Network Models with Unsplittable Node Flows with Application to Unit Train Scheduling

Abstract

We study network models where flows cannot be split or merged when passing through certain nodes, i.e., for such nodes, each incoming arc flow must be matched to an outgoing arc flow of identical value. This requirement, which we call no-split no-merge (NSNM), appears in railroad applications where train compositions can only be modified at yards where necessary equipment is available. This combinatorial requirement is crucial when formulating problems occurring in the unit train business. We propose modeling approaches to represent the NSNM requirement. In particular, we give a linear formulation of the requirement on a single node that describes the convex hull in a lifted space. We present a cut-generating linear program to obtain valid inequalities in the original space of variables, and introduce a polynomial-time procedure to lift strong inequalities of lower-dimensional models into strong inequalities of the original model. In addition, we identify an exponential family of facet-defining inequalities that can be separated efficiently. To evaluate our results computationally, we study a stylized unit train problem. We compare a solution approach based on our results with one that relies on column generation. We then show that our results significantly reduce relaxation times and gaps when compared to leading commercial branch-and-cut software.

Solving Unsplittable Network Flow Problems with Decision Diagrams

“…Many variants of train routing and scheduling problems with different objective functions and set of constraints under deterministic and stochastic conditions have been introduced and vastly studied in the literature; see surveys by Cordeau, Toth, and Vigo (1998), Harrod and Gorman (2010), Lusby et al (2011), Cacchiani andToth (2012), andTurner et al (2016) for different problems classifications and structures. Mixed integer linear and nonlinear programming formulations are among the most frequent exact approaches to model different classes of these problems (Jovanović and Harker 1991, Huntley et al 1995, Sherali and Suharko 1998, Lawley et al 2008, Haahr and Lusby 2017, Davarnia et al 2019. Proposed solution techniques include but are not limited to branch-and-bound methods (Jovanović andHarker 1991, Fuchsberger andLüthi 2007), branch-and-cut frameworks (Zwaneveld, Kroon, andVan Hoesel 2001, Ceselli et al 2008), branch-and-price approaches (Lusby 2008, Lin andKwan 2016), graph coloring algorithms (Cornelsen and Di Stefano 2007), and heuristics (Carey and Crawford 2007, Liu and Kozan 2011, Içyüz et al 2016.…”

“…We study the MIP formulation of the SGUFP based on that of its deterministic counterpart given in Davarnia et al (2019). Consider a network G = (V, A) with node set V := V ∪ {s, t} and arc set A, where s and t are source and sink nodes, respectively.…”

“…We create test instances based on the specification given in Davarnia et al (2019), which is inspired by realistic models. In particular, we consider a base rail network G = (V , A ) where 10% and 30% of the nodes are supply and demand nodes, respectively.…”

“…Given a set of supply, intermediate, and demand locations in a railroad network, the unit train scheduling problem seeks to find optimal routes for unit trains to send flows from supply to demand points with the objective of minimizing the total transportation cost while meeting demand of customers, respecting capacities of tracks, and satisfying no-car attaching/detaching requirements in specific locations. As a result, designing blocking plans to determine locations where cars need to be switched between trains is irrelevant in this problem, unlike scheduling other types of trains (Davarnia et al 2019).…”

“…As a remedy, they develop a time-efficient heuristic that produces good quality solutions. More recently,Davarnia et al (2019) introduce and study the GUFP with application to unit train scheduling. In particular, the authors show how to impose NSNM restrictions in network optimization problems.…”