A two-stage heuristic for disassembly scheduling with assembly product structure

Disassembly scheduling, one of the important operational problems in disassembly systems, is the problem of determining the ordering and disassembly schedules of used or end-of-life products while satisfying the demand of their parts or components over a certain planning horizon. This paper considers products with assembly structure for the objective of minimizing the sum of purchase, set up, inventory holding, and disassembly operation costs, and suggests a two-stage heuristic, in which an initial solution is obtained in the form of the minimal latest ordering and disassembly schedule, and then improved iteratively considering trade-offs among different cost factors. To show the performance of the heuristic, computational experiments were done on the example obtained from the literature and a number of randomly generated test problems, and the results show that the heuristic can give optimal or very near-optimal solutions within very short computation times.     

A branch and bound algorithm for the two-stage assembly scheduling problem

This paper develops a branch and bound algorithm for the two-stage assembly scheduling problem. In this problem, there are m machines at the first stage, each of which produces a component of a job. When all m components are available, a single assembly machine at the second stage completes the job. The objective is to schedule the jobs on the machines so that the maximum completion time, or makespan, is minimized. A lower bound based on solving an artificial two-machine flow shop problem is derived. Also, several dominance theorems are established and incorporated into the branch and bound algorithm. Computational experience with the algorithm is reported for problems with up to 8000 jobs and 10 first-stage machines.     

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 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.     

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 Lagrangean heuristic algorithm for disassembly scheduling with capacity constraints

This paper considers the problem of determining the disassembly schedule (quantity and timing) of products in order to satisfy the demand of their parts or components over a finite planning horizon. The objective is to minimize the sum of set-up, disassembly operation, and inventory holding costs. As an extension of the uncapacitated versions of the problem, we consider the resource capacity restrictions over the planning horizon. An integer program is suggested to describe the problem mathematically, and to solve the problem, a heuristic is developed using a Lagrangean relaxation technique together with a method to find a good feasible solution while considering the trade-offs among different costs. The effectiveness of the algorithm is tested on a number of randomly generated problems and the test results show that the heuristic suggested in this paper can give near optimal solutions within a short amount of computation time.     

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.     

Project scheduling with resource constraints: A branch and bound approach

This paper describes a branch and bound algorithm for project scheduling with resource constraints. The algorithmis based on the idea of using disjunctive arcs for resolving conflicts that are created whenever sets of activities have to be scheduled whose total resource requirements exceed the resource availabilities in some periods. Four lower bounds are examined. The first is a simple lower bound based on longest path computations. The second and third bounds are derived from a relaxed integer programming formulation of the problem. The second bound is based on the Linear Programming relaxation with the addition of cutting planes, and the third bound is based on a Lagrangean relaxation of the formulation. This last relaxation involves a problem which is a generalization of the longest path computation and for which an efficient, though not polynomial, algorithm is given. The fourth bound is based on the disjunctive arcs used to model the problem as a graph.We report computational results on the performance of each bound on randomly generated problems involving up to 25 activities and 3 resources.     

A branch and bound algorithm for bound constrained optimization problems without derivatives

In this paper, we give a new branch and bound algorithm for the global optimization problem with bound constraints. The algorithm is based on the use of inclusion functions. The bounds calculated for the global minimum value are proved to be correct, all rounding errors are rigorously estimated. Our scheme attempts to exclude most uninteresting parts of the search domain and concentrates on its promising subsets. This is done as fast as possible (by involving local descent methods), and uses little information as possible (no derivatives are required). Numerical results for many well-known problems as well as some comparisons with other methods are given.     

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 acyclic subgraph problem

A branch and bound algorithm for the acyclic subgraph problem (feedback are set problem) is described. The branching scheme lexicographically enumerates all permutations, skipping initial segments known by some easy tests not to have any optimal completion. A lower bound for the number of feedback arcs is given by the size of any collection of disjoint cycles. We propose a heuristic algorithm to find a large collection. The size of the problems our branch and bound algorithm can solve varies from 25 to 34 nodes, depending on the nature of the problem.     

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 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 algorithm for the generalized assignment problem

This paper describes what is termed the generalized assignment problem. It is a generalization of the ordinary assignment problem of linear programming in which multiple assignments of tasks to agents are limited by some resource available to the agents. A branch and bound algorithm is developed that solves the generalized assignment problem by solving a series of binary knapsack problems to determine the bounds. Computational results are cited for problems with up to 4 000 0–1 variables, and comparisons are made with other algorithms.This research was partly supported by ONR Contracts N00014-67-A-0126-0008 and N00014-67-A-0126-0009 with the Center for Cybernetic Studies, The University of Texas.     

An improved branch and bound algorithm for single machine scheduling with deadlines to minimize total weighted completion time

We consider a scheduling problem in which n jobs with distinct deadlines are to be scheduled on a single machine. The objective is to find a feasible job sequence that minimizes the total weighted completion time. We present an efficient branch-and-bound algorithm that fully exploits the principle of optimality. Favorable numerical results are also reported on an extensive set of problem instances of 20-120 jobs.     

A branch and bound algorithm with constraint partitioning for integer goal programming problems

Using constraint partitioning and variable elimination, the authors have recently developed an efficient algorithm for solving linear goal programming problems. However, many goal programs require some or all of the decision variables to be integer valued. This paper shows how the new partitioning algorithm can be extended with a modified branch and bound strategy to solve both pure and mixed type integer goal programming problems. A potential problem in combining the partitioning algorithm and the branch and bound search scheme is presented and resolved.     

An interval branch and bound algorithm for bound constrained optimization problems

In this paper, we propose modifications to a prototypical branch and bound algorithm for nonlinear optimization so that the algorithm efficiently handles constrained problems with constant bound constraints. The modifications involve treating subregions of the boundary identically to interior regions during the branch and bound process, but using reduced gradients for the interval Newton method. The modifications also involve preconditioners for the interval Gauss-Seidel method which are optimal in the sense that their application selectively gives a coordinate bound of minimum width, a coordinate bound whose left endpoint is as large as possible, or a coordinate bound whose right endpoint is as small as possible. We give experimental results on a selection of problems with different properties.     

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 solving the multiple-choice knapsack problem

The multiple-choice knapsack problem is a binary knapsack problem with the addition of disjoint multiple-choice constraints. We describe a branch and bound algorithm based on embedding Glover and Klingman's method for the associated linear program within a depth-first search procedure. A heuristic is used to find a starting dual feasible solution to the associated linear program and a ‘pegging’ test is employed to reduce the size of the problem for the enumeration phase. Computational experience and comparisons with the code of Nauss and an algorithm of Armstrong et al. for the same problem are reported.