Shortest path algorithms for knapsack type problems

The group knapsack and knapsack problems are generalised to shortest path problems in a class of graphs called knapsack graphs. An efficient algorithm is described for finding shortest paths provided that arc lengths are non-negative. A more efficient algorithm is described for the acyclic case which includes the knapsack problem. In this latter case the algorithm reduces to a known algorithm.     

Pareto memetic algorithm for multiple objective optimization with an industrial application

Multiple objective combinatorial optimization problems are difficult to solve and often, exact algorithms are unable to produce optimal solutions. The development of multiple objective heuristics was inspired by the need to quickly produce acceptable solutions. In this paper, we present a new multiple objective Pareto memetic algorithm called PMS^MO. The PMS^MO algorithm incorporates an enhanced fine-grained fitness assignment, a double level archiving process and a local search procedure to improve performance. The performance of PMS^MO is benchmarked against state-of-the-art algorithms using 0–1 multi-dimensional multiple objective knapsack problem from the literature and an industrial scheduling problem from the aluminum industry.     

二层多目标最优化问题解集的连通性

研究了二层多目标最优化模型(BLMOP)解集的连通性问题,其中(BLMOP)的上层集值目标函数由下层问题的有效点确定.把(BLMOP)看作成单层的集值函数优化问题,借助集值函数优化问题各种有效解集的连通性的结论,得到了(BLMOP)相应的有效解集连通性的结论.     

Knapsack problems with setups

Knapsack problems with setups find their application in many concrete industrial and financial problems. Moreover, they also arise as subproblems in a Dantzig–Wolfe decomposition approach to more complex combinatorial optimization problems, where they need to be solved repeatedly and therefore efficiently. Here, we consider the multiple-class integer knapsack problem with setups. Items are partitioned into classes whose use implies a setup cost and associated capacity consumption. Item weights are assumed to be a multiple of their class weight. The total weight of selected items and setups is bounded. The objective is to maximize the difference between the profits of selected items and the fixed costs incurred for setting-up classes. A special case is the bounded integer knapsack problem with setups where each class holds a single item and its continuous version where a fraction of an item can be selected while incurring a full setup. The paper shows the extent to which classical results for the knapsack problem can be generalized to these variants with setups. In particular, an extension of the branch-and-bound algorithm of Horowitz and Sahni is developed for problems with positive setup costs. Our direct approach is compared experimentally with the approach proposed in the literature consisting in converting the problem into a multiple choice knapsack with pseudo-polynomial size.     

Zero-one integer programs with few constraints —Lower bounding theory

The zero-one integer programming problem and its special case, the multiconstraint knapsack problem frequently appear as subproblems in many combinatorial optimization problems. We present several methods for computing lower bounds on the optimal solution of the zero-one integer programming problem. They include Lagrangean, surrogate and composite relaxations. New heuristic procedures are suggested for determining good surrogate multipliers. Based on theoretical results and extensive computational testing, it is shown that for zero-one integer problems with few constraints surrogate relaxation is a viable alternative to the commonly used Lagrangean and linear programming relaxations. These results are used in a follow up paper to develop an efficient branch and bound algorithm for solving zero-one integer programming problems.     

带约束条件集值优化问题近似Henig有效解集的连通性

研究了带约束条件集值优化问题近似Henig有效解集的连通性.在实局部凸Hausdorff空间中,讨论了可行域为弧连通紧的,目标函数为C-弧连通的条件下,带约束条件集值优化问题近似Henig有效解集的存在性和连通性.并给出了带约束条件集值优化问题近似Henig有效解集的连通性定理.     

A simple greedy algorithm for a class of shuttle transportation problems

Greedy algorithms for combinatorial optimization problems are typically direct and efficient, but hard to prove optimality. The paper presents a special class of transportation problems where a supplier sends goods to a set of customers, returning to the source after each delivery. We show that these problems with different objective functions share a common structural property, and therefore a simple but powerful generic greedy algorithm yields optimal solutions for all of them.     

Using support vector machines to learn the efficient set in multiple objective discrete optimization

We propose using support vector machines (SVMs) to learn the efficient set in multiple objective discrete optimization (MODO). We conjecture that a surface generated by SVM could provide a good approximation of the efficient set. As one way of testing this idea, we embed the SVM-approximated efficient set information into a Genetic Algorithm (GA). This is accomplished by using a SVM-based fitness function that guides the GA search. We implement our SVM-guided GA on the multiple objective knapsack and assignment problems. We observe that using SVM improves the performance of the GA compared to a benchmark distance based fitness function and may provide competitive results.     

Algorithm robust for the bicriteria discrete optimization problem

We apply Algorithm Robust to various problems in multiple objective discrete optimization. Algorithm Robust is a general procedure that is designed to solve bicriteria optimization problems. The algorithm performs a weight space search in which the weights are utilized in min-max type subproblems. In this paper, we experiment with Algorithm Robust on the bicriteria knapsack problem, the bicriteria assignment problem, and the bicriteria minimum cost network flow problem. We look at a heuristic variation that is based on controlling the weight space search and has an indirect control on the sample of efficient solutions generated. We then study another heuristic variation which generates samples of the efficient set with quality guarantees. We report results of computational experiments.     

Connectedness of Henig Weakly Efficient Solution Set for Set-Valued Optimization Problems

In this paper, we study Henig weakly efficient solutions for set-valued optimization problems. The connectedness of the Henig weakly efficient solution set is proved under the condition that the objective function be a cone-arcwise connected set-valued mapping. As an application of the result, we establish the connectedness of the set of super efficient solutions.     

Connectedness of efficient solutions in multiple criteria combinatorial optimization

In multiple criteria optimization an important research topic is the topological structure of the set X_e of efficient solutions. Of major interest is the connectedness of X_e, since it would allow the determination of Xe without considering non-efficient solutions in the process. We review general results on the subject, including the connectedness result for efficient solutions in multiple criteria linear programming. This result can be used to derive a definition of connectedness for discrete optimization problems. We present a counterexample to a previously stated result in this area, namely that the set of efficient solutions of the shortest path problem is connected. We will also show that connectedness does not hold for another important problem in discrete multiple criteria optimization: the spanning tree problem.     

Connectedness of efficient points in convex and convex transformable vector optimization

Given a convex vector optimization problem with respect to a closed ordering cone, we show the connectedness of the efficient and properly efficient sets. The Arrow–Barankin–Blackwell theorem is generalized to nonconvex vector optimization problems, and the connectedness results are extended to convex transformable vector optimization problems. In particular, we show the connectedness of the efficient set if the target function f is continuously transformable, and of the properly efficient set if f is differentiably transformable. Moreover, we show the connectedness of the efficient and properly efficient sets for quadratic quasiconvex multicriteria optimization problems.     

Scatter tabu search for multiobjective clustering problems

We propose a hybrid heuristic procedure based on scatter search and tabu search for the problem of clustering objects to optimize multiple criteria. Our goal is to search for good approximations of the efficient frontier for this class of problems and provide a means for improving decision making in multiple application areas. Our procedure can be viewed as an extension of SSPMO (a scatter search application to nonlinear multiobjective optimization) to which we add new elements and strategies specially suited for combinatorial optimization problems. Clustering problems have been the subject of numerous studies; however, most of the work has focused on single-objective problems. Clustering using multiple criteria and/or multiple data sources has received limited attention in the operational research literature. Our scatter tabu search implementation is general and tackles several problems classes within this area of combinatorial data analysis. We conduct extensive experimentation to show that our method is capable of delivering good approximations of the efficient frontier for improved analysis and decision making.     

Case-Based Reasoning for Repetitive Combinatorial Optimization Problems, Part II: Numerical Results

This paper presents numerical results from the application of a case-based reasoning approach to several repetitive operations research problems. These experiments are applications of the ideas presented in the previous framework paper, Part I. The three combinatorial optimization problems explored in this paper are the knapsack problem, the travelling salesman problem and the uncapacitated plant location problem. These numerical experiments permit a comparison of the performance of this technique across these three problem classes as well as with the traditional solution algorithms.     

Approximation algorithms for combinatorial fractional programming problems

We are concerned with a combinatorial optimization problem which has the ratio of two linear functions as the objective function. This type of problems can be solved by an algorithm that uses an auxiliary problem with a parametrized linear objective function. Because of its combinatorial nature, however, it is often difficult to solve the auxiliary problem exactly. In this paper, we propose an algorithm which assumes that the auxiliary problems are solved only approximately, and prove that it gives an approximate solution to the original problem, of which the accuracy is at least as good as that of approximate solutions to the auxiliary problems. It is also shown that the time complexity is bounded by the square of the computation time of the approximate algorithm for the auxiliary problem. As an example of the proposed algorithm, we present a fully polynomial time approximation scheme for the fractional 0–1 knapsack problem.     

Case-based reasoning for repetitive combinatorial optimization problems, part I: Framework

This article presents a case-based reasoning approach for the development of learning heuristics for solving repetitive operations research problems. We first define the subset of problems we will consider in this work: repetitive combinatorial optimization problems. We then present several general forms that can be used to select previously solved problems that might aid in the solution of the current problem, and several different techniques for actually using this information to derive a solution for the current problem. We describe both fixed forms and forms that have the ability to change as we solve more problems. A simple example for the 0–1 knapsack problem is presented.     

A robust approach to the chance-constrained knapsack problem

In this paper, the chance-constrained knapsack problem (CKP) is addressed. Relying on robust optimization, a tractable combinatorial algorithm is proposed to solve approximately CKP. For two specific classes of uncertain knapsack problems, it is proved to solve CKP at optimality.     

Complexity of vector optimization problems on graphs

The paper is dedicated to the computation complexity of multi-objective optimization problems on graphs. The classes of multi-objective problems with polynomial complexity or being polynomially reduced to be NP-hard are marked out. The unsolvability of a series of combinatorial multi-objective problems has been set up by means of linear convolution algorithm. The sufficient conditions under which these algorithms are statistically efficient have also been obtained.     

Very Large-Scale Neighborhood Search for the K-Constraint Multiple Knapsack Problem

The K-Constraint Multiple Knapsack Problem (K-MKP) is a generalization of the multiple knapsack problem, which is one of the representative combinatorial optimization problems known to be NP-hard. In K-MKP, each item has K types of weights and each knapsack has K types of capacity. In this paper, we propose several very large-scale neighborhood search (VLSN) algorithms to solve K-MKP. One of the VLSN algorithms incorporates a novel approach that consists of randomly perturbing the current solution in order to efficiently produce a set of simultaneous non-profitable moves. These moves would allow several items to be transferred from their current knapsacks and assigned to new knapsacks, which makes room for new items to be inserted through multi-exchange movements and allows for improved solutions. Computational results presented show that the method is effective, and provides better solutions compared to exact algorithms run for the same amount of time. This paper was written during Dr. Cunha's sabbatical at the Industrial and Systems Engineering Department at the University of Florida, Gainesville as a visiting faculty     

I‐graphs and the corresponding configurations

We consider the class of I‐graphs I(n,j,k), which is a generalization over the class of the generalized Petersen graphs. We study different properties of I‐graphs, such as connectedness, girth, and whether they are bipartite or vertex‐transitive. We give an efficient test for isomorphism of I‐graphs and characterize the automorphism groups of I‐graphs. Regular bipartite graphs with girth at least 6 can be considered as Levi graphs of some symmetric combinatorial configurations. We consider configurations that arise from bipartite I‐graphs. Some of them can be realized in the plane as cyclic astral configurations, i.e., as geometric configurations with maximal isometric symmetry. © 2005 Wiley Periodicals, Inc.