Consistency Cuts for Dantzig-Wolfe Reformulations

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.

Advanced Optimization Models For Bandwidth Provisioning And Routing In Fixed Microwave Backhaul Networks

This paper addresses the problem of determining bandwidth allocation and traffic routes in fixed microwave networks such that overall bandwidth cost is minimized while traffic demands are satisfied with a required reliability level. These networks exhibit high variability in link throughput as modulations schemes are adapted dynamically to ensure acceptable bit-error rate at the receivers according to external conditions such as the weather. First, we formulate an optimal optimization approach based on mixed-integer linear programming, which is subsequently reinforced by inserting problem-specific valid inequalities based on global network capacity and feasible bandwidth/modulation combinations. Then, we introduce a Lagrangian-based heuristic that provides near optimal solutions while reducing drastically the computation time. In comparison to previous work, our experimental results show that our approaches are capable to solve large real-world network instances in an effective manner. Furthermore, the results evaluate the impact of reliability and transported traffic demands on bandwidth cost.

Optimal Pooling, Batching, and Pasteurizing of Donor Human Milk

Human breast milk provides nutritional and medicinal benefits that are important to infants, particularly those who are premature or ill. Donor human milk, collected, processed, and dispensed via milk banks, is the standard of care for infants in need whose mothers cannot provide an adequate supply of milk. In this paper, we focus on streamlining donor human milk processing at nonprofit milk banks. On days that milk is processed, milk banks thaw frozen deposits, pool together milk from multiple donors to meet nutritional specifications of predefined milk types, bottle and divide the pools into batches, and pasteurize the batches using equipment with various degrees of labor requirements. Limitations in staffing and equipment and the need to follow strict healthcare protocols require productive, expedient, and frugal pooling strategies. We formulate integer programs that optimize the batching-pasteurizing decisions and the integrated pooling-batching-pasteurizing decisions by minimizing labor and meeting target production goals. We further strengthen these formulations by establishing valid inequalities for the integrated model. Numerical results demonstrate a reduction in the optimality gap through the strengthened formulation versus the basic integer programming formulation. A case study at Mothers’ Milk Bank of North Texas demonstrates significant improvement in meeting milk type production targets and a modest reduction in labor compared with former practice. The model is in use at Mothers’ Milk Bank of North Texas and has effectively improved their production balance across different milk types.

Stability Analysis of Time-Varying Delay Systems with an Improved Type of LKF

This paper further investigates the problem of stability for a general linear system with time-varying delays. Firstly an improved type of Lyapunov–Krasovskii functional is introduced with integral and nonintegral terms and time-correlation terms. Referring a few existing papers, some valid inequalities mathematical analysis techniques are used in this paper in order to reduce the conservatism of the system. Finally, two examples are presented to demonstrate the advantages of the proposed tactics in this paper.

A Branch-Price-and-Cut Algorithm for the Two-Echelon Vehicle Routing Problem with Time Windows

In this paper, we propose an exact branch-price-and-cut (BPC) algorithm for the two-echelon vehicle routing problem with time windows. This problem arises in city logistics when high-capacity and low-capacity vehicles are used to transport items from depots to satellites (first echelon) and from satellites to customers (second echelon), respectively. The aim is to determine a set of least-cost first- and second-echelon routes such that the load on the routes respect the capacity of the vehicles, each second-echelon route is supplied by exactly one first-echelon route, and each customer is visited by exactly one second-echelon route within its time window. We model the problem with a route-based formulation where first-echelon routes are enumerated a priori, and second-echelon routes are generated using column generation. The problem is solved using BPC. To generate second-echelon routes, one pricing problem per satellite is solved using a labeling algorithm which keeps track of the first-echelon route associated with each (partial) second-echelon route considered. Furthermore, to speed up the solution process, we introduce effective deep dual-optimal inequalities and apply known valid inequalities. We perform extensive computational experiments on benchmark instances and show that our method outperforms a state-of-the-art algorithm. We also conduct sensitivity analyses on the different components of our algorithm and derive managerial insights related to the structure of the first-echelon routes.

Variable and constraint reduction techniques for the temporal bin packing problem with fire-ups

The aim of this letter is to design and computationally test several improvements for the compact integer linear programming (ILP) formulations of the temporal bin packing problem with fire-ups (TBPP-FU). This problem is a challenging generalization of the classical bin packing problem in which the items, interpreted as jobs of given weight, are active only during an associated time window. The TBPP-FU objective function asks for the minimization of the weighted sum of the number of bins, viewed as servers of given capacity, to execute all the jobs and the total number of fire-ups. The fire-ups count the number of times the servers are activated due to the presence of assigned active jobs. Our contributions are effective procedures to reduce the number of variables and constraints of the ILP formulations proposed in the literature as well as the introduction of new valid inequalities. By extensive computational tests we show that substantial improvements can be achieved and several instances from the literature can be solved to proven optimality for the first time.

Robust sustainable multi-period hub location considering uncertain time-dependent demand

This paper presents a mathematical programming model for designing a sustainable continuous-time multi-period hub network considering time-dependent demand. The present model can be used in situations where the distribution of parameters related to the demand function is unknown, and we only can determine the range of changes of these parameters. To model these conditions, we consider interval uncertainty for the demand function parameters. The proposed model is a nonlinear multi-objective model. The objectives of the model cover economic, environmental, and social aspects of sustainability. These objectives include minimizing total costs, minimizing emissions, and maximizing fixed and variable job opportunities. We linearize the model by using some linearization techniques, and then, with the help of Bertsimas and Sim’s method, we construct a robust counterpart of the model. We also present some valid inequalities to strengthen the formulation. To solve the proposed model, we use Torabi and Hassini method. From solving the proposed model, network design decisions and the best time to implement decisions during the planning horizon are determined. To validate the model, we solve a sample problem based on the Turkish dataset and compare the designed network in two cases: in the first case, the demand function parameters take nominal values, and in the second case, the value of these parameters can change up to 20% of their nominal values. The results show that in the second case, the total capacity selected for hubs and hub links is greater than the first case. To investigate changes in objective functions to parameters level of conservatism and probability of constraints violation, we perform sensitivity analysis on these parameters in both single-objective and multi-objective optimization cases and report the results.

Intersection Disjunctions for Reverse Convex Sets

We present a framework to obtain valid inequalities for a reverse convex set: the set of points in a polyhedron that lie outside a given open convex set. Reverse convex sets arise in many models, including bilevel optimization and polynomial optimization. An intersection cut is a well-known valid inequality for a reverse convex set that is generated from a basic solution that lies within the convex set. We introduce a framework for deriving valid inequalities for the reverse convex set from basic solutions that lie outside the convex set. We first propose an extension to intersection cuts that defines a two-term disjunction for a reverse convex set, which we refer to as an intersection disjunction. Next, we generalize this analysis to a multiterm disjunction by considering the convex set’s recession directions. These disjunctions can be used in a cut-generating linear program to obtain valid inequalities for the reverse convex set.

A Repeated Route-then-Schedule Approach to Coordinated Vehicle Platooning: Algorithms, Valid Inequalities and Computation

In “A Repeated Route-then-Schedule Approach to Coordinated Vehicle Platooning: Algorithms, Valid Inequalities and Computation,” Luo and Larson propose a novel repeated route-then-schedule algorithmic framework to efficiently solve a complex vehicle routing and scheduling problem arising in the intelligent transportation system. The goal is to maximize the collective savings of a set of vehicles (especially heavy-duty vehicles) by utilizing the fact that platooning vehicles save energy due to reduced aerodynamic drag. In the algorithm, the original simultaneous route-and-schedule approach is decomposed into the routing stage and scheduling stage with a sophisticated learning-like feedback mechanism to update the presumed fuel cost for each vehicle traversing through each road segment. This leads to an iterative change of objective function in the routing problem and thereby changes the routes that are fed to the scheduling problem. This approach helps identify high-quality solution. The algorithmic framework leads to a very tight formulation of subproblems that can be solved in a timely manner.