Risk-averse Model Research Articles

Two-stage stochastic mixed integer optimization models for power generation capacity expansion with risk measures

We present two-stage stochastic risk averse optimization models for the power generation mix capacity expansion planning in the long run under uncertainty. Uncertainty is described by a set of possible scenarios in the second stage and uncertain parameters are the unit production costs of the existing power plants as well as those of the candidate plants of new technologies among which to choose, the market electricity price, the price of green certificates and the emission permits and the potential market share of the producer. The problem is expressed as a two-stage stochastic integer optimization model subject to technical constraints, market opportunities and budget constraints. First stage variables represent the number of new power plants for each candidate technology to be added to the existing generation mix every year of the planning horizon. Second stage variables are the continuous operation variables of all power plants in the generation mix along the time horizon. We solve the problem of the maximization of the net present value of the expected profits along the time horizon using both a risk neutral approach and different risk averse strategies (conditional value at risk, shortfall probability, expected shortage and first- and second-order stochastic dominance), under different hypotheses of the available budget, analysing the impact of each risk averse strategy on the expected profit. Results show that risk control strongly reduces the possibility of reaching unwanted scenarios as well as providing consistent solutions under different strategies.

Determinants of software piracy under risk aversion: a model with empirical evidence

This paper studies the determinants of software piracy in world economies from a risk avoidance perspective. A risk aversion model for the commercialization of pirated software is developed to account for behavioral elements of risk and uncertainty avoidance among countries’ software pirates (i.e., counterfeiters and suppliers) and test empirically for the effects of country characteristics on piracy levels. Panel regression analysis is conducted to identify the determinants of software piracy using this model on a data set of 87 countries during 2007–2011. The empirical results confirm those obtained in prior research (e.g., the inverted U-shaped relationship between GDP per capita and piracy rates) but divulge that the behavioral-country component capturing the attitudes towards risk of software pirates improves the explanatory power of the statistical regressions after controlling for country performance and institutional factors. We also show that human development and good country governance reduce piracy rates. Besides providing support for our risk aversion-based piracy model and hinting at the need to consider population behavior in policy-making, these findings underline the relevance of human development and country institutions in explaining software piracy rates.

Hybrid manufacturing and remanufacturing lot-sizing problem with stochastic demand, return, and setup costs

This paper presents mixed-integer programming models to hybrid manufacturing and remanufacturing lot-sizing problems in which remanufactured products are treated as new ones, so that both manufactured and remanufactured products compete to meet the demands. Differently from previous studies, we consider an environment with multiple products, both manufacturing and remanufacturing costs, disposal, backlogging, and the inherent uncertainties of demands, return rates of usable products, and setup costs. In order to deal with these uncertainties, we propose a scenario-based two-stage stochastic programming model that assumes production and setup as first-stage decision variables, whereas inventory, disposal, and backlogging are defined as second-stage decision variables. We also analyze a risk-averse model in an attempt to reduce the dispersion of the second-stage costs. The main results of the present study indicate that setup costs for remanufacturing can be decisive in choosing between manufacturing or remanufacturing. Even though remanufacturing costs are lower, the process is still largely dependent on return rates and low storage costs for returned products.

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.

No Claim? Your Gain: Design of Residual Value Extended Warranties Under Risk Aversion and Strategic Claim Behavior

Traditional one-price-for-all extended warranties do not differentiate customers according to their risk attitudes, usage rates, or operating environment. These warranties are priced to cover the cost of high-usage customers who have more failures and are willing to pay a risk premium for their risk aversion. That makes traditional warranties economically unattractive to low-usage customers and those who are less risk averse. These issues can be addressed by residual value warranties, which refund part of the up-front price to customers who have zero or few claims according to a predetermined refund schedule. Residual value warranties may induce strategic claim behavior, since customers may prefer to pay for small failures out of pocket rather than claim failures now and give up potential refunds later. We design and price residual value warranties to maximize expected profits, taking into account strategic claim behavior and risk attitudes. For the constant absolute risk aversion model, we characterize customers’ optimal claim strategy as well as the net value and support cost for residual value warranties. Surprisingly, the total support cost to the service provider, including repair costs and refunds, is lower for more risk-averse customers under the residual value warranties, whereas their willingness to pay is higher. As contingent contracts, residual value warranties can better price discriminate customers than traditional warranties. We identify conditions under which residual value warranties are strictly more profitable than traditional warranties in a homogeneous market, as well as in heterogeneous markets that differ in various dimensions, such as risk attitude, failure rate, and repair cost.

Environmental and Economic Optimization Model for Electric System Planning in Ningxia, China: Inexact Stochastic Risk-Aversion Programming Approach

The main goal of this paper is to provide a novel risk aversion model for long-term electric power system planning from the manager’s perspective with the consideration of various uncertainties. In the proposed method, interval parameter programming and two-stage stochastic programming are integrated to deal with the technical, economics, and policy uncertainties. Moreover, downside risk theory is introduced to balance the trade-off between the profit and risk according to the decision-maker’s risk aversion attitude. To verify the effectiveness and practical application of this approach, an inexact stochastic risk aversion model is developed for regional electric system planning and management in Ningxia Hui Autonomous Region, China. The series of solutions provide the decision-maker with the optimal investment strategy and operation management under different future emission reduction scenarios and risk-aversion levels. The results indicated that pollution control devices are still the main measures to achieve the current mitigation goal and the adjustment of generation structure would play an important role in the future cleaner electricity system with the stricter environmental policy. In addition, the model can be used for generating decision alternatives and helping decision-makers identify desired energy structure adjustment and pollutants/carbon mitigation abatement policies under various economic and system-reliability constraints.

A Risk-Averse Inventory Model with Markovian Purchasing Costs

We study a few dynamic risk-averse inventory models using additive utility functions. We add Markovian behavior of purchasing costs in our models. Such Markovian purchasing costs can reflect a market situation in a global supply chain such as fluctuations at exchange rates or the existence of product spot markets. We provide our problem formulations with finite and infinite MDP (Markovian Decision Process) problems. For finite time models, we first prove (joint) concavity of the model for each state and obtain a (modified) base-stock optimal policy. Then, we conduct comparative static analysis for model parameters and derive monotone properties to the optimal solutions. For infinite time models, we show the existence of stationary base-stock optimal policies and the inheritance of the monotone properties proven at our finite time models.

Risk-Averse Optimization in Two-Stage Stochastic Models: Computational Aspects and a Study

We extend the on-demand accuracy approach of Oliveira and Sagastizabal to constrained convex optimization. The resulting method is applied to risk-averse two-stage stochastic programming problems. We present a survey of risk-averse models. The appropriate oracle is formulated for the case of a conditional value-at-risk constraint. We discuss computational aspects and compare different approaches in a study.

Supply chain network design under uncertainty with new insights from contracts

In this paper, the classical problem of supply chain network design is reconsidered to emphasize the role of contracts in uncertain environments. The supply chain addressed consists of four layers: suppliers, manufacturers, warehouses, and customers acting within a single period. The single owner of the manufacturing plants signs a contract with each of the suppliers to satisfy demand from downstream. Available contracts consist of long-term and option contracts, and unmet demand is satisfied by purchasing from the spot market. In this supply chain, customer demand, supplier capacity, plants and warehouses, transportation costs, and spot prices are uncertain. Two models are proposed here: a risk-neutral two-stage stochastic model and a risk-averse model that considers risk measures. A solution strategy based on sample average approximation is then proposed to handle large scale problems. Extensive computational studies prove the important role of contracts in the design process, especially a portfolio of contracts. For instance, we show that long-term contract alone has similar impacts to having no contracts, and that option contract alone gives inferior results to a combination of option and long-term contracts. We also show that the proposed solution methodology is able to obtain good quality solutions for large scale problems.

A new probability model for insuring critical path problem with heuristic algorithm

Abstract In order to obtain an adequate description of risk aversion for insuring critical path problem, this paper develops a new class of two-stage minimum risk problems. The first-stage objective function is to minimize the probability of total costs exceeding a predetermined threshold value, while the second-stage objective function is to maximize the insured task durations. For general task duration distributions, we adapt sample average approximation (SAA) method to probability objective function. The resulting SAA problem is a two-stage integer programming model, in which the analytical expression of second-stage value function is unavailable, we cannot solve it by conventional optimization algorithms. To avoid this difficulty, we design a new hybrid algorithm by combining dynamic programming method (DPM) and genotype-phenotype-neighborhood based binary particle swarm optimization (GPN-BPSO), where the DPM is employed to find the critical path in the second-stage programming problem. We conduct some numerical experiments via a critical path problem with 30 nodes and 42 arcs, and discuss the proposed risk averse model and the experimental results obtained by hybrid GPN-BPSO, hybrid genetic algorithm (GA) and hybrid BPSO. The computational results show that hybrid GPN-BPSO achieves the better performance than hybrid GA and hybrid BPSO, and the proposed critical path model is important for risk averse decision makers.

No Claim? Your Gain: Design of Residual Value Extended Warranties Under Risk Aversion and Strategic Claim Behavior

Traditional one-price-for-all extended warranties do not differentiate customers according to their risk attitudes, usage rates or operating environment. These warranties are priced to cover the cost of high usage customers who experience more failures and are willing to pay a risk premium for their risk aversion. This makes traditional warranties economically unattractive to low usage customers and those who are less risk averse. Residual value warranties can help address these issues. Residual value warranties refund a part of the upfront price to customers who have zero or few claims according to a pre-determined refund schedule. However, a warranty with residual values may induce strategic claim behavior as customers may prefer to pay for small failures out of pocket rather than to claim failures now and give up potential refunds later. Taking into account the claim behavior and risk attitudes of customers, we design and price residual value warranties. In a constant absolute risk aversion model, we characterize customers’ optimal claim strategy as well as the structure of the net value and support cost for residual value warranties. Surprisingly, the total support cost to the service provider, including repair cost and refund, is lower for more risk-averse customers under the residual value warranties, while their willingness-to-pay is higher. This implies potential profitability of residual value warranties. Moreover, residual value warranties, as contingent contracts, can better price discriminate customers than traditional warranties. We identify conditions under which residual value warranties are strictly more profitable than traditional warranties in a homogeneous market, as well as in heterogeneous markets differing in various dimensions, such as risk attitude, failure rate and repair cost.

Why uncertainty matters: discounting under intertemporal risk aversion and ambiguity

Uncertainty has an almost negligible impact on project value in the standard economic model. I show that a comprehensive evaluation of uncertainty and uncertainty attitude changes this picture fundamentally. The illustration of this result relies on the discount rate, which is the crucial determinant in balancing immediate costs against future benefits, and the single most important determinant of optimal mitigation policies in the integrated assessment of climate change. First, the paper removes an implicit assumption of (intertemporal or intrinsic) risk neutrality from the standard economic model. Second, the paper introduces aversion to non-risk uncertainty (ambiguity). I show a close formal similarity between the model of intertemporal risk aversion, which is a reformulation of the widespread Epstein–Zin–Weil model, and a recent model of smooth ambiguity aversion. I merge the models, achieving a threefold disentanglement between risk aversion, ambiguity aversion, and the propensity to smooth consumption over time.

Time Consistent Behavior Portfolio Policy for Dynamic Mean-Variance Formulation

When we implement a portfolio selection methodology under a mean-risk formulation, it is essential to correctly model investors' risk aversion which may be time-dependent, or even state-dependent during the investment procedure. In this paper, we propose a behavior risk aversion model, which is a piecewise linear function of the current wealth level with a reference point at a preset investment target. Due to the time inconsistency of the resulting multi-period mean-variance model with an adaptive risk aversion, we investigate in this paper the time consistent behavior portfolio policy by solving a nested mean-variance game formulation. We derive semi-analytical time consistent behavior portfolio policy which takes a piecewise linear feedback form of the current wealth level with respect to the discounted investment target.

The study on stage financing model of IT project investment.

Stage financing is the basic operation of venture capital investment. In investment, usually venture capitalists use different strategies to obtain the maximum returns. Due to its advantages to reduce the information asymmetry and agency cost, stage financing is widely used by venture capitalists. Although considerable attentions are devoted to stage financing, very little is known about the risk aversion strategies of IT projects. This paper mainly addresses the problem of risk aversion of venture capital investment in IT projects. Based on the analysis of characteristics of venture capital investment of IT projects, this paper introduces a real option pricing model to measure the value brought by the stage financing strategy and design a risk aversion model for IT projects. Because real option pricing method regards investment activity as contingent decision, it helps to make judgment on the management flexibility of IT projects and then make a more reasonable evaluation about the IT programs. Lastly by being applied to a real case, it further illustrates the effectiveness and feasibility of the model.

Risk Management in Production Planning of Perishable Goods

In food supply chain planning, the trade-off between expected profit and risk is emphasized by the perishable nature of the goods that it has to handle. In particular, the risk of spoilage and the risk of revenue loss are substantial when stochastic parameters related to the demand, the consumer behavior, and the spoilage effect are considered. This paper aims to expose and handle this trade-off by developing risk-averse production planning models that incorporate financial risk measures. In particular, the performance of a risk-neutral attitude is compared to the performance of models taking into account the upper partial mean and the conditional value-at-risk. Insights from an illustrative example show the positive impact of the risk-averse models in operational performance indicators, such as the amount of expired products. Furthermore, through an extensive computational experiment, the advantage of the conditional value-at-risk model is evidenced, as it is able to dominate the solutions from the upper...

Effective location models for sorting recyclables in public management

The recycling of urban solid wastes is a critical point for the “closing supply chains” of many products, mainly when their value cannot be completely recovered after use. In addition to environmental aspects, the process of recycling involves technical, economic, social and political challenges for public management. For most of the urban solid waste, the management of the end-of-life depends on selective collection to start the recycling process. For this reason, an efficient selective collection has become a mainstream tool in the Brazilian National Solid Waste Policy. In this paper, we study effective models that might support the location planning of sorting centers in a medium-sized Brazilian city that has been discussing waste management policies over the past few years. The main goal of this work is to provide an optimal location planning design for recycling urban solid wastes that fall within the financial budget agreed between the municipal government and the National Bank for Economic and Social Development. Moreover, facility planning involves deciding on the best sites for locating sorting centers along the four-year period as well as finding ways to meet the demand for collecting recyclable materials, given that economic factors, consumer behavior and environmental awareness are inherently uncertain future outcomes. To deal with these issues, we propose a deterministic version of the classical capacity facility location problem, and both a two-stage recourse formulation and risk-averse models to reduce the variability of the second-stage costs. Numerical results suggest that it is possible to improve the current selective collection, as well as hedge against data uncertainty by using stochastic and risk-averse optimization models.

Comparative Ross risk aversion in the presence of mean dependent risks

This paper studies comparative risk aversion between risk averse agents in the presence of a background risk. Our contribution differs from most of the literature in two respects. First, background risk does not need to be additive or multiplicative. Second, the two risks are not necessarily mean independent, and may be conditional expectation increasing or decreasing. We show that our order of cross Ross risk aversion is equivalent to the order of partial risk premium, while our index of decreasing cross Ross risk aversion is equivalent to decreasing partial risk premium. These results generalize the comparative risk aversion model developed by Ross for mean independent risks. Our theoretical results are related to utility functions having the n-switch independence property.

Optimization with Multivariate Conditional Value-at-Risk Constraints

For many decision-making problems under uncertainty, it is crucial to develop risk-averse models and specify the decision makers' risk preferences based on multiple stochastic performance measures (or criteria). Incorporating such multivariate preference rules into optimization models is a fairly recent research area. Existing studies focus on extending univariate stochastic dominance rules to the multivariate case. However, enforcing multivariate stochastic dominance constraints can often be overly conservative in practice. As an alternative, we focus on the widely applied risk measure conditional value-at-risk (CVaR), introduce a multivariate CVaR relation, and develop a novel optimization model with multivariate CVaR constraints based on polyhedral scalarization. To solve such problems for finite probability spaces, we develop a cut generation algorithm, where each cut is obtained by solving a mixed-integer problem. We show that a multivariate CVaR constraint reduces to finitely many univariate CVaR constraints, which proves the finite convergence of our algorithm. We also show that our results can be naturally extended to a wider class of coherent risk measures. The proposed approach provides a flexible and computationally tractable way of modeling preferences in stochastic multicriteria decision making. We conduct a computational study for a budget allocation problem to illustrate the effect of enforcing multivariate CVaR constraints and demonstrate the computational performance of the proposed solution methods.

SDDP for some interstage dependent risk-averse problems and application to hydro-thermal planning

We consider interstage dependent stochastic linear programs where both the random right-hand side and the model of the underlying stochastic process have a special structure. Namely, for equality constraints (resp. inequality constraints) the right-hand side is an affine function (resp. a given function b t ) of the process value for the current time step t. As for m-th component of the process at time step t, it depends on previous values of the process through a function h tm . For this type of problem, to obtain an approximate policy under some assumptions for functions b t and h tm , we detail a stochastic dual dynamic programming algorithm. Our analysis includes some enhancements of this algorithm such as the definition of a state vector of minimal size, the computation of feasibility cuts without the assumption of relatively complete recourse, as well as efficient formulas for sharing optimality and feasibility cuts between nodes of the same stage. The algorithm is given for both a non-risk-averse and a risk-averse model. We finally provide preliminary results comparing the performances of the recourse functions corresponding to these two models for a real-life application.

구매가격 변동 하에서 초기재고 재분배를 통한 위험회피 재고모형의 효율화

In traditional inventory models, purchase prices of raw materials are assumed to be fixed and have no effect on the optimal choice of inventory policies. However, when purchase prices fluctuate continuously over time, inventory costs are heavily affected by purchasing prices. Risk-averse inventory model decides order quantity and ordering time by considering not just purchase prices but also the risk from the discrepancy between estimated prices and realized prices. In this paper, we propose a myopic inventory policy which incorporates price risk into deciding ordering time and quantities. While the existing risk-averse model has no mechanism to reallocate inventories already purchased for a specific future period, the revised one reallocates initial inventories of each period to other future periods so that it can avoid purchasing raw materials at high prices. Experimental results demonstrate that the revised model outperforms the existing one in respect of total cost and variability.