Smooth and Flexible Dual Optimal Inequalities

An iterated tabu search heuristic for the Single Source Capacitated Facility Location Problem

A branch-and-price algorithm for the robust single-source capacitated facility location problem under demand uncertainty

In this paper, we introduce a new optimization approach to Entity Resolution. Traditional approaches tackle entity resolution with hierarchical clustering, which does not benefit from a formal optimization formulation. In contrast, we model entity resolution as correlation-clustering, which we treat as a weighted set-packing problem and write as an integer linear program (ILP). In this case, sources in the input data correspond to elements and entities in output data correspond to sets/clusters. We tackle optimization of weighted set packing by relaxing integrality in our ILP formulation. The set of potential sets/clusters can not be explicitly enumerated, thus motivating optimization via column generation. In addition to the novel formulation, we also introduce new dual optimal inequalities (DOI), that we call flexible dual optimal inequalities, which tightly lower-bound dual variables during optimization and accelerate column generation. We apply our formulation to entity resolution (also called de-duplication of records), and achieve state-of-the-art accuracy on two popular benchmark datasets. Our F-DOI can be extended to other weighted set-packing problems.

This work addresses the challenge of stabilizing column generation (CG) via dual optimal inequalities (DOI). We present two novel classes of DOI for the general context of set cover problems. We refer to these as Smooth DOI (S-DOI) and Flexible DOI (F-DOI). S-DOI can be interpreted as allowing for the undercovering of items at the cost of overcovering others and incurring an objective penalty. SDOI leverage the fact that dual values associated with items should change smoothly over space. F-DOI can be interpreted as offering primal objective rewards for the overcovering of items. We combine these DOI to produce a joint class of DOI called Smooth-Flexible DOI (SF-DOI). We apply these DOI to three classical problems in logistics and operations research: the Single Source Capacitated Facility Location Problem, the Capacitated p-Median Problem, and the Capacitated Vehicle Routing Problem. We prove that these DOI are valid and are guaranteed to not alter the optimal solution of CG. We also present techniques for their use in the case of solvingCG with relaxed column restrictions. This work also introduces a CG approach to Multi-Robot Routing (MRR). MRR considers the problem of routing a fleet of robots in a warehouse to collectively complete a set of tasks while prohibiting collisions. We present two distinct formulations that tackle unique problem variants. The first we model as a set packing problem, while the second we model as a set cover problem. We show that the pricing problem for both approaches amounts to an elementary resource constrained shortest path problem (ERCSPP); an NP-hard problem commonly studied in other CG problem contexts. We present an efficient implementation of our CG approach that radically reduces the state size of the ERCSPP. Finally, we present a novel heuristic algorithm for solving the ERCSPP and offer probabilistic guarantees forits likelihood to deliver the optimal solution.

Column generation is one of the most successful approaches for solving large-scale linear programming problems. However, degeneracy difficulties and long-tail effects are known to occur as problems become larger. In recent years, several stabilization techniques of the dual variables have proven to be effective. We study the use of two types of dual-optimal inequalities to accelerate and stabilize the whole convergence process. Added to the dual formulation, these constraints are satisfied by all or a subset of the dual-optimal solutions. Therefore, the optimal objective function value of the augmented dual problem is identical to the original one. Adding constraints to the dual problem leads to adding columns to the primal problem, and feasibility of the solution may be lost. We propose two methods for recovering primal feasibility and optimality, depending on the type of inequalities that are used. Our computational experiments on the binary and the classical cutting-stock problems, and more specifically on the so-called triplet instances, show that the use of relevant dual information has a tremendous effect on the reduction of the number of column generation iterations.

We present a very large-scale neighborhood (VLSN) search algorithm for the capacitated facility location problem with single-source constraints. The neighborhood structures are induced by customer multi-exchanges and by facility moves. We consider both traditional single-customer multi-exchanges, detected on a suitably defined customer improvement graph, and more innovative multicustomer multi-exchanges, detected on a facility improvement graph dynamically built through the use of a greedy scheme. Computational results for some benchmark instances are reported that demonstrate the effectiveness of the approach for solving large-scale problems. A further test on real data involving an Italian factory is also presented.

A column generation approach to capacitated p-median problems

Cost of transportation of goods and services is an interesting topic in today’s society. The Capacitated vehicle routing problem, which is been consider in this research, is one of the variants of the vehicle routing problem. In this research we develop a reinforcement learning technique to find optimal paths from a depot to the set of customers while also considering the capacity of the vehicles, in order to reduce the cost of transportation of goods and services. Our basic assumptions are; each vehicle originates from a depot, service the customers and return to the depot, the vehicles are homogeneous. We solve the CVRP with an exact method; column generation, goole’s operation research tool and reinforcement learning and compare their solutions. Our objective is to solve a large-size of vehicle routing problem to optimality.

We present a new model and a column generation algorithm for the linear minimum cost multicommodity flow problem. The new model has two differences when compared to the classic formulation based on flows on paths: it has an additional set of variables and an additional set of inequalities. The extra variables are associated with flows on circuits where some arcs are reversed. The extra inequalities assure that an optimal solution to the extended model can be converted to an optimal solution to the path formulation. In order to obtain an optimal solution to the extended model, we use column generation. The extra variables are explicitly considered in the restricted master problem, from the beginning of the column generation process. The subproblem remains a set of shortest path problems. Our present work is related to the use of dual–optimal inequalities in the sense that we extend a model with additional primal variables/dual inequalities. However, our extended model also has additional primal inequalities/dual variables. We present computational results of our method in planar instances and compare them with standard column generation, a bundle method, and a general‐purpose solver. For the tested instances, there is an effective improvement in computational time of the column generation method when the extended model is used. © 2007 Wiley Periodicals, Inc. NETWORKS, Vol. 50(1), 3–16 2007

The capacitated p-median problem (CPMP) involves placing p identical facilities in a network and assigning customer nodes to these facilities to satisfy all customer demands with minimal transportation costs. In practical applications, demand and distance parameters are often uncertain during the planning process, leading to infeasible or excessively costly solutions if these uncertainties are disregarded. This paper addresses the robust CPMP (RCPMP), which incorporates demand uncertainty into the problem using the robust optimization paradigm. We propose a general framework to model and solve the RCPMP, considering different polyhedral uncertainty sets, namely the cardinality-constrained and the knapsack sets. We develop exact approaches encompassing compact models, different families of valid inequalities, and branch-and-cut and branch-and-price algorithms, exploring both implemented uncertainty sets and problem structure. Furthermore, we implement an efficient Variable Neighborhood Search (VNS) heuristic to solve these robust variants, which incorporates state-of-the-art algorithms, parallelization techniques, and optimized data structures. Computational experiments using adapted benchmark instances with up to 400 nodes indicate the effectiveness of the proposed approaches. The results show that using parallelization and hash tables within the VNS heuristic promotes significant performance improvements and yields near-optimal solutions for most instances, as well as outperforming the exact approaches in several instances where the optimal solution was not found. Moreover, these results highlight the benefits of using robust solutions in practical settings, especially when considering different uncertainty sets to generate solutions with advantageous trade-offs between cost and risk.

Multi-supply multi-capacitated p-median location optimization via a hybrid bi-level intelligent algorithm

Chemical Reaction Optimization (CRO) is a metaheuristic for solving optimization problems, which mimics the interactions between molecules in a chemical reaction with the purpose of achieving a stable, low-energy state. In the present work, we utilize the CRO metaheuristic to solve, in an efficient manner, the capacitated p-median problem, in order to locate service stations. Results from solving small to medium-sized problems available in the literature, with up to 724 notes and 200 medians, are compared to their optimal or best-known values. Results show that CRO results are comparable, in terms of accuracy and execution time, to many existing successfulmetaheuristics, as well as exact and hybrid methods, having exceeding those in some cases.

The capacitated p-median problem (CPMP) consists of finding p nodes (the median nodes) minimizing the total distance to the other nodes of the graph, with the constraint that the total demand of the nodes assigned to each median does not exceed its given capacity. In this paper we propose a cutting plane algorithm, based on Fenchel cuts, which allows us to considerably reduce the integrality gap of hard CPMP instances. The formulation strengthened with Fenchel cuts is solved by a commercial MIP solver. Computational results show that this approach is effective in solving hard instances or considerably reducing their integrality gap.

Capacitated Clustering Problem (CCP) assigns objects to capacitated clusters. Among various CCPs, this work focuses on the capacitated P-median problem (CPMP). Imposing a strict capacity on each cluster, often leads to inefficient overlap for geographical data, which significantly diminish efficiency. We propose a four-stage clustering algorithm, applying convex polygon boundary perturbation to minimize overlap while satisfying capacity constraints. Necessitating a fixed number of clusters in many applications introduces a challenge in meeting capacity constraint. While most methods open a new cluster for points left unassigned, this paper designs a placement algorithm without allowing new clusters. The proposed algorithm demonstrates superior performance compared to three state-of-the-art approaches on specific instances, achieves similar performance on others, with some results reaching optimality. A novel metric is introduced to compare different methods on the same benchmarks.

An effective VNS for the capacitated p-median problem