Guided construction search metaheuristics for the capacitated p-median problem with single source constraint

In the capacitated p-median problem with single source constraint, also known as the capacitated clustering problem, a given set of n weighted points is to be partitioned into p clusters such that the total weight of the points in each cluster does not exceed a given cluster capacity. The objective is to find a set of p centres that minimizes the total scatter of points allocated to these clusters. In this paper, a (λ, μ)-interchange neighbourhood based on the concept of λ-interchange of points restricted to μ-adjacent clusters is proposed. Structural properties of centres are identified and exploited to derive special data structures for their efficient evaluations. Different search and selection strategies including the variable neighbourhood search descent with respect to μ-nearest points are investigated. The most efficient strategies are then embedded in a guided construction search metaheuristic framework based either on a periodic local search procedure or a greedy random adaptive search procedure to solve the problem. Computational experience is reported on a standard set of benchmarks. The computational experience demonstrates the competitive performance of the proposed algorithms when compared to the best-known procedures in the literature in terms of solution quality and computational requirement.     

A joint production and transportation planning problem with heterogeneous vehicles

We consider a manufacturer's planning problem to schedule order production and transportation to respective destinations. The manufacturer in this setting can use two vehicle types for outbound shipments. The first type is available in unlimited numbers. The availability of the second type, which is less expensive, changes over time. Motivated by some industry practices, we present formulations for three different solution approaches: the myopic solution, the hierarchical solution and the coordinated solution. These approaches vary in how the underlying production and transportation subproblems are solved, that is, sequentially versus jointly or heuristically versus optimally. We provide intractability proofs or polynomial-time exact solution procedures for the sub-problems and their special cases. We also compare the three solution approaches over a numerical study to quantify the savings from integration and explicit consideration of transportation availabilities. Our analytical and numerical results set a foundation and a need for a heuristic to solve the integrated problem. We thus propose a tabu search heuristic, which quickly generates near-optimal solutions.     

Retraction algorithms for solving variational inequalities,pseudomonotone equilibrium problems,and fixed-point problems in Banach spaces

In this paper, using sunny generalized nonexpansive retractions which are different from the metric projection and generalized metric projection in Banach spaces, we present new extragradient and line search algorithms for finding the solution of a J-variational inequality whose constraint set is the common elements of the set of fixed points of a family of generalized nonexpansive mappings and the set of solutions of a pseudomonotone J-equilibrium problem for a J -α-inverse-strongly monotone operator in a Banach space. To prove strong convergence of generated iterates in the extragradient method, we introduce a ? _?-Lipschitz-type condition and assume that the equilibrium bifunction satisfies this condition. This condition is unnecessary when the line search method is used instead of the extragradient method. Using FMINCON optimization toolbox in MATLAB, we give some numerical examples and compare them with several existence results in literature to illustrate the usability of our results.     

Irregular polyomino tiling via integer programming with application in phased array antenna design

A polyomino is a generalization of the domino and is created by connecting a fixed number of unit squares along edges. Tiling a region with a given set of polyominoes is a hard combinatorial optimization problem. This paper is motivated by a recent application of irregular polyomino tilings in the design of phased array antennas. Specifically, we formulate the irregular polyomino tiling problem as a nonlinear exact set covering model, where irregularity is incorporated into the objective function using the information-theoretic entropy concept. An exact solution method based on a branch-and-price framework along with novel branching and lower-bounding schemes is proposed. The developed method is shown to be effective for small- and medium-size instances of the problem. For large-size instances, efficient heuristics and approximation algorithms are provided. Encouraging computational results including phased array antenna simulations are reported.     

Packing circles in the smallest circle: an adaptive hybrid algorithm

The circular packing problem (CPP) consists of packing n circles C _i of known radii r _i , i∈N={1,?…,?n}, into the smallest containing circle ?. The objective is to determine the coordinates (x _i ,?y _i ) of the centre of C _i , i∈N, as well as the radius r and centre (x,?y) of ?. CPP, which is a variant of the two-dimensional open-dimension problem, is NP hard. This paper presents an adaptive algorithm that incorporates nested partitioning within a tabu search and applies some diversification strategies to obtain a (near) global optimum. The tabu search is to identify the n circles’ ordering, whereas the nested partitioning is to determine the n circles’ positions that yield the smallest r. The computational results show the efficiency of the proposed algorithm.     

Constructive heuristics for the uncapacitated continuous location-allocation problem

The multisource location-allocation problem in continuous space is investigated. Two constructive heuristic techniques are proposed to solve this problem. Both methods are based on designing suitable schemes for the generation of the initial solutions. The first considers the furthest distance rule and is enhanced by schemes borrowed from tabu search such as constructing the forbidden regions and freeing strategy. The second considers the discrete solutions found when solving the p-median problem. Some results on existing test problems are presented.     

Solution of the NP-hard total tardiness minimization problem in scheduling theory

The classical NP-hard (in the ordinary sense) problem of scheduling jobs in order to minimize the total tardiness for a single machine 1‖ΣT _j is considered. An NP-hard instance of the problem is completely analyzed. A procedure for partitioning the initial set of jobs into subsets is proposed. Algorithms are constructed for finding an optimal schedule depending on the number of subsets. The complexity of the algorithms is O(n ²Σp _j ), where n is the number of jobs and p _j is the processing time of the jth job (j = 1, 2, …, n).     

All Colors Shortest Path problem on trees

Given an edge weighted tree T(V, E), rooted at a designated base vertex \(r \in V\), and a color from a set of colors \(C=\{1,\ldots ,k\}\) assigned to every vertex \(v \in V\), All Colors Shortest Path problem on trees (ACSP-t) seeks the shortest, possibly non-simple, path starting from r in T such that at least one node from every distinct color in C is visited. We show that ACSP-t is NP-hard, and also prove that it does not have a constant factor approximation. We give an integer linear programming formulation of ACSP-t. Based on a linear programming relaxation of this formulation, an iterative rounding heuristic is proposed. The paper also explores genetic algorithm and tabu search to develop alternative heuristic solutions for ACSP-t. The performance of all the proposed heuristics are evaluated experimentally for a wide range of trees that are generated parametrically.     

A combined local search and integer programming approach to the traveling tournament problem

The traveling tournament problem is a well-known combinatorial optimization problem with direct applications to sport leagues scheduling, that sparked intensive algorithmic research over the last decade. With the Challenge Traveling Tournament Instances as an established benchmark, the most successful approaches to the problem use meta-heuristics like tabu search or simulated annealing, partially heavily parallelized. Integer programming based methods on the other hand are hardly able to tackle larger benchmark instances. In this work we present a hybrid approach that draws on the power of commercial integer programming solvers as well as the speed of local search heuristics. Our proposed method feeds the solution of one algorithm phase to the other one, until no further improvements can be made. The applicability of this method is demonstrated experimentally on the galaxy instance set, resulting in currently best known solutions for most of the considered instances.     

A Cutting Planes Algorithm for the <Emphasis Type="Italic">m</Emphasis>-Salesmen Problem

The existence of reliable and flexible FORTRAN programs for integer linear programming has recently enabled the development of very efficient algorithms for the travelling salesman problem. The main characteristic of these algorithms is the relaxation of most of the constraints of the problem during its solution. The same approach can be used for the solution of the m-salesmen problem in which m salesmen starting from the same city must visit only once n cities at minimum cost. The number of salesmen can be fixed in advance or allowed to vary, upper and lower bounds set on the number of salesmen and even fixed costs associated with the salesmen. The results obtained so far are very encouraging. Problems of up to 100 cities have been solved optimally for the m-travelling salesmen case and other more complex problems are currently under study.     

Computational complexity of the original and extended diophantine Frobenius problem

We deduce the law of nonstationary recursion which makes it possible, for given a primitive set A = {a ₁,...,a _k }, k > 2, to construct an algorithm for finding the set of the numbers outside the additive semigroup generated by A. In particular, we obtain a new algorithm for determining the Frobenius numbers g(a ₁,...,a _k ). The computational complexity of these algorithms is estimated in terms of bit operations. We propose a two-stage reduction of the original primitive set to an equivalent primitive set that enables us to improve complexity estimates in the cases when the two-stage reduction leads to a substantial reduction of the order of the initial set.     

Heuristics for multiple facility location in a network using the variance criterion

In cyclic networks the p-variance location problem is NP-hard, and therefore it is suitable to use heuristic methods to find approximate solutions to the problem. To this end, two strategies are explored, the first using combinatorial search procedures over the vertex set, whereas the second searches for the solution over the entire network. The initial vertex set solutions are generated by using tabu search, variable neighbourhood search and interchange procedures. The heuristics have been tested on instances of up to 30 vertices and 70 edges, and their performances compared.     

A note on a mixed routing and scheduling problem on a grid graph

We consider a particular case of the Fleet Quickest Routing Problem (FQRP) on a grid graph of m × n nodes that are placed in m levels and n columns. Starting nodes are placed at the first (bottom) level, and nodes of arrival are placed at the mth level. A feasible solution of FQRP consists in n Manhattan paths, one for each vehicle, such that capacity constraints are respected. We establish m*, i.e. the number of levels that ensures the existence of a solution to FQRP in any possible permutation of n destinations. In particular, m* is the minimum number of levels sufficient to solve any instance of FQRP involving n vehicles, when they move in the ways that the literature has until now assumed. Existing algorithms give solutions that require, for some values of n, more levels than m*. For this reason, we provide algorithm CaR, which gives a solution in a graph m* × n, as a minor contribution.     

A Heuristic Decomposition Algorithm for Scheduling Problems on Mixed Graphs

We consider a scheduling problem where a set of n jobs has to be processed on a set of m machines and arbitrary precedence constraints between operations are given. Moreover, for any two operations i and j values a _ij >0 and a _ji >0 may be given where a _ij is the minimal difference between the starting times of operations i and j when operation i is processed first. Often, the objective is to minimize the makespan but we consider also arbitrary regular criteria. Even the special cases of the classical job shop problem J//C_max belong to the set of NP-hard problems. Therefore, approximation or heuristic algorithms are necessary to handle large-dimension problems. Based on the mixed graph model we give a heuristic decomposition algorithm for such a problem, i.e. the initial problem is partitioned into subproblems that can be solved exactly or approximately with a small error bound. These subproblems are obtained by a relaxation of a subset of the set of undirected edges of the mixed graph. The subproblems are successively solved and a proportion of the results obtained for one subproblem is kept for further subproblem definitions. Numerical results of the algorithm presented here are given.     

Redundant <Emphasis Type="Italic">τ</Emphasis>-adic expansions I: non-adjacent digit sets and their applications to scalar multiplication

This paper investigates some properties of τ-adic expansions of scalars. Such expansions are widely used in the design of scalar multiplication algorithms on Koblitz curves, but at the same time they are much less understood than their binary counterparts. Solinas introduced the width-w τ-adic non-adjacent form for use with Koblitz curves. This is an expansion of integers \({z = \sum_{i=0}^\ell z_i \tau^i}\) , where τ is a quadratic integer depending on the curve, such that z _i ≠ 0 implies z _w+i-1 = . . . = z _i+1 = 0, like the sliding window binary recodings of integers. It uses a redundant digit set, i.e., an expansion of an integer using this digit set need not be uniquely determined if the syntactical constraints are not enforced. We show that the digit sets described by Solinas, formed by elements of minimal norm in their residue classes, are uniquely determined. Apart from this digit set of minimal norm representatives, other digit sets can be chosen such that all integers can be represented by a width-w non-adjacent form using those digits. We describe an algorithm recognizing admissible digit sets. Results by Solinas and by Blake, Murty, and Xu are generalized. In particular, we introduce two new useful families of digit sets. The first set is syntactically defined. As a consequence of its adoption we can also present improved and streamlined algorithms to perform the precomputations in τ-adic scalar multiplication methods. The latter use an improvement of the computation of sums and differences of points on elliptic curves with mixed affine and López–Dahab coordinates. The second set is suitable for low-memory applications, generalizing an approach started by Avanzi, Ciet, and Sica. It permits to devise a scalar multiplication algorithm that dispenses with the initial precomputation stage and its associated memory space. A suitable choice of the parameters of the method leads to a scalar multiplication algorithm on Koblitz Curves that achieves sublinear complexity in the number of expensive curve operations.     

Parametric AE-solution sets to the parametric linear systems with multiple right-hand sides and parametric matrix equation A(p)X = B(p)

In this paper, the parametric matrix equation A(p)X = B(p) whose elements are linear functions of uncertain parameters varying within intervals are considered. In this matrix equation A(p) and B(p) are known m-by-m and m-by-n matrices respectively, and X is the m-by-n unknown matrix. We discuss the so-called AE-solution sets for such systems and give some analytical characterizations for the AE-solution sets and a sufficient condition under which these solution sets are bounded. We then propose a modification of Krawczyk operator for parametric systems which causes reduction of the computational complexity of obtaining an outer estimation for the parametric united solution set, considerably. Then we give a generalization of the Bauer-Skeel and the Hansen-Bliek-Rohn bounds for enclosing the parametric united solution set which also enables us to reduce the computational complexity, significantly. Also some numerical approaches based on Gaussian elimination and Gauss-Seidel methods to find outer estimations for the parametric united solution set are given. Finally, some numerical experiments are given to illustrate the performance of the proposed methods.     

Fixed Points of Set Functors: How Many Iterations are Needed?

The initial algebra for a set functor can be constructed iteratively via a well-known transfinite chain, which converges after a regular infinite cardinal number of steps or at most three steps. We extend this result to the analogous construction of relatively initial algebras. For the dual construction of the terminal coalgebra Worrell proved that if a set functor is α-accessible, then convergence takes at most α + α steps. But until now an example demonstrating that fewer steps may be insufficient was missing. We prove that the functor of all α-small filters is such an example. We further prove that for β ≤ α the functor of all α-small β-generated filters requires precisely α + β steps and that a certain modified power-set functor requires precisely α steps. We also present an example showing that whether a terminal coalgebra exists at all does not depend solely on the object mapping of the given set functor. (This contrasts with the fact that existence of an initial algebra is equivalent to existence of a mere fixed point.)     

Fast separation for the three-index assignment problem

A critical step in a cutting plane algorithm is separation, i.e., establishing whether a given vector x violates an inequality belonging to a specific class. It is customary to express the time complexity of a separation algorithm in the number of variables n. Here, we argue that a separation algorithm may instead process the vector containing the positive components of x,  denoted as supp(x),  which offers a more compact representation, especially if x is sparse; we also propose to express the time complexity in terms of |supp(x)|. Although several well-known separation algorithms exploit the sparsity of x,  we revisit this idea in order to take sparsity explicitly into account in the time-complexity of separation and also design faster algorithms. We apply this approach to two classes of facet-defining inequalities for the three-index assignment problem, and obtain separation algorithms whose time complexity is linear in |supp(x)| instead of n. We indicate that this can be generalized to the axial k-index assignment problem and we show empirically how the separation algorithms exploiting sparsity improve on existing ones by running them on the largest instances reported in the literature.     

On list incidentor (<Emphasis Type="Italic">k</Emphasis>, <Emphasis Type="Italic">l</Emphasis>)-coloring

A proper incidentor coloring is called a (k, l)-coloring if the difference between the colors of the final and initial incidentors ranges between k and l. In the list variant, the extra restriction is added: the color of each incidentor must belong to the set of admissible colors of the arc. In order to make this restriction reasonable we assume that the set of admissible colors for each arc is an integer interval. The minimum length of the interval that guarantees the existence of a list incidentor (k, l)-coloring is called a list incidentor (k, l)-chromatic number. Some bounds for the list incidentor (k, l)-chromatic number are proved for multigraphs of degree 2 and 4.     

On trees with total domination number equal to edge-vertex domination number plus one

An edge e∈E(G) dominates a vertex v∈V(G) if e is incident with v or e is incident with a vertex adjacent to v. An edge-vertex dominating set of a graph G is a set D of edges of G such that every vertex of G is edge-vertex dominated by an edge of D. The edge-vertex domination number of a graph G is the minimum cardinality of an edge-vertex dominating set of G. A subset D?V(G) is a total dominating set of G if every vertex of G has a neighbor in D. The total domination number of G is the minimum cardinality of a total dominating set of G. We characterize all trees with total domination number equal to edge-vertex domination number plus one.