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Stochastic models are everywhere. In manufacturing, queuing models are used for modeling production processes, realistic inventory models are stochastic in nature. Stochastic models are considered in transportation and communication. Marketing models use stochastic descriptions of the demands and buyer's behaviors. In finance, market prices and exchange rates are assumed to be certain stochastic processes, and insurance claims appear at random times with random amounts.
To each decision problem, a cost function is associated. Costs may be direct or indirect, like loss of time, quality deterioration, loss in production or dissatisfaction of customers. In decision making under uncertainty, the goal is to minimize the expected costs. However, in practically all realistic models, the calculation of the expected costs is impossible due to the model complexity. Simulation is the only practicable way of getting insight into such models. Thus, the problem of optimal decisions can be seen as getting simulation and optimization effectively combined.
The field is quite new and yet the number of publications is enormous. This book does not even try to touch all work done in this area. Instead, many concepts are presented and treated with mathematical rigor and necessary conditions for the correctness of various approaches are stated.
Optimization of Stochastic Models: The Interface Between Simulation and Optimization is suitable as a text for a graduate level course on Stochastic Models or as a secondary text for a graduate level course in Operations Research.



Chapter 1. Optimization

Optimization is the art of finding the best among several alternatives in decision making.
Let S be the set of possible decisions. This set is called the feasible set. The decision variable is denoted by x. If x ε S, then x is called feasible, otherwise infeasible. The net costs caused by decision x are measured by a real valued objective function F(x). The goal is to find the best decision, i.e. the decision with minimal costs. We will always assume that there is one single objective function. The case of several competing objective functions, the multi-criteria decision making problem will not be touched here.
George Ch. Pflug

Chapter 2. Discrete—Event processes

A stochastic process (indexed with time) is a collection of random variables (Z(t)) t ɛℝ+ defined on some probability space (ΩA, IP) which takes values in some state space Z. The parameter t is interpreted as time, the state space Z will be either a finite or denumerable set or the euclidean space ℝd
George Ch. Pflug

Chapter 3. Derivatives

In this chapter we approach the main problem of finding the minimizer of
$$F(x) = \int {H(x,\omega )d{{\mu }_{x}}(\omega )}$$
by discussing various notions of differentiability of parameter integrals.
George Ch. Pflug

Chapter 4. Simulation and sensitivity estimation

We will use the word simulation exclusively for the technique to mimic a random process on a computer. Since a computer is a deterministic machine, true randomness cannot be produced. Instead, one uses algorithms, which produce values which are (to a certain extent) indistinguishable from realizations of genuine random processes.
Georg Ch. Pflug

Chapter 5. Stochastic Approximation

This chapter deals with algorithms for the optimization of simulated systems.In particular we study stochastic variants of the gradient algorithm
$$x_{n + 1} = \,x_n \, - \,a_n \nabla F(x_n )]$$
which was introduced in (1.27) to solve the optimization problem
$$[F(x) = \left\| \begin{gathered} Minimize F(x) \hfill \\ \,x\, \in \,\mathbb{R}^d \hfill \\ \end{gathered} \right.]$$
where F is bounded from below.
George C. Pflug


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