A combinatorial optimization problem (COP) P consists of a collection of its instances. An instance of P can be represented as an ordered pair (F, f), where F is the family of feasible solutions and f is the objective function, which is used to compare two feasible solutions. Very largescale neighborhood (VLSN) search algorithmsearch algorithms have been successfully applied to solve various optimization problems of practical interest. Researchers have used various techniques for developing good neighborhood functions that lead to effective VLSN search algorithms. This includes multi-exchange ejection chains, variable depth methods, integer programming, weighted matching set partitioning and so on. One of the most important theoretical questions in VLSN search (and local search in general) is to find a good bound on the number of improvement steps required to reach termination. In addressing this important theoretical question, Johnson et al. introduced the complexity class PLS.

The crew scheduling problem (CSP) involves assigning crew to trains, while satisfying a variety of Federal Railway Administration (FRA) regulations and trade-union work rules. Train crew work together to move a train from its origin to its destination. As the train travels over its route, it goes through numerous crew districts. In each crew district, the train is manned by an engineer and a conductor who are qualified to operate the train within that district. The objectives of crew scheduling are therefore to assign crew to the trains, while minimizing the cost of operating trains, improving crew quality of life, and satisfying all FRA regulations and work rules.

Abstract The multicommodity flow problem is a generalization of the minimum cost network flow problem. In this problem, several commodities governed by their own network flow constraints share the same underlying network. Given the supplies and demands of the different commodities, the shared capacity of each arc in the network, and cost of flow of each commodity on each arc, the objective is to determine the flow that minimizes the total cost. The multicommodity flow problem has been extensively applied to solve problems in areas such as transportation, telecommunications, and scheduling. In this article, we introduce the multicommodity flow problem, outline some of its applications, and describe efficient methods to solve the problem.

Abstract The shortest path problem (SPP) is a fundamental combinatorial optimization problem that concerns finding the shortest paths between a source and a destination; it is also referred to as a single‐source shortest path problem (SSPP). In some problem settings, shortest paths between all pairs of nodes are desired; these are referred to as all‐pairs shortest path problems (ASPP). The SPP arises in a number of applications, both as a stand‐alone problem and as a subproblem in more complex scenarios. Several applications have been described in the last chapter. In this article, we describe algorithmic methods to solve the SSPP, as well as the ASPP. In real‐life applications, networks vary in their characteristics such as in the presence of negative arcs, directed cycles, and so forth. We present both customized and the most extensively studied algorithms for each scenario, with an analysis of their individual complexities.

Abstract The transportation sector, especially the rail mode, is a very rich source of optimization problems and has been a primary focus of operations researchers. A plethora of articles dedicated to each transportation mode (rail, road, air, and water) can be found in the literature. Ironically, the industries are still using rules of thumb instead of mathematical models for most of the planning and scheduling processes. There are two main reasons for not using the operations research (OR) models: (i) the highly complex nature of the problems, and (ii) the huge gap between the academic (theoretical) and industrial (practical) approaches. With the steep growth in computational capabilities, we have started developing innovative algorithms by shifting the focus from finding optimal‐but‐impractical solutions to finding good‐and‐implementable solutions. We are developing state‐of‐the‐art ideas combining exact approaches such as network flows and mixed integer programming, with heuristics and meta‐heuristics such as the tabu search and the very large‐scale neighborhood (VLSN) search. In this article, we provide an overview of the railroad planning and scheduling problems at the strategic, tactical, and operational levels. We describe the best algorithms successfully implemented at US railroads, which have the potential of saving hundreds of millions of dollars annually.

We study a scheduling problem that belongs to the yard operations component of the railroad planning problems, namely the hump sequencing problem. The scheduling problem is characterized as a single-machine problem with stepwise tardiness cost objectives. This is a new scheduling criterion which is also relevant in the context of traditional machine scheduling problems. We produce complexity results that characterize some cases of the problem as pseudo-polynomially solvable. For the difficult-to-solve cases of the problem, we develop mathematical programming formulations, and propose heuristic algorithms. We test the formulations and heuristic algorithms on randomly generated single-machine scheduling problems and real-life datasets for the hump sequencing problem. Our experiments show promising results for both sets of problems.

Given a solution x* and an a priori estimated cost vector c, the inverse optimization problem is to identify another cost vector d so that x* is optimal with respect to the cost vector d and the deviation of d from c is minimum. In this paper, we consider the inverse spanning tree problem on an undirected graph G = (N, A) with n nodes and m arcs, and where the deviation between c and d is defined by the rectilinear distance between the two vectors (that is, L1 norm). We show that the inverse spanning tree problem can be formulated as the dual of an assignment problem on a bipartite network G0 = (N0, A0) with N0 = N1 ∪ N2 and A0 ⊆ N1 x N2. The bipartite network satisfies the property that |N1| = (n 1), |N2| = (m n + 1), and |A0| = O(nm). In general, |N1| < < |N2|. Using this special structure of the assignment problem, we develop a specific implementation of the successive shortest path algorithm that solves the inverse spanning tree problem in O(n3) time. We also consider the weighted version of the inverse spanning tree problem where we minimize the sum of the weighted deviations of arcs and show that it can be formulated as the dual of the transportation problem. Using a cost scaling algorithm, the transportation problem can be solved in O(n2 m log(nC)), where C denotes the largest arc cost in the data. Finally, we consider a minimax version of the inverse spanning tree problem and show that it can be solved in O(n2) time. * Department of Mathematics, Indian Institute of Technology, Kanpur 208 106, INDIA ** Dept. of Industrial & Management Engg., Indian Institute of Technology, Kanpur 208 106, INDIA *** Sloan School of Management, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Abstract The minimum cost flow problem is a fundamental network flow problem. The objective of this problem is to determine the minimum cost method of transporting units of a goods that originates at one or more points in a network (sources or supplies), with arc capacities to one or more demand points in the network (sinks or demands). In this article we introduce the minimum cost flow problem and its applications, describe the conditions for a feasible flow to be optimal, and discuss three classical algorithms to solve it.

NetworksVolume 57, Issue 1 p. 1-2 Special issue of Networks on optimization in scheduled transportation networks Ravindra K. Ahuja, Corresponding Author Ravindra K. Ahuja [email protected] Department of Industrial & Systems Engineering, University of Florida, Gainesville, FloridaDeptartment of Industrial & Systems Engineering, University of Florida, Gainesville, FloridaSearch for more papers by this authorChristian Liebchen, Christian Liebchen Network Dimensioning, DB Schenker Rail Deutschland AG, Berlin and Mainz Germany, Germany; Institute of Mathematics, Technische Universität Berlin, Berlin, GermanySearch for more papers by this author Ravindra K. Ahuja, Corresponding Author Ravindra K. Ahuja [email protected] Department of Industrial & Systems Engineering, University of Florida, Gainesville, FloridaDeptartment of Industrial & Systems Engineering, University of Florida, Gainesville, FloridaSearch for more papers by this authorChristian Liebchen, Christian Liebchen Network Dimensioning, DB Schenker Rail Deutschland AG, Berlin and Mainz Germany, Germany; Institute of Mathematics, Technische Universität Berlin, Berlin, GermanySearch for more papers by this author First published: 07 December 2010 https://doi.org/10.1002/net.20378Citations: 3AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onEmailFacebookTwitterLinkedInRedditWechat No abstract is available for this article.Citing Literature Volume57, Issue1Special Issue: Optimization in Scheduled Transportation NetworksJanuary 2011Pages 1-2 RelatedInformation

The objective of the classical minimum cost flow problem is to send units of a good that reside at one or more points in a network (sources or supply nodes) with arc capacities to one or more other points in the network (sinks or demand nodes), incurring minimum cost. We develop fast algorithms for previously unstudied specially structured minimum cost flow problems that have applications in many areas, such as locomotive and airline scheduling, repositioning of empty rail freight cars, highway and river transportation, congestion pricing, shop loading, and production planning. First, we consider the case where the n1 supply and n2 demand nodes lie on a circle (or line) (n = n1 + n2) and flow is allowed only in one direction; our algorithm solves this problem in O(n) time. Next, we consider a constrained version of this problem and show that it can be solved in O(n log n2) time. Finally, we consider the version where the nodes lie on a circle (or line), flow is allowed in both directions, and the costs of flow between two nodes in the clockwise and the counterclockwise direction are different; our algorithm solves this problem in O(n log n) time. Our algorithms are based on the successive shortest-path algorithm for the minimum cost flow problem. We exploit the special structure of the problem and use advanced data structures, when required, to achieve short run times.

Railroads face the challenge of competing with the trucking industry in a fastpaced environment. In this respect, they are working toward running freight trains on schedule and reducing travel times. The planned train schedules consist of departure and arrival times at main stations on the rail network. A detailed timetable, on the other hand, consists of the departure and arrival times of each train in each track section of its route. The train dispatching problem aims to determine detailed timetables over a rail network in order to minimize deviations from the planned schedule. We provide a new integer programming formulation for this problem based on a spacetime network; we propose heuristic algorithms to solve it and present computational results of these algorithms. Our approach includes some realistic constraints that have not been previously considered as well as all the assumptions and practical issues considered by the earlier works.

In this paper, we study the curfew planning problem (CPP) encountered by railroads for the maintenance of their railway tracks. The CPP is to design an optimal annual timetable to complete a given set of repairs and replacement jobs (rail work and tie work) on the railway tracks for a set of crews specialized in rail work (rail crew) or tie work (tie crew). We develop the work schedule for each crew such that the disruptions in train routes because of subdivision curfews are minimized. A subdivision is said to be under curfew if any crew is working in it. The solution to the problem must also satisfy several operational and regulatory requirements such as the crew continuity, time windows, the maximum interproject distance travelled by crews, etc. Our paper presents four solution approaches for the CPP: (i) time-space network model (TSNM), (ii) duty-generation model (DGM), (iii) column-generation model (CGM), and (iv) decomposition-based heuristics. We solve each model using CPLEX and present the computational results based on real-life instances.

Scheduled transportation networks give rise to very complex and large-scale networkoptimization problems requiring innovative solution techniques and ideas from mathematical optimization and theoretic

A typical network design problem consists of identifying a subnetwork of a given network that satisfies a set of constraints and minimizes the cost of flow. We consider degree-constrained network design problems where we specify an upper limit on the number of arcs built at each node. We propose two lower bounding techniques using the concept of lower planes. The first lower bound is based on a known lower plane for network design problems; our second lower bound is stronger than the first, yet has the same polynomial computational complexity. We illustrate both lower bounds using a numerical example, test their performance, and present a real-life case study from the area of railroad planning problems. © 2008 Wiley Periodicals, Inc. NETWORKS, 2009

In an incremental optimization problem, we are given a feasible solution x 0 of an optimization problem P, and we want to make an incremental change in x 0 that will result in the greatest improvement in the objective function. In this paper, we study the incremental optimization versions of six well-known network problems. We present a strongly polynomial algorithm for the incremental minimum spanning tree problem. We show that the incremental minimum cost flow problem and the incremental maximum flow problem can be solved in polynomial time using Lagrangian relaxation. We consider two versions of the incremental minimum shortest path problem, where increments are measured via arc inclusions and arc exclusions. We present a strongly polynomial time solution for the arc inclusion version and show that the arc exclusion version is NP-complete. We show that the incremental minimum cut problem is NP-complete and that the incremental minimum assignment problem reduces to the minimum exact matching problem, for which a randomized polynomial time algorithm is known.