A Fuzzy Programming Method for Modeling Demand Uncertainty in the Capacitated Road-Rail Multimodal Routing Problem with Time Windows

Demand uncertainty is an important issue that influences the strategic, tactical, and operational-level decision making in the transportation/logistics/supply chain planning. In this study, we explore the effect of demand uncertainty on the operational-level freight routing problem in the capacitated multimodal transportation network that consists of schedule-based rail transportation and time-flexible road transportation. Considering the imprecise characteristic of the demand, we adopt fuzzy set theory to model its uncertainty and use trapezoidal fuzzy numbers to represent the fuzzy demands. We set multiple transportation orders as the optimization object and employ soft time windows to reflect the customer requirement on on-time transportation. Under the above situation, we establish a fuzzy mixed integer nonlinear programming (FMINLP) model to formulate the capacitated road-rail multimodal routing problem with demand uncertainty and time windows. We first use the fuzzy expected value model and credibility measure based fuzzy chance-constrained programming to realize the defuzziness of the model and then adopt linearization technique to reformulate the crisp model to finally generate an equivalent mixed integer linear programming (MILP) model that can be solved by standard mathematical programming software. Finally, a numerical case is presented to demonstrate the feasibility of the proposed method. Sensitivity analysis and fuzzy simulation are combined to quantify the effect of demand uncertainty on the routing problem and also reveal some helpful insights and managerial implications.