Fuzzy Optimization and Decision Making https://doi.org/10.1007/s10700-018-9289-0 Schedule optimization under fuzzy constraints of vehicle capacity Yanan Zhang 1 · Zhaopeng Meng 1 · Yan Zheng 1 · Anca Ralescu 2 © Springer Science+Business Media, LLC, part of Springer Nature 2018 Abstract The objective of designing timetables for public transportation is twofold: to ensure an efficient use of limited resources and to provide a comfortable ride for passengers. Two models for timetable optimization are investigated in this study. Model 1 uses a crisp constraint on the rate of vehicle capacity usage. Model 2 improves on model 1 by translating the crisp constraint into a fuzzy goal representing passenger satisfaction, and a fuzzy constraint, representing the extent of vehicle usage. Both, the fuzzy goal and the fuzzy constraint, are fuzzy sets on the number of on-board passengers. Heuris- tic methods together with linear programming are proposed for finding the optimal headway. Model 1 selects the largest time interval under the bound on vehicle size. The set of optimal time intervals in model 2 is decided by the simultaneous level cuts of the fuzzy goal and constraint. Experimental results show that fuzzy-set based model 2 is the most flexible and effective way to generate an optimal timetable. Keywords Multi-criteria decision making · Fuzzy optimization · Integer programming · Timetable scheduling · Public transit optimization B Anca Ralescu anca.ralescu@uc.edu Yanan Zhang zhangyn@tju.edu.cn Zhaopeng Meng mengzp@tju.edu.cn Yan Zheng yanzheng@tju.edu.cn 1 School of Computer Software, Tianjin University, Tianjin 300350, China 2 EECS Department, University of Cincinnati, Cincinnati, OH 45221-0030, USA 123