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Deregulation had a significant impact on the U.S. airline industry in the late 1970s. Charter and low-cost airlines such as People Express and Southwest were able to offer seats at a fraction of the price charged by established carriers like Pan Am and American Airlines. Due to their different cost structure, it seemed to be impossible for the big carriers to offer tickets at the same low price. Yet they had to find a way to compete.

In this chapter, we summarize results both on finite and infinite horizon Markov decision processes (MDPs) in a random environment and with an absorbing set. Finite horizon models are used in Chap. 5. In addition, they serve as a starting point for the discussion on sequential utility maximizing decision problems in Chap. 3. The results on infinite horizon models are applied in 4. Since discounting is generally not considered in capacity control models, we focus on the expected total reward criterion.

This chapter deals with decision problems under risk as defined in Knight (1921), i.e. the problem of choosing from a number of options, each of which could give rise to more than one possible outcome with different probabilities. As in Chap. 2, however, the main intention is to clarify notation and to state results that will be used in the following chapters, not to give a complete overview on expected utility theory. For a general introduction and further references, see e.g. the textbooks of Kreps (1988), French (1986), Gollier (2001), or Bamberg and Coenenberg (2006).

We consider a non-stop flight of an airplane with a capacity of that is to depart after a certain time . There are ( ∈ ℕ) booking classes, = 1, . . ., , with associated fares ordered such that . The number of booking periods in [0; ] is given by some external process and might be random. In every booking period , a customer requests a certain number of reservations , for seats of booking class . ( = 0 with = 0 denotes an artificial booking class corresponding to no customer request.) Thus, the ( + 1)st customer request provides information on the number of reservations (the customer is interested in) and the booking class with reward that is offered for each of the reservations. It must be decided how many of these requested reservations should be actually sold.

The two main textbook models of single-resource capacity control are the dynamic and the static capacity control model. Both models fulfill assumptions i) to xi) mentioned in Sect. 1.1.1; they differ only in the assumptions concerning the arrival process. Static capacity control models assume that demand for the different booking classes arrives in non-overlapping periods. Dynamic capacity control models allow passengers to arrive in any order. In turn, they assume demand to be Markovian.

As mentioned before, the assumption of an additive time-separable utility function for all time periods = 0, . . ., , is the one most frequently used in combination with Markov decision processes. Yet, as indicated in Chap. 3, it imposes a special structure of temporal and risk preferences.

In the examples given in Sect. 6.2.2, the add-optimal policies had some rather counter-intuitive properties. Indeed, one might question the assumptions underlying the maximization of expected additive time-separable utility when incorporating risk-aversion into an MDP formulation.

In this chapter, we use an example to show that structural properties known from the risk-neutral setting carry over to the setting of a decision-maker with a concave exponential atemporal utility function even for more general capacity control models.

We have presented (1) an expected revenue maximizing capacity control model that evolves in a random environment and (2) basic single-resource capacity control problems from the perspective of a risk-averse decision-maker.