## Abstract

Revenue Management uses data-driven modelling and optimization methods to decide what to sell, when to sell, to whom to sell, and for which price, in order to increase revenue and profit. Hotel Revenue Management is a very complex context characterized by nonlinearities, many parameters and constraints, and stochasticity, in particular in the demand by customers. It suffers from the *curse of dimensionality* (Bellman 2015): when the number of variables increases (number of rooms, number possible prices and capacities, number of reservation rules and constraints) exact solutions by dynamic programming or by alternative global optimization techniques cannot be used and one has to resort to intelligent heuristics, i.e., methods which can improve current solutions but without formal guarantees of optimality. Effective heuristics can incorporate “learning” (“reactive” schemes) that update strategies based on the past history of the process, the past reservations received up to a certain time and the previous steps in the iterative optimization process. Different approaches can be classified according to the specific *model* considered (stochastic demand and hotel rules), the control mechanism (the *pricing policy*) and the *optimization technique* used to determine improving or optimal solutions. In some cases, model definitions, control mechanism and solution techniques are strongly interrelated: this is the case of dynamic programming, which demands suitably simplified problem formulations. We design a flexible discrete-event simulator for the hotel reservation process and experiment different approaches though measurements of the expected effect on profit (obtained by carefully separating a “training” phase from the final “validation” phase obtained from different simulations). The experimental results show the effectiveness of intelligent heuristics with respect to exact optimization methods like dynamic programming, in particular for more constrained situations (cases when demand tends to saturate hotel room availability), when the simplifying assumptions needed to make the problem analytically treatable do not hold.

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## Appendix A: Optimal prices for sigmoidal acceptance

### Appendix A: Optimal prices for sigmoidal acceptance

This appendix describes the steps to obtain the optimal prices described in the paper.

### A.1 Non-saturated equilibrium price

Here is the derivation of the optimal price value (16) described in Section 4.1.2 in the hypothesis that infinite rooms are available.

We want to find the stationary point of *u* ⋅ *p*_{a}(*u*), where *u* is the price and *p*_{a}(*u*) is the corresponding acceptance probability (15):

by applying the substitutions

and reminding the identity \(\sigma ^{\prime }(s)=\sigma (s)\bigl (1-\sigma (s)\bigr )\), we obtain

After replacing the sigmoid function definition, multiplying by (1 + *e*^{−s})^{2} and simplifying, we are left with

and, multiplying by *e*^{β− 1},

Let us observe that this equation is in the form *x**e*^{x} = *a* for *a* > −*π*/2, whose solution can be analytically expressed as *x* = *W*_{0}(*a*), where *W*_{0}(⋅) is the main branch of Lambert’s function. We get

By replacing the original variables, we finally obtain (16).

### A.2 Dynamic programming optimal price

The derivation of the optimal price policy (23) described in Section 4.1.3 for the dynamic programming technique follows the same steps outlined above.

After replacing (15) into (22), let us perform the following variable substitutions and quantity replacements:

after which we obtain (25), whose solution is, again, (26).

By replacing the original variables from (27), we obtain (23).

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Brunato, M., Battiti, R. Combining intelligent heuristics with simulators in hotel revenue management.
*Ann Math Artif Intell* **88, **71–90 (2020). https://doi.org/10.1007/s10472-019-09651-9

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DOI: https://doi.org/10.1007/s10472-019-09651-9

### Keywords

- Machine learning and intelligent optimization
- Hotel revenue management
- Simulation-based optimization
- Optimization heuristics

### Mathematics Subject Classification (2010)

- 90-04
- 90B50