Scatter search for chemical and bio-process optimization

Scatter search is a population-based method that has recently been shown to yield promising outcomes for solving combinatorial and nonlinear optimization problems. Based on formulations originally proposed in 1960s for combining decision rules and problem constraints such as the surrogate constraint method, scatter search uses strategies for combining solution vectors that have proved effective in a variety of problem settings. In this paper, we develop a general purpose heuristic for a class of nonlinear optimization problems. The procedure is based on the scatter search methodology and treats the objective function evaluation as a black box, making the search algorithm context-independent. Most optimization problems in the chemical and bio-chemical industries are highly nonlinear in either the objective function or the constraints. Moreover, they usually present differential-algebraic systems of constraints. In this type of problem, the evaluation of a solution or even the feasibility test of a set of values for the decision variables is a time-consuming operation. In this context, the solution method is limited to a reduced number of solution examinations. We have implemented a scatter search procedure in Matlab (Mathworks, 2004) for this special class of difficult optimization problems. Our development goes beyond a simple exercise of applying scatter search to this class of problems, but presents innovative mechanisms to obtain a good balance between intensification and diversification in a short-term search horizon. Computational comparisons with other recent methods over a set of benchmark problems favor the proposed procedure.     

Using tabu search to schedule activities of stochastic resource-constrained projects

In this paper, a higher level heuristic procedure “tabu search” is proposed to provide good solutions to resource-constrained, randomized activity duration project scheduling problems. Our adaptation of tabu search uses multiple tabu lists, randomized short-term memory, and multiple starting schedules as a means of search diversification. The proposed method proves to be an efficient way to find good solutions to both deterministic and stochastic problems. For the deterministic problems, most of the optimal schedules of the test projects investigated are found. Computational results are presented which establish the superiority of tabu search over the existing heuristic algorithms.     

A Multiobjective Hybrid Metaheuristic Approach for GIS-based Spatial Zoning Model

This paper presents a multiobjective hybrid metaheuristic approach for an intelligent spatial zoning model in order to draw territory line for geographical or spatial zone for the purpose of space control. The model employs a Geographic Information System (GIS) and uses multiobjective combinatorial optimization techniques as its components. The proposed hybrid metaheuristic consists of the symbiosis between tabu search and scatter search method and it is used heuristically to generate non-dominated alternatives. The approach works with a set of current solution, which through manipulation of weights are optimized towards the non-dominated frontier while at the same time, seek to disperse over the frontier by a strategic oscillation concept. The general procedure and its algorithms are given as well as its implementation in the GIS environment. The computation has resulted in tremendous improvements in spatial zoning.     

“MOSS” multiobjective scatter search applied to non-linear multiple criteria optimization

This paper introduces a multiple criteria scatter search to deal with bounded constrained non-linear continuous vector optimization problems of high dimension, applying a MultiStart Tabu Search (TS) as a diversification generation method, each TS works with its own starting point, recency memory, and aspiration threshold. Frequency memory is used to diversify the search and it is shared between the TS. A Pareto relation is applied in order to designate a subset of the best generated solutions to be reference solutions. A choice function called Kramer Choice function is used to divide the reference solutions in two subsets. The Euclidean distance is used as a measure of dissimilarity in order to find diverse solutions to be combined. Linear combinations of the reference solutions are used as a solution combination method. “Balls” in the decision space and the objective space are used to avoid duplications. Different tabu sets with different tabu tenures are employed in the scatter phase to enhance the diversity of the search. The performance of our approach is compared with Pareto-optimal frontiers and three other state-of-the-art MOEAs for a suite test problems taken from the literature.     

Path-relinking intensification methods for stochastic local search algorithms

Path-relinking is major enhancement to heuristic search methods for solving combinatorial optimization problems, leading to significant improvements in both solution quality and running times. We review its fundamentals and implementation strategies, as well as advanced hybridizations with more elaborate metaheuristic schemes such as tabu search, GRASP, genetic algorithms and scatter search. Numerical examples are discussed and algorithms compared based on their run time distributions.     

Multi-objective tabu search using a multinomial probability mass function

A tabu search approach to solve multi-objective combinatorial optimization problems is developed in this paper. This procedure selects an objective to become active for a given iteration with a multinomial probability mass function. The selection step eliminates two major problems of simple multi-objective methods, a priori weighting and scaling of objectives. Comparison of results on an NP-hard combinatorial problem with a previously published multi-objective tabu search approach and with a deterministic version of this approach shows that the multinomial approach is effective, tractable and flexible.     

Adaptive memory programming for matrix bandwidth minimization

In this paper we explore the influence of adaptive memory in the performance of heuristic methods when solving a hard combinatorial optimization problem. Specifically, we tackle the adaptation of tabu search and scatter search to the bandwidth minimization problem. It consists of finding a permutation of the rows and columns of a given matrix which keeps the non-zero elements in a band that is as close as possible to the main diagonal. This is a classic problem, introduced in the late sixties, that also has a well-known formulation in terms of graphs. Different exact and heuristic approaches have been proposed for the bandwidth problem. Our contribution consists of two new algorithms, one based on the tabu search methodology and the other based on the scatter search framework. We also present a hybrid method combining both for improved outcomes. Extensive computational testing shows the influence of the different elements in heuristic search, such as neighborhood definition, local search, combination methods and the use of memory. We compare our proposals with the most recent and advanced methods for this problem, concluding that our new methods can compete with them in speed and running time.     

An Application of a Multi-Objective Tabu Search Algorithm to a Bicriteria Flowshop Problem

This paper proposes a new tabu search algorithm for multi-objective combinatorial problems with the goal of obtaining a good approximation of the Pareto-optimal or efficient solutions. The algorithm works with several paths of solutions in parallel, each with its own tabu list, and the Pareto dominance concept is used to select solutions from the neighborhoods. In this way we obtain at each step a set of local nondominated points. The dispersion of points is achieved by a clustering procedure that groups together close points of this set and then selects the centroids of the clusters as search directions. A nice feature of this multi-objective algorithm is that it introduces only one additional parameter, namely, the number of paths. The algorithm is applied to the permutation flowshop scheduling problem in order to minimize the criteria of makespan and maximum tardiness. For instances involving two machines, the performance of the algorithm is tested against a Branch-and-Bound algorithm proposed in the literature, and for more than two machines it is compared with that of a tabu search algorithm and a genetic local search algorithm, both from the literature. Computational results show that the heuristic yields a better approximation than these algorithms.     

Protein Conformation of a Lattice Model Using Tabu Search

We apply tabu search techniques to the problem of determining the optimal configuration of a chain of protein sequences on a cubic lattice. The problem under study is difficult to solve because of the large number of possible conformations and enormous amount of computations required. Tabu search is an iterative heuristic procedure which has been shown to be a remarkably effective method for solving combinatorial optimization problems. In this paper, an algorithm is designed for the cubic lattice model using tabu search. The algorithm has been tested on a chain of 27 monomers. Computational results show that our method outperforms previously reported approaches for the same model.     

A tabu search and a genetic algorithm for solving a bicriteria general job shop scheduling problem

This paper deals with a general job shop scheduling problem with multiple constraints, coming from printing and boarding industry. The objective is the minimization of two criteria, the makespan and the maximum lateness, and we are interested in finding an approximation of the Pareto frontier. We propose a fast and elitist genetic algorithm based on NSGA-II for solving the problem. The initial population of this algorithm is either randomly generated or partially generated by using a tabu search algorithm, that minimizes a linear combination of the two criteria. Both the genetic and the tabu search algorithms are tested on benchmark instances from flexible job shop literature and computational results show the interest of both methods to obtain an efficient and effective resolution method.     

An Experimental Evaluation of a Scatter Search for the Linear Ordering Problem

Scatter search is a population-based method that has recently been shown to yield promising outcomes for solving combinatorial and nonlinear global optimization problems. Based on formulations originally proposed in the 1960s for combining decision rules and problem constraints, such as in generating surrogate constraints, scatter search uses strategies for combining solution vectors that have proved effective in a variety of problem settings. In this paper, we present a scatter search implementation designed to find high quality solutions for the NP-hard linear ordering problem, which has a significant number of applications in practice. The LOP, for example, is equivalent to the so-called triangulation problem for input-output tables in economics. Our implementation incorporates innovative mechanisms to combine solutions and to create a balance between quality and diversification in the reference set. We also use a tracking process that generates solution statistics disclosing the nature of combinations and the ranks of antecedent solutions that produced the best final solutions. Extensive computational experiments with more than 300 instances establishes the effectiveness of our procedure in relation to approaches previously identified to be best.     

A Tabu Search Algorithm for the Quadratic Assignment Problem

Tabu search approach based algorithms are among the widest applied to various combinatorial optimization problems. In this paper, we propose a new version of the tabu search algorithm for the well-known problem, the quadratic assignment problem (QAP). One of the most important features of our tabu search implementation is an efficient use of mutations applied to the best solutions found so far. We tested this approach on a number of instances from the library of the QAP instances—QAPLIB. The results obtained from the experiments show that the proposed algorithm belongs to the most efficient heuristics for the QAP. The high efficiency of this algorithm is also demonstrated by the fact that the new best known solutions were found for several QAP instances.     

Tabu Search Based Procedure for Solving the 0-1 MultiObjective Knapsack Problem: The Two Objectives Case

We consider in this paper the solving of 0-1 knapsack problems with multiple linear objectives. We present a tabu search approach to generate a good approximation of the efficient set. The heuristic scheme is included in a redu tion decision space framework. The case of two objectives is developed in this paper. TS principles viewed into the multiobjective context are discussed. According to a prospective way, several variations of the algorithm are investigate. Numerical experiments are reported and compared with available exact efficient solutions. Intuitive justifications for the observed empirical behavior of the procedure and open questions are discussed.     

Improved scatter search for the global optimization of computationally expensive dynamic models

A new algorithm for global optimization of costly nonlinear continuous problems is presented in this paper. The algorithm is based on the scatter search metaheuristic, which has recently proved to be efficient for solving combinatorial and nonlinear optimization problems. A kriging-based prediction method has been coupled to the main optimization routine in order to discard the evaluation of solutions that are not likely to provide high quality function values. This makes the algorithm suitable for the optimization of computationally costly problems, as is illustrated in its application to two benchmark problems and its comparison with other algorithms.     

An effective heuristic for minimising makespan on unrelated parallel machines

Scheduling independent tasks on unrelated machines is a relatively difficult and challenging problem. In this paper, we develop a tabu search based heuristic for minimising makespan for the above problem that can provide good quality solutions for practical size problems within a reasonable amount of computational time. Our adaptation of this tabu search uses hashing to control the tabu restrictions of the search process and requires fewer critical parameters than many of the common tabu search approaches employed for combinatorial optimisation. Hashing is simple to implement and very effective in providing a near-optimal solution. Computational results are presented to demonstrate the effectiveness of the proposed heuristic.     

An introduction to population approaches for optimization and hierarchical objective functions: A discussion on the role of tabu search

Population approaches suitable for global combinatorial optimization are discussed in this paper. They are composed of a number of distinguishable individuals called "agents", each one using a particular optimization strategy. Periods of independent search follow phases on which the population is restarted from new configurations. Due to its intrinsic parallelism and the asynchronicity of the method, it is particularly suitable for parallel computers. Results on two test problems are presented in this paper. The individual search optimization strategies for each agent have been chosen having the basic characteristics of tabu search. This has been done in order to avoid mixing the hypothesized properties of these population approaches with those of more elaborate tabu search strategies, but remarking on its main characteristics. A set of four test problem "landscapes" is discussed and their use to improve and benchmark the results by using tabu search as the individual optimization strategy within a population heuristic is suggested and explored. The application of tabu search to new problem areas, like molecular biology, is also investigated.     

Heuristic algorithms for the cardinality constrained efficient frontier

This paper examines the application of genetic algorithm, tabu search and simulated annealing metaheuristic approaches to finding the cardinality constrained efficient frontier that arises in financial portfolio optimisation. We consider the mean-variance model of Markowitz as extended to include the discrete restrictions of buy-in thresholds and cardinality constraints. Computational results are reported for publicly available data sets drawn from seven major market indices involving up to 1318 assets. Our results are compared with previous results given in the literature illustrating the effectiveness of the proposed metaheuristics in terms of solution quality and computation time.     

Principles of scatter search

Scatter search is an evolutionary method that has been successfully applied to hard optimization problems. The fundamental concepts and principles of the method were first proposed in the 1970s, based on formulations dating back to the 1960s for combining decision rules and problem constraints. In contrast to other evolutionary methods like genetic algorithms, scatter search is founded on the premise that systematic designs and methods for creating new solutions afford significant benefits beyond those derived from recourse to randomization. It uses strategies for search diversification and intensification that have proved effective in a variety of optimization problems.This paper provides the main principles and ideas of scatter search and its generalized form path relinking. We first describe a basic design to give the reader the tools to create relatively simple implementations. More advanced designs derive from the fact that scatter search and path relinking are also intimately related to the tabu search (TS) metaheuristic, and gain additional advantage by making use of TS adaptive memory and associated memory-exploiting mechanisms capable of being tailored to particular contexts. These and other advanced processes described in the paper facilitate the creation of sophisticated implementations for hard problems that often arise in practical settings. Due to their flexibility and proven effectiveness, scatter search and path relinking can be successfully adapted to tackle optimization problems spanning a wide range of applications and a diverse collection of structures, as shown in the papers of this volume.     

Multistart Tabu Search Strategies for the Unconstrained Binary Quadratic Optimization Problem

This paper describes and experimentally compares five rather different multistart tabu search strategies for the unconstrained binary quadratic optimization problem: a random restart procedure, an application of a deterministic heuristic to specially constructed subproblems, an application of a randomized procedure to the full problem, a constructive procedure using tabu search adaptive memory, and an approach based on solving perturbed problems. In the solution improvement phase a modification of a standard tabu search implementation is used. A computational trick applied to this modification – mapping of the current solution to the zero vector – allowed to significantly reduce the time complexity of the search. Computational results are provided for the 25 largest problem instances from the OR-Library and, in addition, for the 18 randomly generated larger and more dense problems. For 9 instances from the OR-Library new best solutions were found.     

Constraint-based search for optimal Golomb rulers

Finding optimal Golomb rulers is an extremely challenging combinatorial problem. The distance between each pair of mark is unique in a Golomb ruler. For a given number of marks, an optimal Golomb ruler has the minimum length. Golomb rulers are used in application areas such as X-ray crystallography, radio astronomy, information theory, and pulse phase modulation. The most recent optimal Golomb ruler search algorithm hybridises a range of techniques such as greedy randomised adaptive search, scatter search, tabu search, clustering techniques, and constraint programming, and obtains optimal Golomb rulers of up to 16 marks with very low success rates. In this paper, we provide tight upper bounds for Golomb ruler marks and present heuristic-based effective domain reduction techniques. Using these along with tabu and configuration checking meta-heuristics, we then develop a constraint-based multi-point local search algorithm to perform a satisfaction search for optimal Golomb rulers of specified length. We then present an algorithm to perform an optimisation search that minimises the length using the satisfaction search repeatedly. Our satisfaction search finds optimal Golomb rulers of up to 19 marks while the optimisation search finds up to 17 marks.