An Interior-Point and Decomposition Approach to Multiple Stage Stochastic Programming

Let us start our discussion with a famous textbook example: the newsboy problem. The story goes like this. Every morning a newsboy has to decide how many newspapers to buy from a newspaper publisher. Let us assume that the publisher sells the newspaper to the boy at the price of $2 each paper, and the boy then sells the newspaper along the street at the price of $5 per copy. In the end of the day, the newsboy may return any unsold copies to the publisher at $1 for each copy. The profit of the newsboy in this business depends, obviously, on the success of the sales and his initial decision on the order quantity. Unfortunately, the problem is that, as it is always the case, one cannot really predict the future with certainty. To make our analysis simple, let us further assume that there are only two possible scenarios: (1) the newsboy can sell 100 copies a day, or, (2) in the case of a boring day, he can only sell 50 copies. Furthermore, let us assume that the chance for a day with some exciting news is lower. Hence, (1) occurs with probability 0.25, and (2) occurs with probability 0.75. In fact, for a person with mathematical background, it is more convenient to introduce a random variable, ω, to denote the demand for this purpose,