We consider the problem of setting safety stock when both the demand in a period and the lead time are random variables. There are two cases to consider. In the first case the parameters of the demand and lead time distributions are known; in the second case they are unknown and must be estimated. For the case of known parameters a standard procedure is presented in the literature. In this paper, examples are used to show that this procedure can yield results that are far from the desired result. A correct procedure is presented. When the parameters are unknown, it is assumed that a simple exponential smoothing model is used to generate estimates of demand in each period and that a discrete distribution of the lead time can be developed from historical data. A correct procedure for setting safety stocks that is based on these two inputs is given for two popular demand models. The approach is easily generalized to other models of demand. Safety stock calculation is simplified when certain normality assumptions are valid. Simulation results in the Appendix indicate when these assumptions about normality are reasonable.inventory/production: stochastic models, inventory/production: simulation, inventory/production: parameter estimation

Duality theory is pervasive in finite dimensional optimization. There is growing interest in solving infinite-dimensional optimization problems and hence a corresponding interest in duality theory in infinite dimensions. Unfortunately, many of the intuitions and interpretations common to finite dimensions do not extend to infinite dimensions. In finite dimensions, a dual solution is represented by a vector of "dual prices" that index the primal constraints and have a natural economic interpretation. In infinite dimensions, we show that this simple dual structure, and its associated economic interpretation, may fail to hold for a broad class of problems with constraint vector spaces that are σ-order complete Riesz spaces (ordered vector spaces with a lattice structure). In these spaces we show that the existence of interior points required by common constraint qualifications for zero duality gap (such as Slater's condition) imply the existence of singular dual solutions that are difficult to find and interpret. We call this phenomenon the Slater conundrum: interior points ensure zero duality gap (a desirable property), but interior points also imply the existence of singular dual solutions (an undesirable property). A Riesz space is the most parsimonious vector-space structure sufficient to characterize the Slater conundrum. Topological vector spaces common to the vast majority of infinite dimensional optimization literature are not necessary.

We consider semi-infinite linear programs with countably many constraints indexed by the natural numbers. When the constraint space is the vector space of all real valued sequences, we show that the finite support (Haar) dual is equivalent to the algebraic Lagrangian dual of the linear program. This settles a question left open by Anderson and Nash [2]. This result implies that if there is a duality gap between the primal linear program and its finite support dual, then this duality gap cannot be closed by considering the larger space of dual variables that define the algebraic Lagrangian dual. However, if the constraint space corresponds to certain subspaces of all realvalued sequences, there may be a strictly positive duality gap with the finite support dual, but a zero duality gap with the algebraic Lagrangian dual.

This paper describes a proposed format to record instances of stochastic programs. It forms part of a larger XML-based schema that is designed to allow the expression of essentially any type of mathematical program within a unifying framework. A wide variety of different linear and nonlinear stochastic programs can be handled, and the paper describes in detail how this is done. Screen captures and sample problems illustrate the use of the schema.

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