Branch and bound algorithms are general methods applied to various combinatorial optimization problems. Recently, parallelizations of these algorithms have been proposed. In spite of the generalzty of these methods, many of the parallelizations have been set up for a specific problem and a specific parallel computer. In this paper, a generalized utility P U B B (Parallelization Utility for Branch and Bound algorithms) is presented. It can be used on a network of workstations and enables us to easily apply parallelized branch and bound algorithms on any specific combinatorial optimization problem. A new selection rule (hybrid selection rule) was implemented during this study. Severalbranch and bound algorithms were experimentally parallelized with PUBB, using up to 111 networked workstations. The results of these experiments show that super-linear speedup in solving time m a y be achieved when the number of processing elements is increased and also indicate that the hybrid selection rule has an advantage over other selection rules.
The facility location problem is one of the well-studied combinatorial optimization problems, and a number of algorithms which could find the optimal solution efficiently have been proposed. The facility location problem which considers the parametrjc analysis on demands has been studied, and this problem can handle the demand uncertainty to some extent. This paper deals with the facility location problem in which the customer's demands are given by the probability distribution func tions. This model can provide more realistic information than the parametric analysis.Firstly, the facility location problem under demand uncertainty is formulated as a mixed integer programming problem. The objective function to be minimized is the expected value of the total cost. To prevent the algorithm from finding such solutions that are very unlikely to satisfy the demand, a constraint is imposed so as to satisfy the total demand with more than a given probabil ity. A branch-and-bound algorithm is proposed to find the exact solution. To find the optimal solution efficiently, a procedure for calculating the lower bound is presented. Moreover, to strength en the lower bound, the Lagrangean relaxation is applied for the constraints on demand. Lastly, to illustrate the efficiency of the proposed method, the computational experience is given. 670T. IEE Japan, Vol.
Abstract. Branch-and-Bound algorithms are general methods applicable to various combinatorial optimization problems. There are two hopeful methods to improve the algorithms. One is development of the algorithms which exploit the structure of each problem. The other is parallelization of the algorithms. These two methods have been studied by different research communities independently. If a well-designed interface separating the two kinds of implementation of the methods clearly could be constructed, it would enable us to adapt latest algorithms or technology easily. In this paper, we propose a small and simple interface design of a generalized system for parallel branch-and-bound algorithms.Key words: parallel processing, combinatorial optimization problem, branchand-bound algorithms. IntroductionBranch-and-bound algorithms are general methods applicable to various combinatorial optimization problems. These algorithms are search-based techniques that enumerate the entire solution space implicitly. In order to perform this enumeration efficiently, it is necessary to limit the number of feasible solutions that need to be explicitly produced. This process can be accomplished by clever algorithms, which exploit the structure of each problem, and many complicated algorithms have been proposed for each problem. We call implementations of such algorithms problem depending implementation. Unfortunately, even if such procedures are applied, some instances of the problems cannot be solved in a reasonable amount of time. Parallelization is one of the most hopeful methods to accelerate the enumeration speed. We consider high-level parallelism of such algorithms. In such kinds of applications of parallelism, generality of the branch-and-bound algorithm framework should be maintained. Therefore, it is reasonable to develop a generalized system that provides the general framework of the parallel branch-and-bound algorithms and some facilities to make use of it. There are several architectures of parallel computers and several ways to map the framework on a architecture of them. Therefore, many implementations of high-level parallelism should be considered. We call an implementation of the parallelism parallelization architecture depending implementation.
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