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A New Family of Valid-Inequalities for Dantzig-Wolfe Reformulation of Mixed Integer Linear Programs In "Consistency Cuts for Dantzig-Wolfe Reformulation," Jens Vinther Clausen, Richard Lusby, and Stefan Ropke present a new family of valid inequalities to be applied to Dantzig-Wolfe reformulations with binary linking variables. They show that, for Dantzig-Wolfe reformulations of mixed integer linear programs that satisfy certain properties, it is enough to solve the linear programming relaxation of the Dantzig-Wolfe reformulation with all consistency cuts to obtain integer solutions. An example of this is the temporal knapsack problem; the effectiveness of the cuts is tested on a set of 200 instances of this problem, and the results are state-of-the-art solution times. For problems that do not satisfy these conditions, the cuts can still be used in a branch-and-cut-and-price framework. In order to show this, the cuts are applied to a set of generic mixed linear integer programs from the online library MIPLIB. These tests show the applicability of the cuts in general. This paper introduces a family of valid inequalities, which we term consistency cuts, to be applied to a Dantzig-Wolfe reformulation (or decomposition) with linking variables. We prove that these cuts ensure an integer solution to the corresponding Dantzig-Wolfe relaxation when certain criteria to the structure of the decomposition are met. We implement the cuts and use them to solve a commonly used test set of 200 instances of the temporal knapsack problem. We assess the performance with and without the cuts and compare further to CPLEX and other solution methods that have historically been used to solve the test set. By separating consistency cuts, we show that we can obtain optimal integer solutions much faster than the other methods and even solve the remaining unsolved problems in the test set. We also perform a second test on instances from the MIPLIB 2017 online library of mixed-integer programs, showing the potential of the cuts on a wider range of problems. [ABSTRACT FROM AUTHOR]

In global liner shipping networks, a large share of transported cargo is transshipped at least once between container vessels, and the total transportation time of these containers depends on how well the corresponding services are synchronized. We propose a problem formulation that integrates service scheduling into the liner shipping network design problem. Furthermore, the model incorporates many industry-relevant modeling aspects: it allows for leg-based sailing speed optimization, it is not limited to simple or butterfly-type services, and it accounts for service-level requirements such as cargo transit time limits. The classic liner shipping network design problem is already a hard problem, and to solve the extended version, we propose a column-generation matheuristic that uses advanced linear programming techniques. The proposed method solves LINER-LIB instances of up to 114 ports and, if applied to the classic liner shipping network design problem, finds new best solutions to all instances, outperforming existing methods reported in the literature. Additionally, we analyze the relevance of scheduling for liner shipping network design. The results indicate that neglecting scheduling and approximating transshipments instead may result in the design of liner shipping networks that underestimate cargo transit times and their implications. [ABSTRACT FROM AUTHOR]

The Traveling Salesman Problem with Pickup and Delivery (TSPPD) is defined on a graph containing pickup and delivery vertices between which there exists a one-to-one relationship. The problem consists of determining a minimum cost tour such that each pickup vertex is visited before its corresponding delivery vertex. In this paper, the TSPPD is modeled as an integer linear program and its polyhedral structure is analyzed. In particular, the dimension of the TSPPD polytope is determined and several valid inequalities, some of which are facet defining, are introduced. Separation procedures and a branch-and-cut algorithm are developed. Computational results show that the algorithm is capable of solving to optimality instances involving up to 35 pickup and delivery requests, thus more than doubling the previous record of 15. [ABSTRACT FROM AUTHOR]

In the pickup and delivery problem with time windows vehicle routes must be designed to satisfy a set of transportation requests, each involving a pickup and delivery location, under capacity, time window, and precedence constraints. This paper introduces a new branch-and-cut-and-price algorithm in which lower bounds are computed by solving through column generation the linear programming relaxation of a set partitioning formulation. Two pricing subproblems are considered in the column generation algorithm: an elementary and nonelementary shortest path problem. Valid inequalities are added dynamically to strengthen the relaxations. Some of the previously proposed inequalities for the pickup and delivery problem with time windows are also shown to be implied by the set partitioning formulation. Computational experiments indicate that the proposed algorithm outperforms a recent branch-and-cut algorithm. [ABSTRACT FROM AUTHOR]