Un nou criteri per optimitzar la resolució de problemes matemàtics

Els problemes de programació matemàtica intenten resoldre processos que tenen diferents possibles solucions, però només una d'elles és l'òptima, la que s'ajusta més a unes condicions prestablertes pel mateix enunciat del problema. Els procediments clàssics permeten trobar solucions òptimes en el cas de problemes convexos i no poden assegurar-ho en cap altre cas. Ara bé, si la convexitat és present en alguna forma, per exemple, quan la funció objectiu pot expressar-se com a diferència de funcions convexes, aleshores es poden descriure nous procediments que permeten calcular solucions òptimes. Un nou ús de la convexitat s'ha près com a eix central en la discriminació de les solucions en el present estudi, per millorar l'eficiència en l'obtenció de les solucions òptimes.Los problemas de programación matemática intentan resolver procesos que tienen diferentes posibles soluciones, pero sólo una de ellas es la óptima, la que se ajusta más a unas condiciones preestablecidas por el mismo enunciado del problema. Los procedimientos clásicos permiten encontrar soluciones óptimas en el caso de problemas convexos y no pueden aseguralo en ningún otro caso. No obstante, si la convexidad está presente en alguna forma, por ejemplo, cuando la función objetivo puede expresarse como una diferencia de funciones convexas, entonces se pueden describir nuevos procedimientos que permiten calcular soluciones óptimas. Un nuevo uso de la convexidad se ha tomado como eje central en la discriminación de las soluciones en el presente estudio, mejorando así la eficiencia en la obtención de las soluciones óptimas.Mathematical programming problems try to solve processes which have different solutions by finding the optimal solution, the one that best fits the pre-established conditions of the problem. The classic procedures work towards finding an optimal solution in the case of convex problems, but cannot guarantee it in any other type of problem. However, if the problem involves some kind of convexity, as for example when the objective function can be expressed as a difference in convex functions, then new procedures making it possible to calculate optimal solutions can be described. A new use of convexity was taken as the central axis in the discrimination of solutions in this study, with the aim of improving the efficiency in obtaining optimal solutions

On the use of biased-randomized algorithms for solving non-smooth optimization problems

Soft constraints are quite common in real-life applications. For example, in freight transportation, the fleet size can be enlarged by outsourcing part of the distribution service and some deliveries to customers can be postponed as well; in inventory management, it is possible to consider stock-outs generated by unexpected demands; and in manufacturing processes and project management, it is frequent that some deadlines cannot be met due to delays in critical steps of the supply chain. However, capacity-, size-, and time-related limitations are included in many optimization problems as hard constraints, while it would be usually more realistic to consider them as soft ones, i.e., they can be violated to some extent by incurring a penalty cost. Most of the times, this penalty cost will be nonlinear and even noncontinuous, which might transform the objective function into a non-smooth one. Despite its many practical applications, non-smooth optimization problems are quite challenging, especially when the underlying optimization problem is NP-hard in nature. In this paper, we propose the use of biased-randomized algorithms as an effective methodology to cope with NP-hard and non-smooth optimization problems in many practical applications. Biased-randomized algorithms extend constructive heuristics by introducing a nonuniform randomization pattern into them. Hence, they can be used to explore promising areas of the solution space without the limitations of gradient-based approaches, which assume the existence of smooth objective functions. Moreover, biased-randomized algorithms can be easily parallelized, thus employing short computing times while exploring a large number of promising regions. This paper discusses these concepts in detail, reviews existing work in different application areas, and highlights current trends and open research lines

A parameter-free approach for solving combinatorial optimization problems through biased randomization of efficient heuristics

This paper discusses the use of probabilistic or randomized algorithms for solving combinatorial optimization problems. Our approach employs non-uniform probability distributions to add a biased random behavior to classical heuristics so a large set of alternative good solutions can be quickly obtained in a natural way and without complex con guration processes. This procedure is especially useful in problems where properties such as non-smoothness or non-convexity lead to a highly irregular solution space, for which the traditional optimization methods, both of exact and approximate nature, may fail to reach their full potential. The results obtained are promising enough to suggest that randomizing classical heuristics is a powerful method that can be successfully applied in a variety of cases

A BRILS metaheuristic for non-smooth flow-shop problems with failure-risk costs

This paper analyzes a realistic variant of the Permutation Flow-Shop Problem (PFSP) by considering a non-smooth objective function that takes into account not only the traditional makespan cost but also failure-risk costs due to uninterrupted operation of machines. After completing a literature review on the issue, the paper formulates an original mathematical model to describe this new PFSP variant. Then, a Biased-Randomized Iterated Local Search (BRILS) algorithm is proposed as an efficient solving approach. An oriented (biased) random behavior is introduced in the well-known NEH heuristic to generate an initial solution. From this initial solution, the algorithm is able to generate a large number of alternative good solutions without requiring a complex setting of parameters. The relative simplicity of our approach is particularly useful in the presence of non-smooth objective functions, for which exact optimization methods may fail to reach their full potential. The gains of considering failure-risk costs during the exploration of the solution space are analyzed throughout a series of computational experiments. To promote reproducibility, these experiments are based on a set of traditional benchmark instances. Moreover, the performance of the proposed algorithm is compared against other state-of-the-art metaheuristic approaches, which have been conveniently adapted to consider failure-risk costs during the solving process. The proposed BRILS approach can be easily extended to other combinatorial optimization problems with similar non-smooth objective functions

Applications of discrete-event simulation to reliability and availability assessment in civil engineering structures

This paper discusses the convenience of predicting, quantitatively, time-dependent reliability and availability levels associated with most building or civil engineering structures. Then, the paper reviews different approaches to these problems and proposes the use of discrete-event simulation as the most realistic way to deal with them, specially during the design stage. The paper also reviews previous work on the use of both Monte Carlo simulation and discrete-event simulation in this area and shows how discrete-event simulation, in particular, could be employed to solve uncertainty in time-dependent structural reliability problems. Finally, a case study is developed to illustrate some of the concepts previously covered in the paper

A parameter-free approach for solving combinatorial optimization problems through biased randomization of efficient heuristics

This paper discusses the use of probabilistic or randomized algorithms for solving combinatorial optimization problems. Our approach employs non-uniform probability distributions to add a biased random behavior to classical heuristics so a large set of alternative good solutions can be quickly obtained in a natural way and without complex con guration processes. This procedure is especially useful in problems where properties such as non-smoothness or non-convexity lead to a highly irregular solution space, for which the traditional optimization methods, both of exact and approximate nature, may fail to reach their full potential. The results obtained are promising enough to suggest that randomizing classical heuristics is a powerful method that can be successfully applied in a variety of cases

Applications of discrete-event simulation to reliability and availability assessment in civil engineering structures

This paper discusses the convenience of predicting, quantitatively, time-dependent reliability and availability levels associated with most building or civil engineering structures. Then, the paper reviews different approaches to these problems and proposes the use of discrete-event simulation as the most realistic way to deal with them, specially during the design stage. The paper also reviews previous work on the use of both Monte Carlo simulation and discrete-event simulation in this area and shows how discrete-event simulation, in particular, could be employed to solve uncertainty in time-dependent structural reliability problems. Finally, a case study is developed to illustrate some of the concepts previously covered in the paper

Applications of discrete-event simulation to reliability and availability assessment in civil engineering structures

This paper discusses the convenience of predicting, quantitatively, time-dependent reliability and availability levels associated with most building or civil engineering structures. Then, the paper reviews different approaches to these problems and proposes the use of discrete-event simulation as the most realistic way to deal with them, specially during the design stage. The paper also reviews previous work on the use of both Monte Carlo simulation and discrete-event simulation in this area and shows how discrete-event simulation, in particular, could be employed to solve uncertainty in time-dependent structural reliability problems. Finally, a case study is developed to illustrate some of the concepts previously covered in the paper

On the use of biased-randomized algorithms for solving non-smooth optimization problems

Soft constraints are quite common in real-life applications. For example, in freight transportation, the fleet size can be enlarged by outsourcing part of the distribution service and some deliveries to customers can be postponed as well; in inventory management, it is possible to consider stock-outs generated by unexpected demands; and in manufacturing processes and project management, it is frequent that some deadlines cannot be met due to delays in critical steps of the supply chain. However, capacity-, size-, and time-related limitations are included in many optimization problems as hard constraints, while it would be usually more realistic to consider them as soft ones, i.e., they can be violated to some extent by incurring a penalty cost. Most of the times, this penalty cost will be nonlinear and even noncontinuous, which might transform the objective function into a non-smooth one. Despite its many practical applications, non-smooth optimization problems are quite challenging, especially when the underlying optimization problem is NP-hard in nature. In this paper, we propose the use of biased-randomized algorithms as an effective methodology to cope with NP-hard and non-smooth optimization problems in many practical applications. Biased-randomized algorithms extend constructive heuristics by introducing a nonuniform randomization pattern into them. Hence, they can be used to explore promising areas of the solution space without the limitations of gradient-based approaches, which assume the existence of smooth objective functions. Moreover, biased-randomized algorithms can be easily parallelized, thus employing short computing times while exploring a large number of promising regions. This paper discusses these concepts in detail, reviews existing work in different application areas, and highlights current trends and open research lines