Risk-averse Criterion Research Articles

Robust inventory routing problem under uncertain demand and risk-averse criterion

Inventory routing problem (IRP), which plays an important role in implementing vendor-managed-inventory strategy, is to determine the optimal timing and quantity of products to be delivered as well as the optimal vehicle delivery routes from the vendor to retailers. Retailers’ demand is usually uncertain and their distributions are ambiguous. We utilize limited historical demand data of retailers to construct a series of scenarios to characterize demand uncertainties. Moreover, we develop a framework to construct ambiguity sets and describe the ambiguities in demand distributions. To cater to the new feature of IRP as being exposed to downside risk and balance the mean and risk level of cost, we incorporate the worst-case mean-conditional value-at-risk (M-CVaR) criterion into the objective function and present a new distributionally robust IRP model. We transform the proposed model into a tractable formulation based on duality theory. A case study demonstrates the feasibility of the proposed method. The results indicate that our model can provide a robust delivery solution for immunizing against the influence of ambiguous demand distributions. In addition, an out-of-sample performance analysis shows that the delivery solution provided by our model can effectively reduce the product shortage rate.

Short-Term Hydro Scheduling considering Risk Aversion in a MINLP Approach

This paper presents an optimization model to help a hydro generating company to schedule its hydroelectric units in the short-term under a competitive environment. The hydro generating company considered has the ultimate goal of maximizing profits, taking also into account some risk aversion criterion. The problem is formulated as a mixed-integer nonlinear programming problem. An application to a real case study is presented. Conclusions are duly drawn.

Risk-averse Contextual Multi-armed Bandit Problem with Linear Payoffs

In this paper we consider the contextual multi-armed bandit problem for linear payoffs under a risk-averse criterion. At each round, contexts are revealed for each arm, and the decision maker chooses one arm to pull and receives the corresponding reward. In particular, we consider mean-variance as the risk criterion, and the best arm is the one with the largest mean-variance reward. We apply the Thompson sampling algorithm for the disjoint model, and provide a comprehensive regret analysis for a variant of the proposed algorithm. For T rounds, K actions, and d-dimensional feature vectors, we prove a regret bound of \(O\left({\left({1 + \rho + {1 \over \rho}} \right)d\,\ln \,T\ln {K \over \delta}\sqrt {dK{T^{1 + 2}}\ln {K \over \delta}{1 \over}}} \right)\) that holds with probability 1 − δ under the mean-variance criterion with risk tolerance ρ, for any \(0 < \in < \frac{1}{2},0 < \delta < 1\). The empirical performance of our proposed algorithms is demonstrated via a portfolio selection problem.

Optimising two-stage robust supplier selection and order allocation problem under risk-averse criterion

This paper studies the supplier selection and order allocation (SS&OA) problem, where risks include a series of disruption scenarios with uncertain probability of occurrence. It is a challenge for industry decision-makers to balance the average cost and the level of risk under the ambiguity set for probabilities. To address this challenge, a two-stage distributionally robust (DR) Mean-CVaR model is presented for the SS&OA problem. A procedure is developed for constructing the ambiguity set, and Polyhedral and Box ambiguity sets are constructed to characterise the uncertain probabilities. The worst-case Mean-CVaR criterion is employed for the second-stage cost within the ambiguity set to trade off the expected cost and CVaR value. Three measures are incorporated to increase the resilience of the supply chain. The proposed robust model is reformulated into two mixed-integer linear programming models. A real case of the Huawei cell phone manufacturer is used to illustrate the validity of the proposed approach in numerical settings. Experimental results show that the new optimising approach can provide a robust SS&OA solution to immunise against the influence caused by uncertain probabilities. By comparative analyses, some management insights are obtained for industry decision-makers.

Two-stage distributionally robust programming based on worst-case mean-CVaR criterion and application to disaster relief management

In this paper, we simultaneously capture several practical features in disaster relief management: integrated facility location, inventory pre-positioning and delivery decisions, relief resource priority, partial probability information of demand and risk-averse criterion. We cast the problem as a two-stage distributionally robust mean-conditional value-at-risk (CVaR) optimization model, for which we derive computationally tractable counterparts under the box and polyhedral ambiguity sets. We further identify the relationship between the proposed distributionally robust model and the traditional two-stage stochastic programming model. We assess the performance of the proposed model by an illustrative small-sized example. From the out-of-sample analysis, we show the superiority of the distributionally robust model compared to the two-stage stochastic programming model in terms of stability. We also implement the proposed model in a realistic large-scale case study of hurricane threats in the southeastern US. We finally achieve the managerial implications and insights of using the distributionally robust optimization method.

Distributionally robust design for bicycle-sharing closed-loop supply chain network under risk-averse criterion

Abstract The requirement of environmental improvement has led to the innovative emergence of shared bicycles. The production and recycling of shared bicycles are a closed logistics network, which can be considered a typical closed-loop supply chain (CLSC) problem. In practice, the CLSC network is influenced by social, economic and environmental factors, which impose high degrees of uncertainty and usually trigger various unanticipated risks, so controlling uncertain parameters becomes a key issue in supply chain decisions. The purpose of this research is to construct a new distributionally robust optimization model for a multi-product, multi-echelon CLSC network, in which the distributions of uncertain transportation cost, demand and the returned product are only partially known in advance. In the proposed model, robust mean-CVaR optimization formulation is employed as the objective function for a trade-off between the expected cost and the risk in the CLSC network. Further, to overcome the obstacle of model solvability resulting from imprecise probability distributions, two kinds of ambiguity sets are used to transform the robust counterpart into its computationally tractable forms. Finally, a case study on a Chinese bicycle-sharing company is addressed to validate the proposed robust optimization model. A comparison study is conducted on the performance between our robust optimization method and the traditional optimization method. In addition, a sensitivity analysis is performed with respect to the risk aversion parameter and the confidence level.

Optimal Hedging Rules for Water Supply Reservoir Operations under Forecast Uncertainty and Conditional Value-at-Risk Criterion

Hedging rules for water supply reservoir operations provide guidelines for balancing the consequences of competing water allocations. When inflow forecast uncertainty is addressed, hedging acts as insurances for offsetting the negative influence of water shortage in the future, especially when drought is anticipated. This study used a risk-averse criterion, the conditional value-at-risk (CVaR), rather than the expected value (EV) criterion, to rationalize water delivery for overcoming the shortcomings of risk-neutral hedging rules in minimizing water shortage impacts in unfavorable realizations, in which actual inflow is less than anticipated. A two-period hedging model with the objective of maximizing the CVaR of total benefits from water delivery and water storage is established, and the optimal hedging rules using first-order optimality condition are analytically derived. Differences in hedging rules under the two criteria are highlighted by theoretical analysis and numerical experiments. The methods are applied to guide the operations of a water supply reservoir, and results show that: (1) the hedging rules under the EV criterion are special cases under the CVaR criterion; (2) water delivery in the current period would be greatly curtailed under the high influence of forecast uncertainty or the significant risk-averse attitude of decision makers; (3) hedging to maximize the CVaR of total benefit is at the cost of reducing the EV of total benefit; and (4) in real-time operations, compared with the hedging policies under the EV criterion, the hedging policies under the CVaR criterion would be more effective when applied to dry and extremely dry hydrological conditions, especially when inflow is overestimated. These implications provide new insights into rationing water supply and risk aversion.

Alternative Generation Sources Portfolio: Optimal Resources Allocation and Risk Analysis Supported by Genetics Algorithms

The natural resources characteristics and current economic factors encourage investments in alternative sources of electric power generation in Brazil. Different technologies can compose a portfolio of generating plants with energetic synergism as a function of the seasonal diversity of their potential production. In such portfolios, it is sought to obtain financial gains by virtue of complementarity generation among candidates sources, under investor's pre-established risk control criteria. From this perspective, our study aims to present an optimization model - supported by genetic algorithms - to define the optimal financial resources allocation for composing renewable sources portfolio (wind, small hydro and biomass cogeneration), given a specified budget and risk-aversion criteria measured by means of the Conditional Value-at-Risk. Case studies involving the cited sources illustrate the application of the model and its potential for supporting analysis and decision making.

Solving Markov decision processes with downside risk adjustment

Markov decision processes (MDPs) and their variants are widely studied in the theory of controls for stochastic discreteevent systems driven by Markov chains. Much of the literature focusses on the risk-neutral criterion in which the expected rewards, either average or discounted, are maximized. There exists some literature on MDPs that takes risks into account. Much of this addresses the exponential utility (EU) function and mechanisms to penalize different forms of variance of the rewards. EU functions have some numerical deficiencies, while variance measures variability both above and below the mean rewards; the variability above mean rewards is usually beneficial and should not be penalized/avoided. As such, risk metrics that account for pre-specified targets (thresholds) for rewards have been considered in the literature, where the goal is to penalize the risks of revenues falling below those targets. Existing work on MDPs that takes targets into account seeks to minimize risks of this nature. Minimizing risks can lead to poor solutions where the risk is zero or near zero, but the average rewards are also rather low. In this paper, hence, we study a risk-averse criterion, in particular the so-called downside risk, which equals the probability of the revenues falling below a given target, where, in contrast to minimizing such risks, we only reduce this risk at the cost of slightly lowered average rewards. A solution where the risk is low and the average reward is quite high, although not at its maximum attainable value, is very attractive in practice. To be more specific, in our formulation, the objective function is the expected value of the rewards minus a scalar times the downside risk. In this setting, we analyze the infinite horizon MDP, the finite horizon MDP, and the infinite horizon semi-MDP (SMDP). We develop dynamic programming and reinforcement learning algorithms for the finite and infinite horizon. The algorithms are tested in numerical studies and show encouraging performance.

Development of an inexact risk-aversion optimization model for regional carbon constrained electricity system planning under uncertainty

Since carbon dioxide (CO2) emission reduction gains global concerns, power industry is one of the main sources of CO2 emission. Thus, carbon cap and trade scheme and the application of carbon capture and storage have been employed to mitigate CO2 emission. In consideration of these technologies and constraints, an inexact risk-aversion model based on interval two-stage stochastic programming (ITSP) and conditional value at risk method (CVaR) for multi-period electric system planning problem was developed in this study. Under the optimization framework, more realistic problems such as the demand growth, technology development and environmental policy changes were taken into consideration. CVaR was employed as a risk aversion criterion to reflect the decision maker’s risk preference. A case study for regional electric system planning was demonstrated to verify the capabilities of this optimal approach. The series results provided decision makers with trade-off between the expected system cost and CVaR. It provided valuable insights to make informed long term electric system planning decisions with respect to CCS technology and political and market uncertainties.

Optimal Control of a Multistate Failure-Prone Manufacturing System under a Conditional Value-at-Risk Cost Criterion

The aim of this paper is to establish the optimality of a hedging-point control policy in a multistate Markovian failure-prone manufacturing system with a risk-averse criterion that is defined as the conditional value-at-risk (CVaR) of the steady-state instantaneous running cost, where the system is subject to a constant single-product demand rate. An explicit expression for the optimal control policy is also obtained for the two-state case. The results are important from both theoretical and practical viewpoints. Indeed, the paper extends the well-known classical theoretical result on the optimality of hedging-point control policies under risk-neutral criteria, which are typically given by long-run average costs, and it develops a flexible and practical method for incorporating risk aversion into cost criteria. The approach presented here can be used to specify optimal control policies in similar manufacturing systems with CVaR criteria.

Optimal trade execution under displaced diffusions dynamics across different risk criteria

We solve a version of the optimal trade execution problem when the mid asset price follows a displaced diffusion (DD). Optimal strategies in the adapted class under various risk criteria, namely value-at-risk (VaR), expected shortfall (ES) and a new criterion called squared asset expectation (SAE), related to a version of the cost variance measure, are derived and compared. It is well known that DDs exhibit dynamics that are in-between arithmetic Brownian motions (ABM) and geometric Brownian motions (GBM) depending of the choice of the shift parameter. Furthermore, DD allows for changes in the support of the mid asset price distribution, allowing one to include a minimum permitted value for the mid price, either positive or negative. We study the dependence of the optimal solution on the choice of the risk aversion criterion. Optimal solutions across criteria and asset dynamics are comparable although differences are not negligible for high levels of risk aversion and low market impact assets. This is illustrated with numerical examples.

Optimal Execution Comparison Across Risks and Dynamics, with Solutions for Displaced Diffusions

We solve a version of the optimal trade execution problem when the mid asset price follows a displaced diffusion. Optimal strategies in the adapted class under various risk criteria, namely value-at-risk, expected shortfall and a new criterion called squared asset expectation (SAE), related to a version of the cost variance measure, are derived and compared. It is well known that displaced diffusions (DD) exhibit dynamics which are in-between arithmetic Brownian motions (ABM) and geometric Brownian motions (GBM) depending of the choice of the shift parameter. Furthermore, DD allows for changes in the support of the mid asset price distribution, allowing one to include a minimum permitted value for the mid price, either positive or negative. We study the dependence of the optimal solution on the choice of the risk aversion criterion. Optimal solutions across criteria and asset dynamics are comparable although differences are not negligible for high levels of risk aversion and low market impact assets. This is illustrated with numerical examples.

Risk-averse profit-based optimal operation strategy of a combined wind farm–cascade hydro system in an electricity market

There is a trend toward direct participation of wind farms in electricity markets. However, wind power is inherently intermittent and cannot be accurately predicted even in short time; thus increasing the imbalance costs paid by wind farm owners. To cope with these problems, some techniques have been proposed in literature including wind farm coupling to hydro units, energy storage facilities, and constructing a virtual power plant (VPP). This paper presents a stochastic profit-based model for day-ahead operational planning of a combined wind farm–cascade hydro system. The generation company (GenCo) that owns the VPP considers a portion of its hydro plants capacity to compensate the wind power forecast errors. The proposed optimization problem is a mixed integer linear programming (MILP), formulated as a two-stage stochastic programming model. The day-ahead scheduling is a here and now decision and the optimal operations of facilities are resources variables. In order to protect the GenCo against low price scenarios and wind power variation, the conditional value at risk (CVaR) is used as the risk aversion criterion. The proposed model is successfully applied to a real case study and the results are presented and discussed. The results are illustrated varying in the risk aversion level and the penalty coefficients for negative/positive imbalances. It is shown that the bidding strategy of the GenCo varies significantly depending on the chosen penalty market mechanism.

Risk-averse profit-based optimal scheduling of a hydro-chain in the day-ahead electricity market

This paper presents a profit-based model for short-term hydro scheduling adapted to pool-based electricity markets. The objective is to determine a feasible and realistic operation of a set of coupled hydro units belonging to a small or medium-size hydroelectric company in order to build the generation bids for the next 24 hourly periods. The company is assumed to be price-taker, and therefore, market prices are considered exogenous variables and modeled via scenarios generated by an Input/Output Hidden Markov Model (IOHMM). In order to be protected against low prices scenarios, two different risk-aversion criteria are introduced in the model: a minimum profit constraint and a minimum conditional Value-at-Risk ( CVaR) requirement, which can be formulated linearly in the context of the optimization problem. In order to ensure a feasible operation, the model takes into account a very detailed representation of the generating units, which includes forbidden discharge intervals, spatial–temporal constraints among cascaded reservoirs, etc. The non-linear relationship among the electrical power, the net-head and the turbine water discharge is treated by means of an under-relaxed iterative procedure where net-heads are successively update until convergence is reached. During each algorithm stage, previous iterations’ information is used to build the input–output curves. This way, the hydro scheduling problem can be formulated as a MILP optimization problem, where unit-commitment decisions are modeled by means of binary variables. The model has been successfully applied to a real-size example case, which is also presented in this paper.

Safe Minimum Standard for Environmental Choices: Old-growth Forest in New South Wales

Conflicts over environmental management are due partly to differences in goals, and partly to differences in ways to characterize goals. The safe minimum standard is a criterion for environmental choice that bridges economic and environmental goals, and provides a characterization that is common to both. The safe minimum standard is a risk-averse criterion which states that society should assure the survival of an ecosystem unless the costs of doing so are unacceptably large. In this application to the debate over management of old-growth forest in New South Wales, the problems of defining "unacceptably large" and of defining ecological scenarios are explored. The local populations of Armidale and Dorrigo in northern New South were surveyed to determine acceptable levels of economic and ecological tradeoffs. A majority of those surveyed were prepared to forgo some regional income to ensure the survival of endangered species

An economic rationale for a system of loglinear asset demand functions

In this paper we derived a system of asset demand functions from an expected utility maximization model, in which the utility function satisfies both the decreasing absolute risk aversion as well as the increasing relative risk aversion criteria. The rates of return are assumed to be distributed as a multivariate log-normal distribution. It is shown that a system of log-linear asset demand functions follows from the exact model as an approximation and performs better than similar types of linear asset demand functions which follow from a negative exponential utility function.