Solving a Multimodal Routing Problem with Pickup and Delivery Time Windows under LR Triangular Fuzzy Capacity Constraints

Abstract

This study models a container routing problem using multimodal transportation to improve its economy, timeliness, and reliability. Pickup and delivery time windows are simultaneously formulated in optimization to provide the shipper and the receiver with time-efficient services, in which early pickup and delayed delivery can be avoided, and nonlinear storage periods at the origin and the destination can be minimized. Furthermore, the capacity uncertainty of the multimodal network is incorporated into the advanced routing to enhance its reliability in practical transportation. The LR triangular fuzzy number is adopted to model the capacity uncertainty, in which its spread ratio is defined to measure the uncertainty level of the fuzzy capacity. Due to the nonlinearity introduced by the time windows and the fuzziness from the network capacity, this study establishes a fuzzy nonlinear optimization model for optimization problem. A chance-constrained linear reformulation equivalent to the proposed model is then generated based on the credibility measure, which makes the global optimum solution attainable by using Lingo software. A numerical case verification demonstrates that the proposed model can effectively solve the problem. The case analysis points out that the formulation of pickup and delivery time windows can improve the timeliness of the entire transportation process and help to achieve on-time transportation. Furthermore, improving the confidence level and the uncertainty level increases the total costs of the optimal route. Therefore, the shipper and the receiver must prepare more transportation budget to improve reliability and address the increasing uncertainty level. Further analysis draws some insights to help the shipper, receiver, and multimodal transport operator to organize a reliable and cost-efficient multimodal transportation under capacity uncertainty through confidence level balance and transportation service and transfer service selection.

Modeling a Multimodal Routing Problem with Flexible Time Window in a Multi-Uncertainty Environment

“…The symbols used to establish the optimization model are given in Table 1, in which they are classified into sets, indices, parameters, and variables. Using these symbols, a fuzzy mixed integer nonlinear programming model (FMINLPM) is constructed based on the modeling structure of Li et al [2] and Ge and Sun [23] in Section 2.2.2 to address the specific MRP. Set of the predecessor nodes to node i,…”

“…After the defuzzification of the optimization objective, we build the chance constraints of Equations ( 13) and ( 14) based on the credibility measure. Compared to the possibility measure (Pos), which represents an extremely optimistic attitude, and the necessity measure (Nec), which represents an extremely pessimistic attitude, this measure expresses a compromised attitude and better fits the real-world decision-making situation [23]. Moreover, only the credibility measure is self-dual, which means that the fuzzy event must hold or fail when its credibility is 1 or 0 [40].…”

“…Although the confidence level belongs to the interval [0, 1.0], the customers usually do not accept a low credibility that is indicated by a low confidence level as they demand reliable transportation under a flexible time window to ensure that the delivery of goods at the destination should not be too early or too late. Therefore, real-world decision-making prefers a interval of [0.5, 1.0] to set the confidence level value [5,23].…”

“…The MRP should be performed before real-world transportation. The working conditions of the multimodal network are influenced by various factors, as summarized by Zhang et al [16], and are, therefore, constantly changing, which makes the prediction of the real-time values of the network parameters impossible [23]. Therefore, the MRP should deal with uncertainty to enhance its reliability and realize the successful transportation of the transportation order.…”

“…(1) There are not enough data in most decision-making cases to fit the probability distribution of uncertain parameters [40,41]. (2) Taking a certain distribution (e.g., normal distribution) to formulate the uncertainty is applicable in theory; however, it might not match the real world [23].…”