A branch and bound algorithm for the matrix bandwidth minimization

In this article, we first review previous exact approaches as well as theoretical contributions for the problem of reducing the bandwidth of a matrix. This problem consists of finding a permutation of the rows and columns of a given matrix which keeps the non-zero elements in a band that is as close as possible to the main diagonal. This NP-complete problem can also be formulated as a labeling of vertices on a graph, where edges are the non-zero elements of the corresponding symmetrical matrix. We propose a new branch and bound algorithm and new expressions for known lower bounds for this problem. Empirical results with a collection of previously reported instances indicate that the proposed algorithm compares favourably to previous methods.     

A branch and bound algorithm for the maximum diversity problem

This article begins with a review of previously proposed integer formulations for the maximum diversity problem (MDP). This problem consists of selecting a subset of elements from a larger set in such a way that the sum of the distances between the chosen elements is maximized. We propose a branch and bound algorithm and develop several upper bounds on the objective function values of partial solutions to the MDP. Empirical results with a collection of previously reported instances indicate that the proposed algorithm is able to solve all the medium-sized instances (with 50 elements) as well as some large-sized instances (with 100 elements). We compare our method with the best previous linear integer formulation solved with the well-known software Cplex. The comparison favors the proposed procedure.     

A branch and bound network approach to the canonical constrained entropy problem

In this paper, we present a branch and bound algorithm for solving the constrained entropy mathematical programming problem. Unlike other methods for solving this problem, our method solves more general problems with inequality constraints. The advantage of the proposed technique is that the relaxed problem solved at each node is a singly constrained network problem. The disadvantage is that the relaxed problem has twice as many variables as the original problem. An application to regional planning is given, and an example problem is solved.     

A best-first branch and bound algorithm for unconstrained two-dimensional cutting problems

This paper is concerned with the problem of unconstrained two-dimensional cutting of small rectangular pieces, each of which has its own profit and size, from a large rectangular plate so as to maximize the profit-sum of the pieces produced. Hifi and Zissimopoulos's recursive algorithm using G and Kang's upper bound is presently the most efficient exact algorithm for the problem. We propose a best-first branch and bound algorithm based upon the bottom-up approach that is more efficient than their recursive algorithm. The proposed algorithm uses efficient upper bound and branching strategies that can reduce the number of nodes that must be searched significantly. We demonstrate the efficiency of the proposed algorithm through computational experiments.     

Some branch and bound techniques for nonlinear optimization

The range of nonlinear optimization problems which can be solved by Linear Programming and the Branch and Bound algorithm is extended by introducing Chains of Linked Ordered Sets and by allowing automatic interpolation of new variables. However this approach involves solving a succession of linear subproblems, whose solutions in general violate the logical requirements of the nonlinear formulation and may lie far from any local or global optimum. The paper describes techniques which are designed to improve the performance of the Branch and Bound algorithm on problems containing chains, and which also yield benefits in integer programming.Each linear subproblem is tightened towards the corresponding nonlinear problem by removing variables which must logically be nonbasic in any feasible solution. This is achieved by a presolve procedure, and also by post-optimal Lagrangian relaxation which tightens the bound on the objective function by assessing the cheapest way to satisfy any violated chain constraints. Frequently fewer subsequent branches are required to find a feasible solution or to prove infeasibility.Formerly of Scicon Ltd.     

A branch and bound method for stochastic global optimization

A stochastic branch and bound method for solving stochastic global optimization problems is proposed. As in the deterministic case, the feasible set is partitioned into compact subsets. To guide the partitioning process the method uses stochastic upper and lower estimates of the optimal value of the objective function in each subset. Convergence of the method is proved and random accuracy estimates derived. Methods for constructing stochastic upper and lower bounds are discussed. The theoretical considerations are illustrated with an example of a facility location problem.     

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.     

A branch and bound algorithm for determining locations of long-term care facilities

This paper focuses on the problem of determining locations for long-term care facilities with the objective of balancing the numbers of patients assigned to the facilities. We present a branch and bound algorithm by developing dominance properties, a lower bounding scheme and a heuristic algorithm for obtaining an upper bound for the problem. For evaluation of the suggested branch and bound algorithm, computational experiments are performed on a number of test problems. Results of the experiments show that the suggested algorithm gives optimal solutions of problems of practical sizes in a reasonable amount of computation time.     

The design of branch and bound algorithms for a class of nonlinear integer programs

This paper is concerned with computational experimentation leading to the design of effective branch and bound algorithms for an important class of nonlinear integer programming problems, namely linearly constrained problems, which are used to model several real-world situations. The main contribution here is a study of the effect of node and branching variable selection and storage reduction strategies on overall computational effort for this class of problems, as well as the generation of a set of adequate test problems. Several node and branching variable strategies are compared in the context of a pure breadth-first enumeration, as well as in a special breadth and depth enumeration combination approach presented herein. Also, the effect of using updated pseudocosts is briefly addressed. Computational experience is presented on a set of eighteen suitably-sized nonlinear test problems, as well as on some random linear integer programs. Some of the new rules proposed are demonstrated to be significantly superior to previously suggested strategies; interestingly, even for linear integer programming problems.     

The design of branch and bound algorithms for a class of nonlinear integer programs

This paper is concerned with computational experimentation leading to the design of effective branch and bound algorithms for an important class of nonlinear integer programming problems, namely linearly constrained problems, which are used to model several real-world situations. The main contribution here is a study of the effect of node and branching variable selection and storage reduction strategies on overall computational effort for this class of problems, as well as the generation of a set of adequate test problems. Several node and branching variable strategies are compared in the context of a pure breadth-first enumeration, as well as in a special breadth and depth enumeration combination approach presented herein. Also, the effect of using updated pseudocosts is briefly addressed. Computational experience is presented on a set of eighteen suitably-sized nonlinear test problems, as well as on some random linear integer programs. Some of the new rules proposed are demonstrated to be significantly superior to previously suggested strategies; interestingly, even for linear integer programming problems.     

An extended branch and bound algorithm for linear bilevel programming

For linear bilevel programming, the branch and bound algorithm is the most successful algorithm to deal with the complementary constraints arising from Kuhn–Tucker conditions. However, one principle challenge is that it could not well handle a linear bilevel programming problem when the constraint functions at the upper-level are of arbitrary linear form. This paper proposes an extended branch and bound algorithm to solve this problem. The results have demonstrated that the extended branch and bound algorithm can solve a wider class of linear bilevel problems can than current capabilities permit.     

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.     

A computational comparison of some branch and bound methods for indefinite quadratic programs

The aim of this paper is to discuss different branch and bound methods for solving indefinite quadratic programs. In these methods the quadratic objective function is decomposed in a d.c. form and the relaxations are obtained by linearizing the concave part of the decomposition. In this light, various decomposition schemes have been considered and studied. The various branch and bound solution methods have been implemented and compared by means of a deep computational test.      

A simplicial branch and bound duality-bounds algorithm for the linear sum-of-ratios problem

In this article, we present and validate a simplicial branch and bound duality-bounds algorithm for globally solving the linear sum-of-ratios fractional program. The algorithm computes the lower bounds called for during the branch and bound search by solving ordinary linear programming problems. These problems are derived by using Lagrangian duality theory. The algorithm applies to a wide class of linear sum-of-ratios fractional programs. Two sample problems are solved, and the potential practical and computational advantages of the algorithm are indicated.     

A branch and bound algorithm for scheduling trains in a railway network

The paper studies a train scheduling problem faced by railway infrastructure managers during real-time traffic control. When train operations are perturbed, a new conflict-free timetable of feasible arrival and departure times needs to be re-computed, such that the deviation from the original one is minimized. The problem can be viewed as a huge job shop scheduling problem with no-store constraints. We make use of a careful estimation of time separation among trains, and model the scheduling problem with an alternative graph formulation. We develop a branch and bound algorithm which includes implication rules enabling to speed up the computation. An experimental study, based on a bottleneck area of the Dutch rail network, shows that a truncated version of the algorithm provides proven optimal or near optimal solutions within short time limits.     

An application of the branch,bound, and remember algorithm to a new simple assembly line balancing dataset

The simple assembly line balancing problem (SALBP) is a well-studied NP-complete problem for which a new problem database of generated instances was published in 2013. This paper describes the application of a branch, bound, and remember (BB&R) algorithm using the cyclic best-first search strategy to this new database to produce provably exact solutions for 86% of the unsolved problems in this database. A new backtracking rule to save memory is employed to allow the BB&R algorithm to solve many of the largest problems in the database.     

A branch and bound algorithm for the robust shortest path problem with interval data

Many real problems can be modelled as robust shortest path problems on interval digraphs, where intervals represent uncertainty about real costs and a robust path is not too far from the shortest path for each possible configuration of the arc costs.A branch and bound algorithm for this problem is presented.     

A branch and bound algorithm for class based storage location assignment

Class-based storage implementation decisions have significant impact on the required storage space and the material handling cost in a warehouse. In this paper, a nonlinear integer programming model is proposed to capture the above. Effects of storage area reduction on order picking and storage space cost are incorporated. A branch and bound algorithm is developed to solve the model. Computational experience with randomly generated data sets and an industrial case shows that branch and bound algorithm is computationally more efficient than a baseline dynamic programming algorithm. It is further observed that the class based policy results in lower total cost of order picking and storage space than the dedicated policy.     

A branch and bound method for the job-shop problem with sequence-dependent setup times

This paper deals with the job-shop scheduling problem with sequence-dependent setup times. We propose a new method to solve the makespan minimization problem to optimality. The method is based on iterative solving via branch and bound decisional versions of the problem. At each node of the branch and bound tree, constraint propagation algorithms adapted to setup times are performed for domain filtering and feasibility check. Relaxations based on the traveling salesman problem with time windows are also solved to perform additional pruning. The traveling salesman problem is formulated as an elementary shortest path problem with resource constraints and solved through dynamic programming. This method allows to close previously unsolved benchmark instances of the literature and also provides new lower and upper bounds.     

A branch and bound algorithm for hybrid flow shop scheduling problem with setup time and assembly operations

A hybrid flow shop scheduling problem (HFSP) with assembly operations is studied in this paper. In the considered problem, a number of products of the same kind are produced. Each product is assembled using a set of several parts. At first, the parts are produced in a hybrid flow shop and then they are assembled in an assembly stage to produce products. The considered objective is to minimize the completion time of all products (makespan). This problem has been proved strongly NP-hard, so in order to solve it, a hierarchical branch and bound algorithm is presented. Also, some lower and upper bounds are developed to increase the efficiency of the proposed algorithm. The numerical experiments are used to evaluate the performance of the proposed algorithm.