# Risk preferences in optimization models

## Consideration of risk preferences in optimization models

While in practice decision-makers are often risk-averse - for various reasons -, mathematical models often assume risk neutrality. On the one hand, this leads to simpler models, but at the same time reduces their acceptance in practice. In the worst case, models are not used at all or their results are more or less arbitrarily changed in order to make "less risky" decisions.

The aim of our research is to capture the risk aversion of decision-makers, which is frequently found in practice, in optimization models of Service Operations Management. In recent years, numerous risk measures have been proposed, particularly in the finance area, which have so far hardly been included in corresponding optimisation models. These include the Conditional Value at Risk (CVaR). CVaR is not only intuitively understandable, but also has numerous desirable theoretical properties.

**Consideration of risk preferences in revenue management**

In addition to maximizing the expected revenue as a classical revenue management objective, more recent approaches in the literature increasingly take into account the variability of the generated revenues. The classical approach is justified with frequent events, since here a convergence to the expected value is given according to the law of large numbers. On the other hand, an optimal policy that takes risk aspects into account is particularly relevant for less frequent events, where a single event already has a high impact on the overall result. An example would be a concert organiser who organises only a few big concerts per year. In this case, the success of a concert has a decisive effect on the annual result and the organiser would probably not be risk neutral, as assumed in the previous models, but on the contrary risk-averse. This just one example why we develop approaches for the integration of risk measures into revenue management.

**Literature**

- Schlosser, R.; Gönsch, J.: Mean-Variance Optimization in Finite Horizon Markov Decision Processes for Revenue Management Applications. 2022. CitationDetails
- Schlosser, R.; Gönsch, J.: Risk-Averse Dynamic Pricing using Mean-Semivariance Optimization. 2022. CitationDetails
- Schur, R.; Gönsch, J.; Hassler, M.: Time-Consistent Risk-Averse Dynamic Pricing. In: European Journal of Operational Research, Vol 277 (2019) No 2, p. 587-603. PDFFull textCitationAbstractDetails
Many industries use dynamic pricing on an operational level to maximize revenue from selling a fixed capacity over a finite horizon. Classical risk-neutral approaches do not accommodate the risk aversion often encountered in practice. When risk aversion is considered, time-consistency becomes an important issue. In this paper, we use a dynamic coherent risk-measure to ensure that decisions are actually implemented and only depend on states that may realize in the future. In particular, we use the risk measure Conditional Value-at-Risk (CVaR), which recently became popular in areas like finance, energy or supply chain management.

A result is that the risk-averse dynamic pricing problem can be transformed to a classical, risk-neutral problem. To do so, a surprisingly simple modification of the selling probabilities suffices. Thus, all structural properties carry over. Moreover, we show that the risk-averse and the risk-neutral solution of the original problem are proportional under certain conditions, that is, their optimal decision variable and objective values are proportional, respectively. In a small numerical study, we evaluate the risk vs. revenue trade-off and compare the new approach with existing approaches from literature.

This has straightforward implications for practice. On the one hand, it shows that existing dynamic pricing algorithms and systems can be kept in place and easily incorporate risk aversion. On the other hand, our results help to understand many risk-averse decision makers who often use “conservative” estimates of selling probabilities or discount optimal prices.

- Gönsch, J.: A Survey on Risk-averse and Robust Revenue Management. In: European Journal of Operational Research, Vol 263 (2017) No 2, p. 337-348. PDFFull textCitationDetails
- Gönsch, J.: Unsicherheiten im Revenue Management. In: Corsten, H.; Roth, S. (Ed.): Handbuch Dienstleistungsmanagement. Vahlen, München 2016, p. 843-862. CitationDetails
- Gönsch, J.; Hassler, M.; Schur, R.: Optimizing Conditional Value-at-Risk in Dynamic Pricing. In: OR Spectrum, Vol 40 (2018) No 3, p. 711-750. PDFFull textCitationAbstractDetails
Many industries use dynamic pricing on an operational level to maximize revenue from selling a fixed capacity over a finite horizon. Classical risk-neutral approaches do not accommodate the risk aversion often encountered in practice. We add to the scarce literature on risk aversion by considering the risk measure conditional value-at-risk (CVaR), which recently became popular in areas like finance, energy, or supply chain management. A key aspect of this paper is selling a single unit of capacity, which is highly relevant in, for example, the real estate market. We analytically derive the optimal policy and obtain structural results. The most important managerial implication is that the risk-averse optimal price is constant over large parts of the selling horizon, whereas the price continuously declines in the standard setting of risk-neutral dynamic pricing. This offers a completely new explanation for the price-setting behavior often observed in practice. For arbitrary capacity, we develop two algorithms to efficiently compute the value function and evaluate them in a numerical study. Our results show that applying a risk-averse policy, even a static one, often yields a higher CVaR than applying a dynamic, but risk-neutral, policy.

## Keywords

Revenue management Dynamic pricing Dynamic programming Risk management Service operations

- Koch, S.; Gönsch, J.; Hassler, M.; Klein, R.: Practical Decision Rules for Risk-Averse Revenue Management using Simulation-Based Optimization. In: Journal of Revenue and Pricing Management, Vol 15 (2016) No 6, p. 468-487. PDFFull textCitationAbstractDetails
In practice, human-decision makers often feel uncomfortable with the risk-neutral revenue management systems’ output. Reasons include a low number of repetitions of similar events, a critical impact of the achieved revenue for economic survival, or simply business constraints imposed by management. However, solving capacity control problems is a challenging task for many risk measures and the approaches are often not compatible with existing software systems. In this paper, we propose a flexible framework for risk-averse capacity control under customer choice behavior. Existing risk-neutral decision rules are augmented by the integration of adjustable parameters. Our key idea is the application of simulation-based optimization (SBO) to calibrate these parameters. This allows to easily tailor the resulting capacity control mechanism to almost every risk measure and customer choice behavior. In an extensive simulation study, we analyze the impact of our approach on expected utility, conditional value-at-risk (CVaR), and expected value. The results show a superior performance in comparison to risk-neutral approaches from the literature.

- Gönsch, J.; Hassler, M.: Optimizing the Conditional Value-at-Risk in Revenue Management. In: Review of Managerial Science (2014) No 8, p. 495-521. PDFFull textCitationAbstractDetails
Many service industries use revenue management to balance demand and capacity. The assumption of risk-neutrality lies at the heart of the classical approaches, which aim at maximizing expected revenue. In this paper, we give a comprehensive overview of the existing approaches, most of which were only recently developed, and discuss the need to take risk-averse decision makers into account. We then present a heuristic that maximizes conditional value-at-risk (CVaR). Although CVaR has become increasingly popular in finance and actuarial science due to its beneficial properties, this risk measure has not yet been considered in the context of revenue management. We are able to efficiently solve the optimization problem inherent in CVaR by taking advantage of specific structural properties that allow us to reformulate this optimization problem as a continuous knapsack problem. In order to demonstrate the applicability and robustness of our approach, we conduct a simulation study that shows that the new approach can significantly improve the risk profile in various scenarios.