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 the one-machine scheduling problem with minimum and maximum time lags

We consider the one-machine scheduling problem with minimum and maximum time lags while minimizing the makespan. This problem typically arises in a manufacturing environment where the next job has to be carried out within a specific time range after the completion of the immediately preceding job. We describe a branch and bound algorithm, based on the input and output of a clique and the relevant propositions, for finding the optimal waiting times. The computational experiments give promising results, showing whether a given instance is feasible or infeasible. With the proposed branch and bound algorithm we can either find an optimal schedule or establish the infeasibility within an acceptable run time.     

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 clique problem

We present a branch and bound algorithm for the maximum clique problem in arbitrary graphs. The main part of the algorithm consists in the determination of upper bounds by graph colorings. Using a modification of a known graph coloring method called DSATUR we simultaneously derive lower and upper bounds for the clique number.

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Zusammenfassung Wir stellen einen Branch and Bound Algorithmus für das Maximum Clique Problem in einem beliebigen Graphen vor. Das Hauptaugenmerk richtet sich dabei auf die Bestimmung oberer Schranken mit Hilfe von Färbungen von Graphen. Es wird eine Modifikation einer bekannten Färbungsmethode, genannt DSATUR, verwendet, mit der sich gleichzeitig obere und untere Schranken für die Cliquezahl erstellen lassen.

     

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.     

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.     

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.     

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.     

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.     

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 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 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 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 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 based heuristic for makespan minimization of washing operations in hospital sterilization services

In this paper, we address the problem of parallel batching of jobs on identical machines to minimize makespan. The problem is motivated from the washing step of hospital sterilization services where jobs have different sizes, different release dates and equal processing times. Machines can process more than one job at the same time as long as the total size of jobs in a batch does not exceed the machine capacity. We present a branch and bound based heuristic method and compare it to a linear model and two other heuristics from the literature. Computational experiments show that our method can find high quality solutions within short computation time.     

Design of planar articulated mechanisms using branch and bound

This paper considers an optimization model and a solution method for the design of two-dimensional mechanical mechanisms. The mechanism design problem is modeled as a nonconvex mixed integer program which allows the optimal topology and geometry of the mechanism to be determined simultaneously. The underlying mechanical analysis model is based on a truss representation allowing for large displacements. For mechanisms undergoing large displacements elastic stability is of major concern. We derive conditions, modeled by nonlinear matrix inequalities, which guarantee that a stable equilibrium is found and that buckling is prevented. The feasible set of the design problem is described by nonlinear differentiable and non-differentiable constraints as well as nonlinear matrix inequalities.To solve the mechanism design problem a branch and bound method based on convex relaxations is developed. To guarantee convergence of the method, two different types of convex relaxations are derived. The relaxations are strengthened by adding valid inequalities to the feasible set and by solving bound contraction sub-problems. Encouraging computational results indicate that the branch and bound method can reliably solve mechanism design problems of realistic size to global optimality.     

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.     

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.     

Calculating safety stocks for assembly systems with random component procurement lead times: A branch and bound algorithm

In this paper, a discrete single-level multi-component inventory control model for assembly systems with random component procurement lead times is considered. The economic order quantity (EOQ) policy is used for a type of finished product. The requirements of the components are constant and cyclic (periodic), and their values per period are deduced from the EOQ for the finished product. The paper focuses on the components safety stock calculation. The objective is to minimise the average holding cost of the components while keeping the desired service level for the finished product. For this, an upper bound, two lower bounds, two dominance properties and an efficient branch and bound algorithm are suggested. Several tests are executed and conclusions are drawn. The proposed model provides a substantial saving for assembly systems with a large number and unreliable delivery of components as in semi-conductor and automotive industries.     

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.