Sensitivity analysis of the knapsack sharing problem: Perturbation of the weight of an item

In this paper, we study the sensitivity analysis of the optimum of the knapsack sharing problem (KSP) to the perturbation of the weight of an arbitrary item. We determine the interval limits of the weight of each perturbed item using a heuristic approach which reduces the original problem to a series of single knapsack problems. A perturbed item belongs either to an optimal class or to a non-optimal class. We evaluate the performance of the proposed heuristic on a set of problem instances of the literature. Encouraging results are obtained.     

Sensitivity analysis of the setup knapsack problem to perturbation of arbitrary profits or weights

The setup knapsack problem can be viewed as a more complex version of the well‐known classical knapsack problem. An instance of such a problem may be defined by a set of n items that is divided into m different classes For each class, only one item is considered as a setup item. The aim of the problem is to pack a subset of items in a knapsack of a predefined capacity that maximizes an objective function. In this paper, we analyze the sensitivity of an optimal solution depending on the variation of the profits or weights of arbitrary items. The optimality of the solution at hand is guaranteed by establishing the sensitivity interval that is characterized by both lower and upper values (called limits). First, two cases are distinguished when varying the profits: perturbation of the profit of an item (either ordinary or setup item) and, variation of the profits of a couple of items containing both setup and ordinary items belonging to the same class. Second, two cases are studied, where the perturbation concerns the weights: the variation is relied on the weight of an item and, the case of the variation of the weights of a subset of items. The established results are first illustrated throughout a didactic example, where both approximate and exact methods are used for analyzing the quality of the proposed results. Finally, an extended experimental part is proposed in order to evaluate the effectiveness of the proposed limits.     

An exact constructive algorithm for the knapsack sharing problem

In this paper, we study the knapsack sharing problem (KSP), a variant of the well-known NP-hard single knapsack problem. We propose an exact constructive tree search that combines two complementary procedures: a reduction interval search and a branch and bound. The reduction search has three phases. The first phase applies a polynomial reduction strategy that decomposes the problem into a series of knapsack problems. The second phase is a size reduction strategy that makes the resolution more efficient. The third phase is an interval reduction search that identifies a set of optimal capacities characterizing the knapsack problems. Experimental results provide computational evidence of the better performance of the proposed exact algorithm in comparison to KSPs best exact algorithm, to Cplex and to KSPs latest heuristic approach. Furthermore, they emphasize the importance of the reduction strategies.     

A probabilistic solution discovery algorithm for solving 0-1 knapsack problem

Abstract

In this paper, a probabilistic solution discovery algorithm is developed to solve the NP-hard 0-1 knapsack problem. The proposed method consists of three steps: strategy development, strategy analysis, and solution discovery. In the first step, Monte Carlo simulation is used to generate the strategies based on a vector defining the probability that each item is included in the knapsack. In the second step, we analyse the capacity imposed by each strategy previously generated and penalise the objective value for those strategies exceeding the capacity of the knapsack. At the last step, a subset of ordered strategies is used to update the vector that defines the probability of choosing each item. Two numerical examples are used to demonstrate the efficiency and the performance of the proposed method.     

求解0-1背包问题的融合贪心策略的回溯算法

0-1背包问题作为经典的NP完全问题一直得到广泛的关注和研究.研究发现,经典回溯算法在解决0-1背包问题时的算法时间复杂度较高,尤其是在物品数量较多时,短时间内不能得到问题的解,导致算法的适用性较差.虽然经典贪心算法和现阶段涌现出的大量新型算法能够极大地缩减算法的运行时间,但普遍是以牺牲算法的准确性为代价的,不能保证可...     

The robust knapsack problem with queries

We consider robust knapsack problems where item weights are uncertain. We are allowed to query an item to find its exact weight,where the number of such queries is bounded by a given parameter Q. After these queries are made, we need to pack the items robustly, i.e., so that the choice of items is feasible for every remaining possible scenario of item weights.The central question that we consider is: Which items should be queried in order to gain maximum profit? We introduce the notion of query competitiveness for strict robustness to evaluate the quality of an algorithm for this problem, and obtain lower and upper bounds on this competitiveness for interval-based uncertainty. Similar to the study of online algorithms, we study the competitiveness under different frameworks, namely we analyze the worst-case query competitiveness for deterministic algorithms, the expected query competitiveness for randomized algorithms and the average case competitiveness for known distributions of the uncertain input data. We derive theoretical bounds for these different frameworks and evaluate them experimentally. We also extend this approach to Γ-restricted uncertainties introduced by Bertsimas and Sim.Furthermore, we present heuristic algorithms for the problem. In computational experiments considering both the interval-based and the Γ-restricted uncertainty, we evaluate their empirical performance. While the usage of a Γ-restricted uncertainty improves the nominal performance of a solution (as expected), we find that the query competitiveness gets worse.     

A virtual pegging approach to the max–min optimization of the bi-criteria knapsack problem

We are concerned with a variation of the knapsack problem, the bi-objective max–min knapsack problem (BKP), where the values of items differ under two possible scenarios. We have given a heuristic algorithm and an exact algorithm to solve this problem. In particular, we introduce a surrogate relaxation to derive upper and lower bounds very quickly, and apply the pegging test to reduce the size of BKP. We also exploit this relaxation to obtain an upper bound in the branch-and-bound algorithm to solve the reduced problem. To further reduce the problem size, we propose a ‘virtual pegging’ algorithm and solve BKP to optimality. As a result, for problems with up to 16,000 items, we obtain a very accurate approximate solution in less than a few seconds. Except for some instances, exact solutions can also be obtained in less than a few minutes on ordinary computers. However, the proposed algorithm is less effective for strongly correlated instances.     

基于猴群算法求解0-1背包问题

0-1背包问题是一个经典的NP完全问题,该问题在实际生活中具有广泛的应用.针对现有算法在求解0-1背包问题时精度不高的缺点,提出了一种诱导因子猴群算法.所给诱导因子猴群算法的基本思想是,在基本猴群算法的爬过程中引入诱导因子,诱导其向上爬行,从而可以逃逸局部最优解,找到全局最优解.在仿真试验中,与已有方法进行比较,结果说明利用所给诱导因子猴群算法求解0-1背包问题是有效的.     

基于动态规划法求解动态o-1背包问题

随机时变背包问题(RTVKP)是一种动态组合优化问题,也是一种典型的NP-hard问题。由于RTVKP问题中物品的价值、重量和背包载重均是动态变化的,导致问题的求解非常困难。在动态规划法基础上,提出了一种求解背包载重随机变化的RTVKP问题的确定性算法,分析了其复杂度和成功求解需要满足的条件。对两个大规模实例的计算表明,该算法是求解RTVKP问题的一种高效算法。     

求解0-1背包问题的二进制狮群算法

针对传统二进制群智能算法求解0-1背包问题易陷入局部最优、收敛速度慢的缺点,提出一种新的解决离散空间问题的二进制狮群算法BLSO。二进制狮群算法对狮王、母狮和幼狮的位置重新定义,引入反置运算、移动算子和学习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处理和充分利用,增强局部搜索能力,进一步提高收敛速度。对9个典型的0-1背包算例进行仿真实验,实验结果表明,该算法不仅可以有效求解0-1背包问题,而且还能够以较快的速度搜索到精度较高的次优解甚至全局最优解,具有较好的稳定性;同时,对高维背包问题的求解与参考算法相比,在寻优时间和精度上更具优势。     

A dynamic programming method with lists for the knapsack sharing problem

In this paper, we propose a method to solve exactly the knapsack sharing problem (KSP) by using dynamic programming. The original problem (KSP) is decomposed into a set of knapsack problems. Our method is tested on correlated and uncorrelated instances from the literature. Computational results show that our method is able to find an optimal solution of large instances within reasonable computing time and low memory occupancy.     

背包问题的最优并行算法

利用分治策略,提出一种基于SIMD共享存储计算机模型的并行背包问题求解算法.算法允许使用O(2^n/4)^1-ε个并行处理机单元,0≤ε≤1,O(2^n/2)个存储单元,在O(2^n/4(2^n/4)^ε)时间内求解n维背包问题,算法的成本为O(2^n/2).将提出的算法与已有文献结论进行对比表明,该算法改进了已有文献的相应结果,是求解背包问题的成本最优并行算法.同时还指出了相关文献主要结论的错误.     

背包问题无存储冲突的并行三表算法

背包问题属于经典的NP难问题，在信息密码学和数论等研究中具有极重要的应用，将求解背包问题著名的二表算法的设计思想应用于三表搜索中，利用分治策略和无存储冲突的最优归并算法，提出一种基于EREW-SIMD共享存储模型的并行三表算法，算法使用O（2^n/4）个处理机单元和O（2^3n/8）的共享存储空间，在O（2^3n/8）时间内求解n维背包问题．将提出的算法与已有文献结论进行的对比分析表明：文中算法明显改进了现有文献的研究结果，是一种可在小于O（2^n/2）的硬件资源上，以小于O（2n/2）的计算时问求解背包问题的无存储冲突并行算法。     

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.     

Problem reduction heuristic for the 0-1 multidimensional knapsack problem

This paper introduces new problem-size reduction heuristics for the multidimensional knapsack problem. These heuristics are based on solving a relaxed version of the problem, using the dual variables to formulate a Lagrangian relaxation of the original problem, and then solving an estimated core problem to achieve a heuristic solution to the original problem. We demonstrate the performance of these heuristics as compared to legacy heuristics and two other problem reduction heuristics for the multi-dimensional knapsack problem. We discuss problems with existing test problems and discuss the use of an improved test problem generation approach. We use a competitive test to highlight the performance of our heuristics versus the legacy heuristic approaches. We also introduce the concept of computational versus competitive problem test data sets as a means to focus the empirical analysis of heuristic performance.     

Development of core to solve the multidimensional multiple-choice knapsack problem

The multidimensional multiple-choice knapsack problem (MMKP) is an extension of the 0–1 knapsack problem. The core concept has been used to design efficient algorithms for the knapsack problem but the core has not been developed for the MMKP so far. In this paper, we develop an approximate core for the MMKP and utilize it to solve the problem exactly.     

Generalized quadratic multiple knapsack problem and two solution approaches

The Quadratic Knapsack Problem (QKP) is one of the well-known combinatorial optimization problems. If more than one knapsack exists, then the problem is called a Quadratic Multiple Knapsack Problem (QMKP). Recently, knapsack problems with setups have been considered in the literature. In these studies, when an item is assigned to a knapsack, its setup cost for the class also has to be accounted for in the knapsack. In this study, the QMKP with setups is generalized taking into account the setup constraint, assignment conditions and the knapsack preferences of the items. The developed model is called Generalized Quadratic Multiple Knapsack Problem (G-QMKP). Since the G-QMKP is an NP-hard problem, two different meta-heuristic solution approaches are offered for solving the G-QMKP. The first is a genetic algorithm (GA), and the second is a hybrid solution approach which combines a feasible value based modified subgradient (F-MSG) algorithm and GA. The performances of the proposed solution approaches are shown by using randomly generated test instances. In addition, a case study is realized in a plastic injection molding manufacturing company. It is shown that the proposed hybrid solution approach can be successfully used for assigning jobs to machines in production with plastic injection, and good solutions can be obtained in a reasonable time for a large scale real-life problem.     

A note on semidefinite relaxation for 0-1 quadratic knapsack problems

We investigate in this note solution properties of semidefinite programming (SDP) relaxation for 0-1 quadratic knapsack problem (QKP). In particular, we focus on the issue of uniqueness of the optimal solution to the SDP relaxation for QKP. We first give a counterexample which shows that the optimal solution to the SDP relaxation for QKP could be non-unique. This is in contrast with the case of unconstrained 0-1 quadratic problems. A necessary and sufficient condition is then derived to ensure the uniqueness of the optimal solution to the SDP relaxation for QKP.     

Bi-dimensional knapsack problems with one soft constraint

In this paper, we consider bi-dimensional knapsack problems with a soft constraint, i.e., a constraint for which the right-hand side is not precisely fixed or uncertain. We reformulate these problems as bi-objective knapsack problems, where the soft constraint is relaxed and interpreted as an additional objective function. In this way, a sensitivity analysis for the bi-dimensional knapsack problem can be performed: The trade-off between constraint satisfaction, on the one hand, and the original objective value, on the other hand, can be analyzed. It is shown that a dynamic programming based solution approach for the bi-objective knapsack problem can be adapted in such a way that a representation of the nondominated set is obtained at moderate extra cost. In this context, we are particularly interested in representations of that part of the nondominated set that is in a certain sense close to the constrained optimum in the objective space. We discuss strategies for bound computations and for handling negative cost coefficients, which occur through the transformation. Numerical results comparing the bi-dimensional and bi-objective approaches are presented.     

Shift-and-merge technique for the DP solution of the time-constrained backpacker problem

We formulate the time-constrained backpacker problem as an extension of the classical knapsack problem (KP), where a ‘backpacker’ travels from a origin to a destination on a directed acyclic graph, and collects items en route within the capacity of his knapsack and within a fixed time limit. We present a dynamic programming (DP) algorithm to solve this problem to optimality, and a ‘shift-and-merge’ DP algorithm to solve larger instances. The latter is an extension of the list-type DP, which has been successful for one-dimensional KPs, to the two-dimensional case. Computational experiments on a series of instances demonstrate advantage of the shift-and-merge technique over commercial MIP solvers.