS can effectively reduce the risk of low-profit/high-cost while minimizing the expected GHG emissions. Meanwhile, the effects of low-probability risk are more likely to be ignored in the risk-neutral solution, and it is necessary to adopt a risk-averse solution to reduce potential worst-case losses.

A Risk-Aversion Approach for the Multiobjective Stochastic Programming Problem

Multiobjective stochastic programming is a field that is well suited to tackling problems that arise in many fields: energy, financial, emergencies, among others; given that uncertainty and multiple objectives are usually present in such problems. A new concept of solution is proposed in this work, which is especially designed for risk-averse solutions. The proposed concept combines the notions of conditional value-at-risk and ordered weighted averaging operator to find solutions protected against risks due to uncertainty and under-achievement of criteria. A small example is presented in order to illustrate the concept in small discrete feasible spaces. A linear programming model is also introduced to obtain the solution in continuous spaces. Finally, computational experiments are performed by applying the obtained linear programming model to the multiobjective stochastic knapsack problem, gaining insight into the behaviour of the new solution concept.

A risk-averse stochastic quadratic model with recourse for supporting irrigation water management in uncertain and nonlinear environments

Agricultural water management system are subject to a variety of uncertainties and nonlinearity, which complicate the decision-making process and exaggerate water-shortage risks. Previously, few studies could tackle multiple uncertainties and the associated system risks in nonlinear problems. In this paper, a novel risk-averse optimization model was developed for supporting irrigation water management to mitigate these problems. Based on a risk measurement tool, Conditional Value-at-Risk (CVaR), a CVaR-based interval fuzzy two-stage stochastic quadratic programming (CIFTSQP) was proposed. CIFTSQP could not only provide risk-averse solutions with recourse in response to extreme economic risk quantified as CVaR and water-shortage risk expressed as fuzzy credibility level, but also handle and quantify the nonlinearity in a stochastic program caused by the economy-of-scale effects of facilities. The proposed method has been applied to an irrigation water management case in the northern China. Results suggested that, in response to severe water shortage, sunflowers would be preferred. Meanwhile, groundwater use in Wuyuan County and surface water consumption of Urad Qianqi and Urad Zhongqi should be reduced. Moreover, desired water-allocation strategies with varied risk-aversion levels were generated under different water inflow levels. Results reveal that the highest economic benefits could be achieved when risk aversion degree reaches 30 %, 40 % and 70 % under the low, medium and high inflow levels, respectively. Results reveal that higher risks would bring higher returns under the low and medium inflow levels, and the highest economic benefits could be achieved when risk aversion degree reaches 70 % under the high inflow level.reveal that higher would bring higher returnsunder the low and medium inflow levels, andthe highest economic benefits could be achieved when risk aversion degree reaches 70 % under the high inflow level. The developed CIFTSQP method provided valuable insights into informed irrigation decision-making with respect to economy-of-scale effects, risk aversion and recourse in an agricultural water allocation system. It can also be applied to address other resource allocation problems under uncertainty and risk.

A data-driven distributionally robust newsvendor model with a Wasserstein ambiguity set

In this paper, we derive a closed-form solution and an explicit characterization of the worst-case distribution for the data-driven distributionally robust newsvendor model with an ambiguity set based on the Wasserstein distance of order We also consider the risk-averse decision with the Conditional Value-at-Risk (CVaR) objective. For the risk-averse model, we derive a closed-form solution for the p = 1 case, and propose a tractable formulation to obtain an optimal order quantity for the p > 1 case. We conduct numerical experiments to compare out-of-sample performance and convergence results of the proposed solutions against the solutions with other distributionally robust models. We also analyze the risk-averse solutions compared to the risk-neutral solutions.

Risk-averse flexible policy on ambulance allocation in humanitarian operations under uncertainty

Proactive ambulance management is constructive to improve the response efficiency for emergency medical service (EMS) systems under uncertainty. In this paper, we present a dynamic optimisation model concerning the ambulance dispatching and relocation. We develop a flexible operation policy driven by the interval rolling to match vehicles with calls in batch. We formulate the problem in Markov Decision Process and incorporate queues to minimise the average response and delay time. Considering the curse-of-dimensionality, we provide a simulation-based empirical dynamic programming with the state aggregation and post-decision state to solve the model. To further accelerate the computational efficiency, a greedy heuristic method is introduced to improve the quality of sampling operations. Then, a risk-averse model is developed based on the stochastic dominance strategy to improve operational reliability. We develop an equivalent linear programming to evaluate concave dominating functions. We test the performance by a numerical case and extract managerial insights for practitioners. Our results show that the proposed flexible and risk-averse solution outperforms the classic model on reducing the delay under uncertain calls. And the improvement is more active during peak hours, when real-time needs exceed available ambulances.

Optimal logistics planning for modular construction using multi-stage stochastic programming

The modular construction method has been adopted extensively by the construction sector for pursuing higher building quality and better project efficiency. However, the employment of this new construction method has not only altered the definition of construction supply chains, but also poses new challenges to the logistics system which has conventionally focused on raw material transportation. This challenge is exacerbated in the transport and inventory aspects when the project is executed in urban settings, owing to the frequent traffic congestion, crowded environment, as well as the bulkiness and delicacy of finished modules. This study develops a multi-stage stochastic programming model for identifying the optimal supply chain configuration for the modular construction method. Site demand is considered to be stochastic, forcing project managers to make several operational decisions at multiple time points during project execution. The developed model can provide the best production, transportation and inventory plans, as well as the most favourable initial inventory preparation schemes. Furthermore, we have proven that the implementation of multi-stage stochastic programming model can yield more economical and risk-averse solutions than the two-stage stochastic programming approach.

Bandits with Global Convex Constraints and Objective

Multiarmed bandit (MAB) is a classic model for capturing the exploration–exploitation trade-off inherent in many sequential decision-making problems. The classic MAB framework, however, only allows “local” constraints on decisions and “sum of rewards” as objective. In many real-world applications, there are multiple complex constraints on resources that are consumed during the entire decision process, and performance may be evaluated through nonlinear utility functions on aggregate rewards. This article presents a new MAB framework that allows such “global” convex constraints and concave objective functions along with new algorithmic techniques with provably near-optimal performance bounds. The authors discuss applications in several domains, such as network revenue management, crowdsourcing, and pay-per-click advertising, which benefit from the new more general framework by admitting richer models and more efficient risk-averse solutions.

Robust Fuzzy chance constraint programming for multi-item EOQ model with random disruption and partial backordering under uncertainty

ABSTRACTIn this paper, a new mathematical model for multi-item Economic Order Quantity model is proposed considering defective supply batches and partial backordering. The proposed model assumes that the supply batches may be defective and can be rejected. The aim is to minimize total inventory costs by determining optimal values of the decision variables including Time interval between successive perfect supply deliveries. Since in real-world situations the main parameters are uncertain, two mathematical programming approaches called: Basic Chance Constraint Programming and Robust Fuzzy Chance Constraint programming are used to handle uncertain parameters. The performance of the mathematical programming models is investigated in a numerical example. The results show that the RFCCP model is able to provide risk-averse solutions comparing to the BCCP model. In the end, sensitivity analyses are performed to determine the effect of any change in the main parameters on objective function value to determine the most critical parameters

A robust fuzzy approach for constrained multi-product economic production quantity with imperfect items and rework process

ABSTRACTThis paper aims to develop a robust multi-item EPQ formulation considering rework process and imperfect items. The mathematical model consists of two objective functions targeting minimization of total inventory costs and the total required warehouse space, respectively. Since, in real-world applications, parameters of the mathematical formulation are due to uncertainty, in this paper, Basic Chance Constraint Programming and Robust Fuzzy Chance Constraint Programming models are developed to handle uncertainties in both objective function and constraints. By solving examples, the superiority of the RFCCP over the BCCP model is showed and its ability to provide risk-averse solutions is ensured. Due to the nonlinearity of the RFCCP model and to provide the decision maker with the Pareto front, two algorithms, namely, Multi-Objective Grey Wolf Optimizer and Multi-Objective Water Cycle Algorithm, are used to optimize the decision variables. The performance of the algorithms is evaluated within different large-size test problems, using diversity, Spacing, number of non-dominated solutions and CPU-time. In the end, to investigate significant differences and to determine the best algorithm ANOVA test is implemented. In the end, we also offer new research directions, e.g. in terms of further ways of uncertainty modelling and risk management.

Pemodelan Rantai Suplai Berbasis Economic Production Quantity dengan Menerapkan Kredit Perdagangan Dua Eselon

<p><em>Implementation of the supply chain is felt more important and very useful in today’s era of industrialization. But in the process of supply chain encountered various risks that may affect the flow of the supply chain so it can not run smoothly, such as interference or imperfectness in transportations. This research was made to propose a mathematical model for a supply chain under the effect of unexpected disruptions in transport. Supplier/manufacturer offers the retailer a trade credit period t<sub>1</sub> , then the retailer offers the customer a credit with a period of t<sub>2</sub> and finally the retailer receives the revenue from t<sub>2</sub> ke T + t<sub>2</sub> , where T is the cycle time at the retailer. Under this situation, the three cases such as T ≤ t<sub>1</sub> ≤ T + t<sub>2</sub> , T ≤ T + t<sub>1</sub> ≤ t<sub>2</sub>, and t<sub>1</sub> ≤ t<sub>2</sub> are discussed. An EPQ (Economic Production Quantity) based model is established and retailer’s optimal replenishment policy is obtained through mathematical theorems. From the results of testing on several cases, the best solution for the two procedures has been obtained, namely between risk neutral and risk averse solutions based on the level or number of items damaged/defective. If the number of items damaged is 2 units, then solution risk neutral is far better than the risk averse solution, whereas if the number of items is damaged </em> <em>3 units, then the risk averse solution is far better than the risk neutral solution.</em></p><p class="HowToCite"> </p>

Mixed robust possibilistic flexible chance constraint optimization model for emergency blood supply chain network design

This research proposes a new tri-objective mathematical model for designing blood supply chain network in emergency situations. The mathematical model aims to minimize total supply chain costs and transportation time between facilities while maximizing total testing reliability of the donated blood in the laboratories. The model considers five echelons including blood donor groups, blood collection facilities, laboratories, blood centers and hospitals. Different transportation means with variant speed and capacity are considered in the model to carry the blood between facilities. Since, most of the main parameters of the mathematical model are tainted with uncertainty in real-world applications, two robust possibilistic flexible chance constraint programming (RPFCCP) and possibilistic flexible chance constraint programming models are developed to provide risk-averse and robust solutions to the decision makers. In addition, the application of the proposed multi-objective mathematical model is investigated in a real-world case study using real data on Iran’s capital, Tehran, which is considered to be a potential place for a destructive earthquake. Using different realizations, the applicability and efficiency of the models are investigated in the case study. The results indicated that the RPFCCP model is able to handle uncertainty in the parameters of the objective function and constraints more efficiently and is able to provide robust and risk-averse solutions for the problem which are resistant to different scenarios.

Allocation flexibility for agribusiness supply chains under market demand disruption

This paper develops a multi-commodity multi-period optimisation model to analyse market demand disruption risk in agribusiness supply chains. It investigates the role of allocation flexibility and the effectiveness of multiple risk management strategies for achieving allocation flexibility. A robust optimisation formulation is used to obtain risk-averse solutions for an objective combining expected profit and risk. Numerical results are presented for a real-life case study of Zespri’s kiwifruit supply chain. The results show that allocation flexibility is effective for mitigating market demand disruption risk. Three proposed risk management strategies, namely diversified demand market, backup demand market and flexible rerouting, improve both expected profit and risk measures. While diversified demand market and backup demand market strategies are equally important for all decision-makers, flexible rerouting is especially significant for less risk-averse decision-makers.

A carbon-constrained stochastic optimization model with augmented multi-criteria scenario-based risk-averse solution for reverse logistics network design under uncertainty

With the increase of the concern from the public for environmental pollution and waste of resources, the value recovery through reuse, repair, remanufacturing and recycling from the end-of-use (EOU) and end-of-life (EOL) products have become increasingly important. Reverse logistics is the process for capturing the remaining value from the EOU and EOL products and also for the proper disposal of the non-reusable and non-recyclable parts. A well-designed reverse logistics system will yield both economic and environmental benefits, so the development of an advanced decision-making tool for reverse logistics system design is of significant importance. The paper presents a novel multi-product multi-echelon stochastic programming model with carbon constraint for sustainable reverse logistics design under uncertainty. Compared with the previous stochastic optimization models in reverse logistics system design, which mainly focuses on the expectation of the optimal value, this paper, however, emphasizes on both optimal value expectation and its reliability in decision-making. Due to this reason, a multi-criteria scenario-based risk-averse solution method is developed based on a latest research in order to obtain the optimal solution with high level of confidence. Later in this paper, the application of the model and the augmented solution method is illustrated and the managerial implications are discussed through the numerical experiment and sensitivity analysis. The result of the study shows that the model can be used for providing decision-makers with a deep insight into the relationship between profit and carbon emission requirement, understanding and resolution of the infeasibility caused by capacity limitation, the use of flexible manufacturing system in reverse logistics, and proper use of the government subsidy as a leverage in reverse logistics design.

The study of the unidirectional quay crane scheduling problem: complexity and risk-aversion

As a special case of the quay crane scheduling problem, the unidirectional quay crane scheduling problem has been received more and more attention. In this paper, we analyze the computational complexity of the unidirectional quay crane scheduling problem and propose a tighter mixed integer programming formulation. Next, we develop three makespan-constrained models to obtain risk-averse solutions to mitigate the impacts of ship instability and solution infeasibility (thus, need to revamp or even reschedule the plans). Comprehensive numerical experiments are designed to investigate the benefits of the tighter formulation and the three optimization models for risk-aversion.

Risk-averse stochastic path detection

We introduce the stochastic path detection problem and propose a risk-averse solution approach. The problem comprises an invader and a protector that both operate on a network with a number of possible source-destination paths. The protector aims at allocating security resources on the network such that the invader’s path is detected with high probability. The invader’s choice of path is known to the protector in terms of a probability distribution reflecting the protector’s beliefs. Errors in these beliefs induce the risk of a low detection probability. We derive a linear programming approximation and leverage the theory of coherent risk measures to consider risk-aversion with respect to errors in the protector’s beliefs. The performance of the resulting risk-averse detection policy is numerically compared with the performance of the risk-neutral policy. We show that the risk-averse policy significantly mitigates the risk of facing a low detection probability in the presence of large errors in the protector’s beliefs.

The Minimum Spanning k-Core Problem with Bounded CVaR Under Probabilistic Edge Failures

This article introduces the minimum spanning k-core problem that seeks to find a spanning subgraph with a minimum degree of at least k (also known as a k-core) that minimizes the total cost of the edges in the subgraph. The concept of k-cores was introduced in social network analysis to identify denser portions of a social network. We exploit the graph-theoretic properties of this model to introduce a new approach to survivable interhub network design via spanning k-cores; the model preserves connectivity and diameter under limited edge failures. The deterministic version of the problem is polynomial-time solvable due to its equivalence to generalized graph matching. We propose two conditional value-at-risk (CVaR) constrained optimization models to obtain risk-averse solutions for the minimum spanning k-core problem under probabilistic edge failures. We present polyhedral reformulations of the convex piecewise linear loss functions used in these models that enable Benders-like decomposition approaches. A decomposition and branch-and-cut approach is then developed to solve the scenario-based approximation of the CVaR-constrained minimum spanning k-core problem for the aforementioned loss functions. The computational performance of the algorithm is investigated via numerical experiments.

$ (Q,r) $ Model with $ CVaR_α $ of costs minimization

In the classical stochastic continuous review, \begin{document}$ (Q,r) $\end{document} model [ 18 , 19 ], the inventory cost \begin{document}$ c(Q,r) $\end{document} has an averaging term which is given as an integral of the expected costs over the different inventory positions during the lead time on any given cycle. The main objective of the article is to study risk averse optimization in the classical \begin{document}$ (Q,r) $\end{document} model using \begin{document}$ CVaR_{α} $\end{document} as a coherent risk measure with respect to the probability distribution of the demand \begin{document}$ D $\end{document} on inventory position costs (the sum of the inventory holding and backorder penality cost), for any given (generic) confidence level \begin{document}$ α∈[0,1) $\end{document} . We show that the objective function is jointly convex in \begin{document}$ (Q,r) $\end{document} . We also compare the risk averse solution and some other solutions in both analytical and computational ways. Additionally, some general and useful results are obtained.

Risk-Averse PDE-Constrained Optimization Using the Conditional Value-At-Risk

Uncertainty is inevitable when solving science and engineering application problems. In the face of uncertainty, it is essential to determine robust and risk-averse solutions. In this work, we consider a class of PDE-constrained optimization problems in which the PDE coefficients and inputs may be uncertain. We introduce two approximations for minimizing the conditional value-at-risk (CVaR) for such PDE-constrained optimization problems. These approximations are based on the primal and dual formulations of CVaR. For the primal problem, we introduce a smooth approximation of CVaR in order to utilize derivative-based optimization algorithms and to take advantage of the convergence properties of quadrature-based discretizations. For this smoothed CVaR, we prove differentiability as well as consistency of our approximation. For the dual problem, we regularize the inner maximization problem, rigorously derive optimality conditions, and demonstrate the consistency of our approximation. Furthermore, we propose a...

Modelos de programação estocástica no planejamento da produção de empresas moveleiras

Esse trabalho aborda um problema de planejamento da produção típico de empresas moveleiras de pequeno porte, em que as demandas e os tempos de preparação dos estágios gargalos são variáveis aleatórias que podem ser aproximadas por um conjunto discreto e finito de cenários ponderados pelas correspondentes probabilidades de ocorrência. O problema com múltiplos cenários é modelado via programação estocástica de dois estágios com recurso. Para controlar a variabilidade dos custos de segundo estágio é proposto um modelo de recurso restrito que gera, progressivamente, um conjunto de soluções menos sensíveis às variações dos cenários, conforme a variabilidade é restringida a uma tolerância dada. Experiências numéricas indicam que, em muitas situações, não é muito dispendioso assegurar soluções aversas ao risco com bons níveis de serviço.