Matheuristics: using mathematics for heuristic design

Matheuristics are heuristic algorithms based on mathematical tools such as the ones provided by mathematical programming, that are structurally general enough to be applied to different problems with little adaptations to their abstract structure. The result can be metaheuristic hybrids having components derived from the mathematical model of the problems of interest, but the mathematical techniques themselves can define general heuristic solution frameworks. In this paper, we focus our attention on mathematical programming and its contributions to developing effective heuristics. We briefly describe the mathematical tools available and then some matheuristic approaches, reporting some representative examples from the literature. We also take the opportunity to provide some ideas for possible future development

MATHEMATICAL PROGRAMMING ALGORITHMS FOR NETWORK OPTIMIZATION PROBLEMS

In the thesis we consider combinatorial optimization problems that are defined by means of networks. These problems arise when we need to take effective decisions to build or manage network structures, both satisfying the design constraints and minimizing the costs. In the thesis we focus our attention on the four following problems: - The Multicast Routing and Wavelength Assignment with Delay Constraint in WDM networks with heterogeneous capabilities (MRWADC) problem: this problem arises in the telecommunications industry and it requires to define an efficient way to make multicast transmissions on a WDM optical network. In more formal terms, to solve the MRWADC problem we need to identify, in a given directed graph that models the WDM optical network, a set of arborescences that connect the source of the transmission to all its destinations. These arborescences need to satisfy several quality-of-service constraints and need to take into account the heterogeneity of the electronic devices belonging to the WDM network. - The Homogeneous Area Problem (HAP): this problem arises from a particular requirement of an intermediate level of the Italian government called province. Each province needs to coordinate the common activities of the towns that belong to its territory. To practically perform its coordination role, the province of Milan created a customer care layer composed by a certain number of employees that have the task to support the towns of the province in their administrative works. For the sake of efficiency, the employees of this customer care layer have been partitioned in small groups and each group is assigned to a particular subset of towns that have in common a large number of activities. The HAP requires to identify the set of towns assigned to each group in order to minimize the redundancies generated by the towns that, despite having some activities in common, have been assigned to different groups. Since, for both historical and practical reasons, the towns in a particular subset need to be adjacent, the HAP can be effectively modeled as a particular graph partitioning problem that requires the connectivity of the obtained subgraphs and the satisfaction of nonlinear knapsack constraints. - Knapsack Prize Collecting Steiner Tree Problem (KPCSTP): to implement a Column Generation algorithm for the MRWADC problem and for the HAP, we need also to solve the two corresponding pricing problems. These two problems are very similar, both of them require to find an arborescence, contained in a given directed weighted graph, that minimizes the difference between its cost and the prizes associated with the spanned nodes. The two problems differ in the side constraints that their feasible solutions need to satisfy and in the way in which the cost of an arborescence is defined. The ILP formulations and the resolution methods that we developed to tackle these two problems have many characteristics in common with the ones used to solve other similar problems. To exemplify these similarities and to summarize and extend the techniques that we developed for the MRWADC problem and for the HAP, we also considered the KPCSTP. This problem requires to find a tree that minimizes the difference between the cost of the used arcs and the profits of the spanned nodes. However, not all trees are feasible: the sum of the weights of the nodes spanned by a feasible tree cannot exceed a given weight threshold. In the thesis we propose a computational comparison among several optimization methods for the KPCSTP that have been either already proposed in the literature or obtained modifying our ILP formulations for the two previous pricing problems. - The Train Design Optimization (TDO) problem: this problem was the topic of the second problem solving competition, sponsored in 2011 by the Railway Application Section (RAS) of the Institute for Operations Research and the Management Sciences (INFORMS). We participated to the contest and we won the second prize. After the competition, we continued to work on the TDO problem and in the thesis we describe the improved method that we have obtained at the end of this work. The TDO problem arises in the freight railroad industry. Typically, a freight railroad company receives requests from customers to transport a set of railcars from an origin rail yard to a destination rail yard. To satisfy these requests, the company first aggregates the railcars having the same origin and the same destination in larger blocks, and then it defines a trip plan to transport the obtained blocks to their correct destinations. The TDO problem requires to identify a trip plan that efficiently uses the limited resources of the considered rail company. More formally, given a railway network, a set of blocks and the segments of the network in which a crew can legally drive a train, the TDO problem requires to define a set of trains and the way in which the given blocks can be transported to their destinations by these trains, both satisfying operational constraints and minimizing the transportation costs

A Polyhedral Study of Mixed 0-1 Set

We consider a variant of the well-known single node fixed charge network flow set with constant capacities. This set arises from the relaxation of more general mixed integer sets such as lot-sizing problems with multiple suppliers. We provide a complete polyhedral characterization of the convex hull of the given set

Design of Heuristic Algorithms for Hard Optimization

This open access book demonstrates all the steps required to design heuristic algorithms for difficult optimization. The classic problem of the travelling salesman is used as a common thread to illustrate all the techniques discussed. This problem is ideal for introducing readers to the subject because it is very intuitive and its solutions can be graphically represented. The book features a wealth of illustrations that allow the concepts to be understood at a glance. The book approaches the main metaheuristics from a new angle, deconstructing them into a few key concepts presented in separate chapters: construction, improvement, decomposition, randomization and learning methods. Each metaheuristic can then be presented in simplified form as a combination of these concepts. This approach avoids giving the impression that metaheuristics is a non-formal discipline, a kind of cloud sculpture. Moreover, it provides concrete applications of the travelling salesman problem, which illustrate in just a few lines of code how to design a new heuristic and remove all ambiguities left by a general framework. Two chapters reviewing the basics of combinatorial optimization and complexity theory make the book self-contained. As such, even readers with a very limited background in the field will be able to follow all the content

Classification of the Existing Knowledge Base of OR/MS Research and Practice (1990-2019) using a Proposed Classification Scheme

This is the author accepted manuscript. The final version is available from Elsevier via the DOI in this recordOperations Research/Management Science (OR/MS) has traditionally been defined as the discipline that applies advanced analytical methods to help make better and more informed decisions. The purpose of this paper is to present an analysis of the existing knowledge base of OR/MS research and practice using a proposed keywords-based approach. A conceptual structure is necessary in order to place in context the findings of our keyword analysis. Towards this we first present a classification scheme that relies on keywords that appeared in articles published in important OR/MS journals from 1990-2019 (over 82,000 articles). Our classification scheme applies a methodological approach towards keyword selection and its systematic classification, wherein approximately 1300 most frequently used keywords (in terms of cumulative percentage, these keywords and their derivations account for more than 45% of the approx. 290,000 keyword occurrences used by the authors to represent the content of their articles) were selected and organised in a classification scheme with seven top-level categories and multiple levels of sub-categories. The scheme identified the most commonly used keywords relating to OR/MS problems, modeling techniques and applications. Next, we use this proposed scheme to present an analysis of the last 30 years, in three distinct time periods, to show the changes in OR/MS literature. The contribution of the paper is thus twofold, (a) the development of a proposed discipline-based classification of keywords (like the ACM Computer Classification System and the AMS Mathematics Subject Classification), and (b) an analysis of OR/MS research and practice using the proposed classification

Evolutionary Computation 2020

Intelligent optimization is based on the mechanism of computational intelligence to refine a suitable feature model, design an effective optimization algorithm, and then to obtain an optimal or satisfactory solution to a complex problem. Intelligent algorithms are key tools to ensure global optimization quality, fast optimization efficiency and robust optimization performance. Intelligent optimization algorithms have been studied by many researchers, leading to improvements in the performance of algorithms such as the evolutionary algorithm, whale optimization algorithm, differential evolution algorithm, and particle swarm optimization. Studies in this arena have also resulted in breakthroughs in solving complex problems including the green shop scheduling problem, the severe nonlinear problem in one-dimensional geodesic electromagnetic inversion, error and bug finding problem in software, the 0-1 backpack problem, traveler problem, and logistics distribution center siting problem. The editors are confident that this book can open a new avenue for further improvement and discoveries in the area of intelligent algorithms. The book is a valuable resource for researchers interested in understanding the principles and design of intelligent algorithms