Presented in this paper is a self-contained analysis of a Markov decision problem that is known as the multi-armed bandit. The analysis covers the cases of linear and exponential utility functions. The optimal policy is shown to have a simple and easily-implemented form. Procedures for computing such a policy are presented, as are procedures for computing the expected utility that it earns, given any starting state. For the case of linear utility, constraints that link the bandits are introduced, and the constrained optimization problem is solved via column generation. The methodology is novel in several respects, which include the use of elementary row operations to simplify arguments.