Several notions of stochastic convexity and concavity and their properties are described in this survey. The notion of sample path stochastic convexity is a refinement of the well used notion of stochastic ordering, and it can be used to construct, on a common probability space, random variables which have desirable convexity (or concavity) properties with probability one. Three open problems from the literature are described. These problems could not be resolved until the introduction of the stochastic convexity notions which are described in this survey. The solutions of these problems illustrate the strength and the usefulness of these notions. Each notion is accompanied by a description of some of its applications. References for more detailed study of these notions are given. Indications of further work in this area are included.1. Introduction. Regular, Sample Path and Strong Stochastic Convexity notions are very valuable in many areas in probability and statistics such as queueing and reliability theory. Consider, for example, the following three open problems: PROBLEM 1. Consider a single stage queueing system at which customers arrive according to a doubly stochastic Poisson (DSP) process. The stochastic intensity of the DSP is a Markov process on {λχ,λ2} (λ t > 0,i = 1,2). The expected time this Markov process spends in state λ t is 0r t ,i =1,2 for some r t > 0, i = 1,2, and θ > 0. The service times of the customers are independent and identically distributed random variables. Let EW(Θ) be the average work load in this DSP/G/1 queueing system. CONJECTURE 1. (Ross 1978). EW(Θ) is a decreasing function of θ.

The polynomiality of nonlinear separable convex (concave) optimization problems, on linear constraints with a matrix with “small” subdeterminants, and the polynomiality of such integer problems, provided the inteter linear version of such problems ins polynomial, is proven. This paper presents a general-purpose algorithm for converting procedures that solves linear programming problems. The conversion is polynomial for constraint matrices with polynomially bounded subdeterminants. Among the important corollaries of the algorithm is the extension of the polynomial solvability of integer linear programming problems with totally unimodular constraint matrix, to integer-separable convex programming. An algorithm for finding a ε-accurate optimal continuous solution to the nonlinear problem that is polynomial in log(1/ε) and the input size and the largest subdeterminant of the constraint matrix is also presented. These developments are based on proximity results between the continuous and integral optimal solutions for problems with any nonlinear separable convex objective function. The practical feature of our algorithm is that is does not demand an explicit representation of the nonlinear function, only a polynomial number of function evaluations on a prespecified grid.

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The supermodular and the symmetric supermodular stochastic orders have been cursorily studied in previous literature. In this paper we study these orders more thoroughly. First we obtain some basic properties of these orders. We then apply these results in order to obtain comparisons of random vectors with common values, but with different levels of multiplicity. Specifically, we show that if the vectors of the levels of multiplicity are ordered in the majorization order, then the associated random vectors are ordered in the symmetric supermodular stochastic order. In the non-symmetric case we obtain bounds (in the supermodular stochastic order sense) on such random vectors. Finally, we apply the results to problems of optimal assembly of reliability systems, of optimal allocation of minimal repair efforts, and of optimal allocation of reliability items. Academic Press

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.

W e are living in an era in which data is generated in huge volume with high velocity and variety. Big Data and technology are reshaping our life and business. Our research inevitably needs to catch up with these changes. In this short essay, we focus on two aspects of supply chain management, namely, demand management and manufacturing. We feel that, while rapidly growing research on these two areas is contributed by scholars in computer science and engineering, the developments made by production and operations management society have been insufficient. We believe that our field has the expertise and talent to push for advancements in the theory and practice of demand management and manufacturing (of course, among many other areas) along unique dimensions. We summarize some relevant concepts emerged with Big Data and present several prototype models to demonstrate how these concepts can lead to rethinking of our research. Our intention is to generate interests and guide directions for new research in production and operations management in the era of Big Data.

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In this paper we give unifying definitions for both hybrid simulation/analytic models and modeling. We present four classes of hybrid simulation/analytic models and give examples of each class, including numerical results for two of the examples Four usages of hybrid simulation/analytic modeling are also presented with examples.