A parallel integer linear programming algorithm

The solution of large scale integer linear programming models is generally dependent, in some way, upon the branch and bound technique, which can be quite time consuming. This paper describes a parallel branch and bound algorithm which achieves super linear efficiency in solving integer linear programming models on a multiprocessor computer. The algorithm is used to solve the Haldi and IBM test problems as well as a system design model.     

Minimizing the sum of job completion times on capacitated two-parallel machines

This paper considers a scheduling problem with two identical parallel machines. One has unlimited capacity; the other can only run for a fixed time. A given set of jobs must be scheduled on the two machines with the goal of minimizing the sum of their completion times. The paper proposes an optimal branch and bound algorithm which employs three powerful elements, including an algorithm for computing the upper bound, a lower bound algorithm, and a fathoming condition. The branch and bound algorithm was tested on problems of various sizes and parameters. The results show that the algorithm is quite efficient to solve all the test problems. In particular, the total computation time for the hardest problem is less than 0.1 second for a set of 100 problem instances. An important finding of the tests is that the upper bound algorithm can actually find optimal solutions to a quite large number of problems.     

A Parallel Interior Point Method and Its Application to Facility Location Problems

We present a parallel interior point algorithm to solve block structured linear programs. This algorithm can solve block diagonal linear programs with both side constraints (common rows) and side variables (common columns). The performance of the algorithm is investigated on uncapacitated, capacitated and stochastic facility location problems. The facility location problems are formulated as mixed integer linear programs. Each subproblem of the branch and bound phase of the MIP is solved using the parallel interior point method. We compare the total time taken by the parallel interior point method with the simplex method to solve the complete problems, as well as the various costs of reoptimisation of the non-root nodes of the branch and bound. Computational results on two parallel computers (Fujitsu AP1000 and IBM SP2) are also presented in this paper.     

On the minimization of total weighted flow time with identical and uniform parallel machines

We consider the NP-hard problem of scheduling jobs on identical parallel machines to minimize total weighted flow time. We discuss the properties that characterize the structure of an optimal solution, present a lower bound and propose a branch and bound algorithm. The algorithm is superior to prior methods presented in the literature. We also extend the algorithm to uniform parallel machines and solve medium-sized problem instances.     

A parallel branch and bound algorithm for solving large asymmetric traveling salesman problems

A parallel branch and bound algorithm that solves the asymmetric traveling salesman problem to optimality is described. The algorithm uses an assignment problem based lower bounding technique, subtour elimination branching rules, and a subtour patching algorithm as an upper bounding procedure. The algorithm is organized around a data flow framework for parallel branch and bound. The algorithm begins by converting the cost matrix to a sparser version in such a fashion as to retain the optimality of the final solution. Computational results are presented for three different classes of problem instances: (1) matrix elements drawn from a uniform distribution of integers for instances of size 250 to 10 000 cities, (2) instances of size 250 to 1000 cities that concentrate small elements in the upper left portion of the cost matrix, and (3) instances of size 300 to 3000 cities that are designed to confound neighborhood search heuristics.     

A new branching rule for the branch and bound algorithm for solving nonlinear integer programming problems

We examine a branch and bound algorithm for solving nonlinear (convex) integer programming problems. In this note we generalize previous results for the quadratic case. The variables are branched in such a way that the number of branch and bound nodes checked in the process is small. Numerical results confirm the efficiency.     

A new branch and bound algorithm for cell formation problem

Cell formation (CF) is the first and the most important problem in designing cellular manufacturing systems. Due to its non-polynomial nature, various heuristic and metaheuristic algorithms have been proposed to solve CF problem. Despite the popularity of heuristic algorithms, few studies have attempted to develop exact algorithms, such as branch and bound (B&B) algorithms, for this problem. We develop three types of branch and bound algorithms to deal with the cell formation problem. The first algorithm uses a binary branching scheme based on the definitions provided for the decision variables. Unlike the first algorithm, which relies on the mathematical model, the second one is designed based on the structure of the cell formation problem. The last algorithm has a similar structure to the second one, except that it has the ability to eliminate duplicated nodes in branching trees. The proposed branch and bound algorithms and a hybrid genetic algorithm are compared through some numerical examples. The results demonstrate the effectiveness of the modified problem-oriented branch and bound algorithm in solving relatively large size cell formation problems.     

Experimental study of scheduling with memory constraints using hybrid methods

In this paper we study divisible load scheduling in systems with limited memory. Divisible loads are parallel computations which can be divided into independent parts processed in parallel on remote computers, and the part sizes may be arbitrary. The distributed system is a heterogeneous single level tree. The total size of processor memories is too small to accommodate the whole load at any moment of time. Therefore, the load is distributed in many rounds. Memory reservations have block nature. The problem consists in distributing the load taking into account communication time, computation time, and limited memory buffers so that the whole processing finishes as early as possible. This problem is both combinatorial and algebraic in nature. Therefore, hybrid algorithms are given to solve it. Two algorithms are proposed to solve the combinatorial component. A branch-and-bound algorithm is nearly unusable due to its complexity. Then, a genetic algorithm is proposed with more tractable execution times. For a given solution of the combinatorial part we formulate the solution of the algebraic part as a linear programming problem. An extensive computational study is performed to analyze the impact of various system parameters on the quality of the solutions. From this we were able to infer on the nature of the scheduling problem.     

Using a Hybrid Genetic-Algorithm/Branch and Bound Approach to Solve Feasibility and Optimization Integer Programming Problems

The satisfiability problem in forms such as maximum satisfiability (MAX-SAT) remains a hard problem. The most successful approaches for solving such problems use a form of systematic tree search. This paper describes the use of a hybrid algorithm, combining genetic algorithms and integer programming branch and bound approaches, to solve MAX-SAT problems. Such problems are formulated as integer programs and solved by a hybrid algorithm implemented within standard mathematical programming software. Computational testing of the algorithm, which mixes heuristic and exact approaches, is described.     

A communication-less parallel algorithm for tridiagonal Toeplitz systems

Diagonally dominant tridiagonal Toeplitz systems of linear equations arise in many application areas and have been well studied in the past. Modern interest in numerical linear algebra is often focusing on solving classic problems in parallel. In McNally [Fast parallel algorithms for tri-diagonal symmetric Toeplitz systems, MCS Thesis, University of New Brunswick, Saint John, 1999], an m processor Split & Correct algorithm was presented for approximating the solution to a symmetric tridiagonal Toeplitz linear system of equations. Nemani [Perturbation methods for circulant-banded systems and their parallel implementation, Ph.D. Thesis, University of New Brunswick, Saint John, 2001] and McNally (2003) adapted the works of Rojo [A new method for solving symmetric circulant tri-diagonal system of linear equations, Comput. Math. Appl. 20 (1990) 61–67], Yan and Chung [A fast algorithm for solving special tri-diagonal systems, Computing 52 (1994) 203–211] and McNally et al. [A split-correct parallel algorithm for solving tri-diagonal symmetric Toeplitz systems, Internat. J. Comput. Math. 75 (2000) 303–313] to the non-symmetric case. In this paper we present relevant background from these methods and then introduce an m processor scalable communication-less approximation algorithm for solving a diagonally dominant tridiagonal Toeplitz system of linear equations.     

Minimizing total tardiness on a single machine with unequal release dates

This study addresses the problem of minimizing total tardiness on a single machine with unequal release dates. Dominance properties established in previous literatures and herein are adopted to develop branch and bound and heuristic procedures. Computational experiments were conducted to evaluate the approaches. The results revealed that the branch and bound algorithm is efficient in solving hard problems and easy problems that involve up to 50 and 500 jobs, respectively. The computational effectiveness of the heuristic is also reported.     

A parallel algorithm for constrained concave quadratic global minimization

The global minimization of large-scale concave quadratic problems over a bounded polyhedral set using a parallel branch and bound approach is considered. The objective function consists of both a concave part (nonlinear variables) and a strictly linear part, which are coupled by the linear constraints. These large-scale problems are characterized by having the number of linear variables much greater than the number of nonlinear variables. A linear underestimating function to the concave part of the objective is easily constructed and minimized over the feasible domain to get both upper and lower bounds on the global minimum function value. At each minor iteration of the algorithm, the feasible domain is divided into subregions and linear underestimating problems over each subregion are solved in parallel. Branch and bound techniques can then be used to eliminate parts of the feasible domain from consideration and improve the upper and lower bounds. It is shown that the algorithm guarantees that a solution is obtained to within any specified tolerance in a finite number of steps. Computational results are presented for problems with 25 and 50 nonlinear variables and up to 400 linear variables. These results were obtained on a four processor CRAY2 using both sequential and parallel implementations of the algorithm. The average parallel solution time was approximately 15 seconds for problems with 400 linear variables and a relative tolerance of 0.001. For a relative tolerance of 0.1, the average computation time appears to increase only linearly with the number of linear variables.     

Execution of logic programs by iterative-deepening A* SLD-tree search

The depth-first search strategy used by PROLOG is supplemented by depth-first iterative-deepening A*, which is complete, i.e. it always succeeds in executing a correct program if enough time and space is available. Iterative-deepening A* is only used for predicates that can be executed faster with iterative-deepening A* instead of depth-first.Since standard iterative-deepening A* is quadratic instead of linear for trees with much one way branching, e.g. most SLD-trees, extrapolation is used to determine a suitable bound on the evaluation function values of nodes that are to be expanded during an iteration. The number of inferences is further reduced by pruning rules such as cutting when the next literal to be evaluated is an alphabetic variant of a literal for which evaluation has failed previously.An advantage of the suggested search strategy is that many more programs can be executed with reasonable efficiency. The main disadvantage is that the programmer either must specify explicitly which strategy to use for a given predicate or provide examples so that the best strategy can be inferred automatically.     

Solving the response time variability problem by means of a psychoclonal approach

The response time variability problem (RTVP) is a combinatorial scheduling problem that has recently appeared in the literature. This problem has a wide range of real life applications in fields such as manufacturing, hard real-time systems, operating systems and network environments. Originally, the RTVP occurs whenever products, clients or jobs need to be sequenced in such a way that the variability in the time between the instants at which they receive the necessary resources is minimized. Since the RTVP is hard to solve, heuristic techniques are needed for solving it. Three metaheuristic—multi-start, GRASP and PSO—algorithms proposed for solving the RTVP in two previous studies have been the most efficient to date in solving non-small instances of the RTVP. We propose solving the RTVP by means of a psychoclonal algorithm based approach. The psychoclonal algorithm inherits its attributes from Maslow’s hierarchy of needs theory and the artificial immune system (AIS) approach, specifically the clonal selection principle. In this paper, we compare the proposed psychoclonal algorithm with the three aforementioned metaheuristic algorithms and show that, on average, the psychoclonal algorithm strongly improves on the results obtained.     

Optimal on-line flow time with resource augmentation

We study the problem of scheduling n jobs that arrive over time. We consider a non-preemptive setting on a single machine. The goal is to minimize the total flow time. We use extra resource competitive analysis: an optimal off-line algorithm which schedules jobs on a single machine is compared to a more powerful on-line algorithm that has ? machines. We design an algorithm of competitive ratio , where Δ is the maximum ratio between two job sizes, and provide a lower bound which shows that the algorithm is optimal up to a constant factor for any constant ?. The algorithm works for a hard version of the problem where the sizes of the smallest and the largest jobs are not known in advance, only Δ and n are known. This gives a trade-off between the resource augmentation and the competitive ratio.We also consider scheduling on parallel identical machines. In this case the optimal off-line algorithm has m machines and the on-line algorithm has ?m machines. We give a lower bound for this case. Next, we give lower bounds for algorithms using resource augmentation on the speed. Finally, we consider scheduling with hard deadlines, and scheduling so as to minimize the total completion time.     

Parallel branch and bound for multidimensional scaling with city-block distances

Multidimensional scaling is a technique for exploratory analysis of multidimensional data. The essential part of the technique is minimization of a multimodal function with unfavorable properties like invariants and non-differentiability. Recently a branch and bound algorithm for multidimensional scaling with city-block distances has been proposed for solution of medium-size problems exactly. The algorithm exploits piecewise quadratic structure of the objective function. In this paper a parallel version of the branch and bound algorithm for multidimensional scaling with city-block distances has been proposed and investigated. Parallel computing enabled solution of larger problems what was not feasible with the sequential version.     

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.     

Investigation of selection strategies in branch and bound algorithm with simplicial partitions and combination of Lipschitz bounds

Speed and memory requirements of branch and bound algorithms depend on the selection strategy of which candidate node to process next. The goal of this paper is to experimentally investigate this influence to the performance of sequential and parallel branch and bound algorithms. The experiments have been performed solving a number of multidimensional test problems for global optimization. Branch and bound algorithm using simplicial partitions and combination of Lipschitz bounds has been investigated. Similar results may be expected for other branch and bound algorithms.     

BBPH: Using progressive hedging within branch and bound to solve multi-stage stochastic mixed integer programs

Progressive hedging, though an effective heuristic for solving stochastic mixed integer programs (SMIPs), is not guaranteed to converge in this case. Here, we describe BBPH, a branch and bound algorithm that uses PH at each node in the search tree such that, given sufficient time, it will always converge to a globally optimal solution. In addition to providing a theoretically convergent “wrapper” for PH applied to SMIPs, computational results demonstrate that for some difficult problem instances branch and bound can find improved solutions after exploring only a few nodes.     

Scheduling on parallel identical machines to minimize total tardiness

This paper focuses on the problem of scheduling n independent jobs on m identical parallel machines for the objective of minimizing total tardiness of the jobs. We develop dominance properties and lower bounds, and develop a branch and bound algorithm using these properties and lower bounds as well as upper bounds obtained from a heuristic algorithm. Computational experiments are performed on randomly generated test problems and results show that the algorithm solves problems with moderate sizes in a reasonable amount of computation time.