Models and Matheuristics for Large-Scale Combinatorial Optimization Problems

Combinatorial optimization deals with efficiently determining an optimal (or at least a good) decision among a finite set of alternatives. In business administration, such combinatorial optimization problems arise in, e.g., portfolio selection, project management, data analysis, and logistics. These optimization problems have in common that the set of alternatives becomes very large as the problem size increases, and therefore an exhaustive search of all alternatives may require a prohibitively long computation time. Moreover, due to their combinatorial nature no closed-form solutions to these problems exist. In practice, a common approach to tackle combinatorial optimization problems is to formulate them as mathematical models and to solve them using a mathematical programming solver (cf., e.g., Bixby et al. 1999, Achterberg et al. 2020). For small-scale problem instances, the mathematical models comprise a manageable number of variables and constraints such that mathematical programming solvers are able to devise optimal solutions within a reasonable computation time. For large-scale problem instances, the number of variables and constraints becomes very large which extends the computation time required to find an optimal solution considerably. Therefore, despite the continuously improving performance of mathematical programming solvers and computing hardware, the availability of mathematical models that are efficient in terms of the number of variables and constraints used is of crucial importance. Another frequently used approach to address combinatorial optimization problems are matheuristics. Matheuristics decompose the considered optimization problem into subproblems, which are then formulated as mathematical models and solved with the help of a mathematical programming solver. Matheuristics are particularly suitable for situations where it is required to find a good, but not necessarily an optimal solution within a short computation time, since the speed of the solution process can be controlled by choosing an appropriate size of the subproblems. This thesis consists of three papers on large-scale combinatorial optimization problems. We consider a portfolio optimization problem in finance, a scheduling problem in project management, and a clustering problem in data analysis. For these problems, we present novel mathematical models that require a relatively small number of variables and constraints, and we develop matheuristics that are based on novel problem-decomposition strategies. In extensive computational experiments, the proposed models and matheuristics performed favorably compared to state-of-the-art models and solution approaches from the literature. In the first paper, we consider the problem of determining a portfolio for an enhanced index-tracking fund. Enhanced index-tracking funds aim to replicate the returns of a particular financial stock-market index as closely as possible while outperforming that index by a small positive excess return. Additionally, we consider various real-life constraints that may be imposed by investors, stock exchanges, or investment guidelines. Since enhanced index-tracking funds are particularly attractive to investors if the index comprises a large number of stocks and thus is well diversified, it is of particular interest to tackle large-scale problem instances. For this problem, we present two matheuristics that consist of a novel construction matheuristic, and two different improvement matheuristics that are based on the concepts of local branching (cf. Fischetti and Lodi 2003) and iterated greedy heuristics (cf., e.g., Ruiz and Stützle 2007). Moreover, both matheuristics are based on a novel mathematical model for which we provide insights that allow to remove numerous redundant variables and constraints. We tested both matheuristics in a computational experiment on problem instances that are based on large stock-market indices with up to 9,427 constituents. It turns out that our matheuristics yield better portfolios than benchmark approaches in terms of out-of-sample risk-return characteristics. In the second paper, we consider the problem of scheduling a set of precedence-related project activities, each of which requiring some time and scarce resources during their execution. For each activity, alternative execution modes are given, which differ in the duration and the resource requirements of the activity. Sought is a start time and an execution mode for each activity, such that all precedence relationships are respected, the required amount of each resource does not exceed its prescribed capacity, and the project makespan is minimized. For this problem, we present two novel mathematical models, in which the number of variables remains constant when the range of the activities' durations and thus also the planning horizon is increased. Moreover, we enhance the performance of the proposed mathematical models by eliminating some symmetric solutions from the search space and by adding some redundant sequencing constraints for activities that cannot be processed in parallel. In a computational experiment based on instances consisting of activities with durations ranging from one up to 260 time units, the proposed models consistently outperformed all reference models from the literature. In the third paper, we consider the problem of grouping similar objects into clusters, where the similarity between a pair of objects is determined by a distance measure based on some features of the objects. In addition, we consider constraints that impose a maximum capacity for the clusters, since the size of the clusters is often restricted in practical clustering applications. Furthermore, practical clustering applications are often characterized by a very large number of objects to be clustered. For this reason, we present a matheuristic based on novel problem-decomposition strategies that are specifically designed for large-scale problem instances. The proposed matheuristic comprises two phases. In the first phase, we decompose the considered problem into a series of generalized assignment problems, and in the second phase, we decompose the problem into subproblems that comprise groups of clusters only. In a computational experiment, we tested the proposed matheuristic on problem instances with up to 498,378 objects. The proposed matheuristic consistently outperformed the state-of-the-art approach on medium- and large-scale instances, while matching the performance for small-scale instances. Although we considered three specific optimization problems in this thesis, the proposed models and matheuristics can be adapted to related optimization problems with only minor modifications. Examples for such related optimization problems are the UCITS-constrained index-tracking problem (cf, e.g., Strub and Trautmann 2019), which consists of determining the portfolio of an investment fund that must comply with regulatory restrictions imposed by the European Union, the multi-site resource-constrained project scheduling problem (cf., e.g., Laurent et al. 2017), which comprises the scheduling of a set of project activities that can be executed at alternative sites, or constrained clustering problems with must-link and cannot-link constraints (cf., e.g., González-Almagro et al. 2020)

Multi-objective model for optimizing railway infrastructure asset renewal

Trabalho inspirado num problema real da empresa Infraestruturas de Portugal, EP.A multi-objective model for managing railway infrastructure asset renewal is presented. The model aims to optimize three objectives, while respecting operational constraints: levelling investment throughout multiple years, minimizing total cost and minimizing work start postponements. Its output is an optimized intervention schedule. The model is based on a case study from a Portuguese infrastructure management company, which specified the objectives and constraints, and reflects management practice on railway infrastructure. The results show that investment levelling greatly influences the other objectives and that total cost fluctuations may range from insignificant to important, depending on the condition of the infrastructure. The results structure is argued to be general and suggests a practical methodology for analysing trade-offs and selecting a solution for implementation.info:eu-repo/semantics/publishedVersio

The extended resource task network: a framework for the combined scheduling of batch processes and CHP plants

The issue of energy has emerged as one of the greatest challenges facing mankind. In an industrial perspective, the development of site utility systems (generally combined heat and power (CHP) systems) for the generation and management of utilities provides a great potential source for energy savings. However, in most industrial sites, a master–slave relationship usually governs this kind of system and limits the potential operating capacity of CHP. To improve the decision-making process, Agha et al. (2010. Integrated production and utility system approach for optimising industrial unit operation. Energy, 35, 611–627) have proposed an integrated approach that carries out simultaneous and consistent scheduling of batch production plants and site utility systems. The modelling of the problem relies on a mixed integer linear programming (MILP) formulation. Nevertheless, although it is a powerful mathematical tool, it still remains difficult to use for non-expert engineers. In this framework, a graphical formalism based on existing representations (STN, RTN) has been developed: the extended resource task network (ERTN). Combined with an efficient and generic MILP formulation, it permits various kinds of industrial problems, including production and consumption of utility flows to be modelled homogenously. This paper focuses on the semantic elements of the ERTN formalism and illustrates their use through representative example

Global supply chains of high value low volume products

Imperial Users onl

On Idle Energy Consumption Minimization in Production: Industrial Example and Mathematical Model

This paper, inspired by a real production process of steel hardening, investigates a scheduling problem to minimize the idle energy consumption of machines. The energy minimization is achieved by switching a machine to some power-saving mode when it is idle. For the steel hardening process, the mode of the machine (i.e., furnace) can be associated with its inner temperature. Contrary to the recent methods, which consider only a small number of machine modes, the temperature in the furnace can be changed continuously, and so an infinite number of the power-saving modes must be considered to achieve the highest possible savings. To model the machine modes efficiently, we use the concept of the energy function, which was originally introduced in the domain of embedded systems but has yet to take roots in the domain of production research. The energy function is illustrated with several application examples from the literature. Afterward, it is integrated into a mathematical model of a scheduling problem with parallel identical machines and jobs characterized by release times, deadlines, and processing times. Numerical experiments show that the proposed model outperforms a reference model adapted from the literature.Comment: Accepted to 9th International Conference on Operations Research and Enterprise Systems (ICORES 2020

Methods to Support the Project Selection Problem With Non-Linear Portfolio Objectives, Time Sensitive Objectives, Time Sensitive Resource Constraints, and Modeling Inadequacies

The United States Air Force relies upon information production activities to gain insight regarding uncertainties affecting important system configuration and in-mission task execution decisions. Constrained resources that prevent the fulfillment of every information production request, multiple information requestors holding different temporal-sensitive objectives, non-constant marginal value preferences, and information-product aging factors that affect the value-of-information complicate the management of these activities. This dissertation reviews project selection research related to these issues and presents novel methods to address these complications. Quantitative experimentation results demonstrate these methods’ significance

Unified Concept of Bottleneck

The term `bottleneck` has been extensively used in operations management literature. Management paradigms like the Theory of Constraints focus on the identification and exploitation of bottlenecks. Yet, we show that the term has not been rigorously defined. We provide a classification of bottleneck definitions available in literature and discuss several myths associated with the concept of bottleneck. The apparent diversity of definitions raises the question whether it is possible to have a single bottleneck definition which has as much applicability in high variety job shops as in mass production environments. The key to the formulation of an unified concept of bottleneck lies in relating the concept of bottleneck to the concept of shadow price of resources. We propose an universally applicable bottleneck definition based on the concept of average shadow price. We discuss the procedure for determination of bottleneck values for diverse production environments. The Law of Diminishing Returns is shown to be a sufficient but not necessary condition for the equivalence of the average and the marginal shadow price. The equivalence of these two prices is proved for several environments. Bottleneck identification is the first step in resource acquisition decisions faced by managers. The definition of bottleneck presented in the paper has the potential to not only reduce ambiguity regarding the meaning of the term but also open a new window to the formulation and analysis of a rich set of problems faced by managers.

Sounds of Silence: a sampling-based bi-criteria harmony search metaheuristic for the resource constrained project scheduling problem with uncertain activity durations and cash flows

In this paper, a new sampling-based bi-criteria hybrid harmony search metaheuristic for the resource-constrained project-scheduling problem (RCPSP) with uncertain activity durations (UAD) and uncertain cash flows (UCF) is proposed, with the total project duration () and the net present value () as objectives. The problem-specific Sounds of Silence (SoS) metaheuristic is an appropriate hybridization of the robust SoS with uncertain activity durations, and the crisp SoS developed for several a primary-secondary (PS) and bi-criteria (BC) project scheduling problems. In order to illustrate the efficiency and stability of the proposed problem-specific SoS, we present detailed computational results for a larger and challenging project instance. Results reveal the fact that the modified and extended SoS is fast, efficient and robust algorithm, which is able to cope successfully with the project-scheduling problems when we replace the traditional crisp parameters with uncertain-but-bounded parameters.

Continuous-Time Formulations for Multi-Mode Project Scheduling

This paper reviews compact continuous-time formulations for the multi-mode resource-constrained project scheduling problem. Specifically, we first point out a serious flaw in an existing start-end-event-based formulation owing to inconsistent mode choices. We propose two options to formulate the missing constraints and we consider an equivalent reformulation with sparser constraint matrix. Second, we formulate an aggregate variant of an existing model that relies on on-off-events and clarify the role of mode consistency issues in such models. Third, we suggest two variants of an existing network flow formulation. We enhance our models by adapting several techniques that have been used previously, e.g., in cases with only a single mode. A large set of benchmark instances from the literature provides the basis for an up-to-date and fair computational study with an out-of-the-box solver package. We compare our models against two models from the literature. Our experiments assert confidently that network flow formulations prevail in the test bed, and they provide a hint on why event-based models become less competitive in multi-mode settings.Comment: 37 page