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     

A Genetic Algorithm for the Multidimensional Knapsack Problem

In this paper we present a heuristic based upon genetic algorithms for the multidimensional knapsack problem. A heuristic operator which utilises problem-specific knowledge is incorporated into the standard genetic algorithm approach. Computational results show that the genetic algorithm heuristic is capable of obtaining high-quality solutions for problems of various characteristics, whilst requiring only a modest amount of computational effort. Computational results also show that the genetic algorithm heuristic gives superior quality solutions to a number of other heuristics.     

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.     

Scatter Search for the 0–1 Multidimensional Knapsack Problem

The evolutionary metaheuristic called scatter search has been applied successfully to optimization problems for several years. In this paper, we apply the scatter search technique to the well-known 0–1 multidimensional knapsack problem. We propose a new relaxation-based diversification generator, which produces an initial population with elite solutions. The computational results obtained for a set of classic and correlated instances clearly show that (1) this generator can also be used as a heuristic for solving the multidimensional knapsack problem; (2) using the population produced by our generator as a starting point for the scatter search algorithm leads to better performance. We also enhance the scatter search algorithm by integrating memory and by using adapted intensification phases. Overall, the results are interesting and competitive compared to other population-based algorithms, such as genetic algorithms.      

Heuristic and Exact Algorithms for the Precedence-Constrained Knapsack Problem

The knapsack problem (KP) is generalized taking into account a precedence relation between items. Such a relation can be represented by means of a directed acyclic graph, where nodes correspond to items in a one-to-one way. As in ordinary KPs, each item is associated with profit and weight, the knapsack has a fixed capacity, and the problem is to determine the set of items to be included in the knapsack. However, each item can be adopted only when all of its predecessors have been included in the knapsack. The knapsack problem with such an additional set of constraints is referred to as the precedence-constrained knapsack problem (PCKP). We present some dynamic programming algorithms that can solve small PCKPs to optimality, as well as a preprocessing method to reduce the size of the problem. Combining these, we are able to solve PCKPs with up to 2000 items in less than a few minutes of CPU time.     

An efficient alorithm for the general multiple-choice knapsack problem (GMKP)

A common problem frequently faced by business firms and individual investors is to select a few investment opportunities from many available possibilities. This problem, in its simplest form, can be modeled as a 0–1 knapsack problem. In a more general investment scenario, however, we obtain a model which is a general knapsack problem with a multiple-choice constraint. To solve this problem, an efficient enumerative algorithm is developed. The algorithm includes an efficient procedure to solve the LP-relaxed problem, a reduction algorithm which may allow the initial fixing of some of the variables, and various other implicit enumeration criteria derived from the group problem. Extensive computational experience illustrates the efficiency of the algorithm and related results.     

Cost Based Filtering for the Constrained Knapsack Problem

We present cost based filtering methods for Knapsack Problems (KPs). Cost based filtering aims at fixing variables with respect to the objective function. It is an important technique when solving complex problems such as Quadratic Knapsack Problems, or KPs with additional constraints (Constrained Knapsack Problems (CKPs)). They evolve, e.g., when Constraint Based Column Generation is applied to appropriate optimization problems. We develop new reduction algorithms for KP. They are used as propagation routines for the CKP with (nlogn) preprocessing time and (n) time per call. This sums up to an amortized time (n) for (logn) incremental calls where the subsequent problems may differ with respect to arbitrary sets of necessarily included and excluded items.     

A dual approach for the continuous collapsing knapsack problem

We formulate and solve a dual version of the Continuous Collapsing Knapsack Problem using a geometric approach. Optimality conditions are found and an algorithm is presented. Computational experience shows that this procedure is efficient.     

0-1背包问题的蜂群优化算法

在项目决策与规划、资源分配、货物装载、预算控制等工作中,提出了0-1背包问题.0-1背包问题是组合优化中的典型NP难题,根据群集智能原理,给出一种基于蜂群寻优思想的新算法—蜂群算法,并针对0-1背包问题进行求解.经实验仿真并与蚁群算法计算结果作对比,验证了算法在0-1背包问题求解上的有效性和更快的收敛速度.     

Tight bounds for periodicity theorems on the unbounded Knapsack problem

Three new bounds for periodicity theorems on the unbounded Knapsack problem are developed. Periodicity theorems specify when it is optimal to pack one unit of the best item (the one with the highest profit-to-weight ratio). The successive applications of periodicity theorems can drastically reduce the size of the Knapsack problem under analysis, theoretical or empirical. We prove that each new bound is tight in the sense that no smaller bound exists under the given condition.     

Cutting and Surrogate Constraint Analysis for Improved Multidimensional Knapsack Solutions

We use surrogate analysis and constraint pairing in multidimensional knapsack problems to fix some variables to zero and to separate the rest into two groups – those that tend to be zero and those that tend to be one, in an optimal integer solution. Using an initial feasible integer solution, we generate logic cuts based on our analysis before solving the problem with branch and bound. Computational testing, including the set of problems in the OR-library and our own set of difficult problems, shows our approach helps to solve difficult problems in a reasonable amount of time and, in most cases, with a fewer number of nodes in the search tree than leading commercial software.     

New lower bounds for bin packing problems with conflicts

The bin packing problem with conflicts (BPC) consists of minimizing the number of bins used to pack a set of items, where some items cannot be packed together in the same bin due to compatibility restrictions. The concepts of dual-feasible functions (DFF) and data-dependent dual-feasible functions (DDFF) have been used in the literature to improve the resolution of several cutting and packing problems. In this paper, we propose a general framework for deriving new DDFF as well as a new concept of generalized data-dependent dual-feasible functions (GDDFF), a conflict generalization of DDFF. The GDDFF take into account the structure of the conflict graph using the techniques of graph triangulation and tree-decomposition. Then we show how these techniques can be used in order to improve the existing lower bounds.     

Exact Algorithm for Concave Knapsack Problems: Linear Underestimation and Partition Method

Integer programming problems with a concave cost function are often encountered in optimization models involving economics of scale. In this paper, we propose an efficient exact algorithm for solving concave knapsack problems. The algorithm consists of an iterative process between finding lower and upper bounds by linearly underestimating the objective function and performing domain cut and partition by exploring the special structure of the problem. The lower bound is improved iteratively via cutting and partitioning the domain. This iteration process converges to the optimality in a finite number of steps. Promising computational results are reported for large-scale concave knapsack problems with up to 1200 integer variables. Comparison results with other existing methods in the literature are also presented. *Research supported by the National Natural Science Foundation of China under Grants 79970107 and 10271073,and the Research Grants Council of Hong Kong under Grant CUHK 4214/01E.     

Tabu Search Based Procedure for Solving the 0-1 MultiObjective Knapsack Problem: The Two Objectives Case

We consider in this paper the solving of 0-1 knapsack problems with multiple linear objectives. We present a tabu search approach to generate a good approximation of the efficient set. The heuristic scheme is included in a redu tion decision space framework. The case of two objectives is developed in this paper. TS principles viewed into the multiobjective context are discussed. According to a prospective way, several variations of the algorithm are investigate. Numerical experiments are reported and compared with available exact efficient solutions. Intuitive justifications for the observed empirical behavior of the procedure and open questions are discussed.     

不可分离凸背包问题的拉格朗日分解和区域分割方法

本文对线性约束不可分离凸背包问题给出了一种精确算法．该算法是拉格朗日分解和区域分割结合起来的一种分枝定界算法．利用拉格朗日分解方法可以得到每个子问题的一个可行解，一个不可行解，一个下界和一个上界．区域分割可以把一个整数箱子分割成几个互不相交的整数子箱子的并集，每个整数子箱子对应一个子问题．通过区域分割可以逐步减小对偶间隙并最终经过有限步迭代找到原问题的最优解．数值结果表明该算法对不可分离凸背包问题是有效的．     

Minimum cycle cover and Chinese postman problems on mixed graphs with bounded tree-width

Let M=(V,E,A) be a mixed graph with vertex set V, edge set E and arc set A. A cycle cover of M is a family C={C₁,…,C_k} of cycles of M such that each edge/arc of M belongs to at least one cycle in C. The weight of C is . The minimum cycle cover problem is the following: given a strongly connected mixed graph M without bridges, find a cycle cover of M with weight as small as possible. The Chinese postman problem is: given a strongly connected mixed graph M, find a minimum length closed walk using all edges and arcs of M. These problems are NP-hard. We show that they can be solved in polynomial time if M has bounded tree-width.     

A real coded genetic algorithm for solving integer and mixed integer optimization problems

In this paper, a real coded genetic algorithm named MI-LXPM is proposed for solving integer and mixed integer constrained optimization problems. The proposed algorithm is a suitably modified and extended version of the real coded genetic algorithm, LXPM, of Deep and Thakur [K. Deep, M. Thakur, A new crossover operator for real coded genetic algorithms, Applied Mathematics and Computation 188 (2007) 895-912; K. Deep, M. Thakur, A new mutation operator for real coded genetic algorithms, Applied Mathematics and Computation 193 (2007) 211-230]. The algorithm incorporates a special truncation procedure to handle integer restrictions on decision variables along with a parameter free penalty approach for handling constraints. Performance of the algorithm is tested on a set of twenty test problems selected from different sources in literature, and compared with the performance of an earlier application of genetic algorithm and also with random search based algorithm, RST2ANU, incorporating annealing concept. The proposed MI-LXPM outperforms both the algorithms in most of the cases which are considered.     

A new approach to factorization of a class of almost-periodic triangular symbols and related Riemann-Hilbert problems

The factorization of almost-periodic triangular symbols, G, associated to finite-interval convolution operators is studied for two classes of operators whose Fourier symbols are almost periodic polynomials with spectrum in the group αZ+βZ+Z (α,β∈]0,1[, α+β>1, α/β∉Q). The factorization problem is solved by a method that is based on the calculation of one solution of the Riemann-Hilbert problem GΦ₊=Φ₋ in L_∞(R) and does not require solving the associated corona problems since a second linearly independent solution is obtained by means of an appropriate transformation on the space of solutions to the Riemann-Hilbert problem. Some unexpected, but interesting, results are obtained concerning the Fourier spectrum of the solutions of GΦ₊=Φ₋. In particular it is shown that a solution exists with Fourier spectrum in the additive group αZ+βZ whether this group contains Z or not. Possible application of the method to more general classes of symbols is considered in the last section of the paper.