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

Enhancing Parallel Cooperative Trajectory Based Metaheuristics with Path Relinking

This paper proposes a novel algorithm combining path relinking with a set of cooperating trajectory based parallel algorithms to yield a new metaheuristic of enhanced search features. Algorithms based on the exploration of the neighborhood of a single solution, like simulated annealing (SA), have offered accurate results for a large number of real-world problems in the past. Because of their trajectory based nature, some advanced models such as the cooperative one are competitive in academic problems, but still show many limitations in addressing large scale instances. In addition, the field of parallel models for trajectory methods has not deeply been studied yet (at least in comparison with parallel population based models). In this work, we propose a new hybrid algorithm which improves cooperative single solution techniques by using path relinking, allowing both to reduce the global execution time and to improve the efficacy of the method. We test here this new model using a large benchmark of instances of two well-known NP-hard problems: MAXSAT and QAP, with competitive results.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Exact and heuristic procedures for the Heijunka-flow shop scheduling problem with minimum makespan and job replicas

In this paper, a new problem of job sequences in a workshop is presented, taking into account non-unit demands for the jobs and whose objective is to minimize the total completion time for all the jobs (Cmax) satisfying a set of restrictions imposed on the problem to preserve the production mix. Two procedures are proposed to solve the new problem: Mixed Integer Linear Programming and a Metaheuristic based on Multistart and Local Search. The two proposed procedures are tested using instance set Nissan-9Eng.I, in both cases giving rise to highly satisfactory performance both in quality of solutions obtained and in the CPU times required. Through a case study of the Nissan engine manufacturing plant in Barcelona, our economic-productive analysis reveals that it is possible to save an average of € 1162.83 per day, manufacturing 270 engines, when we transform the current assembly line into a Heijunka-Flow Shop.Peer ReviewedPostprint (published version

Feature-based tuning of simulated annealing applied to the curriculum-based course timetabling problem

We consider the university course timetabling problem, which is one of the most studied problems in educational timetabling. In particular, we focus our attention on the formulation known as the curriculum-based course timetabling problem, which has been tackled by many researchers and for which there are many available benchmarks. The contribution of this paper is twofold. First, we propose an effective and robust single-stage simulated annealing method for solving the problem. Secondly, we design and apply an extensive and statistically-principled methodology for the parameter tuning procedure. The outcome of this analysis is a methodology for modeling the relationship between search method parameters and instance features that allows us to set the parameters for unseen instances on the basis of a simple inspection of the instance itself. Using this methodology, our algorithm, despite its apparent simplicity, has been able to achieve high quality results on a set of popular benchmarks. A final contribution of the paper is a novel set of real-world instances, which could be used as a benchmark for future comparison

Tuning a Simulated Annealing metaheuristic for cross-domain search

Simulated Annealing is a well known local search metaheuristic used for solving computationally hard optimization problems. Cross-domain search poses a higher level issue where a single solution method is used with minor, preferably no modification for solving characteristically different optimisation problems. The performance of a metaheuristic is often dependant on its initial parameter settings, hence detecting the best configuration, i.e. parameter tuning is crucial, which becomes a further challenge for cross-domain search. In this paper, we investigate the cross-domain search performance of Simulated Annealing via tuning for solving six problems, ranging from personnel scheduling to vehicle routing under a stochastic local search framework. The empirical results show that Simulated Annealing is extremely sensitive to the initial parameter settings leading to sub-standard performance when used as a single solution method for cross-domain search. Moreover, we demonstrate that cross-domain parameter tuning is inferior to domain-level tuning highlighting the requirements for adaptive parameter configurations when dealing with cross-domain search

Hybrid Ant Colony Optimization For Two Satisfiability Programming In Hopfield Neural Network

The representation of 2 Satisfiability problem or 2SAT is increasingly viewed as a significant logical rule in order to synthesize many real life applications. Although there were many researchers proposed the solution of 2SAT, little attention has been paid to the significance of the 2SAT logical rule itself. It can be hypothesized that 2SAT property can be used as a logical rule in the intelligent system. To verify this claim, 2 Satisfiability logic programming was embedded to Hopfield neural network (HNN) as a single unit. Learning in HNN will be inspired by Wan Abdullah method since the conventional Hebbian learning is inefficient when dealing with large number of constraints. As the number of 2SAT clauses increased, the efficiency and effectiveness of the learning phase in HNN deteriorates. Swarm intelligence metaheuristic algorithm has been introduced to reduce the learning complexity of the network. The newly proposed metaheuristic algorithm was enhanced ant colony optimization (ACO) algorithm

Frequency Fitness Assignment: Optimization without Bias for Good Solutions can be Efficient

A fitness assignment process transforms the features (such as the objective value) of a candidate solution to a scalar fitness, which then is the basis for selection. Under Frequency Fitness Assignment (FFA), the fitness corresponding to an objective value is its encounter frequency in selection steps and is subject to minimization. FFA creates algorithms that are not biased towards better solutions and are invariant under all injective transformations of the objective function value. We investigate the impact of FFA on the performance of two theory-inspired, state-of-the-art EAs, the Greedy (2+1) GA and the Self-Adjusting (1+(lambda,lambda)) GA. FFA improves their performance significantly on some problems that are hard for them. In our experiments, one FFA-based algorithm exhibited mean runtimes that appear to be polynomial on the theory-based benchmark problems in our study, including traps, jumps, and plateaus. We propose two hybrid approaches that use both direct and FFA-based optimization and find that they perform well. All FFA-based algorithms also perform better on satisfiability problems than any of the pure algorithm variants

Robust Artificial Immune System in the Hopfield network for Maximum k-Satisfiability

Artificial Immune System (AIS) algorithm is a novel and vibrant computational paradigm, enthused by the biological immune system. Over the last few years, the artificial immune system has been sprouting to solve numerous computational and combinatorial optimization problems. In this paper, we introduce the restricted MAX-kSAT as a constraint optimization problem that can be solved by a robust computational technique. Hence, we will implement the artificial immune system algorithm incorporated with the Hopfield neural network to solve the restricted MAX-kSAT problem. The proposed paradigm will be compared with the traditional method, Brute force search algorithm integrated with Hopfield neural network. The results demonstrate that the artificial immune system integrated with Hopfield network outperforms the conventional Hopfield network in solving restricted MAX-kSAT. All in all, the result has provided a concrete evidence of the effectiveness of our proposed paradigm to be applied in other constraint optimization problem. The work presented here has many profound implications for future studies to counter the variety of satisfiability problem

Incomplete MaxSAT approaches for combinatorial testing

We present a Satisfiability (SAT)-based approach for building Mixed Covering Arrays with Constraints of minimum length, referred to as the Covering Array Number problem. This problem is central in Combinatorial Testing for the detection of system failures. In particular, we show how to apply Maximum Satisfiability (MaxSAT) technology by describing efficient encodings for different classes of complete and incomplete MaxSAT solvers to compute optimal and suboptimal solutions, respectively. Similarly, we show how to solve through MaxSAT technology a closely related problem, the Tuple Number problem, which we extend to incorporate constraints. For this problem, we additionally provide a new MaxSAT-based incomplete algorithm. The extensive experimental evaluation we carry out on the available Mixed Covering Arrays with Constraints benchmarks and the comparison with state-of-the-art tools confirm the good performance of our approaches.We would like to thank specially Akihisa Yamada for the access to several benchmarks for our experiments and for solving some questions about his previous work on Combinatorial Testing with Constraints. This work was partially supported by Grant PID2019-109137GB-C21 funded by MCIN/AEI/10.13039/501100011033, PANDEMIES 2020 by Agencia de Gestio d’Ajuts Universitaris i de Recerca (AGAUR), Departament d’Empresa i Coneixement de la Generalitat de Catalunya; FONDO SUPERA COVID-19 funded by Crue-CSIC-SANTANDER, ISINC (PID2019-111544GB-C21), and the MICNN FPU fellowship (FPU18/02929)