Solving a minimum-power covering problem with overlap constraint for cellular network design

We consider a type of covering problem in cellular networks. Given the locations of base stations, the problem amounts to determining cell coverage at minimum cost in terms of the power usage. Overlap between adjacent cells is required in order to support handover. The problem we consider is NP-hard. We present integer linear models and study the strengths of their continuous relaxations. Preprocessing is used to reduce problem size and tighten the models. Moreover, we design a tabu search algorithm for finding near-optimal solutions effectively and time-efficiently. We report computational results for both synthesized instances and networks originating from real planning scenarios. The results show that one of the integer models leads to tight bounds, and the tabu search algorithm generates high-quality solutions for large instances in short computing time.     

The Attractive Traveling Salesman Problem

In the Attractive Traveling Salesman Problem the vertex set is partitioned into facility vertices and customer vertices. A maximum profit tour must be constructed on a subset of the facility vertices. Profit is computed through an attraction function: every visited facility vertex attracts a portion of the profit from the customer vertices based on the distance between the facility and customer vertices, and the attractiveness of the facility vertex. A gravity model is used for computing the profit attraction. The problem is formulated as an integer non-linear program. A linearization is proposed and strengthened through the introduction of valid inequalities, and a branch-and-cut algorithm is developed. A tabu search algorithm is also implemented. Computational results are reported.     

Multi-neighborhood tabu search for the maximum weight clique problem

Given an undirected graph G=(V,E) with vertex set V={1,??,n} and edge set E?V×V. Let w:V??Z ⁺ be a weighting function that assigns to each vertex i??V a positive integer. The maximum weight clique problem (MWCP) is to determine a clique of maximum weight. This paper introduces a tabu search heuristic whose key features include a combined neighborhood and a dedicated tabu mechanism using a randomized restart strategy for diversification. The proposed algorithm is evaluated on a total of 136 benchmark instances from different sources (DIMACS, BHOSLIB and set packing). Computational results disclose that our new tabu search algorithm outperforms the leading algorithm for the maximum weight clique problem, and in addition rivals the performance of the best methods for the unweighted version of the problem without being specialized to exploit this problem class.     

General solutions to the single vehicle routing problem with pickups and deliveries

The single vehicle routing problem with pickups and deliveries (SVRPPD) is defined on a graph in which pickup and delivery demands are associated with the customer vertices. The problem consists of designing a least cost route for a vehicle of capacity Q. Each customer is allowed to be visited once for a combined pickup and delivery, or twice if these two operations are performed separately. This article proposes a mixed integer linear programming model for the SVRPPD. It introduces the concept of general solution which encompasses known solution shapes such as Hamiltonian, double-path and lasso. Classical construction and improvement heuristics, as well as a tabu search heuristic, are developed and tested over several instances. Computational results show that the best solutions generated by the heuristics are frequently non-Hamiltonian and may contain up to two customers visited twice.     

Integer Programming with Bounded Variables via Canonical Separation

This paper presents a new algorithm for integer programming with bounded variables which is efficient when m < n and when the upper bounds on the variables are small. The main idea is the application of the Balas and Jeroslow canonical hyperplanes and the systematic search of integer points over certain faces of the feasible region. During each iteration the integer points on a certain face are examined, and then this whole face is discarded from the feasible region of a linear programming problem. After a bounded number of iterations, the optimal integer solution is found, if one exists.     

A method for solving the general parametric linear complementarity problem

This paper presents a solution method for the general (mixed integer) parametric linear complementarity problem pLCP(q(θ),M), where the matrix M has a general structure and integrality restriction can be enforced on the solution. Based on the equivalence between the linear complementarity problem and mixed integer feasibility problem, we propose a mixed integer programming formulation with an objective of finding the minimum 1-norm solution for the original linear complementarity problem. The parametric linear complementarity problem is then formulated as multiparametric mixed integer programming problem, which is solved using a multiparametric programming algorithm. The proposed method is illustrated through a number of examples.     

Probability of unique integer solution to a system of linear equations

We consider a system of m linear equations in n variables Ax = d and give necessary and sufficient conditions for the existence of a unique solution to the system that is integer: x ∈ {−1, 1}ⁿ. We achieve this by reformulating the problem as a linear program and deriving necessary and sufficient conditions for the integer solution to be the unique primal optimal solution. We show that as long as m is larger than n/2, then the linear programming reformulation succeeds for most instances, but if m is less than n/2, the reformulation fails on most instances. We also demonstrate that these predictions match the empirical performance of the linear programming formulation to very high accuracy.     

Vehicle routing with cross-docking

Over the past decade, cross-docking has emerged as an important material handling technology in transportation. A variation of the well-known Vehicle Routing Problem (VRP), the VRP with Cross-Docking (VRPCD) arises in a number of logistics planning contexts. This paper addresses the VRPCD, where a set of homogeneous vehicles are used to transport orders from the suppliers to the corresponding customers via a cross-dock. The orders can be consolidated at the cross-dock but cannot be stored for very long because the cross-dock does not have long-term inventory-holding capabilities. The objective of the VRPCD is to minimize the total travel time while respecting time window constraints at the nodes and a time horizon for the whole transportation operation. In this paper, a mixed integer programming formulation for the VRPCD is proposed. A tabu search heuristic is embedded within an adaptive memory procedure to solve the problem. The proposed algorithm is implemented and tested on data sets provided by the Danish consultancy Transvision, and involving up to 200 pairs of nodes. Experimental results show that this algorithm can produce high-quality solutions (less than 5% away from optimal solution values) within very short computational time.     

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.     

Design of manufacturing cells in the presence of alternative cell locations and material transporters

This paper addresses two significant issues in the design of cellular manufacturing (CM) systems: (i) the availability of alternative locations for a cell, and (ii) the use of alternative routes to move part loads between cells when the capacity of the material transporter (MT) employed is limited. In addition, several other important factors in the design of CM systems including machine capacity limitations, batches of part demands, non-consecutive operations of parts, and maximum number of machines assigned to a cell are considered. A nonlinear programming model, comprised of binary and general integer variables, is formulated for the research problem. A higher-level heuristic solution algorithm based upon a concept known as ‘tabu search’ is presented for solving industry-size problems. Six different versions of the heuristic are developed to investigate the impact of long-term memory and the use of fixed versus variable tabu-list sizes. Explicit method-based techniques are developed to convert the original nonlinear programming model into an equivalent mixed (binary)-integer linear programming model in order to test the efficacy of the proposed solution technique for solving small problem instances. The solutions obtained from the heuristics have average deviation of less than 3% of the optimal solutions, and require less than a minute in comparison with optimizing methods that needed 1–10 h of computation time. A carefully designed statistical experiment is used to compare the performance of the heuristics by solving three different problem structures, ranging from four to 30 parts, and three to nine locations. The experiment shows that as the problem size increases, the tabu-search-based heuristic using fixed tabu list size and long-term memory based on minimal frequency strategy is preferred over the other heuristics.     

A column generation based decomposition algorithm for a parallel machine just-in-time scheduling problem

We propose a column generation based exact decomposition algorithm for the problem of scheduling n jobs with an unrestrictively large common due date on m identical parallel machines to minimize total weighted earliness and tardiness. We first formulate the problem as an integer program, then reformulate it, using Dantzig–Wolfe decomposition, as a set partitioning problem with side constraints. Based on this set partitioning formulation, a branch and bound exact solution algorithm is developed for the problem. In the branch and bound tree, each node is the linear relaxation problem of a set partitioning problem with side constraints. This linear relaxation problem is solved by column generation approach where columns represent partial schedules on single machines and are generated by solving two single machine subproblems. Our computational results show that this decomposition algorithm is capable of solving problems with up to 60 jobs in reasonable cpu time.     

A tabu search algorithm for maximum parsimony phylogeny inference

Phylogeny reconstruction is too complex a combinatorial problem for an exhaustive search, because the number of possible solutions increases exponentially with the number of taxa involved. In this paper, we adopt the parsimony principle and design a tabu search algorithm for finding a most parsimonious phylogeny tree. A special array structure is employed to represent the topology of trees and to generate the neighboring trees. We test the proposed tabu search algorithm on randomly selected data sets obtained from nuclear ribosomal DNA sequence data. The experiments show that our algorithm explores fewer trees to reach the optimal one than the commonly used program “dnapenny” (branch-and-bound based) while it generates much more accurate results than the default options of the program “dnapars” (heuristic search based). The percentage of search space needed to find the best solution for our algorithm decreased rapidly as the number of taxa increased. For a 20-taxon phylogeny problem, it needs on average to examine only 3.92 × 10⁻¹⁵% of the sample space.     

Simultaneously scheduling multiple turns for steel color-coating production

This paper investigates a large-scale scheduling problem in the iron and steel industry, called color-coating production scheduling (CCPS). The problem is to generate multiple production turns for the galvanized coils that dynamically arrive from upstream lines within a given scheduling horizon, and at the same time determine the sequence of these turns so that the productivity and product quality are maximized while the production cost and the number of generated turns are minimized. We formulate this problem as a mixed integer nonlinear program and propose a tabu search heuristic to obtain satisfactory solutions. Results on real production instances show that the presented model and heuristic are more effective and efficient with comparison to manual scheduling. A practical scheduling system for CCPS combining the model and heuristic has been developed and successfully implemented in a major iron and steel enterprise in China.     

Scheduling ambulance crews for maximum coverage

This paper addresses the problem of scheduling ambulance crews in order to maximize the coverage throughout a planning horizon. The problem includes the subproblem of locating ambulances to maximize expected coverage with probabilistic response times, for which a tabu search algorithm is developed. The proposed tabu search algorithm is empirically shown to outperform previous approaches for this subproblem. Two integer programming models that use the output of the tabu search algorithm are constructed for the main problem. Computational experiments with real data are conducted. A comparison of the results of the models is presented.     

Optimising a nonlinear utility function in multi-objective integer programming

In this paper we develop an algorithm to optimise a nonlinear utility function of multiple objectives over the integer efficient set. Our approach is based on identifying and updating bounds on the individual objectives as well as the optimal utility value. This is done using already known solutions, linear programming relaxations, utility function inversion, and integer programming. We develop a general optimisation algorithm for use with k objectives, and we illustrate our approach using a tri-objective integer programming problem.     

A distributed computation algorithm for solving portfolio problems with integer variables

A portfolio problem with integer variables can facilitate the use of complex models, including models containing discrete asset values, transaction costs, and logical constraints. This study proposes a distributed algorithm for solving a portfolio program to obtain a global optimum. For a portfolio problem with n integer variables, the objective function first is converted into an ellipse function containing n separated quadratic terms. Next, the problem is decomposed into m equal-size separable programming problems solvable by a distributed computation system composed of m personal computers linked via the Internet. The numerical examples illustrate that the proposed method can obtain the global optimum effectively for large scale portfolio problems involving integral variables.     

A hybrid method combining continuous tabu search and Nelder–Mead simplex algorithms for the global optimization of multiminima functions

Tabu search (TS) is a metaheuristic, which proved efficient to solve various combinatorial optimization problems. However, few works deal with its application to the global minimization of functions depending on continuous variables. To perform this task, we propose an hybrid method combining tabu search and simplex search (SS). TS allows to cover widely the solution space, to stimulate the search towards solutions far from the current solution, and to avoid the risk of trapping into a local minimum. SS is used to accelerate the convergence towards a minimum. The Nelder–Mead simplex algorithm is a classical very powerful local descent algorithm, making no use of the objective function derivatives. A “simplex” is a geometrical figure consisting, in n-dimensions, of (n+1) points. If any point of a simplex is taken as the origin, the n other points define vector directions that span the n-dimension vector space. Through a sequence of elementary geometric transformations (reflection, contraction and extension), the initial simplex moves, expands or contracts. To select the appropriate transformation, the method only uses the values of the function to be optimized at the vertices of the simplex considered. After each transformation, the current worst vertex is replaced by a better one. Our algorithm called continuous tabu simplex search (CTSS) implemented in two different forms (CTSS_single, CTSS_multiple) is made up of two steps: first, an adaptation of TS to continuous optimization problems, allowing to localize a “promising area”; then, intensification within this promising area, involving SS. The efficiency of CTSS is extensively tested by using analytical test functions of which global and local minima are known. A comparison is proposed with several variants of tabu search, genetic algorithms and simulated annealing. CTSS is applied to the design of a eddy current sensor aimed at non-destructive control.     

A heterogeneous cooperative parallel search of branch-and-bound method and tabu search algorithm

In this study, we present a heterogeneous cooperative parallel search that integrates branch-and-bound method and tabu search algorithm. These two algorithms perform searches in parallel and cooperate by asynchronously exchanging information about the best solutions found and new initial solutions for tabu search. The rapid production of a good solution from the tabu search process provides the branch-and-bound process with a better feasible solution to accelerate the elimination of subproblems that do not contain an optimal solution. The new initial solution produced from the subproblem with a least-cost lower bound of the branch-and-bound method suggests the best potential area for tabu search to explore. We use a master-slave model to reduce the complexity of communication and enhance the performance of data exchange. A branch-and-bound process is used as the master process to control the exchange of information and the termination of computation. Several tabu search processes are executed simultaneously as the slave processes and cooperate by asynchronously exchanging information on the best solutions found and the new initial solutions by the master process of branch-and-bound. Based on the computation experiments of solving traveling salesman problems (TSP), the proposed heterogeneous parallel search algorithm outperforms a conventional parallel branch-and-bound method and a conventional parallel tabu search. We also present the computational results showing the efficiency of heterogeneous cooperative parallel search when we use more processors to accelerate search time. Thus, the proposed heterogeneous parallel search algorithm achieves linear accelerations.     

A bicriteria flowshop scheduling with a learning effect

In many situations, a worker’s ability improves as a result of repeating the same or similar tasks; this phenomenon is known as the learning effect. In this paper the learning effect is considered in a two-machine flowshop. The objective is to find a sequence that minimizes a weighted sum of total completion time and makespan. Total completion time and makespan are widely used performance measures in scheduling literature. To solve this scheduling problem, an integer programming model with n² + 6n variables and 7n constraints where n is the number of jobs is formulated. Because of the lengthy computing time and high computing complexity of the integer programming model, the problem with up to 30 jobs can be solved. A heuristic algorithm and a tabu search based heuristic algorithm are presented to solve large size problems. Experimental results show that the proposed heuristic methods can solve this problem with up to 300 jobs rapidly. According to the best of our knowledge, no work exists on the bicriteria flowshop with a learning effect.     

Solving integer programs over monotone inequalities in three variables: A framework for half integrality and good approximations

We define a class of monotone integer programs with constraints that involve up to three variables each. A generic constraint in such integer program is of the form ax−by⩽z+c, where a and b are nonnegative and the variable z appears only in that constraint. We devise an algorithm solving such problems in time polynomial in the length of the input and the range of variables U. The solution is obtained from a minimum cut on a graph with O(nU) nodes and O(mU) arcs where n is the number of variables of the types x and y and m is the number of constraints. Our algorithm is also valid for nonlinear objective functions.Nonmonotone integer programs are optimization problems with constraints of the type ax+by⩽z+c without restriction on the signs of a and b. Such problems are in general NP-hard. We devise here an algorithm, relying on a transformation to the monotone case, that delivers half integral superoptimal solutions in polynomial time. Such solutions provide bounds on the optimum value that can only be superior to bounds provided by linear programming relaxation. When the half integral solution can be rounded to an integer feasible solution, this is a 2-approximate solution. In that the technique is a unified 2-approximation technique for a large class of problems. The results apply also for general integer programming problems with worse approximation factors that depend on a quantifier measuring how far the problem is from the class of problems we describe.The algorithm described here has a wide array of problem applications. An additional important consequence of our results is that nonmonotone problems in the framework are MAX SNP-hard and at least as hard to approximate as vertex cover.Problems that are amenable to the analysis provided here are easily recognized. The analysis itself is entirely technical and involves manipulating the constraints and transforming them to a totally unimodular system while losing no more than a factor of 2 in the integrality.