A Tutorial on Multi-time Scale Optimization Models and Algorithms
Asha Ramanujam, Can Li
TL;DR
This paper surveys multi-time scale optimization models that couple high-level, long-horizon decisions with short-term scheduling across multiple subperiods. It formalizes the MM framework, introduces the Value of the Multi-scale Model (VMM) as a metric to quantify benefits over one-way top-down approaches, and surveys a spectrum of solution methods including full-space, decomposition, metaheuristics, matheuristics, data-driven approaches, and PAMSO. An illustrative capacity-expansion example demonstrates how different algorithms (full-space, Benders, Lagrangian, Dantzig-Wolfe, and PAMSO) perform on a concrete three-day, multi-time-scale problem, with VMM used to quantify improvements relative to a single-scale solution. The work highlights trade-offs between solution quality, computational effort, and scalability, and positions PAMSO as a scalable, transferable approach for large-scale problems. Overall, the tutorial provides a practical, code-backed guide for researchers and practitioners to design and solve multi-time scale optimization problems across engineering domains.
Abstract
Systems across different industries consist of interrelated processes and decisions in different time scales including long-time decisions and short-term decisions. To optimize such systems, the most effective approach is to formulate and solve multi-time scale optimization models that integrate various decision layers. In this tutorial, we provide an overview of multi-time scale optimization models and review the algorithms used to solve them. We also discuss the metric Value of the Multi-scale Model (VMM) introduced to quantify the benefits of using multi-time scale optimization models as opposed to sequentially solving optimization models from high-level to low-level. Finally, we present an illustrative example of a multi-time scale capacity expansion planning model and showcase how it can be solved using some of the algorithms (https://github.com/li-group/MultiScaleOpt-Tutorial.git). This tutorial serves as both an introductory guide for beginners with no prior experience and a high-level overview of current algorithms for solving multi-time scale optimization models, catering to experts in process systems engineering.