A new heuristic algorithm solving the linear ordering problem

The linear ordering problem is an NP-hard combinatorial problem with a large number of applications. Contrary to another very popular problem from the same category, the traveling salesman problem, relatively little space in the literature has been devoted to the linear ordering problem so far. This is particularly true for the question of developing good heuristic algorithms solving this problem.In the paper we propose a new heuristic algorithm solving the linear ordering problem. In this algorithm we made use of the sorting through insertion pattern as well as of the operation of permutation reversal. The surprisingly positive effect of the reversal operation, justified in part theoretically and confirmed in computational examples, seems to be the result of a unique property of the problem, called in the paper the symmetry of the linear ordering problem. This property consists in the fact that if a given permutation is an optimal solution of the problem with the criterion function being maximized, then the reversed permutation is a solution of the problem with the same criterion function being minimized.     

Metastrategy simulated annealing and tabu search algorithms for the vehicle routing problem

The vehicle routing problem (VRP) under capacity and distance restrictions involves the design of a set of minimum cost delivery routes, originating and terminating at a central depot, which services a set of customers. Each customer must be supplied exactly once by one vehicle route. The total demand of any vehicle must not exceed the vehicle capacity. The total length of any route must not exceed a pre-specified bound. Approximate methods based on descent, hybrid simulated annealing/tabu search, and tabu search algorithms are developed and different search strategies are investigated. A special data structure for the tabu search algorithm is implemented which has reduced notably the computational time by more than 50%. An estimate for the tabu list size is statistically derived. Computational results are reported on a sample of seventeen bench-mark test problems from the literature and nine randomly generated problems. The new methods improve significantly both the number of vehicles used and the total distances travelled on all results reported in the literature.     

Lagrangian relaxation guided problem space search heuristics for generalized assignment problems

We develop and test a heuristic based on Lagrangian relaxation and problem space search to solve the generalized assignment problem (GAP). The heuristic combines the iterative search capability of subgradient optimization used to solve the Lagrangian relaxation of the GAP formulation and the perturbation scheme of problem space search to obtain high-quality solutions to the GAP. We test the heuristic using different upper bound generation routines developed within the overall mechanism. Using the existing problem data sets of various levels of difficulty and sizes, including the challenging largest instances, we observe that the heuristic with a specific version of the upper bound routine works well on most of the benchmark instances known and provides high-quality solutions quickly. An advantage of the approach is its generic nature, simplicity, and implementation flexibility.     

On the numerical solution of the linear complementarity problem

The well-known linear complementarity problem with definite matrices is considered. It is proposed to solve it using a global optimization algorithm in which one of the basic stages is a special local search. The proposed global search algorithm is tested using a variety of randomly generated problems; a detailed analysis of the computational experiment is given.     

Violator spaces: Structure and algorithms

Sharir and Welzl introduced an abstract framework for optimization problems, called LP-type problems or also generalized linear programming problems, which proved useful in algorithm design. We define a new, and as we believe, simpler and more natural framework: violator spaces, which constitute a proper generalization of LP-type problems. We show that Clarkson's randomized algorithms for low-dimensional linear programming work in the context of violator spaces. For example, in this way we obtain the fastest known algorithm for the P-matrix generalized linear complementarity problem with a constant number of blocks. We also give two new characterizations of LP-type problems: they are equivalent to acyclic violator spaces, as well as to concrete LP-type problems (informally, the constraints in a concrete LP-type problem are subsets of a linearly ordered ground set, and the value of a set of constraints is the minimum of its intersection).     

Finite line-transitive linear spaces: Theory and search strategies

The paper summarises existing theory and classifications for finite line-transitive linear spaces, develops the theory further, and organises it in a way that enables its effective application. The starting point is a theorem of Camina and the fifth author that identifies three kinds of line-transitive automorphism groups of linear spaces. In two of these cases the group may be imprimitive on points, that is, the group leaves invariant a nontrivial partition of the point set. In the first of these cases the group is almost simple with point-transitive simple socle, and may or may not be point-primitive, while in the second case the group has a non-trivial point-intransitive normal subgroup and hence is definitely point-imprimitive. The theory presented here focuses on point-imprimitive groups. As a non-trivial application a classification is given of the point-imprimitive, line-transitive groups, and the corresponding linear spaces, for which the greatest common divisor gcd（k, v - 1） ≤ 8, where v is the number of points, and k is the line size. Motivation for this classification comes from a result of Weidong Fang and Huffing Li in 1993, that there are only finitely many non-trivial point-imprimitive, linetransitive linear spaces for a given value of gcd（k, v - 1）. The classification strengthens the classification by Camina and Mischke under the much stronger restriction k ≤ 8： no additional examples arise. The paper provides the backbone for future computer-based classifications of point-imprimitive, line- transitive linear spaces with small parameters. Several suggestions for further investigations are made.     

An integer linear programming approach and a hybrid variable neighborhood search for the car sequencing problem

In this paper we present two major approaches to solve the car sequencing problem, in which the goal is to find an optimal arrangement of commissioned vehicles along a production line with respect to constraints of the form “no more than l_ccars are allowed to require a component c in any subsequence of m_cconsecutive cars”. The first method is an exact one based on integer linear programming (ILP). The second approach is hybrid: it uses ILP techniques within a general variable neighborhood search (VNS) framework for examining large neighborhoods. We tested the two methods on benchmark instances provided by CSP_LIB and the automobile manufacturer RENAULT for the ROADEF Challenge 2005. These tests reveal that our approaches are competitive to previous reported algorithms. For the CSP_LIB instances we were able to shorten the required computation time for reaching and proving optimality. Furthermore, we were able to obtain tight bounds on some of the ROADEF instances. For two of these instances the proposed ILP-method could provide new optimality proofs for already known solutions. For the VNS, the individual contributions of the used neighborhoods are also experimentally analyzed. Results highlight the significant impact of each structure. In particular the large ones examined using ILP techniques enhance the overall performance significantly, so that the hybrid approach clearly outperforms variants including only commonly defined neighborhoods.     

The continuous reactive tabu search: Blending combinatorial optimization and stochastic search for global optimization

A novel algorithm for the global optimization of functions (C-RTS) is presented, in which a combinatorial optimization method cooperates with a stochastic local minimizer. The combinatorial optimization component, based on the Reactive Tabu Search recently proposed by the authors, locates the most promising boxes, in which starting points for the local minimizer are generated. In order to cover a wide spectrum of possible applications without user intervention, the method is designed with adaptive mechanisms: the box size is adapted to the local structure of the function to be optimized, the search parameters are adapted to obtain a proper balance of diversification and intensification. The algorithm is compared with some existing algorithms, and the experimental results are presented for a variety of benchmark tasks.     

Decomposition and local search based methods for the traveling umpire problem

The Traveling Umpire Problem (TUP) is a challenging combinatorial optimization problem based on scheduling umpires for Major League Baseball. The TUP aims at assigning umpire crews to the games of a fixed tournament, minimizing the travel distance of the umpires. The present paper introduces two complementary heuristic solution approaches for the TUP. A new method called enhanced iterative deepening search with leaf node improvements (IDLI) generates schedules in several stages by subsequently considering parts of the problem. The second approach is a custom iterated local search algorithm (ILS) with a step counting hill climbing acceptance criterion. IDLI generates new best solutions for many small and medium sized benchmark instances. ILS produces significant improvements for the largest benchmark instances. In addition, the article introduces a new decomposition methodology for generating lower bounds, which improves all known lower bounds for the benchmark instances.     

Efficient local search algorithms for known and new neighborhoods for the generalized traveling salesman problem

The generalized traveling salesman problem (GTSP) is a well-known combinatorial optimization problem with a host of applications. It is an extension of the Traveling Salesman Problem (TSP) where the set of cities is partitioned into so-called clusters, and the salesman has to visit every cluster exactly once.     

Diameter preserving linear maps and isometries, II

We study linear bijections of simplex spacesA(S) which preserve the diameter of the range, that is, the seminorm ϱ(f)=sup{|f(x)−f(y)|:x,yεS}.     

Routing by ranking: A link analysis method for the constrained dial-a-ride problem

The dial-a-ride problem involves the dispatching of a fleet of vehicles in order to transport a set of customers from specific pick-up nodes to specific drop-off nodes. Using a modified version of hyperlink-induced topic search (HITS), we characterize hubs as nodes with many out-links to other hubs and calculate a hub score for each pick-up and drop-off node. Ranking the nodes by hub score gives guidance to a backtracking algorithm for efficiently finding feasible solutions to the dial-a-ride problem.     

Memetic Algorithms: The Polynomial Local Search Complexity Theory Perspective

In previous work (Krasnogor, . In: Studies on the Theory and Design Space of Memetic Algorithms. Ph.D. thesis, University of the West of England, Bristol, UK, 2002; Krasnogor and Smith, IEEE Trans Evol Algorithms 9(6):474–488, 2005) we develop a syntax-only classification of evolutionary algorithms, in particular so-called memetic algorithms (MAs). When “syntactic sugar” is added to our model, we are able to investigate the polynomial local search (PLS) complexity of memetic algorithms. In this paper we show the PLS-completeness of whole classes of problems that occur when memetic algorithms are applied to the travelling salesman problem using a range of mutation, crossover and local search operators. Our PLS-completeness results shed light on the worst case behaviour that can be expected of a memetic algorithm under these circumstances. Moreover, we point out in this paper that memetic algorithms for graph partitioning and maximum network flow (both with important practical applications) also give rise to PLS-complete problems.

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Electronic Supplementary Material The online version of this article (doi:) contains supplementary material, which is available to authorized users.      

The analysis of evolutionary algorithms on sorting and shortest paths problems

The analysis of evolutionary algorithms is up to now limited to special classes of functions and fitness landscapes. E.g., it is not possible to characterize the set of TSP instances (or another NP-hard combinatorial optimization problem) which are solved by a generic evolutionary algorithm (EA) in an expected time bounded by some given polynomial. As a first step from artificial functions to typical problems from combinatorial optimization, we analyze simple EAs on well-known problems, namely sorting and shortest paths. Although it cannot be expected that EAs outperform the well-known problem specific algorithms on these simple problems, it is interesting to analyze how EAs work on these problems. The following results are obtained:– Sorting is the maximization of sortedness which is measured by one of several well-known measures of presortedness. The different measures of presortedness lead to fitness functions of quite different difficulty for EAs.– Shortest paths problems are hard for all types of EA, if they are considered as single-objective optimization problems, whereas they are easy as multi-objective optimization problems.     

The dial-a-ride problem: models and algorithms

The Dial-a-Ride Problem (DARP) consists of designing vehicle routes and schedules for n users who specify pickup and delivery requests between origins and destinations. The aim is to plan a set of m minimum cost vehicle routes capable of accommodating as many users as possible, under a set of constraints. The most common example arises in door-to-door transportation for elderly or disabled people. The purpose of this article is to review the scientific literature on the DARP. The main features of the problem are described and a summary of the most important models and algorithms is provided. This is an updated version of a paper that appeared in 4OR 1:89–101, 2003.     

The radar method: an effective line search for piecewise linear concave functions

The maximization of one-dimensional piecewise linear concave (OPLC) functions arises in the line search associated with the maximization of piecewise linear concave functions (e.g. Kelley cutting plane method). The OPLC line search is usually done by the next-break-point method, where one goes from break point to break point up to the optimum. If the number of break points is large this method will be computationally expensive. One can also use some classical derivative-free line search method as for example the golden section method. Such methods do not take advantage of the OPLC geometry. As an alternative, we propose an improved version of the so-called radar method, which maximizes an OPLC function by maximizing successive outer approximations. We prove superlinear and finite convergence of the radar method. Furthermore, our computational test shows that the radar method is highly effective independently from the number of break points.     

On the structure of multiobjective combinatorial search space: MNK-landscapes with correlated objectives

The structure of the search space explains the behavior of multiobjective search algorithms, and helps to design well-performing approaches. In this work, we analyze the properties of multiobjective combinatorial search spaces, and we pay a particular attention to the correlation between the objective functions. To do so, we extend the multiobjective NK-landscapes in order to take the objective correlation into account. We study the co-influence of the problem dimension, the degree of non-linearity, the number of objectives, and the objective correlation on the structure of the Pareto optimal set, in terms of cardinality and number of supported solutions, as well as on the number of Pareto local optima. This work concludes with guidelines for the design of multiobjective local search algorithms, based on the main fitness landscape features.     

The directional p-median problem: Definition,complexity, and algorithms

An instance of a p-median problem gives n demand points. The objective is to locate p supply points in order to minimize the total distance of the demand points to their nearest supply point. p-Median is polynomially solvable in one dimension but NP-hard in two or more dimensions, when either the Euclidean or the rectilinear distance measure is used. In this paper, we treat the p-median problem under a new distance measure, the directional rectilinear distance, which requires the assigned supply point for a given demand point to lie above and to the right of it. In a previous work, we showed that the directional p-median problem is polynomially solvable in one dimension; we give here an improved solution through reformulating the problem as a special case of the constrained shortest path problem. We have previously proven that the problem is NP-complete in two or more dimensions; we present here an efficient heuristic to solve it. Compared to the robust Teitz and Bart heuristic, our heuristic enjoys substantial speedup while sacrificing little in terms of solution quality, making it an ideal choice for real-world applications with thousands of demand points.