A parallel multi-neighborhood cooperative tabu search for capacitated vehicle routing problems

This paper presents a parallel tabu search algorithm that utilizes several different neighborhood structures for solving the capacitated vehicle routing problem. Single neighborhood or neighborhood combinations are encapsulated in tabu search threads and they cooperate through a solution pool for the purpose of exploiting their joint power. The computational experiments on 32 large scale benchmark instances show that the proposed method is highly effective and competitive, providing new best solutions to four instances while the average deviation of all best solutions found from the collective best results reported in the literature is about 0.22%. We are also able to associate the beneficial use of special neighborhoods with some test instance characteristics and uncover some sources of the collective power of multi-neighborhood cooperation.     

A variable neighborhood search with an effective local search for uncapacitated multilevel lot-sizing problems

In this study, we improved the variable neighborhood search (VNS) algorithm for solving uncapacitated multilevel lot-sizing (MLLS) problems. The improvement is twofold. First, we developed an effective local search method known as the Ancestors Depth-first Traversal Search (ADTS), which can be embedded in the VNS to significantly improve the solution quality. Second, we proposed a common and efficient approach for the rapid calculation of the cost change for the VNS and other generate-and-test algorithms. The new VNS algorithm was tested against 176 benchmark problems of different scales (small, medium, and large). The experimental results show that the new VNS algorithm outperforms all of the existing algorithms in the literature for solving uncapacitated MLLS problems because it was able to find all optimal solutions (100%) for 96 small-sized problems and new best-known solutions for 5 of 40 medium-sized problems and for 30 of 40 large-sized problems.     

The SAT-UNSAT transition for random constraint satisfaction problems

We study threshold phenomena for a large class of random constraint satisfaction problems over finite domains. Our main contribution is a complete classification of the nature (sharp or coarse) of the SAT-UNSAT transition for random Boolean CSPs, which is based on easily decidable properties.     

Statistical search methods for lotsizing problems

This paper reports on our experiments with statistical search methods for solving lotsizing problems in production planning. In lotsizing problems the main objective is to generate a minimum cost production and inventory schedule, such that (i) customer demand is satisfied, and (ii) capacity restrictions imposed on production resources are not violated. We discuss our experiences in solving these, in general NP-hard, lotsizing problems with popular statistical search techniques like simulated annealing and tabu search. The paper concludes with some critical remarks on the use of statistical search methods for solving lotsizing problems.     

Nonmonotone line search for minimax problems

It was recently shown that, in the solution of smooth constrained optimization problems by sequential quadratic programming (SQP), the Maratos effect can be prevented by means of a certain nonmonotone (more precisely, three-step or four-step monotone) line search. Using a well-known transformation, this scheme can be readily extended to the case of minimax problems. It turns out however that, due to the structure of these problems, one can use a simpler scheme. Such a scheme is proposed and analyzed in this paper. Numerical experiments indicate a significant advantage of the proposed line search over the Armijo search.This research was supported in part by NSF Engineering Research Centers Program No. NSFD-CDR-88-03012, by NSF Grant No. DMC-88-15996, and by a grant from the Westinghouse Corporation.     

Variable neighbourhood search for redundancy allocation problems

^** Email: ycliang{at}saturn.yzu.edu.tw^*** Email: s929512{at}mail.yzu.edu.tw^**** Email: s927522{at}mail.yzu.edu.tw A variable neighbourhood search (VNS) algorithm has been developedto solve the redundancy allocation problem (RAP). The VNS methodis perfectly suited to those combinatorial problems with potentialneighbourhood structures, as in the case of the RAP. The moststudied configuration of the RAP is a series system of s-independentk-out-of-n:G subsystems the so-called series–parallelsystem. The RAP is to select the optimal combination and redundancylevels of components to meet system-level constraints. Two typesof objectives are considered in this study—system reliabilitymaximization and system cost minimization. The VNS algorithmis tested on sets of benchmark problems and compared to thebest heuristics in the literature such as tabu search, multipleweighted objective heuristic, ant colony optimization and geneticalgorithm. Computational results show the advantages and benefitsof VNS for solving both types of RAP while considering bothsolution quality and computational efficiency.     

Shifting representation search for hybrid flexible flowline problems

This paper considers the hybrid flexible flowline scheduling problem with a set of additional restrictions and generalizations that are common in practice. These include precedence constraints, sequence dependent setup times, time lags, machine eligibility and release times. There are many potential solution representations for this problem, ranging from simple and compact, to more complex and complete. Typically, when choosing the degree of detail of the solution representation, a tradeoff can be found between efficiency of the algorithm and the size of the search space. Several adaptations of existing methods are introduced (memetic algorithm, iterated local search, iterated greedy), as well as a novel algorithm called shifting representation search (SRS). This new method starts with an iterated greedy algorithm applied to a permutation version of the problem and at a given time, switches to an iterated local search on the full search space. As far as we know, this shift of the solution representation is new in the scheduling literature. Experimental results and statistical tests clearly prove the superiority of SRS compared with classical and existing methods.     

A note on symmetry reduction for circular traveling tournament problems

The traveling tournament problem (TTP) consists of finding a distance-minimal double round-robin tournament where the number of consecutive breaks is bounded. Easton et al. (2001) introduced the so-called circular TTP instances, where venues of teams are located on a circle. The distance between neighboring venues is one, so that the distance between any pair of teams is the distance on the circle. It is empirically proved that these instances are very hard to solve due to the inherent symmetry. This note presents new ideas to cut off essentially identical parts of the solution space. Enumerative solution approaches, e.g. relying on branch-and-bound, benefit from this reduction. We exemplify this benefit by modifying the DFS∗ algorithm of Uthus et al. (2009) and show that speedups can approximate factor 4n.     

A new local search for continuous location problems

This paper presents a new local search approach for solving continuous location problems. The main idea is to exploit the relation between the continuous model and its discrete counterpart. A local search is first conducted in the continuous space until a local optimum is reached. It then switches to a discrete space that represents a discretisation of the continuous model to find an improved solution from there. The process continues switching between the two problem formulations until no further improvement can be found in either. Thus, we may view the procedure as a new adaption of formulation space search. The local search is applied to the multi-source Weber problem where encouraging results are obtained. This local search is also embedded within Variable Neighbourhood Search producing excellent results.     

Direct search for exact solutions to the nonlinear Schrödinger equation

A five-dimensional symmetry algebra consisting of Lie point symmetries is firstly computed for the nonlinear Schrödinger equation, which, together with a reflection invariance, generates two five-parameter solution groups. Three ansätze of transformations are secondly analyzed and used to construct exact solutions to the nonlinear Schrödinger equation. Various examples of exact solutions with constant, trigonometric function type, exponential function type and rational function amplitude are given upon careful analysis. A bifurcation phenomenon in the nonlinear Schrödinger equation is clearly exhibited during the solution process.     

Fast neighborhood search for two- and three-dimensional nesting problems

In this paper we present a new heuristic solution method for two-dimensional nesting problems. It is based on a simple local search scheme in which the neighborhood is any horizontal or vertical translation of a given polygon from its current position. To escape local minima we apply the meta-heuristic method Guided Local Search.     

Global search algorithms for minimum concave-cost network flow problems

We present algorithms for the single-source uncapacitated version of the minimum concave cost network flow problem. Each algorithm exploits the fact that an extreme feasible solution corresponds to a sub-tree of the original network. A global search heuristic based on random extreme feasible initial solutions and local search is developed. The algorithm is used to evaluate the complexity of the randomly generated test problems. An exact global search algorithm is developed, based on enumerative search of rooted subtrees. This exact technique is extended to bound the search based on cost properties and linear underestimation. The technique is accelerated by exploiting the network structure.     

A fast swap-based local search procedure for location problems

We present a new implementation of a widely used swap-based local search procedure for the p-median problem, proposed in 1968 by Teitz and Bart. Our method produces the same output as the best alternatives described in the literature and, even though its worst-case complexity is similar, it can be significantly faster in practice: speedups of up to three orders of magnitude were observed. We also show that our method can be easily adapted to handle the facility location problem and to implement related procedures, such as path-relinking and tabu search. R. F. Werneck: The results presented in this paper were obtained while this author was a summer intern at AT&T Labs Research.     

A variable neighborhood search heuristic for periodic routing problems

The aim of this paper is to propose a new heuristic for the Periodic Vehicle Routing Problem (PVRP) without time windows. The PVRP extends the classical Vehicle Routing Problem (VRP) to a planning horizon of several days. Each customer requires a certain number of visits within this time horizon while there is some flexibility on the exact days of the visits. Hence, one has to choose the visit days for each customer and to solve a VRP for each day. Our method is based on Variable Neighborhood Search (VNS). Computational results are presented, that show that our approach is competitive and even outperforms existing solution procedures proposed in the literature. Also considered is the special case of a single vehicle, i.e. the Periodic Traveling Salesman Problem (PTSP). It is shown that slight changes of the proposed VNS procedure is also competitive for the PTSP.     

Symmetry of minimizers for some nonlocal variational problems

We present a new approach to study the symmetry of minimizers for a large class of nonlocal variational problems. This approach which generalizes the Reflection method is based on the existence of some integral identities. We study the identities that lead to symmetry results, the functionals that can be considered and the function spaces that can be used. Then we use our method to prove the symmetry of minimizers for a class of variational problems involving the fractional powers of Laplacian, for the generalized Choquard functional and for the standing waves of the Davey-Stewartson equation.     

Solutions of two-point boundary value problems for even-order differential equations

The existence of solutions of the two-point boundary value problems consisting of the even-order differential equations

     

An effective linear approximation method for separable programming problems

Among the numerous applications of piecewise linearization methods include data fitting, network analysis, logistics, and statistics. In the early 1950s, a concave function was found to be able to be linearized by introducing 0-1 variables. Most textbooks in Operations Research offer such methods for expressing linear approximations. Various methods of linearization have also been developed in recent literature. Nevertheless, the transformed linear scheme has a severe shortcoming: most standard procedures for linearizing typically involve a large increase in the number of binary variables. Consequently, the gains to be derived from dealing with linear functions are quite likely to be nullified by the increase in the size of the problem.Conventional methods for linearizing a concave function with m break points require m-1 binary variables. However, when m becomes large, the computation will be very time-consuming and may cause a heavy computational burden.This study proposes an effective approach in which only ⌈log₂(m-1)⌉ binary variables are used. The proposed method has the following features: (i) it offers more convenient and efficient means of expressing a piecewise linear function; (ii) fewer 0-1 variables are used; (iii) the computational results show that the proposed method is much more efficient and faster than the conventional one, especially when the number of break points becomes large.     

Combining approximate solutions for linear discrete ill-posed problems

Linear discrete ill-posed problems of small to medium size are commonly solved by first computing the singular value decomposition of the matrix and then determining an approximate solution by one of several available numerical methods, such as the truncated singular value decomposition or Tikhonov regularization. The determination of an approximate solution is relatively inexpensive once the singular value decomposition is available. This paper proposes to compute several approximate solutions by standard methods and then extract a new candidate solution from the linear subspace spanned by the available approximate solutions. We also describe how the method may be used for large-scale problems.     

Generalized monotone line search SQP algorithm for constrained minimax problems

In this article, non-linear minimax problems with general constraints are discussed. By means of solving one quadratic programming an improved direction is yielded and a second-order correction direction can also be at hand via one system of linear equations. So a new algorithm for solving the discussed problems is presented. In connection with a special merit function, the generalized monotone line search is used to yield the step size at each iteration. Under mild conditions, we can ensure global and superlinear convergence. Finally, some numerical experiments are operated to test our algorithm, and the results demonstrate that it is promising.     

Parallel variable neighbourhood search algorithms for job shop scheduling problems

^** Email: msevkli{at}fatih.edu.tr^*** Corresponding author. Email: mehmetaydin{at}acm.org, mehmet.aydin{at}beds.ac.uk Variable neighbourhood search (VNS) is one of the most recentmetaheuristics used for solving combinatorial optimization problemsin which a systematic change of neighbourhood with a local searchis carried out. However, as happens with other metaheuristics,it takes a long time to reach some useful solutions while solvingsome sort of hard combinatorial problems such as job shop scheduling(JSS). Parallelization is one of the most considerable policiesto overcome this matter. In this paper, firstly, a number ofVNS algorithms are examined for JSS problems and then four differentparallelization policies are taken into account to determineefficient parallelization for VNS algorithms. The experimentationreveals the performance of various VNS algorithms and the efficiencyof policies to follow in parallelization. In the end, the unilateral-ringtopology, a noncentral parallelization method, is found as themost efficient policy.