optimal structural policies for ambiguity and risk averse inventory and pricing models this paper discusses multiperiod stochastic joint inventory and pricing models when the decision maker is risk and ambiguity averse we study infinite horizon models with discounted and long run average optimization criteria the main result of this paper is establishing the optimality of stationary s s p policies for the infinite horizon inventory and pricing models

contraction and approximate contraction with an application to multi point boundary value problems abstract the paper deals with some general iterative methods in n metric spaces the obtained results are more explicit than those known in the literature and provides error estimates when the sequence obtained on a floating point system starts oscillating a stopping criteria and error estimates at that stage are provided an application to multi point boundary value problems is also given

relative entropy exponential utility and robust dynamic pricing in the area of dynamic revenue management optimal pricing policies are typically computed on the basis of an underlying demand rate model from the perspective of applications this approach implicitly assumes that the model is an accurate representation of the real world demand process and that the parameters characterizing this model can be accurately calibrated using data in many situations neither of these conditions are satisfied indeed models are usually simplified for the purpose of tractability and may be difficult to calibrate because of a lack of data moreover pricing policies that are computed under the assumption that the model is correct may perform badly when this is not the case this paper presents an approach to single product dynamic revenue management that accounts for errors in the underlying model at the optimization stage uncertainty in the demand rate model is represented using the notion of relative entropy and a tractable reformulation of the robust pricing problem is obtained using results concerning the change of probability measure for point processes the optimal pricing policy is obtained through a version of the so called isaacs equation for stochastic differential games and the structural properties of the optimal solution are obtained through an analysis of this equation in particular i closed form solutions for the special case of an exponential nominal demand rate model ii general conditions for the exchange of the max and the min in the differential game and iii the equivalence between the robust pricing problem and that of single product revenue management with an exponential utility function without model uncertainty are established through the analysis of this equation

robust mean field linear quadratic gaussian games with unknown l 2 disturbance this paper considers a class of mean field linear quadratic gaussian games with model uncertainty the drift term in the dynamics of the agents contains a common unknown function we take a robust

an approximate solution for linear boundary value problems with slowly varying coefficients in this paper an approximate closed form solution for linear boundary value problems with slowly varying coefficient matrices is obtained the derivation of the approximate solution is based on the freezing technique which is commonly used in analyzing the stability of slowly varying initial value problems as well as solving them the error between the approximate and the exact solutions is given and an upper bound on the norm of the error is obtained this upper bound is proportional to the rate of change of the coefficient matrix of the boundary value problem the proposed approximate solution is obtained for a two point boundary value problem and is compared to its solution obtained numerically good agreement is observed between the approximate and the numerical solutions when the rate of change of the coefficient matrix is small

towards a generalized singular value decomposition we suggest a form for and give a constructive derivation of the generalized singular value decomposition of any two matrices having the same number of columns we outline its desirable characteristics and compare it to an earlier suggestion by van loan siam j numer anal 13 1976 pp 76 83 the present form largely follows from the work of van loan but is slightly more general and computationally more amenable than that in the paper cited we also prove a useful extension of a theorem of stewart siam rev 19 1977 pp 634 662 on unitary decompositions of submatrices of a unitary matrix

robust solutions of linear programming problems contaminated with uncertain data optimal solutions of linear programming problems may become severely infeasible if the nominal data is slightly perturbed we demonstrate this phenomenon by studying 90 lps from the well known netlib collection we then apply the robust optimization methodology ben tal and nemirovski 1 3 el ghaoui et al 5 6 to produce robust solutions of the above lps which are in a sense immuned against uncertainty surprisingly for the netlib problems these robust solutions nearly lose nothing in optimality

a unified derivation of several error bounds for newton s process abstract this paper gives a unified derivation of the upper and lower bounds of the errors in newton s process obtained by dennis 4 tapia 27 gragg tapia 6 and recently by potra ptak 20 and miel 15 with the use of different techniques our argument is based upon the kantorovich theorem and the kantorovich recurrence relations from the proof it is concluded that the recent upper and lower bounds of miel are finer than those of potra ptak but do not improve the basic error bounds which are directly obtained from the kantorovich theorem further it is shown that the error bounds obtained with the use of the recurrence relations are the same as those obtained by the majorant principle finally results on the newton method in partially ordered space are surveyed a method is also described for estimating the componentwise errors for computed solutions

two stage robust network flow and design under demand uncertainty we describe a two stage robust optimization approach for solving network flow and design problems with uncertain demand in two stage network optimization one defers a subset of the flow decisions until after the realization of the uncertain demand availability of such a recourse action allows one to come up with less conservative solutions compared to single stage optimization however this advantage often comes at a price two stage optimization is in general significantly harder than single stage optimization for network flow and design under demand uncertainty we give a characterization of the first stage robust decisions with an exponential number of constraints and prove that the corresponding separation problem is np hard even for a network flow problem on a bipartite graph we show however that if the second stage network topology is totally ordered or an arborescence then the separation problem is tractable unlike single stage robust optimization under demand uncertainty two stage robust optimization allows one to control conservatism of the solutions by means of an allowed budget for demand uncertainty using a budget of uncertainty we provide an upper bound on the probability of infeasibility of a robust solution for a random demand vector we generalize the approach to multicommodity network flow and design and give applications to lot sizing and location transportation problems by projecting out second stage flow variables we define an upper bounding problem for the two stage min max min optimization problem finally we present computational results comparing the proposed two stage robust optimization approach with single stage robust optimization as well as scenario based two stage stochastic optimization

improved error bounds for freezing solutions of linear boundary value problems for the error in the freezing solutions of linear boundary value problems we obtain a bound which is sharper than that obtained recently by shahruz and schwartz appl math comput 60 1994 285 comput math appl 28 1994 75 a different freezing technique global freezing is also proposed it is shown that this new technique is easy to implement for numerical computation of the solutions moreover the corresponding solution has an error bound similar to that of the freezing method

numerical methods for determining the stability and response of periodic systems with applications to helicopter rotor dynamics and aeroelasticity abstract this paper reviews and extends some recently developed numerical techniques for analyzing the stability linear response and nonlinear response of periodic systems which are governed by ordinary differential equations with periodic coefficients the main emphasis is on applications aimed at rotor blade aeroelastic stability and response calculations in forward flight first floquer theory is br4iefly reviewed and numerically efficient methods for evaluating the transition matrix at the end of one period are described next the numerical treatment of the linear response problem is discussed finally the numerical solution of the nonlinear response problem is treated using quasilinearization and periodic shooting applications illustrating numerical properties of the methods are described

robust control of partially observable failing systems this paper is concerned with optimal maintenance decision making in the presence of model misspecification specifically we are interested in the situation where the decision maker fears that a nominal bayesian model may be miss specified or unrealistic and would like to find policies that work well even when the underlying model is flawed to this end we formulate a robust dynamic optimization model for condition based maintenance in which the decision maker explicitly accounts for distrust in the nominal bayesian model by solving a worst case problem against a second agent nature who has the ability to alter the underlying model distributions in an adversarial manner the primary focus of our analysis is on establishing structural properties and insights that hold in the face of model miss specification in particular we prove i an explicit characterization of nature s optimal response through an analysis of the robust dynamic programming equation ii convexity results for both the robust va

the statistical foundations of a class of parametric tests for heteroscedasticity abstract in this paper we examine the statistical foundations of the five tests for heteroscedasticity considered by griffiths and surekha 1986 we identify the precise specification of the alternative hypothesis as an important determinant of the power of the tests

analytical study on the motion of a system with varying damping subject to harmonic force to show the possibility of a new kind of a moment shaker the motion of a system composed of a rigid plate constant springs and time varying dampers has been considered when the plate is subjected to harmonic force the vertical displacement at the center of the plate and the rotational displacement of the plate were obtained by numerical integration of the equations of motion the major frequency components of these responses were found through fast fourier transform and the condition under which the plate oscillates with almost a single frequency and a large amplitude was found the existence of the frequency components of the vertical and rotational displacements of the plate has been proved analytically approximate expressions for the amplitude of the vertical and rotational displacements have been derived for a special case

the price of robustness a robust approach to solving linear optimization problems with uncertain data was proposed in the early 1970s and has recently been extensively studied and extended under this approach we are willing to accept a suboptimal solution for the nominal values of the data in order to ensure that the solution remains feasible and near optimal when the data changes a concern with such an approach is that it might be too conservative in this paper we propose an approach that attempts to make this trade off more attractive that is we investigate ways to decrease what we call the price of robustness in particular we flexibly adjust the level of conservatism of the robust solutions in terms of probabilistic bounds of constraint violations an attractive aspect of our method is that the new robust formulation is also a linear optimization problem thus we naturally extend our methods to discrete optimization problems in a tractable way we report numerical results for a portfolio optimization problem a knapsack problem and a problem from the net lib library

robust control of markov decision processes with uncertain transition matrices optimal solutions to markov decision problems may be very sensitive with respect to the state transition probabilities in many practical problems the estimation of these probabilities is far from accurate hence estimation errors are limiting factors in applying markov decision processes to real world problems we consider a robust control problem for a finite state finite action markov decision process where uncertainty on the transition matrices is described in terms of possibly nonconvex sets we show that perfect duality holds for this problem and that as a consequence it can be solved with a variant of the classical dynamic programming algorithm the robust dynamic programming algorithm we show that a particular choice of the uncertainty sets involving likelihood regions or entropy bounds leads to both a statistically accurate representation of uncertainty and a complexity of the robust recursion that is almost the same as that of the classical recursion hence robustness can be added at practically no extra computing cost we derive similar results for other uncertainty sets including one with a finite number of possible values for the transition matrices we describe in a practical path planning example the benefits of using a robust strategy instead of the classical optimal strategy even if the uncertainty level is only crudely guessed the robust strategy yields a much better worst case expected travel time

dynamic pricing without knowing the demand function risk bounds and near optimal algorithms we consider a single product revenue management problem where given an initial inventory the objective is to dynamically adjust prices over a finite sales horizon to maximize expected revenues realized demand is observed over time but the underlying functional relationship between price and mean demand rate that governs these observations otherwise known as the demand function or demand curve is not known we consider two instances of this problem i a setting where the demand function is assumed to belong to a known parametric family with unknown parameter values and ii a setting where the demand function is assumed to belong to a broad class of functions that need not admit any parametric representation in each case we develop policies that learn the demand function on the fly and optimize prices based on that the performance of these algorithms is measured in terms of the regret the revenue loss relative to the maximal revenues that can be extracted when the demand function is known prior to the start of the selling season we derive lower bounds on the regret that hold for any admissible pricing policy and then show that our proposed algorithms achieve a regret that is close to this lower bound the magnitude of the regret can be interpreted as the economic value of prior knowledge on the demand function manifested as the revenue loss due to model uncertainty

pareto efficiency in robust optimization this paper formalizes and adapts the well known concept of pareto efficiency in the context of the popular robust optimization ro methodology for linear optimization problems we argue that the classical ro paradigm need not produce solutions that possess the associated property of pareto optimality and we illustrate via examples how this could lead to inefficiencies and suboptimal performance in practice we provide a basic theoretical characterization of pareto robustly optimal pro solutions and extend the ro framework by proposing practical methods that verify pareto optimality and generate solutions that are pro critically important our methodology involves solving optimization problems that are of the same complexity as the underlying robust problems hence the potential improvements from our framework come at essentially limited extra computational cost we perform numerical experiments drawn from three different application areas portfolio optimization inventory management and project management which demonstrate that pro solutions have a significant potential upside compared with solutions obtained using classical ro methods this paper was accepted by gerard p cachon optimization

on the uncertainty of var of individual risk abstract value at risk var is a prevalent risk measure used in financial risk management the calculation of var relies on the distribution of the potential loss position which is generally unknown in practice in this article we introduce a model of uncertainty for the distribution of a loss variable and investigate the effect on var using a worst scenario approach the proposed model is flexible and can be applied to various types of distributions the robust var and an associated worst scenario measure are identified it is shown that the choice of the loss model is still important when there is an uncertainty model

some early statistical contributions to the theory and practice of linear algebra abstract we outline the history of some of the concepts and techniques of linear algebra which are intimately connected with the development of the method of least squares and related fitting procedures our study concentrates on contributions made during the early years of the nineteenth century but it is not entirely restricted to this period

robust portfolio selection problems in this paper we show how to formulate and solve robust portfolio selection problems the objective of these robust formulations is to systematically combat the sensitivity of the optimal portfolio to statistical and modeling errors in the estimates of the relevant market parameters we introduce uncertainty structures for the market parameters and show that the robust portfolio selection problems corresponding to these uncertainty structures can be reformulated as second order cone programs and therefore the computational effort required to solve them is comparable to that required for solving convex quadratic programs moreover we show that these uncertainty structures correspond to confidence regions associated with the statistical procedures employed to estimate the market parameters finally we demonstrate a simple recipe for efficiently computing robust portfolios given raw market data and a desired level of confidence

robust convex optimization we study convex optimization problems for which the data is not specified exactly and it is only known to belong to a given uncertainty set u yet the constraints must hold for all possible values of the data from u the ensuing optimization problem is called robust optimization in this paper we lay the foundation of robust convex optimization in the main part of the paper we show that if u is an ellipsoidal uncertainty set then for some of the most important generic convex optimization problems linear programming quadratically constrained programming semidefinite programming and others the corresponding robust convex program is either exactly or approximately a tractable problem which lends itself to efficientalgorithms such as polynomial time interior point methods

structured dispersion and validity in linear inference abstract structured matrices i n e e ggee arise in nonstandard linear models where e 1 1 1 n and gg 1 n n their properties are studied including expressions for eigenvalues conditions for positive definiteness and conditioning of be as varies it is shown that if majorizes 0 then the condition numbers are ordered as c c 0 for every condition number c generated by the unitarily invariant matrix norms applications are noted in linear inference and in outlier detection

robust stability and performance of stochastic uncertain systems on an infinite time interval abstract in this paper we consider a robust stability problem for continuous time stochastic uncertain systems the uncertainty in the system is characterized in terms of an uncertain probability distribution on the noise input this uncertainty is assumed to satisfy a certain relative entropy constraint the solution to a specially parametrized risk sensitive performance analysis problem is used to estimate the level of guaranteed performance for the stochastic uncertain system under consideration this solution is obtained by solving an algebraic riccati equation the corresponding performance bound holds for all admissible uncertainties and is nonconservative

intertemporal pricing under minimax regret we consider the pricing problem faced by a monopolist who sells a product to a population of consumers over a finite time horizon customers types differ along two dimensions i their willingness to pay for the product and ii their arrival time during the selling season we assume that the seller knows only the support of the customers valuations and do not make any other distributional assumptions about customers willingness to pay or arrival times we consider a robust formulation of the seller s pricing problem that is based on the minimization of her worst case regret we consider two distinct cases of customers purchasing behavior myopic and strategic customers for both of these cases we characterize optimal price paths for myopic customers the regret is determined by the price at a critical time depending on the problem parameters this critical time will be either the end of the selling season or it will be a time that equalizes the worst case regret generated by undercharging custome

linear unbiased approximators of the disturbances in the standard linear model abstract the least squares residuals from the standard linear model have a variance matrix which is a function of the n q matrix of observations on the regressors we examine two classes of residuals which do not suffer from this defect our first class of residuals luz residuals has a variance matrix which is a scalar multiple of an n n idempotent matrix of rank n q specified by the user and our second class of residuals lus residuals has a variance matrix which is a scalar multiple of the n q n q identity matrix

an approximate solution for homogeneous boundary value problems with slowly varying coefficient matrices abstract in this paper an approximate closed form solution for homogeneous boundary value problems with slowly varying coefficient matrices is obtained the derivation of the approximate solution is based on the freezing technique which is commonly used in analyzing the stability of slowly varying initial value problems as well as solving them the error between the approximate and the exact solutions is given and an upper bound on the norm of the error is obtained this upper bound is proportional to the rate of change of the coefficient matrix of the boundary value problem the proposed approximate solution is obtained for a two point boundary value problem and is compared to its solution obtained numerically good agreement is observed between the approximate and the numerical solutions when the rate of change of the coefficient matrix is small

a review of the development and application of recursive residuals in linear models abstract recursive residuals have been shown to be useful in a variety of applications in linear models unlike the more familiar ordinary least squares residuals or studentized residuals recursive residuals are independent as well as homoscedastic under the model their independence is particularly appealing for use in developing test statistics they are not uniquely defined their values depend on the order in which they are calculated although their properties do not in some applications one can exploit this order dependence coupled with the fact that they are in clear one to one correspondence with the observations for which they are calculated uses for recursive residuals have been suggested in almost all areas of regression model validation regression diagnostics have been constructed from recursive residuals for detecting serial correlation heteroscedasticity functional misspecification and structural change other statistics based on recursive residuals have focused on detection of outli

sufficient conditions for stability of linear time varying systems abstract in this paper we consider sufficient conditions for the exponential stability of linear time varying systems of the form dot x t a t x t t 0 stability guaranteeing upper bounds for different measures of parameter variations are derived

extension of modal analysis to linear time varying systems linear time varying ltv systems have been often dealt with on a case by case basis many well developed concepts and analytic methods of linear time invariant lti systems cannot be applied to ltv systems for example the conventional definition of modal parameters is invalid for ltv systems the first part of this paper explores the possibility of extending the modal concept of ltv systems the discrete time state space model is used to represent ltv systems by analogy to lti systems the pseudo modal parameters are defined using the eigenvalues of the discrete time state transition matrix the paper shows that the pseudo modal parameters preserve certain properties of the conventional modal parameters defined for lti systems the second part of the paper extends a previously developed algorithm to identify the pseudo modal parameters using forced responses and forcing inputs for a general ltv system the input and output hankel matrices formed by an ensemble of data satisfy a matrix facotrization relation the key step of the method is to modify the output matrix in such a way that the range space of the observability matrix can be extracted a robotic manipulator with varying inertia links is used as an example the first part of the numerical simulation illustrates the applications of the pseudo modal parameters the second part of the simulation tests the identification algorithm under different conditions several practical issues are addressed

regret in the newsvendor model with partial information traditional stochastic inventory models assume full knowledge of the demand probability distribution however in practice it is often difficult to completely characterize the demand distribution especially in fast changing markets in this paper we study the newsvendor problem with partial information about the demand distribution e g mean variance symmetry unimodality in particular we derive the order quantities that minimize the newsvendor s maximum regret of not acting optimally most of our solutions are tractable which makes them attractive for practical application our analysis also generates insights into the choice of the demand distribution as an input to the newsvendor model in particular the distributions that maximize the entropy perform well under the regret criterion our approach can be extended to a variety of problems that require a robust but not conservative solution

strong formulations of robust mixed 0 1 programming we introduce strong formulations for robust mixed 0 1 programming with uncertain objective coefficients we focus on a polytopic uncertainty set described by a budget constraint for allowed uncertainty in the objective coefficients we show that for a robust 0 1 problem there is an tight linear programming formulation with size polynomial in the size of an tight linear programming formulation for the nominal 0 1 problem we give extensions to robust mixed 0 1 programming and present computational experiments with the proposed formulations

response of linear slowly varying systems under external excitations abstract an approximate closed form solution for a linear slowly varying system under external excitation is derived the derivation of the solution is based on the technique of freezing slowly varying parameters a technique used in establishing the stability of slowly varying homogeneous systems the error between the approximate and exact solutions is given and an upper bound on the norm of the error in terms of a measure of the rate of change of the system matrix and the norm of the exact solution of the system is derived approximate responses of two particular linear slowly varying systems under external excitation are obtained by the proposed technique good agreement between the approximate and numerically obtained solutions is observed

three theorems with applications to euclidean distance matrices abstract in a recent paper on the theory of euclidean distance matrices gower derived an inequality which characterizes such matrices and challenged his readers to provide a direct proof of this result the present paper represents the author s response to that challenge

stability analysis of nonautonomous difference systems with delaying arguments abstract we present sufficient conditions for the stability of the nonautonomous difference system x k 1 j 0 m a j k x k j k z with m 1 when the n n matrices a j are slowly varying coefficients the proposed approach is based on the generalization of the freezing method for ordinary differential equations the stability conditions are formulated in terms of the corresponding cauchy s function

minimax optimal control of stochastic uncertain systems with relative entropy constraints this paper considers a new class of discrete time stochastic uncertain systems in which the uncertainty is described by a constraint on the relative entropy between a nominal noise distribution and the perturbed noise distribution this uncertainty description is a natural extension to the case of stochastic uncertain systems of the sum quadratic constraint uncertainty description this paper solves problems of worst case robust performance analysis and output feedback minimax optimal controller synthesis in a general nonlinear setting specializing these results to the linear case leads to a minimax linear quadratic gaussian lqg optimal controller this controller is defined by riccati difference equations and a kalman filter like state equation the paper also shows that the minimax lqg problem will have a solution if and only if a corresponding h sup spl infin control problem has a solution a linear example is presented to illustrate the minimax lqg methodology

dynamic pricing with multiple products and partially specified demand distribution we study a dynamic pricing problem with multiple products and infinite inventories the demand for these products depends on the selling prices and on parameters unknown to the seller their value can be learned from accumulating sales data using statistical estimation techniques the quality of the parameter estimates is influenced by the amount of price dispersion however a large amount of variation in the selling prices can be costly since it means that suboptimal prices are used the seller thus needs to balance optimizing the quality of the parameter estimates and optimizing instant revenue i e exploitation and exploration in this study we propose a pricing policy for this dynamic pricing problem the key idea is to use at each time period the price that is optimal with respect to current parameter estimates with an additional constraint that ensures sufficient price dispersion we measure the price dispersion by the smallest eigenvalue of the design matrix and show how a desired growth rate of this eigenvalue can be achieved by a simple quadratic constraint in the price optimization problem we study the performance of our pricing policy by providing bounds on the regret which measures the expected revenue loss caused by using suboptimal prices

robust sensitivity analysis for stochastic systems we study a worst case approach to measure the sensitivity to model misspecification in the performance analysis of stochastic systems the situation of interest is when only minimal parametric information is available on the form of the true model under this setting we post optimization programs that compute the worst case performance measures subject to constraints on the amount of model misspecification measured by kullback leibler divergence our main contribution is the development of infinitesimal approximations for these programs resulting in asymptotic expansions of their optimal values as the divergence shrinks to zero the coefficients of these expansions can be computed via simulation and are mathematically derived from the representation of the worst case models as changes of measure that satisfy a well defined class of functional fixed point equations

point processes and queues martingale dynamics i martingales 1 histories and stopping times 2 martingales 3 predictability 4 square integrable martingales references solutions to exercises chapter i ii point processes queues and intensities 1 counting processes and queues 2 watanabe s characterization 3 stochastic intensity general case 4 predictable intensities 5 representation of queues 6 random changes of time 7 cryptographic point processes references solutions to exercises chapter ii iii integral representation of point process martingales 1 the structure of internal histories 2 regenerative form of the intensity 3 the representation theorem 4 hilbert space theory of poissonian martingales 5 useful extensions references solutions to exercises chapter iii iv filtering 1 the theory of innovations 2 state estimates for queues and markov chains 3 continuous states and nontrivial prehistory references solutions to exercises chapter iv v flows in markovian networks of queues 1 single station the historical results and the filtering method 2 jackson s networks 3 burke s output theorem for networks 4 cascades and loops in jackson s networks 5 independence and poissonian flows in markov chains references solutions to exercises chapter v vi likelihood ratios 1 radon nikodym derivatives and tests of hypotheses 2 changes of intensities a la girsanov 3 filtering by the method of the probability of reference 4 applications 5 the capacity of a point process channel 6 detection formula187 references189 solutions to exercises chapter vi vii optimal control 1 modeling intensity controls 2 dynamic programming for intensity controls complete observation case 3 input regulation a case study in impulsive control 4 attraction controls 5 existence via likelihood ratio references solutions to exercises chapter vii viii marked point processes 1 counting measure and intensity kernels 2 martingale representation and filtering 3 radon nikodym derivatives 4 towards a general theory of intensity references solutions to exercises chapter viii a1 background in probability and stochastic processes 1 introduction 2 monotone class theorem 3 random variables 4 expectations 5 conditioning and independence 6 convergence 7 stochastic processes 8 markov processes references a2 stopping times and point process histories 1 stopping times 2 changes of time and meyer dellacherie s integration formula 3 point process histories references a3 wiener driven dynamical systems 1 ito s stochastic integral 2 square integrable brownian martingales 3 girsanov s theorem references a4 stieltjes lebesgue calculus 1 the stieltjes lebesgue integral 2 the product and exponential formulas references general bibliography

a maximum entropy approach to the newsvendor problem with partial information abstract in this paper we consider the newsvendor model under partial information i e where the demand distribution d is partly unknown we focus on the classical case where the retailer only knows the expectation and variance of d the standard approach is then to determine the order quantity using conservative rules such as minimax regret or scarf s rule we compute instead the most likely demand distribution in the sense of maximum entropy we then compare the performance of the maximum entropy approach with minimax regret and scarf s rule on large samples of randomly drawn demand distributions we show that the average performance of the maximum entropy approach is considerably better than either alternative and more surprisingly that it is in most cases a better hedge against bad results