A large number of problems in production planning and scheduling, location, transportation, finance, and engineering design require that decisions be made in the presence of uncertainty. Uncertainty, for instance, governs the prices of fuels, the availability of electricity, and the demand for chemicals. A key difficulty in optimization under uncertainty is in dealing with an uncertainty space that is huge and frequently leads to very large-scale optimization models. Decision-making under uncertainty is often further complicated by the presence of integer decision variables to model logical and other discrete decisions in a multi-period or multi-stage setting. This paper reviews theory and methodology that have been developed to cope with the complexity of optimization problems under uncertainty. We discuss and contrast the classical recourse-based stochastic programming, robust stochastic programming, probabilistic (chance-constraint) programming, fuzzy programming, and stochastic dynamic programming. The advantages and shortcomings of these models are reviewed and illustrated through examples. Applications and the state-of-the-art in computations are also reviewed. Finally, we discuss several main areas for future development in this field. These include development of polynomial-time approximation schemes for multi-stage stochastic programs and the application of global optimization algorithms to two-stage and chance-constraint formulations.

In this paper we consider stochastic programming problems where the objective function is given as an expected value function. We discuss Monte Carlo simulation based approaches to a numerical solution of such problems. In particular, we discuss in detail and present numerical results for two-stage stochastic programming with recourse where the random data have a continuous (multivariate normal) distribution.

We consider the problem of constructing a portfolio of finitely many assets whose returns are described by a discrete joint distribution. We propose a new portfolio optimization model involving stochastic dominance constraints on the portfolio return. We develop optimality and duality theory for these models. We construct equivalent optimization models with utility functions. Numerical illustration is provided.

A general strong law of large numbers for stochastic programs is established. It is shown that solutions and approximate solutions may not be consistent with the strong law in general, but consistency holds locally, or when the decision space is compact. An additional integrability condition implies the uniform consistency of approximate solutions. The results are applied in the context of linear recourse models. -- 2 -- 1. Introduction The paper examines relations between solutions of a stochastic optimization problem, and the solutions of large sampled versions of the problem. We consider an abstract stochastic program of the form () minimize x2X E P (d) \Gamma f(x; ) \Delta where E P (d) is the expectation operator with respect to the probability measure P over the space \Xi of random elements. The decision space here is taken as a metric space. For a given sequence 1 ; : : : ; n of realizations of the random variable we form the deterministic problem () minimize x2X 1...

This paper addresses a general class of two-stage stochastic programs with integer recourse and discrete distributions. We exploit the structure of the value function of the second stage integer problem to develop a novel global optimization algorithm. The proposed scheme departs from those in the current literature in that it avoids explicit enumeration of the search space while guaranteeing finite termination. Our computational results indicate superior performance of the proposed algorithm in comparison to the existing literature. Keywords: stochastic integer programming, branch and bound, finite algorithms. 1 Introduction Under the twostage stochastic programming paradigm, the decision variables of an optimization problem under uncertainty are partitioned into two sets. The first stage variables are those that have to be decided before the actual realization of the uncertain parameters. Subsequently, once the random events have presented themselves, further design or operational ...

In this paper, stochastic programming problems are viewed as parametric programs with respect to the probability distributions of the random coefficients. General results on quantitative stability in parametric optimization are used to study distribution sensitivity of stochastic programs. For recourse and chance constrained models quantitative continuity results for optimal values and optimal solution sets are proved (with respect to suitable metrics on the space of probability distributions). The results are useful to study the effect of approximations and of incomplete information in stochastic programming.

We establish a verifiable sufficient condition for strong convexity of the expected recourse as a function of the tender variable in a two-stage stochastic program with linear recourse. Generalizing a former result where all components of the second-stage right-hand side vector were random we treat the case where only a subvector of the right-hand side is random. As prerequisite, a refined analysis of the polyhedral complex of lineality regions of the secondstage value function is carried out. The sufficient condition for strong convexity allows to widen the class of recourse models for which certain quantitative results on stability and asymptotic convergence of optimal solutions are valid.

While decisions frequently have uncertain consequences, optimal decision models often replace those uncertainties with averages or best estimates. Limited computational capability may have motivated this practice in the past. Recent computational advances have, however, greatly expanded the range of stochastic programs, optimal decision models with explicit consideration of uncertainties. This paper describes basic methodology in stochastic programming, recent developments in computation, and some practical application examples.

We address the problem of scheduling a flowshop plant with uncertain process- ing times described by discrete probability distributions. The objective is to find a sequence of batches that minimizes the expected makespan. To circumvent the prob- lem of combinatorially explosive state spaces, we propose a novel and rigorous branch and bound algorithm that provides the optimal solution and is based on the result that a lower bound to the expected makespan can be obtained by evaluating over an aggregated probability model. Numerical results for a number of example problems show that the solution times for the proposed method are several orders of magnitude smaller than those for a multiperiod model. In addition, an important extension of this method is proposed for the case of continuous probability distributions of certain forms, using discretization schemes that give excellent approximations.

The prevalent approach to the treatment of processing time uncertainties in production scheduling problems is through the use of probabilistic models. Apart from requiring detailed information about probability distribution functions, this approach also has the drawback that the computational expense of solving these models is very high. In this work, we present a non-probabilistic treatment of scheduling optimization under uncertainty, based on concepts from fuzzy set theory and interval arithmetic, to describe the imprecision and uncertainty in the task durations. We first provide a brief review on the fuzzy set approach, comparing it with the probabilistic approach. We then present MILP models derived from applying this ap-proach to two different problems- flowshop scheduling and new product development process scheduling. Results indicate that these MILP models are computationally tractable for reason-ably sized problems. We also describe tabu search implementations in order to handle larger problems. 1