Berücksichtigung von Risikopräferenzen in Optimierungsmodellen

Während in der Praxis Entscheidungsträger häufig - aus den verschiedensten Gründen - risikoavers sind, wird in mathematischen Modellen häufig Risikoneutralität unterstellt. Dies führt zwar einerseits zu einfacheren Modellen, verringert aber gleichzeitig ihre Akzeptanz in der Praxis. Im schlechtesten Fall werden dann Modelle gar nicht eingesetzt oder ihre Ergebnisse mehr oder weniger willkürlich abgeändert, um "weniger riskante" Entscheidungen zu treffen.

Ziel unserer Forschung ist daher die Abbildung der in der Praxis häufig anzutreffenden Risikoaversion von Entscheidungsträgern in Optimierungsmodellen des Service Operations Management. Vor allem im Finance-Bereich wurden in den vergangenen Jahren zahlreiche Risikomaße entwickelt, die bisher kaum Eingang in entsprechenden Optimierungsmodelle gefunden haben. Hierzu zählt etwa der Conditional Value at Risk (CVaR). Dieser ist nicht nur intuitiv verständlich, sondern besitzt darüber hinaus zahlreiche wünschenswerte theoretische Eigenschaften. 

Berücksichtigung von Risikopräferenzen im Revenue Management

Neben der Maximierung des erwarteten Erlöses als klassische Zielsetzung im Revenue Management berücksichtigen neuere Ansätze in der Literatur auch zunehmend die Variabilität der generierten Erlöse. Der klassische Ansatz ist gerechtfertigt bei häufigen Ereignissen, da hier nach dem Gesetz der großen Zahlen eine Konvergenz zum Erwartungswert gegeben ist. Eine optimale Politik unter Einbeziehung von Risikogesichtspunkten ist hingegen besonders relevant bei selteneren Ereignissen, wobei ein einzelnes Ereignis bereits einen hohen Einfluss auf das Gesamtergebnis besitzt. Beispielhaft zu nennen wäre hier ein Konzertveranstalter, der pro Jahr nur wenige große Konzerte organisiert. In diesem Fall wirkt sich der Erfolg eines Konzerts maßgeblich auf das Jahresergebnis aus und der Veranstalter wäre wohl nicht risikoneutral, wie in den bisherigen Modellen unterstellt, sondern im Gegenteil risikoavers. Im Rahmen der Forschung des Lehrstuhls werden deshalb Ansätze zur Integration von Risikomaßen in Steuerungskonzepte des Revenue Managements entwickelt. 


  • Schlosser, R.; Gönsch, J.: Risk-Averse Revenue Management using Mean-Variance and Mean-Semivariance Optimization. BIB Download Details
  • Schur, R.; Gönsch, J.; Hassler, M.: Time-Consistent Risk-Averse Dynamic Pricing. In: European Journal of Operational Research, Jg. 277 (2019) Nr. 2, S. 587-603. PDF Volltext BIB Download Details

    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, Jg. 263 (2017) Nr. 2, S. 337-348. PDF Volltext BIB Download Details
  • Gönsch, J.: Unsicherheiten im Revenue Management. In: Corsten, H.; Roth, S. (Hrsg.): Handbuch Dienstleistungsmanagement. Vahlen, München 2016, S. 843-862. BIB Download Details
  • Gönsch, J.; Hassler, M.; Schur, R.: Optimizing Conditional Value-at-Risk in Dynamic Pricing. In: OR Spectrum, Jg. 40 (2018) Nr. 3, S. 711-750. PDF Volltext BIB Download Details

    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.


    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, Jg. 15 (2016) Nr. 6, S. 468-487. PDF Volltext BIB Download Details

    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) Nr. 8, S. 495-521. PDF Volltext BIB Download Details

    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.