Combinatorial optimization and metaheuristics

Today, combinatorial optimization is one of the youngest and most active areas of discrete mathematics. It is a branch of optimization in applied mathematics and computer science, related to operational research, algorithm theory and computational complexity theory. It sits at the intersection of several fields, including artificial intelligence, mathematics and software engineering. Its increasing interest arises for the fact that a large number of scientific and industrial problems can be formulated as abstract combinatorial optimization problems, through graphs and/or (integer) linear programs. Some of these problems have polynomial-time (“efficient”) algorithms, while most of them are NP-hard, i.e. it is not proved that they can be solved in polynomial-time. Mainly, it means that it is not possible to guarantee that an exact solution to the problem can be found and one has to settle for an approximate solution with known performance guarantees. Indeed, the goal of approximate methods is to find “quickly” (reasonable run-times), with “high” probability, provable “good” solutions (low error from the real optimal solution). In the last 20 years, a new kind of algorithm commonly called metaheuristics have emerged in this class, which basically try to combine heuristics in high level frameworks aimed at efficiently and effectively exploring the search space. This report briefly outlines the components, concepts, advantages and disadvantages of different metaheuristic approaches from a conceptual point of view, in order to analyze their similarities and differences. The two very significant forces of intensification and diversification, that mainly determine the behavior of a metaheuristic, will be pointed out. The report concludes by exploring the importance of hybridization and integration methods

Common Due-Date Problem: Exact Polynomial Algorithms for a Given Job Sequence

This paper considers the problem of scheduling jobs on single and parallel machines where all the jobs possess different processing times but a common due date. There is a penalty involved with each job if it is processed earlier or later than the due date. The objective of the problem is to find the assignment of jobs to machines, the processing sequence of jobs and the time at which they are processed, which minimizes the total penalty incurred due to tardiness or earliness of the jobs. This work presents exact polynomial algorithms for optimizing a given job sequence or single and parallel machines with the run-time complexities of $O(n \log n)$ and $O(mn^2 \log n)$ respectively, where $n$ is the number of jobs and $m$ the number of machines. The algorithms take a sequence consisting of all the jobs $(J_i, i=1,2,\dots,n)$ as input and distribute the jobs to machines (for $m>1$) along with their best completion times so as to get the least possible total penalty for this sequence. We prove the optimality for the single machine case and the runtime complexities of both. Henceforth, we present the results for the benchmark instances and compare with previous work for single and parallel machine cases, up to $200$ jobs.Comment: 15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computin

A Novel Approach to the Common Due-Date Problem on Single and Parallel Machines

This paper presents a novel idea for the general case of the Common Due-Date (CDD) scheduling problem. The problem is about scheduling a certain number of jobs on a single or parallel machines where all the jobs possess different processing times but a common due-date. The objective of the problem is to minimize the total penalty incurred due to earliness or tardiness of the job completions. This work presents exact polynomial algorithms for optimizing a given job sequence for single and identical parallel machines with the run-time complexities of $O(n \log n)$ for both cases, where $n$ is the number of jobs. Besides, we show that our approach for the parallel machine case is also suitable for non-identical parallel machines. We prove the optimality for the single machine case and the runtime complexities of both. Henceforth, we extend our approach to one particular dynamic case of the CDD and conclude the chapter with our results for the benchmark instances provided in the OR-library.Comment: Book Chapter 22 page

A review of lot streaming in a flow shop environment with makespan criteria

[EN] Purpose: This paper reviews current literature and contributes a set of findings that capture the current state-of-the-art of the topic of lot streaming in a flow-shop. Design/methodology/approach: A literature review to capture, classify and summarize the main body of knowledge on lot streaming in a flow-shop with makespan criteria and, translate this into a form that is readily accessible to researchers and practitioners in the more mainstream production scheduling community. Findings: The existing knowledge base is somewhat fragmented. This is a relatively unexplored topic within mainstream operations management research and one which could provide rich opportunities for further exploration. Originality/value: This paper sets out to review current literature, from an advanced production scheduling perspective, and contributes a set of findings that capture the current state-of-the-art of this topic.This work has been carried out as part of the project “Programación de la Producción con Partición Ajustable de Lotes en entornos de Planificación mixta Pedido/Stock (PP-PAL-PPS)”, ref. GVA/2013/034 funded by Consellería de Educación, Cultura y Deportes de la Generalitat Valenciana.Gómez-Gasquet, P.; Segura Andrés, R.; Andrés Romano, C. (2013). A review of lot streaming in a flow shop environment with makespan criteria. Journal of Industrial Engineering and Management. 6(3):761-770. https://doi.org/10.3926/jiem.553S7617706

Modelling and Scheduling Lot Streaming Flexible Flow Lines

Although lot streaming scheduling is an active research field, lot streaming flexible flow lines problems have received far less attention than classical flow shops. This paper deals with scheduling jobs in lot streaming flexible flow line problems. The paper mathematically formulates the problem by a mixed integer linear programming model. This model solves small instances to optimality. Moreover, a novel artificial bee colony optimization is developed. This algorithm utilizes five effective mechanisms to solve the problem. To evaluate the algorithm, it is compared with adaptation of four available algorithms. The statistical analyses showed that the proposed algorithm significantly outperformed the other tested algorithms

Offline Learning for Sequence-based Selection Hyper-heuristics

This thesis is concerned with finding solutions to discrete NP-hard problems. Such problems occur in a wide range of real-world applications, such as bin packing, industrial flow shop problems, determining Boolean satisfiability, the traveling salesman and vehicle routing problems, course timetabling, personnel scheduling, and the optimisation of water distribution networks. They are typically represented as optimisation problems where the goal is to find a ``best'' solution from a given space of feasible solutions. As no known polynomial-time algorithmic solution exists for NP-hard problems, they are usually solved by applying heuristic methods. Selection hyper-heuristics are algorithms that organise and combine a number of individual low level heuristics into a higher level framework with the objective of improving optimisation performance. Many selection hyper-heuristics employ learning algorithms in order to enhance optimisation performance by improving the selection of single heuristics, and this learning may be classified as either online or offline. This thesis presents a novel statistical framework for the offline learning of subsequences of low level heuristics in order to improve the optimisation performance of sequenced-based selection hyper-heuristics. A selection hyper-heuristic is used to optimise the HyFlex set of discrete benchmark problems. The resulting sequences of low level heuristic selections and objective function values are used to generate an offline learning database of heuristic selections. The sequences in the database are broken down into subsequences and the mathematical concept of a logarithmic return is used to discriminate between ``effective'' subsequences, that tend to lead to improvements in optimisation performance, and ``disruptive'' subsequences that tend to lead to worsening performance. Effective subsequences are used to improve hyper-heuristics performance directly, by embedding them in a simple hyper-heuristic design, and indirectly as the inputs to an appropriate hyper-heuristic learning algorithm. Furthermore, by comparing effective subsequences across different problem domains it is possible to investigate the potential for cross-domain learning. The results presented here demonstrates that the use of well chosen subsequences of heuristics can lead to small, but statistically significant, improvements in optimisation performance

A Digital Twin Framework for Production Planning Optimization: Applications for Make-To-Order Manufacturers

In this dissertation, we develop a Digital Twin framework for manufacturing systems and apply it to various production planning and scheduling problems faced by Make-To-Order (MTO) firms. While this framework can be used to digitally represent a particular manufacturing environment with high fidelity, our focus is in using it to generate realistic settings to test production planning and scheduling algorithms in practice. These algorithms have traditionally been tested by either translating a practical situation into the necessary modeling constructs, without discussion of the assumptions and inaccuracies underlying this translation, or by generating random instances of the modeling constructs, without assessing the limitations in accurately representing production environments. The consequence has been a serious gap between theory advancement and industry practice. The major goal of this dissertation is to develop a framework that allows for practical testing, evaluation, and implementation of new approaches for seamless industry adoption. We develop this framework as a modular software package and emphasize the practicality and configurability of the framework, such that minimal modelling effort is required to apply the framework to a multitude of optimization problems and manufacturing systems. Throughout this dissertation, we emphasize the importance of the underlying scheduling problems which provide the basis for additional operational decision making. We focus on the computational evaluation and comparisons of various modeling choices within the developed frameworks, with the objective of identifying models which are both effective and computationally efficient. In Part 1 of this dissertation, we consider a class of Production Planning and Execution problems faced by job shop manufacturing systems. In Part 2 of this dissertation, we consider a class of scheduling problems faced by manufacturers whose production system is dominated by a single operation

Hyper-heuristic decision tree induction

A hyper-heuristic is any algorithm that searches or operates in the space of heuristics as opposed to the space of solutions. Hyper-heuristics are increasingly used in function and combinatorial optimization. Rather than attempt to solve a problem using a fixed heuristic, a hyper-heuristic approach attempts to find a combination of heuristics that solve a problem (and in turn may be directly suitable for a class of problem instances). Hyper-heuristics have been little explored in data mining. This work presents novel hyper-heuristic approaches to data mining, by searching a space of attribute selection criteria for decision tree building algorithm. The search is conducted by a genetic algorithm. The result of the hyper-heuristic search in this case is a strategy for selecting attributes while building decision trees. Most hyper-heuristics work by trying to adapt the heuristic to the state of the problem being solved. Our hyper-heuristic is no different. It employs a strategy for adapting the heuristic used to build decision tree nodes according to some set of features of the training set it is working on. We introduce, explore and evaluate five different ways in which this problem state can be represented for a hyper-heuristic that operates within a decisiontree building algorithm. In each case, the hyper-heuristic is guided by a rule set that tries to map features of the data set to be split by the decision tree building algorithm to a heuristic to be used for splitting the same data set. We also explore and evaluate three different sets of low-level heuristics that could be employed by such a hyper-heuristic. This work also makes a distinction between specialist hyper-heuristics and generalist hyper-heuristics. The main difference between these two hyperheuristcs is the number of training sets used by the hyper-heuristic genetic algorithm. Specialist hyper-heuristics are created using a single data set from a particular domain for evolving the hyper-heurisic rule set. Such algorithms are expected to outperform standard algorithms on the kind of data set used by the hyper-heuristic genetic algorithm. Generalist hyper-heuristics are trained on multiple data sets from different domains and are expected to deliver a robust and competitive performance over these data sets when compared to standard algorithms. We evaluate both approaches for each kind of hyper-heuristic presented in this thesis. We use both real data sets as well as synthetic data sets. Our results suggest that none of the hyper-heuristics presented in this work are suited for specialization – in most cases, the hyper-heuristic’s performance on the data set it was specialized for was not significantly better than that of the best performing standard algorithm. On the other hand, the generalist hyper-heuristics delivered results that were very competitive to the best standard methods. In some cases we even achieved a significantly better overall performance than all of the standard methods