Volume 2006http://edoc.hu-berlin.de/18452/3632024-03-28T19:26:38Z2024-03-28T19:26:38ZOn Rates of Convergence for Stochastic Optimization Problems Under Non-I.I.D. SamplingHomem-de-Mello, Titohttp://edoc.hu-berlin.de/18452/90242020-03-07T04:14:31Z2006-12-18T00:00:00ZOn Rates of Convergence for Stochastic Optimization Problems Under Non-I.I.D. Sampling
Homem-de-Mello, Tito
http://dx.doi.org/10.18452/8372
Higle, Julie L.; Römisch, Werner; Sen, Surrajeet
In this paper we discuss the issue of solving stochastic optimization problems bymeans of sample average approximations. Our focus is on rates of convergence of estimators of optimal solutions and optimal values with respect to the sample size. Thisis a well studied problem in case the samples are independent and identically distributed (i.e., when standard Monte Carlo is used); here, we study the case where thatassumption is dropped. Broadly speaking, our results show that, under appropriate assumptions, the rates of convergence for pointwise estimators under a sampling schemecarry over to the optimization case, in the sense that convergence of approximatingoptimal solutions and optimal values to their true counterparts has the same rates asin pointwise estimation. Our motivation for the study arises from two types of sampling methods that havebeen widely used in the Statistics literature. One is Latin Hypercube Sampling (LHS),a stratified sampling method originally proposed in the seventies by McKay, Beckman,and Conover (1979). The other is the class of quasi-Monte Carlo (QMC) methods,which have become popular especially after the work of Niederreiter (1992). Theadvantage of such methods is that they typically yield pointwise estimators which notonly have lower variance than standard Monte Carlo but also possess better rates ofconvergence. Thus, it is important to study the use of these techniques in sampling-based optimization. The novelty of our work arises from the fact that, while therehas been some work on the use of variance reduction techniques and QMC methods instochastic optimization, none of the existing work — to the best of our knowledge — hasprovided a theoretical study on the effect of these techniques on rates of convergence forthe optimization problem. We present numerical results for some two-stage stochasticprograms from the literature to illustrate the discussed ideas.
2006-12-18T00:00:00ZConvergent Bounds for Stochastic Programs with Expected Value ConstraintsKuhn, Danielhttp://edoc.hu-berlin.de/18452/90232020-03-07T04:14:31Z2006-12-18T00:00:00ZConvergent Bounds for Stochastic Programs with Expected Value Constraints
Kuhn, Daniel
http://dx.doi.org/10.18452/8371
Higle, Julie L.; Römisch, Werner; Sen, Surrajeet
This article elaborates a bounding approximation scheme for convexmultistage stochastic programs (MSP) that constrain the conditional expectation ofsome decision-dependent random variables. Expected value constraints of this typeare useful for modelling a decision maker’s risk preferences, but they may also ariseas artefacts of stage-aggregation. It is shown that the gap between certain upper andlower bounds on the optimal objective value can be made smaller than any prescribedtolerance. Moreover, the solutions of some tractable approximate MSP give rise to apolicy which is feasible in the (untractable) original MSP, and this policy’s cost differsfrom the optimal cost at most by the difference between the bounds. The consideredproblem class comprises models with integrated chance constraints and conditionalvalue-at-risk constraints. No relatively complete recourse is assumed.
2006-12-18T00:00:00ZAirline Network Revenue Management by Multistage Stochastic ProgrammingMöller, AndrisRömisch, WernerWeber, Klaushttp://edoc.hu-berlin.de/18452/90222020-03-07T04:14:31Z2006-12-14T00:00:00ZAirline Network Revenue Management by Multistage Stochastic Programming
Möller, Andris; Römisch, Werner; Weber, Klaus
http://dx.doi.org/10.18452/8370
Higle, Julie L.; Römisch, Werner; Sen, Surrajeet
A multistage stochastic programming approach to airline network revenue management is presented. The objective is to determine seatprotection levels for all itineraries, fare classes, point of sales of the airlinenetwork and all data collection points of the booking horizon such that theexpected revenue is maximized. While the passenger demand and cance-lation rate processes are the stochastic inputs of the model, the stochasticprotection level process represents its output and allows to control the booking process. The stochastic passenger demand and cancelation rate processesare approximated by a finite number of tree structured scenarios. The scenario tree is generated from historical data using a stability-based recursivescenario reduction scheme. Numerical results for a small hub-and-spoke network are reported.
2006-12-14T00:00:00ZStability of multistage stochastic programs incorporating polyhedral risk measuresEichhorn, AndreasRömisch, Wernerhttp://edoc.hu-berlin.de/18452/90212020-03-07T04:14:31Z2006-12-14T00:00:00ZStability of multistage stochastic programs incorporating polyhedral risk measures
Eichhorn, Andreas; Römisch, Werner
http://dx.doi.org/10.18452/8369
Higle, Julie L.; Römisch, Werner; Sen, Surrajeet
We analyse stability aspects of linear multistage stochastic programs with polyhedral risk measures inthe objective. In particular, we consider sensitivity of the optimal value with respect perturbations ofthe underlying stochastic input process. An existing stability result for multistage stochastic programswith expectation objective is carried forward to the case of polyhedral risk-averse objectives. Beside$L_r$-distances these results also involve filtration distances of the perturbations of the stochasticprocess. We discuss additional requirements for the polyhedral risk measures such that the problemdependent filtration distances can be bounded by problem independent ones. Stability and suchbounds are the basis for scenario tree approximation techniques used in practical problem solving.
2006-12-14T00:00:00ZShape-based Scenario Generation using CopulasKaut, MichalStein, W.http://edoc.hu-berlin.de/18452/90202020-03-07T04:14:31Z2006-12-07T00:00:00ZShape-based Scenario Generation using Copulas
Kaut, Michal; Stein, W.
http://dx.doi.org/10.18452/8368
Higle, Julie L.; Römisch, Werner; Sen, Surrajeet
The purpose of this article is to show how the multivariate structure (the ”shape” of the distribution) can be separated from the marginal distributions when generating scenarios. To dothis we use the copula. As a result, we can define combined approaches that capture shape with one method and handle margins with another. In some cases the combined approach is exact, inother cases, the result is an approximation. This new approach is particularly useful if the shape is somewhat peculiar, and substantially different from the standard normal elliptic shape. But it can also be used to obtain the shape of the normal but with margins from different distribution families, or normal margins with for example tail dependence in the multivariate structure. We provide an example from portfolio management.
2006-12-07T00:00:00ZCutting planes for multi-stage stochastic integer programsGuan, YongpeiAhmed, ShabbirNemhauser, George L.http://edoc.hu-berlin.de/18452/90192020-03-07T04:14:30Z2006-11-21T00:00:00ZCutting planes for multi-stage stochastic integer programs
Guan, Yongpei; Ahmed, Shabbir; Nemhauser, George L.
http://dx.doi.org/10.18452/8367
Higle, Julie L.; Römisch, Werner; Sen, Surrajeet
This paper addresses the problem of finding cutting planes for multi-stage stochastic integer programs.We give a general method for generating cutting planes for multi-stage stochastic integer programs basedon combining inequalities that are valid for the individual scenarios. We apply the method to generatecuts for a stochastic version of a dynamic knapsack problem and to stochastic lot sizing problems. Wegive computational results which show that these new inequalities are very effective in a branch-and-cutalgorithm.
2006-11-21T00:00:00ZA branch-and-cut algorithm for two-stage stochastic mixed-binary programs with continuous first-stage variablesNtaimo, LewisSen, Suvrajeethttp://edoc.hu-berlin.de/18452/90182020-03-07T04:14:30Z2006-10-27T00:00:00ZA branch-and-cut algorithm for two-stage stochastic mixed-binary programs with continuous first-stage variables
Ntaimo, Lewis; Sen, Suvrajeet
http://dx.doi.org/10.18452/8366
Higle, Julie L.; Römisch, Werner; Sen, Surrajeet
This paper presents a branch-and-cut method for two-stage stochastic mixed-integer programming (SMIP) problems with continuous first-stage variables. This method is derived based on disjunctive decomposition(D2) for SMIP, an approach in which disjunctive programming is used to derive valid inequalities for SMIP. The novelty of the proposed method derives from branching on the first-stage continuous domain while the branch-and-bound process is guided by the disjunction variables in the second-stage.Finite convergence of the algorithm for mixed-binary second-stage is established and a numerical example to illustrate the new method is given.Keywords: stochastic programming, disjunctive decomposition, branch-and-bound, branch-and-cut
2006-10-27T00:00:00ZShort-term hydropower production planning by stochastic programmingFleten, Stein-ErikKristoffersen, Trine Kroghhttp://edoc.hu-berlin.de/18452/90172020-03-07T04:14:30Z2006-10-27T00:00:00ZShort-term hydropower production planning by stochastic programming
Fleten, Stein-Erik; Kristoffersen, Trine Krogh
http://dx.doi.org/10.18452/8365
Higle, Julie L.; Römisch, Werner; Sen, Surrajeet
Within the framework of multi-stage mixed-integer linear stochastic programmingwe develop a short-term production plan for a price-taking hydropower plant op-erating under uncertainty. Current production must comply with the day-aheadcommitments of the previous day which makes short-term production planning amatter of spatial distribution among the reservoirs of the plant. Day-ahead marketprices and reservoir inflows are, however, uncertain beyond the current operationday and water must be allocated among the reservoirs in order to strike a balancebetween current profits and expected future profits. A demonstration is presentedwith data from a Norwegian hydropower producer and the Nordic power market atNord Pool.Keywords: OR in energy; hydropower; stochastic programming; scenarios
2006-10-27T00:00:00ZSome remarks on value-at-risk optimizationHenrion, Renéhttp://edoc.hu-berlin.de/18452/90162020-03-07T04:14:30Z2006-10-27T00:00:00ZSome remarks on value-at-risk optimization
Henrion, René
http://dx.doi.org/10.18452/8364
Higle, Julie L.; Römisch, Werner; Sen, Surrajeet
We discuss two observations related to value-at-arisk optimization. First we consider a portfolio problem under an infinite number of value-at-risk inequality constraints (modelling first order stochastic dominance). The random data are assumed to be normally distributed. Although this problem is necessarily non-convex, an explicit solution can be derived. Secondly, we provide a (negative) result on quantitative stability of the value-at-risk under variation of the random variable. Although reduced Lipschitz properties (in the sense of calmness) may hold true at continuously distributed random variables under suitable conditions, the result shows that no full Lipschitz property (more generally: Hölder property at any rate) can hold in the neighbourhood of an arbitrary continuously distributed variable. Even worse, this observation holds true with respect to any probability metric weaker than that of total variation.Keywords: value at risk, stochastic dominance, Hölder continuity, quantile functions
2006-10-27T00:00:00ZRobust solution and risk measures for a supply chain planning problem under uncertaintyPoojari, Chandra A.Lucas, CormacMitra, Gautamhttp://edoc.hu-berlin.de/18452/90152020-03-07T04:14:30Z2006-10-27T00:00:00ZRobust solution and risk measures for a supply chain planning problem under uncertainty
Poojari, Chandra A.; Lucas, Cormac; Mitra, Gautam
http://dx.doi.org/10.18452/8363
Higle, Julie L.; Römisch, Werner; Sen, Surrajeet
We consider a strategic supply chain planning problem formulated as a two-stageStochastic Integer Programming (SIP) model. The strategic decisions include sitelocations, choices of production, packing and distribution lines, and the capacityincrement or decrement policies. The SIP model provides a practical representationof real world discrete resource allocation problems in the presence of future uncertaintieswhich arise due to changes in the business and economic environment. Suchmodels that consider the future scenarios (along with their respective probabilities)not only identify optimal plans for each scenario, but also determine a hedgedstrategy for all the scenarios. We,(1) exploit the natural decomposable structure of the SIP problem through Benders’decomposition,(2) approximate the probability distribution of the random variables using theGeneralised Lambda distribution, and(3) through simulations, calculate the performance statistics and the risk measuresfor the two models, namely the expected-value and the here-and-now.Key words: Supply Chain planning, Stochastic integer Programming, Benders’decomposition, Generalised Lambda distribution, simulation, Genetic algorithm
2006-10-27T00:00:00ZScenario reduction in stochastic programming with respect to discrepancy distancesHenrion, RenéKüchler, ChristianRömisch, Wernerhttp://edoc.hu-berlin.de/18452/90142020-03-07T04:14:29Z2006-10-26T00:00:00ZScenario reduction in stochastic programming with respect to discrepancy distances
Henrion, René; Küchler, Christian; Römisch, Werner
http://dx.doi.org/10.18452/8362
Higle, Julie L.; Römisch, Werner; Sen, Surrajeet
Discrete approximations to chance constrained and mixed-integer two-stage stochastic programs require moderately sized scenario sets. The relevant distances of (multivariate) probability distributions for deriving quantitative stability results for such stochastic programs are B-discrepancies, where the class B of Borel sets depends on their structural properties. Hence, the optimal scenario reduction problem for such models is stated with respect to B-discrepancies. In this paper, upper and lower bounds, and some explicit solutions for optimal scenario reduction problems are derived. In addition, we develop heuristic algorithms for determining nearly optimally reduced probability measures, discuss the case of the cell discrepancy (or Kolmogorov metric) in some detail and provide some numerical experience.
2006-10-26T00:00:00ZOptimal Hedging Strategies for Multi-periodGuarantees in the Presence of Transaction Costs:A Stochastic Programming ApproachFleten, Stein-ErikLindset, Snorrehttp://edoc.hu-berlin.de/18452/90132020-03-07T04:14:29Z2006-10-18T00:00:00ZOptimal Hedging Strategies for Multi-periodGuarantees in the Presence of Transaction Costs:A Stochastic Programming Approach
Fleten, Stein-Erik; Lindset, Snorre
http://dx.doi.org/10.18452/8361
Higle, Julie L.; Römisch, Werner; Sen, Surrajeet
Multi-period guarantees are often embedded in life insurance contracts. In this paper we consider the problem of hedging these multi-period guarantees in the presence of transaction costs. We derive thehedging strategies for the cheapest hedge portfolio for a multi-periodguarantee that with certainty makes the insurance company able tomeet the obligations from the insurance policies it has issued. We findthat by imposing transaction costs, the insurance company reducesthe rebalancing of the hedge portfolio. The cost of establishing thehedge portfolio also increases as the transaction cost increases. Forthe multi-period guarantee there is a rather large rebalancing of thehedge portfolio as we go from one period to the next. By introducingtransaction costs we find the size of this rebalancing to be reduced.Transaction costs may therefore be one possible explanation for whywe do not see the insurance companies performing a large rebalancingof their investment portfolio at the end of each year.
2006-10-18T00:00:00ZA JELS Stochastic inventory model with random demandPanda, G.Khan, D.A.Ray, U.C.http://edoc.hu-berlin.de/18452/90122020-03-07T04:14:29Z2006-10-18T00:00:00ZA JELS Stochastic inventory model with random demand
Panda, G.; Khan, D.A.; Ray, U.C.
http://dx.doi.org/10.18452/8360
Higle, Julie L.; Römisch, Werner; Sen, Surrajeet
A stochastic joint lot size model has been developed in which demand ofthe customer and the stock level of the vendor are assumed to be identicallydistributed continuous random variables. The effective ways for a compromisebetween the vendor and the customer at a common lot size with certain amountof price adjustments are determined and the methodology is explained througha numerical example.Key words: Inventory control programming, Stochastic models, Truncated normaldistribution, Joint economic lot size.
2006-10-18T00:00:00ZStability of ε-approximate solutions to convex stochastic programsRömisch, WernerWets, Roger J.-B.http://edoc.hu-berlin.de/18452/90112020-03-07T04:14:29Z2006-06-21T00:00:00ZStability of ε-approximate solutions to convex stochastic programs
Römisch, Werner; Wets, Roger J.-B.
http://dx.doi.org/10.18452/8359
Higle, Julie L.; Römisch, Werner; Sen, Surrajeet
An analysis of convex stochastic programs is provided if the underlying proba-bility distribution is subjected to (small) perturbations. It is shown, in particular,that ε-approximate solution sets of convex stochastic programs behave Lipschitzcontinuous with respect to certain distances of probability distributions that aregenerated by the relevant integrands. It is shown that these results apply tolinear two-stage stochastic programs with random recourse. Consequences arediscussed on associating Fortet-Mourier metrics to two-stage models and on theasymptotic behavior of empirical estimates of such models, respectively.
2006-06-21T00:00:00ZConvexity of chance constraints with independent random variablesHenrion, RenéStrugarek, Cyrillehttp://edoc.hu-berlin.de/18452/90102020-03-07T04:14:28Z2006-05-15T00:00:00ZConvexity of chance constraints with independent random variables
Henrion, René; Strugarek, Cyrille
http://dx.doi.org/10.18452/8358
Higle, Julie L.; Römisch, Werner; Sen, Surrajeet
We investigate the convexity of chance constraints with independent random variables. It will be shown, how concavity properties of the mapping related to the decision vector have to be combined with a suitable property of decrease for the marginal densities in order to arrive at convexity of the feasible set for large enough probability levels. It turns out that the required decrease can be verified for most prominent density functions. The results are applied then, to derive convexity of linear chance constraints with normally distributed stochastic coefficients when assuming independence of the rows of the coefficient matrix.
2006-05-15T00:00:00ZStochastic programming for optimizing bidding strategies of a nordic hydropower producerFleten, Stein-ErikKristoffersen, Trine Kroghhttp://edoc.hu-berlin.de/18452/90092020-03-07T04:14:28Z2006-04-15T00:00:00ZStochastic programming for optimizing bidding strategies of a nordic hydropower producer
Fleten, Stein-Erik; Kristoffersen, Trine Krogh
http://dx.doi.org/10.18452/8357
Higle, Julie L.; Römisch, Werner; Sen, Surrajeet
From the point of view of a price-taking hydropower producer participating in the day-ahead power market, market prices are highly uncertain. The present paper provides a model for determining optimal bidding strategies taking into account this uncertainty. In particular, realistic market price scenarios are generated and a stochastic mixed-integer linear programming model that takes in both production and physical trading aspects is developed. The idea is to explore the effects of including uncertainty into the optimization model and to compare the stochastic approach to a deterministic one. The model is illustrated with data from a Norwegian hydropower producer and the Nordic power market at Nord Pool.
2006-04-15T00:00:00ZScenario tree modelling for multistage stochastic programsHeitsch, HolgerRömisch, Wernerhttp://edoc.hu-berlin.de/18452/90082020-03-07T04:14:28Z2006-03-31T00:00:00ZScenario tree modelling for multistage stochastic programs
Heitsch, Holger; Römisch, Werner
http://dx.doi.org/10.18452/8356
Higle, Julie L.; Römisch, Werner; Sen, Surrajeet
An important issue for solving multistage stochastic programs consists inthe approximate representation of the (multivariate) stochastic input process inthe form of a scenario tree. In this paper, forward and backward approachesare developed for generating scenario trees out of an initial fan of individualscenarios. Both approaches are motivated by the recent stability result in [15]for optimal values of multistage stochastic programs. They are based on upperbounds for the two relevant ingredients of the stability estimate, namely, theprobabilistic and the filtration distance, respectively. These bounds allow tocontrol the process of recursive scenario reduction [13] and branching. Numericalexperience is reported for constructing multivariate scenario trees in electricityportfolio management.
2006-03-31T00:00:00ZEstimation method of multivariate exponential probabilities based on a simplex coordinates transformOlieman, Niels J.Putten, Bram vanhttp://edoc.hu-berlin.de/18452/90072020-03-07T04:14:28Z2006-03-09T00:00:00ZEstimation method of multivariate exponential probabilities based on a simplex coordinates transform
Olieman, Niels J.; Putten, Bram van
http://dx.doi.org/10.18452/8355
Higle, Julie L.; Römisch, Werner; Sen, Surrajeet
A novel unbiased estimator for estimating the probability mass of a multivariate exponential distribution over a measurable set is introduced and is called the Exponential Simplex (ES) estimator. For any measurable set, the standard error of the ES-estimator is at most the standard error of the well known Monte Carlo (MC) estimator. For non-radially shaped measurable sets, the ES-estimator has a strictly smaller standard error than the MC-estimator. For ray-convex sets, such as convex sets, the ES-estimator can be expressed in a simple analytical form.
2006-03-09T00:00:00ZModels for nuclear smuggling interdictionMorton, David P.Pan, FengSaeger, Kevin J.http://edoc.hu-berlin.de/18452/90062020-03-07T04:14:28Z2006-03-20T00:00:00ZModels for nuclear smuggling interdiction
Morton, David P.; Pan, Feng; Saeger, Kevin J.
http://dx.doi.org/10.18452/8354
Higle, Julie L.; Römisch, Werner; Sen, Surrajeet
We describe two stochastic network interdiction models for thwarting nuclear smuggling.In the first model, the smuggler travels through a transportation network on a path thatmaximizes the probability of evading detection, and the interdictor installs radiationsensors to minimize that evasion probability. The problem is stochastic because thesmuggler’s origin-destination pair is known only through a probability distribution atthe time when the sensors are installed. In this model, the smuggler knows the locationsof all sensors and the interdictor and the smuggler “agree” on key network parameters,namely the probabilities the smuggler will be detected while traversing the arcs of thetransportation network. Our second model differs in that the interdictor and smugglercan have differing perceptions of these network parameters. This model captures thecase in which the smuggler is aware of only a subset of the sensor locations. Forboth models, we develop the important special case in which the sensors can only beinstalled at border crossings of a single country so that the resulting model is definedon a bipartite network. In this special case, a class of valid inequalities reduces thecomputation time for the identical-perceptions model.
2006-03-20T00:00:00ZOn two-stage convex chance constrained problemsErdogan, E.Iyengar, G.http://edoc.hu-berlin.de/18452/90052020-03-07T04:14:27Z2006-03-20T00:00:00ZOn two-stage convex chance constrained problems
Erdogan, E.; Iyengar, G.
http://dx.doi.org/10.18452/8353
Higle, Julie L.; Römisch, Werner; Sen, Surrajeet
In this paper we develop approximation algorithms for two-stage convex chance constrainedproblems. Nemirovski and Shapiro [16] formulated this class of problems and proposed anellipsoid-like iterative algorithm for the special case where the impact function f (x, h) is bi-affine.We show that this algorithm extends to bi-convex f (x, h) in a fairly straightforward fashion.The complexity of the solution algorithm as well as the quality of its output are functions of theradius r of the largest Euclidean ball that can be inscribed in the polytope defined by a randomset of linear inequalities generated by the algorithm [16]. Since the polytope determining ris random, computing r is diffiult. Yet, the solution algorithm requires r as an input. Inthis paper we provide some guidance for selecting r. We show that the largest value of r isdetermined by the degree of robust feasibility of the two-stage chance constrained problem –the more robust the problem, the higher one can set the parameter r. Next, we formulate ambiguous two-stage chance constrained problems. In this formulation,the random variables defining the chance constraint are known to have a fixed distribution;however, the decision maker is only able to estimate this distribution to within some error. Weconstruct an algorithm that solves the ambiguous two-stage chance constrained problem whenthe impact function f (x, h) is bi-affine and the extreme points of a certain “dual” polytope areknown explicitly.
2006-03-20T00:00:00ZGenetic algorithm based technique for solving chance constrained problemsPoojari, Chandra A.Varghese, Bobyhttp://edoc.hu-berlin.de/18452/90042020-03-07T04:14:27Z2006-03-20T00:00:00ZGenetic algorithm based technique for solving chance constrained problems
Poojari, Chandra A.; Varghese, Boby
http://dx.doi.org/10.18452/8352
Higle, Julie L.; Römisch, Werner; Sen, Surrajeet
Management and measurement of risk is an important issue in almost all areas that require decisions to be made under uncertain information. Chance Constrained Programming (CCP) have been used for modelling and analysis of risks in a number of application domains. However, the resulting mathematical problems arenon-trivial to represent using algebraic modelling languages and pose significantcomputational challenges due to their non-linear, non-convex, and the stochasticnature. We develop and implement C++ classes to represent such CCP problems.We propose a framework consisting of Genetic Algorithm and Monte-Carlo simulation in order to process the problems. The non-linear and non-convex nature of theCCP problems are processed using Genetic Algorithm, whereas the stochastic nature is addressed through simulation. The computational investigations have shownthat the framework can effciently represent and process a wide variety of the CCPproblems.
2006-03-20T00:00:00ZContamination for multistage stochastic programsDupacová, Jitkahttp://edoc.hu-berlin.de/18452/90032020-03-07T04:14:27Z2006-03-15T00:00:00ZContamination for multistage stochastic programs
Dupacová, Jitka
http://dx.doi.org/10.18452/8351
Higle, Julie L.; Römisch, Werner; Sen, Surrajeet
Contamination technique will be examined as a possible approach to robustness analysis of results obtained for multistage stochastic linear programs with respect to changes of their structure or input data. We shall focus on the case then the already selected scenario tree gets extended for additional (stress or out-of-sample) scenarios and/or additional stages.
2006-03-15T00:00:00ZCoherent Risk Measures in Inventory ProblemsAhmed, ShabbirCakmak, UlasShapiro, Alexanderhttp://edoc.hu-berlin.de/18452/90022020-03-07T04:14:27Z2006-01-02T00:00:00ZCoherent Risk Measures in Inventory Problems
Ahmed, Shabbir; Cakmak, Ulas; Shapiro, Alexander
http://dx.doi.org/10.18452/8350
Higle, Julie L.; Römisch, Werner; Sen, Surrajeet
We analyze an extension of the classical multi-period, single-item, linear cost inventory problem where the objective function is a coherent risk measure. Properties of coherent risk measures allow us to offer a unifying treatment of risk averse and min-max type formulations. For the single period newsvendor problem, we show that the structure of the optimal solution of the risk averse model is similar to that of the classical expected value problem. For a finite horizon dynamic inventory model, we show that, again, the optimal policy has a similar structure as that of the expected value problem. This result carries over even to the case when there is a fixed ordering cost. We also analyze monotonicity properties of the optimal order quantity with respect to the degree of risk aversion for certain risk measures.
2006-01-02T00:00:00Z