Tree Search Procedure Research Articles

An efficient optimal algorithm for the quantity discount problem in material requirement planning

An optimal algorithm based on branch-and-bound approach is presented in this paper to determine lot sizes for a single item in material requirement planning environments with deterministic time-phased demand and constant ordering cost with zero lead time, where all-units discounts are available from vendors and backlog is not permitted. On the basis of the proven properties of optimal order policy, a tree-search procedure is presented to construct the sequence of optimal orders. Some useful fathom rules have been proven, which make the algorithm very efficient. To compare the performance of this algorithm with the other existing optimal algorithms, an experimental design with various environments has been developed. Experimental results show that the performance of our optimal algorithm is much better than the performance of other existing optimal algorithms. Considering computational time as the performance measure, this algorithm is considered the best among the existing optimal algorithms for real problems with large dimensions (i.e. large number of periods and discount levels).

An exact algorithm for generating homogenous two-segment cutting patterns

An exact algorithm is proposed for generating homogenous two-segment patterns for the constrained two-dimensional guillotine-cutting problems of rectangular items. It is a bottom-up approach combined with branch-and-bound techniques. The stock plate is divided into two segments. Each segment consists of strips of the same length and direction. Only homogenous strips are considered, each of which contains items of the same type. The strip directions of the two segments may be either the same or perpendicular to each other. The algorithm uses a tree-search procedure. It starts from an initial lower bound, implicitly generates all possible segments through the assembly of strips, and constructs possible patterns through the combination of two segments. Tighter bounds are established to discard non-promising segments. The computational results indicate that the algorithm is efficient both in computation time and in material utilization, and is able to deal with relatively large-scale problems.

Cycles and communicating classes in membrane systems and molecular dynamics

We are considering sequential membrane systems and molecular dynamics from the viewpoint of Markov chain theory. The configuration space of these systems (including the transitions) is a special kind of directed graph, called a pseudo-lattice digraph, which is closely related to the stoichiometric matrix. Taking advantage of the monoidal structure of this space, we introduce the algebraic notion of precycle. A precycle leads to the identification of cycles by means of the concept of defect, which is a set of geometric constraints on configuration space. Two efficient algorithms for evaluating precycles and defects are given: one is an algorithm due to Contejean and Devie, the other is a novel branch-and-bound tree search procedure. Cycles partition configuration space into equivalence classes, called the communicating classes. The structure of the communicating classes in the free regime–where all rules are enabled–is analyzed: testing for communication can be done efficiently. We show how to apply these ideas to a biological regulatory system.

Modeling and solving a Crew Assignment Problem in air transportation

A typical problem arising in airline crew management consists in optimally assigning the required crew members to each flight segment of a given time period, while complying with a variety of work regulations and collective agreements. This problem called the Crew Assignment Problem ( CAP) is currently decomposed into two independent sub-problems which are modeled and solved sequentially: (a) the well-known Crew Pairing Problem followed by (b) the Working Schedules Construction Problem. In the first sub-problem, a set of legal minimum-cost pairings is constructed, covering all the planned flight segments. In the second sub-problem, pairings, rest periods, training periods, annual leaves, etc. are combined to form working schedules which are then assigned to crew members. In this paper, we present a new approach to the Crew Assignment Problem arising in the context of airline companies operating short and medium haul flights. Contrary to most previously published work on the subject, our approach is not based on the concept of crew-pairings, though it is capable of handling many of the constraints present in crew-pairing-based models. Moreover, contrary to crew-pairing-based approaches, one of its distinctive features is that it formulates and solves the two sub-problems (a) and (b) simultaneously for the technical crew members (pilots and officers) with specific constraints. We show how this problem can be formulated as a large scale integer linear program with a general structure combining different types of constraints and not exclusively partitioning or covering constraints as usually suggested in previous papers. We introduce then, a formulation enhancement phase where we replace a large number of binary exclusion constraints by stronger and less numerous ones: the clique constraints. Using data provided by the Tunisian airline company TunisAir, we demonstrate that thanks to this new formulation, the Crew Assignment Problem can be solved by currently available integer linear programming technology. Finally, we propose an efficient heuristic method based on a rounding strategy embedded in a partial tree search procedure. The implementation of these methods (both exact and heuristic ones) provides good solutions in reasonable computation times using CPLEX 6.0.2: guaranteed exact solutions are obtained for 60% of the test instances and solutions within 5% of the lower bound for the others.

Métodos exatos baseados em relaxações lagrangiana e surrogate para o problema de carregamento de paletes do produtor

Neste artigo apresentamos métodos exatos, baseados em relaxações Lagrangiana e surrogate, com bom desempenho para resolver o problema de carregamento de paletes do produtor. Este problema consiste em arranjar ortogonalmente e sem sobreposição o máximo número de retângulos de dimensões (l,w) ou (l,w) sobre um retângulo maior (L,W). Os métodos propostos são procedimentos de busca em árvore do tipo branch and bound que, em cada nó, utilizam limitantes derivados de relaxações Lagrangiana e/ou surrogate de uma formulação de programação linear 0-1. Algoritmos de otimização do subgradiente são usados para otimizar estes limitantes. São aplicados ainda testes de redução do problema e heurísticas Lagrangiana e surrogate para se obter boas soluções factíveis na otimização do subgradiente. Testes computacionais foram realizados utilizando exemplos da literatura e exemplos reais de uma transportadora. Os resultados mostram que um dos métodos propostos é competitivo com outros métodos da literatura, incluindo o software GAMS/CPLEX.

Heuristic and exact algorithms for generating homogenous constrained three-staged cutting patterns

An approach is proposed for generating homogenous three-staged cutting patterns for the constrained two-dimensional guillotine-cutting problems of rectangles. It is based on branch-and-bound procedure combined with dynamic programming techniques. The stock plate is divided into segments. Each segment consists of strips with the same direction. Only homogenous strips are allowed, each of which contains rectangles of the same size. The approach uses a tree-search procedure. It starts from an initial lower bound, implicitly generates all possible segments through the builds of strips, and constructs all possible patterns through the builds of segments. Tighter bounds are established to discard non-promising segments and patterns. Both heuristic and exact algorithms are proposed. The computational results indicate that the algorithms are capable of dealing with problems of larger scale. Finally, the solution to a cutting problem taken from a factory that makes passenger cars is given.

A Framework for Optimal Correction of Inconsistent Linear Constraints

The problem of inconsistency between constraints often arises in practice as the result, among others, of the complexity of real models or due to unrealistic requirements and preferences. To overcome such inconsistency two major actions may be taken: removal of constraints or changes in the coefficients of the model. This last approach, that can be generically described as "model correction" is the problem we address in this paper in the context of linear constraints over the reals. The correction of the right hand side alone, which is very close to a fuzzy constraints approach, was one of the first proposals to deal with inconsistency, as it may be mapped into a linear problem. The correction of both the matrix of coefficients and the right hand side introduces non linearity in the constraints. The degree of difficulty in solving the problem of the optimal correction depends on the objective function, whose purpose is to measure the closeness between the original and corrected model. Contrary to other norms, that provide corrections with quite rigid patterns, the optimization of the important Frobenius norm was still an open problem. We have analyzed the problem using the KKT conditions and derived necessary and sufficient conditions which enabled us to unequivocally characterize local optima, in terms of the solution of the Total Least Squares and the set of active constraints. These conditions justify a set of pruning rules, which proved, in preliminary experimental results, quite successful in a tree search procedure for determining the global minimizer.

Scheduling fixtures for New Zealand Cricket

This paper describes the problem faced every year by New Zealand Cricket in scheduling the principal inter-regional fixtures. This is a combinatorial optimization problem with few constraints but many objectives, which are described in detail. A technique known as Subcost-Guided Simulated Annealing is used to solve this problem, producing one or more schedules of high quality. One particular feature of the problem requires great care - the determination of adequate neighbourhoods for such a tightly constrained problem, where most simple changes lead to infeasibility. The approach adopted is to use a complex and unorthodox definition of a perturbation, each one leading to several possible neighbouring solutions which are generated by means of a tree search procedure. The system will be used in practice for the 2003-4 cricket season and beyond.

Branch and win: OR tree search algorithms for solving combinatorial optimisation problems

Currently, most combinatorial optimisation problems have to be solved, if the optimum solution is sought, using general techniques to explore the space of feasible solutions and, more specifically, through exploratory enumerative procedures in trees and search graphs. We propose Branch and Win, a general formulation for understanding and synthesising the different tree search procedures that have been presented in the literature of operations research as well as in that of artificial intelligence. Several general ideas are also presented, whose application allows designing new hybrid search algorithms, in order to implement the procedure.

Recovering Beam Search: Enhancing the Beam Search Approach for Combinatorial Optimization Problems

A hybrid heuristic method for combinatorial optimization problems is proposed that combines different classical techniques such as tree search procedures, bounding schemes and local search. The proposed method enhances the classic beam search approach by applying to each partial solution corresponding to a node selected by the beam, a further test that checks whether the current partial solution is dominated by another partial solution at the same level of the search tree. If this is the case, the latter solution becomes the new current partial solution. This step allows to partially recover from previous wrong decisions of the beam search procedure and can be seen as a local search step on the partial solution. We present here the application to two well known combinatorial optimization problems: the two-machine total completion time flow shop scheduling problem and the uncapacitated p-median location problem. In both cases the method strongly improves the performances with respect to the basic beam search approach and is competitive with the state of the art heuristics.

A heuristic for job shop scheduling to minimize total weighted tardiness

This paper considers the job shop scheduling problem to minimize the total weighted tardiness with job-specific due dates and delay penalties, and a heuristic algorithm based on the tree search procedure is developed for solving the problem. A certain job shop scheduling to minimize the maximum tardiness subject to fixed sub-schedules is solved at each node of the search tree, and the successor nodes are generated, where the sub-schedules of the operations are fixed. Thus, a schedule is obtained at each node, and the sub-optimum solution is determined among the obtained schedules. Computational results on some 10 jobs and 10 machines problems and 15 jobs and 15 machines problems show that the proposed algorithm can find the sub-optimum solutions with a little computation time.

Exact and Approximate Nondeterministic Tree-Search Procedures for the Quadratic Assignment Problem

This article introduces two new techniques for solving the Quadratic Assignment Problem. The first is a heuristic technique, defined in accordance with the Ant System metaphor, and includes as a distinctive feature the use of a new lower bound at each constructive step. The second is a branch-and-bound exact approach, containing some elements introduced in the Ant algorithm. Computational results prove the effectiveness of both approaches.

A dynamic programming based algorithm for the crew scheduling problem

A dynamic programming based algorithm for the crew scheduling problem. There are many papers concerned with the crew scheduling problem in the literature. Most of the research literature views the crew scheduling problem as equivalent of the set covering problem, or as equivalent to the set partitioning problem. In this paper we highlight the fact that the crew scheduling problem is a separate and distinct problem. This leads naturally to a dynamic programming formulation of the problem. Following on from this formulation a new lower bound for the crew scheduling problem is developed based upon dynamic programming. Incorporating this lower bound into a tree search procedure we are able to solve to proven optimality a number of problems involving a relatively large number of tasks.In this paper we consider the crew scheduling problem, that is the problem of assigning K crews to tasks with fixed start and finish times such that each crew does not exceed a limit on the total time it can spend working. This problem is formulated as a problem of finding K time limit constrained vertex disjoint paths which visit all vertices on a network. A lower bound for the problem is found via dynamic programming. This lower bound is improved via a Lagrangean based penalty procedure and subgradient optimisation. Computational results are given for a number of randomly generated problems involving between 50 and 500 tasks.

An improvement of viswanathan and bagchi's exact algorithm for constrained two-dimensional cutting stock

Viswanathan and Bagchi [ Operations Research, 1993, 41(4), 768–776] [1] have proposed a bottom-up algorithm which combines in the nice tree-search procedure Gilmore and Gomory's algorithm, called at each node of the tree, for solving exactly the constrained two-dimensional cutting problem. This algorithm is one of the best exact algorithms known today. In this paper, we propose an improved version of this algorithm by introducing one-dimensional bounded knapsacks in the original algorithm. Then, by exploiting dynamic programming properties, we obtain good lower and upper bounds which lead to significant branching cuts. Finally, the improved version is compared to the standard version of Viswanathan and Bagchi on some small and medium instances.

Constrained two-dimensional cutting: an improvement of Christofides and Whitlock's exact algorithm

Christofides and Whitlock have developed a top-down algorithm which combines in a nice tree search procedure Gilmore and Gomory's algorithm and a transportation routine called at each node of the tree for solving exactly the constrained two-dimensional cutting problem. Recently, another bottom-up algorithm has been developed and reported as being more efficient. In this paper, we propose a modification to the branching strategy and we introduce the one-dimensional bounded knapsack in the original Christofides and Whitlock algorithm. Then, by exploiting dynamic programming properties we obtain good lower and upper bounds which lead to significant branching cuts, resulting in a drastic reduction of calls of the transportation routine. Finally, we propose an incremental solution of the numerous generated transportation problems. The resulting algorithm reveals superior performance to other known algorithms.

The DH/KD algorithm: a hybrid approach for unconstrained two-dimensional cutting problems

We study both weighted and unweighted unconstrained two-dimensional guillotine cutting problems. We develop a hybrid approach which combines two heuristics from the literature. The first one (DH) uses a tree-search procedure introducing two strategies: Depth-first search and Hill-climbing. The second one (KD) is based on a series of one-dimensional Knapsack problems using Dynamic programming techniques. The DH /KD algorithm starts with a good initial lower bound obtained by using the KD algorithm. At each level of the tree-search, the proposed algorithm uses also the KD algorithm for constructing new lower bounds and uses another one-dimensional knapsack for constructing refinement upper bounds. The resulting algorithm can be seen as a generalization of the two heuristics and solves large problem instances very well within small computational time. Our algorithm is compared to Morabito et al.'s algorithm (the unweighted case), and to Beasley's [2] approach (the weighted case) on some examples taken from the literature as well as randomly generated instances.

A tree search algorithm for the crew scheduling problem

In this paper we consider the crew scheduling program, that is the problem of assigning K crews to N tasks with fixed start and finish times such that each crew does not exceed a limit on the total time it can spend working. A zero-one integer linear programming formulation of the problem is given, which is then relaxed in a lagrangean way to provide a lower bound which is improved by subgradient optimisation. Finally a tree search procedure incorporating this lower bound is presented to solve the problem to optimality. Computational results are given for a number of randomly generated test problems involving between 50 and 500 tasks.

A recursive exact algorithm for weighted two-dimensional cutting

Gilmore and Gomory's algorithm is one of the better actually known exact algorithms for solving unconstrained guillotine two-dimensional cutting problems. Herz's algorithm is more effective, but only for the unweighted case. We propose a new exact algorithm adequate for both weighted and unweighted cases, which is more powerful than both algorithms. The algorithm uses dynamic programming procedures and one-dimensional knapsack problem to obtain efficient lower and upper bounds and important optimality criteria which permit a significant branching cut in a recursive tree-search procedure. Recursivity, computational power, adequateness to parallel implementations, and generalization for solving constrained two-dimensional cutting problems, are some important features of the new algorithm.

A new exact algorithm for the vehicle routing problem based onq-paths andk-shortest paths relaxations

We consider the basic Vehicle Routing Problem (VRP) in which a fleet ofM identical vehicles stationed at a central depot is to be optimally routed to supply customers with known demands subject only to vehicle capacity constraints. In this paper, we present an exact algorithm for solving the VRP that uses lower bounds obtained from a combination of two relaxations of the original problem which are based on the computation ofq-paths andk-shortest paths. A set of reduction tests derived from the computation of these bounds is applied to reduce the size of the problem and to improve the quality of the bounds. The resulting lower bounds are then embedded into a tree-search procedure to solve the problem optimally. Computational results are presented for a number of problems taken from the literature. The results demonstrate the effectiveness of the proposed method in solving problems involving up to about 50 customers and in providing tight lower bounds for problems up to about 150 customers.

An exact algorithm for general, orthogonal, two-dimensional knapsack problems

We present a new exact tree-search procedure for solving two-dimensional knapsack problems in which a number of small rectangular pieces, each of a given size and value, are required to be cut from a large rectangular stock plate. The objective is to maximise the value of pieces cut or minimise the wastage. We consider the case where there is a maximum number of times that a piece may be used in a cutting pattern. The algorithm limits the size of the tree search by using a bound derived from a Langrangean relaxation of a 0–1 integer programming formulaton of the problem. Subgradient optimisation is used to optimise this bound. Reduction tests derived from both the original problem and the Lagrangean relaxation produce substantial computational gains. The computational performance of the algorithm indicates that it is an effective procedure capable of solving optimally practical two-dimensional cutting problems of medium size.