Optimal Work-Rest Cycles
© 2025 EPFL
How should a system that wears down during active use and recovers during rest be operated for maximum long-term performance? A new study by Prof. Thomas A. Weber (Chair of Operations, Economics and Strategy, EPFL) develops a precise answer to this question, showing how to balance work and recovery in a way that optimizes average output over time.
Whether the “system” is a machine in need of maintenance, an aircraft between flights, or a person alternating between concentration and breaks, the analysis reveals an optimal rhythm: a cycle with a well-defined point at which to stop working and start resting, and another point to resume work. The findings also show that, even without precise knowledge of how fast performance declines or recovers, a simple rule of thumb can achieve at least half of the best possible performance.
Systems that degrade during operation and recover during inactivity are modeled here as having a “state” that decreases exponentially during work and increases exponentially during rest. For any given minimum required rest length, the optimal policy produces a unique limit cycle — a repeating pattern of work and rest — toward which the system converges, regardless of its initial condition. In the absence of a minimum-break constraint, the optimal operation becomes a “dynamic state of flow,” alternating infinitesimally between work and rest to maintain the best possible steady state.
The paper, titled “Optimal Work-Rest Cycles,” by Prof. Weber, was presented at the 2024 International Conference on Operations Research (OR) and published recently in Lecture Notes in Operations Research. This work builds on a longstanding research program on optimal control theory and its applications in economics, information management, and decision-making under uncertainty [2–8].
Abstract
Assuming an instantaneous benefit proportional to a system’s state we determine the optimal work-rest policy so as to maximize the average cycle benefit, provided the state declines exponentially when the system is active and increases exponentially when it is at rest. Any given lower bound for the length of a rest period determines a unique optimal limit cycle, towards which an optimal state trajectory converges, irrespective of its starting point. The cycle benefit is maximal when the length of the resting period converges to zero while the work-rest split remains nontrivial. For this limiting case, we provide a relatively robust estimate of the system’s unknown time constants.
[1] Weber, T.A. (2025) “Optimal Work-Rest Cycles,” in L. Glomb (Ed.), Operations Research Proceedings 2024, Lecture Notes in Operations Research, Springer, Cham, pp. 89–95. [DOI: 10.1007/978-3-031-92575-7_13]
[2] Weber, T.A. (2011) Optimal Control Theory with Applications in Economics, MIT Press, Cambridge, MA. [DOI: 10.7551/mitpress/9780262015738.001.0001]
[3] Weber, T.A. (2020) “How to Market Smart Products: Design and Pricing for Sharing Markets,” Journal of Management Information Systems, Vol. 37, No. 3, pp. 631–667. [DOI: 10.1080/07421222.2020.1790179]*
[4] Chehrazi, N., Glynn, P., Weber, T.A. (2019) “Dynamic Credit-Collections Optimization,” Management Science, Vol. 65, No. 6, pp. 2737–2769. [DOI: 10.1287/mnsc.2018.3070]*
[5] Weber, T.A., Nguyen, V.A. (2018) “A Linear-Quadratic Gaussian Approach to Dynamic Information Acquisition,” European Journal of Operational Research, Vol. 270, No. 1, pp. 260–281. [DOI: 10.1016/j.ejor.2018.03.003]*
[6] Weber, T.A. (2016) “Optimal Multiattribute Screening,” Ural Mathematical Journal, Vol. 2, No. 2, pp. 87–107. [DOI: 10.15826/umj.2016.2.007]*
[7] Weber, T.A. (2006) “An Infinite-Horizon Maximum Principle with Bounds on the Adjoint Variable,” Journal of Economic Dynamics and Control, Vol. 30, No. 2, pp. 229–241. [DOI: 10.1016/j.jedc.2004.11.006]
[8] Weber, T.A. (2005) “Infinite-Horizon Optimal Advertising in a Market for Durable Goods,” Optimal Control Applications and Methods, Vol. 26, No. 6, pp. 307–336. [DOI: 10.1002/oca.765]
(*Open Access)