An extension of branch-and-bound algorithm for solving sum-of-nonlinear-ratios problem

This paper is concerned with a problem of maximizing the sum of several ratios of functions. We extend an algorithm, which has been designed to solve the sum-of-linear-ratios problem, for solving the sum-of-nonlinear-ratios problem. We also discuss the complexity of the problem and report the results of numerical experiments on the extended algorithm.     

A branch-and-bound algorithm for the coupled task problem

The coupled task problem is to schedule jobs on a single machine where each job consists of two subtasks and where the second subtask has to be started after a given time interval with respect to the first one. The problem has several applications and is NP-hard. In this paper we present a branch-and-bound algorithm for this problem and compare its performance with four integer programming models.     

A branch-and-bound algorithm for the mini-max spanning forest problem

The mini-max spanning forest problem requires to find a spanning forest of an undirected graph that minimizes the maximum of the costs of constituent trees. In a previous work we proved this problem NP-hard. In the current paper we present three lower bounds for this problem and develop a branch-and-bound algorithm to solve the problem exactly. The algorithm is implemented and numerical experiments are conducted on a series of test problems. The experiments compare the performances of the proposed bounds and search strategies in the algorithm as well as investigate the effects of instance characteristics on the behavior of the algorithm. Also, extension of the problem to the case of more than two root vertices as well as to the problem of determining the root locations are discussed.     

A branch-and-bound algorithm for the singly constrained assignment problem

The singly constrained assignment problem (SCAP) is a linear assignment problem (LAP) with one extra side constraint, e.g., due to a time restriction. The SCAP is, in contrast to the LAP, difficult to solve. A branch-and-bound algorithm is presented to solve the SCAP to optimality. Lower bounds are obtained by Lagrangean relaxation. Computational results show that the algorithm is able to solve different types of SCAP instances up to size n = 1000 within short running times on a standard personal computer.     

A branch-and-bound algorithm for the resource-constrained project scheduling problem

We describe a time-oriented branch-and-bound algorithm for the resource-constrained project scheduling problem which explores the set of active schedules by enumerating possible activity start times. The algorithm uses constraint-propagation techniques that exploit the temporal and resource constraints of the problem in order to reduce the search space. Computational experiments with large, systematically generated benchmark test sets, ranging in size from thirty to one hundred and twenty activities per problem instance, show that the algorithm scales well and is competitive with other exact solution approaches. The computational results show that the most difficult problems occur when scarce resource supply and the structure of the resource demand cause a problem to be highly disjunctive.     

A practicable branch-and-bound algorithm for globally solving linear multiplicative programming

To globally solve linear multiplicative programming problem (LMP), this paper presents a practicable branch-and-bound method based on the framework of branch-and-bound algorithm. In this method, a new linear relaxation technique is proposed firstly. Then, the branch-and-bound algorithm is developed for solving problem LMP. The proposed algorithm is proven that it is convergent to the global minimum by means of the subsequent solutions of a series of linear programming problems. Some experiments are reported to show the feasibility and efficiency of this algorithm.     

A genetic algorithm for the two-stage supply chain distribution problem associated with a fixed charge

This paper considers a two-stage distribution problem of a supply chain that is associated with a fixed charge. Two kinds of cost are involved in this problem: a continuous cost that linearly increases with the amount transported between a source and a destination, and secondly, a fixed charge, that incurs whenever there exists a transportation of a non-zero quantity between a source and a destination. The objective criterion is the minimisation of the total cost of distribution. A genetic algorithm (GA) that belongs to evolutionary search heuristics is proposed and illustrated. The proposed methodology is evaluated for its solution quality by comparing it with the approximate and lower bound solutions. Thus, the comparison reveals that the GA generates better solution than the approximation method and is capable of providing solution either equal or closer to the lower bound solution of the problem.     

A simplicial branch-and-bound algorithm for solving quadratically constrained quadratic programs

We propose a branch-and-bound algorithm for solving nonconvex quadratically-constrained quadratic programs. The algorithm is novel in that branching is done by partitioning the feasible region into the Cartesian product of two-dimensional triangles and rectangles. Explicit formulae for the convex and concave envelopes of bilinear functions over triangles and rectangles are derived and shown to be second-order cone representable. The usefulness of these new relaxations is demonstrated both theoretically and computationally.     

A branch-and-bound algorithm for the asymmetrical distance-constrained vehicle routing problem

The aim of this paper is to develop an exact algorithm for the asymmetrical distance-constrained vehicle routing problem. The problem is solved by means of a branch-and-bound tree in which subproblems are modified assignment problems subject to some restrictions. Computational results for problems involving up to 100 nodes are reported.Cet article décrit un algorithme exact pour le probléme de tournées avec contraintes de temps et une matrice de distance asymétrique. On résout le probléme au moyen d'un arbre de “branch and bound” dans lequel les sous-problémes sont des problémes d'affectation généralisée. On présente des résultats numériques pour des problèmes contenant justu'à 100 points de livraison.     

A numerical algorithm for solving the three-dimensional Stefan problem in spatial coordinates fixed to the solution

Presented at the seminar of the Department of Computational Mathematics, November 3, 1983.     

A global optimization algorithm for solving the minimum multiple ratio spanning tree problem

This paper studies the sum-of-ratios version of the classical minimum spanning tree problem. We describe a branch-and-bound algorithm for solving the general version of the problem based on its image space representation. The suggested approach specifically addresses the difficulties arising in the case when the number of ratios exceeds two. The efficacy of our approach is demonstrated on randomly generated complete and sparse graph instances.     

A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization

In this paper, we introduce a new iterative scheme for finding a common element of the set of solutions of an equilibrium problem, the set of common fixed point for a family of infinitely nonexpansive mappings and the set of solutions of the variational inequality for α$\alpha$-inverse-strongly monotone mappings in a Hilbert space. Under suitable conditions, some strong convergence theorems for approximating a common element of the above three sets are obtained. As applications, at the end of the paper we utilize our results to study the optimization problem and some convergence problem for strictly pseudocontractive mappings. The results presented in the paper extend and improve some recent results of Yao and Yao [Y.Y. Yao, J.C. Yao, On modified iterative method for nonexpansive mappings and monotone mappings, Appl. Math. Comput. 186 (2) (2007) 1551–1558], Plubtieng and Punpaeng [S. Plubtieng, R. Punpaeng, A new iterative method for equilibrium problems and fixed point problems of nonlinear mappings and monotone mappings, Appl. Math. Comput. (2007) doi:10.1016/j.amc.2007.07.075], S. Takahashi and W. Takahashi [S. Takahashi, W. Takahashi, Viscosity approximation methods for Equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2006) 506–515], Su, Shang and Qin [Y.F. Su, M.J. Shang, X.L. Qin, An iterative method of solution for equilibrium and optimization problems, Nonlinear Anal. (2007) doi:10.1016/j.na.2007.08.045] and Chang, Cho and Kim [S.S. Chang, Y.J. Cho, J.K. Kim, Approximation methods of solutions for equilibrium problem in Hilbert spaces, Dynam. Systems Appl. (in print)].     

A branch-and-bound algorithm with fuzzy inference for a permutation flowshop scheduling problem

This paper considers an m-machine permutation flowshop scheduling problem of minimizing the makespan. This classical scheduling problem is still important in modern manufacturing systems, and is well known to be intractable (i.e., NP-hard). In fact branch-and-bound algorithms developed so far for this problem have not come to solve large scale problem instances with over a hundred jobs. In order to improve the performance of branch-and-bound algorithms this paper proposes a new dominance relation by which the search load could be reduced, and notices that it is based on a sufficient precondition. This suggests that the dominance relation holds with high possibility even if the precondition approximately holds, thus being more realistic. The branch-and-bound algorithm proposed here takes advantage of this possibility for obtaining an optimal solution as early as possible in the branch-and-bound search. For this purpose this paper utilizes membership functions in the context of the fuzzy inference. Extensive numerical experiments that were executed through Monte Carlo simulations and benchmark tests show that the developed branch-and-bound algorithm can solve 3-machine problem instances with up to 1000 jobs with probability of over 99%, and 4-machine ones with up to 900 jobs with over 97%.     

A combined branch-and-bound and genetic algorithm based approach for a flowshop scheduling problem

In this paper, we study the application of a meta-heuristic to a two-machine flowshop scheduling problem. The meta-heuristic uses a branch-and-bound procedure to generate some information, which in turn is used to guide a genetic algorithm's search for optimal and near-optimal solutions. The criteria considered are makespan and average job flowtime. The problem has applications in flowshop environments where management is interested in reducing turn-around and job idle times simultaneously. We develop the combined branch-and-bound and genetic algorithm based procedure and two modified versions of it. Their performance is compared with that of three algorithms: pure branch-and-bound, pure genetic algorithm, and a heuristic. The results indicate that the combined approach and its modified versions are better than either of the pure strategies as well as the heuristic algorithm.     

A branch-and-bound algorithm to solve the linear ordering problem for weighted tournaments

The linear ordering problem consists in finding a linear order at minimum remoteness from a weighted tournament T, the remoteness being the sum of the weights of the arcs that we must reverse in T to transform it into a linear order. This problem, also known as the search of a median order, or of a maximum acyclic subdigraph, or of a maximum consistent set, or of a minimum feedback arc set, is NP-hard; when all the weights of T are equal to 1, the linear ordering problem is the same as Slater's problem. In this paper, we describe the principles and the results of an exact method designed to solve the linear ordering problem for any weighted tournament. This method, of which the corresponding software is freely available at the URL address http://www.enst.fr/~charon/tournament/median.html, is based upon a branch-and-bound search with a Lagrangean relaxation as the evaluation function and a noising method for computing the initial bound. Other components are designed to reduce the BB-search-tree.     

Lower bound on size of branch-and-bound trees for solving lot-sizing problem

We show that there exists a family of instances of the lot-sizing problem, such that any branch-and-bound tree that solves them requires an exponential number of nodes, even in the case when the branchings are performed on general split disjunctions. This result is of interest since there exists dynamic programming algorithm that solves lot-sizing in polynomial-time. To the best of our knowledge, this is the first study that shows that dynamic programming can be exponentially faster than branch-and-bound.     

A branch-and-bound algorithm for a two-machine flowshop scheduling problem with limited waiting time constraints

In this paper, we consider a two-machine flowshop scheduling problem in which the waiting time of each job between the two machines cannot be greater than a certain time period. For the problem with the objective of minimizing makespan, we identify several dominance properties of the problem and develop a branch-and-bound (B&B) algorithm using the dominance properties. Computational tests are performed on randomly generated test problems for evaluation of performance of the B&B algorithm, and results show that the algorithm can solve problems with up to 150 jobs in a reasonable amount of CPU time.