A parallel two-list algorithm for the knapsack problem

An n-element knapsack problem has 2ⁿ possible solutions to search over, so a task which can be accomplished in 2″ trials if an exhaustive search is used. Due to the exponential time in solving the knapsack problem, the problem is considered to be very hard. In the past decade, much effort has been done in order to find techniques which could lead to practical algorithms with reasonable running time. In 1994, Chang et al. proposed a brilliant parallel algorithm, which needs O(2^n/8) processors to solve the knapsack problem in O(2^n/2) time; that is, the cost of Chang et al.'s parallel algorithm is O(2^5n/8). In this paper, we propose a parallel algorithm to improve Chang et al.'s parallel algorithm by reducing the time complexity to be O(2^3n/8) under the same O(2^n/8) processors available. Thus, the proposed parallel algorithm has a cost of O(2^n/2). It is an improvement over previous literature. We believe that the proposed parallel algorithm is pragmatically feasible at the moment when multiprocessor systems become more and more popular.     

A parallel algorithm for the 0–1 knapsack problem

Research efforts on parallel exact algorithms for the 0–1 knapsack problem have up to now concentrated on solving small problems (at most 1,000 objects) and in many cases results have only been obtained by simulation of the parallel algorithm. After a brief review of a well known sequential branch-and-bound algorithm we discuss a new parallel algorithm for the 0–1 knapsack problem which exploits the potential parallelism that exists during the backtracking steps of the branch-and-bound algorithm. We report results for our parallel algorithm on a transputer network for problems with up to 20,000 objects. The speedup obtained is nearly linear for 2, 4, and 8 processors except when there is a strong correlation between the profit and weight of the objects.     

The two list algorithm for the knapsack problem on a FPS T20

The advent of parallel machines brought about a controverse in the domain of parallel algorithms: is it worth to conceive parallel algorithms for NP-complete problems? In this work we present a parallel implementation of a sequential algorithm from Horowitz and Sahni for the knapsack problem on a FPS T20. Our aim is to establish some experimental results in order to allow comparisons for forthcoming works. The results show that the development of theory and technology yields the computation tractability of very large knapsack problems.     

A hybrid simulated annealing metaheuristic algorithm for the two-dimensional knapsack packing problem

The rectangle knapsack packing problem is to pack a number of rectangles into a larger stock sheet such that the total value of packed rectangles is maximized. The paper first presents a fitness strategy, which is used to determine which rectangle is to be first packed into a given position. Based on this fitness strategy, a constructive heuristic algorithm is developed to generate a solution, i.e. a given sequence of rectangles for packing. Then, a greedy strategy is used to search a better solution. At last, a simulated annealing algorithm is introduced to jump out of the local optimal trap of the greedy strategy, to find a further improved solution. Computational results on 221 rectangular packing instances show that the presented algorithm outperforms some previous algorithms on average.     

改进的差分演化算法求解多维背包问题

多维背包（MKP）是组合优化中一个典型的NP难问题,广泛应用于工程和管理中。提出了一种改进的二进制差分演化算法（Modified Binary Differential Evolution algorithm,MBDE）求解MKP问题,算法关键步骤可分为两部分：二进制群体生成;得到候选可行解。提出了一种有效的衡量商品价值密度的方法用于对二进制个体修正和优化;设计了反向测试搜索和精英局部搜索策略来提高算法探索和开发能力,从而进一步提高了MBDE的求解精度和收敛速度。为验证MBDE算法的有效性,进行了三组实验,并和近期提出的解决MKP问题的其他启发式算法进行了比较,实验结果显示,MBDE算法求解精度更高。从算法运行时间看,求解速度快,非常适合求解大规模的MKP问题。     

An effective hybrid EDA-based algorithm for solving multidimensional knapsack problem

In this paper, an effective hybrid algorithm based on estimation of distribution algorithm (EDA) is proposed to solve the multidimensional knapsack problem (MKP). With the framework of EDA, the probability model is built with the superior population and the new individuals are generated based on probability model. In addition, an updating mechanism of the probability model is proposed and a mechanism for initializing the probability model based on the specific knowledge of the MKP is also proposed to improve the convergence speed. Meanwhile, an adaptive local search is proposed to enhance the exploitation ability. Furthermore, the influences of parameters are investigated based on Taguchi method of design of experiment and the importance of repair operator is also studied via simulation testing and comparisons. Finally, numerical simulation is carried out based on the benchmark instances, and the comparisons with some existing algorithms demonstrate the effectiveness of the proposed algorithm.     

An improved monkey algorithm for a 0-1 knapsack problem

The 0-1 knapsack problem is a classic combinational optimization problem. However, many exiting algorithms have low precision and easily fall into local optimal solutions to solve the 0-1 knapsack problem. In order to overcome these problems, this paper proposes a binary version of the monkey algorithm where the greedy algorithm is used to strengthen the local search ability, the somersault process is modified to avoid falling into local optimal solutions, and the cooperation process is adopted to speed up the convergence rate of the algorithm. To validate the efficiency of the proposed algorithm, experiments are carried out with various data instances of 0-1 knapsack problems and the results are compared with those of five metaheuristic algorithms.     

MOTGA: A multiobjective Tchebycheff based genetic algorithm for the multidimensional knapsack problem

This paper presents a new multiobjective genetic algorithm based on the Tchebycheff scalarizing function, which aims to generate a good approximation of the nondominated solution set of the multiobjective problem. The algorithm performs several stages, each one intended for searching potentially nondominated solutions in a different part of the Pareto front. Pre-defined weight vectors act as pivots to define the weighted-Tchebycheff scalarizing functions used in each stage. Therefore, each stage focuses the search on a specific region, leading to an iterative approximation of the entire nondominated set.     

An improved fruit fly optimization algorithm for solving the multidimensional knapsack problem

This paper presents an improved fruit fly optimization algorithm (IFFOA) for solving the multidimensional knapsack problem (MKP). In IFFOA, the parallel search is employed to balance exploitation and exploration. To make full use of swarm intelligence, a modified harmony search algorithm (MHS) is proposed and applied to add cooperation among swarms in IFFOA. In MHS, novel pitch adjustment scheme and random selection rule are developed by considering specific characters of MKP and FOA. Moreover, a vertical crossover is designed to guide stagnant dimensions out of local optima and further improve the performance. Extensive numerical simulations are conducted and comparisons with other state-of-the-art algorithms verify that the proposed algorithm is an effective alternative for solving the MKP.     

求解背包问题的混合粒子群优化算法

提出一种新的遗传思想：父代的基因决定子代继承某一基因的概率,而不是由单纯的交叉产生子代。根据此思想,提出两种利用遗传概率产生子代的方法,并将它们分别与粒子群优化算法相结合得到两种求解背包问题的混合粒子群优化算法。通过数值实验说明了同样的算法采用遗传策略要比交叉策略寻优性更强,分析了变异概率对算法的影响。     

An efficient algorithm for the collapsing knapsack problem

The collapsing knapsack problem (CKP) is a type of nonlinear knapsack problem in which the knapsack size is a non-increasing function of the number of items included. This paper proposes an exact algorithm for CKP by partitioning CKP to some subproblems, then solving them with the improved expanding-core technique. The proposed algorithm solves the subproblems in the special processing order resulting in the reduction of computing time. Experimental results show that the proposed algorithm is an efficient approach for various random instances of size up to 1000.     

一种基于模式替代的遗传算法解0/1背包问题*

背包问题是一个典型的 NP完全问题。提出一种基于模式替代的遗传算法解0/1背包问题思想,通过收集每代种群中最好的几个个体生成模式来引导种群的搜索方向,以提高遗传算法的搜索速度和寻找最优解的能力。通过仿真数值实验,将该方法与简单遗传算法、贪心算法计算结果比较分析,充分证明了使用基于模式替代遗传算法来求解背包问题的有效性和实用性。     

贪心二进制狮群优化算法求解多维背包问题

针对经典的多约束组合优化问题——多维背包问题(MKP),提出了一种贪心二进制狮群优化(GBLSO)算法。首先,采用二进制代码转换公式将狮群个体位置离散化,得到二进制的狮群算法;其次,引入反置移动算子对狮王位置进行更新,同时对母狮和幼狮位置重新定义;然后,充分利用贪心算法进行解的可行化处理,增强搜索能力并进一步提高收敛速度;最后,对10个MKP典型算例进行仿真实验,并把GBLSO算法与离散二进制粒子群(DPSO)算法和二进制蝙蝠算法(BBA)进行对比。实验结果表明,GBLSO算法是一种有效的求解MKP的新方法,在求解MKP时具有相对良好的收敛效率、较高的寻优精度和很好的鲁棒性。     

Shuffled frog leaping algorithm and its application to 0/1 knapsack problem

This paper proposes a modified discrete shuffled frog leaping algorithm (MDSFL) to solve 01 knapsack problems. The proposed algorithm includes two important operations: the local search of the ‘particle swarm optimization’ technique; and the competitiveness mixing of information of the ‘shuffled complex evolution’ technique. Different types of knapsack problem instances are generated to test the convergence property of MDSFLA and the result shows that it is very effective in solving small to medium sized knapsack problems. Further, computational experiments with a set of large-scale instances show that MDSFL can be an efficient alternative for solving tightly constrained 01 knapsack problems.     

Exact solution of the robust knapsack problem

We consider an uncertain variant of the knapsack problem in which the weight of the items is not exactly known in advance, but belongs to a given interval, and an upper bound is imposed on the number of items whose weight differs from the expected one. For this problem, we provide a dynamic programming algorithm and present techniques aimed at reducing its space and time complexities. Finally, we computationally compare the performances of the proposed algorithm with those of different exact algorithms presented so far in the literature for robust optimization problems.     

求解0-1背包问题的混合贪婪遗传算法

求解0-1背包问题（KP）的最优解的时候,传统遗传算法（GA）的局部求精能力不足而简单局部搜索算法的全局探索能力有限,针对上述问题,将这两个算法整合并提出了混合贪婪遗传算法（HGGA）。在GA全局搜索框架下增加局部搜索模块,并改进传统仅基于物品价值密度的修复算子,增加基于物品价值的贪婪混合选项,从而加速寻优过程。HGGA一方面引导种群在进化的优质解空间中展开精细搜索,另一方面依靠GA的经典操作算子开拓全局搜索空间,从而达到算法求精能力和开拓能力的良好平衡。HGGA分别在三组数据上做了测试,结果表明在第一组15个测试用例中的12个上,HGGA能够百分百找到最优解,成功率达到80%;在第二组小规模数据集上,HGGA的性能明显好于其他同类GA和其他元启发算法;在第三组大规模数据集上,HGGA较其他元启发式算法具有更好的稳定性和高效性。     

求解0-1背包问题的量子蚁群算法

0-1背包问题是组合优化中经典的NP难题,在蚁群算法的基础上结合量子计算提出一种求解0-1背包问题的量子蚁群算法。算法采用量子比特表示信息素,用量子旋转门来更新信息素。大量数据实例的比较测试表明,算法可有效提高蚂蚁算法的性能,减少搜索时间,具有更好的全局寻优能力。     

一种新的求解0-1背包问题的混合算法

该文汲取了蚁群算法（ACA）和抗体免疫克隆算法（AICA）的优点,提出了一种求解0-1背包问题的混合型算法,该算法充分利用了前者的搜索能力和后者的种群多样性。仿真实验对算法的部分参数进行了分析,并与其他文献的算法进行比较,结果表明,该算法是一种具有较高性能的混合优化算法。     

一种求解多维背包问题的混合分布估计算法

针对多维背包问题（MKP）,提出一种基于分布估计算法的混合求解算法,该算法基于优势种群构建概率模型,并基于概率模型采样产生新个体;同时,提出一种基于MKP问题信息的修复机制,有效修复采样后种群中的不可行解．另外,设计了一种自适应的局部搜索操作,以增强算法的局部搜索能力,基于标准测试集的仿真结果和算法比较验证了所提出的混合算法的有效性和鲁棒性．     

求解0-1背包问题的混合蝙蝠算法

针对基本蝙蝠算法易陷入局部最优、收敛速度慢等缺点,对其进行优化研究。基于0-1背包问题的具体特征,在基本蝙蝠算法原有概念和框架的基础上,引入遗传算法中的交叉机制以及反置算子建立全新的位置转移方式和局部搜索规则;加入贪心策略进行解的可行化和充分利用,增强局部搜索能力,加快算法收敛速度,构建全新的混合蝙蝠算法。将混合蝙蝠算法应用于两组0-1背包算例,仿真实验结果优于自适应元胞粒子群算法、基本蝙蝠算法和贪心二进制蝙蝠算法。结果验证了该混合算法求解0-1背包问题的可行性和有效性。